A boundary element method for solving 2-D and 3-D

Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907 www.elsevier.com/locate/cma

A boundary element method for solving 2-D and 3-D static gradient elastic problems Part II: Numerical implementation q K.G. Tsepoura b c

a,b

, S.V. Tsinopoulos a, D. Polyzos

a,b

, D.E. Beskos

c,*

a Department of Mechanical Engineering and Aeronautics, University of Patras, GR-26500 Patras, Greece Institute of Chemical Engineering and High Temperature Chemical Processes, FORTH, GR-26500 Patras, Greece Department of Civil Engineering, Structural Engineering Division, University of Patras, GR-26500 Patras, Greece

Received 28 February 2002; received in revised form 24 March 2003; accepted 24 March 2003

Abstract The boundary element formulation for the static analysis of two-dimensional (2-D) and three-dimensional (3-D) solids and structures characterized by a gradient elastic material behavior developed in the first part of this work, is treated numerically in this second part for the creation of a highly accurate and efficient boundary element solution tool. The discretization of the body is restricted only to its boundary and is accomplished by the use of quadratic isoparametric three-noded line and eight-noded quadrilateral boundary elements for the 2-D and 3-D cases, respectively. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Numerical examples involving a cylindrical bar in tension and a cylinder and a sphere in radial deformation are solved by the proposed boundary element method and the results are found in excellent agreement with the derived by the authors analytical solutions. The bar and sphere problems are solved in a 3-D context, while the cylinder problem is solved in a 2-D context (plane strain). Both the exterior and interior versions of the cylinder and sphere problems are considered. Ó 2003 Elsevier B.V. All rights reserved.

1. Introduction In the first part of this work [1], the boundary element formulation for the static analysis of twodimensional (2-D) and three-dimensional (3-D) solids and structures characterized by a linear gradient elastic material behavior was presented in detail. This formulation includes the boundary integral representation of the displacement and its normal derivative accompanied by all possible boundary conditions,

q

This paper is dedicated to Professor G. Maier on the occasion of his 70th birthday. Corresponding author. Tel.: +30-61-997-654; fax: +30-61-997-812. E-mail address: [email protected] (D.E. Beskos).

*

0045-7825/03/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0045-7825(03)00290-1

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classical and non-classical ones. It also includes the integral representation of the gradient of the displacement as well as the integral expressions for the strain and the stresses (total, Cauchy and double) in the interior of the body. All the necessary kernels in the above integral representations are given explicitly in the Appendices of the first part of this work [1]. The present second part of the work deals with the numerical implementation of the boundary element formulation developed in [1] and its application to representative 2-D and 3-D boundary value problems in order to demonstrate the use and applicability of the method and its high accuracy through comparisons with analytical solutions derived by the authors in this work. The discretization of the body is confined only to its boundary and is accomplished by the use of quadratic isoparametric three-noded line and eight-noded quadrilateral boundary elements for the 2-D and 3-D cases, respectively. Advanced algorithms are employed for the accurate and efficient numerical computation of the singular integrals involved. Numerical examples involving a bar in tension and a cylinder and a sphere in radial deformation are presented to illustrate the method and demonstrate its merits. The bar and the sphere problems are treated as 3-D, while the cylinder problem as a 2-D dimensional one. Both the exterior and interior versions of the cylinder and sphere problems are considered. The organization of this paper is as follows: Section 2 presents a brief presentation of the governing equations and boundary conditions (classical and non-classical) of the gradient elastostatic problem as well as its boundary integral representation. Section 3 describes the discretization of the boundary integral equations and the numerical computation of the singular integrals involved, while Section 4 describes analogous things for the case of the integral equations for strains and stresses in the interior of the elastic body. Numerical examples are presented in detail in Section 5 and the paper closes with a list of conclusions in Section 6. 2. Field equations, boundary conditions and integral representation of the problem This section starts with a brief overview of the linear gradient elasticity theory of Mindlin [2,3] for reasons of completeness. A comprehensive presentation of this theory can be found in Part I [1] of this double paper. Consider a 3-D gradient elastic body of volume V surrounded by a smooth surface S. According to MindlinÕs [2,3] theory in conjunction with the assumption of zero body forces acting on the body, the static equation of equilibrium reads ~Þ ¼ 0 r  ð~s  r  l ð1Þ accompanied by the classical boundary conditions uðxÞ ¼ u0 ; x 2 S1 ; PðxÞ ¼ P0 ; x 2 S2 ;

S1 [ S2 ¼ S

and the non-classical ones ou ¼ q0 ; x 2 S3 ; qðxÞ ¼ on ~^ RðxÞ ¼ ^ nl n ¼ R0 ; x 2 S4 ;

ð2Þ

ð3Þ S3 [ S4 ¼ S;

~ the third order tensor of double where ^ n is the normal unit vector on S, ~s the classical elastic stress tensor, l forces per unit area, P the external surface tractions, R the surface double stresses and P0 , u0 , R0 , and q0 prescribed values. Boundary conditions (2) and (3) are known as classical and non-classical boundary conditions, respectively. Considering the special case for which [4,5] ~s ¼ lðru þ urÞ þ kðr  uÞeI ; ð4Þ

K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

~ ¼ g2 r~s ¼ 2lg2 r~e þ kg2 ðrr  uÞeI ; l

2877

ð5Þ

the governing equation (1) in terms of displacements takes the form lr2 u þ ðk þ lÞrr  u  g2 ½lr2 u þ ðk þ lÞrr  u ¼ 0;

ð6Þ

where k and l are the Lame constants and g the characteristic length of a linear gradient elastic material. The above described boundary value problem for a 2-D or 3-D gradient elastic body with smooth surface S admits an integral representation of the form [1] Z Z e ðx; yÞ  uðyÞ dS þ e ðx; yÞ  qðyÞ dS cðxÞ  uðxÞ þ P R y y S S Z Z e ðx; yÞ  RðyÞ dS ; e ðx; yÞ  PðyÞ dSy þ ¼ Q ð7Þ U y S

S

Z

Z e e ðx; yÞ oP o R ðx; yÞ cðxÞ  qðxÞ þ  uðyÞ dSy þ  qðyÞ dSy onx onx S S Z Z e ðx; yÞ e ðx; yÞ oU oQ ¼  PðyÞ dSy þ  RðyÞ dSy ; onx onx S S

ð8Þ

where x is a field point and y a source point lying on the surface S and cðxÞ is a scalar jump coefficient taking the value 1/2 at the boundary and the value 1 when x is an interior point. The gradient elastostatic fundae and R e , Q e , which appear in the e as well as the remaining kernels P mental solution of the problem U boundary integral equations (7) and (8), are given explicitly in [1]. The evaluation of strains, Cauchy stresses, double stresses and total stresses is accomplished by taking higher order derivatives of Eq. (7) and using the constitutive equations (4) and (5), i.e. Z n o e ðx; yÞ  uðyÞ  r U e ðx; yÞ  PðyÞ dSy cðxÞ  rx uðxÞ þ rx P x S  Z  ouðyÞ e e ¼ rx Q ðx; yÞ  RðyÞ  rx R ðx; yÞ  dSy ony S I n o e ðx; yÞ  uðyÞ dC ; e ðx; yÞ  EðyÞ  rx E þ rx U ð9Þ y C

