A boundary element method for solving 2-D and 3-D static gradient elastic problems

A boundary element method for solving 2-D and 3-D static gradient elastic problems

Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873 www.elsevier.com/locate/cma A boundary element method for solving 2-D and 3-D static gradient...

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Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873 www.elsevier.com/locate/cma

A boundary element method for solving 2-D and 3-D static gradient elastic problems Part I: Integral formulation q D. Polyzos b c

a,b

, K.G. Tsepoura

a,b

, S.V. Tsinopoulos a, D.E. Beskos

c,*

a Department of Mechanical Engineering and Aeronautics, University of Patras, GR26500 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 A boundary element formulation is developed for the static analysis of two- and three-dimensional solids and structures characterized by a linear elastic material behavior taking into account microstructural effects. The simple gradient elastic theory of Aifantis expressed in the framework of MindlinÕs general theory is used to model this material behaviour. A variational statement is established to determine all possible classical and non-classical (due to gradient terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution for both two- and three-dimensional cases is explicitly derived and used to construct the boundary integral representation of the solution with the aid of the reciprocal integral identity especially established for the gradient elasticity considered here. It is found that for a well-posed boundary value problem, in addition to a boundary integral representation for the displacement, a second boundary integral representation for its normal derivative is also necessary. Explicit expressions for interior displacements and stresses in integral form are also presented. All the kernels in the integral equations are explicitly provided. Ó 2003 Elsevier B.V. All rights reserved.

1. Introduction The classical theory of linear elasticity cannot describe satisfactorily the mechanical behavior of linear elastic materials with microstructure, such as polymers, polycrystals or granular materials. In these materials microstructural effects are important and the state of stress has to be defined in a non-local manner. These microstructural effects can be successfully modelled in a macroscopic manner by using higher-order strain gradient, micropolar and couple stress theories. Among those who have developed such theories one can mention Mindlin and co-workers [1–3], Aifantis and co-workers [4–7] and Vardoulakis and co-workers [8,9] in connection with the higher-order strain q

The paper is dedicated to Prof. 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)00289-5

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gradient theories, Eringen and co-workers [10,11] in connection with the micropolar theories and Cosserat [12], Mindlin and Tiersten [13], Koiter [14] and Toupin [15] in connection with the couple-stress theories. From the above theories, the most general and comprehensive is the one due to Mindlin and co-workers [1– 3], while the simplest is the one due to Aifantis and co-workers [4–7]. During the last 15 years or so, a variety of boundary value problems of linear elasticity were solved analytically by employing gradient-elasticity theories of rather simple forms [4–11] and the microstructural effects on the solution were assessed. One can mention here, e.g., static problems dealing with dislocations, fracture mechanics, the half-space under various surface loads, a borehole under pressure, a bar under tension or a beam in bending [5–7,9,16–21] and dynamic problems dealing with wave propagation in beams and the half-space [20,22–25]. It was found that use of these non-classical theories may lead to the elimination of singularities or discontinuities of classical elasticity theory and the capturing of size effects and wave dispersion in cases where this was not possible in the classical elasticity framework. However, analytical methods of solution are restricted to problems of simple geometry and boundary conditions. Realistic engineering problems characterized by complicated geometry and boundary conditions can only be solved by numerical methods, such as the finite element method (FEM) or the boundary element method (BEM). Among the efforts made for the FEM solution of boundary value problems in elastostatics in the framework of strain-gradient elastic behaviour, one can mention the works of Herrmann [26] for the case of the couple-stress elasticity, Nakamura et al. [27], Huang and Liang [28] and Providas and Kattis [29] for the case of the micropolar/Cosserat elasticity, Teneketzis Tenek and Aifantis [30] for the case of the gradient theory of Aifantis and those of Shu et al. [31] and Amanatidou and Aravas [32] for the case of the simple gradient theories of Mindlin. In particular the work in [32] represents the most general and comprehensive treatment of strain-gradient elasticity by the FEM. However, all the above FEM works are restricted to two dimensions. The BEM has also been used for solving two-dimensional (2-D) microstructural elastostatic problems but only in the framework of the micropolar elasticity case. One can mention here the works of Dragos [33], Liang and Huang [34], Huang and Liang [35] and Sladek and Sladek [36]. In this work the BEM in its direct form is employed for the solution of two- and three-dimensional (3-D) elastostatic problems in the framework of a strain-gradient theory, which constitutes a combination of the theories of Mindlin [1–3] and Aifantis [5,6] in the sense that the latter is used in the general framework of the former theory. It can be considered as an extension and generalization of a previous work of some of the present authors (Tsepoura et al. [37]) on the same subject, which was dealing with 3-D problems and restricted to boundary displacements and stresses only. The present work considers both 2- and 3-D problems and provides explicit expressions for the computation of interior displacements, strains and stresses in addition to the boundary ones. The paper is organized as follows: Section 2 deals with the constitutive equations and the boundary conditions. The latter ones are obtained through a variational statement and consist of classical and non-classical ones. Section 3 presents the derivation of the fundamental solution of the problem, Section 4 derives the reciprocal integral identity for gradient elasticity, Section 5 presents the boundary integral representation of the gradient elastostatic problem and finally Section 6 provides integral representations for strains and stresses in the interior of a gradient elastic body. The paper closes with the conclusions and six Appendices, which provide some derivations and explicit expressions for all the kernels of the integral representations.

