The possibility to predict crack patterns on dynamic ...

ISBN XXX-XXX-XX-XXXX-X

Book edited by: Lucas Máximo Alves

State University of Ponta Grossa, Brazil

Chapter I

FRACTAL CONTINUUM MECHANICS

PROPOSTA DE UMA EQUAÇÃO DE CAMPO VETORIAL ELÁSTICO PARA MEIOS CONTÍNUOS COM IRREGULARIDADES Lucas M. Alvesa,b, Luiz A. De Lacerda.b a

GTEME – Grupo de Termodinâmica, Mecânica e Eletrônica dos Materiais, Departamento de Engenharia de Materiais, Setor de Ciências Agrárias e de Tecnologia, Universidade Estadual de Ponta Grossa, Av. Gal. Carlos Calvalcanti, 4748, Campus UEPG/Bloco L – Uvaranas – Ponta Grossa-PR, Brasil, CEP. 84030.000, Cx. Postal 1007, [email protected]; http://dgp.cnpq.br b

LACTEC - Instituto de Tecnologia para o Desenvolvimento, DPEC/DVPE - Divisão de Pesquisas em Estruturas Civis,Centro Politécnico da Universidade Federal do Paraná, Cx. P. 19067,Curitiba, PR, Brazil, [email protected]; http://www.lactec.org.br/pt/

Keywords: fractal dimension, fracture profile, mortar, red ceramic, ruggedness, self-affine surface. Abstract. In this work, a motion and a constitutive equation for brittle material with irregularities (porosity in the volume, ruggedness in the boundary surfaces, that can be fractals or not) is proposed. Numerical solutions of this equation are evaluated by (Difference Finite Method-DFM and Finite Elements Methods-FEM) in a static problem of a stress and strain field at the rugged crack tip. The results are compared with different propositions (Mosolov, Balankin and Yavary) for the stress field at the rugged crack tip. We conclude that the singularity of the stress field at the rugged crack tip have not a unique fractal ruggedness dimension but depends on the position ahead of the crack tip.

Palavras Chave: equação de movimento, equação constitutiva, porosidade, rugosidade, densidade generalizada, fluxo generalizado. Resumo. Neste trabalho propõem-se uma equação de movimento e uma equação constitutiva para materiais frágeis com irregularidades (porosidade no volume, rugosidade nas superfícies de contorno, que podem ser fractais ou não). Soluções numéricas desta equação são avaliadas por (Método de Diferenças Finitas - MDF e Métodos de Elementos Finitos - MEF) em um problema estático de um campo de tensão e deformação na ponta de uma trinca rugosa. Os resultados com e sem rugosidade foram comparados e analisados. Também se comparou os resultados com diferentes propostas da literatura para o campo de tensão na ponta da trinca rugosa. Concluiu-se que a singularidade do campo de tensão na ponta da trinca rugosa não possui uma única dimensão fractal de rugosidade, mas depende da posição na frente da ponta da trinca.

Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar

L. ALVES, L. DE LACERDA

1

INTRODUCTION

A descrição de fenômenos físicos que nos dão a idéia de movimento, como o fluxo de calor, o movimento de uma partícula, a deformação de um corpo, o escoar de um fluido, o crescimento de uma trinca, etc. podem ser unificados pela Teoria Geral do Campo Contínuo Clássico. Essa teoria geral de campo procura unificar os aspectos elementares e reducionistas da mecânica, com os aspectos gerais da termodinâmica. Todo potencial contido em partículas ou ao redor de corpos na forma de campos produzem algum tipo de movimento. Neste sentido sabe-se que todo “movimento” parte da diferença de algum tipo de “potencial” entre dois pontos a qual pode ser chamado de “potencial generalizado”. No caso do fluxo de calor a diferença de potencial corresponde à diferença de temperatura entre dois pontos. No caso do movimento de uma partícula a diferença de potencial mecânica, elétrica ou magnética pode ser atribuída à aplicação de uma força, e assim por diante, na lista dos fenômenos que podem ser unificados pela teoria dos campos escalares e vetoriais e tensoriais. O campo externo ao redor de uma partícula, ou ao redor de um corpo com forma regular de campo interno uniforme em questão, apresentará linhas de mesmo potencial (equipotenciais) que acompanharão o contorno do corpo. As linhas de fluxo desse campo estarão dirigidas para fora na direção perpendicular às linhas equipotenciais e será dado pelo gradiente do seu potencial entre dois pontos, cujo vetor estará apontado na direção normal às linhas de mesmo potencial. Mesmo, quando os campos envolvidos são campos dinâmicos cuja descrição envolve diretamente o conceito de velocidade e de movimento de uma partícula, como no caso de deformações elásticas e/ou plásticas de um sólido, ou taxas de deformações de um fluido, novamente, surge a idéia de campo potencial associado a diferenças de potenciais, as quais são responsáveis pela formação de gradientes de potencial e por sua vez responsáveis pelo movimento das partículas imersas nesses campos. Nesses casos os campos são de potenciais vetoriais ou tensoriais. Mesmo assim, para todas as situações descritas até agora, é possível generalizar o conceito de posição, velocidade, quantidade de movimento, força, potencial, diferença de potencial, etc. com a finalidade de descrever matematicamente o campo clássico de uma forma unificada. No caso de corpos com formas irregulares e se considera que o seu campo interno é uniforme, as linhas equipotenciais externas nos fenômenos exemplificados acima acompanharão as formas irregulares do seu contorno, conforme mostra Figura 1. Até este ponto a complicação da irregularidade dos contornos pode ser trabalhada por soluções numéricas e as correções que surgem em relação ao caso regular são apenas geométricas. Em todo o sentido pode-se sempre adotar que o fluxo deriva do potencial a partir do seu gradiente da seguinte forma:

 kT ( prob. térmico)   J    ( prob. elétrico ) . etc, 

(1)

onde k, e  são as condutividades térmicas e elétricas e T e  são os potenciais térmicos (temperatura) e elétrico, respectivamente. Ou seja, as linhas de fluxos são perpendiculares às linhas equipotenciais, sejam elas de potencial elétrico, térmico ou de outra natureza qualquer dentro dessa mesma classe de fenomenologia.


Mecánica Computacional Vol XXIX, págs. 5107-5131 (2010)

Figura 1.Linhas equipotenciais e de campo em torno de um corpo irregular uniformemente carregado.

Contudo, quando as irregularidades de campo envolvem também a parte física do campo, no sentido de haver campos não uniformes, fontes de campo aleatoriamente dispersas no meio, volumes porosos, efeitos não lineares geométricos e físicos, etc., a solução das equações de campo pode se tornar muito trabalhosa ou analiticamente impraticável. Neste ponto, é preciso recorrer a algum tipo de geometria que possa tratar das descontinuidades geométricas e seus efeitos sobre o valor dos campos dos potenciais. Além do que, é necessário generalizar tanto a descrição geométrica da forma dos objetos e do contorno de suas superfícies, como a descrição geométrica dos campos envolvidos, junto com seus efeitos dinâmicos. Porque, além de uma generalização geométrica e fenomenológica, é preciso também fazer a inserção destas nos modelos praticáveis por uma teoria de campo generalizada. Portanto, um dos objetivos deste artigo é formular uma teoria do contínuo que envolva a rugosidade da superfície de contorno e o fração volumétrica irregular efetiva ou fração volumétrica deformada do interior do volume do corpo. De forma mais direta, as conseqüências desta nova formulação teórica serão utilizadas no estudo da Mecânica da Fratura. A teoria do campo contínuo clássica utiliza a geometria euclidiana na descrição dos fenômenos de transferência de calor, massa, momentum, etc. Com esta geometria é possível descrever apenas os fenômenos que acontecem em formas regulares sem considerar os efeitos da rugosidade das superfícies, ou da porosidade do interior dos volumes. Mesmo quando as formas são cheia de detalhes geométricos, utilizam-se modelos numéricos e cálculos aproximados (Blyth 2003, Xie 2003, Hyun 2004). Na fratura, por exemplo, esta quase nunca acontece sem o surgimento de superfícies rugosas. Os casos de fratura com superfícies lisas aparecem normalmente em processos de clivagem de monocristais e no interior de alguns grãos do material. No caso de fratura de materiais policristalinos com considerável rugosidade (cuja ponta da trinca rugosa interage no processo de fratura) o uso da geometria euclidiana deixa a desejar. Pois os resultados não são completos e os cálculos não são exatos, e ainda não se consegue explicar diversos fenômenos da fratura quase-estática e dinâmica em que a rugosidade é presente (Fineberg 1991, 1992, Xie, 1995; Alves 2005, Alves et al 2010). A descrição matemática do crescimento da curva J-R de resistência a fratura, por exemplo, só pode ser explicado se for levado em conta o aparecimento da rugosidade durante o processo de propagação ou crescimento de uma trinca (Alves 2001, Alves et al 2010). Isto significa que o modelo geométrico da rugosidade precisa ser incluído no cálculo analítico da integral-J de Eshelby-Rice (Su et al , 2000; Weiss, 2001; Rupnowski, 2001; Alves 2001, Alves et al 2010). Um outro fenômeno na fratura que envolve o surgimento de rugosidade é a propagação de trincas rápidas, onde surgem instabilidades com ramificação de trincas e oscilações na velocidade de crescimento da trinca, a partir de uma velocidade crítica (Fineberg 1991, 1992). Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar


1.1 O surgimento de teorias do campo contínuo com a inclusão de irregularidades A presença de irregularidades de forma e de microestrutura na superfície e no interior de materiais sujeitos as fenomenologias é uma realidade na natureza e também nos materiais desenvolvidos pelo homem. Não é de agora que existe a necessidade de se tratar com as irregularidades e os defeitos presentes em um material. Para isso têm surgido ao longo dos anos tópicos específicos das ciências exatas que tratam de irregularidades geométricas e microestruturais nas superfícies e no interior dos materiais (Bammann, 1982; Forest, 1998; Trovalusci, 1998; Duda, 2007; Engelbrecht, 2009). Desde que surgiu a teoria fractal, grande tem sido os esforços em descrever as formas irregulares da natureza como também o seu efeito sobre os fenômenos físicos e químicos e nos materiais (Hornbogen 1989, Panin, 1992, Lazarev, 1993). Hornbogen (1989) enumerou diferentes aspectos da microestrutura de materiais que podem ser tratados pela geometria fractal. Panagiotopoulos (1992) propôs a utilização da geometria fractal na descrição da estrutura dos sólidos irregulares, Panin (1992) descreveu os fundamentos da meso-mecânica de um meio com estruturas, Tarasov (2005) descreveu uma mecânica do contínuo para meios fractais utilizando o calculo fracional. Yavari (2006) descreveu as leis de equilíbrio covariantes espaciais e materiais na elasticidade. Outras abordagens estão sendo elaboradas por diversos cientistas e publicadas na literatura especializada, e dizem respeito a uma teoria especificamente fractal envolvendo o cálculo fracional em múltipla escala (Dyskin, 2005, Carpinteri, 2009). Dyskin (2005) têm publicado vários trabalhos no sentido de utilizar a teoria fractal e o calculo fracional para descrever uma mecânica de múltipla escala. Carpinteri (2009) et al utilizaram o calculo fracional como uma forma de incluir a teoria fractal na descrição de fenômenos de elasticidade e fratura envolvendo a rugosidade e o efeito de escala. Todas estas são propostas de uma mecânica que possa tratar inclusivamente a irregularidade de forma e de microestrutura no seu contexto matemático.



1.2 Importância da inclusão da rugosidade na teoria do campo contínuo clássica Análises geométricas de superfícies rugosas de fratura em materiais específicos, como madeira, vidro, cimento, argila, demonstram que as rugosidades nesses materiais são características de cada tipo de material (Morel 1998, Ponson 2006, Alves 2004). Os aspectos geométricos de superfícies rugosas de fratura em argamassa de cimento, por exemplo, apresentam semelhança entre si. Assim como as superfícies rugosas de fratura obtidas em argilas ou tijolos de argilas também possuem aspectos semelhantes entre si, diferindo, contudo, dos aspectos geométricos das superfícies de fratura do cimento. Ou seja, cada tipo de material define uma classe de superfícies de fratura, cujos aspectos geométricos são semelhantes para as superfícies de fratura da mesma classe de material. Isto nos ajuda a pensar que a rugosidade deve depender do tipo de material e deve ser incluída na equação constitutiva do mesmo para o estudo dos fenômenos do contínuo (calor, elasticidade, fratura, etc). Portanto, vamos fazer algumas modificações na equação de movimento e nas equações constitutivas básicas começando com a teoria do campo escalar (calor, eletrostática) depois passando para a teoria do campo vetorial (elasticidade, eletrodinâmica, fluidos, etc.) até a fratura de materiais frágeis elasticamente lineares. A idéia de se fazer estas modificações de forma evolutiva, em grau de complexidade do fenômeno (campos escalares primeiro e depois campos vetoriais) é para poder se aprender com os resultados que cada modificação pode fornecer. Isto permitirá obter a melhor consistência possível na descrição matemática dos fenômenos do contínuo, que envolve a participação da rugosidade no processo, tanto de transferência de calor como nos processos de mecânicos de elasticidade e fratura. Procura-se descrever os fenômenos físicos em termos de uma descrição matemática mais autêntica, que leva em conta os aspectos irregulares ou a rugosidade das estruturas. Uma metodologia de transcrição dos fenômenos descritos na geometria euclidiana para uma geometria irregular torna-se necessária. A idéia básica consiste em trocar os comprimentos, áreas e volumes projetados, isto é, lisos (denotado neste trabalho pelo subscrito “0”) pelos comprimentos, áreas e volumes que apresentam rugosidades reais. Matematicamente, isto significa passar, simplesmente, o contorno liso d 0 para o contorno rugoso, d , da seguinte forma: d  ( x, y ) 

d d 0 . d 0

(2)

usando apenas umas simples transformação de coordenadas pela regra da cadeia. Esta simples transformação matemática é a causa de grandes mudanças no paradigma das superfícies rugosas, introduzindo uma nova visão para os fenômenos da mecânica da fratura, como foi visto nas secções anteriores. Ela também poderá ser bem aproveitada nos Métodos Numéricos de Elementos de Contorno, por exemplo, na simulação de uma trinca rugosa, como será visto nos resultados deste trabalho. Nas outras áreas da Mecânica do Contínuo tais como Calor, Elasticidade, Fluidos fica a proposta para trabalhos de futuros. A solução encontrada por alguns cientistas (Irwin 1948, Muskhelisvili 1954, Barenblatt 1962, Rice 1968) para se descrever alguns fenômenos que estão associados a geometrias irregulares (comprimentos, superfícies e volumes) foi utilizar uma relação energética entre a superfície irregular e a superfície de projeção euclidiana de tal forma que esta é escrita como:

dU dU    . dA0 dA


(3)


 onde U é a energia envolvida na superfície de área rugosa A , e na superfície projetada de  geometria euclidiana de área A0 . Relações deste tipo pressupõem que a superfície irregular não influencia no fenômeno. Isto pode ser visto se expressarmos a equação acima da seguinte forma:  dU dU dA     . (4) dA0 dA dA0 Observe que comparando a expressão Eq. (3) com a Eq. (4) vemos que a relação entre a   área rugosa ou irregular A e a área de projeção A0 é igual unidade para o caso em que a   equivalência energética é considerada. Contudo, quando a relação entre a dA / dA0 é diferente da unidade, a equivalência energética Eq. (3) não é válida. Nesta situação há duas alternativas;  (i) ou, se reescreve as relações diretamente em termos da geometria irregular A construindose uma nova fenomenologia, (ii) ou, se mantém a equivalência energética na forma da relação   Eq. (4) com o termo dA / dA0  1 . Dependendo da fenomenologia e de sua larga aplicabilidade, uma ou outra, alternativa é necessária. Neste trabalho, optamos por fazer as correções necessárias da teoria fenomenológica com base na geometria euclidiana, acrescentando-se nas   derivadas o termo de correção dA / dA0  1 em todas as equações. Neste sentido, nós vamos desenvolver os cálculos que serão úteis na descrição dos fenômenos que envolvem potenciais escalares e vetoriais para problemas de Elementos de Contorno Rugoso, os quais serão descritos pela geometria fractal. Portanto, observa-se a necessidade de haver uma teoria matemática do contínuo com irregularidades (poros, rugosidade, etc.) que siga um método de solução do problema irregular diretamente a partir das equações diferenciais governantes do campo clássico com irregularidades. Desta forma, uma transição do meio contínuo regular clássica para o meio irregular se faz necessária, a qual será vista nesse artigo. Observa-se, então, que é necessário modificar a teoria do campo contínuo desde a teoria dos campos escalares até os campos tensoriais passando pela teoria do calor, teoria da elasticidade, fratura, por exemplo, para envolver na sua descrição matemática o efeito da rugosidade descrevendo-a e explicando o seu surgimento com seus efeitos e conseqüências. Portanto, vamos agora neste artigo propor as modificações desejadas na teoria do campo contínuo de forma a incluir a rugosidade das superfícies e o fração volumétrica irregular efetiva ou fração volumétrica deformada dos volumes, onde estaremos designando esses meios materiais com o nome de Meios Irregulares. Neste artigo propomos uma modificação espacial e material das leis da mecânica do continuo através de volumes porosos e superfícies rugosas, considerando que essas irregularidades geométricas introduzem uma “transformação covariante” na mecânica do contínuo clássica que pode ser fractal ou não. Neste sentido a transformação é introduzida por um tensor ξ  " F " (1) de rugosidade responsável por um certo tipo de “estiramento” ou variação da superfície rugosa em relação a superfície média aparente projetada no espaço euclidiano, onde:  dA dA ξ   (5)  nˆ  nˆ0  . dA0 dA0 1

o tensor de estiramento na mecânica do contínuo é comumente denotado pela letra F em negrito Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar


A nossa proposta não se limita a uma irregularidade fractal, podendo ser este apenas um dos modelos a serem utilizados na teoria proposta. Após esta pequena (ou breve) introdução ao problema de potenciais com rugosidade, vamos elaborar um desenvolvimento matemático geral para a teoria da rugosidade e do fração volumétrica irregular efetiva ou fração volumétrica deformada .



2

FUNDAMENTOS TEÓRICOS DA MECÂNICA DOS MEIOS IRREGULARES

Para descrever o processo de dissipação de energia em um caminho rugoso é necessário a partir de agora, considerar que as funções vetoriais e escalares que definem as superfícies   A  A  x, y  e os volumes V  V  x, y, z  irregulares, respectivamente. Essas são funções descritas por algum modelo (como o modelo fractal, por exemplo) capaz de fornecer funções analíticas e diferenciáveis nas vizinhanças dos pontos genéricos de coordenadas P  P  x, y, z  , a fim de que seja possível calcular as grandezas que se propõem, tais como, rugosidade e porosidade.

Figura 2. Vetores normais a uma quina suave e a um “bico” ou quina brusca.

Ao contrário de utilização das funções fractais não-diferenciáveis onde se utiliza o cálculo fracional e a teoria da renormalização para contornar o problema da não diferenciabilidade, nosso intento é evitar a não-diferenciabilidade dessas funções. Vamos considerar que sempre é possível definir um vetor normal em “bicos” e quinas e que as cúspides são consideradas inexistentes na escala natural dos fenômenos, conforme mostra a Figura 2. Do contrário uma teoria que envolve sub-diferenciais para definir uma família de vetores normais em “bicos” e “quinas” torna-se necessária. Mas esta proposta é mais complexa e sai fora da proposta desse nosso modelo. 2.1 A Teoria mecânica dos meios irregulares em outras áreas A fundamentação teórica das transformações matemáticas das equações de diversos fenômenos descritos em termos da geometria euclidiana, para uma geometria irregular (fractal ou não), passa por uma abordagem do entendimento das densidades e dos fluxos generalizados em termos dessa nova geometria. Com isso é preciso estabelecer quais transformações matemáticas são necessárias em termos das coordenadas, dos comprimentos, das áreas das superfícies e dos volumes irregulares. È importante elaborar um tratamento matemático da Elasticidade e Fratura em objetos com superfícies rugosas ou com múltiplos vazios no seu volume para aplicação em problemas de contato térmico, convecção, porosidade, deformações mecânicas e fratura. Na elasticidade, P. D. Panagiotopoulos (1992) percebeu a necessidade de reformular a Mecânica dos Sólidos com base na teoria fractal. Paralelamente na Mecânica da Fratura, Arash Yavari (2000, 2002, 2006) reformulou o campo elástico ao redor de uma trinca usando o escalonamento fractal e descobriu novos modos de fratura e uma equação para a curva J-R fractal. Todos estes esforços vêm corroborar a idéia da existência de um novo campo a ser pesquisado e desenvolvido na ciência que unirá em um único ramo os problemas do campo contínuo com as irregularidades físicas e geométricas. Dentro do contexto desse trabalho observou-se que todas as correções feitas ao campo clássico (escalar, vetorial ou tensorial), quer em problemas de fluxo de calor, de elasticidade e de fratura poderiam ser incluídas em um único contexto de uma Mecânica do Contínuo de Meios Irregulares. Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar


2.2 Características básicas da microestrutura de um meio irregular Entende-se por irregularidades quaisquer acréscimos físicos ou geométricos feitos ao meio contínuo tais como: poros, rugosidades superficiais, inclusão de partículas, zonas plásticas, zonas fundidas, trincas internas, etc. Nesta nova roupagem a Mecânica dos Meios Irregulares se reduz a Mecânica do Contínuo quando as irregularidades não existem. Por outro lado a teoria fractal se insere neste contexto quando a opção pela modelagem das irregularidades for utilizando modelos fractais por causa da invariância por transformação de escalas dessas irregularidades. A Mecânica dos Meios Irregulares poderá neste sentido incluir a Mecânica dos Meios Desordenados quando as equações do continuo Irregular forem transformadas em equações discretas com irregularidades.

Figura 3. Diferentes tipos de defeitos e irregularidades presentes num material que agem como concentradores de tensão e influenciam na formação da superfície de fratura.

Na microestrutura de um material sólido, por exemplo, encontra-se diferentes tipos de defeitos, entre eles estão, as inclusões, os precipitados, as discordâncias, microtrincas, fraturas, etc. conforme mostra a Figura 3. Todas essas irregularidades básicas e/ou geométricas podem ser devidamente incluídas na teoria do campo contínuo clássico, na forma de defeitos pontuais, lineares, superficiais e volumétricos, desde que uma representação matemática apropriada seja elaborada de forma a descrever a cinética ou a dinâmica destes defeitos. Para isso vamos iniciar a nossa proposta incluindo na Mecânica do Contínuo apenas a influência geométrica dos defeitos. Para se obter uma teoria matemática de campo com irregularidades vamos definir algumas grandezas necessárias a obtenção da equação de movimento generalizada.



2.3 Densidades e potenciais generalizadas em termos de geometrias irregulares (rugosas ou porosas) A hipótese de um meio contínuo permite transformar as grandezas da Mecânica Clássica e da Mecânica dos Sólidos em densidades generalizadas, fazendo as grandezas originais se tornarem em grandezas por unidade de volume. Desta forma, uma grandeza X qualquer, que





pode ser massa, M , momento linear, p , Força, F , Energia, U , etc., deverá ser transformada na sua respectiva densidade da seguinte forma:

X . V 0 V

 X  lim

(6)

  onde X  m, p, F ,U , etc (massa, momento, força, energia, etc.) que podem ser grandezas escalares, vetoriais, tensoriais, etc.; ou seja qualquer coisa pode ser utilizada para definir uma densidade generalizada da seguinte forma:

X 

dX dX dm dX   X   . dV dm dV dM

(7)

Considere o seguinte volume irregular encapsulado, ou inscrito, dentro de um volume euclidiano regular aparente, conforme mostra a Figura 4.

Figura 4. Volume irregular V encapsulado, ou inscrito, dentro de um volume euclidiano regular aparente Vo.

Este volume aparente pode ser qualquer sólido ou forma regular que apresente um volume definido.



2.4 O conceito escalar do fração volumétrica irregular efetiva ou fração volumétrica deformada Definindo as densidades generalizadas  e o em termos da geometria euclidiana e irregular (que pode ser fractal, ou não), respectivamente, temos:

 Xo 

dX . dV0

(8)

e a densidade dentro do volume irregular (rugoso ou poroso) é dada por:

X 

dX . dV

(9)

mas pela regra da cadeia podemos escrever:

 Xo 

dX dV

 dV   dV0

 . 

(10)

Logo, a expressão da densidade euclidiana em termos da densidade no volume irregular (rugoso ou poroso) pode ser expressa como:

 Xo   X

dV . dV0

(11)

observe que o seguinte termo é válido para a conservação da massa, quando a grandeza , X  m é dada por esta. obtendo:

0 dV0   dV  dm .

(12)

diz que a massa dentro do volume considerado permanece constante: Chamando de fração volumétrica irregular efetiva ou fração volumétrica deformada  ao seguinte termo:



dV . dV0

(13)

logo temos que:

 Xo   X  .

(14)

as densidades reais e a aparente estão relacionadas uma com a outra pelo fração volumétrica irregular efetiva ou fração volumétrica deformada local. Esta última equação será utilizada dentro de outras equações que se seguem para se fazer as correções necessárias para os termos de rugosidade de superfícies e volumes.



2.5 Conceito tensorial de rugosidade O conceito de rugosidade que descreveremos, a seguir, permite o desenvolvimento de uma teoria de campo contínuo contendo irregularidades onde os teoremas fundamentais (Teorema da Divergência e Teorema de Gauss, etc) podem ser inseridos nesta teoria, de forma análoga a teoria de campo contínuo clássico regular (escalar, vetorial ou tensorial) da qual já estamos acostumados.

Figura 5. Rugosidade de uma linha ou de uma superfície em relação a uma projeção média lisa de referência.

Consideremos uma linha ou superfície rugosa se esta apresenta qualquer desvio ou “deformação” relativa a uma reta ou a um plano médio de projeção dito liso, isto é, sem rugosidade, conforme mostra a Figura 5. Observe que a rugosidade pode ser localizada ou distribuída. Quando esta “rugosidade” for do tipo volumétrica, isto é, se achar no interior de um volume qualquer, chamá-la-emos de porosidade volumétrica. Pois em termos de uma generalização dimensional esta idéia é consistente com a teoria fractal, por exemplo. É preciso não confundir a rugosidade tensorial que será definida a seguir com o tensor   gradiente de deformação F  x / X da Mecânica do Contínuo Clássica. Este último é um dos tensores que transforma as coordenadas não deformadas em coordenadas deformadas. É necessário obter uma expressão matemática que defina a rugosidade de forma local, ou seja, dependente das coordenadas. Neste sentido, podemos definir e equacionar a rugosidade de uma superfície irregular da seguinte forma:  dA dA (15)       nˆ  nˆ0  . dA0 dA0   o símbolo " " denota o produto tensorial entre dois vetores, ou seja, a  b   aib j   Aij que  é uma matriz correspondendo a um tensor de ordem 2, onde dA é o elemento de área sobre a  superfície rugosa e dA0 é o elemento de área sobre a superfície lisa.   Observe que o elemento de área rugosa dA e o elemento de área sobre a superfície lisa dA0 podem ser escritos como:  dA  dAnˆ . (16)  dA0  dA0 nˆ0 ou seja, estes elementos de superfícies estão relacionados ao vetor normal em cada ponto das superfícies rugosa e projetada, respectivamente. Portanto, cada um destes elementos depende das coordenadas das superfícies: Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar


  A  A  x, y  .   A0  A0  x, y 

(17)

A operação diferencial em Eq. (17), na verdade, dá origem a um “tensor de rugosidade” que pode ser chamado de “gradiente de superfície” e este por sua vez está relacionado ao tensor de curvatura ou a variação do vetor normal com a posição sobre a superfície rugosa (Mariano, 2003). O tensor de rugosidade em 3D pode ser escrito explicitamente em coordenadas cartesianas como:  2 A   xx  2 A      yx  2 A   zx

2 A xy 2 A yy 2 A zy

2 A   xz  2 A  . yz  2 A   zz 

(18)

A Eq. (17) pode ser reescrita como:

  A    .dA0 .

(19)

S

Esta relação nos ajuda a obter valores de área rugosas em termos da rugosidade. 2.6 Taxas, fluxos e equações de movimento generalizados em termos de geometrias rugosas Na natureza algumas grandezas dinâmicas podem ser representadas por meio de taxas e fluxos generalizados. Entre eles está a taxa e o fluxo de massa, o fluxo de calor, o fluxo de momento linear, etc., que atravessa um corpo, por exemplo. Definimos a taxa de uma determinada grandeza X como sendo:

  dX . X dt

(20)

a derivada temporal de uma grandeza X define uma grandeza chamada de derivada material no contínuo. De forma geral o fluxo de uma grandeza generalizada X que atravessa uma área  infinitesimal, dAo , em um intervalo infinitesimal de tempo, dt, é definido como:  dX d  dX  J Xo      . dA0 dA0  dt 

(21)

Esta grandeza, X , é de natureza geral e pode ser um escalar (massa M, carga elétrica q,   calor Q, energia U entropia S, etc.) ou um vetor (momento p , velocidade v , etc.) ou um tensor (tensão  , Polarização P , etc.). O sobrescrito “ 0 ” indica que a geometria considerada é a geometria euclidiana regular.



2.7 O Fluxo de generalizado, JX0, através de uma superfície rugosa   Considere a superfície irregular A e a sua respectiva projeção A0 no plano euclidiano, pelas quais passam o fluxo de alguma grandeza X  X 0 , pois sua medida absoluta não depende da rugosidade da superfície.  Seja J Xo o fluxo generalizado da grandeza X  d  dX  J Xo    0  . dA0  dt    onde A0 é a área de projeção euclidiana de A , conforme mostra a Figura 6.

(22)

Figura 6. Fluxo através de uma superfície irregular A contida em uma superfície euclidiana regular aparente



projetada A0 .

Vamos a partir de agora definir o fluxo generalizado, J X 0 , das grandezas generalizadas, X 0 consideradas anteriormente, como sendo: J X0 

d  dX 0  dA .   dA  dt  dA0

(23)

   onde X 0  m, p0 , F0 ,U 0 , etc . Desde que dX/dt é uma derivada material para as grandezas X = Q (calor), q (carga elétrica), C (concentração), p (momento), etc. O fluxos correspondentes podem ser definidos      como J X  J Q (fluxo de calor), J q (fluxo de corrente elétrica), J m (fluxo de massa), J p

(fluxo de momento = pressão + tensão tangencial), etc, dados respectivamente pelas lei de Fourier, Ohm, Fick, Newton, etc.   Normalmente o fluxo J X 0 está associado a uma densidade  X 0 e a uma velocidade v X 0 do processo ou fenômeno em questão:

   d  dX 0  dA  J X0       J X 0   X 0 vX 0 . dA  dt  dA0

(24)

Observa-se que mesmo uma densidade (como no caso da tensão mecânica que é uma densidade de energia por unidade de volume) esta também pode ser o fluxo de uma outra grandeza (no caso da tensão mecânica é um fluxo de momento). Assim densidades, fluxos e velocidades generalizadas estão associadas umas as outras. Logo para o caso onde se escolhe a  grandeza X 0  p0 correspondendo ao momento linear da partícula tem-se que o seu fluxo correspondente é dado por: Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar


    d  dp0  dA J p0       J p0  σ 0 . dA  dt  dA0

(25)

 onde J p0  σ0 é também relacionado ao tensor das tensões o qual será utilizado mais adiante.   Veja também que na escolha da grandeza X 0  p0 igual ao momento linear o fluxo J p0 do momento linear corresponde à densidade volumétrica de forças ou ao gradiente do tensor das tensões:   (26) f X 0   F0  J p0 . ou seja todas essa são formas de expressar a mesma grandeza ou a relação entre elas.   Se a área A0 que o fluxo J Xo atravessa é a área de projeção euclidiana, para passar esta equação para a descrição irregular (fractal ou não) basta incluir a derivada em relação a área de superfície rugosa da seguinte forma:   d  dX 0  dA J Xo    (27)   . dA  dt  dA0 Observe que na Eq. (27) manteve-se a descrição fenomenológica em termos da área projetada onde apenas a correção da rugosidade foi acrescentada. Desta forma o fluxo em termos da superfície de área rugosa é dado de forma análoga a Eq. (22), como:  d  dX  JX    0 . dA  dt 

(28)

Observe que embora as grandezas X  X 0 sejam equivalentemente as mesmas, os seus   fluxos generalizados J X e J Xo não são. Portanto, podemos escrever Eq. (27) da seguinte forma:    dA J Xo  J X  . dA0

(29)

onde definindo-se o tensor rugosidade   dado em Eq. (17) por:  dA dA       nˆ  nˆ0  . dA0 dA0 logo, a equação do fluxo atuante na superfície rugosa é:   J Xo  J X   .


(30)

(31)


2.8 A equação de movimento generalizada Para descrever o processo de dissipação de energia em um caminho rugoso é necessário: I) Postular que a energia para criar uma superfície no caminho rugoso é a mesma que a energia no caminho projetado, L0 , então, UVo  UV e  0   (esta é a equivalência previamente proposta por Irwin). II) Postular que os formalismos matemáticos da Mecânica do Contínuo são invariantes, isto é, são os mesmos em um caminho de rugoso e em um caminho projetado. Mantendo-se a relação entre as taxas temporais da grandeza X inalterada, a partir de Eq. (28) e Eq. (29) temos que:     dX (32)   J Xo dA0   J X .dA . dt Definindo-se o divergente como sendo:  d  dX .J Xo   dV0  dt

 . 

(33)

 d  dX .J X   dV  dt

 . 

(34)

e

Aqui é importante observar como o teorema da divergência pode ser escrito a partir de Eq. (29) e Eq. (32) em termos de volumes que envolvem ou não rugosidades ou irregularidades, fractais, obtendo-se:    J dA   . J (35)  Xo 0  Xo dV0 . e 

J

X

  .dA   .J X dV .

(36)

Substituindo Eq. (35) e Eq. (36) em Eq. (22) ou Eq. (32), temos:   dX   .J Xo dV0   .J X .dV . dt

(37)

Trocando a ordem das derivadas Eq. (33) e Eq. (34) pela regra de Schwartz para funções com derivadas contínuas podemos escrever:  d  dX  .J Xo   . dt  dV0 

(38)

 d  dX  .J X   . dt  dV 

(39)

e

substituindo Eq. (8) e Eq. (9) nas Eqs. (38) e (39), respectivamente, temos:  d .J Xo   Xo . dt

(40)



e

 d .J X   X . dt

(41)

Escrevendo a Eq. (38) ou (40) da continuidade em termos da Eqs. (29) e (11)   dA  d  dV  .  J X  nˆ  nˆ0     X .  dA0  dt  dV0 

(42)

ou em termos do tensor rugosidade   dado na Eq. (17) e do fração volumétrica irregular efetiva ou fração volumétrica deformada  dada na Eq. (13) temos:

 d . J X      X   . dt





(43)

Definimos a equação da continuidade na nova roupagem geométrica para várias fenomenologias que dependem de geometrias irregulares. Neste conjunto de fenomenologias estão os fenômenos: da difusão, transferência de calor, escoamento viscoso, deformação de sólidos, mecânica da fratura, eletromagnetismo, etc. A equação de movimento Eq. (43) pode ser ainda generalizada, porque as forças de superfícies sempre podem ser escritas como divergentes de fluxos, da seguinte forma:   (44)  f S  . J X   .





logo temos:

 onde,



  d fV   f S    X   . dt

 fV , é a somatória da densidade de forças de volume, e

densidade das forças de superfície.


(45)



 f S é a somatória da


2.9 Modificação da equação constitutiva de potenciais vetoriais – caso elástico Para os propósitos deste trabalho vamos estudar os fenômenos de potencial vetorial como a teoria da elasticidade e a mecânica da fratura. Robert Hooke descobriu a relação entre tensão e deformação de um material elástico. Na versão generalizada de sua lei para o campo de tensão-deformação para meios irregulares com porosidade devemos ter:





J X 0  . X 0 I    X 0  T  X 0 .

(46)

Observe que embora a expressão original da Lei de Hooke seja escrita em termos dos gradiente de deformação aqui ela foi escrita de forma generalizada em termos das densidade que além de incluir originalmente o gradiente de deformação inclui também o efeito das irregularidades geométricas. Logo, com a correção das irregularidades (porosidades) temos:

   dV  dV  dV   T  J Xo  .   X  I      X     X  . dV0  dV0  dV0      

(47)

explicitando a operação do gradiente sobre os termos entre parêntesis temos:  J Xo   . X       X 

 dV   dV      X    I   dVo   dV0  

 dV   dV  T    X       X dV dV  0  0

.

(48)

  dV  T  dV     X      dV0   dV0  

ou seja, a equação do fluxo de campo escalar com irregularidades atuante no volume poroso: J Xo   . X   X   I     X   X    T  X   X  T  

.

(49)

reorganizando os termos dessa equação tem-se: J Xo      . X  I     X  T  X     X            T    .   Energética

(50)

Geométrica

ou para as superfícies rugosas





J Xo  . X I     X  T  X    .

(51)

Neste conjunto de fenomenologias que seguem a equação da continuidade estão os fenômenos, do escoamento viscoso, a deformação de sólidos, mecânica da fratura, eletromagnetismo, etc. Tomando o divergente da equação (50) temos:





.J Xo  .     . X  I     X  T  X     X           T    .

(52)

ou



.J Xo    .  . X  I  .   X  T  X       . X  I     X  T  X  





.  X            T   

. (53)

ou ainda .J Xo          . X   .   X        . X      X  T  X   . (54)  X       .     .  T     . X           T   

Portanto, .J Xo          . X   .   X        . X      X  T  X   . (55)   X     2   .      . X        

Finalmente, .J Xo          . X   .   X        X  T  X    X     2   .      . X   2       


.

(56)


2.10 Equação do potencial vetorial para porosidades no domínio Para se escrever uma equação de movimento para o campo elástico com irregularidades, podemos substituir Eq. (14) na Eq. (47) ou a Eq. (49) e Eq. (11) na Eq. (40) temos:

    dV  dV  dV    d  dV  T  . .   X I           X   X      X . dV0  dV0  dV0    dt  dVo      

(57)

ou desenvolvendo os operadores internos a partir de da Eq. (57)   .  . X  

 dV   dV       X  I    dV0   dV0   

  .    X  

.

(58)

 d   dV   dV   dV  dV  T T  dV      X      X    X        X  dV0   dV0   dV0   dV0   dV0    dt 

para  ,   cte temos:   dV   dV   . . X   I  X    I   dV0   dV0    .    dV   dV        dV dV d dV T T  .  X    X      X    X      X  dV dV dV dV dt dV  0  0  0  0   0  

(59)

ou . I. X   X  I    .  X    X   T  X    X T   

. d  X  dt

(60)

executando o cálculo do gradiente e do Laplaciano com irregularidades no domínio tem-se:

  I.  . X   2. X    X .        .   X    2 X    2T  X   2T  X T    X  2T   

. d X  dt

(61)

ou

  I.  . X   2. X    X .      .   X   2 X    X .     .  T  X    d .      X   T  X   2. X T    X .T   dt

(62)

ou agrupando em termos semelhantes tem-se:   .    .     .        .      .     .        T

X

X

T

X

Energética

Geométrica

X



. (63)

d      X    X   2. X     . X   . X       X      dt T

T

Termos de Interação



ou

       .   .         2  .     X

X

Energética

X



Geométrica

.

d      X  T  X   2      . X     X      dt

(64)

Termos de Interação

Esta é uma proposta de equação de movimento para um meio elástico com irregularidades. Utilizando-se a equivalência entre rugosidade e o fração volumétrica irregular efetiva ou fração volumétrica deformada dado na Eq. (13) para descrever o potencial vetorial  X na superfície do material. Observe que a parte energética possui forma análoga à parte geométrica, ou seja, isoladamente as soluções são análogas, a menos do termo de interação. È certo que a solução de uma equação do tipo mostrada na Eq. (63) é muito complexa, por isso precisamos recorrer a métodos alternativos ou aproximados. Uma das alternativas é acrescentar correções do termo de porosidade ponto a ponto no domínio a partir da solução primitiva sem irregularidades (problema euclidiano), ou seja, corrige-se a solução do problema elástico sem irregularidades acrescentando-se termos de correções ponto a ponto no domínio para se obter a solução com irregularidades. Outra alternativa é corrigir a solução sem irregularidades com modelos geométricos fractais desde que a geometria do problema seja fractal que possa aceitar tais correções. Para o caso estático tem-se d   X   / dt  0 , logo:

  2

X

 



  .   X   .  T  X     .      .     .   T     X 

.

(65)

     X  T  X   2. X     . X   . X T    0

ou

 



  .   X   .  T  X        2  .      X  .      X  T  X   2      . X   0 2

X

(66)

Portanto,

       .   .        2   .    X

X

     X  T  X   2      . X   0


X



.

(67)


2.11 Equação do potencial vetorial para as superfícies rugosas Para se escrever uma equação de movimento para o campo elástico com irregularidades, podemos substituir a Eq. (51) na Eq. (43) e obter: d .  . X     X  T  X       X   .   dt





(68)

Logo

    . X   . X .     .     X  T  X       X  T  X  .  

.

d  X   dt

(69)

ou reescrevendo temos:

    . X   .     X  T  X    d      X  T  X   . X  .     X   dt

.

(70)

.

(71)

E finalmente para  ,   cte temos:

    . X     .   X   . T  X    d      X  T  X   . X  .     X   dt

Esta é uma proposta de equação de movimento para um meio elástico com rugosidades na superfície. Para o caso estático tem-se d   X   / dt  0 , logo:

    . X   

 .   X   .  T  X  2



  .

(72)

     X  T  X   . X  .   0 Reescrevendo esta equação temos:



  . X    .   X   .  T  X     X    X   . X T

  .   0 .

(73)

 

O que resulta em duas equações separadas: Uma para problema elástico sem rugosidade:    . X    .   X   .  T  X      X  T  X   . X k  0 .



 



(74)

e outra apenas para a rugosidade:

 .   k    0 .

(75)

Isto nos faz acreditar que o problema que apresenta apenas rugosidade na superfície com interior do domínio sólido, pode ser resolvido apenas com funções de correção Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar


geométrica, como no caso de uma trinca rugosa como será mostrado mais adiante nos resultados. A solução da Eq. (75) é do tipo:  (76)     0 exp k .rn .





Isto significa que o efeito da rugosidade sobre o campo do potencial em questão é atenuado  exponencialmente à medida que um observador se afasta da borda rugosa  rn  0  para o   interior do domínio do campo  rn    , onde rn é o raio vetor tomando na direção normal a cada ponto sobre a superfície rugosa.



2.12 Solução das Equações do Potencial Vetorial com Irregularidades Para solucionar parte dessa equação, nós devemos considerar a equivalência entre a rugosidade superficial e a porosidade mostrada na equação representada a seguir:  X   . X  I   

( 77)

Considerando que a rugosidade possui uma dependência dada por (75) podemos utilizar essa dependência da seguinte forma:  .  1  k    .  .  

( 78)

Supondo que de forma análoga a rugosidade, a densidade do potencial  X 0 também se comporta da seguinte forma:   X ( 79)  k   X 1 X . X Logo substituindo (70) em (68) tem-se:      I    k  0 .

( 80)

     Ik    k .

( 81)

ou

Usando o resultado (67) tem-se:       Ik   0 exp  k .rn k .





( 82)

Aplicando a técnica do fator integrante tem-se:       exp k .r   exp k .r  Ik  exp k .r  0 exp  k .rn k .

 

 

 



onde podemos reescrever o lado esquerdo e o lado direito como:       exp k .r     0 exp  k .  rn  r   k .  

 



( 83)

( 84)

ou ainda integrando dos dois lados temos:       exp k .r     0 exp  k .  rn  r   k .dr .

( 85)

Logo o efeito da porosidade sobre o campo pode ser expressa como:         exp k .r   0 exp  k .  rn  r   k .dr .

( 86)

  



Observe que a integral no lado esquerdo corresponde a uma das representações da função delta de Dirac. Logo, se os limites de integração envolvem um domínio que vai desde  ,  , a solução da equação ( 86) será:  ( 87)    0 exp k .rn .







Este resultado mostra que assim como a rugosidade o efeito da porosidade sobre o campo se esvaece exponencialmente à medida que se afasta da periferia da irregularidade de domínio (poro) na direção de regiões regulares. 2.13 Solução das Equações do Fluxo Vetorial com Irregularidades Reescrevendo (74) em função de (46) temos:   .J X  k .J X  0 .

( 88)

De forma análoga a (67) temos:    J X  J 0 exp k .rn .





 Podemos reescrever o produto escalar k .r na superfície como sendo:   1 k .rn     .  .rn .





( 89)

( 90)

1

Observe que o inverso do tensor de rugosidade   aparece no expoente operando sobre um vetor .  que seleciona na operação do divergente apenas as componentes nas direções puras, evitando as direções “cisalhantes” onde se mistura as componentes do tensor de  rugosidade   . Observe com isso que se o vetor k for nulo obtém-se de volta o problema euclidiano de superfície lisa.



3

SIMULAÇÃO COMPUTACIONAL E METODOLOGIA

Neste artigo vamos definir o problema sem irregularidades (será chamado de P1) e com irregularidades (será chamado de P2) e vamos solucioná-los por pelos menos dois métodos numéricos (Diferenças Finitas e Elementos Finitos) para obter um crivo numérico seguro que possa garantir a correta solução desses problemas. Depois procuraremos inferir a solução P2 a partir de P1 por alguma aproximação ou correção a qual chamaremos esse de problema equivalente (será chamado de PE). Desta forma poderemos obter o conhecimento da influência das irregularidades em um problema elástico.



4

RESULTADOS COMPUTACIONAIS

Analisando-se agora o efeito da rugosidade de uma trinca sobre o campo de tensão elástico em um material frágil, observa-se o que era esperado pelas Eqs. (74) e (75), ou seja, uma perturbação geométrica na intensidade do campo ao redor da falha, mas que se atenua à medida que se afasta da rugosidade da trinca na direção do interior do material conforme mostra a Figura 7 para a componente  XX .

Figura 7. Imagem do resultado da simulação numérica do campo de tensão sem rugosidade. Sigma XX sem rugosidade e com rugosidade.

Para grandes distancias longe do efeito da rugosidade superficial espera-se que os valores do campo elástico para o caso sem e com rugosidade se aproxime um do outro. O mesmo observa-se para as outras componentes do campo como no caso do cisalhamento dado pela intensidade da componente  XY , conforme mostra a Figura 8.



Figura 8. Imagem do resultado da simulação numérica do campo de tensão sem rugosidade. Sigma XY sem rugosidade e com rugosidade.

A componente do campo que apresentou menor variação quanto a sua forma foi a componente  YY , conforme mostra a Figura 9. Mesmo para as direções principais  1 e  2 as formas do campo sofrem o mesmo tipo de perturbação na sua intensidade devido a geometria rugosa da trinca. Quando se gráfica a intensidade da componente  YY do campo em função do raio na frente da trinca para um ângulo   0 obtém-se os gráficos da Figura 10. Esta figura mostra o campo de tensão sofreu um efeito na sua intensidade devido a geometria rugosa da trinca.

Figura 9. Imagem do resultado da simulação numérica do campo de tensão sem rugosidade. Sigma YY sem rugosidade e com rugosidade.



Figura 10. Gráfico da intensidade do campo de tensão na ponta da trinca em função da distancia r sem e com rugosidade.

Uma vez que a rugosidade afastada da região de campo de tensão mais intensa, a rugosidade só influencia o campo de tensão  yy se esta acontecer na direção perpendicular ao raio de curvatura na ponta de uma trinca. Tensao YY na frente da trinca sem rugosidade

1,40E+08 1,20E+08

Tensao YY

1,00E+08

CT1

8,00E+07

CT2 CT3

6,00E+07

CT4 4,00E+07

CT5 CT6

2,00E+07 0,00E+00 0,00E+00 -2,00E+07

5,00E-01

1,00E+00

1,50E+00

2,00E+00

2,50E+00

Raio r para teta=0

Figura 11. Gráfico da intensidade do campo de tensão na ponta da trinca em função do raio de curvatura sem rugosidade.