Z n o e ðU Þ ðx; yÞ  PðyÞ dSy e ðP Þ ðx; yÞ  uðyÞ  T T cðxÞ  ~sðxÞ þ S  Z  ðQ Þ e ðR Þ ðx; yÞ  ouðyÞ dSy e ¼ T ðx; yÞ  RðyÞ  T ony S I n o e ðE Þ ðx; yÞ  uðyÞ dCy ; e ðU Þ ðx; yÞ  EðyÞ  T þ T

ð10Þ

C

Z n o 2 e ðP Þ ðx; yÞ  uðyÞ  rx T e ðU Þ ðx; yÞ  PðyÞ dSy ~ rx T cðxÞ  lðxÞ þ g S  Z  2 e ðQ Þ ðx; yÞ  RðyÞ  rx T e ðR Þ ðx; yÞ  ouðyÞ dSy ¼g rx T ony S I n o e ðU Þ ðx; yÞ  EðyÞ  rx T e ðE Þ ðx; yÞ  uðyÞ dCy ; þ g2 rx T C

ð11Þ

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K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

cðxÞ  ~sðxÞ  g2

Z n o e ðP Þ ðx; yÞ  uðyÞ  r2 T e ðU Þ ðx; yÞ  PðyÞ dSy r2x T x S

¼ g

2

Z 

e ðQ Þ ðx; yÞ r2x T

S

 RðyÞ 

e ðR Þ ðx; yÞ r2x T

 ouðyÞ  dSy ony

I n o e ðU Þ ðx; yÞ  EðyÞ  r2 T e ðE Þ ðx; yÞ  uðyÞ dCy ; g r2x T x 2

ð12Þ

C

cðxÞ  e r ðxÞ þ ¼

Z  S

þ

Z n

 o e ðU Þ ðx; yÞ  g2 r2 T e ðP Þ ðx; yÞ  g2 r2 T e ðP Þ ðx; yÞ  uðyÞ  T e ðU Þ ðx; yÞ  PðyÞ dSy T x x

S

 ouðyÞ  ðQ Þ ðR Þ 2 2 e ðQ Þ 2 2 e ðR Þ e e T ðx; yÞ  g rx T ðx; yÞ  RðyÞ  T ðx; yÞ  g rx T ðx; yÞ  dSy ony

I n

 o e ðE Þ ðx; yÞ  g2 r2 T e ðU Þ ðx; yÞ  g2 r2 T e ðU Þ ðx; yÞ  EðyÞ  T e ðE Þ ðx; yÞ  uðyÞ dCy ; ð13Þ T x x

C

where all the kernels appearing in Eqs. (9)–(13) are explained and given in [1].

3. Boundary element formulation and direct numerical evaluation of singular and hypersingular integrals In this section, the boundary element formulation and solution procedure of both 3-D and 2-D static gradient elastic problems described by Eqs. (7) and (8) are presented in detail. 3.1. Three-dimensional case The goal of the boundary element methodology is to solve numerically the well-posed boundary value problem constituted by the system of the two integral equations (7) and (8) and the boundary conditions (2) and (3). To this end, the smooth surface S is discretized into E eight-noded quadrilateral and/or six-noded triangular quadratic continuous isoparametric boundary elements. For a nodal point k, the discretized integral equations (7) and (8) have the form AðeÞ Z 1 Z 1 E X X 1 e ðxk ; ye ðn ; n ÞÞN a ðn ; n ÞJ ðn ; n Þ dn dn  ue uðxk Þ þ P 1 2 1 2 1 2 1 2 a 2 1 1 e¼1 a¼1

þ

AðeÞ Z E X X

1

e¼1 a¼1

¼

AðeÞ Z E X X e¼1 a¼1

þ

1

1

1

e ðxk ; ye ðn ; n ÞÞN a ðn ; n ÞJ ðn ; n Þ dn dn  qe R 1 2 1 2 1 2 1 2 a

e ðxk ; ye ðn1 ; n2 ÞÞN a ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2  Pe U a

1 1

1

1

Z

1

AðeÞ Z E X X e¼1 a¼1

Z

1

Z

1

1

e ðxk ; ye ðn ; n ÞÞN a ðn ; n ÞJ ðn ; n Þ dn dn  Re ; Q 1 2 1 2 1 2 1 2 a

ð14Þ

K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

2879

AðeÞ Z 1 Z 1 E X e ðxk ; ye ðn ; n ÞÞ X 1 oP 1 2 qðxk Þ þ N a ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2  uea 2 on x 1 1 e¼1 a¼1 AðeÞ Z E X X

þ

1

e¼1 a¼1 AðeÞ Z E X X

¼

þ

e¼1 a¼1

1

1

e ðxk ; ye ðn ; n ÞÞ oR 1 2 N a ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2  qea onx

e ðxk ; ye ðn1 ; n2 ÞÞ oU N a ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2  Pea onx

1

1

AðeÞ Z E X X

1

1

Z

1

1

e¼1 a¼1

Z

1

Z

1

1

e ðxk ; ye ðn ; n ÞÞ oQ 1 2 N a ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2  Rea ; onx

ð15Þ

where AðeÞ is the number of nodes of the current element e (A ¼ 8 or 6 for quadrilateral or triangular elements, respectively), N a (a ¼ 1; 2; . . . ; A) the shape functions of a typical quadrilateral or triangular quadratic element, J the corresponding Jacobian of the transformation from the global ðX1 ; X2 ; X3 Þ to the local co-ordinate system ðn1 ; n2 Þ and uea , qea , pea and Rea are the nodal values of the corresponding field functions. Adopting now a global numbering for the nodes, each pair ðe; aÞ is associated to a number b and the integral equations (14) and (15) are written as L L L L X X X 1 k X e k  Pb þ e k  ub þ e k  qb ¼ e k  Rb ; u þ H K L G b b b b 2 b¼1 b¼1 b¼1 b¼1

ð16Þ

L L L L X X X 1 k X ~ k  ub þ e k  qb ¼ e k  Pb þ f q þ T V W kb  Rb ; S b b b 2 b¼1 b¼1 b¼1 b¼1

ð17Þ

where L is the total number of nodes and ek ¼ H b

ek ¼ K b

ek ¼ G b

ek ¼ L b

ek ¼ S b

Z

1

Z

1

Z

1

1

Z

1

Z

1

1

Z

1

1

1

1

Z

1

Z

1 1

1

Z

1

1

1

Z

1

1

e k e a P ðx ; y ðn1 ; n2 ÞÞN ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2

;

ð18Þ

e ðxk ; ye ðn ; n ÞÞN a ðn ; n ÞJ ðn ; n Þ dn dn R 1 2 1 2 1 2 1 2

;

ð19Þ

;

ð20Þ

;

ð21Þ

ðe;aÞ!b

ðe;aÞ!b

e ðxk ; ye ðn1 ; n2 ÞÞN a ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2 U

ðe;aÞ!b

e ðxk ; ye ðn ; n ÞÞN a ðn ; n ÞJ ðn ; n Þ dn dn Q 1 2 1 2 1 2 1 2

ðe;aÞ!b

e ðxk ; ye ðn ; n ÞÞ oP 1 2 a N ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2 onx

ðe;aÞ!b

;

ð22Þ

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K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

ek T b

Z

¼

1

Z

1

Z

ek ¼ V b

f W kb

1

1

Z

1

Z

¼

e ðxk ; ye ðn ; n ÞÞ oR 1 2 a N ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2 onx

