Mechanics of Advanced Materials and Structures ...

This article was downloaded by: [Indian Institute of Technology Roorkee] On: 20 June 2015, At: 22:33 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Mechanics of Advanced Materials and Structures Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/umcm20

Evaluation of Mechanical Properties using Spherical Ball Indentation and Coupled FE–EFG Approach a

a

a

A. S. Shedbale , I. V. Singh , B. K. Mishra & Kamal Sharma

a

a

Department of Mechanical and Industrial Engineering, Indian Institute of Technology Roorkee, Uttarakhand, India Phone: +91-1332-285888, Fax: +91-1332-285665 Accepted author version posted online: 01 Jun 2015.

Click for updates To cite this article: A. S. Shedbale, I. V. Singh, B. K. Mishra & Kamal Sharma (2015): Evaluation of Mechanical Properties using Spherical Ball Indentation and Coupled FE–EFG Approach, Mechanics of Advanced Materials and Structures, DOI: 10.1080/15376494.2015.1029171 To link to this article: http://dx.doi.org/10.1080/15376494.2015.1029171

Disclaimer: This is a version of an unedited manuscript that has been accepted for publication. As a service to authors and researchers we are providing this version of the accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proof will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to this version also.

PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

ACCEPTED MANUSCRIPT Evaluation of Mechanical Properties using Spherical Ball Indentation and Coupled FE– EFG Approach A. S. Shedbale, I. V. Singh, B. K. Mishra, Kamal Sharma Department of Mechanical and Industrial Engineering,

Downloaded by [Indian Institute of Technology Roorkee] at 22:33 20 June 2015

Indian Institute of Technology Roorkee, Uttarakhand, India Phone: +91-1332-285888, Fax: +91-1332-285665, Email: [email protected]

ABSTRACT In the present work, the mechanical properties of pressure tube material (Zr–Nb2.5) are evaluated using coupled FE–EFG approach. Penalty approach is used to impose contact constraints and non–penetration condition at the interface. An efficient node–to–segment algorithm is employed to model the contact behavior. An updated Lagrangian approach is used to model the large deformation. Loading and unloading response of the indentation process is analyzed using von–Mises and GTN plasticity models. In multiple indentations, the indentation depth is progressively increased up to maximum specified limit with partial unloading. Load– indentation depth curves are used to extract the flow properties of the material. Keywords: Mechanical properties; coupled FE–EFG approach; penalty approach; spherical indentation; contact modeling; large deformation

1

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 1. INTRODUCTION The structural performance of any component under a given loading and environment mainly depends upon its mechanical behavior. The mechanical behavior of any component or structure is directly related to the mechanical properties. Thus, the proper evaluation of mechanical


properties of the materials is quite important from the design and performance point of view. The properties of the materials are mainly evaluated through experimental testing, which is quite expensive and time consuming task. To overcome this issue, many attempts were made in past to numerically assess the mechanical behavior of the materials. The numerical modeling of contact between two non–conforming bodies is very challenging task for the mechanics community. Many problems related to metal forming (rolling, extrusion, pressing), crack propagation, powder compaction, material characterization (destructive and non–destructive testing) involve contact. Spherical indentation is a unique method to assess the mechanical properties of the materials in a non–destructive manner. In past, many theories and models have been developed for the measurement of basic mechanical properties through indentation technique. In 1908, Mayer (1908) developed a relationship between mean pressure and impression diameter to evaluate the yield strength of the materials. In 1951, Tabor (1951) proposed an empirical relationship to find the representative strain within the plastic region through spherical ball indentation. Tabor‟s relation was found close to the test observations when indentation process becomes fully plastic. Over the years, the technological advancement and a need of evaluating the mechanical properties on small scale have led an increased interest in development of methods based on instrumented indentation experiments. Several methods based on indentation techniques were

2

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT proposed by the researchers to measure hardness and Young‟s modulus (Doerner and Nix, 1986; Oliver and Pharr, 1992; Dao et al., 2001), plastic flow properties (Giannakopoulos and Suresh, 1999; Chollacoop et al. 2003; Cao and Lu, 2004; Lee et al., 2010), residual stress (Suresh and Giannakopoulos, 1998; Zeng and Chiu, 2001; Lee et al., 2005; Dias et al., 2010) and fracture


toughness (Haggag and Nanstad, 1989; Haggag et al., 1990; Haggag, 1993). The indentation process is inherently quite complex and nonlinear which involves the nonlinear (elasto–plastic) material behavior, large deformation and variable contact condition. Thus, in order to overcome the difficulties faced by analytical approaches to model the ball indentation, it becomes necessary to use the powerful numerical techniques such as finite element method. In past few decades, some research efforts have been made to study the indentation–induced elastic–plastic deformation within the finite element framework. Hardy et al. (1971) carried out finite element analysis of elastic–plastic half–space indentation and found the effect of applied contact load on stress field and contact pressure distribution. A finite element code with constant strain triangle and grid expansion technique was employed by Follansbee and Sinclair (1984) to improve the computational performance of elastic–plastic indentation. The results were found in excellent agreement with analytical solution based on classical Hertz theory and experimental testing. Giannakopoulos et al. (1994) obtained a constitutive relation between normal load and indentation depth for elastic as well as elastic–plastic material behavior in Vicker‟s indentation process using FEA. Many researchers derived the dimensionless relations based on the constitutive model proposed by Johnson (1985) and presented the numerical results in the form of dimensionless parameters for elastic–plastic indentation. Komvopoulos and Ye (2001) analyzed the indentation of an elastic–perfectly plastic half space with a rigid sphere using finite

3

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT element approach. They derived the dimensionless constitutive relationship for the mean contact pressure and contact area, and observed that the mean contact pressure is equal to 2.9 times the yield strength in the fully plastic deformation regime. The finite element based indentation mechanics models have been used by many researchers (Bhattacharya and Nix, 1988; Ye and