2. Constitutive equations and boundary conditions In this section, the equation of equilibrium and the corresponding boundary conditions that should be satisfied by any linear elastic material with microstructure described in the gradient elastic theory of Ai-

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fantis [5,6], are derived and presented in detail. Since AifantisÕ theory is a special case of MindlinÕs strain gradient theory [18], the derivation of both, the equation of equilibrium and the boundary conditions is accomplished by taking first the variation of the strain energy defined by Mindlin [1–3] and then inserting the constitutive equation proposed by Aifantis [5,6]. Consider a linear elastic body of volume V surrounded by a surface S. This body is characterized by a microstructure, which is modeled macroscopically by the gradient of the deformation. The geometry of this body is described with the aid of the unit normal vector ^n on S, and a Cartesian coordinate system with its origin located interior to V . According to MindlinÕs [1–3] strain gradient theory the stored strain energy depends upon both strain and strain gradient, i.e.,  Z  Z 321 . ~s : ~e þ ð~ U¼ ðsij eij þ lijk oi ejk Þ dV ; ð1Þ lÞ .. r~e dV ¼ V

V

where ~s and ~e are the classical second order symmetric stress and strain tensors, respectively, r is the ~ is a third order tensor with its 27 components lijk representing double forces per gradient operator and l unit area. The first subscript of lijk indicates the normal vector on the surface on which the double stresses act, while the other two have the same significance as the corresponding ones of the classical stress tensor sij . It should be noted that double stresses contribute only to the potential energy and to the boundary conditions of the problem without giving any resultant stress or couple vector at any surface of the studied gradient elastic body. Finally, the double and triple dots in Eq. (1) indicate dyad and triad inner products, respectively, according to the rule ða  bÞ : ðc  dÞ ¼ ðb  cÞða  dÞ; . ða  b  mÞ .. ðl  c  dÞ ¼ ðm  lÞðb  cÞða  dÞ;

ð2Þ

where a, b, c, d, m, l are vectors in three dimensions, while  denotes dyadic product and the symbol ðÞ means ða  b  cÞ321 ¼ c  b  a:

321

ð3Þ

Taking into account that ~e ¼ 12 ðru þ urÞ, the variation of the strain energy U of the body, given by Eq. (1) can be written in terms of the displacement vector u as Z . dU ¼ ð4Þ ½~s : rdu þ ð~ lÞ321 .. rrdu dV : V