Tensao YY na frente da trinca com rugosidade 1,40E+08 1,20E+08

Tensao YY

1,00E+08

CT1

8,00E+07

CT2 CT3

6,00E+07

CT4 4,00E+07

CT5

2,00E+07

CT6

0,00E+00 0,00E+00 -2,00E+07

5,00E-01

1,00E+00

1,50E+00

2,00E+00

2,50E+00

Raio r para teta=0

Figura 12. Gráfico da intensidade do campo de tensão na ponta da trinca em função do raio de curvatura com rugosidade.



5

DISCUSSÃO

5.1 Aspecto geral do campo de tensão ao redor de uma trinca rugosa Classicamente, um meio elástico frágil possui um campo tensorial assintótico, cujo expoente de singularidade, fornece uma função homogênea para o campo de tensão na ponta de uma trinca em função da distância r na frente da trinca (Hutchinson 1968; Rice & Rosengren, 1968) do tipo:

 I , II ,III  r  ~

K I , II , III r

n

.

(91)

n 1

onde n  1 , é o grau de homogeneidade da função do campo de deformação do meio elástico dado pela lei de Hooke Modelos da literatura têm discutido a singularidade do campo de tensão de uma trinca rugosa fractal (Balankin, 1994; Mosolov 1991, 1992, 1993, Yavari, 2002). Mosolov (1991, 1992) foi o primeiro a fazer conjecturas sobre o campo de tensão de uma trinca fractal rugosa em uma meso-escala. Mosolov (1993) sugeriu que se esse campo elástico deveria possuir um expoente de singularidade fracionário associado a dependência assintótica com a distância r na frente da trinca dado pela expressão:

 I , II ,III  r  ~

K I , II , III r

.

(92)

onde:



2  DB H  . 2 2

(93)

Yavari (2002) discutiu a expressão proposta por Mosolov (1993) e também a proposta de Balankin (1994), Yavari (2002) acredita que a expressão do campo pode tensão na ponta de uma trinca fractal deve satisfazer a seguinte expressão:

K  ~ I . r

(94)

onde:



2H 1 . 2H

(95)

e ainda Yavari (2002) afirma que singularidade deste campo depende do modo de fratura imposto sobre a trinca. Quando se utiliza um mapeamento do campo elástico ao redor de uma trinca a partir de uma expressão obtida por uma solução analítica do campo de tensão observa-se que uma mudança no expoente   1/ 2 para um outro expoente  dado pela Eq. (95) muda-se brutalmente os aspecto geral do campo elástico ao redor de uma falha, conforme mostra a Figura 13.



Figura 13. Campo de Tensão

 yy

no modelo fractal para o Modo I de Fratura com singularidade 1/ r

Modelo de Mosolov   0.2; 0, 4; 0,5 na parte superior e



.

  0,5; 0, 6; 0,8 na parte inferior.

Esta figura mostra como varia o aspecto do campo de tensão elástico ao redor de uma falha se a singularidade muda. 95-100 S43

85-90

S39 S37

80-85

S35 S33 S31 S29 S27 S25 S23 S21

91

88

85

82

79

76

73

70

67

64

61

58

55

52

49

46

43

40

37

34

31

28

25

22

19

16

13

60-65 55-60 50-55 45-50

35-40

S13

30-35 25-30

S7

20-25

S5

15-20

S1 7

65-70

40-45

S3 10

70-75

S15

S9

4

75-80

S19 S17

S11

1

90-95

S41

10-15 5-10 0-5

Figura 14. Exemplo preliminar de pontas de rugosidade penetrando em regiões intensas da vizinhança do campo escalar ou vetorial de tensão da ponta principal gerando outras zonas plásticas (cardióide para tensão plana) simulado pelo método de Diferenças Finitas para um campo escalar.

Contudo, resultados preliminares do nosso estudo numérico do campo de tensão na ponta de um entalhe com rugosidade senoidal para o problema elástico sem propagação de trinca, mostra que o campo de tensão possui uma dependência assintótica com a posição r na frente da trinca que varia desde uma valor  máximo que depende do material até uma valor  mínimo igual a 1/ 2 , que corresponde ao valor clássico. Ou seja, nosso estudo preliminar, Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar


mostra que a singularidade do campo de tensão na ponta da trinca varia como se fosse um multifractal e que não há um único expoente fractal na ponta da trinca, mas este varia com a posição r . Um outro resultado preliminar simulado por Diferenças Finitas e Elementos Finitos, mostra que pontas de rugosidade que penetram regiões intensas da vizinhança dos campos escalares ou vetoriais de tensão da ponta principal da trinca geram outras zonas plásticas (cardióide para tensão plana) conforme mostra a Figura 14.



6

CONCLUSÕES

Os resultados analíticos obtidos a partir das equações do campo elástico com irregularidades geométricas, apresentados no capítulo III, mostram que uma irregularidade, quer na superfície (rugosidade), quer no interior do domínio (porosidade), produz uma perturbação no campo que se esvai exponencialmente à medida que se afasta da irregularidade para o interior do meio. O fato desse resultado de decaimento da perturbação do campo ser exponencial com a distância, é porque as irregularidades consideradas são estáticas, no espaço e no tempo. Caso contrário, se elas se movessem, ou se elas oscilassem de tamanho no tempo, sua perturbação dinâmica interagiria diretamente com o campo produzindo efeitos não-lineares produzindo ondas elásticas de tensão/deformação que se propagariam pelo meio. Todos esses resultados apontados pela descrição analítica do problema parecem confirmar a intuição tida anteriormente. Pois se uma trinca é a inserção de uma perturbação geométrica estática no campo elástico, que produz uma singularidade no campo à medida que o raio de curvatura tende a zero. Torna plausível pensar que se essa “perturbação” ou irregularidade geométrica possuam efeitos esvanescente uma vez que ela é estática. Para o caso dinâmico, que embora não tenha sido simulado, nem estudado e nem simulado nesse trabalho pode ser inferido a priori como sendo de uma perturbação dinâmica que interagem como campo de na forma de ondas elásticas, podendo levar as instabilidades já apontada experimentalmente por alguns autores (Fineberg, 1991, 1992). i) É possível unificar os critério de fratura: 1) monocristal, 2 ) Inglis,3) Griffith,4) Irwin acrescentado um termo multiplicativo de rugosidade que unifica todos eles. ii) Carregamento simétrico produz campo simétrico iii) Portanto como a rugosidade de uma face da trinca é a complementar da outra, campos assimétricos são mais afetados pela rugosidade do que campos simétricos. iv) As tensões SIGMAYY, SIGMA1, são pouco afetadas pela rugosidade da trinca em qualquer caso, v) Mas as tensões SIGMAXX e SIGMA2 são mais afetadas pela rugosidade da trinca. vi) Rugosidade que possuem pontas que penetram o campo de tensão produzem efeitos análogos a ponta principal (formação de leminiscata para tensão plana e cardióide para tensão plana). Para uma trinca rugosa, observa-se que novas cardióides surgem além daquela da trinca principal, quando protuberâncias dessa rugosidade penetram dentro de uma região mais intensa do campo de tensão na ponta da trinca. vii) Isto os leva a concluir que a rugosidade permite uma forma alternativa de dissipação que pode levar a bifurcação da trinca em processo de altas taxas de deformação para trincas rápidas, por exemplo. Este resultado corrobora o que já havia sido demonstrado experimentalmente por Fineberg (1992). viii) A variação do raio de curvatura alarga o campo de tensão mantendo o padrão de variação de intensidade. ix) A singularidade dom campo de tensão na ponta da trinca não possui uma única dimensão fractal de rugosidade mas depende da posição na frente da ponta da trinca.



REFERENCES Alves Lucas Máximo, Silva Rosana Vilarim da, Mokross Bernhard Joachim, The influence of the crack fractal geometry on the elastic plastic fracture mechanics. Physica A: Statistical Mechanics and its Applications. 295, 1/2:144-148, 12 June 2001. Alves Lucas Máximo, Fractal geometry concerned with stable and dynamic fracture mechanics. Journal of Theorethical and Applied Fracture Mechanics, 44/1:44-57, 2005. Alves, Lucas Máximo, Silva, Rosana Vilarim da, Lacerda, Luiz Alkimin De, Fractal modeling of the J-R curve and the influence of the rugged crack growth on the stable elastic-plastic fracture mechanics, Engineering Fracture Mechanics, 77:2451-2466, 2010. Alves – Alves, Lucas Máximo; et al., Verificação de um Modelo Fractal do Perfil de Fratura de Argamassa de Cimento, 48º Congresso Brasileiro de Cerâmica, realizado no período de 28 de junho a 1º de julho de 2004, em Curitiba – Paraná. Alves, Lucas Máximo; Lacerda, Luiz Alkimin De, Application of a generalized fractal model for rugged fracture surface to profiles of brittle materials , artigo em preparação, 2010. Bammann, D. J. and Aifantis, E. C., On a proposal for a Continuum with Microstructure, Acta Mechanica, 45:91-121, 1982. Balankin , A.S and P. Tamayo, Revista Mexicana de Física 40, 4:506-532, 1994. Barenblatt, G. I. The mathematical theory of equilibrium cracks in brittle fracture, Advances in Applied Mechanics, 7:55-129, 1962. Blyth, M. G. , Pozrikidis, C., Heat conduction across irregular and fractal-like surfaces, International Journal of Heat and Mas Transfer, 46: 1329-1339, 2003. Carpinteri, A.; Puzzi, S., Complexity: a new paradigm for fracture mechanics, Frattura ed Integrità Strutturale,10, 3-11, 2009, DOI:10.3221/IGF-ESIS.1001 Dyskin, A. V., Effective characteristics and stress concetrations in materials with self-similar microstructure, International Journal of Solids and Structures, 42:477-502, 2005 Duda, Fernando Pereira; Souza, Angela Crisina Cardoso, On a continuum theory of brittle materials with microstructure, Computacional and Applied Mathemathics, 23, 2-3:327-343, 2007. Engelbrecht, J., Complexity in Mechanics, Rend. Sem. Mat. Univ. Pol. Torino, 67, 3:293-325, 2009 Fineberg, Jay; Gross; Steven Paul; Marder, Michael and Swinney, Harry L. Instability in dynamic fracture, Physical Review Letters, 67, 4:457-460, 22 July 1991. Fineberg, Jay; Steven Paul Gross, Michael Marder, and Harry L. Swinney, Instability in the propagation of fast cracks. Physical Review B, 45, 10:5146-5154 (1992-II), 1 March, 1992. Forest, S. Mechanics of generalized continua: construction by homogenization, J. Phys. IV, France, 8:39-48, 1998. Hyun, S. L.; Pei, J. –F.; Molinari, and Robbins, M. O., Finite-element analysis of contact between elastic self-affine surfaces, Physical Review E, 70:026117, 2004. Hornbogen, E.; Fractals in microstructure of metals; International Materials Reviews, 34. 6:277-296, 1989. Hutchinson, J.W., Plastic Stress and Strain Fields at a Crack Tip., J. Mech. Phys. Solids, 16:337-347, 1968. Irwin, G. R., “Fracture Dynamics”, Fracturing of Metals, American Society for Metals, Cleveland, 147-166, 1948. Lazarev, V. B., Balankin, A. S. and Izotov, A. D., Synergetic and fractal thermodynamics of inorganic materials. III. Fractal thermodynamics of fracture in solids, Inorganic materials, 29, 8:905-921,1993. Copyright © 2010 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar


Mariano Paolo Maria o, Influence of the material substructure on crack propagation: a unified treatment, arXiv:math-ph/0305004v1, May 2003. Morel, Sthéphane, Jean Schmittbuhl, Juan M.Lopez and Gérard Valentin, Size effect in fracture, Phys. Rev. E, 58, 6, Dez 1998. Mosolov, A. B., Zh. Tekh. Fiz. 61, 7, 1991. (Sov. Phys. Tech. Phys., 36, 75, 1991). Mosolov, A. B. and F. M. Borodich Fractal fracture of brittle bodies during compression, Sovol. Phys. Dokl., 37, 5:263-265, May 1992. Mosolov, A. B., Mechanics of fractal cracks in brittle solids, Europhysics Letters, 24, n. 8:673678, 10 December 1993. Muskhelisvili, N. I., Some basic problems in the mathematical theory of elasticity, Nordhoff, The Netherlands, 1954. Panagiotopoulos, P.D. Fractal geometry in solids and structures, Int. J. Solids Structures, 29, 17:2159-2175, 1992. Panin, V. E., The physical foundations of the mesomechanics of a medium with structure, Institute of Strength Physics and Materials Science, Siberian Branch of the Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, 4:5-18, Plenum Publishing Corporation, 305 - 315, April, 1992. Ponson, L., D. Bonamy, H. Auradou, G. Mourot, S. Morel, E. Bouchaud, C. Guillot, J. P. Hulin, Anisotropic self-affine properties of experimental fracture surfaces, arXiv:condmat/0601086, 1, 5 Jan 2006. Rice, J. R., A path independent integral and the approximate analysis of strain concentrations by notches and cracks, Journal of Applied Mechanics, 35:379-386, 1968. Rupnowski, Przemysław; Calculations of J integrals around fractal defects in plates, International Journal of Fracture, 111: 381–394, 2001. Su, Yan; LEI, Wei-Cheng, International Journal of Fracture, 106:L41-L46, 2000. Tarasov, Vasily E. Continuous medium model for fractal media, Physics Leters A 336:167174, 2005.. Trovalusci, P. and Augusti, G., A continuum model with microstructure for materials with flaws and inclusions, J. Phys. IV, France, 8:353-, 1998. Xie, Heping; Effects of fractal cracks, Theor. Appl. Fract. Mech., 23:235-244, 1995. Xie, J. F., S. L. Fok and A. Y. T. Leung, A parametric study on the fractal finite element method for two-dimensional crack problems, International Journal for Numerical Methods in Engineering, 58:631-642, 2003. (DOI: 10.1002/nme.793) Yavari, Arash, The fourth mode of fracture in fractal fracture mechanics, International Journal of Fracture, 101:365-384, 2000. Yavari, Arash, The mechanics of self-similar and self-affine fractal cracks, International Journal of Fracture, 114:1-27, 2002, Yavari, Arash, On spatial and material covariant balance laws in elasticity, Journal of Mathematical Physics, 47, 042903:1-53, 2006. Weiss, Jérôme; Self-affinity of fracture surfaces and implications on a possible size effect on fracture energy, International Journal of Fracture, 109: 365–381, 2001.


Chapter II

ELASTIC LINEAR FRACTAL STRESS FIELD AROUND A RUGGED CRACK

A fractal model of the stress field around a rough crack Lucas Máximo Alves, Marcio Ferreira Hupalo, Selauco Vurobi Júnior, Luiz Alkimin de Lacerda State University of Ponta Grossa – UEPG, Sector of Agricultural Sciences and Technologies, Department of Materials Engineering

Abstract The use of fractal measure theory to model fracture phenomena is of great interest in the mathematical development of Fractal Fracture Mechanics. However, the fractal models for fracture proposed by several authors presuppose a fractal crack with an infinite range of self-affine scales that cannot be verified experimentally. It is known that a real crack has limits imposed by a finite number of fractal scales ranging from an upper fractal cut-off scale to a lower fractal cut-off scale. In a real fracture, the finite limits of fractal scales are always compatible with the minimum microcrack and with the length of the macroscopic crack in the sample. Thus, the lower fractal cut-off scale is given by the characteristic size of a microcrack, while the length of the macroscopic crack on the specimen indicates the upper fractal cut-off scale. Therefore, fractal models of fracture phenomena require a more realistic approach that takes into account the limits of scales and singularity exponents, and hence, the fractal scales that are actually found in a real fracture. This chapter presents a model of a stress and displacement field around a rough crack tip in brittle and ductile materials, using a pre-fractal. This approach allows for a more realistic fractal scale of a real fracture. A special stress vector was defined for rough surfaces, and it was found that to consider the pre-fractal also results in a different asymptotic limit for the singularity of the stress field at the crack tip in brittle materials. The asymptotic limit obtained here differs from the order of singularity presented by other authors. Therefore, other consequences, such as fracture instability, are also included in the mathematical model presented here. A generalization of the stress fields for cases of brittle and ductile materials is proposed. Changes in fracture geometry are caused by the roughness at the crack tip, as shown by the mapping of simulated stress fields around a crack tip. The fractal stress field functions are mapped at different orders of magnitude of singularity, indicating that the qualitative aspects of these fields, alone, do not suffice to determine which model presents the best fit to experimental results. Therefore, the model is validated based on quantitative experimental measurements.

Keywords: stress field, fractal fracture mechanics, fractal dimension, Hurst dimension, stress intensity factor

1. Introduction In classical fracture mechanics theory, a brittle material has an asymptotic stress field whose singularity exponent depends on the distance from the crack tip, which Hutchinson [1] defines as:

 I , II , III  r  ~

K I , II , III r

n

 p / n  1

(1)

n 1

where n  1 is the degree of homogeneity of the distribution of the elastic stress field given by Hooke’s Law. This singularity is discussed in the literature on rough fractal cracks. Mosolov [2,3] attempted to explain crack growth under compression based on the fractality of cracks. He was the first researcher to study and theorize about changes in the stress field singularity magnitude according to the fractality of rough crack surfaces in mesoscale dimensions. He showed that the stress at the crack tip of a fractal crack under uniform compressive stress is unique. Mosolov [2,3,4,5] stated that the singularity in the stress field in front of a rough fractal crack should be corrected by means of a fractional exponent. He also suggested that the elastic field ahead of the crack tip should have a fractional singularity exponent associated to the asymptotic dependence on the distance, represented by [2-3]:

 I , II , III  r  ~

K I , II , III r n n 1

 p / n  1   I , II , III  r  ~

K I , II , III r

(2)

This is because the stress field is primarily responsible for generating the noise of a crack or fracture surface in a solid body subjected to the action of a strain load, which is transferred for the formation of new surfaces through the release of elastic or elastic-plastic energy in the fracture, as follows:

  r   G  L   2 eff

(3)

After Mosolov’s observation [2,3,4,5], Borodich [6,7,8] began to develop a fracture criterion involving the roughness of a crack based on fractal theory. These two authors established the mathematical relationships between the elastic stress field around a crack or fracture surface and the fractal roughness exponent of a fracture surface, using the dependence of the fractional exponents of singularity of the field at the crack tip and the fractional dependence of the fractal exponents of the scaling of fracture surfaces. Using the Griffith criterion, and considering the fact that the actual length of a fractal crack is larger than its apparent size, they came up with the correct asymptotic expression for a self-similar fractal crack in Mode I. They did this by comparing the stress field, as shown in the following equation:

U L  L  ~ r 2 L2  U  ~ 2 eff LD

(4)

where D is the so-called Hausdorff-Besicovitch fractal dimension [9,10] of a self-similar crack. Note that there are many definitions for the fractal dimension. All these definitions give the same fractal dimension for a self-similar fractal. Cherepanov and Balankin [11,12,13] attempted to define a relationship between the exponent  suggested by Mosolov [2,3] and the fractal topology of a rough crack. Later, Balankin [12,13] published a series of papers proposing a complete modification of fracture mechanics, from linear elasticity theory to nonlinear elastic fracture mechanics. Based on a dimensional analysis such as that shown in (4), Balankin [13] found in self-affine fractal cracks the same degree of singularity of the stress field as that proposed by Mosolov for a self-similar fractal crack. The relationship proposed by Balankin [12, 13] was as follows:

 I , II , III  r  ~

K I , II , III r

 2  DB H  2  2 p / d  1 p/   3  DB  H p / d  2  2 2

(5)

where DB is the so-called Hausdorff-Besicovitch [9,10] fractal box dimension for a selfsimilar crack. Thus, Mosolov and Balankin established mathematical relationships between the elastic stress field around the crack and the roughness parameter of the fracture surface. It should be noted that Equation (5) results in   H 2 in the two aforementioned cases. The consequences of this mathematical result for fracture mechanics could not be fully confirmed through experimental results. On the other hand, it was also not possible to establish a relationship between Balankin’s model [12,13] and the other fundamental consequences he proposed for fractures. Realizing the limitations of the Balankin model [12, 13], Yavari [14,15,16] proposed a new approach to stress field singularity exponents and developed his model to its ultimate consequences. He even proposed two new fracture modes and discrete fracture mechanics that combined the fractal with quantum aspects of the fracture described by Pugno and Ruoff [17]. Yavari [14,15,16] also discussed the earlier expressions proposed by Mosolov [7] in (2), as well as the model proposed by Balankin [12]. However, unlike Balankin [12, 13], he found a relationship between the singularity exponent using (4) and substituted the boxdimension DB by the divider dimension, DD  d H . He argued that the stress field at the fractal crack tip should satisfy the following expression:

 r  ~

K I , II , III r

 2  DD 2 H  1  p/ d 1  2H p/  2  3  DD  3 H  2 p / d  2  2 2H

(6)

Yavari [14,15] then proposed a systematic approach for calculating the magnitude

of stress singularity for fractal cracks, using the force lines method, which is applicable to all fracture modes. He considered the three classical fracture modes for fractal cracks and made a more in-depth investigation of the problem of stress singularity at the crack tip. In contrast, however, this chapter proposes a new approach to the problem of stress field singularity, considering a geometrical instability coefficient,  L , to correct the Cauchy stress field around a crack, which, in Euclidean geometry for a fractal stress field, is defined as follows:

 

0 L

(7)

where the new singularity exponent,  , for the stress field is given by:

 I , II , III  r  ~

K I , II , III r

p/ 

3  2H p / d  1, 2 2

(8)

Equations (7) and (8) are consistent with the experimental measurements and mathematical developments made to date [18,19,20,21,22,23,24,25]. These mathematical results are considered the most acceptable and realistic corrections and are therefore adopted and developed throughout this chapter. The proposal presented here is based on the relationship between the rough crack length and its Euclidean projection, which is given by Alves [18,19,20,21,22,23,24,25], i.e., the rough crack length is considered to be given by L ~ l0 n1 H , where l0 is the minimum permissible size for the advance of a crack. Therefore, it was also considered that the box dimension, D B , is able to portray the irregularities associated with microregions of strain around a two-dimensional (2D) crack and the threedimensional (3D) fracture surface. In terms of the crack length or area of the fracture surface, our proposal differs from that of Yavari [14,15,16], who considered that the rough 1

crack length is of the type: L ~ l0 n H , using the divider dimension DD , which represents only the irregularities on the noisy line of the crack or on the surface fracture, without considering the effects of strain in the microregions around the crack.

2. Theoretical development: analysis of the influence of roughness on the stress field at the crack tip The roughness of a real crack extends within a finite limit ranging from a lower to an upper fractal cut-off scale. The lower fractal cut-off scale is given by the characteristic size of a microcrack, while the length of the macroscopic crack on the specimen indicates the upper fractal cut-off scale. To model the real movement of the crack tip, one must consider a infinitesimal element of length of a rough crack with its corresponding flat Euclidean projection, as shown in Figure 1. Thus, all the quantities previously considered and associated with the projected length of a crack must be rewritten in terms of the actual rough crack length. Hence, the stress field will be modified by some function of the type:

   0 f  L L0 

(9)

Consequently, all the other quantities of fracture mechanics will also be modified analogously. As can be seen in Figure 1, a crack grows with variations in the coordinates of its tip that follow a rough path, having both a microscopic and a macroscopic velocity, which produce the surface roughness and its macroscopic path, respectively. .

Figure 1. Rough crack with an energy-equivalent flat projection. Figure adapted from Ohr [26].

Figure 2. Influence of surface roughness of a crack and the appearance of the stress field at its tip, shown by a series of electron micrographs depicting the interaction between a crack and a molybdenum grain boundary. As the crack approaches the boundary, a small crack is nucleated and eventually joins the main crack. Figure adapted from Ohr [26].

Figure 2 presents experimental results obtained by Ohr in 1985 [26]. These results demonstrated experimentally that the aspect of the stress field in front of a crack appears deformed because of its roughness. Also, according to Xin [27], inclined crack models show that the appearance of the stress field ahead of the crack tip is distorted relative to a noninclined crack. This supports the proposal of this chapter that roughness affects both the intensity of the field as its geometric aspect.

The geometric instability coefficient of a crack with a fractal roughness, which is given in (7), can be generalized based on the following arguments: (i) experimental observations; (ii) previous arguments about the influence of roughness on intensity; (iii) geometric aspect of the stress field at the crack tip; and (iv) the coherent mathematical development; and a correction can be proposed for the mathematical solution of the rough stress field, using a quantity  L . This quantity, which gives the ratio between rough and smooth lengths for global influence, multiplied by the ratio of their differences for local influence, can be called the geometric instability coefficient of the crack, as follows:

L 

L dL L dL 0 0

(10)

global local

This coefficient reflects the interaction between the local and global aspects of the crack length. Considering a self-affine fractal crack, according to Alves et al. [18,23,22], the coefficient of geometric instability, which depends on the roughness at the crack tip, is given as follows:

 L  H , l0 , L0  

l  L dL 1   1   2  H   0  L0 dL0 2   L0  

2H 2

  

(11)

Since there is a relationship between the distance r in front of the crack and the fractality at its tip in the region of the concentrated stress field, the quantities r and rough length L are closely related, following the “principle of equivalence of the closed crack.” Hence, it is possible that the stress field intensities,  , begin to depend on both r and L , to the point that a single dependency arises between these quantities, as shown by Yavari [14,15]. However, this is only possible if the asymptotic limits of the crack proposed by Yavari [14,15] actually occur in a real crack; otherwise, a simple roughness behaves exactly like a local field strength modifier and not like a change in the degree of singularity. In

 2  H  l0

regions

L0 

2 H 2

near

the

crack

tip

( L, L0 ~ r  x, y   0 ),

the

term

 1 , hence, equation (11) can be written as:

 L  H , l0 , L0  

l  L dL  2  H  0  L0 dL0  L0 

2H 2

(12)

The geometric instability coefficient of a crack in its complex form can be defined as:

l   L  z, H    2  H   0  z

2H 2

(13)

where the Cartesian coordinates x and y originate at the crack tip. The coefficient of i

instability of the fracture surface in its polar form, z  re , can also be written as:

l   L  H , r,    2  H   0  r

2H 2

 cos  2  2H    i sen  2  2H   

(14)

Therefore, one can propose that the complex function of Westergaard, Z [28], which depends on z  x  iy , should be modified by adding the coefficient of instability of the fracture surface given in (10) as:



1/2 L

l   z, H    2  H   0  z

2H  2 2

(15)

i

Its polar form, z  re , can also be considered as:



1/ 2 L

l  H , r ,    2  H   0  r

2H 2 2

   2  2H     2  2H       i sen    cos   2 2     

(16)

Figure 3. Complex map of the variation in fractal roughness of the crack given by equation (16) for: a) b) H  0.0 with      ; H  0.2 with 0.8    0.8 ;c) H  0.4 with 0.6    0.6 ; d) H  0.5 with  0.75    0.75 ; e) H  0.6 with 0.4    0.4 ; f)

H  0,8 with 0.2    0.2 e l0  0, 001 , for the interval of 50  r  50 .

The graphic of  L1/2 as a function of x  iy is shown in Figure 3. In this figure, note that there is a forbidden area with a value of r  l0 and r  L0 , and also an oscillation angle of the crack,  min     max , which varies according to the value of the Hurst exponent H . For the value H  1.0 that corresponds to a smooth crack, the oscillation angles are zero,  min     max  0 , because there is no angle variation in a smooth crack. This is in line with the intuitive idea that a crack generally oscillates around the direction of propagation. The results shown in Figure 3 confirm a previous finding that a rough crack has limit angles of propagation for opening of its angular oscillations [25]. Now, in this chapter, these limit angles have been related with the Hurst exponent of crack roughness, where the entropy for a rough line is given by:

    h0 h0l0 S  k  ln  H cos     H L cos      0   0 0 

(17)

for      , where   1  H . The coefficient of instability of the complex fractal,

 L1/2 , shows that the stress field will be affected by the degree of singularity and by the intensity mapping across the plane stress around the crack.

3. Modeling the fractal stress field around a rough crack There are three independent movements that correspond to the three fundamental fracture modes, as pointed out by Irwin [29]. These basic fracture modes are usually called Mode I, Mode II, and Mode III, respectively, and any fracture mode in a cracked body may be described by one of the three basic modes, or by combinations thereof. The Airy-Westergaard function for the stress field with fractal roughness can be determined based on the foregoing arguments about the influence of roughness on the intensity and the geometrical aspect of the stress field at the crack tip. Based on the mathematical development performed here, we propose that the Westergaard complex function,  , which depends on z  x  iy , should be modified by adding the parameter

 L1/ 2 

L

L0  dL dL0 , as follows:

 I , II , III  z,  L  z , H   

K I , II , III 1/ 2

 2 z   z, H  

(18)

L

where the three parameters K I , K II and K III , are called stress intensity factors that correspond to the opening, sliding, and tearing (anti-plane shearing) fracture modes, respectively. These expressions indicate that the stresses have an inverse square root singularity at the crack tip and that the stress intensity factors K I , K II and K III measure the intensities of the singular stress fields of opening, in-plane shearing, and anti-plane

shearing, respectively. The stress intensity factor, a new concept in solids mechanics, plays an essential role in the study of the fracture strength of cracked solids. Substituting (15) in (18), one has:

K I , II , III

 I , II , III  H , z  

1 2 H 2 2   l   2 z  2  H   0   z  

(19)

and grouping similar terms, one has:

I , II , III

K I , II , III

 l0  H, z    2  2  H  l0  z 

3 2 H 2

(20)

Considering the above conditions, we propose the following stress function due to the dominant term of the stress function around the tip of a fractal crack. Therefore, the potential function  , depends on z  x  iy and the fractal exponent of the crack is

  3  2H  2 i

,

By replacing the complex variable z for its polar form, according to Euler, z  re where r and  are the polar coordinates measured from the crack tip, one can rewrite the

function  as:

K I , II , III

 l0  I , II , III  H , z     2  2  H  l0  r  where I , II , III  z  and Í , II , III  z  

3 2 H 2

  3  2H       3  2 H      i sen    cos  2 2      

dI , II , III dz

(21)

are the Westergaard stress functions, which are

holomorphic, i.e., analytical, and satisfy the Cauchy-Riemann conditions.

3.1. Calculation of the stress field with fractal roughness 3.1.1. Solution of stress fields and displacement around the tip of a fractal crack in Mode I  GIC  K IC  In Mode I, one finds a two-dimensional tensile stress at infinity N :

 x  , t x    , t y  0   xx    ,  xy  0 N  .  y  , t x  0, t y      yy   xx ,  xy  0

(22)

Note that we have applied a constant load stress in the x direction in the infinite, which has no corresponding forces. This particular tensile stress is introduced to simplify the boundary condition at infinity for a uniform stress state,   . An additional stress,   , is produced in the x direction by this particular tensile stress. This stress is constant because it acts on the crack plane and is therefore not affected by the internal boundary conditions that would otherwise be imposed on the surfaces of the crack. This extraneous stress can later be subtracted from the solution, if desired [32]. Analogous to what Westergaard discussed regarding various mode I crack problems, these problems involving fractal roughness can be solved as follows:

  r ,  , H   Re I  r ,  , H   y Im I  r ,  , H   3  2H   d d   and I  z   I , I  z   I 2 dz dz  

where y  r sin 

(23) and Í  z  

d I dz

are the

Westergaard stress functions, which are holomorphic or analytic and satisfy the CauchyRiemann conditions. If the elasticity problem can be arranged so that the crack of interest extends over a line segment of the x axis

 y  0 ,

according to Westergaard (1939), the stresses and

displacements can be obtained from the stress function   z  , as

 xx  Re I  z   y Im I ´ z   2 A  yy  Re I  z   y Im I ´ z 

(24)

 xy   y Re I ´ z  Note that this particular Westergaard formulation is restricted to solutions that have the properties of  x   y and  xy  0 along the x -axis  y  0  . Therefore, the biaxial stress boundary condition at infinity  x   y    is needed in order to apply the technique to the Westergaard Mode problem - I. In addition,

4Gu   k  1 Re   2 y Im  4Gu   k  1 Im   2 y Re 

(25)

where Re and Im denote the real and imaginary parts of a complex function, and parameter k is defined individually for plane stress and plane strain by:

 3  v  / 1  v  plane stress k  . plane strain 3  4v The term

(26)

k 1 that appears in COD calculations can be simplified to 2 E´ , where 4G

E´ E / 1  2  corresponds to plane strain and E´ E to plane stress, and also



E . 2 1  v 

(27)

This term was used to relate  with E . These quantities, in turn, can be substituted in equations (24), resulting in the asymptotic solution associated to the stresses, which is valid for both plane stress and plane strain in mode I loading. By applying equations in (24)), one has:

 xx 

 yy 

 xy 

KI

 l0    2  2  H   l0  r 

KI

3 2 H 2

 l0    2  2  H   l0  r 

 l0    2  2  H   l0  r  KI

3 2 H 2

32 H 2

 3  2H     3  2H    5  2 H    cos    1  sin    sin    2 2 2        

(28)

 3  2 H     3  2H    5  2H    cos    1  sin    sin    2 2 2        

(29)

 3  2H      3  2H     5  2H    cos    sen    cos    2 2 2        

(30)

where K If   L  L0 ,  , H  

K I l01 H

(31)

2  H 

and

 zz  v  xx   yy  .

(32)

Similarly, following the same procedure, in plane stress and plane strain loading conditions, the asymptotic displacements around the crack tip or deflections

u  u  , r ,  , a  in Mode I are determined explicitly from equation (25), as:

ux 

K If 8

  3  2H    5  2H    2 r  2k  1 cos     cos    , 2 2      

(33)

ux 

K If 8

  3  2H     5  2H    2 r  2k  1 sin     sin    . 2 2      

(34)

4. Results and analysis of the mapped stress field of a rough fractal fracture To better illustrate the results obtained by Mosolov [2-3], Yavari [14,15] and Alves, the classical fractal and stress fields at the tip of a crack in loading mode I are mapped in this chapter to compare them qualitatively with the approach of other authors.

4.1. Mapping based on analytical results of fractal fractures, varying Yavari’s exponent of singularity for Mode I loading This section describes the calculation of the stress field for mode I loading according to the model proposed by Yavari:



2  DD 2 H  d  2 2H

(35)

where DD  d H for d  1, 2 is the divider dimension. Substituting the values of H  0.0,0.2, 0.4, 0.5, 0.6, 0.8 and 1.0 for the stress field given by:

 xx   yy   xy  f

 H 1

where K I  l0

2H

K If 

cos     sin  cos   1 



cos     sin  sin   1 

 2 r  K If

 2 r  K If



 2 r 

(36)

 sin  cos   1 

a

Figure 4 to Figure 6 illustrate the stress fields around a crack tip with fractal singularity order: and to   -1.5-0.25;0.0;0.167;0.375 0.5 corresponding H  0.0,0.2, 0.4, 0.5, 0.6, 0.8 and 1.0 . Note that the stress field obtained for

H  0.0, 0.2, 0.4, 0.5 does not show a result compatible with the reality of the stress field of a crack tip.

Figure 4. Stress field Model

 xx

of the fractal model for Mode I fracture with singularity

1 / r

. Yavari

H  0.0,0.2, 0.4, 0.5, 0.6, 0.8 and 1.0 .

Figure 5. Stress field

 xy


1 / r

. Alves Model

H  0.0,0.2, 0.4, 0.5, 0.6, 0.8 and 1.0 . The idea of determining the degree of singularity of the stress field at the fractal crack tip proposed by Mosolov [2-3], and later by Yavari [14,15], starts from the field



expression. This, in turn, depends on the radius vector r , which measures the distance from the crack tip to any point within the material. Mosolov simply generalizes the exponent that correlates the stress  ij with r by changing the value of the exponent given by 1/  n  1 for

n

integer to a fractional exponent   . Balankin [13], and later Yavari

[14,15], made this generalization and found the singularity exponent based on a dimensional analysis. However, these methodologies should be further refined, since the results of the numerical simulation of the stress field around a rough crack reported by Alves et al. [19] indicate that this field is only slightly changed starting from the field around a smooth crack.

Figure 6. Stress field Model

 yy


1 / r

. Alves

H  0.0,0.2, 0.4, 0.5, 0.6, 0.8 and 1.0 .

The calculations performed by Yavari [14,15] are mathematically correct, but do not correspond to physical reality in the interval for Hurst exponent of 0  H  0.5 , because he considered the roughness exponent as ~ 1 H , and hence, the singularity order  can be negative, causing this singularity to disappear, as shown in the mapping for H  0.0 to H  0.5 in Figure 4 to Figure 6. He considered the rough crack as a true fractal with selfaffinity in the range of scale from l0  r   C ; therefore, the asymptotic limits are different from a crack whose real fractality extends within a range of scales of width. It is known from experimental observation that cracks are not actually mathematical fractals and are called pre-fractals, since their self-affinity is contained within a limited range of scales:  min  l0 L0     max  L0 L0 , where l0 is the minimum crack length and L0 is the

macroscopic crack length. Thus, Alves et al. [19] propose a correction for the field around a rough crack, based on the development performed in this work.

4.2. Mapping of the analytical results of fractal fractures, with variation of the singularity exponent by Alves This section describes the calculation of the stress field for mode I loading according to the model proposed in this chapter, for



3  2H 2

(37)

where H is the Hurst exponent. Substituting the values of H  0.0 , 0.2, 0.4, 0.5, 0.6 and 0.8 for the stress field given by (28),(29) and (30) results in the maps shown in Figure 7 to Figure 9. These figures show the stress fields around a crack tip with fractal singularity order corresponding to   1.5;1.3;1.1;1.0;0.9 and 0.7 . Note that the result of the stress field obtained for H  0.0,0.2,0.4,0.5 is compatible with the reality of the stress field of a crack.

 xx of the fractal model for Mode I fracture l0  0.0001 , H  0.0 , 0.2, 0.4, 0.5, 0.6 and 0.8 .

Figure 7. Stress field model for

with singularity

1 / r

. Alves

The stress field for H  1.0 is not shown in Figure 7 to Figure 9 because, in all the models presented here, the value of   0.5 corresponds to the classical Euclidean field.

Figure 8. Stress field for

 xy


l0  0.0001 , H  0.0 , 0.2, 0.4, 0.5, 0.6

Figure 9. Stress field model for

 yy

and

1 / r

. Alves model

0.8 .


l0  0.0001 , H  0.0 , 0.2, 0.4, 0.5, 0.6

and

0.8 .

1 / r

. Alves

5. Discussion A comparison of the fractal model proposed by Alves et al. [23] and the model proposed by Mosolov-Borodich [2,6,7,3,8] and Yavari [14,15], published in the literature, leads to the following conclusions about these two theories. In the Mosolov-Borodich model [2,6,7,3,8], cracks are treated as true mathematical fractals that extend within an infinite range of scales: 1      . However, it is known from observation of nature that cracks are not actually mathematical fractals and are called pre-fractals, since their self-affinity extends only within a range of scales:  min  l0 L0     max  L0 L0 . In the fractal model proposed by Alves et al. [23], a crack is considered a pre-fractal. In considering the fractal in the model proposed by MosolovBorodich [2,6,7,3,8], Yavari [14,15], we use the coordinated function of the crack, which is a non-differentiable curve (or the Lebesgue derivative and integral, in this case). On the other hand, the fractal model proposed by Alves et al. [23], considering the fractal, is that the real length function of a crack L  f  L0  is a differentiable function, since it does not use the coordinated function (i.e., it avoids the problem of non-differentiability, since any length function is differentiable because it results in an integral, even if the coordinate function of the points is not). In the model of Yavari a non-root square was took at the equations (15), (16) and (18) until (21), whereas in the model proposed by Alves et al. [23] an exact root square was  took to maintain the symmetric dependence of the radius vector r , which measures the distance from the crack tip to any point within the material. The model of Mosolov-Borodich [2,6,7,3,8], Yavari 14,15] and Carpinteri [30,31], at the limit of small scales, uses the renormalization theory to calculate infinitesimals in order to satisfy the Griffith criterion. Conversely, the fractal model proposed by Alves et al. [23], at the limit of small scales, uses the calculation on the  min scale to satisfy the Griffith criterion (this avoids the unnecessary complication of using a highly advanced theory). In the model of Yavari [14,15], the rough crack length is calculated based on the assumption that L ~ l01/ H . The Yavari [14,15] model divides the rough crack length into intervals H : 0  H  1 / 2 (for ductile materials) and H :1 / 2  H  1 (for brittle materials), while the fractal model proposed by Alves et al. [23], the model of rough crack length alternates continuously between brittle and ductile, thus satisfying what is observed in practice. In the Mosolov Borodich [2,6,7,3,8], Yavari [14,15] model, the J-R curve is only locally independent of the path, because the integration of its radius depends on the form r  D . Conversely, in the fractal model proposed by Alves et al. [23], the J-R curve is totally independent of the path because its integration does not depend on either the radius r or the rough crack length, but on the roughness,   dL / dL0 which is outside of the J-integral. In the other aspects of the fracture theory for fractals, the models agree with each other.

5.1. Influence of the local roughness of a fracture surface on the stress field at the crack tip The graph in Figure 10 was created by comparing the values of the Hurst exponent for the models proposed here by Alves with those of the Yavari model [14,15]. This graph clearly shows a discontinuity in the value of the singularity of the field 1 / r   1 2 . Considering that the Alves model is valid for brittle materials and the Yavari model [14,15] is valid for ductile materials, the point 1 / r   1 2 on the horizontal axis corresponds to the threshold of brittle-to-ductile transition, which divides the graph in Figure 10 into two distinct regions. When a ductile material undergoes stresses before fracturing, it produces defects that interact with pre-existing defects, causing it to become brittle before it fractures. If the material is subjected to a high loading rate it can harden even further, creating more defects and producing a higher level of stress than it displayed its initial stage. This strain hardening, which is the result of the pile-up and interaction of these defects, mainly 

dislocations, undoubtedly changes the degree of singularity, 1 / r , of the stress field due to the change in the degree of homogeneity (exponent  ) of energy per unit volume at the crack tip. In a region at the crack tip that has been subjected to sudden loading, an increase 

in the degree of singularity, 1 / r , of the stress field produces a fracture with some roughness, depending on the loading rate. Therefore, these effects may generate differences in the surface roughness of the newly formed crack. In other words, as the loading rate increases, the stress level at the crack tip increases because of the increase in the degree of 

singularity 1 / r , thus increasing the tendency of the ductile material to generate an increasingly less rough crack  H  0  . On the other hand, a brittle material does not undergo strain hardening before it breaks. The material breaks upon reaching ultimate failure. In this case, roughness is simply an effect of the interaction between the crack and the microstructure of the material, where the crack may be intergranular, transgranular or mixed, depending on the material’s internal mechanical strength relative to the energy imposed by the load as the crack propagates. However, if the loading rate on a brittle material is increased, roughness may increase due to the nonlinear effect of the interaction of the stress field on the microstructure of the material. In this case, cracks may appear rougher  H  0  in response to the increase in loading rate, and the loading level increases due to the increase in the degree of 

singularity 1 / r . Figure 10 can therefore explain the behavior of the degree of singularity 

of the field, 1 / r , around a crack as a function of its roughness, based on the surface roughness exponent,  , of the fracture that propagates under different loading conditions or according to different types of materials. In Figure 10, the horizontal axis represents the 

level of singularity 1 / r and the vertical axis represents the degree of roughness, given by the Hurst exponent, H . At H  0 , namely 0  H  1 2 , the crack is rougher (hackle),

while at H  1 , namely 1 2  H  1 , the crack is smoother (mirror). In the region of

H  1 2 there is a mixed zone.

Figure 10. Plot of the Hurst exponent as a function of stress field singularity, calculated based on the brittle fracture predicted by Alves and on the ductile fracture predicted by Yavari

In a more ductile material, as the stress level increases in response to the emergence of defects in the crack tip, i.e., as the material hardens, this stress level increases due to the 

increase in the degree of singularity 1 / r , with a rapid growth of exponent H . This means that, a ductile materials become embrittled, they produce an increasingly less rough crack ( H  1 ) until they reach the threshold given by the peak of the curve. After reaching this threshold, the behavior is reversed. In other words, most brittle materials, which require a higher stress level to fracture compared to ductile materials, show a tendency to produce an increasingly rough crack ( H  0 ) in response to increasing stress levels at the crack tip. 

This means that the degree of singularity 1 / r of these materials increases in response to an increase in the loading rate. Hence, as can be seen in Figure 10, the threshold of ductile-tobrittle transition is at its peak when H  1 . The left side of this figure shows a fracture response in the form of crack roughness in a ductile material, while the right side shows a fracture response in the form of crack roughness in a brittle material.

6. Conclusions – The proposed – model describes the stress field around a crack tip in brittle materials. – The model proposed by Mosolov-Balankin-Yavari is insufficient to portray field intensity variations around the crack tip due to roughness in the interval for Hurst exponent of 0  H  0.5 . – The analysis of stress field fractal models based on the coefficient of geometric instability,

 L  r ,  , H  , proved feasible for the definition of a more realistic fractal model. – In the model presented here, the exponent  proposed by Mosolov is given by:

  3  2H  2 . – In the model proposed by Mosolov-Balankin-Yavari, the analysis of the coefficient of geometric instability,  L  r ,  , H  with variation of the Hurst coefficient,

 0  H  1 , did

not present a realistic range of roughness. – The fractal behavior of the stress field can be characterized as a fractal within a finite range of scales (  min  l0 L0     max  L0 L0 ). – The asymptotic limit of the singularity of the stress field can lead to other results if the range of scales is not considered. – The model can provide a narrower reliability limit for the fracture stress of brittle materials.