Komvopoulos, 2003) to characterize the mechanical behavior of layered materials. Park and Pharr (2004) and Song and Komvopoulos (2013) analyzed the indentation response in terms of dimensionless parameters to interpret the post–yield behavior consisting of various deformation regimes. Apart from numerical simulations and experimental testing, many attempts have been made to extract elastic–plastic material properties based on reverse analysis using finite element approach. Knapp et al. (1999) developed an approach to interpret the elastic modulus and hardness of thin films by nano–indentation. Nakamura et al. (2000) proposed a new procedure based on inverse analysis to determine the properties of functionally graded materials using instrumented indentation. Dao et al. (2001) established the forward and reverse algorithms based on dimensionless analysis to extract the elasto–plastic material properties using sharp indentation. Nayebi et al. (2002) carried out an inverse analysis to determine the yield strength and hardening exponent based on regression functions. The reverse analysis technique was also used by the other researchers to measure the material properties (Chollacoop et al., 2003; Cao and Lu, 2004; Chen et al., 2007; Lee et al., 2010). So far, FEM has been widely used to perform the structural analyses including indentation. The analyses of structures involving small deformation can be done effectively using FEM but in case of large deformation, it induces an error due to extensive element distortion. Thus, the

4

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT meshfree methods become an ideal choice for the evaluation of mechanical properties using spherical ball indentation as they were successfully used in many complex problems including plasticity, static and dynamic fracture (Rabczuk et al., 2004; Rabczuk and Eibl, 2006; Rabczuk and Belytschko, 2007; Rabczuk et al., 2007; Zhuang et al., 2012; Cai et al., 2013; Pathak et al.,


2014). Among various meshfree methods (Nguyen et al., 2008) developed in last two decades, the element free Galerkin (EFG) method has been widely used for analyzing various engineering and science problems (Singh et al., 2011; Pant et al., 2013). In spite of its various advantages over FEM, it was found computationally quite expensive. In order to tackle this issue, the EFG method was coupled with FEM (Belytschko et al., 1995; Rabczuk et al., 2006; Kumar et al., 2014a; Kumar et al., 2014b; Shedbale et al., 2014) to exploit the benefits of both the methods. Thus, in the present work, a coupled FE-EFG approach is extended further to analyze the elasto– plastic response of the material during spherical indentation. The frictionless contact model proposed by Wriggers (2002) is employed to model the contact between rigid indenter and specimen. A well-known node to segment (NTS) approach is employed using the penalty approach. A displacement controlled technique is adopted to simulate an axisymmetric problem. The constitutive equations applicable for the nonlinear behavior of the materials under large deformation are used to determine the deformation during indentation process. Based on associative flow rule of plasticity, the computational elasto-plastic constitutive equations are derived for two yield criteria viz. von-Mises and GTN. The constitutive equations are extended further to account the three different hardening rules (isotropic, kinematic and mixed) for evaluating the post yield material response. In case of multiple indentations, the load–indentation

5

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT depth curves are used to calculate yield strength and ultimate strength based on various evolution equations (Haggag et al., 1990; Haggag, 1993; Lee et al., 2005). The paper work is summarized as follows: Section 2 provides the modeling using coupled FE–EFG approach. Overview of finite deformation theory along with the formulation of elasto–


plastic constitutive equations is presented in section 3. In section 4, the node to segment (NTS) technique is presented for frictionless contact based on penalty approach. In section 5, the numerical simulations are presented to demonstrate the applicability of the coupled FE–EFG approach for the modeling of large deformation. Finally, some concluding remarks are given in section 6. 2. MODELING WITH COUPLED FE–EFG METHOD In coupled FE–EFG (Belytschko et al., 1995; Kumar et al., 2014a; Kumar et al., 2014b) method, the transition elements are used to couple FE domain with EFG domain as shown in Fig. 1. In the transition region, the displacement approximation is obtained by interpolating between FE h displacement and EFG displacement. An approximate displacement u (x) that must be

consistent in the transition region, is expressed by following relation,

uh (x)  1  R(x) uFE (x)  R(x)uEFG (x)

(1)

where u FE (x) and u EFG (x) are the FE and EFG displacement approximations respectively and

R(x) is a ramp function defined using FE shape functions N I (x) , R(x)   N I (x), x EFG

(2)

I

6

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT From the above equation, the ramp function is equal to the sum of FE shape functions associated with transition element nodes on the EFG boundary  EFG . The transition shape functions are developed by substituting FE and EFG displacement approximations in Eq. (1), nen

nsn

nsn

I 1

I 1

I 1

u (x)  [1  R(x)]  N I (x) u I  R(x)  I (x) u I   N I (x) u I


h

(3)

where, nen is the number of element nodes, nsn is the number of nodes in the support or domain of influence of x ,  I (x) are EFG shape functions. The transition shape functions N I (x) are given as,

N I (x)  [1  R(x)] N I (x)  R(x) I (x)

(4)

Therefore, the shape functions of coupled EF–EFG method  I (x) are summarized as,

 N I (x),   I (x)   I (x),  N (x),  I

x  FE x  EFG x  TE

(5)

3. GOVERNING EQUATIONS It has been observed that the yield stress associated with a material may vary upon plastic loading. For certain class of materials, the yield stress in compression is different than tension. In case of elasto–plastic materials, the yield surface may change in size, shape and position, and these phenomena can be accounted by incorporating the various hardening rules. Therefore, different hardening rules are considered to model the material hardening behavior. This section provides a brief review of the governing equations for the large deformation elasto-plastic analysis of an axisymmetric problem. Some elasto–plastic constitutive relations for

7

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT von–Mises and GTN plasticity models is presented followed by a brief description of isotropic and kinematic hardening rules. 3.1 Overview of the Finite Deformation Theory In this work, the updated Lagrangian formulation is used to account for the finite deformation


(Reddy, 2009). Consider the equilibrium of the body under applied surface tractions t at the boundary  and body forces b over the domain  , the following expression is obtained using the principle of virtual work,





σ :  ε d   t  u d   b  u d 

(6)