Utilizing the identities [38] . ~ÞT : ru þ ð~ r  ½ð~ lÞ132 : ru ¼ ðr  l lÞ321 .. rru; ~Þ  u ¼ ½r  ðr  l ~Þ  u þ ðr  l ~ÞT : ru; r  ½ðr  l r  ð~s  uÞ ¼ ðr  ~sÞ  u þ ~s : ru; with T denoting transposition, and the symmetry relation ~; lijk ¼ likj or ð~ lÞ132 ¼ l expression (4) takes the form Z   ~Þ  du ½r  ð~s r  l ~Þ  du þ r  ½~ dU ¼ r  ½ð~s r  l l : rðduÞ dV : V

ð5Þ

ð6Þ

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Employment of the divergence theorem transforms Eq. (6) into Z Z Z ~Þ  du dS þ ^n  l ~ : rðduÞ dS: ~Þ  du dV þ ^ dU ¼ ½r  ð~s r  l n  ð~s r  l V

S

ð7Þ

S

However, as it is mentioned in Mindlin [1], the last integral of Eq. (7) contains the function rðduÞ, which is not independent of du on S. Only its normal component ^n  rðduÞ is independent of du on S. Splitting the gradient operator into tangential and normal parts on the surface S, the last integral of Eq. (7) can be written as    Z Z  o ^ ^ ~ : rðduÞ dS ¼ ~ : rS þ ^ nl nl ðduÞ dS; ð8Þ n on S S or Z

^ ~ : rðduÞ dS ¼ nl

Z

S

~^ ð^ nl nÞ½^ n  rðduÞ dS þ S

Z

~Þ : rS ðduÞ dS; ð^n  l

ð9Þ

S

where rS is the surface gradient defined as n^ nÞ  r; rS ¼ ðeI ^

ð10Þ

with eI denoting the unit tensor. With the aid of the identities [38] T

T

~Þ  du ¼ ½rS  ð^ ~Þ  du þ ð^ ~Þ : rS ðduÞ; rS  ½ð^ nl nl nl ~þ^ ~Þ ¼ ðrS ^ nÞ : l n  ½rS  ð~ nl lÞ213 ; rS  ð^

ð11Þ

~, the last integral of Eq. (9) receives the ~ÞT ¼ ^n  l and recalling the symmetry relation (5), which means ð^n  l form Z Z n h i o ~ þ ^n  ½rS  ð~ ~Þ : rS ðduÞ dS ¼ ~Þ  du ðrS ^nÞ : l ð^ nl rS  ½ð^ nl lÞ213  du dS: ð12Þ S

S

However, as it is proved in Appendix A(a), the first term of the integrand of the right hand side of Eq. (12) can be expressed as ~  du: ~Þ  du ¼ ^ ~  duÞ þ ½ðrS ^nÞð^n  ^nÞ : l rS  ½ð^ ð13Þ nl n  rS ½^ n ð^ nl R ~  duÞ dS vanishes when the surface Also, as it is proved in Appendix A(b), the integral S ^n  rS ½^n ð^n  l S is smooth, while it obtains the form Z XI ^ ^  ^nÞ : l ~  duÞ dS ¼ ~k  dug dC; n  rS ½^ n ð^ nl fkðm ð14Þ S

Ca

Ca

for non-smooth boundaries, where Ca are the edge lines formed by the intersection of two surface portions ^ ¼ ^s ^ S1 and S2 of S, m n with ^s being the tangential vector to Ca , and the brackets k  k indicate that the enclosed quantity is the difference between the values on the surface portions S1 and S2 . For 2-D problems R ~  duÞ dS is always equal to zero (Appendix A(c)). n  rS ½^ the integral S ^ n ð^ nl Inserting Eqs. (13) and (14) into (12) and the resulting integral into Eq. (9), the variation of the strain energy given by Eq. (7) takes eventually the form

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

dU ¼

Z Z

~Þ  du dV þ ½r  ð~s r  l

V

Z S

~  ^nÞ  ½^n  rðduÞ dS ð^ nl !