7. References 1. Hutchinson, J.W., " Plastic Stress and Strain Fields At a Crack Tip." J. Mech. Phys. Solids, 16, 337-347 (1968). 2. Mosolov, A. B., Zh. Tekh. Fiz. V. 61, N. 7, 1991. (Sov. Phys. Tech. Phys. , V. 36, 75, 1991). 3. Mosolov, A. B., Mechanics of Fractal Cracks In Brittle Solids, Europhysics Letters, Vol. 24, N. 8, P. 673-678, 10 December 1993. 4. Gol’dshteˇin R V and Mosolov A B 1991 Sov. Phys. Dokl.36 603–5 5. Gol’dshteˇin R V and Mosolov A B 1992 J. Appl. Math. Mech. 56 563–71 6. Mosolov, A. B. and F. M. Borodich Fractal Fracture of Brittle Bodies During Compression, Sovol. Phys. Dokl., Vol. 37, N. 5, P. 263-265, May 1992. 7. Mosolov AB, Borodich FM. Fractal fracture of brittle bodies during compression. Sov Phys Dokl 1992;37(5):263–5

8. Borodich, F. M., “Some Fractals Models of Fracture,” J. Mech. Phys. Solids, Vol. 45, N. 2, P. 239-259, 1997. 9. Besicovitch AS. Sets of fractional dimensions (IV): On rational approximation to real numbers. Classics on fractals. G. A. Edgar. Boston, Addison-Wesley Reading: 161-168; 1993b. 10. Besicovitch AS, Ursell, HD. Sets of fractional dimensions (V): On dimensional numbers of some continuous curves. Classics on fractals. G. A. Edgar. Boston, Addison-Wesley Reading: 171-179; 1993c. 11. Cherepanov, Genady P., Alexander S. Balankin, and Vera S. Ivanova. "Fractal fracture mechanics—a review." Engineering Fracture Mechanics 51.6 (1995): 997-1033. 12. Balankin , A.S and P. Tamayo, Revista Mexicana de Física 40, No. 4, Pp. 506-532, 1994. 13. Balankin, Alexander S., Engineering Fracture Mechanics, Vol. 57, N. 2/3, Pp.135-203, 1997. 14. Yavari, Arash, the Fourth Mode of Fracture In Fractal Fracture Mechanics, International Journal of Fracture, Vol. 101, 365-384, 2000. 15. Yavari, Arash, the Mechanics of Self-Similar and Self-Afine Fractal Cracks, International Journal of Fracture, Vol. 114, 1-27, 2002. 16. Yavari, Arash, Generalization of Barenblatt's Cohesive Fracture Theory for Fractal Cracks, Fractals, World Scientic Publishing Company, Vol. 10, No. 2 (2002) 189-198. 17. Pugno N, Ruoff R. Quantized fracture mechanics. Philos Mag 2004;84:2829–45. 18. Alves, Lucas Máximo. Fractal geometry concerned with stable and dynamic fracture mechanics. Journal of Theoretical and Applied Fracture Mechanics. 2005;Vol 44(1), pp. 44-57. 19. Alves, Lucas Máximo. “Modelagem e Simulação do Campo Contínuo com Irregularidades: Aplicações em Mecânica da Fratura com Rugosidade,” Tese de Doutorado, PPGMNE-CESEC-UFPR-Curitiba-Paraná, 2011. 20. Alves, Lucas Máximo, Foundations of Measurement Fractal Theory for the Fracture Mechanics, Edited by Alexander Belov, Intech-Open, http://dx.doi.org/10.5772/51813 21. Alves, Lucas Máximo et al, Analytical fractal model for rugged fracture surface of brittle materials, Engineering Fracture Mechanics paper accepted for publishing in 17 May 2016, Final version published online: 8-JUN-2016, (2016), pp. 232-255 DOI information: 10.1016/j.engfracmech.2016.05.015; http://www.sciencedirect.com/science/article/pii/S0013794416302375 22. Alves, Lucas Máximo; Rosana Vilarim da Silva, Bernhard Joachim Mokross, the Influence of the Crack Fractal Geometry on the Elastic Plastic Fracture Mechanics. Physica A: Statistical Mechanics and Its Applications. Vol. 295, N. 1/2, P. 144-148, 12 June 2001. 23. Alves, Lucas Máximo; Rosana Vilarim da Silva, Luiz Alkimin de Lacerda, Fractal

Modeling of the J-R Curve and the Influence of the Rugged Crack Growth on the Stable Elastic Plastic Fracture Mechanics, Engineering Fracture Mechanics, 77, Pp. 2451-2466, 2010. 24. Alves, Lucas Máximo and de Lacerda, Luiz Alkimin, Fractal Fracture Mechanics Applied to Materials Engineering; In: Applied Fracture Mechanics, Edited by Alexander Belov, Intech-Open, http://dx.doi.org/10.5772/52511 25. Alves, Lucas Máximo, de Lacerda, Luiz Alkimin, Souza, Luiz Antônio et al.Modelo Termodinâmico para uma Linha Rugosa, apresentação de seminário na Semana da PósGraduação em Métodos Numéricos em Engenharia, Universidade Estadual de Ponta Grossa · 08/August 2010, DOI: 10.13140/RG.2.1.4309.4881. 26. S.M. Ohr and J. Narayan. Electron Microscope Observation of Shear Cracks in Stainless Steel Single Crystals. Phil. Mag. A. 1980;41, pp. 81–89. 27. Xin, G., Hangong, W., Xingwu, K., Liangzhou, J.. Analytic solutions to crack tip plastic zone under various loading conditions.European Journal of Mechanics A/Solids.2010;29, pp.738-745. 28. Westergaard, H., 1939. Bearing pressures and cracks. J. Appl. Mech. A, 49–53. 29. Irwin G. R., “Fracture Dynamics,” Fracturing of Metals. Cleveland, OH: American Society for Metals; p. 147–166, 1948. 30. Carpinteri, A; Chiaia, B.; Cornetti, P., A fractal theory for the mechanics of elastic materials Materials Science and Engineering, A365, p. 235–240, 2004. 31. Carpinteri, A.; Puzzi, S., Complexity: a New Paradigm For Fracture Mechanics, Frattura Ed Integrità Strutturale,10, 3-11, 2009, Doi:10.3221/Igf-Esis.1001. 32. Unger, David J. "Introduction,” Analytical Fracture Mechanics, Academic Press, 1995.

Chapter III

FOUNDATIONS OF MEASUREMENT FRACTAL THEORY FOR THE FRACTURE MECHANICS



    

1. Introduction A wide variety of natural objects can be described mathematically using fractal geometry as, for example, contours of clouds, coastlines , turbulence in fluids, fracture surfaces, or rugged surfaces in contact, rocks, and so on. None of them is a real fractal, fractal characteristics disappear if an object is viewed at a scale sufficiently small. However, for a wide range of scales the natural objects look very much like fractals, in which case they can be considered fractal. There are no true fractals in nature and there are no real straight lines or circles too. Clearly, fractal models are better approximations of real objects that are straight lines or circles. If the classical Euclidean geometry is considered as a first approximation to irregular lines, planes and volumes, apparently flat on natural objects the fractal geometry is a more rigorous level of approximation. Fractal geometry provides a new scientific way of thinking about natural phenomena. According to Mandelbrot [1], a fractal is a set whose fractional dimension (Hausdorff-Besicovitch dimension) is strictly greater than its topological dimension (Euclidean dimension). In the phenomenon of fracture, by monotonic loading test or impact on a piece of metal, ceramic, or polymer, as the chemical bonds between the atoms of the material are broken, it produces two complementary fracture surfaces. Due to the irregular crystalline arrangement of these materials the fracture surfaces can also be irregular, i.e., rough and difficult geometrical description. The roughness that they have is directly related to the material microstructure that are formed. Thus, the various microstructural features of a material (metal, ceramic, or polymer) which may be, particles, inclusions, precipitates, etc. affect the topography of the fracture surface, since the different types of defects present in a material can act as stress concentrators and influence the formation of fracture surface. These various microstructural defects interact with the crack tip, while it moves within the material, forming a totally irregular relief as chemical bonds are broken, allowing the microstructure   

20 Applied Fracture Mechanics

to be separated from grains (transgranular and intergranular fracture) and microvoids are joining (coalescence of microvoids, etc..) until the fracture surfaces depart. Moreover, the characteristics of macrostructures such as the size and shape of the sample and notch from which the fracture is initiated, also influence the formation of the fracture surface, due to the type of test and the stress field applied to the specimen. After the above considerations, one can say with certainty that the information in the fracture process are partly recorded in the "story" that describes the crack, as it walks inside the material [2]. The remainder of this information is lost to the external environment in a form of dissipated energy such as sound, heat, radiation, etc. [30, 31]. The remaining part of the information is undoubtedly related to the relief of the fracture surface that somehow describes the difficulty that the crack found to grow [2]. With this, you can analyze the fracture phenomenon through the relief described by the fracture surface and try to relate it to the magnitudes of fracture mechanics [3 , 4 , 5 , 6 , 7, 8, 9 - 11, 12, 13]. This was the basic idea that brought about the development of the topographic study of the fracture surface called fractography. In fractography anterior the fractal theory the description of geometric structures found on a fracture surface was limited to regular polyhedra-connected to each other and randomly distributed throughout fracture surface, as a way of describing the topography of the irregular surface. Moreover, the study fractographic hitherto used only techniques and statistical analysis profilometric relief without considering the geometric auto-correlation of surfaces associated with the fractal exponents that characterize the roughness of the fracture surface. The basic concepts of fractal theory developed by Mandelbrot [1] and other scientists, have been used in the description of irregular structures, such as fracture surfaces and crack [14 ], in order to relate the geometrical description of these objects with the materials properties [15 ]. The fractal theory, from the viewpoint of physical, involves the study of irregular structures which have the property of invariance by scale transformation, this property in which the parts of a structure are similar to the whole in successive ranges of view (magnification or reduction) in all directions or at least one direction (self-similarity or self-affinity, respectively) [36]. The nature of these intriguing properties in existing structures, which extend in several scales of magnification is the subject of much research in several phenomena in nature and in materials science [16 , 17 and others]. Thus, the fractal theory has many contexts, both in physics and in mathematics such as chaos theory [18], the study of phase transitions and critical phenomena [19, 20, 21], study of particle agglomeration [22], etc.. The context that is more directly related to Fracture Mechanics, because of the physical nature of the process is with respect to fractal growth [23, 24, 25, 26]. In this subarea are studied the growth mechanisms of structures that arise in cases of instability, and dissipation of energy, such as crack [27, 28] and branching patterns [29]. In this sense, is to be sought to approach the problem of propagation of cracks.

 Foundations of Measurement Fractal Theory for the Fracture Mechanics 21

The fractal theory becomes increasingly present in the description of phenomena that have a measurable disorder, called deterministic chaos [18, 27, 28]. The phenomenon of fracture and crack propagation, while being statistically shows that some rules or laws are obeyed, and every day become more clear or obvious, by understanding the properties of fractals [27, 28].

2. Fundamental geometric elements and measure theory on fractal geometry In this part will be presented the development of basic concepts of fractal geometry, analogous to Euclidean geometry for the basic elements such as points, lines, surfaces and fractals volumes. It will be introduce the measurement fractal theory as a generalization of Euclidean measure geometric theory. It will be also describe what are the main mathematical conditions to obtain a measure with fractal precision.

2.1. Analogy between euclidean and fractal geometry It is possible to draw a parallel between Euclidean and fractal geometry showing some examples of self-similar fractals projected onto Euclidean dimensions and some self-affine fractals. For, just as in Euclidean geometry, one has the elements of geometric construction, in the fractal geometry. In the fractal geometry one can find similar objects to these Euclidean elements. The different types of fractals that exist are outlined in Figure 1 to Figure 4.

2.1.1. Fractais between 0 D 1 (similar to point) An example of a fractal immersed in Euclidean dimension I d 1 1 with projection in d 0 , similar to punctiform geometry, can be exemplified by the Figure 1.

 Fractal immersed in the one-dimensional space where D

0,631 .

This fractal has dimension D 0,631 . This is a fractal-type "stains on the floor." Other fractal of this type can be observed when a material is sprayed onto a surface. In this case the global dimension of the spots may be of some value between 0 D 1 .

2.1.2. Fractais between 1 D 2 (similar to straight lines) For a fractal immersed in a Euclidean dimension I d 1 2 , with projectionin d 1 , analogous to the linear geometry is a fractal-type peaks and valleys (Figure 2). Cracks may also be described from this figure as shown in Alves [37]. Graphs of noise, are also examples of linear fractal structures whose dimension is between 1 D 2 .


 Fractal immersed in dimension d = 2. rugged fractal line.

2.1.3. Fractals between 2 D 3 (similar to surfaces or porous volumes) For a fractal immersed in a Euclidean dimension, I d 1 3 with projection in d 2 , analogous to a surface geometry is fractal-type "mountains" or "rugged surfaces" (Figure 3). The fracture surfaces can be included in this class of fractals.

 Irregular or rugged surface that has a fractal scaling with dimension D between 2

D

3.

 Comparison between Euclidean and fractal geometry. D , d and D f represents the topological, Euclidean and fractal dimensions, of a point, line segment, flat surface, and a cube, respectively


Making a parallel comparison of different situations that has been previously described, one has (Figure 4)

2.2. Fractal dimension (non-integer) An object has a fractal dimension, D , d dimension which is immersed, when: F

D

d 1 I , where I is the space Euclidean

D

L0

(1)

F L0

where L0 is the projected length that characterizes an apparent linear extension of the fractal , is the scale transformation factor between two apparent linear extension, F L0

is a

function of measurable physical properties such as length, surface area, roughness, volume, etc., which follow the scaling laws, with homogeneity exponent is not always integers, whose geometry that best describe, is closer to fractal geometry than Euclidean geometry. These functions depend on the dimensionality, I , of the space which the object is immersed. Therefore, for fractals the homogeneity degree n is the fractal dimension D (non-integer) of the object, where is an arbitrary scale. Based on this definition of fractal dimension it can be calculates doing: F( Lo )

D

(2)

F ( Lo ) taking the logarithm one has ln D

F ( Lo ) F ( Lo )

(3)

ln( )

From the geometrical viewpoint, a fractal must be immersed into a integer Euclidean dimension, I d 1 . Its non-integer fractal dimension, D , it appears because the fill rule of the figure from the fractal seed which obeys some failure or excess rules, so that the complementary structure of the fractal seed formed by the voids of the figure, is also a fractal. For a fractal the space fraction filled with points is also invariant by scale transformation, i.e.: P( Lo )

F( Lo ) F( Lo )

1 N( Lo )

(4)

Thus, D

where P L0

P L0 ou N L0

D

is a probability measure to find points within fractal object

(5)


Therefore, the fractal dimension can be calculated from the fllowing equation: ln N L0

D

(6)

ln

If it is interesting to scale the holes of a fractal object (the complement of a fractal), it is observed that the fractal dimension of this new additional dimension corresponds to the Euclidean space in which it is immersed less the fractal dimension of the original.

2.3. A generalized monofractal geometric measure Now will be described how to process a general geometric measure whose dimension is any. Similarly to the case of Euclidean measure the measurement process is generalized, using the concept of Hausdorff-Besicovitch dimension as follows. Suppose a geometric object is recovered by -dimensional, geometric units, uD , with extension, k and k , where is the maximum -dimensional unit size and is a positive real number. Defining the quantity: MD

, ,{ k }

k

(7)

k

Choosing from all the sets

k

, that reduces this summation, such that: MD

, {

inf k}

k

(8)

k

The smallest possible value of the summation in (8) is calculated to obtain the adjustment with best precision of the measurement performed. Finally taking the limit of tending to zero, 0 , one has: MD ( )

limM D ( , )

(9)

0

The interpretation for the function M D is analogous to the function for a Euclidean measure of an object, i.e. it corresponds to the geometric extension (length, area, volume, etc.) of the set measured by units with dimension, . The cases where the dimension is integer are same to the usual definition, and are easier to visualize. For example, the calculation of M D for a surface of finite dimension, D 2 , there are the cases: -

-

For 1 D 2 measuring the "length" of a plan with small line segments, one gets MD , because the plan has a infinity length, or there is a infinity number of line segment inside the plane. For, 2 D 2 measuring the surface area of small square, one gets M D Ad 2 A0 . Which is the only value of where M D is not zero nor infinity (see Figure 5.) For 3 D 2 measuring the "volume" of the plan with small cubes, one gets M D 0 , because the "volume" of the plan is zero, or there is not any volume inside the plan.


 Measuring, M D units uD for D 1, 2, 3 .

of an area A with a dimension, D

2 made with different measure

Therefore, the function, M D possess the following form 0 para MD ( )

M para para

D D.

(10)

D

That is, the function M D only possess a different value of 0 and defining a generalized measure

at a critical point

D

2.4. Invariance condition of a monofractal geometric measure Therefore, for a generalized measurement there is a generalized dimension which the measurement unit converge to the determined value, M , of the measurement series, according to the extension of the measuring unit tends to zero, as shown in equations equações (9) and (10), namely:

MD

, ,{ k }

k

MDo

D

(11)

k

where

is the Euclidean projected extension of the fractal object measured on

-

dimensional space Again the value of a fractal measure can be obtain as the result of a series. One may label each of the stages of construction of the function M D i.

as follows:

the first is the measure itself. Because it is actually the step that evaluates the extension of the set, summing the geometrical size of the recover units. Thus, the extension of the set is being overestimated, because it is always less or equal than tthe size of its coverage. ii. The next step is the optimization to select the arrangement of units which provide the smallest value measured previously, i.e. the value which best approximates the real extension of the assembly. iii. The last step is the limit. Repeat the previous steps with smaller and smaller units to take into account all the details, however small, the structure of the set.


As the value of the generalized dimension is defined as a critical function, M D it can be concluded, wrongly, that the optimization step is not very important, because the fact of not having all its length measured accurately should not affect the value of critical point. The optimization step, this definition, serves to make the convergence to go faster in following step, that the mathematical point of view is a very desirable property when it comes to numerical calculation algorithms.

2.5. The monofractal measure and the Hausdorff-Besicovitch dimension In this part we will define the dimension-Hausdorf Besicovicth and a fractal object itself. The basic properties of objects with "anomalous" dimensions (different from Euclidean) were observed and investigated at the beginning of this century, mainly by Hausdorff and Besicovitch [32,34]. The importance of fractals to physics and many other fields of knowledge has been pointed out by Mandelbrot [1]. He demonstrated the richness of fractal geometry, and also important results presented in his books on the subject [1, 35, 36]. The geometric sequence, S is given by: S

Sk

onde k

0,1,2,....

(12)

k

represented in Euclidean space, is a fractal when the measure of its geometric extension, given by the series, M k satisfies the following Hausdorf-Besicovitch condition:

Md ( k )

( d) k

k

Nd (

k) k

(d) 0

k

0;

D

MD ;

D,

;

(13)

D

where: d is the geometric factor of the unitary elements (or seed) of the sequence represented geometrically. : is the size of unit elements (or seed), used as a measure standard unit of the extent of the spatial representation of the geometric sequence. N : is the number of elementary units (or seeds) that form the spatial representation of the sequence at a certain scale : the generalized dimension of unitary elements D : is the Hausdorff-Besicovitch dimension.

2.6. Fractal mathematical definition and associated dimensions Therefore, fractal is any object that has a non-integer dimension that exceeds the topological dimension ( D I , where I is the dimension of Euclidean space which is immersed) with some invariance by scale transformation (self-similarity or self-affinity), where for any continuous contour that is taken as close as possible to the object, the number of points N D , forming the fractal not fills completely the space delimited by the contour, i.e., there is


always empty, or excess regions, and also there is always a figure with integer dimension, I , at which the fractal can be inscribed and that not exactly superimposed on fractal even in the limit of scale infinitesimal. Therefore, the fraction of points that fills the fractal regarding its Euclidean coverage is different of a integer. As seen in previous sections - 2.2 - 2.5 in algebraic language, a fractal is a invariant sequence by scale transformation that has a Hausdorff-Besicovitch dimension. According to the previous section, it is said that an object is fractal, when the respective magnitudes characterizing features as perimeter, area or volume, are homogeneous functions with non-integer. In this case, the invariance property by scaling transformation (self-similar or self-affinity) is due to a scale transformation of at least one of these functions. The fractal concept is closely associated to the concept of Hausdorff-Besicovitch dimension, so that one of the first definitions of fractal created by Mandelbrot [36] was: Fractal by definition is a set to which the Haussdorf-Besicovitch dimension exceeds strictly the topological dimension". One can therefore say that fractals are geometrical objects that have structures in all scales of magnification, commonly with some similarity between them. They are objects whose usual definition of Euclidean dimension is incomplete, requiring a more suitable to their context as they have just seen. This is exactly the Hausdorff-Besicovitch dimension. A dimension object, D , is always immersed in a space of minimal dimension I d 1 , which may present an excessive extension on the dimension d , or a lack of extension or failures in one dimension d 1 . For example, for a crack which the fractal dimension is the dimension in the range of 1 D 2 the immersion dimension is the dimension I 2 in the case of a fracture surface of which the fractal dimension is in the range 2 D 3 the immersion dimension is the I 3 . When an object has a geometric extension such as completely fill a Euclidean dimension regular, d , and still have an excess that partially fills a superior dimension I d 1 , in addition to the inferior dimension, one says that the object has a dimension in excess, de given by de D d where D is the dimension of the object. For example, for a crack which the fractal dimension is in the range 1 D 2 the excess dimension is de D 1 , in the case of a fracture surface of which the fractal dimension is in the range of 2 D 3 the excess dimension is de D 2 . If on the other hand an object partially fills a Euclidean regular dimension, I d 1 certainly this object fills fully a Euclidean regular dimension, d , so that it is said that this object has a lack dimension d fl I D d 1 D , where de 1 d fl . For example, for a crack which the fractal dimension is the range of 1 D 2 the lack dimension is d fl 2 D . In the case of a fracture surface of which the fractal dimension is the range of 2 D 3 the lack dimension is d fl 3 D .

2.7. Classes and types of fractals One of the most fascinating aspects of the fractals is the extremely rich variety of possible realizations of such geometric objects. This fact gives rise to the question of classification,


and the book of Mandelbrot [1] and in the following publications many types of fractal structures have been described. Below some important classes will be discussed with some emphasis on their relevance to the phenomenon of growth. Fractals are classified, or are divided into: mathematical and physical (or natural) fractals and uniform and non-uniform fractals. Mathematical fractals are those whose scaling relationship is exact, i.e., they are generated by exact iteration and purely geometrical rules and does not have cutoff scaling limits, not upper nor lower, because they are generated by rules with infinity interactions (Figure 6a) without taking into account none phenomenology itself, as shown in Figure 6a. Some fractals appear in a special way in the phase space of dynamical systems that are close to situations of chaotic motion according to the Theory of Nonlinear Dynamical Systems and Chaos Theory. This approach will not be made here, because it is another matter that is outside the scope of this chapter.

 Example of branching fractals, showing the structural elements, or elementary geometrical units, of two fractals. a) A self-similar mathematical fractal. b) A statistically self-similar physical fractal.

Real or physical fractals (also called natural fractals) are those statistical fracals, where not only the scale but all of fractal parameters can vary randomly. Therefore, their scaling relationship is approximated or statistical, i. e., they are observed in the statistical average made throughout the fractal, since a lower cutoff scale, min , to a different upper cutoff scale max (self-similar or self-affine fractals), as shown in Figure 6b. These fractals are those which appear in nature as a result of triggering of instabilities conditions in the natural processes [24] in any physical phenomenon, as shown Figure 6b. In these physical or natural fractals the extension scaling of the structure is made by means of a homogeneous function as follows: F

~

d D

,

(14)

where d is the Euclidean dimension of projection of the fractal and D is the fractal dimension of self-similar structure.


It is true that the physical or real fractals can be deterministic or random. In random or statistical fractal the properties of self-similarity changes statistically from region to region of the fractal. The dimension cannot be unique, but characterized by a mean value, similarly to the analysis of mathematical fractals. The Figure 6b shows aspects of a statistically selfsimilar fractal whose appearance varies from branch to branch giving us the impression that each part is similar to the whole. The mathematical fractals (or exact) and physical (or statistical), in turn, can be subdivided into uniform and nonuniform fractal. Uniforms fractals are those that grow uniformly with a well behaved unique scale and constant factor, , and present a unique fractal dimension throughout its extension. Non-uniform fractals are those that grow with scale factors i ' s that vary from region to region of the fractal and have different fractal dimensions along its extension. Thus, the fractal theory can be studied under three fundamental aspects of its origin: 1. 2. 3.

From the geometric patterns with self-similar features in different objects found in nature. From the nonlinear dynamics theory in the phase space of complex systems. From the geometric interpretation of the theory of critical exponents of statistical mechanics.

3. Methods for measuring length, area, volume and fractal dimension In this section one intends to describe the main methods for measuring the fractal dimension of a structure, such as: the compass method, the Box-Counting method, the Sand-Box Method, etc. It will be described, from now, how to obtain a measure of length, area or fractal volume. In fractal analysis of an object or structure different types of fractal dimension are obtained, all related to the type of phenomenon that has fractality and the measurement method used in obtaining the fractal measurement. These fractal dimensions can be defined as follows.

3.1. The different fractal dimensions and its definitions A fractal dimension D f in general is defined as being the dimension of the resulting measure of an object or structure, that has irregularities that are repeated in different scales (a invariance by scale transformation). Their values are usually noninteger and situated between two consecutive Euclidean dimensions called projection dimension d of the object and immersion dimension, d 1 , i.e. d D f d 1 . In the literature there is controversy concerning the relationship between different fractal dimensions and roughness exponents. The term "fractal dimension" is used generically to refer to different fractional dimensions found in different phenomenologies, which results in formation of geometric patterns or energy dissipation, which are commonly called fractals [1]. Among these patterns is the growth of aggregates by diffusion (DLA - Diffusion Limited


Aggregation), the film growth by ballistic deposition (BD), the fracture surfaces (SF), etc.. The fractal dimensions found in these phenomena are certainly not the same and depend on both the phenomenology studied as the fractal characterization method used. Therefore, to characterize such phenomena using fractal geometry, a distinction between the different dimensions found is necessary. Among the various fractal dimensions one can emphasize the Hausdorff-Besicovitch dimension, DHB , which comes from the general mathematical definition of a fractal [32, 33,34]. Other dimensions are the dimension box, DB , the roughness dimension or exponent Hurst, H , the Lipshitz-Hölder dimension, , etc.. Therefore, a mathematical relationship between them needs to be clearly established for each phenomenon involved. However, is observed, then that relationship is not unique and depends not only on phenomenology, but also the characterization method used. Therefore, the phenomenological equation of the fracture phenomenon can also, in theory, provide a relationship between fractal dimension and roughness exponent of a fracture surface, as happens to other phenomenologies. In this study, there was obtained a fractal model for a fracture surface, as a generalization of the box-counting method. Thus, will be discussed the relationship between the local and global box dimension and the roughness dimension, which are involved in the characterization of a fracture surface, and any other dimension necessary to describe a fractal fracture surface.

3.1.1. Compass methods and divider dimension, DD The divider dimension DD is defined from the measure of length of a roughened fractal line, for example, when using the compass method. This measure is obtained by opening a compass with an aperture and moving on the line fractal to obtain the value of the line length rugosa (see Figure 7). The different values of the rough line length due to the compass aperture determines the dimension divider.

 Compass method applied to a rugged line.

For a fractal rough line the divider dimension can be defined as:


ln

L .

DD ln

(15)

L0

where L0 is the projected length obtained from the rugged fractal length L

 Compass method applied on a line noise or a rough self-affine fractal.

Several methods for determining the fractal dimension based on the compass method, among them stand out the following methods: the Coastlines Richardson Method, the Slit Island Method, etc.

3.2. Methods of measurement for determining the fractal dimension of a structure There are basically two ways to recover an object with boxes for fractal dimension measuring. In the first method, boxes of different sizes extending from a minimum size min until to a maximum size max , from a fixed origin recovering the whole object at once time. In the second case, one side of the recovering box is kept fixed, and with a minimum size ruler, min , then recovers the figure by moving the boundary of that recovering from the minimum min to maximum size max of the object. The first method is known as a method Box-Counting exemplified in Figure 9 and the second method is known as Sand-box, shown in Figure 10. The advantage of the second over the first is that it detects the changes in dimension D with the length of the object. If the object under consideration has a local dimension for boxes with size 0 , unlike the global dimension, , it is said that the object is self-affine fractal. Otherwise the object is said self-similar. These two main methods of counts of structures which may lead to determination of the fractal dimension of an object [38].

3.2.1. Box-counting method by static scaling of the elements in a fractal structure The Box-Counting method, comes from the theory of critical phenomena in statistical mechanics. In statistical mechanics there is an analogous mathematical method to describing


phenomena which have self-similar properties, permitting scale transformations without loss of generality in the description of physical information of the phenomenon ranging from quantities such as volume up to energy. However, in the case described here, the BoxCounting method is performed filling the space occupied by a fractal object with boxes of arbitrary size , and count the number N of these boxes in function its size, (Figure 9 and Figure10). This number N of boxes is given as follows: N

C

D

(16)

Plotting the data in a log log graph one obtains from the slope of the curve obtained, the fractal dimension of the object. In the Box-Counting method (Figure 9), a grid that recover the object is divided into nk L0 / k boxes of equal side k and how many of these boxes that recovering the object is counted. Then, varies the size of the boxes and the counting is retraced, and so on. Making a logarithm graph of the number N k of boxes that recovering the object in function of the scale for each subdivision k k / L0 , one obtains the fractal dimension from the slope of this plot. Note that in this case the partition maximum is reached when, N L0 / k k L0 / l0 , where Lmax L0 is the projected crack length l0 is the length of the shortest practicable ruler.

 Fragment of a crack on a testing sample showing the variation of measurement of the crack length L with the measuring scale, k variável and Lk L0 (fixed), with k / L0 for a partition, k sectioning done for counting by one-dimensional Box-Counting scaling method.


Therefore, the number N k

k

depending on the size,

k

, of these boxes is given as follows:

D

Nk

k

(17)

k max

In the Figure 9 is illustrated the use of this method in a fractal object. Are present different grids, or meshes, constructed to recover the entire structure, whose fractal dimension one wants to know. The grids are drawn from an original square, involving the whole space occupied by the structure. At each stage of refinement of the grid L0 (the number of equal parts in the side of the square is divided) are counted the number of squares N L0 which contain part of the structure. Repeatedly from the data found, is constructed the graph of log L0 log N L0 . If the graph thus obtained is a straight line, then the fractal behavior of the structure has self-similarity or statistical self-affinity whose dimension D is obtained by calculating the slope of the line. For more compact structure, it is recommended to make a statistical sampling, that is, the repeat the counting of the squares N L0 for different squares constructed from the gravity center (counting center) of the in the structure. Thus, one obtains a set of values N L0 for another set of values L0 . These data must be statistically treated to obtain the value of fractal dimension, " D " . From the viewpoint of experimental measurement, one can consider using different methods of viewing the crack to obtain the fractal dimension, such as optical microscopy, electron microscopy, atomic force microscope, etc.., Which naturally have different rules k and therefore different scales of measurement k ,. The fractal dimension is usually calculated using the Box-Counting shown in Figure 9, i.e. by varying the size of the measuring ruler k and counting the number of boxes, N k that recover the structure. In the case of a crack the fractal dimension is obtained by the following relationship:

D

ln N ln(lo / Lo )

(18)

The description of a crack according to the Box-Counting method follows the idea shown in Figure 9, which results in: D

ln 57 ln(1 / 40)

1.096 .

(19)

The same result can be obtained using the Box-Sand method, as shown in Figure 10.

3.2.2. The sand-box counting method of the elements by static scaling of a fractal structure The Sand-box method consists in the same way as the Box-Counting method, to count the number of boxes, N u , but with fixed length, u , as small as possible, extending gradually


up the boundary count until to reach out to the border of the object under consideration. This is done initially by setting the counting origin from a fixed point on the object, as shown in Figure10. This method seems to be the most advantageous, as well as to establish a coordinate system, or a origin for calculating the fractal dimension, it also allows, in certain cases, to infer dynamic data from static scaling, as shown by Alves [47].

 Fragment of a crack on test specimen showing the variation of measurement of the crack length L with the measuring scale, k variável , and k l0 (fixed), with k / L0 for a partition Lk sectioning done for counting by one-dimensional Sand-Box scaling method.

In the Sand-Box method (Figure10), the figure is recovered with boxes of different sizes Lk , no matter the form, which can be rectangular or spherical, however, fixed at a any point " O " on figure called origin, from which the boxes are enlarged. It is counted the number of elementary structures, or seeds, which fit within each box. Plotting the graph of log N k log k min Lk in the same manner as in the above method the fractal dimension is obtained. Note that in this case the maximum partition is achieved when L0 is the projected crack length and min l0 it is N Lk k L0 l0 , where L min the length of the lower measuring ruler practicable.

3.2.3. The global and local box dimensions To define the box dimension, DB , is assumed that all the space containing the fractal is recovered with a grid (set of -dimensional units juxtaposed in the same shape and size, )


with maximum size, max , which inscribes the fractal object. Defining the relative scale, on the grid size, max , as being given by:

max

countting the number of boxes N( ) that have at least one point of the fractal. The box dimension is therefore defined as: DB

lim

0

ln N ( ) ln

(21)

At this point, there are two ways to obtain the actual value of the measure, or taking the limit when 0 and allows that the dimension D fits the end value of N ( ) , or it is considered a linear correlation in value of ln N ( ) ln , which D is the slope of the line, and this defines the measure independently of the scale. In the case of numerical estimation, one can not solve the limit indicated in the equation (21). Then, DB is obtained as a slope, ln N( ) ln when it is small. The value N ( ) is obtained by an algorithm known as Box-Counting. Self-affine fractals requiring different variations in scale length for different directions. Therefore, one can use the Box-Counting method with some care being taken, in the sense that the box dimension DB to be obtained has a crossing region between a local and global measure of the dimensions. From which follows that for each region is used the following relationships:

lim N L0

L0 l0

lim N L0

L0 l0

l0

0

DBg

p / L0

L0 s

(22)

p / L0

L0 s

(23)

for a global measurement

l0

0

DBl

where L0s is the threshold saturation length which the fractal dimension changes its behavior from local to global stage. For measurement, generally, for any self-affine fractal structure the local fractal dimension is related to the Hurst exponent, H , as follow, DBl

d 1 Hq

1

(24)

At this point, one observes that for a profile the relationship DBl 2 H commonly used, only serves for a local measurements using the box counting method. While for global


measures one can not establish a relationship between DBg and H . For the global fractal dimension, Dg d and I d 1 the Euclidean dimension where the fractal is embedded one has d

DBg

d 1

(25)

Some textbooks on the subject show an example of calculation of local and global fractal dimension of self-affine fractals, obtained by a specific algorithm [18, 22, 23, 26, 38,39]. In crossing the limit of fractal dimension local Dl to global Dg , there is a transition zone called the "crossover", and the results obtained in this region are somewhat ambiguous and difficult to interpret [39]. However, in the global fractal dimension, the structure is not considered a fractal [42 , 43].

3.2.4. The Relationship between box dimension and HausdorffBesicovitch dimensions The mathematical definition of generalized dimension of Haussdorff-Besicovitch need a method that can measure it properly to the fractal phenomenon under study. Some authors [23, 40, 44, 45] have discussed the possibility of using the Box-Counting method as one of the graphical methods which obtains a box dimension DB , very close to generalized Haussdorff Besicovitch, DHB , i.e. [44]:

DB

DHB

(26)

In this sense the box dimension, DB is obtained for self-asimilar fractals that may be rescaled for the same variation in scales lengths in all directions by using the relationship:

N L0

L0 l0

DB

(27)

where l0 is the grid size used and L0 is the apparent size of the fractal to be characterized. The analytical calculation of the Hausdorff dimension is only possible in some cases and it is difficult to implement by computation. In numerical calculation, is used another more appropriate definition, called box dimension, DB , which in the case of dynamic systems, has the same value of the Haussdorff dimension, D [44]. Thus, it is common to call them without distinction as fractal dimensions, D as will be shown below. All the definitions related to fractal exponents that are shown here, and all numerical evaluation of these, always calculates the inclination of some amount against on a logarithmic scale. The two definitions of, Hausdorff-Besicovitch Dimension, DH and Box-Dimension, DB are allocated the same amount, but in a way somewhat different from each other. In inaccurate way, one can think that the connection between the two is done considering that:


MD

d

D ~N

,

(28)

by analogy with equation (13), i.e. approximating to the geometric extension of the object by the number of boxes (of the same size) necessary to recover it. But, since the definition of the box dimension there is no optimization step, and its value is directly dependent on N (which is not the case with the Hausdorff dimension) in practice one has often the geometric extension is overestimated, particularly for large, i. e. upper limit 1 and thus DB D . However, for the lower limit, i.e. 0 , the Hausdorff-Besicovitch dimensions, DH and the box dimension, DB are equal, becoming valid the measure of geometric extension process, M D at box counting algorithm. Considering from (28) that:

N

~

D

d

D d 1 ,

(29)

D

d

(30)

and that

N

~

max

max

D d 1

Therefore, dividing (29) by (30) has: D

N N taking

max

d

~ max

D d 1

(31)

max

the total grid extension that recover the object, one has: max

1

(32)

D d 1

(33)

From as early as (31) D

N

d

Substituting (33) in (28) has:

MD

D ~

D

,

(34)

This equation is analogous to the fundamental Richardson relationship for a fractal length.

4. Crack and rugged fracture surface models The two main problematics of mathematical description of Fracture Mechanics are based on the following aspects: the surface roughness generated in the process and the field stress/strain applied to the specimen. This section deals with the fractal mathematical description of the first aspect, i.e., the roughness of cracks on Fracture Mechanics, using fractal geometry to model its irregular profile. In it will be shown basic mathematical


assumptions to model and describe the geometric structures of irregular cracks and generic fracture surfaces using the fractal geometry. Subsequently, one presents also the proposal for a self-affine fractal model for rugged surfaces of fracture. The model was derived from a generalization of Voss [48] (1) equation and the model of Morel [49] for fractal self-affine fracture surfaces. A general analytical expression for a rugged crack length as a function of the projected length and fractal dimension is obtained. It is also derived the expression of roughness, which can be directly inserted in the analytical context of Classical Fracture Mechanics. The objectives of this section are: (i) based geometrical concepts, extracted from the fractal theory and apply them to the CFM in order to (ii) construct a precise language for its mathematical description of the CFM, into the new vision the fractal theory. (iii) eliminate some of the questions that arise when using the fractal scaling in the formulation of physical quantities that depend on the rough area of fracture, instead of the projected area, in the manner which is commonly used in fracture mechanics. (iv) another objective is to study the way which the fractal concept can enrich and clarify various aspects of fracture mechanics. For this will be done initially in this section, a brief review of the major advances obtained by the fractal theory, in the understanding of the fractography and in the formation of fracture surfaces and their properties. Then it will be done, also, a mathematical description of our approach, aiming to unify and clarify aspects still disconnected from the classical theory and modern vision, provided by fractal geometry. This will make it possible for the reader to understand what were the major conceptual changes introduced in this work, as well as the point from which the models proposed progressed unfolding in new concepts, new equations and new interpretations of the phenomenon.

4.1. Application of fractal theory in the characterization of a fracture surface In this section one intends to do a brief history of the fractography development as a fractal characterization methodology of a fractal fracture surface.

4.1.1. Geometric aspects and observations extracted from the quantitative fractography of irregular fracture surface The technique used for geometric analysis of the fracture surface is called fractography. Until recently it was based only on profilometric study and statistical analysis of irregular surfaces [50]. Over the years, after repeated observations of these surfaces at various magnifications, was also revealed a variety of self-similar structures that lie between the micro and macro-structural level, characteristic of the type of fracture under observation. Since 1950 it is known that certain structures observed in fracture surfaces by microscopy, showed the phenomenon of invariance by magnification. Such structures recently started to be described in a systematic way by means of fractal geometry [51, 25]. This new approach allows the description of patterns that at first sight seem irregular, but keep an invariance by 1

Voss present a fractal description for the noise in the Browniano mouvement


scale transformation (self-similarity or self-affinity). This means that some facts concerning the fracture have the same character independently of the magnification scale, i.e. the phenomenology that give rise to these structures is the same in different observation scales. The Euclidean scaling of physical quantities is a common occurrence in many physical theories, but when it comes to fractality appears the possibility to describe irregular structures. The fracture for each type of material has a behavior that depends on their physical, chemical, structural, etc. properties. Looking at the topography and the different structures and geometrical patterns formed on the fracture surfaces of various materials, it is impossible to find a single pattern that can describe all these surfaces (Figure 11), since the fractal behavior of the fracture depends on the type of material [52]. However, the fracture surfaces obtained under the same mechanical testing conditions and for the same type of material, retains geometric aspects similar of its relief [53] (see Figure 12). This similarity demonstrates that exist similar conditions in the fracture process for the same material, although also exist statistically changinga from piece to piece, constructed of the same material and under the same conditions [54; 55]. Based on this observation was born the idea to relate the surface roughness of the fracture with the mechanical properties of materials [50].

 Various aspects of the fracture surface for different materials: (a) Metallic B2CT2 sample, (b) Polymeric, sample PU1.0, with details of the microvoids formation during stable crack propagation, (c) Ceramic [56].

4.1.2. Fractal theory applied to description of the relief of a fracture surface Let us now identify the fractal aspects of fracture surfaces of materials in general, to be obtained an experimental basis for the fractal modeling of a generic fracture surface. The description of irregular patterns and structures, is not a trivial task. Every description is related to the identification of facts, aspects and features that may be included in a class of phenomena or structure previously established. Likewise, the mathematical description of the fracture surface must also have criteria for identifying the geometric aspects, in order to identify the irregular patterns and structures which may be subject to classification. The criteria, used until recently were provided by the fractográfico study through statistical


analysis of quantities such as average grain size, roughness, etc. From geometrical view point this description of the irregular fracture surface, was based, until recently, the foundations of Euclidean geometry. However, this procedure made this description a task too complicated. With the advent of fractal geometry, it became possible to approach the problem analytically, and in more authentic way.

 Fracture surfaces of different parts mades with the same material, a) Lot A9 b) Lot A1 [56 1999].

Inside the fractography, fractal description of rugged surfaces, has emerged as a powerful tool able to describe the fracture patterns found in brittle and ductile materials. With this new characterization has become possible to complement the vision of the fracture phenomenon, summarizing the main geometric information left on the fracture surface in just a number, " D ", called fractal dimension. Therefore, assuming that there is a close relationship between the physical phenomena and fractal pattern generated as a fracture surface, for example, the physical properties of these objects have implications on their geometrical properties. Thinking about it, one can take advantage of the geometric description of fractals to extract information about the phenomenology that generated it, thereby obtaining a greater understanding of the fracture process and its physical properties. But before modeling any irregular (or rough) fracture surface, using fractal geometry, will be shown some of the difficulties existing and care should be taken in this mathematical description.

4.2. Fractal models of a rugged fracture surface A fracture surface is a record of information left by the fracture process. But the Classical Fracture Mechanics (CFM) was developed idealizing a regular fracture surface as being smooth and flat. Thus the mathematical foundations of CFM consider an energy equivalence between the rough (actual) and projected (idealized) fracture surfaces [57]. Besides the mathematical complexity, part of this foundation is associated with the difficulties of an accurate measure of the actual area of fracture. In fact, the geometry of the crack surfaces is usually rough and can not be described in a mathematically simple by Euclidean geometry


[52]. Although there are several methods to quantify the fracture area, the results are dependent on the measure ruler size used [56]. Since the last century all the existing methods to measure a rugged surface did not contribute to its insertion into the analytical mathematical formalism of CFM until to rise the fractal geometry. Generally, the roughness of a fracture surface has fractal geometry. Therefore, it is possible to establish a relationship between its topology and the physical quantities of fracture mechanics using fractal characterization techniques. Thus, with the advent of fractal theory, it became possible to describe and quantify any structure apparently irregular in nature [1]. In fact, many theories based on Euclidean geometry are being revised. It was experimentally proved that the fracture surfaces have a fractal scaling, so the Fracture Mechanics is one of the areas included in this scientific context.

4.2.1. Importance of fracture surface modeling The mathematical formalism of the CFM was prepared by imagining a fracture surface flat, smooth and regular. However, this is an mathematical idealization because actually the microscopic viewpoint, and in some cases up to macroscopic a fracture surface is generally a rough and irregular structure difficult to describe geometrically. This type of mathematical simplification above mentioned, exists in many other areas of exact sciences. However, to make useful the mathematical formalism developed over the years, Irwin started to consider the projected area of the fracture surface [57] as being energetically equivalent to the rugged surface area. This was adopted due to experimental difficulties to accurately measure the true area of the fracture, in addition to its highly complex mathematics. Although there are different methods to quantify the actual area of the fracture [56], its equationing within the fracture mechanics was not considered, because the values resulting from experimental measurements depended on the "ruler size" used by various methods. No mathematical theory had emerged so far, able to solve the problem until a few decades came to fractal geometry. Thus, modern fractal geometry can circumvent the problem of complicated mathematical description of the fracture surface, making it useful in mathematical modeling of the fracture. In particular, it was shown experimentally that cracks and fracture surfaces follow a fractional scaling as expected by fractal geometry. Therefore, the fractal modeling of a irregular fracture surface is necessary to obtain the correct measurement of its true area. Therefore, fracture mechanics is included in the above context and all its classical theory takes into account only the projected surface. But with the advent of fractal geometry, is also necessary to revise it by modifying its equations, so that their mathematical description becomes more authentic and accurate. Thus, it is possible to relate the fractal geometric characterization with the physical quantities that describe the fracture, including the true area of irregular fracture surface instead of the projected surface. Thought this idea was that Mandelbrot and Passoja [58] developed the fractal analysis by the "slit island method ". Through this method, they sought to correlate the fractal dimension with the physical wellknown quantities in fracture mechanics, only an empirical way. Following this pioneering


work, other authors [3 , 4 , 5 , 6, 7 , 8, 11, 12, 13, 59 ] have made theoretical and geometrical considerations with the goal of trying to relate the geometrical parameters of the fracture surfaces with the magnitudes of fracture mechanics, such as fracture energy, surface energy, fracture toughness, etc.. However, some misconceptions were made regarding the application of fractal geometry in fracture mechanics. Several authors have suggested different models for the fracture surfaces [60-63]. Everyone knows that when it was possible to model generically a fracture surface, independently of the fractured material, this will allow an analytical description of the phenomena resulting the roughness of these surfaces within the Fracture Mechanics. Thus the Fracture Mechanics will may incorporate fractal aspects of the fracture surfaces explaining more appropriately the material properties in general. In this section one propose a generic model, which results in different cases of fracture surfaces, seeking to portray the variety of geometric features found on these surfaces for different materials. For this a basic mathematical conceptualization is needed which will be described below. For this reason it is done in the following section a brief bibliographic review of the progress made by researchers of the fractal theory and of the Fracture Mechanics in order to obtain a mathematical description of a fracture surface sufficiently complete to be included in the analytical framework of the Mechanics Fracture.

4.2.2. Literature review - models of fractal scaling of fracture surfaces Mosolov [64] and Borodich [3 ] were first to associate the deformation energy and fracture surface involved in the fracture with the exponents of surface roughness generated during the process of breaking chemical bonds, separation of the surfaces and consequently the energy dissipation . They did this relationship using the stress field. Mosolov and Borodich [64, 3 ] used the fractional dependence of singularity exponents of this field at the crack tip and the fractional dependence of fractal scaling exponents of fracture surfaces, postulating the equivalence between the variations in deformation and surface energy. Bouchaud [62] disagreed with the Mosolov model [64] and proposed another model in terms of fluctuations in heights of the roughness on fracture surfaces in the perpendicular direction to the line of crack growth, obtaining a relationship between the fracture critical parameters such as KIC and relative variation of the height fluctuations of the rugged surface. In this scenario has been conjectured the universality of the roughness exponent of fracture surfaces because this did not depend on the material being studied [63]. This assumption has generated controversy [61] which led scientists to discover anomalies in the scaling exponents between local and global scales in fracture surfaces of brittle materials. Family and Vicsék [39, 65] and Barabasi [66] present models of fractal scaling for rugged surfaces in films formed by ballistic deposition. Based on this dynamic scaling Lopez and Schimittibuhl [67, 68] proposed an analogous model valid for fracture surfaces, where they observed in your experiments anomalies in the fractal scaling, with critical dimensions of transition for the behavior of the roughness of these surfaces in brittle materials. In this sense Lopez [67, 68] borrowed from the model of Family and Vicsék [39, 65] analogies that could be applied to the rough fracture surfaces.


4.2.3. The fractality of a crack or fracture surface By observing a crack, in general, one notes that it presents similar geometrical aspects that reproduce itself, at least within a limited range of scales. This property called invariance by scale transformation is called also self-similarity, if not privilege any direction, or selfaffinity, when it favors some direction over the other. Some authors define it as the property that have certain geometrical objects, in which its parts are similar to the whole in in successive scales transformation. In the case of fracture, this takes place from a range of minimum cutoff scale , min until a maximum cutoff scale, max , contrary to the proposed by Borodich [3], which defines an infinite range of scales to maintain the mathematical definition fractal. In the model proposed in this section, one used the fractal theory as a form closer to reality to describe the fracture surface with respect to Euclidean description. This was done in order to have a much better approximatation to reality of the problem and to use fractal theory as a more authentic approach.

 Self-similarity present in a pine (fractal), with different levels of scaling, k.