1

1

1

Z

1

1

1

1

;

ð23Þ

;

ð24Þ

ðe;aÞ!b

e ðxk ; ye ðn1 ; n2 ÞÞ oU a N ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2 onx

ðe;aÞ!b

e ðxk ; ye ðn ; n ÞÞ oQ 1 2 a N ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2 onx

:

ð25Þ

ðe;aÞ!b

Collocating Eqs. (16) and (17) at all nodal points L, one obtains the following linear system of algebraic equations "

1e Iþ 2

e H

e K 1e e IþT 2

e S

#    e u  ¼ G e q V

   e P L  ; f R W

ð26Þ

e, T e, L e, K e, S e, G e, V e and f where matrices H W contain all the submatrices given by Eqs. (18)–(25), respectively. Applying the boundary conditions (2) and (3) and rearranging Eq. (26), one produces the final linear system of algebraic equations of the form e  X ¼ B; A

ð27Þ

where the vectors X and B contain all the unknown and known nodal components of the boundary fields, respectively. When b 6¼ k, integrals (18)–(25) are non-singular and can be easily computed numerically by Gauss quadrature, utilizing, as in the present work a 6  6 integration points scheme. In case b ¼ k, the integrals (20), (21) and (24) are also non-singular, while the remaining integrals (19), (25), (18), (23) and (22) become singular with the first two being weakly singular integrals of order Oð1=rÞ, the next two strongly singular (CPV) integrals of order Oð1=r2 Þ and the last one a hypersingular integral of order Oð1=r3 Þ. In the present work, the singular integrals are evaluated with high accuracy by applying a methodology for direct treatment of CPV and hypersingular integrals in a unified manner as proposed by Guiggiani et al. [6] and Huber et al. [7], respectively. According to this methodology, a local polar co-ordinate (q; h) system, centered at the singularity, is introduced and the aforementioned singular integrals (19), (25), (18), (23) and (22) are expressed in that system and take the form Z h2 Z q^ k e e ðq; hÞ dq dh; Kk ¼ K ð28Þ 0

h1

f W kk ¼

Z

h2

Z

Z

h2

Z

¼

Z

h2 h1

q^ 0

h1

ek T k

f W ðq; hÞ dq dh;

ð29Þ

0

h1

ek ¼ H k

q^

Z 0

q^

! Z h2 q^ðhÞ ~ hðhÞ e ~ H ðq; hÞ  hðhÞ ln dq dh þ dh; bðhÞ q h1

ð30Þ

! Z h2 q^ðhÞ ~tðhÞ e ðq; hÞ  ~tðhÞ ln T dq dh þ dh; q bðhÞ h1

ð31Þ

K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

ek ¼ S k

Z

h2

h1

Z 0

q^

2881

! " #) Z h2 ( q^ðhÞ 2~ sðhÞ 1~sðhÞ cðhÞ 1 2 1 e ~sðhÞ ln dq dh þ þ dh; S ðq; hÞ  2   ~sðhÞ bðhÞ q q ðbðhÞÞ2 q^ðhÞ h1 ð32Þ

~ ðq; hÞ are the integrands of the integrals (19), (25), (18), (23) e ðq; hÞ, f e ðq; hÞ, T e ðq; hÞ and S where K W ðq; hÞ, H and (22), respectively, expressed in polar coordinates, while all the remaining parameters are given in Appendix A. The obtained integrals (28)–(32) are now regular and can be easily computed by Gauss quadrature. In the present work, 6  6 integration points for the first four integrals and 8  8 integration points for the fifth integral are employed. 3.2. Two-dimensional case For a 2-D static gradient elastic problem described by the system of the two integral equations (7) and (8) and the boundary conditions (2) and (3), the smooth surface S is discretized into E three-noded quadratic isoparametric line boundary elements. For a nodal point k, the discretized integral equations (7) and (8) have the form AðeÞ Z 1 AðeÞ Z 1 E X E X X X 1 e e ðxk ; ye ðnÞÞN a ðnÞJ ðnÞ dn  qe k k e a e uðx Þ þ P ðx ; y ðnÞÞN ðnÞJ ðnÞ dn  ua þ R a 2 1 1 e¼1 a¼1 e¼1 a¼1 AðeÞ Z 1 AðeÞ Z 1 E X E X X X e ðxk ; ye ðnÞÞN a ðnÞJ ðnÞ dn  Re ; e ðxk ; ye ðnÞÞN a ðnÞJ ðnÞ dn  Pe þ ¼ Q ð33Þ U a a e¼1 a¼1

1

e¼1 a¼1

1

AðeÞ Z 1 AðeÞ Z 1 E X E X e ðxk ; ye ðnÞÞ e ðxk ; ye ðnÞÞ X X 1 oP oR k a e qðx Þ þ N ðnÞJ ðnÞ dn  ua þ N a ðnÞJ ðnÞ dn  qea 2 onx onx 1 1 e¼1 a¼1 e¼1 a¼1 AðeÞ Z 1 AðeÞ Z 1 E X E X e ðxk ; ye ðnÞÞ X X e ðxk ; ye ðnÞÞ oU oQ ¼ N a ðnÞJ ðnÞ dn  Pea þ N a ðnÞJ ðnÞ dn  Rea ; on on x x 1 1 e¼1 a¼1 e¼1 a¼1

ð34Þ where AðeÞ is the number of nodes of the current element e (A ¼ 3 for quadratic isoparametric line elements), N a ða ¼ 1; 2; 3Þ the shape functions of a typical quadratic isoparametric line element, J the corresponding Jacobian of the transformation from the global ðX1 ; X2 Þ to the local co-ordinate system n and uea , qea , pea and Rea are the nodal values of the corresponding field functions. Adopting a global numbering for the nodes, each pair ðe; aÞ is associated to a number b and the integral equations (33) and (34) are written as L L L L X X X 1 k X e k  Pb þ e k  ub þ e k  qb ¼ e k  Rb ; u þ H K L G b b b b 2 b¼1 b¼1 b¼1 b¼1

ð35Þ

L L L L X X X 1 k X e k  ub þ e k  qb ¼ e k  Pb þ f q þ T V W kb  Rb ; S b b b 2 b¼1 b¼1 b¼1 b¼1

ð36Þ

where L is the total number of nodes and Z 1 e k k e a e P ðx ; y ðnÞÞN ðnÞJ ðnÞ dn Hb ¼ 1

ðe;aÞ!b

;

ð37Þ

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K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

ek ¼ K b ek ¼ G b ek ¼ L b

ek S b

¼

e ðxk ; ye ðnÞÞN a ðnÞJ ðnÞ dn R

1

1

Z

1

Z

1

1

Z

e k e a Q ðx ; y ðnÞÞN ðnÞJ ðnÞ dn

¼

f W kb ¼

Z

1

1

Z

1

1

Z

1

1

ð38Þ

;

ð39Þ

;

ð40Þ

ðe;aÞ!b

ðe;aÞ!b

e ðxk ; ye ðnÞÞ oP a N ðnÞJ ðnÞ dn onx

1

; ðe;aÞ!b

e ðxk ; ye ðnÞÞN a ðnÞJ ðnÞ dn U

1

1

ek ¼ T b

ek V b

Z

;