Initially, reference configuration and current configuration are same hence Cauchy stress σ and second Piola–Kirchhoff stress S are equal. In the updated Lagrangian description, the deformation process is described by three configurations i.e. reference configuration C0 , previous known deformed configuration C1 and current deformed configuration C2 . During each incremental step, all the state variables are defined with respect to the state at the beginning of the increment, and at the end of each increment, the variables are updated with respect to the state at the end of the increment. Let X and x represents the material point position vector in the Lagrange frame and Eulerian frame respectively. The deformation gradient F  x X represents the mapping of X to x . For general formulation, it is necessary to define the Jacobian of deformation gradient as

J  F . Green–Lagrange strain tensor E , defined in the reference configuration, is related to deformation gradient through E  1 2  FT F  I  . The linear and nonlinear strain components of E are denoted by E L and E NL respectively. At the end of the increment, a reference configuration

8

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT C0 must be changed to configuration C1 and the second Piola–Kirchhoff stress S is transformed

to the Cauchy stress σ as σ  FSFT J . The constitutive relation between stress and strain increments is given by S  Dep E . By using the above relations, the equilibrium equation defined in equation (6) takes the


following form,

  D E :  E d   ep





σ :  EL d   σ :  ENL d   t  u d   b  u d 





(7)

The above equation can be written in terms of elemental matrices as, Ku  Fint  Fext with K  K mat  K geo

(8)

where, K mat   BT Dep B d and K geo   G T MσG d are the material tangent stiffness matrix 



and geometric stiffness matrix respectively, Fint   BTσˆ d is the internal force vector with σˆ 

as Cauchy stress vector, M σ is the matrix of Cauchy stress components, B , G are matrices of shape function derivatives. For a 2–D axi–symmetric problem, the displacement field and coordinate field are given as, u  ur

uz  , x   r T

z

T

(9)

Using Voigt notation, σ and ε can be written as, σ   rr  zz    rz  , ε   rr  zz   rz  T

T

(10)

For 2–D axisymmetric problem, the matrices B , G and M σ are defined as,   I ,r B  0

0

 I ,z

 I ,z  I ,r

I r 

  I ,r , G  0   0 T

0

 I ,z

0

 I ,z

0

 I ,r

9

I r 

T

0 

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT  rr   zr Mσ   0   0  0

 rz  zz 0 0 0

0 0

0 0

 rr  zr 0

 rz  zz 0

0  0  0   0    

3.2 Elasto–Plastic Constitutive Equations Downloaded by [Indian Institute of Technology Roorkee] at 22:33 20 June 2015

A micromechanics based model for porous materials showing the role of hydrostatic stress on plastic yield and void growth was developed by Gurson (1977a, b). An approximate yield function for a porous plastic material having void volume fraction f is given as,

   σ, α,  f , f     f 

2

  3q p  2   2 q1 f cosh  2   1   q1 f   0   2 f 

where p    13  tr  ζ  is the hydrostatic stress,  

 3 2  ζ : ζ

(11)

is the von–Mises equivalent

stress and  f is the flow stress of matrix material. The ζ and ζ represents the effective stress and stress deviator respectively, and are defined as, ζ  σ  α , ζ  σ  α

(12)

where, α is the back stress, σ and σ are Cauchy stress and its stress deviator respectively. The parameters q1 and q2 were introduced by Tvergaard (1981, 1982). The modification related to overall void growth due to nucleation of new voids and growth of existing voids were suggested by Tvergaard and Needleman (1984) and Chu and Needleman (1980).

10

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT An associative plastic–flow rule is assumed in the present model. In this model, the direction of plastic flow coincides with the growth of the yield surface. According to the normality condition of plastic flow, the plastic strain increment is expressed as,

dε pl  d

 σ

(13)


where, d is the positive plastic flow multiplier. A nonlinear evolution function proposed by Chaboche (1989) is used to describe the isotropic hardening.



 f  pl    y 0  Q 1  exp b pl 



(14)

where,  y 0 is the initial yield stress, Q corresponds to the amplitude of exponential function, b denotes coefficient of decay and 

pl

is the microscopic equivalent plastic strain.

The kinematic hardening behavior is taken into account by the following nonlinear evolution law (Armstrong and Frederick, 1966),

2 dα  C dε pl   d pl α 3

(15)

where, C and  are material dependent parameters referred as hardening and saturation parameters respectively. In von–Mises plasticity modeling, the yield function is obtained by neglecting the effect of porosity from GTN model. Thus, the resulting yield function can be obtained from equation (11) as,

   σ, α,  f     f 

2

   1  0 

(16)

4. CONTACT MODELING

11

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT In contact problems, constraints are expressed in terms of condition of impenetrability and tangential slip at the contact surface. The normal contact condition refers to the impenetration condition whereas the tangential slip refers to the friction or frictionless behavior at contact surface. In the present study, a simple and efficient algorithm is employed to impose the contact


constraints using the penalty approach (Wriggers, 2002). The contact surfaces are assumed frictionless. The NTS technique is employed for the modeling of large deformation. The penetration function g N associated with the node k on the contactor body is defined by the inequality,

g N   xk  x1   nˆ  0

(17)

where, nˆ denotes the normal to the segment 1–2 of target body, x k , x1 and x 2 are the spatial coordinate vectors corresponding to contactor node k , node 1 and node 2 of the target segment 1–2 ( Fig. 2). The spatial coordinate x can be defined as x  X  u where X is the material coordinate vector and u is the displacement vector. The traction vector at the point of contact is t   pN nc with normal contact pressure pN and unit normal vector n c to the contact surface  c . Thus, the standard contact condition g N  0,

pN  0, g N pN  0 is considered for the frictionless contact.