To understand clearly the statements of the preceding paragraph, one can use the pine example shown in Figure 13. It is known that any stick of a pine is similar in scale, the other branches, which in its turn are similar to the whole pine. The relationship between the scales mentioned above, in case of pine, can be obtained considering from the size of the lower branch (similar to the pine whole) until the macroscopic pine size. Calling of min l0 , the size of the lower branch and max L0 , the macroscopic size of whole pine one may be defined cutoff scales lower and upper (minimum and maximum), subdivided, therefore, the pine in discrete levels of scales as suggested the structure, as follows:

lo min

Lo

where an intermediate scale

lk k

Lo k

min

Lo max

Lo

k

max

k

1

static

case L0

dynamic case L0

L0 max L0 (t )

(35)

can also be defined as follows: lk . Lo

(36)


The magnitude k represents the scaling ratio which depicts the size of any branch with length, lk , in relation to any pine whole. l0 is related to the Mishnaevsky minimum size for a crack which is shown in section - 4.2.6 and L0 L0 max is the maximum leght if the fracture already been completed. Similarly it is assumed that the cracks and fracture surfaces also have their scaling relations, like that represented in equations (35) and (36). In the continuous cutoff scale levels, lower and upper (minimum and maximum), are thus defined as follows: lo min

Lo

l Lo

Lo max

Lo

1

(37)

Note that the self-similarity of the pine so as the crack self-affinity, although statistical, is limited by a lower scale min as determined by the minimum size, l0 , and a upper scale max , given by macroscopic crack size, L0 . From the concepts described so far, it is verified that the measuring scale k to count the structure elements is arbitrary. However, in the scaling of a fracture surface, or a crack profile, follows a question: Which is the value of scale k to be properly used in order to obtain the most accurate possible measurement of the rugged fracture surface? There is a minimum fracture size that depends only on the type of material? Surely the answer to this question lies in the need to define the smallest size of the fractal structure of a crack or fracture surface, so that its size can be used as a minimal calibration measuring ruler(2). Since an fracture surface or crack, is considered a fractal, first, it is necessary to identify in the microstructure of the material which should be the size as small as possible a of a rugged fracture, i.e. the value of lmin . This minimal fracture size, typical of each material, must be then regarded as an elementary structure of the formation of fractal fracture, so defining a minimum cutoff scale, min , for the fractal scaling, where min l0 L0 , where l0 it is a planar projection of lmin . In practice, from this value the minimum scale of measurement, l0 L0 one defines a minimum ruler size min , for this case, equal to the value of the min plane projection the smallest possible fracture size, i.e. min l0 . Thus, the fractal scaling of the fracture surface, or crack, may be done by obtaining the most accurate possible value of the rough length, L . However, the theoretical prediction of the minimum fracture, lmin , must be made from the classical fracture mechanics, as will be seen below.

4.2.4. Scaling hierarchical limits Mandelbrot [58] pointed out in his work that the fracture surfaces and objects found in nature, in general, fall into a regular hierarchy, where different sizes of the irregularities 2

This must be done so that the measurement scales are not arbitrary and may depend on some property of the material.


described by fractal geometry, are limited by upper and lower sizes, in which each level is a version in scale of the levels contained below and above of these sizes. Some structures that appear in nature, as opposed to the mathematical fractals, present the property of invariance by scale transformation (self- similarity of self-affinity) only within a limited range of scale transformation ( min max ) . Note in Figure 13 that this minimal cutoff scale min , one can find an elementary part of the object similar to the whole, that in iteration rules is used as a seed to construct the fractal pattern that is repeated at successive scales, and the maximum cutoff scale D one can see the fractal object as a whole. One must not confuse this mathematical recursive construction way, with the way in which fractals appear in nature really. In physical media, fractals appears normally in situations of local or global instability [24], giving rise to structures that can be called fractals, at least within a narrow range of scaling ( min max ) as is the case of trees such as pine, cauliflower, dendritic structures in solidification of materials, cracks, mountains, clouds, etc. From these examples it is observed that, in nature, the particular characteristics of the seed pattern depends on the particular system. For these structures, it is easy to see that that fractal scaling occurs from the lowest branch of a pine, for a example, which is repeated following the same appearance, until the end size of the same, and vice versa. In the case of a crack, if a portion of this crack, is enlarged by a scale, , one will see that it resembles the entire crack and so on, until reaching to the maximum expansion limit in a minimum scale, min , in which one can not enlarge the portion of the crack, without losing the property of invariance by scaling transformation (self-similar or self-affinity). As the fractal growth theory deals with growing structures, due to local or global instability situations [24], such scaling interval is related to the total energy expended to form the structure. The minimum and maximum scales limit is related to the minimum and maximum scale energy expended in forming the structure, since it is proportional to the fractal mass. The number of levels scaling, k , between min and max depends on the rate at which the formation energy of fractal was dissipated, or also on the instability degree that gave rise to the fractal pattern.

4.2.5. The fractal geometric pattern of a fracture and its measurement scales Considering that the fracture surface formed follows a fractal behavior necessarily also admits the existence of a geometric pattern that repeats itself, independent of the scale of observation. The existence of this pattern also shows that a certain degree of geometric information is stored in scale, during the crack growth. Thus, for each type of material can be abstract a kind of geometric pattern, apparently irregular with slight statistical variations, able to describe the fracture surface. Moreover, for the same type of material is necessary to observe carefully the enlargement or reduction scales of the fracture surface. For, as it reduces or enlarges the scale of view, are found pattern and structures which are modified from certain ranges of these scales. This can be seen in Figure 14. In this figure is shown that in an alumina ceramic, whose ampliation of one of its grains at the microstructure reveals an underlying structure of the


cleavage steps, showing that for different magnifications the material shows different morphologies of the surface of fracture.

 Changings in pattern of irregularities with the magnification scale on a ceramic alumina, Lot A8 [56].

To approach this problem one must first observe that, what is the structure for a scale becomes pattern element or structural element to another scale. For example, to study the material, the level of atomic dimensions, the atom that has its own structure (Figure 15a) is the element of another upper level, i.e., the crystalline (Figure 15b). At this level, the cleavage steps formed by the set of crystalline planes displaced, in turn, become the structural elements of microsuperfície fracture in this scale (Figure 15c). At the next level, the crystalline, is the microstructural level of the material, where each fracture microsurface becomes the structural member, although irregular, of the macroscopic rugged fracture surface, as visible to the naked eye, as is shown diagrammatically in Figure 15d. Thus, the hierarchical structural levels [69] are defined within the material (Figure 15), as already described in this section.

 Different hierarchical structural levels of a fracture in function of the observation scale; a) atomic level; b) crystalline level (cleavage steps); c) microstructural level (fracture microsurfaces) and d) macrostructural level (fracture surface).

Based on the observations made in the preceding paragraph, it is observed that the fractal scaling of a fracture surface should be limited to certain ranges of scale in order to maintain


the mathematical description of the same geometric pattern (atom, crystal, etc.) , which is shown in detail in section - 4.2.3. Although it is possible to find a structural element, forming a pattern, at each hierarchical level, it should be remembered that each type of structure has a characteristic fractal dimension. Therefore, it is impossible to characterize all scale levels of a fracture with only a single fractal dimension. To resolve this problem one can uses a multifractal description. However, within the purpose of this section a description monofractal provides satisfactory results. For this reason, it was considered in the first instance, that a more sophisticated would be unnecessary. Considering the analytical problem of the fractal description, one must establish a lower and an upper observation scale, in which the mathematical considerations are kept within this range. These scales limits are established from the mechanical properties and from the sample size, as will be seen later. Obviously, a mathematical description at another level of scale, should take into account the new range of scales and measurement rules within this other level, as well as the corresponding fractal dimension. As already mentioned, the description of the rough fracture surface can be performed at the atomic level, in cleavage steps level (crystalline) or in microstructural level (fracture microsurfaces), depending on the phenomenological degree of detail that wants to reach. This section will be fixed at the microstructural level (micrometer scale), because it reflects the morphology of the surface described by the thermodynamic view of the fracture. This means that the characteristics lengths of generated defects are large in relation to the atomic scale, thus defining a continuous means that reconciles in the same scale the mechanical properties with the thermodynamic properties. Meanwhile, the atomic level and the level of cleavage steps is treated by molecular dynamics and plasticity theory, respectively, which are part areas.

4.2.6. The calibration problem of a fracture minimum size as a "minimal ruler size" of their fractal To answer the previous question, about the minimum fracture size, Mishnaevsky [70] proposes a minimum characteristic size, a , given by the size of the smallest possible microcrack, formed at the crack tip (or notch) as a result of stress concentration in the vicinity of a piling up dislocations in the crystalline lattice of the material, satisfying a condition of maximum constriction at the crack tip, where: (38)

a ~ ko nb ,

where k0 is a proportionality coefficient. n is the number of dislocations piling up that can be calculated by: n

l (1 b

)

,

(39)

where v is the Poisson's ratio, l is the length of the piling up of dislocations, is the normal or tangential stress, is the shear modulus and b is the Burgers vector. Substituting (39) in (38) one has;


a~

kon l (1

)

.

(40)

Mishnaevsky equates with mathematical elegance, the crack propagation as the result of a "physical reaction" of interaction of a crack size, L0 with a piling up of dislocations, nb , forming a microtrinca size, a , i.e.;

Lo where a

Lo e nb

nb

Lo

a ,

(41)

Lo .

Mishnaevsky proposes a fractal scaling for the fracture process since the minimum scale, given by the size a , until the maximum scale, given by the macroscopic size crack, L0 . As a consequence for the existence of a minimum fracture size, recently has arosen a hypothesis that the fracture process is discrete or quantized (Passoja, 1988, Taylor et al., 2005; Wnuk, 2007). Taylor et al. (2005) conducted mathematical changes in CFM to validate this hypothesis. Experimental results have confirmed that a minimum fractures length is given by: l0 ~

2 Kc

.

(42)

0

where Kc is the fracture toughness, material fracture.

0

is the stress of the yielding strength before the

4.2.7. Fractal scaling of a self-similar rough fracture surface or profile A mathematical relationship between the extension of the self-similar contour and a extension of its projection is calculated as follows. Being A the surface extension of the fractal contour, given by a self-similar homogenous function with fractional degree, D , where: A

D

Au

.

(43)

A0 is the plane projection extension, given by a self-similar homogeneous function with integer degree, d , in accordance with the expression: d

A0

Au

,

(44)

d where, Au is the unit area of measurement, whose values on the rugged and plane surface are the same. Thus the relationships (43) and (44) can be written in the same way as the equations (43) and (44). Therefore, by dividing these equations, one has:

A

A0

d D

.

(45)


An illustration of the relationship (43), (44) and (45) can be seen in Figure 16.

 Rugged surface formed by a homogeneous function A , with frational degee D , whose planar projection, A0 is a homogeneous function of integer degree d , showing the unit surface area Au .

The rugged fracture surface, may be considered to be a homogeneous function with frational degree, D , ni.e.: A

Ak

D k

,

(46)

and its planar projection, may be considered as a homogeneous function with integer degree d 2 , i.e.:

A0

Ar

d r

.

(47)

The index k was chosen to designate the irregular surface at a k -level of any magnification or reduction. The index r has been chosen to designate the smooth (or flat) surface at a r level, and the index, 0 , was chosen to designate the projected surface corresponding to rugged surface, at the k -level. Considering that, for k r and k r , the area unit, Ak and Ar , are necessarily of equal value and dividing relationships (46) and (47) , one has:

A( k )

A0

d D k

.

(48)

The equation (48), means that the scaling performed between a smooth and another irregular surface, must be accompanied by a power term of type k d D . Thus, there is the fractal scaling, which relates the two fracture surfaces in question: a rugged or irregular surface, which contains the true area of the fracture and regular surface, which contains the projected area of the fracture. From now on will be obtained a relationship between the rugged and the projected profile of the fracture in analogous way to equation (250) for a thin flat plate (Figure 17a


and Figure 17b) with thickness e written as:

0 . In this case the area of rugged surface can be

A

Le ,

(49)

 Scaling of a rugged profile of a fracture surface or a crack, using the Mishnaevsky minimum size as a "measuring ruler"; a) in the case of a crack is a non-fractal straight line, where D d 1 ; b) in the case of tortuous fractal crack, with its projected crack length, where d D d 1 .

and the area of the projected surface as A0

L0 e ,

(50)

According to the equation (48) the valid relationship is: L( k )

Lo

d D k

,

where, L( k ) is the measured crack length on the scale measured on the same scale, in a growth direction.

(51) k

, L0 is the projected crack length

4.2.8. The self-similarity relationship of a fractal crack The fracture is characterized from the final separation of the crystal planes. This separation has a minimum well-defined value, possibly given by theory Mishnaevsky Jr. (1994). If it is considered that below of this minimum value the fracture does not exist, and above it the crack is defined as the crystal planes moving continuously (and the formed crack tip penetrates the material), so that an increasing number of crystal planes are finally separated. One can in principle to use this minimum microscopic size as a kind of ruler (or scale) for the measurement of the crack as a whole( 3), i.e. from the start point from which the crack grows until its end characterized by instantaneous process of crack growth, for example. The above idea can be expressed mathematically as follows:

3

During or concurrently with its propagation, in a dynamic scaling process, or not


L

L0

d D

(52)

,

dividing the entire expression (52) above by the minimum Mishnaevsky size one has: L a

L0

d D

a

,

(53)

or N

N0

d D

,

(54)

where N

L a : is the number of crack elements a on the non-projected crack

N0

L0 a : is the number of cracl elements a on the crack projected

and yet: (55)

a L0 , where: : is the scaling factor of the fractal crack d : is the Euclidean dimension of the crack projection D : is the crack fractal dimension.

Within this context the number of microcracks that form the macroscopic crack is given by:

N

a Lo

D

.

(56)

In this context (in Mishnaevsky model), the above expression is volumetric and admits cracks branching generated in the fracture process with opening and coalescence of microcracks. However, he continue equating the process in a one-dimensional way reaching an expression for the crack propagation velocity. A complete discussion of this subject, using a self-affine fractal model to be more realistic and accurate, can be done in another research paper. The answer to the question about what should be the best scale to be used for fractal fracture scaling is then given as follows: being the limit of the crack length Lk in any scale, given by Lk L (actual size) as well as lk lmin , the value of the minimum size ruler, l0 it must be equal to the minimum crack size, a (4), given by Mishnaevsky [70], through its energy balance for the fracture of a single monocrystal of the microstructure of a material. The physical reason for this choice is because the Mishnaevsky minimum size is determined by a It is possible that this minimal ruler size be very low than the scale used in fractal characterization of the fracture surfaces. However, it must to be the smallest possible size for a microcrack. 4


energy balance, from which the crack comes to exist, because below this size, there is no sense speak of crack length. Therefore, the scale that must be considered is given by: min

a L0 ,

(57)

where a is given by relation (40). Therefore, the statistical self-similarity or self-affinity of a fracture surface, or a crack is limited by a cutoff lower scale min , determined by the minimum critical size, l0 a , and a cutoff upper scale max , given by the macroscopic crack length, L0 . In two dimensions, the problem of existence of a minimum scale size (possibly given by the Mishnaevky minimum size), leads to abstraction of a microsurface with minimum area, whose shape will be investigated further, in Appendices, in terms of the number of stress concentrators nearest existing within a material.

4.3. Model of self-affine fracture surface or profiles In this section one intend to present the development of fractal models of self-similar surfaces. From a rough fracture surface can be extracted numerous profiles also rough on the crack propagation direction. However, in this section is considered only one profile, which is representative of the entire fracture surface (Figure 18). The plane strain condition admits this assumption. Because, although the fracture toughness varies along the thickness of the material to a plastic zone reduced in relation to material thickness, it can be considered a property. This means that it is possible to obtain a statistically rough profile, equivalent to other possible profiles, which can be obtained within the thickness range considered by plane strain conditions.

 Statistically equivalent profiles along the thickness of the material

In order also equivalent to this, it is also possible to obtain an average projected crack length as a result of an average of the crack size along the thickness of the material thickness within the range considered by plane strain, for the purpose of calculations in CFM, it is considered


this average size as if it were a single projected crack length, as recommended by the ASTM E1737-96 [71]. Therefore, inwhat follows, is effected by reducing or lowering the dimensional degree of relationship from the two-dimensional case, shown above, for the one-dimensional case, as follows:

A x, y

L x .

(58)

thus, for a self-affine fractal one has:

L( x x)

H x

L( x),

(59)

where

H

2 D

(60)

is the Hurst exponent measuring the profile ruggedness. In one-dimensional case the fracture surface is a profile whose length L is obtained from measuring the projected length, L0 , as illustrated below in Figure 19.

4.3.1. Calculation of the rugged crack length as a function of its projected length Considering a profile of the fracture surface as a self-affine fractal, analogous to the fractal of Figure 19, which perpendicular directions have the same physical nature the Voss [48] equation to the Brownian motion can be generalized(5) to obtain rugged crack length L , depending on the projected crack length, L0 . Figure 19 illustrates one of the methods for fractal measuring. This measure can be obtained by taking boxes or rectangular portions, based L0 and height H 0 on the crack profile, and recovering up this profile, within these boxes, with "little boxes" (recovering units) with small sizes, l0 and h0 , respectively (Figure 19). Instead of little boxes is also possible to use other shapes(6) compatible with the object to be measured. Then makes the counting of the little boxes (or recovering units) needed to recover the extension of the rugged crack, centered in the box L0 H0 . The number of these little boxes (or recovering units) of size r in function of the boxes extension (or parts), L0 H 0 , provides the fractal dimension, as shown in section 3 - Methods for Measuring Length, Area, Volume and Fractal Dimension. Assume that the rectangular little boxes (or recovering units) of microscopic size, r , recover the entire crack length, L inside the box with greater length, L0 H 0 . The number of little boxes (recovering unit) with sides of l0 h0 needed to recover a crack in the horizontal direction, inside the box (or stretch) of rectangular area L0 H 0 , for the self-affine fractal can be obtained by the expression: Voss [48], modeled the noise plot of the frational Brownian motion , where in the y-direction, he plots the amplitude, VH, and in the x-direction, he plots the time, t. 6 Some authors used balls 5


 Self-affine fractal of Weierstrass-Mandelbrot, where k 1 / 4 and Dx 1.5 and H 0.5 , used to represent a fracture profile (Family, Fereydoon; Vicsek, Tamas Dynamics of Fractal Surfaces, World Scientific, Singapore , 1991, p.7).

L0

Nv where

0 v

l0

in vertical direction

L0 is the crack horizontal projection and

v

(61)

is the vertical scaling factor.

Considering that the self-affine fractal extends in the horizontal direction along L0 , and oscillates in the perpendicular direction, i.e. in the vertical direction, the number of little boxes N h , with size, l0 in the horizontal direction, are gathered to form the projected length L0 , while vertically the number of little boxes N v , with size h0 , overlap each other, increasing (as power law) this number in comparison to the number of little boxes gathered horizontally. Therefore, for the vertical direction with a projection H0 , the box sides L0 H 0 , an expression for the number of boxes (or units covering) can be writen as: Nh

H0 h0

H h

in horizontal direction .

where H is the Hurst exponent, H0 is the total variation in height ( lo h is the scale transformation factor in the horizontal direction. Therefore, for the corresponding rugged crack length (real) can writes: L

N vr

L , the stretch

(62) H0

Lo ) and

L0

H 0 one

(63)


where r is equal to the rugged crack length on a microscopic scale, as a function of extension of the little boxes l0 h0 by:

r

l0 2

h0 2

(64)

where l0 and h0 are the microscopic sizes of the crack length in horizontal and vertical directions, respectively. Substituting (64) in (63), one has: L

N v l02

h02

(65)

L0 l0 2 l0

h0 2

(66)

substituting (61) in (65), one has:

L

Since that in the fracture process, the scales in orthogonal directions are the same physical nature, one can choose v l0 / L0 , and one can writes from (62) that: h

Nh being necessarily N h

H0 l0

L0 l0

H

(67)

N v , one has: L0 l0

H0 l0

H

L0 l0

(68)

rewriting the equation (66), one has:

L

h0 l0

L0 1

2

(69)

writing h0 from (68), as:

h0

H0

L0 l0

H 1

.

(70)

Eliminating in (69) the dependence of h0 , by substituting (70) in (69), one has:

L

L0 1

H0 l0

2

L0 l0

2 H 1

(71)


The curve length in the stretch, L0 H 0 considering the Sand-Box method [38] whose counting starts from the origin of the fractal, can be written as: L L, L0 L0 and H 0 H 0 hence the equation (71) shall be given by:

L

H0

L0 1

2

l0

L0

2 H 1

,

l0

(72)

whose the plot is shown in Figure 20. Note that the lengths L0 and H0 correspond to the projected crack length in the horizontal and vertical directions, respectively. Applying the logaritm on the both sides of equation (72) one obtains an expression that relates the fractal dimension with the projected crack length:

ln Df

ln L / l0 ln L0 / l0

1

1 2

H0 l0

2

L0 l0

ln L0

2 H 1

(73)

 Graph of the rugged length L in function of the projected length L0 , showing the influence of height, H0 , of the boxes in the fractal model of fracture surface: a) in the upper curves is observed the effect of H0 as it tends to unity ( H0 1.0 ), b) in the lower curves, that appearing almost overlap, is observed the effect of H0 as it tends to zero ( H0 0 ).

The graph in Figure 20 shows the influence of the boxes height H 0 on the rugged crack length, L , as a function of the projected crack length, L0 . Note that for boxes of low height ( H 0 0 ), in relation to its projected length, L0 , the lower curves (for H 0 0.01,0.001,0.0001 ), denoted by the letter " b ", almost overlap giving rise to a linear relation between these lengths (Figure 21). While for boxes of high height ( H 0 1.0 ) in relation to its projected length, L0 , the relation between the lengths become each more distinct from the linear relationship for the same exponent roughness, H .


 Counting boxes (or strechts) with rectangular sizes Lo x H o where the boxes that recovers the profile have different extensions in the horizontal and vertical directions.

Making up the counting boxes (or stretch) with rectangular sizes Lo x H o where the boxes recovering the profile have different extensions in the horizontal and vertical directions respectively, i.e., H o lo the equation (72). Is simplified to: L

Lo 1

lo

2H 2

.

Lo

(74)

which plot is shown in Figure 22. The graph in Figure 22 shows the influence of the roughness dimension on the rugged crack length, L , in function of the projected length, L0 . Note that for H0 1.0 , corresponding to a smooth surface, the relation between the rugged and projected length becomes increasingly linear. While for H 0 0 , which corresponds to a rougher surface, the the relation between the rugged and projected length becomes increasingly non-linear.

 Graph of the rugged length, L , in function of the projected length, L0 , showing the influence of the Hurst exponent H , in the fractal model of the fracture surface.

Note that for Lo

H o , one has, from the equation (62) and (68) the following relationship: Lo

ho

lo Lo

H

,

(75)


which is a self-similar relation between the projected crack length, L0 , and height of the little box, h0 . This relationship shows that all self-affine fractal, in the approximation of a small scale, has a local self-similarity forming a fractal substructure, when is considered square portions, L0 L0 , instead of rectangular portions, L0 H0 . It important to observe that L0 denotes the distance between two points of the crack (the projected crack length). The self-affine measure, L of L0 , in the fractal dimension, D , is given by (72). l0 is the possible minimum length of a micro-crack, which defines the scale l0 / L0 under which the crack profile is scrutinized, as discussed in previous section and will be discussed after in the section #.5.4.5. The Hurst exponent, H , is related to D by (60). In the study of a self-affine fractal there are two extremes limits to be verified. One is the limit at which the boxes height is high in relation to its projected length, L0 , i.e. ( H0 L0 ), which is also called local limit. The other limit is one in which the boxes height is low in relation to its projected length, L0 , i.e., ( H0 h0 ) which is called global limit. It will be seen now each one of this limits case contained in the expression (72). Case 1 : The self-similar or local limit of the fractality Taking the local limit of the self-affine fractal measure as given by (72), i. e. for the case where, H 0 L0 l0 , one has: L

Lo

lo Lo

H 1

(76)

where L Lo

2 H

lo H

1

(77)

constant

This equation is analogous to self-similar mathematical relationship only that the exponent is 1 H instead of D 1 , which satisfies the relation H 2 D [3 , 40, 51 , 70]. According to these results it is observed that the relation (77) has a commitment to the Hurst exponent of the profiles on the considered observation scale l0 / L0 . It is observed that the consideration of a minimum fracture size l01 over a region, one must consider the local dimension of the fracture roughness on this scale. Similarly, if the considerations of a minimum fracture size are made in a scale that involves several regions, l02 this should take into account the value of the roughness global dimension on this scale, so that: (2 H1 )l01 although l01

l02 e H1

H1 1

(2 H 2 )l02

H2 1

constant ,

(78)

H2 .

Case 2: The self-affine or global limit of fractality Taking the global limit of the self-affine fractal measure given by (72), i.e. for the case in which: H0 l0 L0 . Therefore the length L is independently of H and D 1 , so


L

Lo

(79)

It must be noted that the ductile materials by having a high fractality have a crack profile which can be better fitted by the equation (76), while brittle materials by having a low fractality will be better fitted by the equation (79) corresponding the classical model, i.e., a flat geometry for the fracture surface. Furthermore, the cleavage which occurs on the microstructure of ductile materials tend to produce a surface, where L L0 , which could be called smooth. However, this cleavage effect is just only local in these materials and therefore the resulting fracture surface is actually rugged.

4.3.2. Local Ruggedness of a fracture surface Defining the local roughness of a fracture surface, as: dA dAo

A

A0 dA0 .

(80)

where A is the rugged surface and A0 is the projected surface. In the case of a rugged crack profile,one has: dL dLo

L

L0 dL0

(81)

using (74) in (81), one has that: l 1 (2 H ) o Lo 1

lo Lo

2H 2

2H 2

(82)

From (81) note that when there is no roughness on surfaces (flat fracture) one has that: L L0 , thus dL dLo The quantity

1.

(83)

dL seems be a good definition of ruggedness unlike the definition where dLo

the ruggedness is given by

L / L/ / [56, 57] (where L/ /

L0 M cos , see Figure 23) does not

satisfy the requirement intuitive of the ruggednes when L0M is only inclined with respect to L0 , while maintaining, L0

L0M , as shown Figure 23.


 Schematization of a rugged surface which is inclined with respect to its projection.

The ruggedness must depend on infinitesimally of the projected length and its relative orientation to it. In this case, the surface roughness by the usual definition adds an error equal to the angle secant , or L L/ /

L0 M L/ /

L L0 M

L 1 . L0 M cos

(84)

However, by the definition proposed herein, when only one inclines a smooth surface against dL dL dLoM to the horizontal, one has: L L0 M and L0 M L0 and again, 1. dLo dLoM dLo Within this philosophy will be considered as rugged any surface that presents in an dL infinitesimal portion a variation of their contour such that 1 , therefore has to be: dLo dL dLo

1.

(85)

4.3.3. Preliminary considerations on the proposed model Considering a fractal model for the fracture surface given by the equation:

L

Lo 1

Ho lo

2

lo Lo

2H 2

,

(86)

it is possible to describe its ruggedness in order to include it in the mathematical formalism CFM to obtain a Fractal Fracture Mechanics (FFM). This derivative of equation (86) defines a fractal surface ruggedness, which for the case of a self-affine crack which grows with, H0 l0 , is given by:


1 (2 H )

1

lo Lo

lo Lo

2H 2

2H 2

1.

(87)

such modifications were added to equations of the Irregular Fracture Mechanics to obtain a Fractal Fracture Mechanics as described below.

4.3.4. Comparison of fractal model with experimental results In Figure 24 and Figure 25, a good agreement is observed in the curve fitting of equation (72) and equation (73) to the fractal analyses of the mortar specimen A 2 side1 and the red ceramic specimen A 8 , respectively.

 a) Fractal analysis of mortar specimen A2 side 1 Fractal dimension x Projected length, L0 ; b)Fractal analysis of mortar specimen A2 side 1 - rugged length L x projected length, L0

 a) Fractal analysis of red clay A8 side 1 Fractal dimension x Projected length, L0 ; b) Fractal analysis of red ceramic specimen A8 - rugged length L x projected length, L0


5. Conclusions i.

It is possible, in principle, mathematically distinguish a crack in different materials using geometric characteristics which can be portrayed by different values of roughness exponents in the relations (72) and (74). ii. The fractal model of the rugged crack length, L in function of the projected crack length, L0 , suggested by Alves [72 73 ,74 , 75 ] seems have a good agreement with experimental results. This results allowed us to consolidate the model previously published in the literature on fracture [72 , 73 , 75 ]. iii. The rugged crack length is a response to its interaction with the microstructure. of the material. Therefore, mathematically is possible to portray the rugged peculiar behavior of a crack using fractal geometry iv. The mathematical model presents a wealth (mathematical richness) that can still be explored in terms of determining the minimum crack length, l0 for each material and the fractal dimension as a function of test parameters and material properties. v. The mathematical model is sensitive to variations in the behavior of the crack length it is a linear or logarithmic with the projected crack length. Comparing the experimental results with the model proposed in this chapter, it is concluded that one of the more important results obtained here are the equations (72), (74) and (82) leading to finding that the fracture surfaces of the materials analyzed are indeed (actually) self-affine fractals. Starting from this verification it becomes feasible to consider the fractal model of rugged fracture surface and its ruggedess inside the equations of the classical fracture mechanics, according to equation (74) and (82). As there is a close relationship between phenomenology and structure formed by virtue of its fractal geometry, the understanding of the formation processes of these dissipative structures, as the cracks, should be derived from their mathematical analysis, as the close relationship between the phenomenology of the formation process of dissipative structures and their fractal geometry. Therefore, the mathematical description of fractal structures must exceed a simple geometrical characterization, in order to correlate the pattern formed in the process of energy dissipation with the amount of energy dissipated in the process that generated it. Thus, it is possible to use the fractal geometry in order to understand other more and more complex processes inside the fracture mechanics. Therefore, the various mechanisms responsible by the crack deviation and by the formation of the rugged fracture surface can then, from the fractal model, be quantified in the fractal analysis of this surface. The idea of obtaining a relationship between L and L0 comes the need to maintain the present formalism used by the CFM, showing that fractal geometry can greatly contribute to the continued advancement of this science. On the other hand, we are interested in developing a Fractal Thermodynamic for a rugged crack that will be related to the CFM and the Classical Fracture Thermodynamics when the crack ruggedness is neglected or the crack is considered smooth.


Author details Lucas Máximo Alves GTEME Grupo de Termodinâmica, Mecânica e Eletrônica dos Materiais, Departamento de Engenharia de Materiais, Setor de Ciências Agrárias e de Tecnologia, Universidade Estadual de Ponta Grossa, Brazil

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[38] Bunde, Armin Shlomo Havlin (1994) Fractals In Science. Springer-Verlag. [39] Family, Fereydoon; Vicsek, Tamás (1991) Dynamics of Fractal Surfaces. Singapore: World Scientific Publishing Co. Pte. Ltd. P.7-8, P. 73-77. [40] Feder, Jens (1989) Fractals, New York: Plenum Press. [41] Barnsley, Michael (1988) Fractals Everywhere, Academic Press, Inc, Harcourt Brace Jovanovich Publishers. [42] Milman V. Yu., Blumenfeld R., Stelmashenko N. A. and Ball R. C. (1993) Phys. Rev. Lett. 71 , 204. [43] Milman, Victor Y.; Nadia A. Stelmashenko and Raphael Blumenfeld (1994) Fracture Surfaces: a Critical Review of Fractal Studies and a Novel Morphological Analysis of Scanning Tunneling Microscopy Measurements, Progreess In Materials Science. 38: 425-474. [44] Yamaguti, Marcos (1992). Doctoral Thesis - Universidade de São Paulo [45] Allen, Martin; Gareth J. Brown; Nick J. Miles (1995) Measurements of Boundary Fractal Dimensions: Review of Current Techniques. Powder Technology. 84:1-14. [46] Richardson, L. F. (19161) The Problem of Contiguity: An Appendix To Statistics of Deadly Quarrels. General Systems Yearbook. (6): 139-187. [47] Alves, Lucas Máximo (1998) Escalonamento Dinâmico da Fractais Laplacianos Baseado No Método Sand-Box, In: Anais Do 42o Cong. Bras. de Cerâmica, Poços de Caldas de 3 a 6 de Junho,. Artigo Publicado neste Congresso Ref.007/1. [48] Voss, Richard F. (1991) In: Family, Fereydoon. and Vicsék, Tamás, editors. Dynamics of Fractal Surfaces. Singapore: World Scientific. pp. 40-45. [49] Morel, Sthéphane, Jean Schmittbuhl, Elisabeth Bouchaud and Gérard Valentin (2000) Scaling of Crack Surfaces and Implications on Fracture Mechanics Arxiv:CondMat/0007100 6 Jul 2000, vol. 1, Or Phys. Rev. Lett. 21 August 85(8). [50] Underwood, Erwin E. and Kingshuk Banerji (1996) Quantitative Fractography. Engineering Aspectes of Failure and Failure Analysis - Asm - Handbook - Vol. 12, Fractography - the Materials Information Society (1992). Astm 1996 , pp. 192-209 [51] Dauskardt, R. H.; F. Haubensak and R. O. Ritchie (1990) On the Interpretation of the Fractal Character of Fracture Surfaces; Acta Metall. Matter. 38(2): 143-159. [52] Underwood, Erwin E. and Kingshuk Banerji (1986) Fractals In Fractography, Materials Science and Engineering, Ed. Elsevier. 80: pp. 1-14. [53] Underwood, Erwin E. and Kingshuk Banerji (1996) Fractal Analysis of Fracture Surfaces. Engineering Aspectes of Failure and Failure Analysis - Asm - Handbook - Vol. 12, Fractography - the Materials Information Society (1992), Astm 1996. pp. 210-215 [54] Alves, L. M. ; Chinelatto, Adilson Luiz ; Chinelatto, Adriana Scoton Antonio ; Prestes, Eduardo (2004) Verificação de um modelo fractal de fratura de argamassa de cimento. In: Anais do 48º Congresso Brasileiro de Cerâmica, 28 de Junho a 1º de Julho de 2004, Em Curitiba Paraná. [55] Alves, L. M. ; Chinelatto, Adilson Luiz ; Chinelatto, Adriana Scoton Antonio ; Grzebielucka, Edson Cezar (2004) Estudo do perfil fractal de fratura de cerâmica vermelha. In: Anais do 48º Congresso Brasileiro de Cerâmica, 28 de Junho a 1º de Julho de 2004, Em Curitiba Paraná. [56] Dos Santos, Sergio Francisco (1999) Aplicação Do Conceito de Fractais Para Análise Do Processo de Fratura de Materiais Cerâmicos, Master Dissertation, Universidade Federal


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A GEOMETRIA DA FRATURA Research · August 2015

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A GEOMETRIA DA FRATURA

Lucas Máximo Alves Av. Carlos Cavalcanti, s/n, Uvaranas,CEP - 84030-000, Caixa Postal 1007, Fone/Fax: (042) 2203055, [email protected], Centro Interdisciplinar de Pesquisas em Materiais, Universidade Estadual de Ponta Grossa-PR, - CIPEM/UEPG-PR.

RESUMO Trincas e superfícies de fraturas são estruturas que possuem uma geometria irregular que difere da geometria euclidiana. Estas estruturas podem ser descritas pelos conceitos extraídos da geometria fractal aplicados à fratura. O objetivo deste trabalho é apresentar uma linguagem precisa para a descrição da mecânica da fratura, na nova visão da teoria fractal. Esta linguagem aplicada diretamente à fenomenologia da fratura, permitirá entender os processos de formação de trincas e microtrincas com padrões ramificados ou não. Padrões fractais são encontrados desde a fratura estável até o processo de fragmentação. Palavras-chaves: trabalho de fratura, energia de superfíce, curva-R, materiais frágeis, dimensão fractal. INTRODUÇÃO A fractografia trata da caracterização das trincas e superfícies de fratura através de observações em microscópios. Com estas observações feitas ao longo dos anos, percebeu-se que, as trincas e superfícies de fratura, apresentam aspectos geométricos similares, (1,2)

independentes da escala de magnificação

. Esta idéia, fêz com que se admitisse que algum

tipo de padrão, aparentemente irregular, se conserva. A partir daí, as trincas e surpefícies de fratura tem sido estabelecidas através de inúmeras medidas experimentais(2,4-6,9-11), como sendo objetos geométricos que apresentam um escalonamento fracionário ou “fractal”, conforme definido por Benoit Mandelbrot(1). A geometria fractal trata da descrição de superfícies e padrões de crescimento irregulares, (1,2)

que até então só eram estudados pela topografia

. No caso particular da fratura, este estudo é

feito pela fractografia(2), por meio da análise estatística e das técnicas de perfilometria de superfícies(3). Com o surgimento da geometria fractal(1), a caracterização topográfica e (2)

fractográfica tem recebido um novo impulso . O principal responsável por esta nova abordagem foi o próprio Mandelbrot, que também criou alguns dos métodos de análise de superfícies, (3)

baseado na medida da dimensão fractal . Entre estes métodos, destaca-se o método das ilhas

cortadas(4), o qual tem sido amplamente utilizado na caracterização de superfíces irregulares de uma forma geral. (4)

Apesar de ter criado o método das ilhas cortadas, Mandelbrot e Passoja , procuraram relacionar a dimensão fractal com as grandezas já conhecidos da mecânica da fratura, apenas de uma forma empírica. Na sequência, alguns autores(5-7,10), têm feito considerações teóricas ou geométricas, no sentido de tentar relacionar os parâmetros geométricos das superfícies de fratura, com as grandezas da mecânica da fratura, tais como: energia de fratura, energia de superfície, tenacidade a fratura, etc. No entanto, algumas confusões têm sido feitas. O objetivo deste trabalho é sanar algumas das dúvidas que surgem, quando se utiliza o escalonamento fractal em relação a quantidades que dependem da área rugosa de fratura e da área projetada, da forma como é comumente usada. FUNDAMENTOS TEÓRICOS Seguramente o fenômeno físico que gera um fractal, como uma superfície de fratura por exemplo, está estreitamente relacionado as propriedades físicas destes objetos, e estas por sua vez, tem implicações nas suas propriedades geométricas. Pensando nisso, pode-se tirar proveito da descrição geométrica e extrair informações do fenômeno que gera estes objetos, além de um (8)

entendimento maior das suas propriedades físicas . Em primeiro lugar, deve-se começar com a definição de função homogênea dada por Euler, que constitue a base de todo o escalonamento fractal. De acordo com o teorema de Euler para funções homogêneas de grau n qualquer, uma transformação de escala, k (min  k  max), numa função F(c) deste tipo resulta em: -n

F(kc) = k F(c)

(0  k  1)

(A)

Este resultado significa, que o valor de uma função numa escala, F(c), está relacionado com o valor desta mesma função numa outra escala, F(kc) por uma relação entre as escalas k, elevada a uma potência n que corresponde ao grau de homogeneidade da função. Embora Euler tenha considerado casos de funções com n inteiro, o teorema é válido para um valor de n qualquer. Um fractal, é um objeto que apresenta a propriedade da autosimilaridade, obedecendo a um tipo de escalonamento que pode ser fracionário, ou não. No primeiro caso, o grau de (2)

homegeneidade n da função descrita em (A) não é inteiro. Mandelbrot mostrou que as trincas e as superfícies de fratura são estruturas geométricas fractais que satisfazem o teorema de Euler.

Até o advento dos fractais, toda a mecânica da fratura era tratada utilizando-se apenas a superfíce projetada. Porém, com a geometria fractal, é necessário rever a mecânica da fratura relacionando as grandezas já estabelecidas com a área verdadeira da superfície irregular (ou rugosa). A superficie de fratura rugosa Ak, pode ser considerada como sendo uma função -D

homogênea de grau D, ou seja, Ak = k Au e a sua projeção no plano Ar, como sendo uma função homogênea de grau E = 2, ou seja, Ar = k-EAu. Como as áreas unitárias Au são necessariamente iguais, dividindo-se estas relações tem-se: E-D

Ak = Ark

(B)

O índice k foi escolhido de forma a designar a superfície irregular num nível k qualquer. O índice r, foi escolhido para designar a superfície regular de projeção correspondente a superfície rugosa k. A relação (B), significa que o escalonamento realizado entre uma superfíce regular e uma E-D

outra irregular, deve ser acompanhado de um termo de potência do tipo k . Desta forma, temse o escalonamento fractal, que relaciona as duas superfícies de fratura em questão: a superfície irregular ou rugosa, que contém a área verdadeira da fratura e a superfície regular, que contém a área projetada da fratura. semente ou gerador ak k=0

iniciador To = Tr

k=1 T1  A1 Tr k=2 aeff

ar

Tk  Ak

Tr  Ar

Figura - 1. Fractal auto-afim, do movimento Browniano fracional, onde  = 1/4 e D = 1.0, para três níveis de escalonamento, utilizado para representar uma trinca.

Por outro lado, para uma fina placa plana (Figura - 2 e 3) de espessura e 0 , onde Ak = e.Tk ou Ar = e.Tr, vale a relação: Tk = Tr kE-D

(C)

onde: Tk : tamanho medido da trinca na escala k; Tr: é o tamanho projetado da trinca numa determinada direção. A Figura –1, mostra uma fractal do auto-afim, do movimento Browniano fracional, onde  = 1/4 e D = 1.0, para três níveis de escalonamento. Este fractal será utilizado para representar uma trinca. As linhas inclinadas deste fractal representa os diferentes planos cristalinos fraturados na propagação de uma trinca. As trincas e superfícies de fratura são objetos fractais auto-afins. Fractais auto-afins, são aqueles que possuem projeção em figuras regulares tais como retas, planos, etc. E apresentam escalonamentos anisotrópicos nas diferentes direções x e y, por exemplo, possuindo também (7)

diferentes dimensões fractais Dx e Dy nestas direções. A curva de Koch , não é um fractal autoafim, por que em um nível de escalonamento k  , ela não apresentam projeção sobre um plano ou sobre uma linha, como é de se esperar para uma superfície de fratura, ou para uma simples trinca. Esta é a principal razão porque este fractal não pode ser usado como representativo das trincas ou superfícies de fratura. Para se efetuar o escalonamento fractal correto das superfícies de fratura, é preciso lembrar, que um objeto fractal possue um iniciador e uma semente, conforme mostra a Figura - 1. Fazendo-se uma analogia entre a curva da Figura - 1 e a trinca das Figuras - 2 e 3, vê-se que o iniciador, pode ser a própria superfície de projeção na qual a superfície irregular esta apoiada. E sobre esta superfície projetada, constroe-se a superfície irregular, recobrindo-se a primeira com a semente de área ak, nas mais diversas escalas desde a escala mínima min (k = 1) que corresponde a escala da própria semente fractal, até a escala máxima máx (k = kmáx), que corresponde a escala do comprimento final da trinca. Fazendo-se desta forma, no final a superfície irregular será reconstruída, com um número de sementes ou elementos da estrutura dado por: E-D

Nk = Ak /ak = (Ar/ak) k

(D)

No caso de uma trinca (Figura -1), o número de estruturas lineares em que se consegue subdividir a trinca é dado por:

Nk = Tk /ck = (Tr/ck) kE-D

(E)

onde: Nk: é o número de estruturas na partição com tamanho ck ou na escala k; D: dimensão fractal; ck: tamanho do segmento característico, ou da régua de medida da trinca (Figura -1); k = ck/Tr: tamanho da partição ou escala de medida. Portanto a rugosidade Q da superfície de fratura que também é uma grandeza que pode ser escalonada, é definida como: E-D

Qk = Ak /Ar = Nk/Nr = k

(F)

Para a determinação do número de estruturas de um fractal, existem dois métodos principais de contagens, que podem levar a determinação da dimensão fractal de um objeto(3). O primeiro é o método chamado Box-counting, exemplificado na Figura - 2. O segundo é o método Sand-box exemplificado na Figura - 3. No primeiro método (Figura - 2), subdivide-se o objeto em nk = Tr/ck caixas iguais de lado ck e conta-se quantas destas caixas cobrem o objeto. Em seguida, varia-se o tamanho das caixas e refaz-se a contagem. Fazendo-se o gráfico do logaritmo do número Nk de caixas que cobrem o objeto pela escala de cada subdivisão (k = ck/Tr), obtém-se a partir da inclinação deste gráfico a dimensão fractal. Do ponto de vista da medida experimental, pode-se pensar em usar diferentes métodos de visualização da trinca para a obtenção da dimensão fractal, tais como: microscópio ótico, microscópio eletrônico, microscópio de força atômica, etc., os quais apresentam naturalmente diferentes réguas cK e consequentemente diferentes escalas de medida k. N1

K = cK/Tr

Tr = To

N2

T1 c1

N3

T2 c2

Nk

T3 c3

N*

Tk .. ck

T = T ...........c  = c*

Figura - 2. Trecho de uma trinca sobre um corpo de prova, mostrando, a variação da medida do comprimento Tk da trinca com a escala de medida K = cK/Tr ou partição ck variável (Tr = cte), na contagem pelo método Box-Counting.

No segundo método (Figura - 3), levando-se em conta o tamanho infinitesimal da régua de medida c* do método anterior, cobre-se a figura com caixas de tamanhos Tr diferentes, não importando a forma, que podem ser retangulares ou esféricas, porém, fixadas em um ponto “O” qualquer sobre a figura, denominado origem, a partir do qual as caixas são ampliadas. Conta-se o número Nk de estruturas elementares, ou sementes, que cabem dentro de cada caixa. Fazendo-se o gráfico de logNk x log(c*/Tr) obtém-se, da mesma forma que no método anterior a dimensão fractal. Do ponto de vista experimental, é preciso escolher um único método de medida, para a fixação do tamanho da régua c*, a partir daí, são tomadas diferentes extensões da trinca, para a variação da escala de medida r, uma vez que o tamanho da régua ou ou partição ck = c* se mantém fixa. N1

N2

N3

Nk

Tr3 T3 r = c*/Tr

N* Trk Tk

T = T

Tr2 T2

c* Tr

Tr1 T1

O

O

O

O

O

Figura - 3. Trecho de uma trinca sobre um corpo de prova, mostrando, a variação da medida do comprimento Tk da trinca com a escala de medida r = c*/Tr, para uma partição fixa c* = cte (Tr = variável), na contagem pelo método Sand-Box. Dos métodos descritos anteriormente, verifica-se que a escala k ou r de medida para contagem dos elementos de estrutura é arbitrária. Porém, no escalonamento da superfície de fratura segue uma pergunta: Qual é o valor da escala  que deve ser utilizada corretamente? A resposta é dada da seguinte forma: sendo o limite do tamanho da trinca Tk numa escala qualquer, dado por Tk  T = T (tamanho real) assim como ck  c= c*. O valor de c* pode ser o tamanho crítico da trinca encontrado por Griffith no seu balanço de energia para fratura de um monocristal. Pois a auto-similaridade estatística de uma trinca está limitada por uma escala inferior min determinada pelo tamanho critico e por uma escala superior máx, dado pelo tamanho macroscópico da trinca. A razão física para esta escolha, é porque o tamanho crítico de Griffith é determinado por um balanço de energia, a partir do qual a trinca passa a existir. Pois abaixo deste tamanho, não há sentido em se falar em comprimento de trinca. Portanto, a escala que deve ser considerada é aquela dada por:

min = c*/Tr 

(G)

Relações de Escalonamento. Sabe-se que a energia de superfície, é a grandeza mais diretamente relacionada com a forma geométrica das trincas ou superfícies de fratura. Desta forma, pode-se tentar relacionar as diversas grandezas da mecânica da fratura, com a forma geométrica de sua superfície. Ou seja, a fratura de um grupo de planos cristalinos, quaisquer com energia de superfície eff de suas ligações, é dado por: 2eff = ueff /aeff

(H)

onde: aeff: é a área irregular de um conjunto de planos cristalinos como aqueles representados pela semente fractal da Figura –1. ueff: é a energia necessária para fraturar tais planos cristalinos lnN D1 - E lnN1 Dk - E lnNk=2

Dm - E Região I

Região II

Dmáx- E III

lnNmáx=3 1

m

k=2

máx=3 ln

Figura - 4. Gráfico do escalonamento fractal de uma trinca, mostrando as três regiões de dimensão fractal diferentes. [6,9]

Numa fratura normalmente tem-se diferentes regiões

, particularmente para o caso de [9]

uma fratura provocada por um identador, Tsai e Mecholsky

apontam três, as quais foram

denominadas de: espelhada, mista e fatiada. Estas diferentes regiões podem apresentar dimensões fractais diferentes, de forma análoga ao gráfico da Figura - 4. A confusão que normalmente acontece, é em relação ao escalonamento entre as grandezas rugosas e projetadas, pois não se distingue que região da fratura esta sendo considerada(5-7). A semente fractal que pode representar as três regiões da Figura – 4., rigorosamente deveriam ser diferentes, mas se o material for considerado como sendo homogêneo e isotrópico, a diferenciação entre as dimensões fractais é justificada apenas pelos regimes de propagação, conforme mostra o gráfico genérico de uma curva-R crescente (Figura - 5), que representa a

história da dificuldade que a trinca teve para se propagar. Nesta figura, apresenta-se os diferentes regimes de propagação de uma trinca representados por uma curva-R crescente. Neste gráfico está implicitamente relacionado, três situações: o regime inicial o intermediário e o regime final de propagação de uma trinca. Ainda nesta figura, os diferentes valores representativos de R, obtidos a partir da superfície projetada (regular) e apresentados sobre o eixo vertical do gráfico da curva-R, podem relacionados com as diferentes energias de superfícies definidas sobre a área irregular.