o~ l 213 ~Þ ^n  ½rS  ð~ ^ n  ðrS  l lÞ  du dS on S Z XI ^ ^ ^ ^  ^nÞ : l ~ ~ ~k  dug dC: þ ððrS  nÞð^ n  nÞ : l ðrS nÞ : lÞ  du dS þ fkðm

þ

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^ n  ~s ð^ n^ nÞ :

S

Ca

ð15Þ

Ca

The variation of the work done by the external forces in V is due to body forces f acting on the body, as well as to external surface tractions P, surface double stresses R and surface jump stresses E acting on its surface and reads [1–3] Z Z Z XI dW ¼ f  du dV þ R  ½^ n  rðduÞ dS þ P  du dS þ fE  dug dC: ð16Þ V

S

S

Ca

Ca

In view of the fact that dU ¼ dW , relations (15) and (16) imply that the equation of equilibrium for a 2-D or 3-D gradient elastic body is ~Þ þ f ¼ 0; r  ð~s r  l

ð17Þ

accompanied by the classical boundary conditions PðxÞ ¼ ^ n  ~s ð^ n^ nÞ :

o~ l 213 ~ ¼ P0 ; ~Þ ^ ~ ðrS ^nÞ : l ^ n  ðrS  l n  ½rS  ð~ lÞ þ ðrS  ^nÞð^n  ^nÞ : l on ð18Þ

and/or u ¼ u0 ; and the non-classical ones ~^ R¼^ nl n ¼ R0 and=or

ou ¼ q0 ; on

ð19Þ

^ ^ ~k ¼ E0 ; E ¼ kðm nÞ : l where P0 , u0 , R0 , q0 and E0 denote prescribed values. Mindlin [2], considering isotropic materials and a special case of his theory where the macroscopic strain coincides to microdeformation, proposed a modification of HookeÕs law expressed by the following relations ~ ¼ ~s þ ~s; r ~s ¼ 2l~e þ kðr   uÞeI ; ~e ¼ ðr uþ urÞ=2;

ð20Þ

~s ¼ ½2lc3 r2~e þ kc1eI r2 ðr   uÞ þ kc2 rrðr  uÞ ; ~ is the total stress tensor, ~s and ~s are the so-called by Mindlin, Cauchy stress where r2 is the Laplacian, r tensor and relative stress tensor, respectively, and ~e is the strain tensor. The total stresses are correlated to strains and strain gradients through five independent material constants, i.e., k, l, c1 , c2 and c3 with the first two being the well-known LameÕ constants. A simpler and mathematically more tractable constitutive equation is that proposed by Aifantis and co~ with the relative stresses ~s according to the relations workers ([4–6]) and correlates the double stress tensor l

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~ ¼ g2 r~s; l ~s ¼ r  l ~ ¼ g2 r2~s;

ð21Þ

where g2 is the volumetric strain gradient energy coefficient, the only constant which relates the microstructure with the macrostructure. It is easy to see [18] that this simple theory can be obtained as a special case of that of Mindlin if one sets c1 ¼ c2 ¼ g2 and c3 ¼ 0 in Eq. (20). Adopting the above simple theory of Aifantis and inserting the constitutive Eq. (21) into Eq. (17) one obtains the following equation of motion of a gradient elastic continuum in terms of the displacement field  u: u þ ðk þ lÞrr   u g2 r2 ðlr2  u þ ðk þ lÞrr  uÞ þ f ¼ 0: lr2 

ð22Þ

3. 2-D and 3-D gradient elastic fundamental solutions In this section the 2-D and 3-D static fundamental solutions of an infinitely extended gradient elastic material are explicitly derived. These fundamental solutions are defined as the solution of the partial differential equation I~ u ðrÞ ¼ dðx yÞeI ;