ð41Þ

;

ð42Þ

;

ð43Þ

ðe;aÞ!b

e ðxk ; ye ðnÞÞ oR N a ðnÞJ ðnÞ dn onx

ðe;aÞ!b

e ðxk ; ye ðnÞÞ oU a N ðnÞJ ðnÞ dn onx

ðe;aÞ!b

e ðxk ; ye ðnÞÞ oQ N a ðnÞJ ðnÞ dn onx

:

ð44Þ

ðe;aÞ!b

Collocating Eqs. (35) and (36) at all nodal points L and applying the boundary conditions (2) and (3) one obtains the final linear system of algebraic equations of the form e  X ¼ B; A ð45Þ where the vectors X and B contain all the unknown and known nodal components of the boundary fields, respectively. When b 6¼ k, integrals (37)–(44) are non-singular and can be easily computed numerically by Gauss quadrature, utilizing, as in the present work a six integration points scheme. In case b ¼ k, the integrals (39), (40) and (43) are also non-singular, while the remaining integrals (38), (44), (37), (42) and (41) become singular with the first two being weakly singular integrals of order Oðln rÞ, the next two strongly singular (CPV) integrals of order Oð1=rÞ and the last one a hypersingular integral of order Oð1=r2 Þ. The singular integrals are evaluated with high accuracy by applying a methodology for direct treatment of CPV. This direct evaluation makes use of a limiting process in the singular part of the kernels and then a semianalytical integration is performed on a local co-ordinate system of the element, which has the origin at the singular point. More details about the numerical treatment of the strongly and hypersingular integrals for 2-D and 3-D case can be found in the work of Guiggiani [8]. The hypersingular integrals are evaluated by following essentially the same steps as for the 3-D case. According to the above methodology the aforementioned singular integrals (38), (44), (37), (42) and (41) take the form Z 1 ek ¼ e ðn; nÞ dn; K K ð46Þ k 1

f W kk ¼

Z

1 1

f W ðn; nÞ dn;

ð47Þ

K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

ek ¼ H k

ek T k

¼

ek ¼ S k

Z

1

1

Z

1

1

Z

1

1

2883

! Z 1 ~ðnÞ 1 h e dn; dn þ ~ hðnÞ H ðn; nÞ  jn  nj 1 jn  nj

ð48Þ

! Z 1 ~tðnÞ 1 e ðn; nÞ  dn; T dn þ ~tðnÞ jn  nj jn  nj 1

ð49Þ

! Z 1 Z 1 es 2 ðnÞ es 1 ðnÞ 1 1 e dn þ ~s2 ðnÞ  dn; dn þ ~s1 ðnÞ S ðn; nÞ  2 2 jn  nj jn  nj 1 jn  nj 1 jn  nj

ð50Þ

where the integrals in Eqs. (46) and hand side of the Eqs. (48)–(50) are R 1 (47) and the first integrals R 1 in the right 2 regular. The singular integrals 1 ð1=jn  njÞ dn and 1 ð1=jn  nj Þ dn are evaluated analytically. The values of these integrals as well as all the remaining parameters in Eqs. (48)–(50) are given in Appendix B. The obtained integrals (46)–(50) are now regular and can be easily computed by Gauss quadrature.

4. Integral formulations for interior displacements, strains and stresses and their numerical treatment Following the boundary element procedure described in the previous section, the boundary displacements u, tractions P, normal gradient of displacements q and double tractions R are numerically evaluated at all the boundary nodes. As soon as the boundary values u, P, q and R are known, the interior values of displacements can be easily obtained by Eq. (7) with the jump coefficient being cðxÞ ¼ 1. Thus, adopting the notation of Eq. (35), the displacement vector ui at any interior point xi is given by the relation uðxi Þ ¼

L X

e i  Pb  G b

b¼1

L X

e i  ub þ H b

b¼1

L X

e i  Rb  L b

b¼1

L X

e i  qb : K b

ð51Þ

b¼1

Similarly the evaluation of the internal strains is accomplished with the aid of Eq. (19) (with cðxÞ ¼ 1), which for an interior point xi is written in the form L X

rx uðxi Þ ¼

e i  Pb  rx G b

b¼1

L X b¼1

e i  ub þ rx H b

L X b¼1

e i  Rb  rx L b

L X

e i  qb ; rx K b

ð52Þ

b¼1

where ei ¼ rx H b

ek ¼ rx K b

ei ¼ rx G b

ei ¼ rx L b

Z

1

Z

1

Z

1

1

Z

1

Z

1

1

1

1 1

Z

1

Z

1

1 1

Z

1

1

e ðxi ; ye ðn ; n ÞÞN a ðn ; n ÞJ ðn ; n Þ dn dn rx P 1 2 1 2 1 2 1 2

;

ð53Þ

e i e a rx R ðx ; y ðn1 ; n2 ÞÞN ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2

;

ð54Þ

;

ð55Þ

ðe;aÞ!b

ðe;aÞ!b

e ðxi ; ye ðn1 ; n2 ÞÞN a ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2 rx U

ðe;aÞ!b

e i e a rx Q ðx ; y ðn1 ; n2 ÞÞN ðn1 ; n2 ÞJ ðn1 ; n2 Þ dn1 dn2

ðe;aÞ!b

ð56Þ

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for the 3-D case and Z 1 e i i e a e rx H b ¼ rx P ðx ; y ðnÞÞN ðnÞJ ðnÞ dn 1

Z

ei ¼ rx K b

1

Z

ei ¼ rx G b ei ¼ rx L b

1

1

1

Z

1

1

e ðxi ; ye ðnÞÞN a ðnÞJ ðnÞ dn rx R

;

ð57Þ

;

ð58Þ

;

ð59Þ

ðe;aÞ!b

ðe;aÞ!b

e ðxi ; ye ðnÞÞN a ðnÞJ ðnÞ dn rx U e i e a rx Q ðx ; y ðnÞÞN ðnÞJ ðnÞ dn

ðe;aÞ!b

ð60Þ ðe;aÞ!b

for the 2-D case. ~, relative stresses ~s and total stresses e Finally, the interior Cauchy stresses ~s, double stresses l r at the point xi are obtained from the integral equations (10)–(13), respectively, for cðxÞ ¼ 1 and written according to the notation adopted in the previous section as follows: ~sðxi Þ ¼

L X

e ðGi Þ  Pb  T b

b¼1

~ðx Þ ¼ g l

2

e ðHi Þ  ub þ T b

b¼1

( i

L X

L X

e ðGi Þ rx T b

( ~sðx Þ ¼ g

2

L X b¼1

e r ðxi Þ ¼

L h X

b

P 

L X

e ðHi Þ rx T b

e ðGi Þ r2x T b

b

P 

L X

L X

e ðKi Þ  qb ; T b

ð61Þ

b¼1

b

u þ

b¼1

L X

e ðLi Þ rx T b

b

R 

L X

b¼1

e ðHi Þ r2x T b

b

u þ

b¼1

L X

) e ðKi Þ rx T b

q

b

ð62Þ

;

b¼1

e ðLi Þ r2x T b

b

R 

b¼1

L X

) e ðKi Þ r2x T b

q

b

;