Rmáx  2máx R  2k  2wof

Ro  2eff

0

Comprimento projetado da trinca (T)

Tr

Figura - 5. Gráfico de uma curva-R crescente em função do comprimento da trinca, mostrando as relações diretas entre as energias de superfícies e os parâmetros desta curva. Para que seja feito na prática, um escalonamento correto sobre a superfíce de fratura experimentalmente obtida, a semente, deve ser adequadamente escolhida, afim de que ela seja representativa do grupo de ligações químicas quebradas durante a propagação da trinca. De forma que a condição de homogeneidade seja satisfeita, quando se escalonar toda a superfície. Como eff está associado a semente aeff do escalonamento fractal, ela pode ser pensada como sendo a energia de superfície inicial de um gráfico de curva R versus o comprimento da trinca (Figura - 5), onde eff = o + p, onde p é a energia de superfície devido a deformação plástica. Para se escalonar a expressão (H), a fim de se encontrar quanto vale a energia de superfície média que envolve toda a superfície de fratura, deve-se em primeiro lugar verificar se trata-se de uma superfície irregular (rugosa) ou regular (lisa). Considerando que, eff corresponde a energia de superfície de um grupo de ligações químicas, que estão em planos cristalinos genéricos, conforme descreve a semente da Figura -1, então a superfície associada com esta energia é do tipo irregular, isto é, rugosa. Para se escalonar esta grandeza, isto é eff, com uma grandeza correspondente a energia de superfície projetada de toda a área de fratura como 2wof = UT/ArT por exemplo, (UT: é a energia total gasta para

fraturar toda a superfície de área projetada ArT) deve-se transformar eff em Ro = ueff/ar da seguinte forma: as energias de superfície fraturada rugosa, ueff, e projetada, ur, são necessariamente iguais portanto tem-se : Roar = 2effaeff projetada

(I)

verdadeira

onde: Ro: é a energia de superfície projetada no início da propagação da trinca e corresponde ao valor inicial da curva-R; eff: é a energia de superfície da superfície fraturada no início da propagação da trinca. Multiplicando-se os dois lados de (H) por 1E-D1 tem-se : E-D1

Roar1

E-D1

= 2effaeff1

(J)

Usando-se o resultado (B) tem-se : Ro = 2eff1E-D1

(K)

A curva-R é definida sobre a superfície de fratura projetada. Logo, o escalonamento entre wof e Ro deve seguir uma transformação de escala de homogeneidade inteira igual a E. Portanto, Ro se escalona com wof da seguinte forma: Ro = 2wof (2/1)E

(L)

onde: wof = rT: é a energia de superfície média do trabalho total de fratura sobre a área regular ( ou projetada). Logo substituindo (L) em (K) tem-se: E-D1

wof = eff2

(M)

No caso representado na Figura 2 ou 3 onde Nk  NT (onde NT é o numero total de estruturas lineares até romper o corpo de prova) tem-se que: Ar  ArT, r  rT = wof e eff  logo, o lado esquerdo de (M) torna-se portanto: E-Dm

wof = m

(N)

onde: : é a energia de superfície média do trabalho total de fratura sobre a a área irregular T

T

(ou verdadeira) ; Ar = Tre: é a área total da superfície projetada de fratura (área regular); Aeff = Te: é a área total da superfície real de fratura (área irregular). Analogamente, haverá escalonamento entre as grandezas R e 2k ; Rmáx e 2máx. Fazendo-se todas as possíveis transformações de escala, entre as grandezas conhecidas da mecânica da fratura, que se

relaciona diretamente com a superfície de fratura, pode-se construir uma tabela de escalonamento de acordo com os gráficos das Figuras - 4 e 5, conforme mostram as Tabelas - I (7)

e II. Das relações apresentadas acima, algumas delas são também sugeridas por Rodrigues . Porém, ele cometeu um engano ao afirmar que suas relações sugerem um eff definido por todos os planos cristalográficos de clivagem de um simples cristal, ou pela fratura do estado vítreo correspondente. Porque uma vez que suas relações seguem de um eff definido por (H), onde aeff é uma área irregular, a afirmação acima é contraditória. Caso não seja feito esta distinção em um mínimo de três níveis de escalonamento diferentes, corre-se o risco de enganosamente tratar o caso de energia de superfície de curvas-R planas, como sendo um caso geral, onde Ro = R = Rmáx e eff = k = máx = , e a relação entre estas grandezas envolve apenas uma única dimensão fractal D. Tabela - I. Relações de escalonamento entre grandezas irregulares (rugosas) e regulares (projetadas). Grandezas Projetadas (-E) (liso ) Ro R = 2r Rmáx = 2 wof

Grandezas Irregulares (-D) (rugoso ) 2eff 2k 2máx 2 E-D1

E-Dk

1 2E-D1 E-D1 3 E-D1 m

E-Dmáx

1 2E-Dk E-Dk 3 E-Dk m

E-Dm

1 2E-Dmáx E-Dmáx 3 E-Dmáx m

1 2E-Dm E-Dm 3 E-Dm m

As grandezas associadas a primeira linha e a primeira coluna da Tabela - I, podem ser colocadas numa outra tabela (Tabela - II). Tabela - II. Relação entre as grandezas regulares (projetadas) e irregulares (rugosas) da mecânica da fratura Grandezas Taxa de energia elástica liberada Energia de superfície (irregular) Parâmetros da curva-R (projetada) Tenacidade a fratura

Diferentes níveis de escalonamentos das grandezas em questão GIco G Gmáx E-D1

2eff 1 Ro

2 Ico/E

K

E-Dk

2k 2

E-Dmáx

2máx3

Rmáx

R = 2r 2

K IR/E

2 Imáx/E

K

E-Dm

2 m

= 2 wof 2

/E

A regra básica para o escalonamento está resumida na Tabela -I. Conforme já foi visto anteriormente, para uma superfície regular, o escalonamento envolve um potência inteira E = 2 e para uma superfície irregular, o escalonamento envolve uma potência não inteira D (E

 D  E + 1). O escalonamento feito entre uma grandeza regular e uma irregular envolve a potência E - D, conforme a região em consideração. DISCUSSÕES A primeira questão que pode ser discutida neste trabalho, é aquela da escala mínima min de medida, dada supostamente pela teoria de Griffith. É necessário portanto, que seja feito medidas experimentais rigorosas, para a comprovação desta suposição. No entanto, (12)

observações realizadas por Rodrigues

apontam fortemente para este fato, o que também é

esperado por outros autores(10) . Como conseqüência, é possível que haja uma “quantização” da fratura dada pela escala de Griffith. Esta questão também tem sido levantada por Passoja(11), porém não se deve ser tão rigoroso neste ponto. O que a teoria fractal demonstra, antes de qualquer afirmação precipitada sobre uma “quantização na escala”, é que existe um limite físico superior máx dado pelas dimensões do corpo e também um limite inferior min para a fratura acontecer. Sendo este último, um fator determinante de uma régua inferior de medida c*, que pode ser usado na determinação do comprimento exato da trinca e que não necessariamente, implica numa “quantização” da fratura, como aquela que existe em outros ramos da Física. Dentro ainda da questão do escalonamento, uma vez que existe um limite inferior para o tamanho crítico, dado por: cK = c*, a partir do qual a propagação da trinca ocorre, resolve-se o problema da arbitrariedade na escolha da escala de medida, passando de K = cK/Tr, para um valor bem determinado dado por: min = c*/Tr. Desta forma, a determinação das propriedades físicas de um material, usando-se o escalonamento fractal, torna-se consistente com a teoria clássica da mecânica da fratura, além de sugerir que experimentalmente se busque confirmar a existência do limite mínimo de escalonamento sugerido acima. A explicação da variação da energia de fratura, ou do valor de R com a dimensão fractal D, e não com a escala, segue do raciocínio anterior. Uma vez que se considera um material homogêneo, tendo a princípio o mesmo tamanho crítico c* para diferentes regiões deste material, a dimensão fractal representa os diferentes regimes de propagação demonstrado ao longo de toda curva-R, quando esta é diferente de uma curva-R plana. Sendo este caso, melhor representado por uma relação multifractal(15) dada por: q

Dq = lim0 1/(q -1)ln pk /ln

(O)

onde: pk = Ak/AT, e q é um índice que generaliza todos as possíveis dimensões encontradas em um multifractal.

Por outro lado, em situações em que a dimensão fractal da fratura não varia com o aumento da energia de fratura, é preciso lembrar que: 1) Nos processos de transformação de fase induzida pela ponta da trinca, como o caso de compósitos de alumina-zircônia metaestável(12), a energia liberada na fratura, deve estar registrada na forma de microtrincas, deformação plástica, etc. que pode não ter sido computada na direção de propagação da trinca principal. 2) Sistemas nos quais o volume de uma zona de processo, ou a quebra de fibras é envolvido, tem-se duas alternativas: ou se considera uma escala inferior c* fixa, para a medida do comprimento da trinca em todo o processo, incluindo os diferentes mecanismos de liberação de energia, com diferentes dimensões fractais. Ou escolhe-se diferentes escalas inferiores r, com diferentes tamanhos c* para cada região da propagação, e fixa-se a dimensão fractal D, considerando-se a variação no valor de R como sendo um transiente na escala de fratura, para que o valor escalonado seja correto. Apesar desta última alternativa parecer de acordo com a questão levantada acima, ela não é recomendável, visto que a propriedade de homogeneidade do material dever ser preservada na quantificação de um único valor de c*. Esta grandeza portanto, a princípio, não apresenta nenhuma razão física para que em outra região do material possua valor diferente. O que deve ser feito portanto, é uma avaliação correta da energia de fratura registrada no material, incluindo a porção que se encontra fora da direção de propagação da trinca principal, e que por razões de dificuldade experimentais não é computada junto com esta. Ainda é bom lembrar que o resultado apresentado por Mott(13) para a fratura dinâmica, leva em conta que todo o saldo de energia acima do valor critico de Griffith, é transformado em energia cínética de propagação, que no final das contas aparece registrado sob a forma de superfície fraturada. Mesmo para o caso de trincas extremamente instáveis (14), acontece a busca de modos alternativos de dissipação da energia, que em última instância acaba sendo registrado no material sob a forma de fratura em microtrincas ramificadas, que muitas vezes não está incluído na computação energética que leva em consideração o comprimento da trinca principal. Necessita-se portanto da criação de métodos experimentais que possam medir toda á área verdadeiramente fraturada no interior de um material. CONCLUSÕES As relações de escalonamentos usadas neste trabalho, constituem-se em aproximações, cuja validade, está associada a trincas e superfícies de fratura, que apresentam uma estrutura homogênea de acordo com o teorema de Euler. No caso das trincas ou superfícies de fratura apresentarem, diferentes dimensões fractais para diferentes regiões, a homogeneidade é

portanto localizada. Assim conclui-se que, para o caso de materiais que apresentam curva-R não-plana, a superfície de fratura apresenta diferentes dimensões fractais, para cada região desta curva. Pode-se então dizer que ela é um multifractal, devendo ser caracterizada de uma forma geral pela relação multifractal(15) dada em (O). AGRADECIMENTOS O autor agradece ao Prof. Dr. José Anchieta Rodrigues pelas valiosas discussões sobre o assunto. REFERÊNCIAS BIBLIOGRÁFICAS 1. Mandelbrot, B. B., The Fractal Geometry of Nature, Freeman - New York 1982. 2. ASM - Handbook - Vol. 12, Fractography - The materials information society (1992) 3. Allen, Martin; Brown, Gareth J.; Miles, Nick J. Powder Technology 84 (1995) 1-14. 4. Mandelbrot, B. B.; Passoja, D. E.; Paullay, A. J., Nature (London), 308 [5961] 721-22 (1984). 5. Melcholsky, J. J.; Pasoja, D. E.; et al, J. Am. Ceram. Soc. 72 [1] 60-65 (1989). 6. Long, Q. Y.; Suqin, L.; Lung, C. W., J. Phys. D: Appl. Phys. 24 (1991) 602-607. 7. Rodrigues, J. A.; Pandolfelli, V. C. Materials Research, v. 1, n. 1, 47-52, 1998. o 8. Alves, L. M. “Uma teoria estastistica fractal para a curva-R”, In: Anais do 41 Congresso Brasileiro de Cerâmica, 1998. São Paulo-SP 9. Tsai, Y. L. and Mecholsky J. J. Journ. Mater Res., Vol. 6, No. 6, Jun 1991. 10. TANAKA, M. Journal of Materials Science, 31 (1996) 749-755 11. Passoja, D. E. Advances in Ceramics, Vol. 22: Fractography of Glasses and Ceramics, 1988,101-126. 12. Rodrigues J. A. comunicação pessoal. 13. Mott, N. F., Engineering 165 (1947) 16-18. 14. Fineberg, Jay, Gross, Steven P. et al, Phys. Rev. B, v. 45, n. 10, 1 March 1992, 51465154. 15. Xie, Heping et al, Direct fractal measurements and ..., Phys. Lett. A 242 (1998) 41-50.

THE FRACTURE GEOMETRY ABSTRACT The irregular structures that are not show a euclidean geometry, such as: cracks and fracture surfaces, can be described by fractal geometry concepts applied to fracture mechanics. The purpose of this work is to show an accurate language to describe the fracture mechanics into the new view of the fractal theory. In order to applied this language straightforward to fracture phenomenology to understand the process of cracks and microcracks formation that show branched patterns or not. Which this patterns are found from the quasistatic fracture until fragmentation processs. Key words: fracture work, surface energy, curve-R, brittle materials, fractal dimension.

Chapter IV

ANALYTICAL FRACTAL MODEL FOR RUGGED FRACTURE SURFACE

Engineering Fracture Mechanics 162 (2016) 232–255

Contents lists available at ScienceDirect

Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech

Analytical fractal model for rugged fracture surface of brittle materials Lucas M. Alves a,⇑, A.L. Chinelatto b, Edson Cezar Grzebielucka b, Eduardo Prestes b, L.A. de Lacerda c a GTEME – Grupo de Termodinâmica, Mecânica e Eletrônica dos Materiais, Departamento de Engenharia de Materiais, Setor de Ciências Agrárias e de Tecnologia, Universidade Estadual de Ponta Grossa, Av. Gal. Carlos Calvalcanti, 4748, Campus UEPG/Bloco CIPP – Uvaranas, Ponta Grossa, PR CEP. 84030.900, Brazil b Departamento de Engenharia de Materiais, Setor de Ciências Agrárias e de Tecnologia, Universidade Estadual de Ponta Grossa, Av. Gal. Carlos Calvalcanti, 4748, Campus UEPG/Bloco CIPP – Uvaranas, Ponta Grossa, PR CEP. 84030.900, Brazil c LACTEC – Instituto de Tecnologia para o Desenvolvimento, Universidade Federal do Paraná - PPGMNE, Centro Politécnico da Universidade Federal do Paraná, Cx. P. 19067, Curitiba, PR, Brazil

a r t i c l e

i n f o

Article history: Received 28 August 2015 Received in revised form 15 May 2016 Accepted 17 May 2016 Available online 26 May 2016 Keywords: Fractal dimension Fracture profile Mortar Heavy clay Roughness Self-affine surface

a b s t r a c t The fractal modeling of a rugged fracture surface has received different purposes. However none definitive model for the most of materials has been reached. Therefore, a general selfaffine fractal model is proposed for fracture surfaces and applied to heavy clay and mortar. An analytical expression for the rugged crack length is obtained for application on fractal fracture mechanics. Stereoscopic images are obtained for each tested specimens. Image processing filters are used to extract the rugged profile of the cracks. The box-counting and sand-box methods are used on the crack profile to obtain the local and the global roughness exponents. Specimens prepared under different conditions validated the model. Mortars and heavy clay specimens were characterized by measuring their modulus of rupture and the rugged crack profile under 3-point bending tests. A good agreement between the model and the experimental results was observed. A strong correlation between the fractal dimension and the sintering temperature for heavy clay specimens was verified. The results also showed that the increasing rugged crack length of the profile of the fractured mortar specimens is well correlated with the increase in water/cement ratio. These results validate the application of the proposed model for estimating the fracture strength of brittle materials. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The fracture surface is a record of the information left by the fracture process. Generally, the rugged fracture surface profile has fractal geometry, so it is possible to establish a relationship between its topology and physical quantities of the fracture mechanics using fractal characterization techniques. However, the classical fracture mechanics (CFM) was developed idealizing a flat, smooth, and regular fracture surface, as the geometry of crack surfaces is usually rugged and cannot be easily described by the Euclidean geometry [1]. In this sense, the mathematical basis of CFM considers an energetic equivalent between the rugged and the projected fracture surfaces [2]. Besides the mathematical complexity, part of this

⇑ Corresponding author. E-mail addresses: [email protected] (L.M. Alves), [email protected] (A.L. Chinelatto), [email protected] (L.A. de Lacerda). http://dx.doi.org/10.1016/j.engfracmech.2016.05.015 0013-7944/Ó 2016 Elsevier Ltd. All rights reserved.

L.M. Alves et al. / Engineering Fracture Mechanics 162 (2016) 232–255

233

Nomenclature

Latin characters 0 index to denote measurements taken on the projected plane a minimal area of fracture surface DA area of fracture surface A1; A2; A3; Ar1; . . . ; Ar30 mortar and ceramic samples B1; B2; B3; Br1; . . . ; B30 mortar and ceramic samples C1; C2; C3; Cr1; . . . ; Cr30 mortar and ceramic samples d infinitesimal increment D1; D2; D3; Dr1; . . . ; Dr30 mortar and ceramic samples DB fractal box dimension Df fractal dimension divider dimension DD DBC fractal dimension measured by box-counting method e0 fractal cell size or minimal crack length in transversal direction E elastic or Young’ s modulus crack length measured in transversal direction E0 DE0 variation of crack length measured in transversal direction f general functions g general functions or subscript index for global quantity G elastic energy release rate h0 fractal cell size or minimal crack length in vertical direction H Hurst’s exponent H0 plane projected crack height DH0 variation in the plane projected crack height K multiplicative factor Kx multiplicative factor in x-direction multiplicative factor in y-direction Ky l subscript index for local quantity L rugged crack length LBC rugged crack length measured by box-counting method lx ; ly ; lz rulers length or scale in the x-direction, y-direction, z-direction, respectively l0 fractal cell size or minimal crack length in propagation direction L0 plane projected crack length measured in propagation direction L0S saturation plane projected crack length measured in propagation direction Griffith’s critical crack length L0C DL rugged crack length DL0 distance between two points of the crack (the projected length of the crack) DL0C critical crack length NL number of units of the crack length in longitudinal or propagation direction NT number of units of the crack length in transversal direction NV number of units of the crack length in vertical or height direction number of units of the crack length in the growth direction Nx Ny number of units of the crack length in the perpendicular direction sat subscript index indicate saturation Y0 shape function x; y; z spatial coordinates Dx; Dy; Dz, incremental measure length in x-direction, y-direction, z-direction Dzx ; Dzy fluctuation or roughness mean squares or incremental measure of crack height in x-direction, y-direction, respectively w width of specimen test Greek letters exponent b d incremental measure @ partial derivative eL ; eT horizontal and vertical scale of the fractal scaling ce ; cp elastic and plastic specific energy surface, respectively kx amplification factor scale in x-direction

234

ky n fg ; fl fx ; fy

rf t v


amplification factor scale in y-direction, local roughness global and local roughness exponent, respectively roughness exponent in x-direction, y-direction, respectively fracture stress dynamic roughness exponent Poisson’s ratio

background is associated with the difficulties of making an accurate measurement of the real fracture area. Although there are several methods capable of quantifying the fracture area, the results are dependent on the size of the ruler of measurement [3], which did not contribute for its insertion in the CFM in the past. With the increasing interest in fractal theory, it became possible to describe and quantify almost any apparently irregular structure in nature [4]. In fact, many theories based on Euclidean geometry are being reviewed. It has been experimentally proved that fracture surfaces have fractal scaling, so the fracture mechanics is one of the scientific areas included in this context. The work herein deals with the mathematical description of the roughness of cracks in the fracture mechanics, using the fractal geometry to model its irregular profiles. Considering that the fracture surface is a record of the information left by the fracture process, it may be possible to establish a relationship between the topology of this surface and the physical quantities of the fracture mechanics using fractal characterization techniques. Mandelbrot et al. [5] developed the ‘‘Island Slit” method by searching for correlations between fractal dimensions and well-known physical quantities of fracture mechanics. Following this pioneering work, other authors also made theoretical and geometrical considerations with a similar purpose [6–14], as described briefly below. 1.1. Brief review of fractal scaling models of fracture surfaces Mosolov [15] and Borodich [14] were the pioneers to associate the strain with the surface energies involved in the fracture process with the roughness exponents of the surfaces produced during the breaking process and the splitting of the crack surfaces. They related these quantities using the singularity exponent of the stress field at the crack tip and the fractionary dependence of the fractal scaling exponent in fracture surfaces, postulating the equivalence between the changes in the strain and the surface energies. Bouchaud and Bouchaud [16] proposed an alternative model from Mosolov [15] who showed fracture parameters in terms of the height fluctuations of the rugged fracture surfaces, whose fluctuations are perpendicular to the crack propagation direction. Bouchad proved the relation between the fracture critical parameters as fracture toughness K IC and the relative fluctuations in the height of the rugged surface. The universality of the roughness exponent for fracture surfaces had been assumed, as it did not depend on the type of material tested [17]. However, this assumption generated a lot of controversy [18], which led to the discovery of anomalies [19] in the scaling exponents for the local and the global scales in the fracture surfaces of brittle materials. Family and Vicsek [20] and Barabási and Stanley [21] showed a fractal scaling model for rugged surfaces formed in ballistic deposition films. This fractal scaling model was an inspiration for subsequent models of fractal surfaces present in other phenomenologies. Since then, the modeling of fracture surfaces was done based on the models of Family and Vicsek [22], receiving different supplements [19,23,24]. Based on this work, López and Schmittbuhl [25] proposed an analogous model for the fracture surfaces, which accounted for anomalies in the fractal scaling, with critical crossover dimensions for the transition in the roughness behavior of these surfaces in brittle materials. Morel et al. [26] observed experimental anomalies in the roughness of wood fracture surfaces, and using the fractal model of López et al. [19], they determined the roughness and the dynamic exponent values. They also showed that the anomalies could only be explained by a dynamic scaling fractal model involving local and global roughness exponents [25]. From the initial ideas of López and Schmittbuhl [25] and following their fundamental hypothesis, Morel et al. [27] proposed a G–R curve model for fracture surfaces in wood. Morel applied this model to describe the relief presented by the rugged fracture surfaces whose topology is characterized by local and global roughness exponents, relating them with the behavior of the elastic energy release rate and fracture resistance, called ‘‘G–R curve” [28] for a brittle material. He generalized the López and Schmittbuhl model including situations that have size effect [27]. This present work presents a generalization of the previous models, elaborating a fractal scaling in the longitudinal direction similar to that in the transversal direction for the crack propagation direction in a coherent way. The fractal surfaces always exceeds the Euclidean dimensions of its projection and does not fulfill the immediately superior Euclidean dimensions in which it is immersed. For instance, a fracture profile has a fractal dimension in the interval 1 6 D 6 2, so it exceeds the dimensions of a smooth line, but it does not fulfill those of a smooth surface. Morel et al. [24] shows that the fractal nature of a fluctuation Dz in a direction must be scaled using lengths and parameters of the same direction, although there can be other factors related to the perpendicular direction that can be coupled to the model. In this sense, it is adopted the fol lowing relationship for the roughness mean square Dzx ðlx ; xÞ and Dzy ly ; y in the transversal and longitudinal directions, respectively:


(

f

Dxfg fl lxl Dxfg 8 f f g l f < Dy t lyl Dz y l y ; y ¼ : ftg Dy Dz x ð l x ; xÞ ¼

235

if lx Dx if lx Dx if ly Dy

:

ð1Þ

if ly Dy

where x; y are the coordinates, lx ; ly are the used ruler lengths, Dx; Dy is the interval of measure of roughness, Dzx ; Dzy is the height variation in the transversal and longitudinal directions on the fracture surface respectively, and fg ; fl ; t are the global, local and dynamic roughness exponents, respectively. It is also shown that from a general and analytical definition of a rugged surface, whose height z ¼ f ðx; yÞ is a fluctuation function of the coordinates of the mean plane of the projected surface, it is possible to establish a general expression that involves the local and the global self-affine fractal scaling for the rugged fracture surface. Many researchers have developed fractal models for the fracture surface, by employing the roughness dimension and introducing it to the CFM [16,29–32]. In particular, the work of Alves [32] introduced a self-affine fractal model of a fracture surface in the mathematical formalism of the CFM through a local roughness derivative term. However, it lacked a more detailed explanation about the origin of that model. It is shown in the present paper the basic mathematical premises of this self-affine fractal model derived from Voss [33], who presented a fractal description for the noise in Brownian motions. To calculate the rugged dimensions and define the model, the ‘‘box-counting” method is used [34]. A general expression for the rugged crack length as a function of the projected length and fractal dimensions is shown. Also, a roughness expression is derived, which can be directly introduced into classical fracture mechanics. Despite the generality of the model presented in the work herein, applications were focused on brittle material and experimental tests were carried out for its validation. Mortar and heavy clay specimens were prepared under different conditions and characterized by measuring their modulus of rupture and the rugged crack profile after a 3-point bending test. Obtained results showed that the fractal dimension was capable of representing the roughness of the fracture surface of the different materials. A strong correlation between the Lipfshit–Hölder exponent for the longitudinal and the vertical directions, and the sintering temperature were verified on the heavy clay material. The results also showed that the increasing rugged crack length of the profile of the fractured mortar specimens is well correlated with the increasing water/cement ratio. Overall, the application of the proposed self-affine fractal model was validated for estimating the fracture strength of brittle materials. 2. Analytical model for rough surfaces Consider a rough surface z ¼ f ðx; yÞ, as shown in Fig. 1. The area DA of the surface can be parameterized and calculated from Greenberg [35],

Z

DA ¼

DE0 þdE0

Z

DE0

DL0 þdL0

D L0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 @f @f þ 1þ dxdy: @x @y

ð2Þ

If this rough surface is discretized with a mesh lx by ly elements with l0 6 lx 6 dL0 and e0 6 ly 6 dE0 , a box-counting method can be used to define a relationship between the surface height fluctuations and the area within the limits dL0 and dE0 . Thus, for a non-differentiable rough surface the rough area can be approximately calculated from the following:

Z

DA ffi since

@f @x

DE0 þdE0

Z

DE0

ffi DDzxx and

DL0 þdL0

D L0 @f @y

ffi

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dzx ðx; yÞ 2 Dzy ðx; yÞ 2 þ 1þ dxdy lx ly

ð3Þ

D zy . Dy

The area da of each projected element lx by ly is given by the following:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ffi Dz x Dzy 2 da lx ly 1 þ þ ¼ lx ly n lx ly

ð4Þ

It is observed that the term n ¼ da=lx ly represents the local roughness of the rugged surface and the whole area DA is given by, Ny X Nx X DA daij ¼ dL0 dE0 j¼1 i¼1

"

y X Nx 1 X n Nx Ny j¼1 i¼1 ij

N

#

ð5Þ

with N x ¼ dLlx0 and N y ¼ dEly0 . Analogously, the length of a rough profile can be calculated with

DL

" # Nx Nx X 1 X dli ¼ dL0 ni Nx i¼1 i¼1

which is a function of the projected length dL0 in the Euclidean space.

ð6Þ

236


Fig. 1. (a) Generic rough surface z = f(x, y); (b) close view of the rough surface limited by dL0 and dE0 .

3. The self-affine fractal model of a rugged fracture surface Fracture surfaces are considered fractal objects with statistical invariance by scale transformation (self-similarity or selfaffinity). A fractal is an object whose measure of geometrical extension depends on the size of the measurement ruler and has a so-called Hausdorff–Besicovitch dimension [21,36–39]. Consider now a generic rugged fracture surface (Fig. 1) with self-affine properties [4,17]. As its morphology can be characterized by its fractal nature [5,8,14,40], a scale transformation along the orthogonal directions x and y with scale factors kx and ky , respectively, results in the following mathematical relationship: f

zðkx x; ky yÞ ¼ kfxx kyy zðx; yÞ

ð7Þ

where fx is the roughness exponent of the surface in direction x and fy is the rough exponent of the surface in direction y. The self-affinity property allows formulation of Eq. (7) in an uncoupled form and in each perpendicular direction, one has the following:

zx ðkx x; yÞ ¼ kfxx zðx; yÞ

ð8Þ

f

zy ðx; ky yÞ ¼ kyy zðx; yÞ 3.1. Computing the area of the rugged fracture surface

The height fluctuations Dzðx; yÞ of the fracture surface can be correlated with their fractal behavior in a window of observation in directions x and y in the following manner,

Dzx ðkx x; yÞ ¼ kfxx Dzðx; yÞ f

Dzy ðx; ky yÞ ¼ kyy Dzðx; yÞ

ð9Þ

To calculate the area effectively, it is useful to uncouple the mutual dependence of the transversal and the longitudinal fractal properties of the height fluctuations. The following general scaling relationships for the height fluctuations are proposed, dividing Eq. (9) by the unit scale sizes lx and ly , respectively, one obtains Dzx ðlx ;xÞ lx

Dz

Dzy ðly ;yÞ ly

f

kxx lx

Dz

f

kyy ly

¼ Dlxz

fx

¼ Dlyz

Dx lx

fy Dy ly

ð10Þ


237

Fig. 2. The changing fractal dimension of a fracture profile of a brittle material.

3.2. Local and global roughness dimensions Measurements of the fractal dimension Df of a fracture surface profile in a brittle material with the sand-box method [41] show an asymptotic behavior with the growing crack length L0 . Fig. 2 shows this behavior, where the fractal dimension decreases from an initial local value Df > 1 with L0 L0S and L0S being the saturation length, to a global value Df 1 for L0 L0S . In this sense, the rough length L of the profile is not dependent on the measuring scale lx when L0 L0S , being closer in size to the projected length L L0 [25]. To cope with this behavior, Eqs. (10) can be adapted according to these intervals, and the roughness fluctuations in the transversal and the longitudinal directions [24,26,27] are, respectively, given by

8 fx > < Dlxx Dzx ðlx ; xÞ ffi K y f > : Dx gx

if x xsat

8 fy > < Dlyy Dz y l y ; y ffi K x f > : Dy gy

if y ysat

lx

ly

ð11Þ

if x xsat

if y ysat

:

ð12Þ

where K x ; K y are functions that relates the interaction of the roughness of one direction to another direction. Note the cross terms in this equation where the roughness in one direction affects the roughness in the other direction. Although the model separates contributions from every direction, the fracture occurs at the same time in both directions. Thus the fibers or particles in the microstructure to break up, suffer local strains that have normal and tangential components affecting perpendicular directions. fgx and fgy are the global rough fractal exponents in directions x and y, respectively. 3.3. Relationship between transversal and longitudinal roughness exponents As the self-affine crack presents rough exponents with local and global aspects, the following expressions can be used to account for the roughness transition from local to global scales in x and y directions [24,26,27].

fx ¼ fgx flx

ð13Þ

fy ¼ fgy fly

ð14Þ

Considering, for example, that the crack propagates in y direction, the rough exponent fy differs from fx by a dynamic factor 1=t [22],

238


fy ¼ fx =t ¼ b

ð15Þ

where b characterizes crack propagation in y direction along time. Thus, Eqs. (11) and (12) become,

8 ðfgx flx Þ > > < Dl x x Dzx ðlx ; xÞ ffi K y f > Dx gx > :

if x xsat

8 ðfgy fly Þ > > < Dl y y Dzy ly ; y ffi K x f > Dy gy > :

if y ysat

ð16Þ

if x xsat

lx

ð17Þ

if y ysat

ly

3.4. Defining the quantities K x and K y The dependency of quantities K x and K y must be analyzed. In isotropic materials, the functions that define the self-affine fractal scaling have the same nature in the transversal and the longitudinal directions. Therefore, the height fluctuations are defined by isomorphic1 functions in both directions, and

@f @f $ @x @y

ð18Þ

which results in, Ky lx Ky lx

ðfgx flx Þ Dx lx

fgx Dx lx

$

Kx ly

$ Klyx

ðfgx tflx Þ Dy ly

fgxt Dy ly

if DA0 DA0sat

ð19Þ

if DA0 DA0sat

Assuming the scale relationship used by Morel et al. [24,26–28],

1t Dx Dy ¼ lx ly

ð20Þ

Based on Eq. (19), it is possible to express K x and K y as functions of the same scaling factors and roughness exponents such as,

Kx ¼ K Ky ¼ K

ðfgx flx Þ Dx lx

ð21Þ

ðfgx f lx Þ t Dy ly

where K is a scaling ratio defined in the z direction. Substituting Eq. (21) in Eqs. (16) and (17), one has,

Dz x ð l x ; xÞ ffi K

Dy ly

8 > Dx ðfgx flx Þ lx Þ > ðfgx f < t lx

if x xsat

lx

if x xsat

fgx > > : Dx

ð22Þ

and

8 ðfgx flx Þ t > ðfgx flx Þ > < Dl y Dx y Dz y l y ; y ffi K fgy > lx > : Dy t ly

if y ysat if y ysat

3.5. Computing the K factor Consider a section of a rough fracture surface as illustrated in Fig. 3. 1

$ this symbol is used to denote isomorphic functions.

ð23Þ


239

Fig. 3. Scheme of a window lx ; ly on a rough fracture surface showing the minimal crack lengths e0 ; l0 for the transversal and the longitudinal directions.

Height fluctuations also depend on the scaling of vertical direction, suggesting that K depends on scaling between directions z and x and directions z and y of the crack. In this way, to calculate K ¼ f ðh0 ; DH0 Þ as a power law function of parameters in directions x; y, and z, one must analyze the crack profile on the planes xz or yz, as illustrated in Fig. 4. Considering square units with sizes l0 h0 , covering the crack length DL0 within the area DL0 DH0 , and e0 h0 , covering the crack thickness DE0 within the area DE0 DH0 , the number of necessary units to cover the self-affine profile in each direction is obtained from, fly

ðlongitudinalÞ

ð24Þ

flx

ðtransv ersalÞ

ð25Þ

NL ¼ N V eL

NT ¼ NV eT

where N V is the number of minimal boxes on the z-perpendicular direction to the plane of the crack. flx ; fly is the local roughness exponent, related to the called divider dimension as fl ¼ D1D [42], DE0 is the transversal projection of the crack, DH0 is its vertical projection,

eT and eL are the transversal and the longitudinal scale factors, and e0 6 lx 6 DE0 and l0 6 ly 6 DL0 .

Fig. 4. Self-affine fractal of Weierstrass–Mandelbrot, where eek ¼ 1=4, Dx = 1.5, and H ¼ 0:5, is used to represent a fracture profile. Source: Modified from FAMILY and VICSEK [20, p. 7].

240


Taking a cut plane of the fracture surface in the longitudinal direction the profile of Fig. 4 is obtained. In the profile shown in Fig. 4, the basic measuring units l0 , and h0 are different in the longitudinal y, and the vertical z directions, respectively. If h0 is the minimum scaling factor in vertical direction, one has,

NV ¼

DH 0 ðv erticalÞ h0

Choosing

ð26Þ

eL ¼ eT and NL ¼ NT for counting the number of cells in the profile at the xz and the yz planes, one has

DH 0 DH 0 f f

eL ly ¼

eT lx h0 h0

ð27Þ

fly flx DH 0 l0 DH 0 e0

¼ h0 DL 0 h0 DE 0

ð28Þ

or

Thus,

l0 DL0

e0 ¼ DE 0

fflx

ly

ð29Þ

For each xz and yz plane across the crack surface, there is a projected crack profile. The number of cells covered by the fractal scaling in each of these planes in the horizontal and the vertical directions is given by,

Nxz L ffi Nxz V ffi and

Nyz T ffi Nyz V ffi

fy 9 > =DL fy DH 0 0 ffi

> h0 DH 0 ; l0

D L0 l0

ð30Þ

h0

fx 9 > =DE fx DH 0 0 ; ffi

> e0 h0 DH 0 ;

D E0 e0

ð31Þ

h0

respectively. From Eqs. (30) and (31) and

t ¼ fx =fy , one has,

9 DL0 =DL 1=t DE ffi DhH00 > l0 0 0 ffi fx > l e0 0 DE0 DH 0 ; ffi h0 e0

fy

ð32Þ

which is equivalent to Eq. (20). Therefore, from Eq. (31) for K h0 , one has

K ¼ DH 0

e0 DE0

flx

ð33Þ

Substituting Eq. (33) in Eqs. (22) and (23), one can now drop the x subscript of all equations. Therefore the height fluctuations are as follows:

8 fl ðfg fl Þ ðfg tfl Þ > > @f Dy Dx > < @x ffi Dlzx x ¼ DlHx 0 DeE00 lx ly fl fg > > > Dy Dx : @f ffi Dzx ¼ DH0 e0 lx lx lx D E0 @x ly

ðfg fl Þ t

if x xsat

ð34Þ

if x xsat

and

8 fl ðfg fl Þ ðfg tfl Þ > Dz @f Dy > Dx < @y ffi lyy ¼ DlHy 0 DeE00 ly lx

if y ysat

fg > > : @f ffi Dzy ¼ DH0 e0 fl Dx ðfg fl Þ Dy t lx ly ly DE0 @y ly

if y ysat

ð35Þ

Substituting Eqs. (34) and (35) in Eq. (3) for lx ¼ e0 and ly ¼ l0 , the area DA is given by:

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> 2 2fl 2ðfg fl Þ 2ðfgtfl Þ 2 > > R DE0 þdE0 R DL0 þdL0 DH 0 DH0 e0 Dy Dx > þ 1 þ dxdy if DA0 DA0sat > DL0 e0 l0 e0 DE0 l0 < D E0 DA ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> 2 2fl 2ðfg fl Þ 2ftg > 2 R R > e0 Dy > Dx > DDEE0 þdE0 DDLL0 þdL0 1 þ DlH0 þ DeH0 dxdy if DA0 DA0sat : e0 DE0 l0 0 0 0 0

ð36Þ


241

This equation is in agreement with the model presented by Morel et al. [24] and can be seen as a generalization. Alternatively, Eq. (36) for the rugged area can be rewritten like,

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> 2 2fl 2ðfg fl Þ 2ðfgtfl Þ 2 > > DH 0 DH 0 e0 D E0 D L0 > D A þ 1 þ if DA0 DA0sat > l0 e0 DE0 e0 l0 < 0 DA ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

>

2fg > > > DA 1 þ DH0 2 þ DH0 2 e0 2fl DE0 2ðfg fl Þ DL0 t > if DA0 DA0sat : 0 l0 e0 DE0 e0 l0

ð37Þ

3.6. Crack profile model For the analysis of a crack profile DE0 ffi e0 , Eq. (37) becomes:

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> 2 2ðfgtfl Þ 2 > > DH 0 DH 0 DL0 > D L þ 1 þ if DL0 DL0s > < 0 l0 e0 l0 DL ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> 2fg > > > DL 1 þ DH0 2 þ DH0 2 DL0 t > if DL0 DL0s : 0 l0 e0 l0

ð38Þ

Observe that the lengths DL0 and DH0 are not necessarily equal and correspond to the projected lengths of the crack, in the horizontal and the vertical directions, respectively. To describe a growing crack, one can define a fixed origin coincident with the crack onset at the left (or right) side of the counting box. The size of the counting box can be adjusted to cover the growing crack. This is the basis of the sand-box method [41] for crack length measurement. Mathematically, this is defined with DL ¼ L; DL0 ¼ L0 and the rugged length is given by,

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> 2 2ðfgtfl Þ 2 > > DH0 L0 > þ DeH00 if L0 L0s > < L0 1 þ l0 l0 L¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> 2 2ftg > 2 > L0 > > L0 1 þ DlH0 þ DeH0 if L0 L0s : l0 0 0

ð39Þ

Applying the logarithm on the both sides of Eq. (39) one obtains an expression that relates the fractal dimension to the projected crack length:

Df

8 > > > > > > > > > > > > > :

(

ln

2

( ln

1 þ 12

1þ

1þ

DH0 l0

2 þ

DH 0 e0

ln L0

DH0 l0

2 þ

DH 0 e0

ln L0

2 2ðfgtfl Þ

)

L0 l0

)

2 2ftg

if L0 L0s

ð40Þ

L0 l0

if L0 L0s

It is important to point out that DL0 denotes the projected distance between two points of the crack (the projected crack length) and l0 is the minimum possible length of a microcrack, which defines the scale l0 =L0 under which the crack profile is scrutinized. Fig. 5 shows the influence of the height DH0 of the box on the rugged crack length measure, as a function of the projected length L0 , for L0 L0S and characteristic values for the parameters fl ¼ 0:8; fg ¼ 1:2; t ¼ 4:0, and l0 ¼ 0:1. Considering a fixed roughness exponent fl ¼ 0:8, a linear relationship between L and L0 is seen in lower boxes ðDH0 ! 0Þ, while in higher boxes ðDH0 ! 1Þ a nonlinear one is observed. Fig. 6 shows the influence of the roughness dimension of the rugged crack length, as a function of the projected length L0 . Observe that for fl ! 1, which corresponds to a flatter profile, the relationship between the rugged and the projected lengths is more linear. Whereas for fl ! 0, corresponding to a more rugged profile, the relationship between the rugged and the projected lengths is increasingly nonlinear. In near-flat cracks, the box height can be chosen as DH0 ¼ l0 ¼ e0 , and Eq. (39) simplifies to,

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2ðfgtfl Þ > > L > > if L0 L0s < L0 1 þ 2 l00 L¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2ftg > > > if L0 L0s : L0 1 þ 2 Ll00

ð41Þ

242


Fig. 5. Rugged length, L as a function of the projected length, L0 and the varying the box height, DH0 , width l0 ¼ 0:1, representing the box height (rectangular box DL0 – DH0 Þ with local roughness dimension, fl ¼ 0:8, global roughness dimension fg ¼ 1:2, and dynamic exponent, t ¼ 4:0.

Fig. 6. Rugged length L as a function of the projected length L0 and the varying the local roughness dimension, fl representing the box height for flat cracks with rectangular boxes DH0 ¼ 1:0; l0 ¼ 0:1, global roughness dimension, fg ¼ 1:2, and dynamic exponent, t ¼ 4:0.

In analogous way to the case of Eq. (39), i.e., applying the logaritm on the both sides of Eq. (41) one obtains an expression that relates the fractal dimension to the projected crack length,

Df

8 > > > > > > > > > > > > > > > :1 þ 1 2

" ln

2ðfgtfl Þ

1þ2

ln L0

" ln

t

2fg

1þ2

#

L0 l0

#

if L0 L0s

ð42Þ

L0 l0

ln L0

if L0 L0s

In the study of a self-affine fractal, two limits can be verified. The first is called ‘‘local limit,” when the height of the boxes is increased compared to its length ðDH0 ! L0 Þ. The second is the ‘‘global limit,” when the height of the boxes is small compared to its length ðDH0 ! h0 Þ. These limits are presented as follows.

243


3.7. Case 1: The initial apparent self-similar limit of the local fractality The local self-similar limit of the crack fractal measure can be obtained from Eq. (39), considering DH0 ¼ L0 l0 ,

L ffi L0

1ðfg tfl Þ l0 L0

ð43Þ

It is a self-similar relationship between the projected crack length Lo and the height ho of the unit box. This expression shows that all self-affine crack fractals can represent a local self-similarity when the square areas L0 L0 are considered instead of the rectangular ones L0 DH0 . This is commonly observed in the onset of cracks [12,13,43], where one can extract the following result, ðfg fl Þ

L ðfg fl Þ

L0

ffi l0

t

1

¼ cte

ð44Þ

t

which is a constant relationship between the micro- and macroscales. This equation is analogous to the self-similar mathematical relationship described by different authors [7–9,14,44]. 3.8. Case 2: The global self-affine limit of the fractality On the other side, the global self-affine limit of the crack fractal measure can be obtained from Eq. (39), considering DH0 ¼ l0 L0 ,

L ffi L0

ð45Þ

Ductile materials presenting a higher fractality possess a crack profile that can be better adjusted by Eq. (43), while brittle materials presenting a lower fractality are better adjusted by Eq. (45), which is closer to the classical fracture mechanics model with a flat fracture profile. 3.9. The roughness of a fracture profile The classical definition of roughness in the fracture mechanics is given by n ¼ L=L0 [2,3]. Alternatively, a local roughness is defined in this work considering a limiting approach for the counting box dimensions,

n lim

DL0 !l0

LðL0 þ DL0 Þ LðL0 Þ DL0

ð46Þ

In practice, Eq. (46) can be seen as the derivative of the rugged length L ¼ f ðL0 Þ with respect to its projected length Lo , written as

n

dL dL0

ð47Þ

This definition provides a full characterization of fractal profiles and can be useful to describe the crack growth phenomena. Eq. (47) can be used to calculate the local roughness of the profile, if an expression that relates the rugged length L to the projected length L0 is known. For instance, for the monofractal case with a square counting box, the measure of the rugged crack length is given by Eq. (39). Therefore, the expression for the local roughness is given by,

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ðfgtfl Þ u DH 0 L0 2 2ðfgtfl Þ u fg fl l0 l0 DH 0 L0 t þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n¼ 1þ P1 l0 l0 t 2 2ðfgtfl Þ DH 0 L0 1 þ l0 l0

ð48Þ

or simply

1þ n¼

ðfg fl Þ

2ðfg fl Þ 2 t DH 0 L0 þ 1 t l0 l0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P1 2 2ðfgtfl Þ DH0 L0 1 þ l0 l0

ð49Þ

4. Materials and methodology The mortar and heavy clay brittle specimens were prepared under different conditions and molded and tested in laboratory to obtain the rugged fracture surfaces. The specimens were characterized by measuring their modulus of rupture under 3-point bending tests.