The length l , unit tangent vector tˆ and unit normal vector nˆ for the target segment 1–2 can be defined as,

l  x2  x1 ,

1 tˆ   x 2  x1  , l

nˆ  e3  tˆ

(18)

12

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT With these quantities, the relative position of contactor node k on the target segment 1–2 and the minimum distance g N between contactor node and target segment can be obtained as,

 

g N  xk  1    x1   x2   nˆ

1  xk  x1   tˆ , l

(19)


The contributions due to the contact constraints are added into the weak form of equilibrium equation. For an active contact surface  c , the contact constraints are obtained using penalty approach. The penalty term  c at the contact interface is obtained as,

c 

1 2



 N  g N  dc ,

N  0

2

c

(20)

where,  N is the penalty parameter and g N is the penetration function which defines the active constraints. Further, equation (20) can be written in terms of nodal displacement du (between contactor body & target body) and normal shape functions, N N as,

c 

1 T  du  N NT  N N N  du  2

(21)

The normal stiffness matrix at the contact interface can be obtained as,

K con  N NT  N N N

(22)

The normal shape functions N N may be expressed as N N   nˆ  nˆ  N . The shape functions at the contact interface N C are defined as,  NC    

1

0

 1   

0



0

0

1

0

 1   

0



   

(23)

5. NUMERICAL ANALYSIS

13

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT In this section, the results obtained through indentation process simulation using single and multiple indentations are presented. First, a convergence study is performed then the parameters evaluation procedure for material models is briefly discussed. Further, the procedure for the evaluation of material flow properties based on the results of multiple indentations is discussed.


5.1 Geometrical and Computational Model A schematic of an elastic–plastic half–space along with rigid spherical indenter and boundary conditions is depicted in Fig. 3. An axisymmetric two–dimensional formulation is employed. The spherical indenter is modeled as rigid surface using series of nodes at its periphery. The domain is discretized with the help of four node isoparametric quadrilateral elements and EFG nodes as shown in Fig. 4. The purpose of using EFG nodes under the indenter is to reduce the error which may be caused due to extensive distortion in case of regular finite elements. Nodes on the axis of symmetry are constrained in radial direction whereas the nodes on the bottom are constrained in axial direction. A very fine mesh is used under the indenter to ensure the numerical accuracy whereas a progressively coarser mesh is used away from the indenter to reduce the computational time. A displacement controlled technique is adopted to simulate the indentation process. The loading process of indentation is achieved by imposing a downward displacement (in negative z– direction) on the indenter. Subsequently, an unloading indentation process is achieved by applying an upward displacement on the indenter. The load computation is achieved by summing the reaction forces at the contact nodes on the specimen. The interface between rigid sphere and deformable specimen is assumed to be frictionless. Convergence study is carried out for two mesh sizes i.e. mesh–1 (135x155 nodes) and mesh–2 (192x215 nodes). In this analysis, the size

14

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT of transient elements is taken same as the size of elements in the EFG region. Hence, the size of transient elements for mesh-1 is more as compared to mesh-2. The results show that neraly same load–indentation depth curves are obtained as shown in Fig. 5 for both the mesh sizes. Thus, the size of transient elements has got negligible effect on the results obtained.


5.2 Parameters Input and Analysis The simulations are performed on a specimen made of Zr–Nb2.5 (pressure tube material, specimen ID „A‟, Sharma et al., 2011a) with a spherical indenter of 1.46 mm diameter. The properties of specimen (Northwood et al., 1975; Sharma et al., 2011a) and indenter (Sharma et al., 2011b) are as follows, For specimen (pressure tube material): Young‟s modulus ( Es ) = 97 GPa Poisson‟s ratio ( s ) = 0.341 Initial yield strength ( y 0 ) = 606 MPa Ultimate strength ( uts ) = 811 MPa For rigid indenter (Tungsten carbide): Young‟s modulus ( Ei ) =645 GPa Poisson‟s ratio ( i ) =0.26 The evolution equations of isotropic and kinematic hardening are used to model the material p n behavior. The flow curve is usually expressed by the well–known power law  t  K ( ) . A plot

between  t and  p is given in Sharma et al. (2011a) for Zr–Nb2.5. The values of the strength coefficient, K = 1266 MPa and strain hardening exponent, n = 0.132 are obtained using

15

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT regression analysis. The isotropic hardening behavior is modeled by a nonlinear evolution function in equation (14). Thus, it is necessary to obtain the parameters Q and b defined in the evolution equation (14). A number of data points are generated using power law then a best fit curve is obtained from Chaboche model using an initial yield stress of 606 MPa. The best fit


curve as shown in Fig. 6 is expressed by following equation,



 f  606  349 1  exp 25.51 pl 



(24)

On comparing equation (14) with equation (24), the values of Q  349 MPa and b  25.51 are obtained. Further, in order to model the kinematic hardening behavior of the material, it is necessary to evaluate the values of parameters C and  defined in the nonlinear evolution equation (15). The relations C   Q and   b are used to determine the values of C and  (Metzger, 2009). For small indentation depth, the results of the ball indentation simulation should agree with the prediction of classical Hertz theory. For pure elastic deformation, the relation between load and indentation depth, obtained from the classical Hertz theory, is expressed as (Johnson, 1985),

P

3 2 2D E  h  2 3

(25)

where, P is the normal load, D is the diameter of the indenter, h is the indentation depth and

E  is the equivalent Young‟s modulus expressed as, 1 1  s2 1  i2   E Es Ei

(26)

For sufficiently small deformation, the results obtained by coupled FE-EFG approach are compared with the analytical solution (Hertz theory, equation (25)) as shown in Fig. 7. A close

16

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT agreement between the simulation and analytical results validates the model and also verifies the suitability of the mesh used. 5.3 Numerical Analysis of Multiple Indentations and Property Extraction Automated ball indentation (ABI) is a semi–destructive technique used for the measurement of


mechanical properties of materials. It is based on multiple indentations by a spherical indenter at the same location of the specimen. A schematic representation of indentation profile in ABI is given in Fig. 8, where ht represents total indentation depth, h p plastic indentation depth, d t represents total indentation diameter and d p represents plastic indentation diameter. A typical load–indentation depth plot is shown in Fig. 9. From ABI test, a stress–strain curve is obtained using measured load–indentation depth data. The numerical simulations are performed to obtain the load–indentation depth response for both the plasticity models i.e. von–Mises and GTN. In the simulations, a very small displacement increment of 0.3 µm is given to the indenter in a single step so as to achieve the total depth of 0.1 mm with intermediate partial unloading. During unloading, the redistribution of stresses is assured through convergence check after each step of unloading. During simulations, it is observed that the partial unloading does not produce any effect on the distribution of equivalent plastic distribution. Thus, in case of von–Mises material model with kinematic hardening, the contour plots of equivalent plastic strain at initial and intermediate loading step are presented in Fig. 10 whereas Figure 11 presents a contour plot of equivalent plastic strain at the final load step. Figure 11 also shows that the plastic region is spread in the EFG region only. The load– indentation depth curves obtained using von–Mises material model for different hardening rules are compared with the experimental data in Fig. 12. From Fig. 12, it is observed that the results