ð63Þ

b¼1

L h i i X b b e ðGi Þ  g2 r2 T e i e ðHi Þ  g2 r2 T e i T T b b x ðGb Þ  P  x ðHb Þ  u

b¼1

þ

e ðLi Þ  Rb  T b

b¼1

b¼1

i

L X

L h X

b¼1

i

b e ðLi Þ  g2 r2 T e i T b x ðLb Þ  R 

b¼1

L h X

i b e ðKi Þ  g2 r2 T e i T b x ðKb Þ  q :

ð64Þ

b¼1

5. Numerical examples In this section three groups of characteristic static gradient elastic problems with analytical solutions obtained by the authors are also solved numerically to illustrate the accuracy of the proposed 2-D and 3-D boundary element methodology. 5.1. Tension of a bar The first numerical example deals with a gradient elastic bar of length L subjected to a constant axial tensile stress. In this example zero axial strains at the two ends of the bar are considered to be the nonclassical boundary conditions. Thus, all the boundary conditions of the problem read

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2885

Fig. 1. Geometry of the thick solid cylinder in axial tension.

rðLÞ ¼ T0 ;

ð65Þ

qðxÞjx¼L ¼ 0;

ð66Þ

where u and r are the axial displacement and stress, respectively and T0 is the applied axial load at the two ends of the bar. The analytical solution of this one-dimensional problem has the form [9] uðxÞ ¼

T0 T0 g ðejxj=g  ejxj=g Þ jxj þ 2E coshðL=gÞ E

jxj 6 L;

ð67Þ

where E is the YoungÕs modulus. In order to demonstrate the accuracy of the proposed BEM, the above-described one-dimensional (1-D) problem has been solved utilizing a 3-D model. According to this model, the axial bar in tension is modeled by a thick solid cylinder of height 2L ¼ 2:4a and diameter D ¼ 8:4a, as shown in Fig. 1. Due to the restriction of the present numerical implementation to analyze gradient elastic bodies with smooth surfaces, the edges of the cylinder have been modeled as smooth surfaces with a curvature R ¼ 0:2a. The discretization consists of 268 quadratic quadrilateral boundary elements was restricted to one quarter of the cylinder because of symmetry. The problem has been solved for a ¼ 0:5 and T0 =E ¼ 1 and the axial displacement ux , the strain ex , Cauchy stress sxx , double stress lxxx and total stress rxx have been evaluated and displayed in Figs. 2–6 as functions of the distance x for different values of the material characteristic length g. As it is evident, the obtained numerical results are in an excellent agreement with the analytical ones derived by Eq. (67). 5.2. Radial deformation of a sphere In the present subsection two internal problems concerning the radial deformation of a solid sphere and two external problems dealing with the uniform deformation of a spherical cavity are numerically solved. Consider a solid sphere of radius a subjected to a radial displacement u0 (classical boundary condition), while the normal displacement gradient vanishes at the boundary (non-classical boundary condition), i.e., uðrÞjr¼a ¼ u0^r; qr ðrÞjr¼a

our ðrÞ ¼ ¼ 0; on r¼a

ð68Þ ð69Þ

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Fig. 2. Axial displacement ux of the thick solid axisymmetric cylinder versus distance x for various values of g.

Fig. 3. Axial strain ex of the thick solid cylinder versus distance x for various values of g.

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Fig. 4. Axial Cauchy stresses sxx of the thick solid cylinder versus distance x for various values of g.

Fig. 5. Double stresses lxxx of the thick solid cylinder versus distance x for various values of g.

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Fig. 6. Axial total stress rxx of the thick solid cylinder versus distance x for various values of g.

where ur is the radial displacement and r the distance from the center of the sphere. This problem can be easily solved analytically and its solution, as obtained by the present authors, has the form    coshðr=gÞ 2 sinhðr=gÞ ^r; ð70Þ þg u ¼ Ar þ C  g r2 r where A¼

2gu0 a coshða=gÞ  2g2 u0 sinhða=gÞ  u0 a2 sinhða=gÞ ; a½3ga coshða=gÞ þ 3g2 sinhða=gÞ þ a2 sinhða=gÞ

ð71Þ

C¼

u0 a2 : ½3ga coshða=gÞ þ 3g2 sinhða=gÞ þ a2 sinhða=gÞ

ð72Þ

The above problem has also been solved numerically by the present BEM for the values of u0 ¼ 1, a ¼ 1 and m ¼ 0. Due to the symmetry of the problem, only one octant of the sphere needs to be discretized utilizing 38 quadratic quadrilateral boundary elements. The radial displacement ur , the strain er as well as the double stresses lrrr as functions of the distance r for different values of the internal characteristic length g have been evaluated. As it is apparent in Figs. 7–9, the results are in excellent agreement with those obtained analytically by using Eq. (70). The same sphere is also subjected to an external pressure P0 radially applied, while the double stresses R vanish at the boundary, i.e. PðrÞjr¼a ¼ P0^r;

ð73Þ

RðrÞjr¼a ¼ 0:

ð74Þ

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2889

Fig. 7. Radial displacement ur versus radial distance of the solid sphere of radius a, for various values of g. The classical boundary condition is uðaÞ ¼ u0^r and the non-classical one qðaÞ ¼ 0.

Fig. 8. Radial strain er versus radial distance of the solid sphere of radius a, for various values of g. The classical boundary condition is uðaÞ ¼ u0^r and the non-classical one qðaÞ ¼ 0.

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Fig. 9. Double stresses lrrr versus radial distance of the solid sphere of radius a, for various values of g. The classical boundary condition is uðaÞ ¼ u0^r and the non-classical one qðaÞ ¼ 0.

This problem can be easily solved analytically and its solution has the form of Eq. (70) with A¼

P0 ð1  2mÞð1 þ mÞ ; Eð1 þ mÞ

C ¼ 0:

ð75Þ ð76Þ

For the boundary conditions (73) and (74) the analytical solution shows that the gradient elastic sphere exhibits an elastic behavior, independent of the material characteristic length. Assuming a ¼ 1, P0 =E ¼ 1 and m ¼ 0 and discretizing only one octant of the sphere the radial displacements as well as the radial strain for three values of g are evaluated. Both displacement and strain radial fields are depicted in Figs. 10 and 11, respectively, as functions of r and compared to the corresponding free of g analytical ones. As it is evident from Figs. 10 and 11 the agreement between numerical and analytical solutions is excellent. Next, consider the problem of a spherical cavity of radius a embedded into an infinite gradient elastic 3-D space and subjected to a radial displacement u0 . The classical boundary conditions of this problem are uðrÞjr¼a ¼ u0^r;

ð77Þ

uðrÞjr!1 ¼ 0

ð78Þ

and the non-classical is assumed to be our ðrÞ qr ðrÞjr¼a ¼ ¼ 0: on r¼a

ð79Þ

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2891

Fig. 10. Radial displacement ur versus radial distance of the solid sphere of radius a, for various values of g. The classical boundary condition is PðaÞ ¼ P0^r and the non-classical one RðaÞ ¼ 0.

Fig. 11. Radial strain er versus radial distance of the solid sphere of radius a, for various values of g. The classical boundary condition is PðaÞ ¼ P0^r and the non-classical one RðaÞ ¼ 0.