244


4.1. Mortar specimens To prepare the mortar specimens, Portland cement ASTM type V (high initial resistance) and natural sand with controlled grain sizes [45] were used. The Portland cement ASTM was used to compare the results obtained in the work herein with the values obtained by other authors in the technical literature [31]. Four weight proportions of cement, sand, and water ðc=s=wÞ were applied, as shown in Table 1. It can be seen that from types A to D that water/cement and sand/cement ratios increase. Twelve specimens in total with standard sizes ð40 40 160 mmÞ were prepared according to ASTM C 348 [46] and mixed according to ASTM C 305 [47]. For the specimens of Table 1, the relative content of cement was decreasing from A to D, while the proportion of water was increased. That was made to compensate the increase in surface area of sand excess in order to get an effect in porosity to obtain a more rugged surface in the fracture testing. After seven days of curing, 3-point bending tests were carried out using an EMIC DL10000 to evaluate their modulus of rupture. Those results are shown in Table 2. 4.2. Heavy clay specimens A homogenized dry clay powder with 5% added water was used to prepare ninety heavy clay specimens in total of dimensions 60:2 20:3 mm and thickness of 4 mm, approximately. This thickness was a result of a conforming process of nearly 8.5 g of clay under compressive forces applied in two stages – an initial load of 20 kN, followed by a period of rest and a final load of 30 kN. After that, the specimens were dried along one day and sinterized for two hours in an electrical furnace under temperatures of 800, 900 and 1000 °C (see Table 3). After sintering, the specimens were weighed, and their dimensions were measured, and taken for the 3-point bending tests. Their average modulus of rupture is shown in Table 4.

Table 1 Weight proportions of cement, sand and water used for molding the mortar specimens. Mortar type

Weight proportion c/s/w

Number of specimens

A1, A2, A3 B1, B2, B3 C1, C2, C3 D1, D2, D3

1:1:0.30 1:2:0.45 1:3:0.60 1:4:0.75

3 3 3 3

Table 2 Average modulus of rupture of mortar specimens obtained after seven days curing. Modulus of rupture (MPa) Mortar type

Number of specimens

Average stress

Error

A1, A2, A3 B1, B2, B3 C1, C2, C3 D1, D2, D3

3 3 3 3

9.6 7.8 6.3 4.6

±0.7 ±0.9 ±0.2 ±0.5

Table 3 Heavy clay specimens prepared with different sintering temperatures. Specimen type

Sintering temperature (°C)

Number of specimens

Ar1; Ar2; . . . Ar30 Br1; Br2; . . . Br30 Cr1; Cr2; . . . Cr30

800 900 1000

30 30 30

Table 4 Average modulus of rupture of heavy clay specimen types. Modulus of rupture (MPa) Specimen type

Number of specimens

Average stress

Error

Ar1; Ar2; . . . Ar30 Br1; Br2; . . . Br30 Cr1; Cr2; . . . Cr30

30 30 30

2.0 4.2 9.0

±0.2 ±1.3 ±1.0


245

4.3. Crack profiles After the rupture tests, the fractured surfaces of the specimens were selected for digital imaging with an optical stereomicroscope (Leica MZ6) connected to a digital CCD camera (KODO KC-512NT). For each fractured side of each specimen, a complete fracture surface image was formed by juxtaposing the images. Through software analysis (MOCHA Image Analysis v1.2.10) the crack boundary profiles were obtained. Figs. 7–10 show the crack profiles from specimens mortar D2 side 1, mortar D2 side 2, heavy clay Ar8, and heavy clay Br3, respectively. The fracture surfaces were digitalized and characterized with the box-counting and sand-box methods to obtain the local and the global roughness exponents. It can be seen that all profiles can be well covered by a rectangular box for the application of the box-counting method. The bottom side of the box (L0 Þ represents the thickness of each specimen. The rugged profile length and other model parameters were calculated after fitting the analytical expression.

5. Experimental results The proposed fractal model was used to describe the roughness of ruptured mortar and heavy clay specimens. The following specimens had their two sides modeled: mortars A2; B2; C1; D2 and heavy clay Ar8; Br3; Cr25. For each profile, the roughness dimensions were obtained applying the box-counting and sand-box methods and fitting the proposed monofractal model given by Eq. (39). These results are listed in Table 5, represented by DBC ; LBC ; L0 ; L0s and Table 6 repre-

Fig. 7. Fracture profile of mortar specimen D2 – side 1: digital images side by side and crack profile from processed image with a comparative total ruler length equal to 1.0 mm.

Fig. 8. Fracture profile of mortar specimen D2 – side 2: digital images side by side and crack profile from processed image with a comparative total ruler length equal to 1.0 mm.

246


Fig. 9. Fracture profile of heavy clay specimen Ar8: digital images side by side and crack profile from the processed image with a comparative total ruler length equal to 1.0 mm.

sented by DH0 ; l0 ; fl ; fg ; b and t. The fitting parameters L0 ffi 40 mm for mortar and L0 ffi 4 mm for heavy clay ceramic, l0 and fl in Eq. (39) are also shown in Table 6. Despite the different roles in the morphological description of the fracture surface in x and y directions, experimental measurements showed that the roughness in these two directions has similar scaling properties for very different materials [48]. A unique roughness exponent is then generally considered and it has been claimed that it has a universal value of fl ¼ 0:8 [17]. This behavior has been confirmed for a large variety of experimental situations [23,49]. However, other studies have also shown that a different value for fl ¼ 0:5 may be applicable [50–52]. These two values of the roughness exponent are connected to the length scale at which the crack is examined. In particular, at small length scales, one observes a roughness exponent fl ¼ 0:5, whereas at large length scales, the larger value fl ¼ 0:8 is found. Results presented in Table 6 show rough exponent values between 0.59 and 0.99. The value of l0 for mortar D2 specimen is very small because its roughness global dimension is very next of unity. This means that the crack profile is almost a straight line. In this case a straight line can be scrutinized by any l0 value that is infinitely small. Fig. 11 shows the asymptotic behavior of the fractal dimension with the growing crack length L0 , measured with sand-box method. The local L0 L0S and global L0 L0S ranges are depicted in the figure. In Figs. 12 and 13, a good agreement is observed in the curve fitting of Eq. (40) to the fractal analyses of the mortar specimen A2 side1 and the heavy clay specimen A8, respectively. In Figs. 14 and 15, a good agreement is observed in the curve fitting of Eq. (39) to the fractal analyses of the mortar specimen A2 side1 and the heavy clay specimen Ar8, respectively. In Figs. 16 and 17, a comparison of the rugged length L projected length L0 of each crack profile is made for mortars and heavy clay, respectively. The general behavior is consistent within each group of results. This trend is observed in the LBC =L0 ratios in Table 5. In Fig. 18, the local roughness dL=dL0 was plotted versus the square value of the modulus of rupture r2f , for the mortar specimens. A linear trend can be observed where the values plotted are correlated for a R-square correlation coefficient of 0.7012.

247


Fig. 10. Fracture profile of heavy clay specimen Br3: digital images side by side and crack profile from the processed image with a comparative total ruler length equal to 1.0 mm.

Table 5 Results from fractal analysis using box-counting method of brittle specimens. Specimen

DBC

LBC (mm)

L0 (mm)

LBC =L0

L0s (mm)

Mortar A2 side 1 Mortar A2 side 2 Mortar B2 side 1 Mortar B2 side 2 Mortar C1 side 1 Mortar C1 side 2 Mortar D2 side 1 Mortar D2 side 2 Heavy clay Ar8 Heavy clay Br3 Heavy clay Cr25

1.0500 ± 0.0001 1.0534 ± 0.0001 1.0431 ± 0.0005 1.0380 ± 0.0001 1.0607 ± 0.0003 1.0504 ± 0.0001 1.0240 ± 0.0002 1.0514 ± 0.0001 1.1511 ± 0.0004 1.2292 ± 0.0001 1.1627 ± 0.0001

59.2064 ± 0.0001 61.0022 ± 0.0002 65.8381 ± 0.0003 58.5383 ± 0.0002 67.1098 ± 0.0008 72.9876 ± 0.0004 69.3074 ± 0.0003 73.6647 ± 0.0005 11.4017 ± 0.0003 12.7726 ± 0.0002 13.7940 ± 0.0008

37.2288 ± 0.0001 39.1765 ± 0.0006 38.2453 ± 0.0001 37.8984 ± 0.0001 35.3309 ± 0.0008 40.0000 ± 0.0003 40.0000 ± 0.0003 38.3006 ± 0.0005 3.5490 ± 0.0004 3.950 ± 0.001 3.7148 ± 0.0004

1.59034 1.55711 1.72147 1.54461 1.89947 1.82469 1.73269 1.92333 3.21282 3.23357 3.71326

3.616 ± 0.001 3.0123 ± 0.0003 8.5710 ± 0.0003 8.7261 ± 0.0003 9.9510 ± 0.0006 13.700 ± 0.001 12.1460 ± 0.0004 8.2860 ± 0.0002 0.5400 ± 0.0004 0.0728 ± 0.0001 0.7232 ± 0.0003

In Fig. 19 it is shown the increasing water/cement ratio in the mortar specimens in function of the decreasing modulus of rupture. Comparing this result with Fig. 18 it is observed that the water/cement ratio affects directly the roughness of the fracture profile. The nonlinear fitting of this curve shown a result given by the following equation:

r2f ¼ r2f 0 eax where

ð50Þ

r2f ¼ 237:59 ðGPa2 Þ, a ¼ 3:1205 and x = water/cement ratio for a R-square correlation coefficient value of 0.9515.

248


Table 6 Results from fitting the model for brittle specimens. Specimen

DH0 (mm)

l0 (mm)

fl ¼ 1=DD

fg

b

t

Mortar A2 side 1 Mortar A2 side 2 Mortar B2 side 1 Mortar B2 side 2 Mortar C1 side 1 Mortar C1 side 2 Mortar D2 side 1 Mortar D2 side 2 Heavy clay Ar8 Heavy clay Br3 Heavy clay Cr25

0.3740 ± 0.0003 0.3635 ± 0.0001 0.7071 ± 0.0001 0.3040 ± 0.0003 0.84250 ± 0.0003 0.83470 ± 0.0004 1.4160 ± 0.0001 1.4618 ± 0.0003 4.1542 ± 0.0003 1.6998 ± 0.0001 3.2733 ± 0.0001

1.3249 ± 0.0004 2.2526 ± 0.0003 0.0800 ± 0.0003 0.7610 ± 0.0001 0.8261 ± 0.0001 0.0623 ± 0.0004 (1.6040 ± 0.0001)E6 (3.75 ± 0.0001)E10 0.1376 ± 0.0003 0.3641 ± 0.0001 0.1555 ± 0.0001

0.800 ± 0.001 0.800 ± 0.001 0.5954 ± 0.0001 0.6148 ± 0.0001 0.8613 ± 0.0001 0.8705 ± 0.0001 0.9927 ± 0.0001 0.9854 ± 0.0001 0.7062 ± 0.0001 0.7322 ± 0.0001 0.6579 ± 0.0001

1.0652 1.0652 1.109095 1.051606 1.0000 1.0000 1.0016 1.0031 1.179023 1.171628 1.197972

0.86097 0.92510 0.402303 0.96519 2.233964 2.294475 1.321737 1.257799 0.287048 0.091712 0.227454

0.0991 ± 0.0001 0.0997 ± 0.0001 0.1945 ± 0.0001 0.3555 ± 0.0001 0.0621 ± 0.0001 0.0564 ± 0.0001 0.0067 ± 0.0001 0.0141 ± 0.0001 0.1531 ± 0.0001 0.1554 ± 0.0001 0.1829 ± 0.0001

Fig. 11. Determination of crack length saturation by fractal analysis of mortar specimen A2 side 1 – fractal dimension projected length, L0 .

Fig. 12. Fractal analysis of mortar specimen A2 side 1 – fractal dimension, D versus projected length, L0 . Source: Ó 2012 Alves LM. Published in [short citation] under CC BY 3.0 license. Available from: http://dx.doi.org/10.5772/51813 [52, p. 61].


249

Fig. 13. Fractal analysis of heavy clay specimen Ar8 side 1 – fractal dimension, D versus projected length, L0 . Source: Ó 2012 Alves LM. Published in [short citation] under CC BY 3.0 license. Available from: http://dx.doi.org/10.5772/51813 [52, p. 61].

Fig. 14. Fractal analysis of mortar specimen A2 side 1 – rugged length, L versus projected length, L0 . Source: Ó 2012 Alves LM. Published in [short citation] under CC BY 3.0 license. Available from: http://dx.doi.org/10.5772/51813 [52, p. 61].

6. Discussion 6.1. The relation with the Lopez–Morel fractal model of fracture surface Morel et al. [24,26–28] describe the variations of roughness with the distance x measured from the notch in the transversal direction to the crack propagation. In fact, observing the microstructural aspects of rugged fracture surfaces of woods used in their work [53], it appears as a ‘‘wave beach” or as a ‘‘brazilian roof,” where in the direction of crack propagation there are almost no fluctuations in the height of the rugged surface. In spite of this fact, to adapt his scaling to the experimental reality, prefactors A and B were presented whose nature depends on the material and are obtained by fitting of experimental curves. This means that if the material to be modeled presents fluctuations of height in the growth direction, these prefactors must be explained mathematically in terms of the relations of fractal scaling for the distances of rugged length in

250


Fig. 15. Fractal analysis of heavy clay specimen Ar8 – rugged length, L versus projected length, L0 . Source: Ó 2012 Alves LM. Published in [short citation] under CC BY 3.0 license. Available from: http://dx.doi.org/10.5772/51813 [52, p. 61].

Fig. 16. Comparative plot of the rugged crack length, L projected crack length, L0 for the sides A and B among main mortars specimen profiles.

the direction parallel to the crack growth. The model represents the dependencies in the transversal x and the longitudinal y directions in a unique mathematical term. However, for materials such as glass, alumina and mortar, that present fluctuations in the transversal and the longitudinal directions, the model cannot portray authentically the fluctuations in the height observed in the fracture surface of these materials in both directions. The work herein, it has been shown that the dependence of the fluctuations in the height Dzðxðt Þ; yðtÞÞ can be decoupled and written in a simple way as,

DzðxðtÞ; yðt ÞÞ ¼ f ðxðt ÞÞ g ðyðt ÞÞ

ð51Þ

without changes in the final result and obtaining the Morel’s model [24,26–28] as a particular case. From Eq. (36), it is possible to reduce dimensionally the model of a fracture surface proposed by Morel et al. [24,26–28] to a fracture profile model, neglecting the fluctuations of the roughness in the transversal crack propagation direction, as in the case of wood fracture surfaces considering a fracture propagation in the transversal direction to the fibers. In that case the fracture surface is given by:


251

Fig. 17. Comparative plot of the rugged crack length, L projected crack length, L0 among main heavy clay specimen profiles.

Fig. 18. Plot of square of modulus of rupture versus the derivative of the rugged crack length, dL=dL0 .

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2fl 2ðfg fl Þ 2ðfgtfl Þ > R R > D E þdE D L þdL e0 Dy 0 0 0 0 > Dx > 1 þ DeH00 dxdy if DA0 DA0sat < DE0 DL0 lx DE0 ly DA ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 2fl 2ðfg fl Þ 2ftg > > R DE0 þdE0 R DL0 þdL0 e0 Dy Dx > 1 þ DeH00 dxdy if DA0 DA0sat : DE0 DL0 lx DE0 ly

ð52Þ

where DA0 ¼ dL0 dE0 and DA0sat ¼ L0s E0 . For minimal subdivisions of the profile in the transversal and the longitudinal directions, i.e., lx ¼ e0 and ly ¼ l0 , Eqs. (34) and (35) are rewritten as,

Dz ¼

8

f f fl ðfg fl Þ Dy ð g t l Þ > > Dx < DeH0 DeE0 e l 0

0

0

0

fg > > : DH0 e0 fl Dx ðfg fl Þ Dy t e0

DE0

e0

l0

:

ð53Þ

252


Fig. 19. Plot of modulus of rupture versus the water/cement ratio for a mortar cured by seven days.

Substituting Eq. (53) in (52) one has:

8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 2 2fl 2ðfg fl Þ 2ðfgtfl Þ > R > D L þdL e0 DE0 Dy 0 0 > > 1 þ DeH00 dy if DA0 DA0sat < dE0 DL0 D E0 e0 l0 DA ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 2fl 2ðfg fl Þ 2ftg > R DL þdL > e0 DE0 Dy > dy if DA0 DA0sat : dE0 DL00 0 1 þ DeH00 D E0 e0 l0

ð54Þ

Eq. (52) is analogous to the approximations presented by Morel et al. [24] considering the crack propagation direction as being the y direction and the constants A; B, and nðyÞ in their equation given by,

A ¼ DeH00 B¼

e0 D E0

fl ðfg fl Þ 1 e0

E0

ð55Þ

1=t

ly

nðyÞ ¼ E0

1=t Dy ly

The saturation length of the fracture surface in the transversal crack propagation is given by,

nðysat Þ ¼ E0

Dy L0S

1=t

¼ E0

ð56Þ

6.2. Analysis of the values obtained by the fitting The proposed fractal model presented a good fitting agreement with the experimental results, allowing to obtain the Hurst roughness dimension of the profiles. The plots of the rugged crack length of the fracture profiles versus the projected length shown in Fig. 16 for mortar and in Fig. 17 for heavy clay were fitted, using the multiparameter nonlinear minimal squares fitting method for curves, and Levenberg–Marquardt Iteration Algorithm, with the model represented by Eq. (39). Some plots of the rugged crack length as a function of the projected crack length show a clear transition between the local and the global regimes for the crack growth. This is particularly observed in Fig. 15, after the crack length L0 ffi 0:75 mm, approximately. Beyond this length, the relationship between the rugged and the projected crack lengths becomes quasilinear. The saturation length L0S was determined in two different ways. The first way was made directly by fitting from the plots of the rugged length as a function of the projected length. The second way was made by a tangent line from the plot of fractal dimension as a function of the projected crack length by sand-box method. In this second method, a straight tangent line is drawn from the value 2.0 in the vertical axis (which represents the maximal fractal limit dimension of a rugged line), until the horizontal axis, passing by a unique tangent point of the curve fractal dimension versus projected crack length, Df L0 . From Table 5 it is possible to observe that the saturation length, L0S for mortar is larger than heavy clay. This result means that the rugged crack length of the mortar is smother than the heavy clay which can be easily verified comparing Figs. 7 and 8 with Figs. 9 and 10.


253

The local and the global roughness dimensions determined in this work, for the mortar and the heavy clay specimens, are compatible with results in the literature [24,26–28,30,31,52,53]. However, for some specimens, the values seem to exceed the expected number, as in the case of mortar C1 (13.699 for the sides 1 and 2) and D2 (12.1456 for the sides 1 and 2). The fractal box-dimension, DB shown in Table 5 compared to the rugged global dimension, fg in Table 6 presents compatibles results one with other. The mortar fractal dimensions are in the interval between 1.0 and 1.1 and the heavy clay has its fractal dimension in the interval between 1.15 and 1.23. This results is in agreement with the more rugged aspect of the crack in the heavy clay as shown in Figs. 9 and 10. Therefore, it is possible to distinguish mathematically a crack in this two materials using its geometrical characteristics which can be portraited by different values of the exponents in the relations (39) and (41). A linear relation is observed between the square of modulus of rupture r2f and the values of the local roughness of the crack given by n ¼ dL=dL0 , for the mortar specimens, as shown in Fig. 18. A linear fitting is shown in that figure. Observe that the data more below of the linear fitting presented a constant displacement compared to data closer to the straight line. Thus it was due to systematic deviation of the points in the roughness values. Some authors [20,22,42,50,53] suggest a relation between the local, global, and Hurst exponents, for the tridimensional case, such as:

1H

3b ; 2

ð57Þ

where b ¼ f=t and in this case b 1 þ 2H. Considering that this equation is also valid for b ¼ fg fl =t, then

3 fg fl =t 1H 2

ð58Þ

and one finds an approximate relation for the dynamic exponent t given by:

t

fg fl 1 þ 2H

ð59Þ

If the local roughness dimension fl is equal to Hurst exponent H ¼ 2 DB ¼ 1=DD where DB is the box-dimension and DD is the divider dimension. Then one has the following relation between the obtained exponents,

fg fl 1 þ 2fl

ð60Þ

fg fl 1 H

ð61Þ

t where

and

t

1H 1 þ 2H

ð62Þ

Values presented in Table 6 were derived from this expression. Values of t are lower than those in the literature. This was expected as the tests were conducted to a fast rupture instead of crack growth. 6.3. Mechanical results The increasing water/cement ratio in the mortar specimens resulted in a decreasing modulus of rupture as shown in Fig. 19. Comparing this result with Fig. 18 it is observed that these results are also well correlated with the roughness of the crack profile. In Tables 3 and 4 for heavy clay, the increasing sintering temperature resulted in higher modulus of rupture. However, the roughness cannot be associated due to the small number of temperatures values of tested specimens. From the digitalized images of mortar and heavy clay, it can be observed that the fracture surface of the heavy clay presented bigger tortuosity compared to the mortar material. This is also evident in the fitting results where the rugged length is near 3.5 times the projected length for heavy clay, and only 1.7 for mortar. It can also be seen in Figs. 14 and 15 that for each type of material the slopes of the asymptotic curves are in very close agreement with the model proposed. This work presents several future perspectives in terms of correlating the rugged profiles with the material granulometry, chemical composition, and cement curing time. The mathematical model can still be explored for understanding the minimal crack length and the fractal dimension according to the mechanical testing parameters and material properties.

254


7. Conclusions A general self-affine fractal model was presented and applied to brittle fracture surfaces. From the proposed equation, an analytical expression for the rugged profile length was derived. Also, an analytical expression for the local roughness was derived, which can be directly introduced into the classical fracture mechanics. Comparing the fractal box-dimension, DB shown in Table 5 and the rugged global dimension, fg shown in Table 6 for the mortar and heavy clay it was possible to distinguish mathematically a more rugged crack in this two materials using its geometrical characteristics which can be portraited by different values of those exponents in the relations (39) and (41). The small ratio between roughness dimension for transversal and longitudinal directions, t, presented in Table 6 and given by Eqs. (15), (60) and (62) shown that these two directions are weakly coupled for the analyzed materials, mortar and heavy clay. The rugged crack length is a response to the interaction of the crack tip with the microstructure of the material. By the model presented in work herein it is possible mathematically to portray the peculiar rough behavior of a crack in Portland cement mortar and red ceramics using fractal geometry. The experimental technique of obtain crack profiles, proved very able to present satisfactory results that are very close to reality. A good agreement between the fractal model and the experimental results was observed. A strong correlation between the fractal dimension and the sintering temperature was verified. The results also showed that the increasing rugged crack length of the profile of the fractured mortar specimens is well correlated with the rising water/cement ratio. Acknowledgments The authors acknowledge contributions of Interdisciplinary Laboratory of Ceramic Materials – LIMAC-CIPE-UEPG, Prof. Dr. Vicente Campitelli (Civil Engineering Laboratory) and PIBIC/CNPq/UEPG. 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J Am Ceram Soc 1989;72(1):60–5. [8] Heping X. The fractal effect of irregularity of crack branching on the fracture toughness of brittle materials. Int J Fract 1989;41(4):267–74. [9] Chelidze T, Gueguen Y. Evidence of fractal fracture. Int J Rock Mech Min Sci Geomech Abstr 1990;27(3):223–5. [10] Lin GM, Lai JKL. Fractal characterization of fracture surfaces in a resin-based composite. J Mater Sci Lett 1993;12(7):470–2. [11] Nagahama H. A fractal criterion for ductile and brittle fracture. J Appl Phys 1994;75(6):3220–2. [12] Lei W, Chen B. Fractal characterization of some fracture phenomena. Eng Fract Mech 1995;50(2):149–55. [13] Tanaka M. Fracture toughness and crack morphology in indentation fracture of brittle materials. J Mater Sci 1996;31(3):749–55. [14] Borodich FM. Some fractal models of fracture. J Mech Phys Solids 1997;45:239–59. http://dx.doi.org/10.1016/S0022-5096(96)00080-4. [15] Mosolov AB. Mechanics of fractal cracks in brittle solids. Europhys Lett 1993;24:673–8. http://dx.doi.org/10.1209/0295-5075/24/8/009. [16] Bouchaud E, Bouchaud JP. Fracture surfaces: apparent roughness, relevant length scales, and fracture toughness. Phys Rev B 1994;50(23):17752. [17] Bouchaud E, Lapasset G, Planes J. Fractal dimension of fractured surfaces: a universal value? EPL (Europhys Lett) 1990;13(1):73. [18] Bouchaud E. Scaling properties of cracks. J Phys: Condens Matter 1997;9(21):4319–44. [19] López JM, Rodríguez MA, Cuerno R. Superroughening versus intrinsic anomalous scaling of surfaces. Phys Rev E 1997;56(4):3993. [20] Family F, Vicsek T. Dynamics of fractal surfaces. World Scientific; 1991. [21] Barabási AL, Stanley HE. Fractal concepts in surface growth, vol. 83. Cambridge University Press; 1995. [22] Vicsek T, Meakin P, Family F. Scaling in steady-state cluster–cluster aggregation. Phys Rev A 1985;32(2):1122. [23] Schmittbuhl J, Roux S, Berthaud Y. Development of roughness in crack propagation. EPL (Europhys Lett) 1994;28(8):585. [24] Morel S, Schmittbuhl J, Bouchaud E, Valentin G. Scaling of crack surfaces and implications for fracture mechanics. Phys Rev Lett 2000;85(8):1678. [25] López JM, Schmittbuhl J. Anomalous scaling of fracture surfaces. Phys Rev E 1998;57(6):6405. [26] Morel S, Schmittbuhl J, López JM, Valentin G. Anomalous roughening of wood fractured surfaces. Phys Rev E 1998;58(6):6999. [27] Morel S, Bouchaud E, Schmittbuhl J, Valentin G. R-curve behavior and roughness development of fracture surfaces. Int J Fract 2002;114(4):307–25. [28] Morel S, Bouchaud E, Valentin G. Size effect in fracture: roughening of crack surfaces and asymptotic analysis. Phys Rev B 2002;65(10):104101. [29] Alves LM, Da Silva R, Mokross B. Influence of crack fractal geometry on elastic–plastic fracture mechanics. Phys A Stat Mech Appl 2001;295:144–8. http://dx.doi.org/10.1016/S0378-4371(01)00067-X. [30] Mourot G, Morel S, Valentin G. Comportement courbe-r d’un matériau quasi-fragile (le bois): Influence de la forme des specimens d’essai. Colloque_MATERIAUX 2002:1–4. [31] Mourot G, Morel S, Bouchaud E, Valentin G. Scaling properties of mortar fracture surfaces. Int J Fract 2006:39–54. http://dx.doi.org/10.1007/s10704005-3471-4. [32] Alves LM. Fractal geometry concerned with stable and dynamic fracture mechanics. Theor Appl Fract Mech 2005;44:44–57. http://dx.doi.org/10.1016/ j.tafmec.2005.05.004. [33] Voss RF. Dynamics of fractal surfaces. Singapore: World Scientific; 1991. [34] Vicsek T. Fractal growth phenomena. Singapore: World Scientific; 1992. [35] Greenberg MD. Curves, surfaces, and volumes. Advanced engineering mathematics. Prentice-Hall; 1998. [36] Barnsley MF. Fractals everywhere. New York: Academic Press; 1988. [37] Besicovitch AS. On the sum of digits of real numbers represented in the dyadic system (on sets of fractional dimensions II). In: Edgar GA, editor. Classics on fractals. Boston: Addison-Wesley Reading; 1993. p. 155–9.


255

[38] Besicovitch AS. Sets of fractional dimensions (IV): on rational approximation to real numbers. In: Edgar GA, editor. Classics on fractals. Boston: Addison-Wesley Reading; 1993. p. 161–8. [39] Besicovitch AS, Ursell HD. Sets of fractional dimensions (V): on dimensional numbers of some continuous curves. In: Edgar GA, editor. Classics on fractals. Boston: Addison-Wesley Reading; 1993. p. 171–9. [40] Dauskardt R, Haubensak F, Ritchie R. On the interpretation of the fractal character of fracture surfaces. Acta Metall Mater 1990;38(2):143–59. [41] Bunde A, Havlin S. Fractals in science. Springer-Verlag New York, Inc; 1995. [42] Katsuragi H. Multiscaling analysis on rough surfaces and critical fragmentation [Ph.D. dissertation]. Fukuoka, Japan: Kyushu University; 2004. [43] Lung CW. Fractals and the fracture of cracked metals. Fractals in physics. Elsevier Science Publishers; 1985. p. 189–92. [44] Mishnaevsky Jr LL. A new approach to the determination of the crack velocity versus crack length relation. Fatigue Fract Eng Mater Struct 1994;17 (10):1205–12. [45] NBR. Areia normal para ensaio de cimento. ABNT, Normas Técnicas. 7214; 1982. [46] ASTM C348-97. Standard Test Method for Flexural Strength of Hydraulic-Cement Mortars. West Conshohocken, PA: ASTM International; 1997. www. astm.org. [47] ASTM C 305-06. Standard Practice for Mechanical Mixing of Hydraulic Cement Pastes and Mortars of Plastic Consistency. PA, West Conshohocken: 1994. www.astm.org. [48] Plouraboué F, Kurowski P, Hulin JP, Roux S, Schmittbuhl J. Aperture of rough cracks. Phys Rev E 1995;51(3):1675. [49] Måløy KJ, Hansen A, Hinrichsen EL, Roux S. Experimental measurements of the roughness of brittle cracks. Phys Rev Lett 1992;68(2):213–5. [50] Milman VY, Blumenfeld R, Stelmashenko NA, Ball RC. Comment on ‘‘Experimental measurements of the roughness of brittle cracks”. Phys Rev Lett 1993;71(1):204. [51] Alves LM, da Silva RV, De Lacerda LA. Fractal modeling of the J–R curve and the influence of the rugged crack growth on the stable elastic–plastic fracture mechanics. Eng Fract Mech 2010;77:2451–66. http://dx.doi.org/10.1016/j.engfracmech.2010.06.006. [52] Alves LM. Foundations of measurement fractal theory for the fracture mechanics. In: Belov A, editor. Applied Fracture Mechanics. Croatia: Intech; 2012, ISBN 978-953-51-0897-9. http://dx.doi.org/10.5772/51813. [53] Ponson L, Bonamy D, Auradou H, Mourot G, Morel S, Bouchaud E, et al. Anisotropic self-affine properties of experimental fracture surfaces. Int J Fract 2006;140(1–4):27–37.

Chapter V

FRACTAL MODEL FOR ELASTIC LINEAR RUGGED FRACTURE MECHANICS



    

1. Introduction The Classical Fracture Mechanics (CFM) quantifies velocity and energy dissipation of a crack growth in terms of the projected lengths and areas along the growth direction. However, in the fracture phenomenon, as in nature, geometrical forms are normally irregular and not easily characterized with regular forms of Euclidean geometry. As an example of this limitation, there is the problem of stable crack growth, characterized by the J-R curve [1, 2]. The rising of this curve has been analyzed by qualitative arguments [1, 2, 3, 4] but no definite explanation in the realm of EPFM has been provided. Alternatively, fractal geometry is a powerful mathematical tool to describe irregular and complex geometric structures, such as fracture surfaces [5, 6]. It is well known from experimental observations that cracks and fracture surfaces are statistical fractal objects [7, 8, 9]. In this sense, knowing how to calculate their true lengths and areas allows a more realistic mathematical description of the fracture phenomenon [10]. Also, the different geometric details contained in the fracture surface tell the history of the crack growth and the difficulties encountered during the fracture process [11]. For this reason, it is reasonable to consider in an explicit manner the fractal properties of fracture surfaces, and many scientists have worked on the characterization of the topography of the fracture surface using the fractal dimension [12, 13]. At certain point, it became necessary to include the topology of the fracture surface into the equations of the Classical Fracture Mechanics theory [6, 8, 14]. This new Fractal Fracture Mechanics (FFM) follows the fundamental basis of the Classical Fracture Mechanics, with subtle modifications of its equations and considering the fractal aspects of the fracture surface with analytical expressions [15, 16]. The objective of this chapter is to include the fractal theory into the elastic and plastic energy released rates G0 and J 0 , in a different way compared to other authors [8, 13, 14, 17, 18, 19]. The non-differentiability of the fractal functions is avoided by developing a differentiable   


analytic function for the rugged crack length [20]. The proposed procedure changes the classical G0 , which is linear with the fracture length, into a non-linear equation. Also, the same approach is extended and applied to the Eshelby-Rice non-linear J-integral. The new equations reproduce accurately the growth process of cracks in brittle and ductile materials. Through algebraic manipulations, the energetics of the geometric part of the fracture process in the J-integral are separated to explain the registered history of strains left on the fracture surfaces. Also, the micro and macroscopic parts of the J-integral are distinguished. A generalization for the fracture resistance J-R curve for different materials is presented, dependent only on the material properties and the geometry of the fractured surface. Finally, it is shown how the proposed model can contribute to a better understanding of certain aspects of the standard ASTM test [15].

2. Literature review of fractal fracture mechanics 2.1. Background of the fractal theory in fracture mechanics Mandelbrot [21] was the first to point out that cracks and fracture surfaces could be described by fractal models. Mecholsky et al. [12] and Passoja and Amborski [22] performed one of the first experimental works reported in the literature, using fractal geometry to describe the fracture surfaces. They sought a correlation of the roughness of these surfaces with the basic quantity D called fractal dimension. Since the pioneering work of Mandelbrot et al. [23], there have been many investigations concerning the fractality of crack surfaces and the fracture mechanics theory. They analyzed fracture surfaces in steel obtained by Charpy impact tests and used the "slit island analysis" method to estimate their fractal dimensions. They have also shown that D was related to the toughness in ductile materials. Mecholsky et al. [12, 24] worked with brittle materials such as ceramics and glass-ceramics, breaking them with a standard three point bending test. They calculated the fractal dimension of the fractured surfaces using Fourier spectral analysis and the "slit island" method, and concluded that the brittle fracture process is a self-similar fractal. It is known that the roughness of the fracture surface is related to the difficulty in crack growth [25] and several authors attempted to relate the fractal dimension with the surface energy and fracture toughness. Mecholsky et al. [24] followed this idea and suggested the dependence between fracture toughness and fractal dimension through

KIC

E D* a0

1/ 2

(1)

where E is the elastic modulus of the material, a0 is its lattice parameter, D * D d is the fractional part of the fractal dimension and d is the Euclidean projection dimension of the fracture.

Fractal Fracture Mechanics Applied to Materials Engineering 69

Mu and Lung [26] suggested an alternative equation, a power law mathematical relation between the surface energy and the fractal dimension. It will be seen later in this chapter that both suggestions are complementary and are covered by the model proposed in this work.

2.2. The elasto-plastic fracture mechanics There have been several proposals for including the fractal theory into de fracture mechanics in the last three decades. Williford [17] proposed a relationship between fractal geometric parameters and parameters measured in fatigue tests. Using Willifords proposal Gong and Lai [27] developed one of the first mathematical relationships between the J-R curve and the fractal geometric parameters of the fracture surface. Mosolov and Borodich [32] established mathematical relations between the elastic stress field around the crack and the rugged exponent of the fracture surface. Later, Borodich [8, 29] introduced the concept of specific energy for a fractal measurement unit. Carpinteri and Chiaia [30] described the behavior of the fracture resistance as a consequence of its self-similar fractal topology. They used Griffiths theory and found a relationship between the G-curve and the advancing crack length and the fractal exponent. Despite the non-differentiability of the fractal functions, they were able to obtain this relationship through a renormalizing method. Bouchaud and Bouchaud [31] also proposed a formulation to correlate fractal parameters of the fracture surface. Yavari [28] studied the J-integral for a fractal crack and showed that it is path-dependent. He conjectured that a J-integral fractal should be the rate of release of potential energy per unit of measurement of the fractal crack growth. Recently, Alves [16] and Alves et al. [20] presented a self-affine fractal model, capable of describing fundamental geometric properties of fracture surfaces, including the local and global ruggedness in Griffith´s criterion. In their formulations the fractal theory was introduced in an analytical context in order to establish a mathematical expression for the fracture resistance curve, putting in evidence the influence of the crack ruggedness.

3. Postulates of a fracture mechanics with irregularities To adapt the CFM, starting from the smooth crack path equations to the rugged surface equations, and using the fractal geometry, it is necessary to establish in the form of postulates the assumptions that underlie the FFM and its correspondence with the CFM. I. Admissible fracture surfaces Consider a crack growing along the x-axis direction (Figure 1), deviating from the x-axis path by floating in y-direction. The trajectory of the crack is an admissible fractal if and only if it represents a single-valued function of the independent variable x. II. Scale limits for a fractal equivalence of a crack The irregularities of crack surfaces in contrast to mathematical fractals are finite. Therefore, the crack profiles can be assumed as fractals only in a limited scale l0 L0 L0 max [36]. The


lower limit l0 is related to the micro-mechanics of the cracked material and the upper limit L0 max is a function of the geometric size of the body, crack length and other factors.

 Rugged crack and its projection in the plan of energetic equivalence.

III. Energy equivalence between the rugged crack surface and its projection Irwin apud Cherepanov et al. [36] realized the mathematical complexity of describing the fracture phenomena in terms of the complex geometry of the fracture surface roughness in different materials. For this reason, he proposed an energy equivalence between the rough surface path and its projection on the Euclidean plane. In the energetic equivalence between rugged and projected crack surfaces it is considered that changes in the elastic strain energy introduced by a crack are the same for both rugged and projected paths, U L0

UL

(2)

where the subscript " 0 " denotes quantities in the projected plane. Consequently, the surface energy expended to form rugged fracture surfaces or projected surfaces are also equivalent, U

0

U

(3)

IV. Invariance of the equations Consider a crack of length L and the quantities that describe it. Assuming the existence of a geometric operation that transforms the real crack size L to an apparent projected size L0 , the length L may be described in terms of L0 by a fractal scaling equation, as presented in a previous chapter. It is claimed that the classical equations of the fracture mechanics can be applied to both rugged and projected crack paths, i.e., they are invariant under a geometric transformation between the rugged and the projected paths. In the crack wrinkling operation (smooth to rough) it is desired to know what will be the form of the fracture mechanics equations for the rough path as a function of the projected length L0 , and their behavior for different roughness degrees and observation scales.


V. Continuity of functions It is considered that the scalar and vector functions that define the irregular surfaces A

A x , y are described by a model (as the fractal model) capable of providing analytical

and differentiable functions in the vicinity of the generic coordinate points P

P x , y , z , so

that it is possible to calculate the surface roughness. Thus, it is always possible to define a normal vector in corners. VI. Transformations from the projected to rugged path equations As a consequence of the previous two postulates, it can be shown using the chain rule that the relationship between the rates for projected and rugged paths are given by df L0 dL0

df L dL dL dL0

(4)

This result is used to transform the equations from the rugged to the projected path.

4. Energies in linear elastic fracture mechanics for irregular media The study of smooth, rough, fractal and non-fractal cracks in Fracture Mechanics requires the development of their respective equations of strain and surface energies.

4.1. The elastic strain energy UL for smooth, rugged and fractal cracks Consider three identical plates of thickness t , with Youngs modulus E´, subjected to a stress , each of them cracked at its center with a smooth, a rugged and a fractal crack as shown in Figure 2. The area of the unloaded elastic energy due to the introduction of the crack with length Ll is Al where ml

ml L2l

(5)

is the shape factor for the smooth crack. The accumulated elastic energy is 2

Ue

2 E'

dV

(6)

Thus, the elastic energy released by the introduction of a smooth crack with length Ll is Ul

ml t

l

2 2 Ll

2 E'l

(7)

For an elliptical crack the unloaded region can be considered almost elliptical and the shape factor is ml , thus


Ul

t

l

2 2 Ll

(8)

2 E 'l

 Griffith model for the crack growth introduced in a plate under

stress: a) flat crack and

initial length Ll with increase dLl in size; b) rugged crack and initial length L with increase dL in size; c) fractal crack, showing increase dL in size.

Analogously, the area of the unloaded elastic energy due to the introduction of a rugged crack of length L is given by A

m * L2

(9)

where m * is a shape factor for the rugged crack. Thus, the elastic energy released by the introduction of a rugged crack with length L is 2 2

UL

m*t

L

r

2 E'

(10)

Considering that the rugged crack is slightly larger than its projection, then L

L0

(11)

Consequently, the change of elastic strain energy from the point of view of the projected length L0 can be expressed as: 2

U L0 where

0

r

.

m*t

0

L0 2

2 E'

(12)


4.2. A self-affine fractal model for a crack - LEFM To take the roughness into account, it will be inserted in the CFM equations a self-affine fractal model developed in a previous chapter of this book.

4.2.1. The relationship between strain energies for rugged UL and projected UL0 cracks in terms of fractal geometry The crack length of the self-affine fractal can be expressed as

L

H0

L0 1

2

2 H 1

L0

l0

(13)

l0

where H 0 is the vertically projected crack length and the unloading fractal area of the elastic energy can be expressed as a function of the apparent length, m0 L0 2

A0

(14)

And results that 2

U L0

2m * t

r

2 E '0

1

2

H0

2 H

l0

l0

2H 2

L0 dL0

L0

(15)

Therefore, the elastic energy released by the introduction of a crack length L0 is 2

U L0

where

0

r

1

H0

2

l0

l0

0

m*t

L0 2

(16)

2 E '0

2H 2

L0

Observe that equation (12) is recovered from equation (16) applying the limits H 0 and H

1.0 with

r

0

and E '

l0

L0

E '0 .

To understand the effect of crack roughness on the change of elastic strain energy, one may consider postulates III and IV, thus

U Lo

UL

m*

2 r

2E '

L0 2

1

H0 l0

2

l0 L0

2H 2

(17)

It can be noticed that for H 1 , which corresponds to a smoother surface, the relationship between the strain energy and the projected length L0 is more linear. While for H 0 ,


which corresponds to a rougher surface, this relationship is increasingly non-linear. This is reasonable since the more ruggedness, more elastic strain per unit of crack length.

4.2.2. Relationship between the applied stress on the rough and projected crack lengths Comparing (8), (10) and (12), one has U L0

m*

U Ll

(18)

UL

Then, from postulate III, i.e., the following relationship is valid only for the situation of free loading without crack growth. 2

2

0

r

E '0

E'

2

L L0

(19)

Using equation (13) in (19), one has the resilience as a function of the projected length L0 0

2

E0

1 2 1 2 E

2

H0 l0

l0 L0

2H 2

(20)

Or, the rugged length L can be written in terms of the projected length L0 , thus E' E '0

L

0

(21)

L0

r

Since the elasticity modulus is independent of the crack path, one has 0 L0

(22)

rL

Substituting equation (13) in equation (22), one has the relationship between stresses on the rough and projected surfaces,

0

r

2

1

H0 l0

2

l0 L0

2H 2

1/ 2

(23)

This last result is still incomplete since it is not valid for crack propagation. For its correction it will be considered that the elastic energy released rate G can be expressed as a function of G0 according to equation (4).

4.2.3. The surface energy U 0 for smooth, rugged and projected cracks in accordance with fractal geometry The surface energy of a smooth and a rugged crack are, respectively, given by


U

l

2 Llt

U

l

2 ltLl

U

L

2 Lt

L

2 r tL

l

(24)

r

(25)

and

U

Using equation (11), the surface energy of the projected length L0 is given by

where

0

r

U

0

2 tL0

U

0

2 0tL0

0

(26)

. The surface energy equation (25) can be rewritten in terms of the projected

length L0 of a self-affine fractal crack

U

0

H0

2 r tL0 1

2

2H 2

l0

l0

(27)

L0

To see the influence of crack roughness on the surface energy, one may consider postulates III and IV, thus

U

0

U

2 r L0 2

1

H0 l0

2

2H 2

l0 L0

(28)

5. Stable or quasi-static fracture mechanics to the rough path In this section, a review of the conceptual changes introduced by Irwin (1957) in Griffith's theory (1920) is presented considering an irregular fracture surface, taking into account the postulates previously proposed. The purpose of this section is to use the mathematical formalism of Linear Elastic Fracture Mechanics for stable growth of smooth cracks, generalizing it to the case of an irregular rough crack.

5.1. The Griffith energy balance in terms of fractal geometry According to Griffith´s energy balance, one has

dUT

d Ui U L

F U

0

(29)

whilst F UL

U

(30)


Where UT is the total energy, U i is the initial potential elastic energy, F is the work done by external forces, U L is the change of elastic energy stored in the body caused by the introduction of the crack length L0 and U is the energy released to form the fracture surfaces. One can now add the contributions of

U L0 and

U

0

to reproduce Griffith´s energy

F

(31)

balance in a fractal vision. In other words, UT

Ui

UL

U

and

d (U dLl i

2

L0 2 1 2E

H0 l0

2

l0 L0

2H 2

2 Lo 2

1

H0 l0

2

l0 L0

2H 2

F) 0

(32)

This new result is shown in Figure 3, which is analogous to the traditional Griffith energy balance graphs, but distorted due to the roughness of the fracture surface. Observe that for a reference total energy value the roughness of the crack surface tends to increase the critical size of the fracture L0C compared to a material with a smooth fracture LlC

LC . This is

due to the roughness being a result of the interaction of the crack with the microstructure of the material.

 Griffith´s energy balance in the view of the fractal geometry of fracture surface roughness.

5.2. The modification of Irwin in Griffith´s energy balance theory for smooth, rugged and projected cracks Irwin found from Griffith´s instability equation, given by (29), that this instability should take place by varying the crack length, so


d U dL i

UL

U

F

(33)

0

which can be rewritten as d ( F UL ) dL

dU

(34)

dL

since U i is constant. On the left hand side of equation (34), dF dL dU L dL is the amount of energy that remains available to increase crack extension by an amount dL . On the right hand side of equation (34), dU dL is the surface energy that must be released to form the rugged crack surfaces. This energy is the crack growth resistance. Deriving equation (30) with respect to the projected crack length L0 , one has d (F UL ) dL0

dU

(35)

dL0

Considering postulate II, one can apply the derivation chain rule and obtain d dL (F UL ) dL dL0

dU dL dL dL0

(36)

Considering the following cases: i.

Fixed grips condition with F

constant : since U L

U L0

m*

2 2 L0

2 E ' decreases with

the crack length, and using equations (10) and (25) in (36), one can derive m * r2 L E'

2 r.

(37)

Or, by using equations (17) and (26) in (35), one finds m * r 2 L0 1 2E '

ii.

2 H

H0 l0

2

l0 L0

2H 2

2 0.

Condition of constant loading or stress, where necessarily UL

UL0

m*

2 2 L0

(38)

F

2 U L , since

2 E ' increases with the work of external forces, and using

equations (10) and (25) in (36), one can find m * r2 L E'

2 r.

(39)


Or, by using equations (17) and (26) in (35), one has m*

2 r

2E '

L0

1

2 H

2

H0 l0

l0 L0

2H 2

2 0.

(40)

Irwin defined the elastic energy released rate G and the fracture resistance R in equation (34), like

d( F U L ) dL

G

(41)

and dU

R

.

dL

(42)

These definitions can be extended to the terms in equation (35), so G0

R0

d( F U L0 )

G0

dL0

(43)

and R0

dU

0

dL0

.

Notice that the proposal made by Irwin extended the concept of specific energy

(44)

eff

to

the concept of R-curve given by equation (42), allowing to consider situations where the microstructure of the material interacts with the crack tip. In this way, it is assumed that the surface energy is dependent on the direction of crack growth. Finally, using equations (41) and (42) in (36), the Griffith-Irwin criterion is obtained, G

dL dL0

R

dL . dL0

(45)

5.3. Comparative analysis between smooth, projected and rugged fracture quantities Based on the results of the previous section, further analyses of the magnitudes of the Fracture Mechanics are performed in order to obtain a mathematical reformulation for an irregular or rugged Fracture Mechanics.


5.3.1. Relationship between the elastic energy released rate rates for smooth, projected and rugged cracks Using the chain rule, it is possible to write G0 in terms of G , G0

G

dL dL0

(46)

The energetics equivalence between the rugged surface and its projection establishes that the energy per unit length along the rugged path is equal to the energy per unit length along the projected path. Notice that

since dL dL0

dU L0

dU L

dL0

dL

(47)

1 , therefore, G0

G.