17

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT obtained using kinematic hardening model are quite close to the experimental results as compared to other hardening models. In case of GTN model, the simulations are carried out assuming q1  1.5 and q2  1 with an initial porosity f  0.5% . Figure 13 shows a comparison of the load–indentation depth curves


obtained using various hardening rules for GTN model with the experimental results. From these plots, it is clear that the results obtained by kinematic hardening model are more close to the experimental results as compared to other hardening models. Finally, the results obtained using kinematic hardening rule for both the plasticity models are compared with the experimental results in Fig. 14. From these results, it is observed that the results obtained by von–Mises model using kinematic hardening rule are more close to the experimental results. The above load–indentation depth curves are further processed to determine the flow properties using the evolution equations (Haggag et al., 1990; Haggag, 1993). The last nine cycles of the load–indentation depth curves are used to extract the yield strength and ultimate strength of the material. 5.3.1 Calculation of yield strength From the spherical geometric configuration (Fig. 8), the total indentation diameter d t is obtained as (Haggag et al., 1993), dt  2 ht D  ht2

(27)

18

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT where, D is the indenter diameter and ht is the total depth of indentation computed for each loading cycle. Following the Meyer‟s relation, the data points from all loading cycles are fit by regression analysis as, P dt2  A  dt D 

m2

(28)


where, P is the applied indentation load, m is the Meyer‟s coefficient and A is a material parameter. Again, m and A are obtained by regression analysis using equation (28) for different values of dt D and P dt2 . The yield strength ( y ) of the material is calculated using the obtained value of A and is expressed by the relation given below,

 y  m A

(29)

where,  m is a material–constant which is equal to 0.2652 (Chatterjee et al., 2011). 5.3.2 Calculation of ultimate strength The ultimate strength  uts  of material is evaluated by the following expression (Haggag et al., 1993),

 uts  K (n / e)n

(30)

where, e  2.718 , K and n represent the strength coefficient and strain hardening exponent respectively which are determined through regression analysis using the following power law equation,

 t  K  p 

n

(31)

where,  t and  p are true stress and true plastic strain respectively. The equations used to determine  t and  p are given as,

19

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT  p  0.2  d p D 

(32)

 t  4P  d p2

(33)

where, the plastic indentation diameter d p and the constraint factor  are calculated from the


following relations (Haggag et al., 1993),



 

 2.735P Es1  Ei1 D hp2  0.25d p2 dp   hp2  0.25d p2  hp D 

 1.12    1.12   ln    max 



  

1

3

(34)

 1 1    27   27

(35)

where,

   p Es 0.43 t

(36)

 max  2.87* m

(37)

   max  1.12 ln  27 

(38)

In equation (37), a parameter  m is proportional to strain rate sensitivity of the specimen. The value of  m  1 is used for the calculation (Haggag et al., 1993). In equation (38), “ln” represents the natural logarithm. Based on the above mentioned procedure, the values of true stress and true plastic strain are obtained for each unloading cycle. A comparison of true stress–true plastic strain curves with the literature is given in Fig. 15 for von–Mises model with different hardening rules. It is observed that the results obtained using kinematic hardening model are close to the experimental results as

20

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT compared to other hardening models. Similar observations are made in case of GTN model as shown in Fig. 16. The results obtained using both the plasticity models for kinematic hardening rule are compared in Fig. 17. It is clear that the results obtained by von–Mises model using kinematic hardening rule are in good agreement with the experimental results. The complete


procedure followed for the calculation of flow properties from load–indentation depth curves is given in the flow charts (Fig. 18 and Fig. 19). The values of true stress and true strain presented in Fig. 15 and Fig. 16 are used to obtain the values of K and n using regression analysis. A comparison of the best fit values of K and n with the corresponding experimental values (Sharma et al., 2011a) is presented in Table 1. It is observed that the values obtained using kinematic hardening rule are found close to the experimental values for both von–Mises and GTN models. Using the equations (29) and (30), the values of yield strength and ultimate strength are calculated for both the plasticity models with different hardening rules. These calculated values are compared with the experimental results obtained by Automated Ball Indentation (ABI) test and conventional test (Sharma et al., 2011a), and are presented in Table 2 and Table 3 for von– Mises and GTN models respectively. From Table 2, it is observed that the results obtained using the kinematic hardening model are fund close to the experimental results obtained using ball indentation test whereas the results obtained using the isotropic hardening model are found close to the conventional test results. A similar conclusion is derived from Table 3 for the GTN model. 6. CONCLUSIONS In this study, a coupled FE–EFG approach is used to evaluate the material properties by simulating spherical ball indentation process. In the coupled approach, a small region under the

21

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT indenter is modelled by EFG method while the rest of the region is modelled by FEM. The transition region between FE and EFG regions is modeled by shape functions consisting of both FE and EFG shape functions. A penalty based NTS technique is employed to model the frictionless contact. Large deformation is modeled by updated Lagrangian approach whereas the


material nonlinearity is modeled using three different hardening rules. Load–indentation depth curves are used to extract the flow properties. Based on the present simulations, the following conclusions are drawn,  In the elastic regime, the results obtained by coupled approach are found quite close to the Hertz theory.  The load–depth curves obtained by coupled approach using two plasticity models and three hardening rules are found in good agreement with the literature.  Only a minor variation in the material behavior is observed when different hardening rules are used.  The results obtained using von–Mises material model with the kinematic hardening rule are found close to the experimental results obtained through ABI test.  The computed values of strength coefficient and strain hardening coefficient deviate marginally from the experimental values.  A small deviation is found in calculated values of yield strength and ultimate strength with the experimental results. The difference in the numerical results obtained through the reverse analysis is found less than 10%.