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The analytical solution of this problem, as obtained by the present authors, has the form   rffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p u ¼ B 2 þ DGðrÞ ^r; with GðrÞ ¼ K3 ðr=gÞ r 2ðr=gÞ 2

ð80Þ

and B¼

D¼

u0 a3 dG dr r¼a

2GðrÞjr¼a þ a dGðrÞ dr

;

ð81Þ

r¼a

2u0 : 2GðrÞjr¼a þ a dG dr r¼a

ð82Þ

The radial displacement and strain fields as well as the radial double stress of this boundary value problem obtained numerically via the proposed BEM for a ¼ 1, P0 =E ¼ 1, m ¼ 0 and various values of the length scale parameter g are presented in Figs. 12–14, respectively. Again, the agreement between numerical and analytical results is excellent. Finally, the same spherical cavity is subjected to an external pressure P0 radially applied at infinity, while the double stresses R vanish at the boundary. Thus, the boundary conditions of the problem read PðrÞjr¼a ¼ P0^r;

ð83Þ

RðrÞjr¼a ¼ 0:

ð84Þ

Fig. 12. Radial displacement ur versus radial distance of the spherical cavity of radius a, for various values of g. The classical boundary conditions are uðaÞ ¼ u0^r, uðrÞjr!1 ¼ 0 and the non-classical one qðaÞ ¼ 0.

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Fig. 13. Radial strain er versus radial distance of the spherical cavity of radius a, for various values of g. The classical boundary conditions are uðaÞ ¼ u0^r, uðrÞjr!1 ¼ 0 and the non-classical one qðaÞ ¼ 0.

Fig. 14. Double stresses lrrr versus radial distance of the spherical cavity of radius a, for various values of g. The classical boundary conditions are uðaÞ ¼ u0^r, uðrÞjr!1 ¼ 0 and the non-classical one qðaÞ ¼ 0.

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Fig. 15. Radial displacement ur versus radial distance of the spherical cavity of radius a, for various values of g. The classical boundary conditions are PðaÞ ¼ P0^r, uðrÞjr!1 ¼ 0 and the non-classical one RðaÞ ¼ 0.

This problem can be easily solved analytically and its solution has the form of Eq. (80) with B¼

P0 ð1  2mÞð1 þ mÞa3 ð6g3 þ 6g2 a þ 3ga2 þ a3 Þ ; 2E½3g3 ð3 þ 4mÞ þ 3g2 ð3 þ 4mÞa þ 3gð1 þ 2mÞa2 þ ð1 þ 2mÞa3 

D¼

pffiffiffiffiffiffiffi 6ea=g P0 ð1 þ 2mÞð1 þ mÞ g=ra4 pffiffiffiffiffiffiffi : Ep r=g½3g3 ð3 þ 4mÞ þ 3g2 ð3 þ 4mÞa þ 3gð1 þ 2mÞa2 þ ð1 þ 2mÞa3 

ð85Þ

ð86Þ

Assuming a ¼ 1, P0 =E ¼ 1 and m ¼ 0 the radial displacements, the radial strains and the radial double stresses versus r and for three values of g have been evaluated by the proposed BEM and the excellent agreement of the results with the corresponding analytical values is apparent in Figs. 15–17. 5.3. Radial deformation of a cylinder In this subsection, four problems dealing with the radial deformation of cylinders and cylindrical cavities are studied in order to further assess the accuracy of the proposed BEM in its 2-D version (conditions of plane strain). The first problem concerns a solid cylinder subjected to a radial displacement u0 (classical boundary condition), while the normal displacement gradient vanishes at the boundary (non-classical boundary condition), i.e., uðrÞjr¼a ¼ u0^r; qr ðrÞjr¼a ¼

our ðrÞ ¼ 0; on r¼a

ð87Þ ð88Þ

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2895

Fig. 16. Radial strain er versus radial distance of the spherical cavity of radius a, for various values of g. The classical boundary conditions are PðaÞ ¼ P0^r, uðrÞjr!1 ¼ 0 and the non-classical one RðaÞ ¼ 0.

Fig. 17. Double stresses lrrr versus radial distance of the spherical cavity of radius a, for various values of g. The classical boundary conditions are PðaÞ ¼ P0^r, uðrÞjr!1 ¼ 0 and the non-classical one RðaÞ ¼ 0.

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where ur is the radial displacement and r the distance from the center of the cylinder. This problem can be easily solved analytically and its solution, as obtained by the present authors, has the form u ¼ fAr þ CI1 ðr=gÞg^r;

ð89Þ

where A¼

u0 ½I0 ða=gÞ þ I2 ða=gÞ ; 2aI2 ða=gÞ

C¼

gu0 ; aI2 ða=gÞ

ð90Þ ð91Þ

while, In ðÞ is the modified Bessel function of the first kind and nth order. The above problem has also been solved numerically by the 2-D version of the proposed BEM for u0 ¼ 1, a ¼ 0:6 and m ¼ 0. The discretization of one fourth of the circular boundary involved 37 quadratic line elements. The radial displacement ur , the strain er and the double stresses lrrr as functions of the distance r for different values of the internal characteristic length g have been evaluated. The results, as it is evident in Figs. 18–20, are in excellent agreement with those obtained analytically by using Eq. (89) in conjunction with Eqs. (90) and (91). The same cylinder is subjected to an external radially applied pressure P0 , while the double stresses R vanish at the boundary. This means that the classical and non-classical boundary conditions of the problem are, respectively PðrÞjr¼a ¼ P0^r;

ð92Þ

RðrÞjr¼a ¼ 0:

ð93Þ

Fig. 18. Radial displacement ur versus radial distance of the solid cylinder of radius a, for various values of g. The classical boundary condition is uðaÞ ¼ u0^r and the non-classical one qðaÞ ¼ 0.

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2897

Fig. 19. Radial strain er versus radial distance of the solid cylinder of radius a, for various values of g. The classical boundary condition is uðaÞ ¼ u0^r and the non-classical one qðaÞ ¼ 0.

Fig. 20. Double stresses lrrr versus radial distance of the solid cylinder of radius a, for various values of g. The classical boundary condition is uðaÞ ¼ u0^r and the non-classical one qðaÞ ¼ 0.

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This problem can be easily solved analytically and its solution has the form of Eq. (89) with A¼

P0 ð1  2mÞð1 þ mÞ ; Eð1 þ mÞ

C ¼ 0:

ð94Þ ð95Þ

For the boundary conditions (92) and (93) the analytical solution shows that the gradient elastic cylinder exhibits an elastic behavior, independent of the material characteristic length. Assuming a ¼ 0:6, P0 =E ¼ 1 and m ¼ 0, the radial displacements as well as the radial strains for three values of g are evaluated. Both radial displacement and strain fields are depicted in Figs. 21 and 22, respectively, as functions of r and compared to the corresponding free of g analytical ones. As it is evident from these figures, the agreement between the solutions is excellent. Next, consider the problem of a circular cavity of radius a embedded into an infinitely extended gradient elastic space, which is subjected to an external radial displacement u0 . The classical boundary conditions of the problem are uðrÞjr¼a ¼ u0^r;

ð96Þ

uðrÞjr!1 ¼ 0

ð97Þ

and the non-classical one is assumed to be our ðrÞ ¼ 0: qr ðrÞjr¼a ¼ on r¼a

ð98Þ

Fig. 21. Radial displacement ur versus radial distance of the solid cylinder of radius a, for various values of g. The classical boundary condition is PðaÞ ¼ P0^r and the non-classical one RðaÞ ¼ 0.