(48)

The elastic energy released rates for the projected and rugged paths are, respectively G0

dU L0

2

m*

dL0

0

L0

E '0

(49)

and

G

dU L dL

m * r2L . E'

(50)

Combining these expressions and including, for comparison, the elastic energy released rate for a smooth path, one has for infinitesimal crack lengths, G0

Gl

dLl m * dL0

G

dL dL0

(51)

Considering that the smooth crack length is equal to the projected crack length, one has G0

Gl

m*

G

dL dL0

(52)

Observe that the difference between the elastic energy released rate for the smooth, rugged and projected cracks is the ruggedness added on crack during its growth. Using a thermodynamic model for the crack propagation, it can be concluded that a rugged crack dissipates more energy than a smooth crack propagating at the same speed.


The elastic energy released rate G0 can be written in terms of a fractal geometry,

G0

m * r2 H L0 1 (2 H ) 0 E' l0

2

l0 L0

2H 2

(53)

5.4. The crack growth resistance R for smooth, projected and rough paths Considering a plane strain condition, crack growth resistance for a smooth crack is given by Rl

dU

l

(54)

dLl

Substituting equation (24) in equation (54), one finds Rl

2

(55)

l

Observe that if the fracture path is smooth, the specific surface energy

l

is a cleavage

surface energy and does not necessarily depend on the crack length. This model is only valid for brittle crystalline materials where the plastic strain at the crack tip does not absorb sufficient energy to cause dependence between fracture toughness and crack length. Similarly, for a rugged crack, the fracture resistance to propagation is given by R

2

r

(56)

The concept of fracture growth resistance for the projected surface is given by

R0

dU dL0

(57)

and substituting equation (26) in equation (57), one has R0

2

0

(58)

Again, this model is valid for ideally brittle materials where there is almost no plastic strain at the crack tip. It basically corresponds to the model presented by Griffith, with a modified interpretation introduced by Irwin with the G R curve concept.

5.5. Relationship between rugged R and projected R0 fracture resistances Using the chain rule, and admitting Irwin´s energetic equivalence represented by equation (3), the projected fracture resistance can be written on the basis of the resistance of the real surface,


R0

R

dL dL0

(59)

where dL / dL0 is derived from equation (13),

dL dL0

2

H0 1 (2 H ) l0 H0 l0

2 1

2

l0 L0

2H 2

1

2H 2

l0 L0

(60)

Therefore, the crack growth resistance ( R -curve), which is defined for a flat projected surface, is given substituting equation (56) and equation (60) in equation (59),

R0

2

2

H 1 (2 H ) 0 l0 r

H0 l0

2 1

2

l0 L0

2H 2

(61)

2H 2

l0 L0

5.6. Final remarks about equivalent quantities of smooth, rugged and projected fracture surfaces It is important to emphasize that the energetic equivalence between the rugged surface crack path and its projection was considered such that the developed equations of the Fracture Mechanics for the flat plane path are still valid in the absence of any roughness. However, if a flat and smooth fracture Ll is considered with the same length of a projected fracture L0 , the energetic quantities and their derivatives have the following relationship, U Ll

U L0

dU L

dU L0

dLl

dL0

Gl

G0

(62)

Rl

R 0,

(63)

and U

l

U

dU 0

dLl

l

dU

0

dL0

which have produced conflicting conclusions in the literature [37, 38, 46]. Since the energy for the smooth length Ll0 is smaller than the energy for the projected L0 or rough L lengths, one has U Ll

UL

Gl

G

dL dL0

(64)


and U

l

U

R

R

l

dL dL0

(65)

In postulate III it was assumed that the rugged crack path satisfies the same energetic conditions of the plan path, but in the LEFM this roughness is not taken into account, causing discrepancies between theory and experiments. For example, it has not been possible to explain by an analytical function in a definitive way the growth of the G R curve. The proposed introduction of the term dL / dL0 allows correcting this problem.

6. The elastic-plastic fractal fracture mechanics The non-linear elastic plastic energy released rate J0 for a crack of plane projected path can be extended from the Irwin-Orowan approach. They introduced the specific energy of plastic strain p on the elastic energy released rate G0 to describe the fracture phenomenon with considerable plastic strain at the crack tip. Thus, it is possible to define the elastic plastic energy released rate in an analogous way to the definition of the elastic energy released rate, Jo

d( F UVo )

(66)

dLo

where UVo is the volumetric strain energy given by the sum of the elastic and plastic ( U pl ) contributions to the strain energy in the material.

6.1. Influence of ruggedness in elastic plastic solids with low ductility Considering elastic plastic materials with low ductility where the effect of the plastic term is small compared to the elastic term, one can define a crack growth resistance as J Ro

KRo 2 f ( v) E

,

(67)

where f ( v) is a function that defines the testing condition. For plane stress f v for plane strain f v

1 , and

1 v 2 and K Ro is the fracture toughness resistance curve.

Due to the ruggedness, the crack grows an amount dL

dL0 and correcting equation (59),

one has Ro Similarly,

dU dL dL dLo

2

e

p

dL . dLo

(68)


d( F UV ) dL . dL dLo

Jo

(69)

The energy balance proposed by Griffith-Irwin-Orowan, for stable fracture, is Jo

Ro .

(70)

Therefore, for plane stress or plane strain conditions, one can write from equation (61) that, J Ro

2

e

K Ro 2 f ( v) E

dL dLo

p

(71)

Thus, 2

KRo

e

E dL . f ( v) dLo p

(72)

Knowing that fracture toughness is given by 2

KCo

e

p

E

f ( v)

,

(73)

one has, K Ro

KCo

dL . dLo

(74)

From the Classical Fracture Mechanics, the fracture resistance for the loading mode I, is given by K IRo

where Yo

Lo w

Lo w

f

Lo ,

(75)

is a function that defines the shape of the specimen (CT, SEBN, etc) and the

type of test (traction, flexion, etc), and L0

Yo

L0C , then K IR 0

f

is the fracture stress. Considering the case when

KIC 0 and the fracture toughness for the loading mode I is given by K ICo

Yo

Loc w

f

Loc .

(76)

Therefore, from equation (72) the fracture toughness curve for the loading mode I is given by


K IRo

dL . dLo

K ICo

(77)

Substituting equation (75) and equation (76) in equation (77), one has dL dLo

Yo 2

2

Lo

f

w

2

L o f ( v)

e

E

p

(78)

,

Observe that according to the right hand side of equation (78), the ruggedness dL dL0 is determined by the condition of the test (plane strain or stress), the shape of the sample (CT, SEBN, etc), the type of test (traction, flexion, etc) and kind of material. Considering the fracture surface as a fractal topology, one observes that the characteristics of the fracture surface listed above in equation (78) are all included in the ruggedness fractal exponent H. Substituting equation (60) in equation (71), one obtains 1 J Ro

2

e

H0 l0

2 H

p

H0 l0

1 which is non-linear in the crack extension

2

2

l0 L0

l0 L0

2H 2

2H 2

.

(79)

L0 . It corresponds to the classical equation (70)

corrected for a rugged surface with Hurst's exponent H. Experimental results [1, 2] show that J0 and the crack resistance R0 rise non-linearly and it is well known that this rising of the J-R curve is correlated to the ruggedness of the cracked surface [3, 4].

6.2. The J0 Eshelby-Rice integral for rugged and plane projected crack paths The J-integral concept of Eshelby-Rice is a non-linear extension of the definition given by Irwin-Orowan, for the linear elastic plastic energy released rate. In this context the potential energy 0 is defined as

WdV0

0 V0

(80)

T .uds , C

where W the energy density integral in the in the volume V0 encapsulated by the boundary C with tractions T and displacements u , and s is the distance along the boundary C , as shown in Figure 4. Accordingly, J0

d 0 dL0

d dL0

WdV0 V

T .uds C

(81)


where dL0 is the incremental growth of the crack length. In the two-dimensional case, where the fracture surface is characterized by a crack with length body, one has dV

L0 and a unit thickness

dxdy and

J0

d

0

W

dL0

For a fixed boundary C , d dL0

V

dx dy dL0

T. C

u ds . L0

(82)

d dx , and the J0 -integral for the plane projected crack

path can be written only in terms of the boundary, J0

Wdy V

T. C

u ds. x

(83)

 Boundary around to the rugged crack tip where is defined the J-Integral [43].

Now, the J-R Eshelby-Rice integral theory is modified to include the fracture surface ruggedness. Initially, equation (82) is rewritten, J0

W V

dx dL dy dL dL0

T. C

u dL ds . L dL0

(84)

From postulate IV, the new J-integral on the rugged crack path is given by J

d dL

W V

dx * dy * dL

T. C

u ds L

(85)

where the * symbol represents coordinates with respect to the rugged path. So, in an analogous way to the J-integral for the projected crack path given by equation (85), since d dL d dx * , one has J

Wdy * V

T. C

u ds. x*

(86)


Returning to equation (82) and considering postulate III along with the derivative chain rule and substituting equation (85), one has J0

d dL dL dL0

dx * dy * dL

W V

T. C

u dL ds . L dL0

(87)

Comparing (84) with equation (87) and considering that the rugged crack is a result of a transformation in the volume of the crack, analogous to the bakers´ transformation of the projected crack over the Euclidian plane, it can be concluded that ( x*, y *) dxdy ( x , y)

dx * dy *

dx * dL dy * dL dLo

dx dL dy dL dLo

(88)

which show the equivalence between the volume elements, dV

dx * dy *

dxdy.

(89)

Therefore, the ruggedness dL / dL0 of the rugged crack path does not depend on the volume V, nor on the boundary C and nor on the infinitesimal element length ds or dy . Thus, it must depend only on the characteristics of the rugged path described by the crack on the material. Finally, the integral in equation (84) can be written as J0

W V

dx dy dL

where the infinitesimal increment dx / dL

T. C

cos

i

u dL ds L dL0

(90)

accompanies the direction of the rugged

path L , as show in Figure 4. Thus, J

Wdy cos

T.

i

V

C

u cos i ds. x

(91)

Observe that the J-integral for the rugged crack path given by equation (91) differs from the J-integral for the plane projected crack path given by equation (83) by a fluctuating term, cos i inside the integral. It can be observed that the energetic and geometric parts of the fracture process are separated and put in evidence the influence of the ruggedness of the material in the elastic plastic energy released rate, J0

J

dL . dL0

(92)

It must be pointed out that this relationship is general and the introduction of the fractal approach to describe the ruggedness is just a particular way of modeling.


6.3. Fractal theory applied to J-R curve model for ductile materials This section includes the formalism of fractal geometry in the EPFM to describe the roughness effects on the fracture mechanical properties of materials. For this purpose the classical expression of the elastic-plastic energy released rate was modified by introducing the fractality (roughness) of the cracked surface. With this procedure the classical expression (49) of LEFM, linear with the crack length, is changed into a non-linear equation (53), which reproduces with precision the quasi-static crack propagation process in ductile materials. Observe that the quasi-static crack growth condition is obtained with Griffith fracture criterion, doing J 0 R0 and dJ 0 / dL0 dR0 / dL0 . In this case, it is concluded that the J-R curve is given by Griffith criterion J affine crack with H0

2

in equations (92) and (59). Therefore, for a self-

eff

l0 , one has

J0

2

2H 2

l 1 (2 H ) 0 L0 eff

H0 l0

2 1

2

(93)

2H 2

l0 L0

This model shows in unambiguous way how different morphologies (roughness) are correlated with the J-R curve growth. Given the energy equivalence between rough and projected surfaces for the crack path, the J-R curve increases due to the influence of the roughness, which has not been computed previously with the classical equations of EPFM. The J-integral on the rugged crack path is a specific characteristic of the material and can be considered as being proportional to JC [15], on the onset of crack extension, since in this case it has the rugged crack length greater than the projected crack length L

L0 . Thus,

dL . dLo

J o ~ JC

(94)

Substituting the fractal crack model proposed in equation (60), one has

1

H0 l0

2 H

J o ~ JC 1

H0 l0

2

2

lo Lo

lo Lo

2H 2

2H 2

,

(95)

corroborating that the surface specific energy is related to the critical fracture resistance. JC ~ 2

e

p

.

(96)


6.3.1. Case 1. Ductile self-similar limit The local self-similar limit can be calculated applying the condition H0

L0

l0 in

equation (79), obtaining

J Ro or, with D

2

e

p

2 H

lo Lo

H 1

(97)

2 H , one has

J0

l 2 eff D 0 L0

1 D

.

(98)

This result corresponds to the one found by Mu and Lung [26, 37] for ductile materials. Equation (98) is shown in Figure 5, where J-R curves are calculated for different values of the fractal dimension D . 2

eff

= 10.0KJ / m2 is adopted and L0 l0 is the crack length in l0

units. This figure shows very clearly how the surface morphology (characterized by D ) determines the shape of the J-R curve at the beginning of the crack growth.

 J-R curves calculated according to the projected crack length L0 , for a fracture of unit thickness, and fractal dimensions D

1.0,1.1,1.3,1.5,1.7 and 2.0 with 2

e

10 KJ / m2 .

In Figure 6, J-R curves with fractal dimension D 1.3 are calculated according to the projected length L0 for different measuring rulers l0 , showing how the morphology of rugged surface cracks is best described for small values of l0 , causing the pronounced rising of J-R curve. Figure 6 and equation (98) show that the initial crack resistance is correlated to the surface morphology characterized by dimension D , in accordance with the literature. The self-similar limit of J-R curve, given by equation (98), is valid only for regions near the onset of the crack growth in brittle materials ( H 0 L0 ). This is due to the hardening of the material, which gives rise to ruggedness of the fracture surface.


In the case of ductile materials, the length of the work hardening zone H0 affects an increasingly greater area of the material as the crack propagates, but the self-similar limit H0

L0

l0 is still valid.

 J-R curves calculated in function of the projected crack length L0 with different ruler lengths

l0

0.0001,0.001,0.01,0.1 and 1.0mm , for a fracture of unit thickness, fractal dimension D

and 2

e

1.3

10 KJ / m2 .

However, in the case of brittle materials (ceramics), after the initial stage of hardening, the crack maintains this state in a region of length H 0 , very short if compared to the crack length L0 , generating a self-similar fractal structure only when the crack length L0 is small, in the order of l0 , i.e., H0

L0

l0 . When the crack length L0 becomes much larger than the

initial size of the hardening region H 0 present at the onset of crack growth, the self-similar limit is not valid, and the self-affine (or global) limit of fracture becomes valid.

6.3.2. Case 2. Brittle self-affine limit It is easy verify that in stable crack growth, where J 0 one has dL / dL0

1 when L

R0 , using equations (59) and (79),

. The global self-affine limit of J 0 can be calculated

applying the condition when the observation scale corresponds to a rather small amplitude of the crack, similar in size to the crack increment, i.e., when H 0 l0 L0 in equation (79), resulting the linear elastic expression J0 where J 0

2

(99)

eff

G0 and GRo

2

e

p.

(100)


This result corresponds to a classic one in Fracture Mechanics, which is the general case valid for brittle materials as glass and ceramics.

7. Experimental analyses 7.1. Ceramic, metallic and polyurethane samples The analyzed ceramic samples were produced by Santos [19] and Mazzei [41]. The raw material used for its production was an alumina powder A-1000SG by ALCOA with 99% purity. Specimens of dimensions 52mm 8mm 4mm were sintered at 1650 °C for 2 hours, showing average 7 mm grain sizes. Their average mechanical properties are shown in Table 1 with elastic modulus E = 300 GPa and rupture stress f 340 MPa . The analyzed metallic samples were multipass High Strength Low Alloy (HSLA) steel weld metals and standard DCT specimens. HSLA are divided in two groups based on the welding process utilized and the microstructural composition. The first group (A1 and A2 welds) is composed of C-Mn Ti-Killed weld metals and were joined by a manual metal arc process. The second group (B1 and B2 welds), joined by a submerged arc welding process, is also a C-Mn Ti-Killed weld metal, but with different alloying elements added to increase the hardenability. Mechanical properties of both welds and DCT metals are listed in Table 1.

Material Sample

f E JIC(exp)(KJ/m2)L0C(exp)(mm) KIC (MPa.m1/2) (MPa) (GPa)

Ceramic Alumina 340

Metals

H (exp)

300

0,030

0.4956

424,2477056

0,7975 0,0096

A1CT2 516,00 1,34

291,60

0,48256

635,3313677

0,71

0,01

A2SEB2 537,00 3,63

174,67

0,36264

573,1747828

0,77

0,01

B1CT6 771,00 16,64

40,61

0,22634

650,1446157

0,77

0,02

B2CT2 757,00 1,96

99,22

0,26553

691,3971955

0,58

0,05

DCT1 554,001,7197

227,00

0,40487

624,8021278

-

DCT2 530,001,6671

211,47

0,3995

593,7576222

-

DCT3 198,750,3902

318,00

1,00000

352,2752029

-

PU0,5 40,70

0.8 0.0

8,10

0,29951

39,47980593

0,47 ± 0,07

PU1,0 40,70

0.8 0.0

3,00

0,23685

35,10799599

0,50 ± 0,05

Polymers

 Data extracted from experimental testing of J-R curves obtained by compliance method.

The analyzed polymeric samples are a two-component Polyurethane, consisting of 1:1 mixture of polyol and prepolymer. The polyol was synthesized from oil and the prepolymer from diphenyl methane diisocyanate (MDI). Their mechanical properties are shown in Table 1.


7.2. Fracture tests A standard three-point bending test was performed on alumina specimens, SE(B), notched plane. Low speed and constant prescribed displacement 1 mm/min was employed to obtain stable propagation. The R-curve was obtained using LEFM equations and fracture results are shown in Table 1. The fracture toughness evaluation of metallic samples was executed using the J-integral concept and the elastic compliance technique with partial unloadings of 15% of the maximum load. For weld metals the J-R curve tests were performed by the compliance and multi-test techniques. Tests were executed in a MTS810 (Material Test System) system at ambient temperature, according to standard ASTM E1737-96 [15]. A single edge notch bending SENB and compact tension CT were used. One J-R curve for each tested specimen was retrieved and fracture results are shown in Table 1. To obtain the fractured surfaces of polymeric materials, fracture toughness tests were performed by multiple specimen technique using the concept of J-R curve according to ASTM D6068-2002 [42]. However, these tests were different from the ones used for weld metals, due to the viscoelasticity of the polymers. The used nomenclatures PU0,5 and PU1,0 mean the loading rate used during the test, 0,5 mm/min and 1,0 mm/min, respectively. Fracture results are shown in Table 1.

7.3. Fractal analyses of fractured specimens The fractured surfaces of ceramic samples were obtained with a Rank Taylor Hobson profilometer (Talysurf model 120) and an HP 6300 scanner. The fractal analyses to obtain the Hurst dimensions were made by methods, such as Counting Box, Sand Box and Fourier transform. The fracture surface analysis of metallic and polymeric samples were executed using scanning electronic microscopy SEM and the analyses to obtain the Hurst exponents were made with the Contrast Islands Fractal Analysis. Fractal dimension results are shown in the last column of Table 1.

7.4. G-R and J-R curve tests and fitting with self-similar and self-affine fractal models A characteristic load-displacement result in the Alumina ceramic sample is shown in Figure 7. Observe that the stiffness of the material at the first deflection region is constant, corresponding to the elastic modulus of the material. However, as the crack propagates, the stiffness varies significantly. The corresponding G-R curve test is shown in Figure 8. It can be seen that at the onset of crack growth ( L0 L0C ), the behavior of this material is self-similar, as previously discussed. However, the results in the wider range of crack lengths ( L0C

L0

L0 max ) show

that this material behave according to the self-affine model. Finally, at the end of G-R curve


( L0

) the behavior is explained by the influence of the shape function Y L0 / w used in

the testing methodology [41].

 Load (X) versus displacement (u) for a G-R curve test in a ceramic sample [41].

J-R curves obtained from standard metallic specimens provided by ASTM standard testing are shown in Figure 9 along with the fitting with the proposed fractal models. Fitting results with these samples, named DCT1, DCT2 and DCT3, are a consistent validation of the applied fractal models. The fitting results of the self-similar and self-affine models coincide and are not distinguishable in Figure 9.

 G-R curve fitted with the self-similar model (equation (97)) and the self-affine model (equation (100)) for the Alumina sample [41].


 J-R curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for steel samples DCT1, DCT2 and DCT3 [43].

Typical testing results performed to obtain J-R curves of metallic weld materials are shown in Figure 10 and Figure 11. In all results, J-R curves measured experimentally were fitted using models given by equations (93) and (97), where the factor 2 e p was obtained by adjusting the l0 and H values for each different sample, by the self-similar and the selfaffine models. The J-R curves for the tested polymeric specimens are shown in Figure 12 and Figure 13. Reasonably good results were obtained despite the greater dispersion of data.

 J-R curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for HSLA-Mn/Ti steel (sample A1CT2).


 J-R curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for HSLA-Mn/Ti steel (sample B2CT2) killed with titanium and other alloy elements to increase hardenability [43].

 J-R curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for the poliurethane polymer PU0,5.


 J-R curve fitted with the self-similar model shown in equation (97) and the self-affine model shown in equation (93) for the poliurethane polymer PU1,0.

After the experimental J-R curves were fitted using equation (79) and equation (97), values of 2 eff , H and l0 were determined and are shown in Table 2 and Table 3. With J R0 2 eff , the value of the crack size L0

eff

was calculated and it corresponded to the specific surface

energy. Using the experimental values of J IC , L0C and H given in Table 1, the values of the constants in the last column of Table 2 and Table 3 were calculated.

MateSample rial CeraAlumi-na mic

2

eff

KJ / m2

0,0301871

H theo

1,000

l0 mm 0,2493645

L0

C1 2

eff

l0 2 H

1/ H 1

eff

2 H l0H

1

JC LC H

1

= constant

0,2493645

1,00000

0,03018707

A1CT2

283,247

0,417

0,018 1,00944

0,459079

1,57411

445,862579

A2SEB2

187,639

0,208

0,057 0,82912

0,396956

2,07868

390,042318

B1CT6

40,514

0,573

0,038 0,51758

0,225086

1,89071

76,600193

Metals B2CT2

101,204

0,592

0,0041 0,64484

0,278764

1,68407

170,433782

DCT1

230,843

0,426

0,91887

0,416893

1,65219

381,397057

DCT2

209,127

0,461

0,87082

0,391328

1,65806

346,745868

DCT3

317,819

0,393

2,18249

0,999062

1,00057

318,000000

PU0,5

17,4129

0,476

2,88612

1,291434

0,87464

15,230001

PU1,0

2,95252

0,503

0,51653

0,229374

2,079

6,138287

Polymers

 Fitting data of J-R curves with the self- similar model [43].


A good level of agreement is seen between measured Hursts exponents H at Table 1 and theoretical ones shown in Table 2 and Table 3. Larger differences in metals can be attributed to the quality of the fractographic images, which did not present well defined Contrast Islands.

Material Sample 2 Ceramic Alumina

Metals

Polymers

KJ / m2

eff

H theo

L0

l0 mm

C1 2

eff

l0 2 H

1/ H 1

eff

2 H l0H

1

JC LC H

1

= constant

0,0301871

1,000

0,2493645

0,2493645

1,00000

0,03018707

A1CT2

160,640

0,609

0,24422

0,105004

2,413408

387,700806

A2SEB2

102,750

0,442

0,31002

0,140040

2,993092

307,535922

B1CT6

22,980

0,700

0,08123

0,033873

2,757772

63,385976

B2CT2

57,978

0,705

0,10304

0,042893

2,529433

146,651006

DCT1

129,850

0,599

0,23309

0,100540

2,511844

326,184445

DCT2

118,850

0,624

0,20167

0,086294

2,512302

298,592197

DCT3

178,810

0,612

0,5282

0,226901

1,778386

318,000000

PU0,5

7,500

0,664

0,56541

0,238775

1,618852

12,150370

PU1,0

1,690

0,649

0,10898

0,046244

2,938220

4,971102

 Fitting data of J-R curves with the self- affine model [43].

7.5. Complementary discussion The proposed fractal scaling law (self-affine or self-similar) model is well suited for the elastic-plastic experimental results. However, the self-similar model in brittle materials appears to underestimate the values of specific surface energy eff and the minimum size of the microscopic fracture l0 , although not affecting the value of the Hurst exponent H . For a self-affine natural fractal such as a crack, the self-similar limit approach is only valid at the beginning of the crack growth process [39], and the self-affine limit is valid for the rest of the process. It can be observed from the results that the ductile fracture is closer to selfsimilarity while the brittle fracture is closer to self-affinity. Equation (79) represents a self-affine fractal model and demonstrates that apart from the coefficient H , there is a certain "universality" or, more accurately, a certain "generality" in the J-R curves. This equation can be rewritten using a factor of universal scale,

f (2

e

p , J0 )

Jo 2(2

energetic

e

1 (2 H ) p)

2 1

l0 / L0 , as

2H 2

2H 2

g( , H )

(101)

geometric

which is a valid function for all experimental results shown in Figure 14. It shows the existent relation between the energetic and geometric components of the fracture resistance


of the material. The greater the material energy consumption in the fracture, straining it plastically, the longer will be its geometric path and more rugged will be the crack.

 Generalized J-R curves for different materials, modelled using the self-affine fractal

l0 L0 of the crack length [43].

geometry, in function of the scale factor

In the self-similar limit

l0

L0

H0 , equation (97) is applicable and the energetic and

geometric components are put in evidence in the equation below, J0

(2

p )(2

eo

H)

energetic

l0 L0

H 1

(102)

geometric

From equation (102), an expression can be derived which results in a constant value associated to each material, J0 L0 macroscopic

H 1

(2

e0

p )(2

H )l0 H

1

( const )material

(103)

microscopic

It is possible to conclude that the macroscopic and microscopic terms on the left and righthand sides of equation (103) are both equal to a constant, suggesting the existence of a fracture fractal property valid for the beginning of crack growth, and justified experimentally and theoretically. These constant values were calculated for each point in each J-R curve for the tested materials. The average value for each material is listed in the last column of Table 2 and Table 3. Observe that this new property is uniquely determined by the process of crack growth, depending on the exponent H , the specific surface energy 2 e p and the minimum crack length l0 .


This new constant can be understood as a "fractal energy density" and it is a physical quantity that takes into account the ruggedness of the fracture surface and other physical properties. Its existence can explain the reason for different problems encountered when defining the value of fracture toughness K IC . This constant can be used to complement the information yielded by the fracture toughness, which depends on several factors, such as the thickness B of the specimen, the shape or size of the notch, etc. To solve this problem, ASTM E1737-96 [15] establishes a value for the crack length a (approximately 0.5 a / W 0.7 and, B 0.5W , where W is the width of the specimen) for obtaining the fracture toughness K IC , in order to maintain the small-scale yielding zone. As shown in equation (103), a relationship exists between the specific surface energy 2 and the minimum crack size l0 in the considered observation scale

eff

l0 / L0 . In Figure 15,

it can be observed that the consideration of a minimum size for the fracture l01 on a grain should mean the effective specific energy of the fracture 2

eff 1

in this scale. In a similar way,

the consideration of a minimum size of fracture in a different scale, like one that involves several polycrystalline grains l02 , l03 etc.., should take into account the value of an effective specific energy in this other scale, 2

eff 2 ,2 eff 3

, etc., in such a way that

 Microstructural aspects of the observation scale with different l0 ruler sizes, for the fractal scaling of fracture [43].

2 although l01

l02

ef 1 (2

l03 and 2

H1 )lo1H1 eff 1

2

1

eff 2

2 2

eff 2 (2

eff 3

H 2 )lo 2 H 2

1

const ,

(104)

. So, the constant does not depend on the

single rule of measurement l0 used in the fractal model, but it depends on the kind of material used in the testing. Another interpretation of equation (102) can be made by splitting the elastic and plastic terms,


J0

H 1

l0 L0

2 e (2 H )

p (2

elastic

For the particular situation where J0

l0 L0

H)

H 1

,

(105)

plastic

J IC and

L0

L0C , it can be derived from equation

(97),

J IC

2

e

p

2 H

l0

H 1

(106)

L0C

and from equation (72),

KIC

(2

e

p ) E(2

lo

H)

H 1

(107)

Loc

Therefore, using the fact that once the experimental value of J IC is determined and the fitting of J-R curve has already yielded the values 2

e

p , l0

and H for the material, the

value L0C can be calculated. Fracture Mechanics science was originally developed for the study of isotropic situations and homogeneous bodies. At the microscopic level, the elastic material is modeled considering Einsteins solid harmonic approximation where Hooke's law is employed for the force between the chemical bonds of the atoms or molecules [48]. Therefore, the elastic theory is used to make linear approximations and it does not involve micro structural effects of the material. At the mesoscopic level the equation of energy used for the fracture does not take into account effects at the atomic scale involving non-homogeneous situations [47]. Based on the arguments of the last paragraphs, it becomes clear why Herrmman et al. [49] needed to include statistical weights, as a crack growth criterion, for the break of chemical bonds in fracture simulations, as a form of portraying micro structural aspects of the fracture (defects) when using finite difference and finite element methods in computational models. At the macroscopic level, on the other hand, Griffiths theory uses a thermodynamic energy balance. It is important to remember that the linear elastic theory of fracture developed by Irwin and Westergaard and the Griffiths theory are differential theories for the macroscopic scale, which means they are punctual in their local limit. These two approaches involve the micro structural aspects of the fracture, since they take a larger infinitesimal local limit than the linear elastic theory at the atomic and mesoscopic scales. This infinitesimal macroscopic scale is big enough to include 1015 particles as the lower thermodynamic limit, where the physical quantity Fracture Resistance (J-R Curve) portrays aspects of the interaction of the crack with the microstructure of the material.


In this chapter, Classical Fracture Mechanics was modified directly using fractal theory, without taking into account more basic formulations, such as the interaction force among particles, or Lamés energy equation in the mesoscopic scale as a form to include the ruggedness in the fracture processes. The use of the fractality in the fracture surface to quantify the physical process of energy dissipation was approached with two different proposals. The first was given by Mu and Lung [26, 37], who proposed a phenomenological exponential relation between crack length and the elastic energy released rate in the following form GIC

GI 0

1 D

,

(108)

where is the length of the measurement rule. The second proposal was given by Mecholsky et al. [24] and Mandelbrot et al. [23], who suggested an empirical relation between the fractional part of the fractal dimension D * and fracture toughness KIC , KIC ~ A D * where A

1/ 2

(109)

E0 l0 is a constant and E0 is the stiffness modulus and l0 is a parameter that has

a unit length (an atomic characteristic length). The elastic energy released rate is then given by, G0 where G0C

El0 D *

(110)

2 K IC / E is the critical energy released rate.

The authors cited above used the Slit Island Method in their measurements of the fractal dimension D and it is important to emphasize that both proposals have plausible arguments, in spite of their mathematical differences. Observe that in the proposal of Mu and Lung [26, 37] the fractal dimension appears in the exponent of the scale factor, while in the proposal of Mecholsky et al. [24] and Mandelbrot et al. [23] the fractal dimension appears as a multiplying term of the scale factor. The mathematical expression proposed in this work, equation (93) and equation (97), for the case J0 G0 , is compatible with the two proposals above and can be seen as a unification of these two different approaches in a single mathematical expression. In other words, the two previous proposals are complementary views of the problem according to the expression deduced in this chapter. A careful experimental interpretation must be done from results obtained in a J-R curve test. The authors mentioned above worked with the concept of G , valid for brittle materials, and not with the concept of J valid for ductile materials. The experimental results show that for the case of metallic materials the fitting with their expressions are only valid in the initial development of the crack because of the self-similar limit, while self-affinity is a general characteristic of the whole fracture process [39].


The plane strain is a mathematical condition that allows defining a physical quantity called K IC , which doesn't depend on the thickness of the material. The measure of an average crack size along the thickness of the material, according to ASTM E1737-96 [15], is taken as an average of the crack size at a certain number of profiles along the thickness. In this way, any self-affine profile, among all the possible profiles that can be obtained in a fracture surface, are statistically equivalent to each other, and give a representative average for the Hurst exponent. The crack height (corresponding to the opening crack test CTOD) follows a power law with the scales, h l0 L0 and can be written as, v H0

L0

h0

l0

1 H

(111)

This relation shows that, while the measurement of the number of units of the crack length Nh L0 l0 in the growth direction grows linearly, the number of units of the crack height units N v

H0 l0 grows with a power of 1 H . If it is considered that the inverse of the

number of crack increments in the growth direction N h

1

l0

L0 is also a measure of

strain of the material, as the crack grows, and considering that the number of crack height increments can be a measure of the amount of the piling up dislocation, in agreement with equation (111), then the normal stress is of the type [44, 45] ~

H

(112)

Observe that this relation shows a homogeneity in the scale of deformations, similar to the power law hardening equation [34]. This shows that the fractal scaling of a rugged fracture surface is related to the power law of the hardening. It is possible that the fractality of the rugged fracture surface is a result of the accumulation of the pilled up dislocations in the hardening of the material before the crack growth. In all three situations (metallic, polymer and ceramic) the presence of microvoids, or other microstructural defects, cooperate with the formation of ruggedness on the fracture surface. This ruggedness on the way it was modeled records the "history" of crack growth being responsible for the difficulty encountered by the crack to propagate, thus defining the crack growth resistance. In EPFM literature, the rising of J-R curve for a long time has been associated with the interposition of plane stress and plane strain conditions generating the unique morphology of the fracture surface ruggedness [1, 2]. In metals this rising has been associated with the growth and coalescence of microvoids [2]. However, the Fractal EPFM has proposed that the morphology of the fracture surface, characterized by parameters of fractal geometry, explains in a simple and direct way the rising of the J-R curves. The success of fracture fractal modeling between the J-R curve and the exponent H can be attributed to the following fact: a fracture occurs only after a process of hardening in the


material, even minimal. Such a process follows a power law [35], self-similar [33], of the stress applied, with the strain , as shown in equation (166). It is therefore possible to associate the elasto-plastic energy released rate J which is an energetic quantity with the applied stress , which is an energy density, and the fracture length L0 with strain, and l / l and the ruggedness exponent H with the strain hardening exponent " n " [15]. As the strain hardening occurs before the onset of crack growth, it is evident that its physical result appears registered in the fracture surface in terms of ruggedness, created in the process of crack growth. This process of crack growth admits a fractal scaling in terms of the projected surface L0 , so it is possible that the effect of its prior work hardening is responsible for the further self-affinity of fracture valid at the beginning of crack growth. This is because in the limit of the beginning of crack growth, the fractal scaling relationship is a self-similar power law, analogous to the power law hardening relationship [8, 33]. The technical standards ASTM E813 [40] and ASTM E1737-96 [15] suggest an exponential fitting of the type J0

C1 L0

C2

(113)

for the J-R curves. They do not supply any explanation for the nature of the coefficients for this fitting. However, by comparing equation (113) with equation (97), it can be concluded that C1

2

eff

2 H l0H

1

and C2

1 H , which explains the physical nature of this

parameters;

8. Conclusions The theory presented in this chapter introduces fractal geometry (to describe ruggedness) in the formalism of classical EPFM. The resulting model is consistent with the experimental results, showing that fractal geometry has much to contribute to the advance of this particular science. It was shown that the rising of the J-R curve is due to the non-linearity in Griffith-IrwinOrowan's energy balance when ruggedness is taken into account. The idea of connecting the morphology of a fracture with physical properties of the materials has been done by several authors and this connection is shown in this chapter with mathematical rigor. It is important to emphasize that the model proposed in this chapter illuminates the nature of the coefficients for the fitting proposed by the fractal model, which is the true influence of ruggedness in the rising of the J-R curve. The application of this model in the practice of fracture testing can be used in future, since the techniques for obtaining the experimental parameters, l0 , H , and eff can be accomplished with the necessary accuracy. The method for obtaining the J-R curves proposed in this chapter does not intend to substitute the current experimental method used in Fracture Mechanics, as presented by the ASTM standards. However, it can give a greater margin of confidence in experimental


results, and also when working with the microstructure of the materials. For instance, in search of new materials with higher fracture toughness, once the model explains micro and macroscopically the behavior of J-R curves. It is well known that the fracture surfaces in general are multifractal objects [9] and the treatment presented here applies only to monofractals surfaces. However, for purposes of demonstrating the ruggedness influence on the phenomenology of Fracture Mechanics, through the models presented in this chapter, the obtained results were satisfactory. The generalization by multifractality is a matter to be discussed in future work.

Author details Lucas Máximo Alves GTEME Grupo de Termodinâmica, Mecânica e Eletrônica dos Materiais, Departamento de Engenharia de Materiais, Universidade Estadual de Ponta Grossa, Uvaranas, Ponta Grossa PR, Brazil Luiz Alkimin de Lacerda LACTEC Instituto de Tecnologia para o Desenvolvimento, Departamento de Estruturas Civis, Centro Politécnico da Universidade Federal do Paraná, Curitiba PR, Brazil

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[9] Xie, H.; Wang, J-A.; Stein, E. (1998) Direct Fractal Measurement and Multifractal Properties of Fracture Surfaces, Physics Letters A. 242: 41-50. [10] Herrmann, H.J.; Stéphane, R. (1990) Statistical Models For the Fracture of Disordered Media, Random Materials and Processes. In: Series Editors: H. Eugene Stanley and Etienne Guyon editors. Amsterdam: North-Holland. [11] Rodrigues, J.A.; Pandolfelli, V.C (1998) Insights on the Fractal-Fracture Behaviour Relationship. Materials Research. 1(1): 47-52. [12] Mecholsky, J. J.; Passoja, D.E.; Feinberg-Ringel, K.S. (1989) Quantitative Analysis of Brittle Fracture Surfaces Using Fractal Geometry, J. Am. Ceram. Soc. 72(1): 60-65. [13] Tanaka, M. (1996) Fracture Toughness and Crack Morphology in Indentation Fracture of Brittle Materials. Journal of Materials Science. 31: 749-755. [14] Xie, H. (1989) The Fractal Effect of Irregularity of Crack Branching on the Fracture Toughness of Brittle Materials. International Journal of Fracture. 41: 267-274. [15] ASTM E1737 (1996) Standard Test Method For J-Integral Characterization of Fracture Toughness. pp.1-24. [16] Alves, L.M. (2005) Fractal Geometry Concerned with Stable and Dynamic Fracture Mechanics. Journal of Theoretical and Applied Fracture Mechanics. 44(1): 44-57. [17] Williford, R. E. (1990) Fractal Fatigue. Scripta Metallurgica et Materialia. 24: 455-460. [18] Chelidze, T.; Gueguen, Y. (1990) Evidence of Fractal Fracture, (Technical Note) Int. J. Rock. Mech Min. Sci & Geomech Abstr. 27(3): 223-225. [19] Dos Santos, S.F. (1999) Aplicação do Conceito de Fractais para Análise do Processo de Fratura de Materiais Cerâmicos, Dissertação de Mestrado, Universidade Federal de São Carlos, São Carlos. [20] Alves, L.M.; Silva, R.V.; Mokross, B.J. (2001) The Influence of the Crack Fractal Geometry on the Elastic Plastic Fracture Mechanics. Physica A: Statistical Mechanics and Its Applications. 295(1/2): 144-148. [21] Mandelbrot, B.B. (1977) Fractals: Form Chance and Dimension, San Francisco, Cal-USA: W. H. Freeman and Company. [22] Passoja, D.E.; Amborski, D.J. (1978) In Microsstruct. Sci. 6: 143-148. [23] Mandelbrot, B.B.; Passoja, D.E.; Paullay, A.J. (1984) Fractal Character of Fracture Surfaces of Metals, Nature (London), 308 [5961]: 721-722. [24] Mecholsky, J.J.; Mackin, T.J.; Passoja, D.E. (1988) Self-Similar Crack Propagation In Brittle Materials. In: Advances In Ceramics, Fractography of Glasses and Ceramics, the American Ceramic Society, Inc. J. Varner and V. D. Frechette editors. Westerville, Oh: America Ceramic Society 22: pp. 127-134. [25] Rodrigues, J.A.; Pandolfelli, V.C. (1996) Dimensão Fractal e Energia Total de Fratura. Cerâmica 42(275). [26] Mu, Z.Q.; Lung, C.W. (1988) Studies on the Fractal Dimension and Fracture Toughness of Steel, J. Phys. D: Appl. Phys. 21: 848-850. [27] Gong, B.; Lai, Z.H. (1993) Fractal Characteristics of J-R Resistance Curves of Ti-6Al-4V Alloys, Eng. Fract. Mech. 44(6): 991-995. [28] Yavari, A. (2002) The Mechanics of Self-Similar and Self-Afine Fractal Cracks, Int. Journal of Fracture. 114: 1-27.


[29] Borodich, F. M. (1994) Fracture energy of brittle and quasi-brittle fractal cracks. Fractals in the Natural and Applied Sciences(A-41), Elsevier, North-Holland, 6168. [30] Carpinteri, A.; Chiaia, B. (1996) Crack-Resistance as a Consequence of Self-Similar Fracture Topologies, International Journal of Fracture, 76: 327-340. [31] Bouchaud, E.; Bouchaud, J.P. (1994) Fracture Surfaces: Apparent Roughness, Relevant Length Scales, and Fracture Toughness. Physical Review B, 50(23): 1775217755. [32] Mosolov, A.B.; Borodich, F.M. (1992) Fractal Fracture of Brittle Bodies During Compression, Sovol. Phys. Dokl., May. 37(5): 263-265. [33] Mosolov, A.B. (1993) Mechanics of Fractal Cracks In Brittle Solids, Europhysics Letters, 10 December. 24(8): 673-678. [34] Anderson, T.L. (1995) Fracture Mechanics, Fundamentals and Applications. CRC Press, 2th Edition. [35] Kanninen, M.F.; Popelar, C.H. (1985) Advanced Fracture Mechanics, the Oxford Engineering Science Series 15, Editors: A. Acrivos, et al. Oxford: Oxford University Press. Chapter 7, p. 437. [36] Cherepanov, G.P.; Balankin, A.S.; Ivanova, V.S. (1995) Fractal fracture mechanicsA review. Engineering Fracture Mechanics, 51(6): 997-1033. [37] Lung, C.W.; Mu, Z.Q. (1988) Fractal Dimension Measured with Perimeter Area Relation and Toughness of Materials, Physical Review B, 38(16): 11781-11784. [38] Lei, W.; Chen, B. (1995) Fractal Characterization of Some Fracture Phenomena, Eng. Fract. Mechanics. 50(2): 149-155. [39] Mandelbrot, B.B. (1991) Self-affine Fractals and Fractal Dimension. In: Family, Fereydoon. and Vicsék, Tamás editors. Dynamics of Fractal Surfaces. Singapore: World Scientific. pp.19-39. [40] ASTM E813, (1989) Standard Test Method For Jic, A Measure of Fracture Toughness. [41] Mazzei, A.C.A. (1999) Estudo sobre a determinação de curva-R de compósitos cerâmicacerâmica. Tese de Doutorado, DEMA-UFScar. [42] ASTM D6068 - 10 (2002) Standard Test Method for Determining J-R Curves of Plastic Materials, crack growth resistance, fracture toughness, JR curves, plastics, 96. [43] Alves, L.M.; Da Silva, R.V.; De Lacerda, L.A. (2010) Fractal Modeling of the J-R Curve and the Influence of the Rugged Crack Growth on the Stable Elastic-Plastic Fracture Mechanics, Engineering Fracture Mechanics, 77, pp. 2451-2466. [44] Zaiser, M.; Grasset, F.M.; Koutsos, V.; Aifantis, E.C. (2004) Self-Affine Surface Morphology of Plastically Deformed Metals, Phys. Rev. Lett. 93: 195507. [45] Weiss, J. (2001) Self-Affinity of Fracture Surfaces and Implications on a Possible Size Effect on Fracture Energy. International Journal of Fracture. 109: 365381. [46] Mishnaevsky Jr, L. (2000) Optimization of the Microstructure of Ledeburitic Tool Steels: a Fractal Approach. Werkstoffkolloquium (MPA, University of Stuttgart). [47] Fung, Y.C. (1969) A first course in continuum mechanics. N. J: Prentice-Hall, INC, Englewood Criffs. [48] Holian, B.L.; Blumenfeld, R.; Gumbsch, P. (1997) An Einstein Model of Brittle Crack Propagation. Phys. Rev. Lett. 78: 7881, DOI: 10.1103/PhysRevLett.78.78.


[49] Herrmann, H.J., Kertész, J.; De Arcangelis, L. (1989) Fractal Shapes of Deterministic Cracks, Europhys. Lett. 10(2): 147-152.