22

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT REFERENCES Armstrong, P. J., Frederick, C. O., A mathematical representation of the multiaxial Bauschinger effect, Central Electricity Generating Board, Report RD/B/N 731, 1966. Belytschko, T., Organ, D., Krongauz, Y., A coupled finite element–element–free Galerkin


method, Computational Mechanics, vol. 17, pp. 186–195, 1995. Cai, Y., Zhuang, X., Zhu, H., A generalized and efficient method for finite cover generation in the numerical manifold method, International Journal of Computational Methods, vol. 10, 1350028, 2013. Cao, Y.P., Lu, J., A new method to extract the plastic properties of metal materials from an instrumented spherical indentation loading curve, Acta Materialia, vol. 52, pp. 4023–4032, 2004. Chaboche, J. L., Constitutive equations for cyclic plasticity and cyclic viscoplasticity, International Journal of Plasticity, vol. 5, pp. 247–302, 1989. Chatterjee, S., Madhusoodanan, K., Singh, R.N., Panwar, S., Kayal, J.N., Nondestructive measurement of mechanical properties of Zr 2.5 wt% Nb pressure tube, 21st International Conference on Structural Mechanics in Reactor Technology (SMiRT 21), New Delhi, India, Div–VIII, 2011. Chen, X., Ogasawara, N., Zhao, M., Chiba, N., On the uniqueness of measuring elastoplastic properties from indentation: the indistinguishable mystical materials, Journal of the Mechanics and Physics of Solids, vol. 55, pp. 1618–1660, 2007. Chollacoop, N., Dao, M., Suresh, S., Depth–sensing instrumented indentation with dual sharp indenters, Acta Materialia, vol. 51, pp. 3713–3729, 2003.

23

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Chu, C. C., Needleman, A., Void nucleation effects in biaxially stretched sheets, Journal of Engineering Materials and Technology, vol. 102, pp. 249–256, 1980. Dao, M., Chollacoop, N., Van Vliet, K.J., Venkatesh, T.A., Suresh, S., Computational modeling of the forward and reverse problems in instrumented sharp indentation, Acta Materialia, vol.


49, pp. 3899–3918, 2001. Dias, A.M.S., Godoy, G.C.D., Determination of stress–strain curve through Berkovich indentation testing, Materials Science Forum, vols. 636–637, pp. 1186–1193, 2010. Doerner, M.F., Nix, W.D., A method for interpreting the data from depth–sensing indentation instruments, Journal of Materials Research, vol. 1, pp. 601–609, 1986. Follansbee, P.S., Sinclair, G.B., Quasi–static normal indentation of an elasto–plastic half–space by a rigid sphere – I. Analysis, International Journal of Solids and Structures, vol. 20, pp. 81–91, 1984. Giannakopoulos, A.E., Larsson, P. L., Vestergaard, R., Analysis of Vickers indentation, International Journal of Solids and Structures, vol. 31, pp. 2679–2708, 1994. Giannakopoulos, A.E., Suresh, S., Determination of elastoplastic properties by instrumented sharp indentation, Scripta Materialia, vol. 40, pp. 1191–1198, 1999. Gurson, A.L., Continuum theory of ductile rupture by void nucleation and growth: Part I–Yield criteria and flow rules for porous ductile media, Journal of Engineering Materials and Technology, vol. 99, pp. 2–15, 1977a. Gurson, A.L., Porous rigid–plastic materials containing rigid inclusions–Yield function, plastic potential, and void nucleation, Fracture, vol. 2, pp. 357–364, 1977b.

24

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Haggag, F.M., Nanstad, R.K., Estimating fracture toughness using tension or ball indentation tests and a modified critical strain model, Innovative Approaches to Irradiation Damage and Failure Analysis, vol. 170, pp. 41–46, American Society of Mechanical Engineers, New York, 1989.


Haggag, F.M., Nanstad, R.K., Hutton, J.T., Thomas, D.L., Swain, R.L., Use of automated ball indentation testing to measure flow properties and estimate fracture toughness in metallic materials, Applications of Automation Technology to Fatigue and Fracture Testing, ASTM STP 1092, pp. 188–208, 1990. Haggag, F.M., In–situ measurements of mechanical properties using novel automated ball indentation system, Small Specimen Test Techniques Applied to Nuclear Reactor Vessel Thermal Annealing and Plant Life Extension, ASTM STP 1204, pp. 27–44, 1993. Hardy, C., Baronet, C.N., Tordion, G.V., The elasto–plastic indentation of a half–space by a rigid sphere, International Journal for Numerical Methods in Engineering, vol. 3, pp. 451– 462, 1971. Johnson, K. L., Contact Mechanics, Cambridge University Press, 1985. Komvopoulos, K., Ye, N., Three–dimensional contact analysis of elastic–plastic layered media with fractal surface topographies, ASME Journal of Tribology, vol. 123, pp. 632–640, 2001. Knapp, J.A., Follstaedt, D.M., Myers, S.M., Barbour, J.C., Friedmann, T.A., Finite element modeling of nano–indentation, Journal of Applied Physics, vol. 85, pp. 1460–1474, 1999. Kumar, S., Singh, I.V., Mishra, B.K., A coupled finite element and element–free Galerkin approach for the simulation of stable crack growth in ductile materials, Theoretical and Applied Fracture Mechanics, vol. 70, pp. 49–58, 2014a.

25

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Kumar, S., Singh, I.V., Mishra, B.K., A multigrid coupled (FE–EFG) approach to simulate fatigue crack growth in heterogeneous materials, Theoretical and Applied Fracture Mechanics, vol. 72, pp. 121–135, 2014b. Lee, H., Lee, J.H., Pharr, G.M., A numerical approach to spherical indentation techniques for


material property evaluation, Journal of the Mechanics and Physics of Solids, vol. 53, pp. 2037–2069, 2005. Lee, J.H., Kim, T., Lee, H., A study on robust indentation techniques to evaluate elastic–plastic properties of metals, International Journal of Solids and Structures, vol. 47, pp. 647–664, 2010. Metzger, M., Improved implementation of an algorithm for nonlinear isotropic/kinematic hardening in pressure–dependent elastoplasticity, Master Thesis, 2009. Meyer, E., Contribution to the Knowledge of Hardness and Hardness Testing, Zeitschrift Des Vereines Deutscher Ingenieure, vol. 52, pp. 740–844, 1908. Nakamura, T., Wang, T., Sampath, S., Determination of properties graded materials by inverse analysis and instrumented indentation, Acta Materialia, vol. 48, pp. 4293–4306, 2000. Nayebi, A., Abdi, R.E., Bartier, O., Mauvoisin, G., New procedure to determine steel mechanical parameters from the spherical indentation technique, Mechanics of Materials, vol. 34, pp. 243–254, 2002. Northwood, D.O., London, I.M., Bahen, L.E., Elastic constants of zirconium alloys, Journal of Nuclear Materials, vol. 55, pp. 299–310, 1975.