ð23Þ

where d is the Dirac d-function, x is the point where the displacement field ~u due to a unit force applied at a point y should be obtained, r ¼ jx yj and I is the linear operator I  lr2 þ ðk þ lÞrr  g2 r2 ðlr2 þ ðk þ lÞrrÞ:

ð24Þ

According to the Helmholtz decomposition applied to dyadic fields [39], the fundamental solution ~u ðrÞ can be decomposed into irrotational and solenoidal parts as e ðrÞ; ~ u ðrÞ ¼ rruðrÞ þ rr AðrÞ þ r r G

ð25Þ

e ðrÞ a dyadic function. where uðrÞ is a scalar function, AðrÞ a vector function and G Substituting Eq. (25) into Eq. (23) and taking into account the relation r2 ðcðrÞÞ ¼ dðrÞ;

ð26Þ

where 8 1 1 > < ln cðrÞ ¼ 2p r > : 1 4pr

for 2-D; ð27Þ for 3-D;

and the identity r2 ¼ grad div rot rot:

ð28Þ

Eq. (23) takes the form rr½ðk þ 2lÞðr2 uðrÞ g2 r4 uðrÞÞ þ rr ½ðk þ 2lÞðr2 AðrÞ g2 r4 AðrÞÞ

h  i e ðrÞ g2 r4 G e ðrÞ ¼ rrðcðrÞÞ r r ðcðrÞeI Þ: þ r r l r2 G

ð29Þ

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e ðrÞ, respectively, as well as the Taking into account the irrotational and solenoidal nature of uðrÞ and G fact that AðrÞ is a vector of r, it is easy to see one that Eq. (29) is satisfied identically if AðrÞ ¼ 0 and uðrÞ, e ðrÞ are solutions of the equations G ðk þ 2lÞ½r2 uðrÞ g2 r4 uðrÞ ¼ cðrÞ; h i e ðrÞ g2 r4 G e ðrÞ ¼ cðrÞeI : l r2 G

ð30Þ ð31Þ

e ðrÞ functions, which satisfy Eqs. (30) and (31), respectively, For a 3-D problem, the scalar uðrÞ and tensor G have the form (Appendix B)   r=g 1 r g2 C1 2e ð32Þ uðrÞ ¼ þ g þ ; 4pðk þ 2lÞ 2 r r r e ðrÞ ¼ 1 G 4pl



 r=g r g2 2e eI C2 eI ; þ g 2 r r r

ð33Þ

where C1 and C2 are constants both equal to zero, since for g2 ¼ 0, u and G should give, via Eq. (25), the classical fundamental solution. e ðrÞ obtain the form For a 2-D problem, the scalar uðrÞ and the tensor G  2   1 r r uðrÞ ¼ ðln r 1Þ þ g2 ln r þ g2 K0 ; ð34Þ 2pðk þ 2lÞ 4 g  2   e ðrÞ ¼ 1 r ðln r 1Þ þ g2 ln r þ g2 K0 r eI ; G 2pl 4 g

ð35Þ

where K0 ðÞ is the modified Bessel of the second kind and zero order. Inserting Eqs. (32)–(35) into Eq. (25) and considering A ¼ 0, the fundamental solution of Eq. (23) takes the final form h i 1 ~ Wðr; m; gÞeI X ðr; gÞ^r  ^r ; ð36Þ u ðr; l; m; gÞ ¼ 16plð1 mÞ where m is the Poisson ratio, ^r the unit vector in the direction r ¼ y x and X , W scalar functions given by the relations   2 1 6g2 6g 6g 2 r=g X ðr; gÞ ¼ þ 3 ð37Þ e ; þ 2þ r r r r r3   2   1 g2 g g r=g Wðr; m; gÞ ¼ ð3 4mÞ þ 2ð1 2mÞ 3 þ þ e r r r3 r2   2  2  g g g 1 r=g þ 4ð1 mÞ 3 e þ 2þ ; 3 r r r r for three dimensions (3-D) and   8g2 r X ðr; gÞ ¼ 2 þ 2 4K2 ; g r Wðr; m; gÞ ¼ 2ð3 4mÞ ln r þ