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2899

Fig. 22. Radial strain er versus radial distance of the solid cylinder of radius a, for various values of g. The classical boundary condition is PðaÞ ¼ P0^r and the non-classical one RðaÞ ¼ 0.

The analytical solution of this problem, as obtained by the present authors, has the form   1 u ¼ B þ DK1 ðr=gÞ ^r; r

ð99Þ

where

  1 K2 ða=gÞ B ¼ u0 a 1 þ ; 2 K0 ða=gÞ

ð100Þ

gu0 ; aK0 ða=gÞ

ð101Þ

D¼

with Kn ðÞ being the modified Bessel function of the second kind and nth order. The radial displacement and strain fields as well as the radial double stresses of this boundary value problem taken numerically via the proposed BEM for a ¼ 0:6, P0 =E ¼ 1, m ¼ 0 and various values of the length scale parameter g are presented in Figs. 23–25, respectively. As it is evident the agreement between numerical and analytical results is excellent. Finally, the same cylindrical cavity is subjected to an external radially applied at infinity, pressure P0 , while the double stresses R vanish at the boundary. Thus, the boundary conditions of the problem read PðrÞjr¼a ¼ P0^r;

ð102Þ

RðrÞjr¼a ¼ 0:

ð103Þ

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Fig. 23. Radial displacement ur versus radial distance of the cylindrical cavity of radius a, for various values of g. The classical boundary conditions are uðaÞ ¼ u0^r, uðrÞjr!1 ¼ 0 and the non-classical one qðaÞ ¼ 0.

Fig. 24. Radial strain er versus radial distance of the cylindrical cavity of radius a, for various values of g. The classical boundary conditions are uðaÞ ¼ u0^r, uðrÞjr!1 ¼ 0 and the non-classical one qðaÞ ¼ 0.

K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

2901

Fig. 25. Double stresses lrrr versus radial distance of the cylindrical cavity of radius a, for various values of g. The classical boundary conditions are uðaÞ ¼ u0^r, uðrÞjr!1 ¼ 0 and the non-classical one qðaÞ ¼ 0.

Fig. 26. Radial displacement ur versus radial distance of the cylindrical cavity of radius a, for various values of g. The classical boundary conditions are PðrÞjr¼a ¼ P0^r, uðrÞjr!1 ¼ 0 and the non-classical one RðaÞ ¼ 0.

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Fig. 27. Radial strain er versus radial distance of the cylindrical cavity of radius a, for various values of g. The classical boundary conditions are PðaÞ ¼ P0^r, uðrÞjr!1 ¼ 0 and the non-classical one RðaÞ ¼ 0.

Fig. 28. Double stresses lrrr versus radial distance of the cylindrical cavity of radius a, for various values of g. The classical boundary conditions are PðaÞ ¼ P0^r, uðrÞjr!1 ¼ 0 and the non-classical one RðaÞ ¼ 0.

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2903

This problem can be easily solved analytically and its solution has the form of Eq. (99) with B¼

P0 ð1 þ 2mÞð1 þ mÞa4 ½3K1 ða=gÞ þ K3 ða=gÞ ; 2E½gð1 þ 3mÞaK0 ða=gÞ þ ð2g2 ð5 þ 7mÞ þ ð1 þ 3mÞa2 ÞK1 ða=gÞ

D¼

4g2 P0 ð1 þ 2mÞð1 þ mÞa : E½gð1 þ 3mÞaK0 ða=gÞ þ ð2g2 ð5 þ 7mÞ þ ð1 þ 3mÞa2 ÞK1 ða=gÞ

ð104Þ

ð105Þ

Assuming a ¼ 0:6, P0 =E ¼ 1 and m ¼ 0, the radial displacement and strain fields as well as the radial double stresses are depicted in Figs. 26–28 as functions of r and compared to the corresponding analytical ones. As it is evident from these figures the agreement between the solutions is excellent.

6. Conclusions A boundary element method for solving 2-D and 3-D static, gradient elastic problems has been developed and used to solve a number of boundary value problems in order to illustrate the method and demonstrate its merits. The discretization of the body is confined only to its surface and is accomplished with the aid of quadratic isoparametric line and surface boundary elements for 2-D and 3-D cases, respectively. The computation of the singular integrals involved is done with the aid of highly accurate advanced algorithms. The highly accuracy of the proposed boundary element method is demonstrated by the excellent agreement of the numerical solution with the analytical one obtained by the authors herein for a variety of 2-D and 3-D boundary value problems involving bars, cylinders and spheres.

Acknowledgements All the authors acknowledge with thanks the support of the Greek Institute of Governmental Scholarships (I.K.Y) through the program IKYDA 2002 (Scientific cooperation between the University of Patras, Greece and the Ruhr-University Bochum, Germany). The first and third authors also gratefully acknowledge the support of the Karatheodory program for basic research offered by the University of Patras. Appendix A Let n be the local coordinates of the singular point. After the transformation into polar coordinates (q; h), the Taylor expansions of all the involved functions are given as follows: n1 ¼ n1 þ q cos h;

ðA:1Þ

n2 ¼ n2 þ q sin h; "

# ox ox r ¼ yðn1 ; n2 Þ  xðn1 ; n2 Þ ¼ q cos h þ sin h on1 n¼n on2 n¼n 2 3 2 2 2 2 2 o x cos h o x o x sin h cos h sin h þ 5 þ Oðq3 Þ þ þ q2 4 2 2 on1 on2 n¼n 2 on1 on22 n¼n

2

n¼n

3

¼ qAðhÞ þ q BðhÞ þ Oðq Þ;

ðA:2Þ

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  ðA  BÞ rn ¼ jrjn ¼ qn A 1 þ nq þ Oðqnþ2 Þ; A2

n ¼ 1; 2; 3; . . .

A ¼ jAj;

ðA:3Þ ðA:4Þ

  r A B ðA  BÞ ^r ¼ ¼ þ q A þ Oðq2 Þ ¼ d0 þ qd1 þ Oðq2 Þ; r A A A3  1 1 1 ¼ þO ; r2 q2 A2 q  1 1 3ðA  BÞ 1 ¼  þ O : r3 q3 A3 q 2 A5 q Expansion of the interpolation functions " # oN a oN a a a N ðnÞ ¼ N ðnÞ þ q cos h þ sin h þ Oðq2 Þ ¼ N0a þ qN1a þ Oðq2 Þ: on1 n¼n on2 n¼n Expansion of the Jacobian " # oJ oJ JðnÞ ¼ JðnÞ þ q cos h þ sin h þ Oðq2 Þ ¼ J0 þ qJ1 þ Oðq2 Þ: on1 n¼n on2 n¼n

ðA:5Þ

ðA:6Þ

ðA:7Þ

ðA:8Þ

ðA:9Þ

Expanded form of the Jacobian magnitude J ¼ jJj ¼ jJ0 j þ q

ðJ0  J1 Þ þ Oðq2 Þ ¼ J0 þ qJ1 þ Oðq2 Þ: jJ0 j

Expanded form of the normal vector   J J0 J0 J1 ^ ny ¼ ¼ þ q n0y þ q^n1y þ Oðq2 Þ:  J0 2 þ Oðq2 Þ ¼ ^ J J0 J0 J0