26

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M., Meshless methods: A review and computer implementation aspects, Mathematics and Computers in Simulation, vol. 79(3), pp. 763–813, 2008. Oliver, W.C., Pharr, G.M., An improved technique for determining hardness and elastic modulus


using load and displacement sensing indentation, Journal of Materials Research, vol. 7, pp. 1564–1583, 1992. Pant, M., Singh, I.V., Mishra, B.K., A novel enrichment criterion for modeling kinked cracks using element free Galerkin method, International Journal of Mechanical Sciences, vol. 68, pp. 140-149, 2013. Pathak, H., Singh, A., Singh, I.V., Fatigue crack growth simulations of homogeneous and bimaterial interfacial cracks using element free Galerkin method, Applied Mathematical Modeling, vol. 38, pp. 3093–3123, 2014. Park, Y.J., Pharr, G.M., Nanoindentation with spherical indenters: finite element studies of deformation in the elastic–plastic transition regime, Thin Solid Films, vols. 447–448, pp. 246–250, 2004. Rabczuk, T., Belytschko, T., Xiao, S.P., Stable particle methods based on Lagrangian kernels, Computer Methods in Applied Mechanics and Engineering, vol. 193, pp. 1035–1063, 2004. Rabczuk, T., Eibl, J., Modeling dynamic failure of concrete with meshfree particle methods, International Journal of Impact Engineering, vol. 32, pp. 1878–1897, 2006. Rabczuk, T., Belytschko, T., A three dimensional large deformation meshfree method for arbitrary evolving cracks, Computer Methods in Applied Mechanics and Engineering, vol. 196, pp. 2777– 2799, 2007.

27

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Rabczuk, T., Areias, P.M.A., Belytschko, T., A simplified meshfree method for shear bands with cohesive surfaces, International Journal for Numerical Methods in Engineering, vol. 69, pp. 993–1021, 2007. Rabczuk, T., Xiao, S. P., Sauer, M., Coupling of meshfree methods with finite elements: Basic


concepts and test results, Communications in Numerical Methods in Engineering, vol. 22, pp. 1031–1065, 2006. Reddy, J. N., An introduction to nonlinear finite element analysis, Oxford University Press, pp. 327–353, 2009. Sharma, K., Singh, P.K., Bhasin, V., Vaze, K.K., Application of automated ball indentation for property measurement of degraded Zr2.5Nb, Journal of Minerals and Materials Characterization and Engineering, vol. 10, pp. 661–669, 2011a. Sharma, K., Bhasin, V., Vaze, K.K., Ghosh, A.K., Numerical simulation with finite element and artificial neural network of ball indentation for mechanical property estimation, Sadhana, vol. 36, pp. 181–192, 2011b. Singh, I.V., Mishra, B.K., Pant, M., An enrichment based new criterion for the simulation of multiple interacting cracks using element free Galerkin method, International Journal of Fracture, vol. 167(2), pp. 157-171, 2011. Shedbale, A.S., Singh, I.V., Mishra, B.K., A numerical comparison of plasticity models and hardening rules for modeling crack growth in ductile materials, Strength of Materials, 2014 (Accepted).

28

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Song, Z., Komvopoulos, K., Elastic–plastic spherical indentation: Deformation regimes, evolution of plasticity, and hardening effect, Mechanics of Materials, vol. 61, pp. 91–100, 2013. Suresh, S., Giannakopoulos, A.E., New method for estimating residual stresses by instrumented


sharp indentation, Acta Materialia, vol. 46, pp. 5755–5767, 1998. Tabor, D., The Hardness of Metals, Clarendon Press, Oxford, 1951. Tvergaard, V., Influence of voids on shear band instabilities under plane strain conditions, International Journal of Fracture, vol. 17, pp. 389–407, 1981. Tvergaard, V., On localization in ductile materials containing spherical voids, International Journal of Fracture, vol. 18, pp. 237–252, 1982. Tvergaard, V., Needleman, A., Analysis of the cup–cone fracture in a round tensile bar, Acta Metallurgica, vol. 32, pp. 157–169, 1984. Wriggers, P., Computational Contact Mechanics, John Wiley, New York, 2002. Ye, N., Komvopoulos, K., Indentation analysis of elastic–plastic homogeneous and layered media: criteria for determining the real material hardness, ASME Journal of Tribology, vol. 125, pp. 685–691, 2003. Zeng, K., Chiu, C.H., An analysis of load–penetration curves from instrumented indentation, Acta Materialia, vol. 49, pp. 3539–3551, 2001. Zhuang, X., Augarde, C., Mathisen, K., Fracture modelling using meshless methods and level sets in 3D: framework and modelling, International Journal for Numerical Methods in Engineering, vol. 92, pp. 969–998, 2012.