Theoretical and Applied Fracture Mechanics 44 (2005) 44–57 www.elsevier.com/locate/tafmec

Fractal geometry concerned with stable and dynamic fracture mechanics L.M. Alves

*

GTEME-Grupo de Termodinaˆmica, Mecaˆnica e Eletroˆnica dos Materiais, Departamento de Engenharia de Materiais, Setor de Cieˆncias Agra´rias e de Tecnologia, Universidade Estadual de Ponta Grossa—Caixa Postal 1007, Av. Gal. Carlos Calvalcanti, 4748, Campus UEPG/Bloco L—Uvaranas,CEP 84030.000 Ponta Grossa, Parana´, Brazil Available online 26 July 2005

Abstract Fractal modeling of the rugged crack geometry is considered for the stable and dynamic fracture mechanics characterizing the morphology of a fracture surface and the influence of its growth. It is shown that the fractal dimension has a strong influence on the rising of the R-curve in brittle materials. For the unstable Griffith–MottÕs approach or dynamical crack growth the fractal dimension has a strong influence on the velocity limit of the crack growth. It is also shown that the limit of crack velocity lowers with increasing surface ruggedness (higher fractal dimension D = 2 H) explaining the intangibility of the Rayleigh wave velocity by the cracks. Ó 2005 Elsevier Ltd. All rights reserved. PACS: 46.50.+a; 62.20.Mk; 05.45.Gg; 47.52.+j; 46.30.Nz; 05.45.Df; 61.43H Keywords: Fast crack growth; Fracture surface; Time delay; Ruggedness; Self-affine surface

1. Introduction The classical fracture mechanics (CFM) quantifies growth, velocity and dissipation energy of crack growth in terms of Euclidean geometry, i.e., in terms of projected lengths, areas and velocities along the growth direction. It is well established that cracks are fractal objects [1,2]. The *

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physical properties of crack growth can be treated in an explicit way by accounting for the fractal properties of the crack surface. However, the CFM has used the fractal geometry just as fractographic characterization technique [1,2] of the fracture surface. This mesoscopic characterization facilitates the understanding of the fracture surface roughness process formation. It is possible to relate the energetic greatness from the classical stable theory with the roughness surface [3–6]. For example, CFM characterizes stable crack growth by the

0167-8442/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2005.05.004

L.M. Alves / Theoretical and Applied Fracture Mechanics 44 (2005) 44–57

R-curve [7]. In the case of fixed grip condition (zero work done by the load on the sample) the R-curve rises with increasing projected lengths of the fracture, L0. This rising has been analyzed by qualitative arguments [7–10] but no definite explanation in the realm of CFM has been provided. In this paper, by introducing concepts of fractal geometry, it is shown in an unambiguous way how different morphologies (rugosities) influence the rising of the R-curve. Another problem in CFM has been concerned with the velocity of crack growth in dynamic fracture processes. Fineberg et al. performed experiments on fast crack growth [11,12] for different brittle materials, such as PMMA, soda-lime glass, etc., revealing many new aspects, defiant to the fracture dynamic theory, related to unstable crack growth. They observed the existence of a critical velocity starting from which instability begins and measured the correlation between the fluctuation in crack growth velocity and the ruggedness of the generated surfaces. It is shown experimentally in [11,12], the existence of a correlation between the variation of the ruggedness surface, DL, and the projected crack growth velocity, v0. It was found in [13] that a power law relation exits concerning the foregoing physics that was explained by using fractals. This experiments have shown that crack growth starts at an initial velocity [11,12] and, as the crack length increases, the velocity increases up to an upper limit value. This velocity is about half the Rayleigh wave velocity cR [11–14] contradicting any model based on a velocity independent fracture energy [15]. In this sense it was elaborated in [16] that an instability criterion can be advanced by introducing the principle of maximum energy dissipation rate as an energy criterion for crack dynamics formulation. This led to the upper limit of the crack velocity in brittle materials close to experimental values. The velocities are obtained by measuring the (projected) crack length in at time, without considering the rugosity of the surface. It is obvious that the velocity changes direction as the crack propagates. However, the work in [16] did not mention the fractal aspect of the fracture surface generation as a phenomenological mechanism to explain the correlation between the fluctuation in the crack

45

growth velocity and the ruggedness of the fracture surface. On the other hand, the work in [19] showed a mesoscopic formalism that involves the fundamental fractals and dynamics aspects of the fracture process through a mathematical expression of the crack velocity growth that can be used to explain the mechanism of the instability. Therefore, the fluctuation of the crack direction (and consequently of the velocity) along a projected direction is taken into account via fractal geometry. It is shown that the fractal dimension of the crack profile plays an important role on the velocity limit of crack growth. With all these different aspects of the stable and dynamic fractures theory, it can be concluded that it is possible to mathematically include some physical aspects that relate the irregularity of surfaces created with the dissipation energy process developed by the crack. This was accomplished by considering a elastodynamic formulation from the point of view of the surface formation. The fractal theory was introduced in order to inter-relate the classical stable and dynamic theory [20,21] in conjunction with the approach in [22,23] for considering the macroscopic and atomistic aspect of the problem. Other researchers have studied this problem by simulating fracture in the non-linear physics field and the computational simulation [24]. The generalization of the classical fracture mechanics by fractal geometry is done basically through the relation between the projected area or projected length, L0, and the rugged area or rugged length, L that corresponds to the true area in the fracture process. In this sense all the classical equations can be corrected by fractal geometry [3]. To the dynamic case a fractal relation between the projected surface growth velocity, v0, and roughness surface growth velocity, v, can be obtained from the differentiation by the time of the relation between the crack length projected, L0 and rugged, L, respectively, mentioned above. Inserting these relations in the dynamic formalism one gets a fracture dynamical theory to the case where the roughness surface is computed. Therefore, it is possible from experimental measurements to verify the validity of the expressions between the projected (smooth) and the true (roughness) greatness developed in the formalism mentioned above. Other

46


similar discussions to that presented in this paper can be found in [18,25–27]. They will be discussed and compared with this work. This work relates the fractographic fractal characterization of the surface fracture (rugged profile and fractal dimension) with the static and dynamic process of energy dissipation and crack instability. Described is the elastic–plastic energy release rate G0 in terms of the fractal dimension and relating it with the crack length, L0, (or surface fracture area), and with the crack growth velocity, v0. Using a modified SlepyanÕs maximum dissipation energy principle, the elastodynamic energy release rate GD0 is related to the rugged crack growth velocity, v. This explains the effect of ruggedness on the upper limit of the crack growth velocity.

U ceff 2ðL0 ceff Þ

ð2Þ

where L0 is the length of a crack introduced in a plate of unit thickness as shown in Fig. 1(a). ceff is a property of the material. Substitution of Eq. (2) in Eq. (1) yields R0 ¼ 2ceff

ð3Þ

In order to have the limit between stable and unstable fractures, Griffith–IrwinÕs energy balance approach requires that [7,15] G0 P R 0

ð4Þ

where G0 is the linear elastic–plastic energy release rate per unit thickness, i.e.,

2. Stable crack growth—the R-curve The analysis that follows is based on GriffithÕs energy balance for a stable crack growth [7,15]. For the analysis of dynamic fracture growth, MottÕs extension of GriffithÕs energy balance [15] is used together with SlepyanÕs proposed principle of maximum energy rate dissipation [16]. The crack resistance, R0, per unit thickness is defined as [7] dU ceff R0 dL0

surface energy of the material, ceff = ce + cp, by the surface area of the crack (two surfaces), length L0. Therefore

ð1Þ

Uceff is the elastic–plastic energy surface which it is equal to the product of the specific elastic–plastic

G0 ¼

dðF U L0 Þ dL0

ð5Þ

F is the work performed by external forces on the sample and U L0 is the change in the elastic–plastic strain energy caused by the introduction in the sample of a crack with length L0. For plane strain [7] pr2 L20 ð6Þ 2E For constant displacement of the specimen edges (the so-called fixed grip condition) the external forces do not perform work during the crack U L0 ¼

Fig. 1. Schematic of crack growth (a) line crack, (b) self-affine fractal model for crack growth.


extension process. Therefore dF/dL0 = 0 and the energy release rate is given by [7] dU L0 pr2 L0 ð7Þ G0 ¼ ¼ dL0 E The instability condition therefore becomes pr2 L0 P 2ceff ð8Þ E where Eqs. (3), (4) and (7) were used. The critical crack length L0c is given by L0c ¼

2ceff E pr2f

ð9Þ

where rf is the stress under which the specimen fractures. Eq. (9) shows that the critical length, L0c, depends on the material being tested (YoungÕs modulus, E, and specific surface energy, ceff) and the experimental set-up which yielded Eq. (6) for the change in the strain energy, U L0 , caused by the introduction of the crack with length L0. For homogeneous brittle materials, the specific surface energy, ceff do not depend on the direction of the crack growth. In the original criterion formulated by Griffith no distinction was made between rugged fracture surface, L, or projected fracture surface, L0. Therefore the definition proposed in Eq. (1) assumes implicitly that L0 = L and hence R ¼ R0 2ceff

ð10Þ

The R-curve is obtained experimentally by plotting G0 and R0 against the projected crack length L0. Instability occurs at G0 = R0 and dG0/ dL0 = dR0/dL0 [7]. Eq. (7) shows that G0 has a linear dependence in the crack length, L0, whereas experimental results [7,8] show that the crack resistance, R0, rises in a non-linear way. It is well known that this rising of the R-curve is correlated to ruggedness (morphology) of the cracked surface [9,10]. Therefore, it is necessary to model the fracture surface to include mathematically its ruggedness in the classical formalism of the fracture mechanics to know its influence on the stable and dynamic dissipation energy process. 2.1. The fractal model of the fracture surface CFM quantifies crack surfaces and crack profiles without considering its ruggedness, i.e., all

47

physical properties are measured and quantified along the projected areas and projected lengths and have been identified above by the subscript zero. The fracture surface is a self-affine fractal. Therefore, the morphology of a fractured surface may be characterized by its fractal nature [1,2,17,18]. The mathematical relation that describes the self-affinity property is given by Lðkx x; ky yÞ ¼ kx kHy Lðx; yÞ

ð11Þ

where H = 2 D is the Hurst exponent that measures the ruggedness of the profile. D is the fractal dimension. The one-dimensional case of a fracture surface is a rugged profile whose length L from the measure of L0 [3], is shown in Fig. 1(b). L0 denotes the distance between two points of the crack (the projected length of the crack), where from the measure L, of L0, at fractal dimension D is given by [1,3,17–19] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2H H0 l0 ð12Þ L ¼ L0 1 þ l0 L0 where H0 is the vertical size of the grid. For a small crack length, L0, one can approximate L0 H0 and therefore it has [3], sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H 2ffi l0 ð13Þ L ¼ L0 1 þ L0 where l0 is the length which defines the scale l0/L0 under which the crack profile is scrutinized [1,17– 19,28]. In this way the ruggedness of a fractal fracture surface can be expressed as a function of Eq. (12) as 2H 2 l0 1 þ ð2 H Þ dL L0 n ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ 2H 2ffi dL0 l0 1þ L0 A fractal measure is obtained taking an ndimensional grid of boxes (or cells) and covering the fractal object, of dimension, D, (n 6 D 6 n + 1) with this grid and counting and marking how many of these cells of size, l0, of the grid, intercept the object (Fig. 1(b)). The fractal dimension (also called box dimension) is the exponent

48


that relates the number of marked cells in the grid covering function of the extension of the cells scaled by the total size of the grid, l0/L0. Therefore, to relate the energy release rate with the ruggedness of the fracture surface it is necessary that the fractal measure of crack growth be accompanied instantaneously. The Box-Counting method cannot be applied in this case because it can only be used after all of the crack was formed. But the Sand-Box method can be used to represent the crack growth. In this case it is possible to suppose that the ruggedness happens simultaneously with fractal measure, if it is imagined the border of the boxes of counting, used in the Sand-Box method at the beginning of the crack growth and the opposite box borders in the crack tip. Therefore, the projected crack length, L0, is extended instantaneously with the crack tip becoming a function of the time, L0 = L0(t) (Fig. 1(b)). There are experimental techniques that accompany the crack growth. This technique uses optical instruments. Considering that the Sand-Box method is applicable to crack growth. It is possible to admit that the ruggedness happens instantaneously. Therefore, the crack growth can be mathematically described to finalities of the modeling by this counting technique. What happens is that usually one uses the Box-Counting methods in their mathematical reasoning. Only this method can be applied after all of the crack was formed. Therefore this reasoning does the interpretation of the ruggedness in terms of the fractal scaling not simultaneous with the crack growth process. Taking the local limit of the fractal measure where the boxes that cover this profile have the same extension in the direction horizontal and vertical direction, one can approximate Eq. (13) by H 1 l0 L ¼ L0 ð15Þ L0 This equation is analogous to the self-similar mathematical relation only that the exponent is (1 H) instead (D 1). It is necessarily valid only in the beginning of the R-curve. The R-curve is measured on the plane projected fracture surface, L0, and not on the ruggedness fracture surface, L. The term introduced into Eq. (13) makes the necessary corrections between these two surfaces.

2.2. The relationship between crack resistance (R-curve) and crack geometry Because of the mathematical complexity of the rugged surface, the CFM adopted an energetic equivalence between the real fracture surface and the projected surface making U L ¼ U L0 and Uc = Uc0. Using fractal aspects of the fracture surface a more realistic expression to the energy of the rugged surface can be rescued just substituting relation Eq. (13) in Eqs. (2) and (1) and substituting Eq. (13) in Eqs. (6) and (5) as shown as follow. It is necessary to establish a connection between R and R0. R is the energy necessary to create a unit of the rugged crack. In CFM this energy is referred to the length projected on the crack growth direction, allowing us to conclude that R = R0. The crack resistance of the material, R0, as is usually used in the experimental tests of brittle materials is concerned with the energy necessary to create two surfaces in the projected direction, but not in the rugged surface. Therefore, the actual expression has to be corrected by fractal terms corresponding to the fractal dimension, D, of the rugged surface, i.e. R0 ¼

dU ceff dU ceff dL ¼ dL0 dL dL0

ð16Þ

where the subscript zero in R0 for the ruggedness crack path has been dropped. Therefore, R0 ¼ 2ceff

dL dL0

ð17Þ

where 2ceff is the resistance offered by the material for an increase of the fracture of length, L, by a non-projected amount of length, dL, whereas R0 is the resistance offered by the material for an increase of the fracture of projected length, L0, by a projected amount of length, dL0, (Fig. 1). Substituting the result of the derivation of Eq. (14) into Eq. (17) it is obtained 2H 2 l0 1 þ ð2 H Þ L0 R0 ¼ 2ceff sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H 2ffi l0 1þ L0

ð18Þ


R0 is the resistance of the material to the crack growth along the growth direction. Eq. (18) corresponds to Eq. (3) but now corrected for a rugged surface having rugged dimension, H. 2.3. R-curve in terms of fractal geometry The elastic–plastic energy release rate as defined by Eq. (7), G0, is derived from the change in energy of the stress field U L0 , Eq. (6), when the size of the introduced crack is L0 + dL0 instead of L0. Actually the crack grew to an amount of dL P dL0 and Eq. (1) has to be corrected, i.e., G0 ¼

dðF U L0 Þ dðF U L0 Þ dL ¼ dL0 dL dL0

ð19Þ

where the subscript zero in G0 for the ruggedness crack path has been dropped. Therefore, G0 ¼ G

dL dL0

ð20Þ

Defining, G, as being the elastic energy released rate on the rugged crack path. The Griffith–Irwin criterion of fracture can be expressed for the rugged crack path also, as ð21Þ

GPR

Therefore, the energy balance proposed by Griffith–Irwin, instead of Eq. (4), using Eqs. (16) or (17), becomes G0 P 2ceff

dL dL0

ð22Þ

Substituting the result of the derivation of Eq. (13), given by (14), into Eq. (22) results in 2H 2 l0 1 þ ð2 H Þ L0 G0 P 2ceff sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð23Þ 2H 2ffi l0 1þ L0 For a stable crack growth, G0 = R0, Eq. (23) becomes 2H 2 l0 1 þ ð2 H Þ L0 G0 ¼ 2ceff sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð24Þ 2H 2ffi l0 1þ L0

49

This is the condition for a stable crack growth. It corresponds to Eq. (4) but now corrected for a rugged surface having the fractal dimension D = 2 H. (i) In the case of the self-affine limit, where l0 = H0 L0, there prevails G0 ¼ 2ceff

ð25Þ

and Eq. (3) is reproduced (ii) In the case of the self similar limit, where L0 = H0 l0 there results H 1 l0 ð26Þ G0 ¼ 2ceff ð2 H Þ L0 A rearrangement of this equation gives G0 2c ð2 H Þ ¼ eff 1H ¼ constant 1H L0 l0

ð27Þ

This is a new result yielded by the inclusion of the fractal geometry into the classical fracture mechanics that it is confirmed experimentally and indirectly by the norm ASTM 1737-1996, when this fit the J–R curve by a relation given by J 0 ¼ C 1 LC2 0 , considering the case of brittle materials when J0 = G0. In Fig. 2 G–R-curves are calculated by Eq. (18) or (24) taking different values for ruggedness dimension, H = 2 D. 2ceff is taken equal to 10.0 J/m2 which is a value compatible with brittle materials. L0 is the crack length in units of l0 = 0.01 mm. In accordance with the literature [10] Fig. 2 and Eqs. (18), (24) or Eq. (13) shows that initial crack resistance depends on the morphology of the surface characterized by D.

Fig. 2. Resistance curves for l0 = 0.01 mm and ceff = 5 J/m2.

50


jected crack length). In order to have unstable fracture, it is required that [7,15,30] GD0 P C0 ðv0 Þ

ð29Þ

Making the necessary corrections of the fractal ruggedness in Eqs. (28) and (29) for the dynamic case as done in an analogous way in Eqs. (19) and (20) it is observed that G D0 ¼

Fig. 3. Resistance curves for H = 2 D = 0.6 and ceff = 5 J/m2.

In the CFM literature [7,8,15], the rising of the R-curve has for long time been associated with the interplay of plane-strain conditions and threedimensional effect [7,8,15], generating a peculiar surface morphology. However, CFM has not provided a definite analysis of the rising of the R-curve. It is shown that fracture surface morphology characterized by fractal geometry parameters explains in a simple and straightforward way the rising shape of the R-curve. In Fig. 3 R-curves for ceff = 5 J/m2 with ruggedness dimension, H = 2 D = 0.6, are calculated as a function of the projected crack length, L0, for different measuring rulers, l0, showing that the rugged morphology of the cracked surface is enhanced for smaller values of l0, bringing about an accentuated rising of the R-curve.

3. Unstable crack growth The dynamic energy release rate, GD0 , is given by [7,15] GD0 ¼

d½F ðU L0 T 0 Þ dL0

ð30Þ

Defining, GD, as being the elastodynamic energy released rate on the rugged crack path. Griffith–MottÕs criterion of fracture can be expressed for the rugged crack path also, as ð31Þ

GD P CðvÞ

where C is the dynamical work of the fracture for the rugged crack path. Therefore, the energy balance proposed by Grifith–MottÕs, for dynamical case, instead of Eq. (29) becomes GD0 P Cðv0 Þ

dL dL0

ð32Þ

From Eq. (32) it is possible to explain the achievement of the limit crack velocity for values lower than the Rayleigh wave velocity as it will be shown as follows. 3.1. The relationship between crack velocity and crack geometry The limitation to the projected crack growth velocity can be seen to occur at a value lower than the Rayleigh wave velocity. This can be clearly explained by the energy consumption due the ruggedness, dL/dL0, between the real rugged crack path, L, formed in the instability process, and the projected crack path, L0 along the crack growth direction in the following way:

ð28Þ

where F is the work performed by the external forces on the sample, U L0 is the change in the elastic–plastic strain energy caused by the introduction of a crack with length L0 into the sample and T0 is the kinetic energy of the crack growth. L0 is the distance between two points of the crack (or pro-

d½F ðU L0 T 0 Þ dL dL dL0

v0 ¼ v

dL0 dL

ð33Þ

as n = dL/dL0 P 1 to v = cR there results v0 6 c R

ð34Þ

However, it is stressed for the dynamical problem the crack resistance, C0, and therefore the specific


51

elastic–plastic surface energy of the material, ceff, are dependent on the crack velocity v0. This assumption allows, at least in principle, to obtain the crack velocity as a function of the time. However, it is difficult to identify some kind of stable connection between the effective surface energy, ceff, and the crack velocity. Besides this conceptual difficulty, experimental observations [11–14,16,29] show that the crack growth process is usually characterized by two periods: in the first period, the crack velocity, v0 increases and the energy release, GD0 , per unit area is almost constant. In the second period, the crack velocity, v0, is constant but the energy release, GD0 , increases. It is clear that the classical energy criterion (Eq. (4)) is not valid in the second period of the crack growth. Therefore, the corrections done above using the fractal ruggedness are not sufficient to explain the dependence of the work of fracture, C0(v0), with the velocity although. To circumvent these difficulties, an energy dissipation criterion [16] was proposed that accounts for the excess of the energy flux M0 as

but it was now modified to include the ruggedness, dL/dL0, of the crack surface, by the substitution of Eq. (17), instead of Eq. (3) into Eq. (35). It is important to note that in Eq. (37) the dynamical energy release rate, GD0 , is the dynamic energy necessary to release in order to increase the crack length by an amount, dL0, which is the projected length of actual crack growth, dL. Therefore, the specific crack resistance, R = 2ceff in Eqs. (35) was corrected by Eq. (17) since the actual crack length is dL instead of dL0. Therefore, SlepyanÕs maximum dissipation energy principle, including instability and ruggedness effects, expressed in Eq. (36), can be now written in terms of fractal geometry as Z Z dL M 0 ðv0 Þ ds ¼ GD0 2ceff ð38Þ v0 ds dL0 C C

M 0 ðv0 Þ ¼ ðGD0 2ceff Þv0

The dynamic energy release rate, GD0 , can be expressed in terms of the stable energy release rate, G0, and the straight line crack velocity, v0 [37], by

ð35Þ

It has a straight line velocity, v0, that maximizes in the following integral: Z Z M 0 ðv0 Þ ds ¼ ðGD0 2ceff Þv0 ds ð36Þ C

C

where C is an arbitrary contour around the tip of the crack. However, this formulation, in spite of explaining the achievement of the limit crack velocity for values lower than the Rayleigh wave velocity, does not include the effect of the ruggedness directly. 3.2. The maximum energy dissipation rate principle in terms of fractal geometry For a material that presents a non-linear crack resistance, R0, this principle can be expressed in a more general way as dL M 0 ðv0 Þ ¼ ðGD0 R0 Þv0 ¼ GD0 2ceff v0 dL0 ð37Þ

we obtain the following integrand, in Eq. (38), to be maximized Z dL dI ¼ d GD0 2ceff v0 ds ¼ 0 ð39Þ dL0 C

dL dL 0 ¼ v0 dL 1 cR dL0 2ceff

GD0

ð40Þ

where cR is the Rayleigh velocity. In Fig. 4 the dynamic work of fracture C0(v0) is plotted for the elastodynamic condition, GD0 = C0(v0), in accord with Eq. (29), for different values of the ruggedness, dL/dL0. In the instability phenomenon, observed by Fineberg–Gross [11], the fracture surface presents different values for its local ruggedness, dL/dL0, along the crack growth. Therefore, the experimental C0(v0)-curve will have values that fluctuate between the different curves shown in Fig. 4 while the crack length increases [37]. The elastodynamic energy released rate, GD0 , that maximizes the excess of the energy flux, M0, along the projected crack length, L0, in the integral expression given by, Eq. (38), yields for a

52


v0 ¼ cR

2ceff f ðt þ sÞ dL dL0 1 G0 dL0 dL

ð45Þ

or sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H 2 l0 1þ L0 v0 2c f ðt þ sÞ dL ¼ 1 eff 2H 2 G0 dL0 cR l0 1 þ ð2 H Þ L0 ð46Þ

Fig. 4. Dynamic work of fracture versus normalized crack growth velocity, for different ruggedness dL/dL0.

maximum dissipation the following straight line crack velocity, v0 [37], in the following way: v0 2c dL dL0 ð41Þ ¼ 1 eff cR GD0 dL0 dL According to the approach in [7,15] and from Eqs. (9) and (7), the relation between 2ceff and G0 [7] can be shown as 2ceff L0c ¼ G0 L

ð42Þ

On the other hand, the elastodynamic energy release rate GD0 [16] is related to the elastic–plastic energy release rate G0 as a function of the time g(t). But, because of the relaxation effects at the crack tip [38] consider that this function g(t) should be substituted by another function of the time, t, similar to that suggested by Slepyan, which can be written as f(t + s) in the following way: GD0 ¼ G0 gðt þ sÞ ¼

G0 f ðt þ sÞ

Observe that the fractal ruggedness mathematical term outside of the parentheses in Eq. (46) is responsible for the lower of the maximum limit of crack growth velocity for values inferior to the Rayleigh wave velocity while the temporal term inside of the parenthesis is responsible for the fluctuation on the absolute value of the velocity. In Fig. 5 the normalized straight line velocity v0/cR is calculated as a function of the projected crack length, L0, for different Hurst exponents, H, and ruler length, l0 = 0.01 mm, L0c = 0.25 mm and f(t + s) = 1. The ruggedness dimension, H, of the profile of the fracture yields different values for the ratio v0/cR once a constant velocity is achieved. In Fig. 6 the normalized straight line crack velocity v0/cR is calculated for different materials characterized by the critical length L0c, Eq. (9), with ruler length l0 = 0.01 mm and f(t + s) = 1. For comparison the same ruggedness dimension (Hurst exponent), H = 0.6, has been used in the calculation. In Fig. 7 the normalized straight line crack

ð43Þ

therefore the relation between 2ceff and GD0 [7] can be expressed as R0 2c dL 2c f ðt þ sÞ dL ¼ eff ¼ eff G0 dL0 GD0 GD0 dL0 2 L0c rf ¼ f ðt þ sÞ L r

ð44Þ

When r = rf the crack starts growing and Eq. (41) becomes

Fig. 5. Crack growth velocity versus crack length for L0c = 0.25 mm, l0 = 0.01 mm and f(t + s) = 1.


Fig. 6. Crack growth velocity versus crack length for H = 2 D = 0.6, l0 = 0.01 mm and f(t + s) = 1.

Fig. 7. Crack growth velocity versus crack length for H = 2 D = 0.6, L0c = 0.25 mm and f(t + s) = 1.

velocity, v0/cR, is calculated as a function of the projected crack length, L0, for different ruler lengths, with Hurst exponent, H = 0.6 and critical length L0c = 0.25 mm and f(t + s) = 1, showing its effect on the measure of the initial crack growth velocity, v0i.

4. Discussion 4.1. Stable crack growth The crack resistance to fracture, R0, defined by Eq. (1) is supposed to be a property of the material

53

(see Eq. (3)). Experimentally it is shown that it rises with crack growth [7–10]. In metals this increase has been associated with growth and coalescence of microvoids [7]. In this paper it is shown that in brittle materials this increase is associated with the morphology of the surface characterized by the fractal dimension D. The mechanism by which surface irregularities are introduced into the cracked surface are very complex and not well understood and depends on innumerous factors such as inhomogeneites, microcracks, grain boundaries, instabilities induced by the stress field at the crack tip, etc. Therefore, fractal geometry is a powerful resort allowing a simplified analysis of a problem with such complexity. The shape of the R-curve is also influentiated by several factors of particular fracture testing, such as: thickness and geometry of the specimen, position or shape of notch. The model developed concerning the R-curve leaves out the influence of these factors, but only consider the geometrical mechanism of tenacification. To obtain a more realistic plot of the R-curve taking into account the influence of these factors, it is necessary to introduce the mathematical aspects of the fracture testing into Eq. (23) by the G0 calculation. This G0 term corresponds to the real energy released rate by the specimen. It depends on the deviation angle, h, of the crack growth direction from the projected direction. It determines consecutively the morphology or the rugosity of the fracture surface. The dependence of the G0 value with the crack projected length can yield the saturation or the decrease of the R-curve found in real fracture testing. The modification of the fracture mechanics by the fractal scaling suggests that the model presented in this paper is non-linear. This model reproduces very well the elastic–plastic results of the J–Rcurve as the approximation done in [3]. Therefore, with this modification the energy necessary to propagate a fracture is not anymore proportional to the fractured area. The success of the fractal modeling of fracture mechanics can be attributed to the fact that the fracture happens only after the hardening. This process admits a power law between the stress and strain whose practical result appears registered in the fracture surface in terms of the ruggedness created in the process. This

54


ruggedness by its time admits a fractal scaling in terms of the projected surfaces. This was shown in Eq. (14). In Fig. 2 it is shown in an unequivocal way how different fractal dimensions D influence the rising of the R-curve. In Fig. 3 it is shown that for a given ruggedness dimension, e.g., H = 2 D = 0.6, how the size of the measuring ruler influences this rising. For smaller rulers the rising is more accentuated than for larger rulers due to the increasing lengths of the actual fracture. The derivation presented in this paper differs from the treatment done by Chelidze et al. [32]. In order to introduce the ruggedness those authors equate for a crack of unit width G0L0 = GL (see Eq. (4) of Ref. [32]). This is incorrect since by definition, Eq. (1) of Ref. [32] or Eq. (5) in this paper, the proper equation should read G0 dL0 ¼ GdL

ð47Þ

which is an approximation of Eq. (15) of this paper with the following result: G0 dL0 ¼ GD

1D l0 dL0 L0

ð48Þ

and not Eq. (4) of [32]. The apparent similarity between Eq. (48) and the equation proposed by Chelidze [32] is a stumbling to a careful conceptual interpretation that must be done. The idea to correlate the fracture properties with the ruggedness is not new. But this correlation do not have been accomplished correctly. The use of the differential dL/dL0 instead of the simple rate L/L0 as done in [32] constitute in a different interpretation in terms of the fractal characterization and measure methods. The derivation presented differs from the treatment given in [33]. Instead of Eq. (26) of this paper, these authors based on phenomenological arguments propose the following expression for the energy release rate which in the notation adopted in this paper becomes G0 ¼ 2ceff e1D

ð49Þ

where e is the yardstick length. It must be remarked that Eq. (49) does not depend on the scale L0/l0.

Another proposition based on empirical arguments can be found in [34] which in our notation reads ð50Þ G0 ¼ Ea0 D Since G0c ¼ K 2IC =E, where KIC is the fracture toughness and E is YoungÕs modulus. a0 is a parameter having the unit of length. In contrast to Eqs. (19), (49) and (50) are empirical and have no rigorous mathematical basis. They have been used widely in the literature [4,5,35,36]. Other approaches have been done to relate the energy released rate G with the fractal geometry. The work in [25] discusses the different models [4–6,33]. The same was found in [26] for the correlation as in [2] between GIC, KIC, DKth, in relation to the impact energy and the increment of fractal dimension, D = 1 D. Recently, it was shown in [27] that a correlation between the roughness (fractal dimension) and fracture toughness can be established. The tougher material had the higher fractal dimension. It is important to observe that the fractal dimension, D, with the critical values of the material parameters such as, GIC, KIC, etc. can be related to the ruggedness defined by Eq. (14) with the G–R curve of energy released rate, G0, and crack resistance, R0, given by Eq. (24), or yet the toughness, given by the general relation, i 1h G0 ¼ ð1 v2 ÞðK 2I0 þ K 2II0 Þ þ ð1 þ v2 ÞK 2III0 ð51Þ E where from Eq. (24) one can have for the Mode-I loading the K R0 -curve, given by vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2H 2 u l0 u 1 þ ð2 H Þ u L 0 ð52Þ K IR0 ¼ u u2ceff E sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H 2ffi u l0 t 1þ L0 and for the critical value of KIR = KIC one can have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2H 2 u l0 u 1 þ ð2 H Þ u L 0c K IC0 ¼ u ð53Þ u2ceff E sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2H 2ffi u l0 t 1þ L0c


where L0c, is GriffithÕs critical size of fracture. From Eq. (27) for a self-similar case, one can have K IC qffiffiffiffiffiffiffiffiffiffi ¼ L1H 0c

sffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G0c E 2ceff Eð2 H Þ ¼ 1H L0c l1H 0

¼ constant

55

is due to an increase in the rugosity of the surface characterized by the fractal dimension D.

5. Conclusions ð54Þ

4.2. Unstable crack growth It is well known that unstable cracks propagates well below RayleighÕs limit velocity, cR [11,14,29]. Different approaches have been proposed to explain the apparent anomaly but so far no consensus has been achieved [31]. In this paper SlepyanÕs principle of maximum energy dissipation expressed by the maximization of Eq. (38), and with the approximate relation between GD0 and G0, Eq. (44), obtained in [16] is used. By this calculation the saturation or limit velocity approaches with increasing crack length, L0, the value v0/cR ! 0.5, as is shown in Fig. 5 for H = 1.0. If the crack is rugged the crack resistance has to be corrected by Eq. (17). In Fig. 5, it is shown how increasing the ruggedness dimension, H, change the behavior of the crack velocity. Besides achieving the limit velocity at larger lengths, L0, the limits have different values for different H values. It is important to notice that crack growth starts at L0 = L0c. Therefore the smaller the ruggedness dimension, H, the larger will be the initial velocity as can be checked by Eq. (46). In Fig. 6 it is shown how different materials, characterized by the critical crack length, L0c, have different saturation velocities. From Eq. (9) it is noticed that the higher the YoungÕs modulus E and the higher the ratio of ceff =r2f , the lower will be the limit of the growth velocity v0/cR. It can be observed that the saturation of the curves, v0/cR ! 0.5, in Fig. 5 is due only to the SlepyanÕs maximum energy dissipation rate principle. The fractality D just changes v0/cR to values lower than 0.5 but does not cause v0/cR < 1. Besides achievement of constant velocity v0 below RayleighÕs velocity cR, observations showed the existence of a correlation between surface morphology (branching) and velocity instabilities [29]. Therefore, the lowering in the v0/cR curves

The model for stable crack growth using the fractal geometry could explain quantitatively: (i) For a stable crack growth the rising of the Rcurve is strongly influenced by the crack morphology characterized by the fractal dimension. (ii) The increase of the R-curve and the dependence of its shape with the rugosity of fracture surface. (iii) The dependence of the initial fracture resistance value, R0c with the minimal critical length l0. The model for unstable or dynamical crack growth could explain qualitatively: (iv) For unstable or dynamical crack growth the limit of crack velocity lowers with increasing surface ruggedness (higher fractal dimension D). (v) Intangibility of the Rayleigh velocity by the crack. (vi) The existence of the critical velocity, v0c, where the energy flux to the crack tip stops to increase when this value is ultra passed. (vii) The values to the critical velocity do not depend on the critical length l0 used to measure the crack, which can be interpreted as a minimal critical length to initiate the fracture. If one consider that the fracture surface has not only one fractal dimension, D, but there also exists a little fluctuation of the fractal measure around a mean value, because the instability process, the plots of the crack velocity can exhibit a fluctuation on the crack velocity analogue that reported by Fineberg [11] and Sharon [29]. (viii) It is important to notice that in this paper the surface morphology has been characterized by only one fractal dimension. It is well known that fractured surfaces are multifractal

56


objects [18] and the treatment here presented only applies for monofractal surfaces. This multifractality might be correlated with the crack length L0, and be responsible for velocity fluctuation as observed in [11,29]. These studies including a generalization for multifractality will be the subject of future research. Acknowledgements The author thanks Prof. Bernhard Joachim Mokross, Prof. Leon Mishnaevsky Jr. and Prof. Jose´ A. Rodrigues for helpful discussions and special acknowledges to Dr. George C. Sih for his opportune corrections of the original manuscripts. Lucas Ma´ximo Alves thanks the Brazilian program PICT/CAPES for concession of a scholarship and for the financial support of this research. References [1] R.H. Dauskardt, F. Haubensak, R.O. Ritchie, On the interpretation of the fractal character of fracture surfaces, Acta Metall. Mater. 38 (2) (1990) 143–159. [2] B.B. Mandelbrot, D.E. Passoja, A.J. Paullay, Fractal character of fracture surfaces of metals, Nature 308 (5961) (1984) 721–722. [3] A. Lucas Ma´ximo, R.V. da Silva, B.J. Mokross, The influence of the crack fractal geometry on the elastic plastic fracture mechanics, Physica A: Stat. Mech. Appl. 295 (1–2) (2001) 144–148. [4] H. Nagahama, A fractal criterion for ductile and brittlefracture, J. Appl. Phys. 75 (6) (1994) 3220–3222. [5] W. Lei, B. Chen, Eng. Mater. Mech. 50 (2) (1995) 149–155. [6] M. Tanaka, J. Mater. Sci. 31 (1996) 749–755. [7] H.L. Ewalds, R.J.H. Wanhill, Fracture Mechanics, Edward Arnold Publishers, London, 1993. [8] J.M. Kraff, A.M. Sullivan, R.W. Boyle, Effect of dimensions on fast fracture instability of notched sheets, in: Proc. of the Cracks Propagation Symp. Cranfield, 1962 (The College of Aeronautics, Cranfield, England, 1962), vol. 1. pp. 8–28. [9] Peter L. Swason et al., Crack-interface grain bridging as a fracture resistance mechanism in ceramics: I. experimental study on alumina, J. Am. Soc. 70 (4) (1987) 279–289. [10] H. Hu¨bner, W. Jillek, Subcritical crack extension and crack resistance in polycrystalline alumina, J. Mater. Sci. 12 (1) (1977) 117–125. [11] J. Fineberg, S.P. Gross, M. Marder, et al., Instability in dynamic fracture, Phys. Rev. Lett. 67 (4) (1991) 457–460.

[12] J. Fineberg, S.P. Gross, M. Marder, et al., Instability in the propagation of fast cracks, Phys. Rev. B 45 (10) (1992) 5146–5154. [13] J.F. Boudet, S. Ciliberto, V. Steinberg, Dynamics of crack propagation in brittle materials, J. Phys. II France 6 (1996) 1493–1516. [14] K. Ravi-Chandar, W.G. Knauss, Int. J. Fract. 26 (1984) 141. [15] T.L. Anderson, Fracture Mechanics, Fundamental and Applications, second ed., CRC Press, Boca Raton, FL, 1995. [16] L.I. Slepyan, Principle of maximum energy dissipation rate in crack dynamics, J. Mech. Phys. Solids 4 (1993) 1019– 1033. [17] F.M. Borodich, J. Mech. Phys. Solids 45 (1997) 239– 259. [18] H. Xie, J. Wang, E. Stein, Phys. Lett. A 242 (1998) 41– 50. [19] L.L. Mishnaevsky, Fatigue Fract. Engng. Mater. Struct. 17 (1994) 1205–1212. [20] N.F. Mott, Brittle fracture in mild steel plates, Engineering 165 (1948) 16–18. [21] L.B. Freund, Dynamic Fracture Mechanics, Cambridge University Press, New York, 1990. [22] A.A. Griffith, The phenomena of rupture in flow in solids, Philos. Trans. R. Soc. London (Mech. Eng.) A221 (1920) 163–198. [23] G.R. Irwin, Fracture Dynamics, Fracturing Metals, American Society for Metals, Cleveland, 1948, p. 147–166. [24] F.F. Abraham, D. Brodbeck, R.A. Rafey, W.E. Rudge, Instability dynamics of fracture: a computer simulation investigation, Phys. Rev. Lett. 73 (2) (1994) 272– 275. [25] E. Charkaluk, M. Bigerelle, A. Iost, Fractals and fracture, Eng. Fract. Mech. 61 (1) (1998) 119–139. [26] Z.G. Wang, D.L. Cheng, X.X. Jiang, S.H. Ai, C.H. Shih, Relationship between fractal dimension and fatigue threshold value in dual-phase steels, Scripta Met. 22 (1988) 827– 832. [27] M.A. Issa, M.A. Issa, Md.S. Islam, A. Chudnovsky, Fractal dimension—a measure of fracture roughness and toughness of concrete, Eng. Fract. Mech. 70 (1) (2003) 123–137. [28] J. Feder, Fractals, Plenum Press, New York, 1989. [29] E. Sharon, S.P. Gross, J. Fineberg, Local crack branching as a mechanism for instability in dynamic fracture, Phys. Rev. Lett. 74 (25) (1995) 5096–5099. [30] L.B. Freund, J. Elast. 2 (1972) 341–349. [31] B. Lawn, Fracture of brittle solids, second ed. Cambridge Solid States Science Series, Cambridge Press, Cambridge, 1993. [32] T. Chelidze, Y. Gueguen, Int. J. Rock. Mech. Min. Sci. Geomch. Abstr. 27 (3) (1990) 223–225. [33] Z.Q. Mu, C.W. Lung, J. Phys. D: Appl. Phys. 21 (1988) 848–850. [34] J.J. Mecholsky, D.E. Passoja, K.S. Feinberg-Ringel, J. Am. Ceram. Soc. 72 (1) (1989) 60–65.

L.M. Alves / Theoretical and Applied Fracture Mechanics 44 (2005) 44–57 [35] H. Xie, The fractal effect of irregularity of crack branching on the fracture toughness of brittle materials, Int. J. Fract. 41 (1989) 267–274. [36] G.M. Lin, J.K.L. Lai, J. Mater. Sci. Lett. 12 (1993) 470– 472. [37] L.M. Alves, A new fractal thermodynamic principle to the energy dissipation in the fracture dynamics, in preparation.

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[38] L.M. Alves, R.F.M. Lobo, A chaos and fractal dynamic approach to the fracture mechanics, in: The Logistics Map and the Route to Chaos: From the Beginning to Modern Applications, Proceedings of Verhulst 200 Congress on Chaos, 16–18 September, 2004, Brussels, Belgium, Springer, in press.

UMA TEORIA ESTATÍSTICA FRACTAL PARA A FRATURA EM MATERIAIS FRÁGEIS - Parte I

Lucas Máximo Alves Al. Nabuco de Araújo ,469,Uvaranas,CEP - 84031-510, Caixa Postal 1007, Fone/Fax: (042) 223-9355 Ramal-38, [email protected], Centro Interdisciplinar de Pesquisas em Materiais, Universidade Estadual de Ponta Grossa-PR, CIPEM/UEPG-PR.

RESUMO Definindo-se uma probabilidade local de fratura, e fazendo-se considerações estatísticas análogas ao (1) formalismo matemático feito por Weibull porém para a propagação de uma trinca, foi possível chegar a uma expressão análitica que descreve a curva-R em materiais frágeis. Usando-se a descrição fractal para a geometria da trinca, foi possível, junto com a descrição anterior, relacionar analiticamente a dimensão fractal com a curva-R e com a energia total de fratura, além de outras grandezas que descrevem a fratura frágil.

Sendo que: Ti  T (tamanho real) assim como ci c*, onde c* pode ser o tamanho crítico da trinca encontrado por Griffith, pois a estatística auto-similaridade de uma trinca está limitada por uma escala inferior min determinada pelo tamanho critico e por uma escala superior máx dado pelo tamanho macroscópico da trinca. O número de estruturas lineares em que se consegue subdividir a trinca (Figura -1) é dado por: E-D

Ni = Ti /ci = (Tr/ci) i

(D)

onde: Ni: é o número de estruturas na partição com tamanho ci ou na escala i

Palavras-chaves: probabilidade de fratura materiais frágeis, dimensão fractal

N1

N2

N3

Ni

N*

INTRODUÇÃO De acordo com o teorema de Euler para funções homogêneas de grau n qualquer, uma transformação de escala i (min  i  max) numa função deste tipo resulta em: F(ic) = i-nF(c)

Tr

c1

c2

T1

T2

c3 T3

ci ....Ti

c* .................T

(A)

(2)

Mandelbrot mostrou que as trincas e as superfícies de fratura são estruturas geométricas fractais que satisfazem o teorema de Euler. Considerando-se a superficie de fratura rugosa, como sendo uma função -D homogenea de grau D, ou seja, Ao = i Au e a sua projeção no plano, como sendo uma função homogênea de grau E = 2, ou seja, Ar = i-HAu. Como as areas unitárias Au são necessáriamente iguais, dividindo-se estas relações tem-se : E-D

Ao = Ari

(B)

Por outro lado, para uma fina placa plana (Figura 1) de espessura e 0 , onde Ao = Tie ou Ar = Tre, vale a relação: Ti = Tr iE-D

(C)

onde: Ti : tamanho medido da trinca na escala i Tr: é o tamanho projetado da trinca numa determinada direção i = ci/Tr: tamanho da partição ou escala de medida. D: dimensão fractal ci: tamanho do segmento característico, ou da régua de medida da trinca (Figura -1).

Figura - 1. Trecho de uma trinca sobre um corpo de prova, mostrando, a variação da medida do comprimento Ti da trinca com a escala de medida ou partição ci. Como as energias das superfícies fraturada Uo e projetada Ur são necessariamente iguais tem-se : 2rAr = 2oAo

(E)

onde: r: é a energia de superfície projetada o: é a energia de superfície da superfície fraturada. Multiplicando-se os dois lados de (E) por iE-D temse : 2rAriE-D = 2oAoiE-D

(F)

Usando-se o resultado (2) tem-se : r = oiE-D

(G) (3)

Esta relação tem sido sugerida por Rodrigues . No caso de Ni  NT (onde NT é o numero total de estruturas lineares até romper o corpo de prova) tem-se T que: Ar  Ar , r rT e o oT logo o lado esquerdo de (G) torna-se portanto:

E-D

rT = oT 

(H)

onde : rT: é a energia de superfície regular do trabalho total de fratura. oT: é a energia de superfície irregular do trabalho total de fratura. T Ar = Tre: é a área total da superfície projetada de fratura (área regular) AoT = Te: é a área total da superfície real de fratura (área irregular). rT = oT E-D

(H)

As relações (H) e (I) são também sugeridas por (3) Rodrigues .

C ci+1

e i+1 r B oi

r ci A

Af

i

b Figura - 3. Célula para cálculo da probabilidade geométrica local de propagação de uma trinca. portanto: pri = ciseni /ri

O MODELO Probabilidade geométrica local de trincamento pri e probabilidade de fratura de um corpo trincado Pf. Um corpo sob tensão, pode ou não resistir a carga aplicada e vir a fraturar. A trinca formada, pode permanecer estática ou apresentar um crescimento subcrítico ou estável, devido aos fenômenos de corrosão ou de temperatura, ou ainda pode se romper catastroficamente. Estas são condições extremas que precisam ser incluidas na teoria fractal aplicada a fratura, para que se possa envolver um espectro cada vez maior de casos existentes na prática. Pode-se considerar três regiões de comportamento de propagação de trincas conforme mostra a Figura - 2. e escrever uma probabilidade geométrica local de trincamento ou fratura pri. região subcrítica

região estacionária pri(c) = ?

fratura catastrófica

Figura - 2. Condições extremas de crescimento de trincas e de fratura, onde pri(c) é a probabilidade local de fratura em função do tamanho de um trecho da trinca). Subdividindo-se a trinca da Figura -1. em pequenas células de tamanho r conforme mostra a Figura 3, será determinado qual deve ser a probabilidade local de um corpo produzir uma próxima trinca ci+1 com inclinação i+1 que resulte em fratura local, uma vez que ele tenha produzido uma trinca ci com inclinação i. Para isso, devese examinar a Figura - 3. Supondo-se que o material é subdividido em NrT = L/r células quadradas de tamanho r fixo, onde se encontra uma trinca conforme a descrição da Figura - 3, a probabilidade geométrica local de fratura da célula, ou seja a probabilidade que a célula de tamanho r trinque completamente, uma vez havendo uma trinca de tamanho ci variável, é dada por: pri = Uo/Ur = (ci /ri)oi /ri

Para um corpo extenso de tamanho L, após Ni trincas de tamanho ci a probabilidade de falha será dada por: Pf =

Ni

pri =



Ni



i 1

C ci+1

ciseni/ri

(P)

i 1

C e

ci+2

e i+2

i+1 B

B

r

ci i+1 ci

Af

i A

r* ci

Af i A

b b Figura - 4. Reparticionamento da trinca para cálculo da probabilidade local de propagação de uma trinca.

região catastrófica

cresc. subcr. de trincas

(O)

(N)

mas ri é a projeção de oi conforme mostra a Figura -3,

Observe que por causa da expressão (D) o número de estruturas depende da partição feita ao longo de toda a trinca. Porém, é sempre possível encontrar um número Ni = N* (Figura - 4), tal que acima desta partição a fractalidade desaparece dentro de uma escala inferior determinada por ci = c* (que pode ser o tamanho crítico da trinca determinada pela teoria de Griffith), fazendo com que a trinca deixe de ser estatisticamente auto-similar (também existe uma escala superior onde a trinca deixa de ser estatisticamente auto-similar). Neste caso, é sempre posível encontrar um ci = c* fixo, desde que este seja escolhido numa dimensão tal, que a área real de fratura ou extensão real da trinca T seja dada da seguinte forma: Ai = eTi =

Ni



eci  Ao = eT = N*ec*

(Q)

i1

para   min onde a área unitária de fratura ao é dada por: ao = e c*, e por outro lado, o volume que contém a fratura é dado por: V = ebTr =

Ni

 i1

ebl  V = Niebl = N*ebc* (R)

Observe que b não precisa ser indexado, pois pode ser escolhido arbitráriamente, desde que envolva os extremos de deflexão da trinca. Note que, bmáx corresponderá ao comprimento S do corpo de prova. Para r  r*, deve-se impor necessariamente que r* = c*. Sendo, v = ebc*, das expressões (16) e (18) tem-se que: N* = Ao/ao = V/v  N* = T/c*

(T)

onde a trinca é reparticionada até que se encontre um número N* de estruturas, tal que ci possa ser fixado em c*, e retomando-se novos valores de i = i* conforme mostra a Figura - 4. Neste caso a expressão (D) fica: N* = T /c* = (Tr/c*) *E-D

p*ri = sen*i

N* = T/c* = (Tr/c*) *E-D

(V)

E-D

Pf = exp[Tr(1 - *

N*



seni*

(X)

e a probabilidade de sobrevivência é dada por: Ps =

 i 1

qri* =

N*

(1 - pri*)=

 i 1

N*

(1 - seni*)



Pf(c*) = N*(c*)/N*T(c*) = N*r/N*rT

onde: N*(c*): é o numero de defeitos conectados pela trinca real Mas N*T(c*) é dado de forma análoga por: N*T = (L/c*)E-D

(AH)

onde: N*T: é o número de defeitos que devem ser conectados pela trinca real para que haja a ruptura completa do corpo. logo a probabilidade de fratura Pf é dada por: (AI)

onde: L: dimensão do corpo de prova Tr: dimensão linear projetada da trinca

(Y)

Cálculo da Curva-R. A curva de resistência a propagação de uma trinca, em (J/m2) é definda como:

escrevendo-se, porém:

R = dUf/dAf

N*

[1 - (1 - seni*)]



(Z)

i 1

considerando-se que as deflexões da trinca são pequenas, o ou seja, i está em torno de 90 logo, 1 - seni

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