ðA:10Þ

ðA:11Þ

Expanded form of the dot products ð^ nx  ^rÞ ¼ ð^ nx  d0 Þ þ qð^ nx  d1 Þ þ Oðq2 Þ ¼ ð^ nx  ^rÞ0 þ qð^nx  ^rÞ1 þ Oðq2 Þ;

ðA:12Þ

h i n0y  d1 Þ þ ð^ n1y  d0 Þ þ Oðq2 Þ ¼ qð^ny  ^rÞ1 þ Oðq2 Þ; ð^ ny  ^rÞ ¼ q ð^

ðA:13Þ

ny Þ ¼ ð^ nx  ^ n0y Þ þ qð^ nx  ^ n1y Þ þ Oðq2 Þ ¼ ð^nx  ^ny Þ0 þ qð^nx  ^ny Þ1 þ Oðq2 Þ; ð^ nx  ^

ðA:14Þ

rS  ^ ny ¼ r S  ^ n0y þ OðqÞ ¼ ðrS  ^ ny Þ0 þ OðqÞ;

ðA:15Þ

~ e ðq; hÞ ¼ tðhÞ þ Oð1Þ; T q

ðA:16Þ

K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

2905

~ e ðq; hÞ ¼ hðhÞ þ Oð1Þ; H q

ðA:17Þ

2 1 e ðq; hÞ ¼ ~sðhÞ þ ~sðhÞ þ Oð1Þ: S 2 q q

ðA:18Þ

Appendix B Let n be the local coordinate of the singular point. The Taylor expansions of all the involved functions are given as follows: d ¼ n  n;

ðB:1Þ

sgnðdÞ ¼ jdj=d; r ¼ yðnÞ  xðnÞ ¼

2 dx d2 x ðn  nÞ þ    ¼ dAðnÞ þ d2 BðnÞ þ Oðd3 Þ; ðn  nÞ þ dn n¼n 2 dn2 n¼n

  ðA  BÞ r ¼ jrj ¼ jdj A 1 þ nd þ Oðdnþ2 Þ; A2 n

n

n

n

n ¼ 1; 2; 3; . . . ;

ðB:2Þ

ðB:3Þ

A ¼ jAj;

ðB:4Þ

  r A B ðA  BÞ ^r ¼ ¼ sgnðdÞ þ jdj A þ Oðd2 Þ ¼ d0 þ dd1 þ Oðd2 Þ; r A A A3

ðB:5Þ

1 1 ¼ þ Oð1Þ; r dA

ðB:6Þ

1 1 2ðA  BÞ ¼ 2  þ Oð1Þ: 2 2 r dA4 dA Expansion of the interpolation functions dN a ðnÞ a a ðn  nÞ þ    ¼ N0a þ dN1a þ Oðd2 Þ: N ðnÞ ¼ N ðnÞ þ dn

ðB:7Þ

ðB:8Þ

n¼n

Expansion of the Jacobian JðnÞ ¼ JðnÞ þ d

d2 xðnÞ þ Oðd2 Þ ¼ J0 þ qJ1 þ Oðd2 Þ: dn2 n¼n

ðB:9Þ

Expanded form of the Jacobian magnitude J ¼ jJj ¼ jJ0 j þ d

ðJ0  J1 Þ þ Oðd2 Þ ¼ J0 þ dJ1 þ Oðd2 Þ: jJ0 j

Expanded form of the normal vector   J J0 J0 J1 ^ n0y þ d^n1y þ Oðd2 Þ: ny ¼ ¼ þ d  J0 2 þ Oðd2 Þ ¼ ^ J J0 J0 J0

ðB:10Þ

ðB:11Þ

2906

K.G. Tsepoura et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2875–2907

Expanded form of the dot products nx  d0 Þ þ dð^ nx  d1 Þ þ Oðd2 Þ ¼ ð^ nx  ^rÞ0 þ dð^nx  ^rÞ1 þ Oðd2 Þ; ð^ nx  ^rÞ ¼ ð^

ðB:12Þ

h i ð^ ny  ^rÞ ¼ d ð^n0y  d1 Þ þ ð^ n1y  d0 Þ þ Oðd2 Þ ¼ dð^ny  ^rÞ1 þ Oðd2 Þ;

ðB:13Þ

ð^ nx  ^ ny Þ ¼ ð^ nx  ^ n0y Þ þ dð^ nx  ^ n1y Þ þ Oðd2 Þ ¼ ð^nx  ^ny Þ0 þ dð^nx  ^ny Þ1 þ Oðd2 Þ;

ðB:14Þ

ny ¼ r S  ^ n0y þ OðdÞ ¼ ðrS  ^ ny Þ0 þ OðdÞ; rS  ^

ðB:15Þ

~ e ðn; nÞ ¼ tðnÞ þ Oð1Þ; T d

ðB:16Þ

~ e ðn; nÞ ¼ hðnÞ þ Oð1Þ; H d

ðB:17Þ

2 1 e ðn; nÞ ¼ ~sðnÞ þ ~sðnÞ þ Oð1Þ; S 2 d d Z 1 1n 1 ; dn ¼ ln 1  n 1 jn  nj

Z

1 1

1 jn  nj

2

dn ¼

2 ; 1  n2

ðB:18Þ

ðB:19Þ

ðB:20Þ

when jnj 6¼ 1 and Z

1 1

Z

1 1

1 dn ¼ sgnðdÞ ln j2J ðnÞj; jn  nj 1

1 c dn ¼   sgnðdÞ; 2 2 jn  nj ½J ðnÞ 2

ðB:21Þ

ðB:22Þ

when jnj ¼ 1, where c¼

1 ox o2 x  2 : 2 on on n¼n

ðB:23Þ

References [1] D. Polyzos, K.G. Tsepoura, S.V. Tsinopoulos, D.E. Beskos, A boundary element method for solving 2-D and 3-D static gradient elastic problems. Part I: Integral formulation, Comp. Methods Appl. Mech. Engrg. 192 (26–27) (2003) 2845–2873. [2] R.D. Mindlin, Microstructure in linear elasticity, Arch. Rat. Mech. Anal. 10 (1964) 51–78. [3] R.D. Mindlin, Second gradient of strain and surface tension in linear elasticity, Int. J. Solids Struct. 1 (1965) 417–438. [4] S. Altan, E.C. Aifantis, On the structure of the mode III crack-tip in gradient elasticity, Scripta Metall. Mater. 26 (1992) 319–324. [5] C.Q. Ru, E.C. Aifantis, A simple approach to solve boundary value problems in gradient elasticity, Acta Mechanica 101 (1993) 59– 68.

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[6] M. Guiggiani, A. Gigante, A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method, J. Appl. Mech. ASME 57 (1990) 906–915. [7] O. Huber, A. Lang, G. Kuhn, Evaluation of the stress tensor in 3D elastostatics by direct solving of hypersingular integrals, Comp. Mech. 12 (1993) 39–50. [8] M. Guiggiani, Formulation and numerical treatment of the boundary integral equations with hypersingular kernels, in: V. Sladek, J. Sladek (Eds.), Singular Integrals in Boundary Element Methods, Computational Mechanics Publications, Southampton, 1998, pp. 85–124 (Chapter 3). [9] K.G. Tsepoura, S. Papargyri-Beskou, D. Polyzos, D.E. Beskos, Static and dynamic analysis of a gradient elastic bar in tension, Arch. Appl. Mech. 72 (2002) 483–497.