29

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Table 1: A comparison of calculated values of K and n with the input values

Type of hardening

Calculated

Calculated

(von–Mises)

(GTN)

K

n

K


Kinematic 1275.43 0.1318 1238.07

Input

n

K

n

0.1282

Mixed

1224.49 0.1268 1200.28

0.1249

Isotropic

1178.58 0.1232 1153.94

0.1209

30

1266

0.13 2

% error

% error

(von–Mises)

(GTN)

K

n

K

n

0.75

0.15 2.21 2.89

3.28

3.93 5.19 5.34

6.91

6.61 8.85 8.42

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Table 2: A comparison of  y and  uts obtained by coupled FE–EFG method with the reference values for von–Mises model

Calculated Type of

Experimental (ABI)

(MPa)


 uts

Kinematic

656.19

855.91

Mixed

631.21

830.47

Isotropic

607.76

804.93

Conventional

Absolute %

error

(MPa)

error

(MPa)

hardening

y

Absolute %

y

658

 uts

862

31

y

 uts

0.27

0.71

4.07

3.66

7.63

6.62

y

606

 uts

811

y

 uts

8.28

5.54

4.16

2.40

0.29

0.75

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT Table 3: A comparison of  y and  uts obtained by coupled FE–EFG method with the reference values for GTN model

Calculated Type of

Experimental (ABI)

(MPa)


 uts

Kinematic

640.82

836.99

Mixed

621.76

816.86

Isotropic

599.58

792.06

Conventional

Absolute %

error

(MPa)

error

(MPa)

hardening

y

Absolute %

y

658

 uts

862

32

y

 uts

2.61

2.90

5.51

5.24

8.88

8.11

y

606

 uts

811

y

 uts

5.75

3.21

2.60

0.72

1.06

2.34

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT


EFG node

FE node

Fig. 1: Coupling of FE-EFG using transition element

33

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

Contactor body


Target segment

Fig. 2: Modeling of contact constraints in normal direction

34

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT z Axis of symmetry


∆

ur = 0

Indenter

Specimen

r uz = 0 Fig. 3: A schematic of the specimen under spherical indenter along with the boundary conditions

35

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

Transition region


EFG region

FE region

Refined mesh region

Fig. 4: A typical discretized mesh used for spherical indentation modeling

36

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

700 Mesh 1 (135x155 nodes)

600

Mesh 2 (195x215 nodes)

Load (N)


500 400 300 200 100

0 0

0.01

0.02

0.03 0.04 0.05 Indentation depth (mm)

0.06

0.07

Fig. 5: A convergence study for two meshes

37

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 950

900

True stress (MPa)


850

800 pl

y = 606 + 349(1-e-25.51 )

750

700

Power law Chaboche model

650

600

0.01

0.02

0.03

0.04

0.05 0.06 0.07 True plastic strain

0.08

0.09

0.1

Fig. 6: Chaboche model fit for power law

38

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 7 6 Coupled FE-EFG

Load (N)


5

Hertz analysis

4 3 2 1 0 0

0.3

0.6

0.9 1.2 Indentation depth (µm)

1.5

1.8

2.1

Fig. 7: Load vs indentation depth plot in the elastic deformation regime

39

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

Material pile-up

Reference surface


Indentation profile after unloading Indentation profile during loading Fig. 8: Schematic representation of indentation process during loading and unloading

40

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

4th cycle


Load (P)

3rd cycle

2nd cycle 1st cycle

Indentation depth (h)

Fig. 9: Schematic representation of load-penetration depth for cyclic loading

41

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 0.7

1.4

0.6

1.2


0.5

1

0.4

0.8

0.3

0.6

0.2

0.4

0.1

0.2

0

0

(a)

(b)

Fig. 10: Equivalent plastic strain at (a) initial load step and (b) intermediate load step for von-Mises model with kinematic hardening

42

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

1.8 1.6


1.4 1.2

EFG region

1 0.8 0.6 0.4

Transition elements

0.2 0

Fig. 11: Equivalent plastic strain at final load step for von-Mises model with kinematic hardening

43

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 1600 Kinematic hardening 1400

Mixed hardening Isotropic hardening

1200

Experimental (Sharma et al., 2011)

Load (N)


1000 800 600 400 200 0 0

0.02

0.04

0.06

0.08

0.1

0.12

Indentation depth (mm)

Fig. 12: A comparison of load-indentation depth curves for von-Mises model with different hardening rules

44

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

1600 Kinematic hardening 1400

Mixed hardening Isotropic hardening

1200

Experimental (Sharma et al., 2011) Load (N)


1000 800 600 400 200 0 0

0.02

0.04

0.06

0.08

0.1

0.12

Indentation depth (mm)

Fig. 13: A comparison of load-indentation depth curves for GTN model with different hardening rules

45

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 1600 von-Mises (Kinematic hardening) 1400 GTN (Kinematic hardening) 1200


Load (N)


1000 800 600 400 200 0 0

0.02

0.04 0.06 0.08 Indentation depth (mm)

0.1

0.12

Fig. 14: A comparison of load-indentation depth curves obtained using kinematic hardening for von-Mises and GTN models

46

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 1400 1200

True stress (MPa)


1000 800 Kinematic hardening

600

Mixed hardening 400 Isotropic hardening 200


0 0.05

0.06

0.07

0.08

0.09

0.1

0.11

True plastic strain

Fig. 15: A comparison of true stress-true plastic strain curves for von-Mises model with different hardening rules

47

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 1400 1200

True stress (MPa)


1000 800 Kinematic hardening

600

Mixed hardening 400 Isotropic hardening 200 0 0.05


0.06

0.07

0.08

0.09

0.1

0.11

True plastic strain

Fig. 16: A comparison of true stress-true plastic strain curves for GTN model with different hardening rules

48

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

1400 1200

True stress (MPa)


1000 800 600

von-Mises (Kinematic hardening)

400

GTN (Kinematic hardening) Experimental (Sharma et al., 2011)

200 0 0.05

0.06

0.07

0.08

0.09

0.1

0.11

True plastic strain

Fig. 17: A comparison of true stress - true plastic strain curves obtained using kinematic hardening for von-Mises and GTN models

49

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

START

Input data


Determine

Store data as

No

Is required cycles reached…? Yes Determine , by regression analysis

Determine

Output data

END Fig. 18: Flow chart for calculation of

50

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT START Input data

Initial guess for


Calculate

, as

No Yes Calculate

Calculate

Store data as

Calculate

Calculate

Yes

No

Is required cycles reached…?

No

Determine , by regression analysis

Determine

Yes Output data

Fig. 19: Flow chart for the calculation of

51

END

ACCEPTED MANUSCRIPT