    4g2 r r 2ð3 4mÞK 2K ; 0 2 2 g g r

ð38Þ

ð39Þ

ð40Þ

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for two dimensions (2-D) with K0 ðÞ and K2 ðÞ being the modified Bessel functions of the second kind and zero and second order, respectively. For the gradient coefficient g being equal to zero, one can easily prove that ( 2 for 2-D; 1 X ðrÞ ¼ ð41Þ for 3-D; r ( 2ð3 4mÞ ln r for 2-D; 1 ð42Þ Wðr; mÞ ¼ for 3-D; ð3 4mÞ r which are the expressions of the classical elastostatic fundamental solution [40]. Utilizing the expansions r r2 r3 r4 e r=g ¼ 1 þ þ ; 2 3 g 2!g 3!g 4!g4 " #   n 2 n n r p 2 2þn ð1=gÞ r2 n 2 ð1=gÞ r Kn þ þ  ¼ g 2 sinðnpÞ C½1 n

C½2 n

# " n 2þn p 2 n ð1=gÞ 2 2 n ð1=gÞ r2 n r  ; þ 2 sinðnpÞ C½1 þ n

C½2 þ n

ð43Þ

where C is the gamma function, it is easy to prove that both functions X and W given by relations (37), (39), and (38), (40), respectively, are regular with respect to the distance r ! 0 according to the asymptotic relations X ðrÞ ¼ Oðln rÞ; Wðr; mÞ ¼ Oð1Þ for 2-D case; X ðrÞ ¼ OðrÞ; Wðr; mÞ ¼ Oð1Þ for 3-D case:

ð44Þ

4. Reciprocal integral identity for gradient elasticity It is well known that in the framework of linear elastic theory, BettiÕs reciprocal identity [40] is an essential integral relation for the derivation of integral representations of linear elastic boundary value problems. The goal of the present section is the analytical derivation of a new reciprocal identity valid for the present gradient elastic case. Consider the vector ~  u r ~  u; w¼r

ð45Þ

~ and u are the total stress tensor and the displacement field of a gradient elastic continuum body of where r volume V and surface S, respectively, and ð~ r; uÞ, ð~ r ; u Þ are two deformation states of the same body. In view of Eqs. (20a) and (21a), the divergence of w can be written as ~Þ  u ð~s r  l ~ Þ  u : r  w ¼ r  ½ð~s r  l

ð46Þ

Recalling the identity r  ½~s  u ¼ ðr  ~sÞ  u þ ~s : ru

ð47Þ

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Eq. (46) obtains the form ~Þ : ru ð~s r  l ~ Þ : ru: r  w ¼ ½r  ð~s þ ~sÞ  u ½r  ð~s þ ~s Þ  u þ ð~s r  l

ð48Þ

However, as it is proved in Appendix C(a) ~s : ru ~s : ru ¼ 0:

ð49Þ

Thus, relation (48) takes the form ~ Þ  u ðr  l ~Þ : ru þ ðr  l ~ Þ : ru: ~Þ  u ½r  ð~s r  l r  w ¼ ½r  ð~s r  l The Gauss divergence theorem reads Z Z ^ n  w dS; r  w dV ¼ V

ð50Þ

ð51Þ

S

and in view of Eqs. (45) and (50), takes the form Z Z ~Þ  u ½r  ð~s r  l ~ Þ  ug dV fðr  l ~Þ : ru ðr  l ~ Þ : rug dV f½r  ð~s r  l V V Z    ~Þ  u ½^ ~ Þ  ug dS: ¼ f½^ n  ð~s r  l n  ð~s r  l

ð52Þ

S

Taking into account that both fields ð~ r; uÞ and ð~ r ; u Þ satisfy the equilibrium equation (17) with body forces f and f  , respectively, the integral relation (52) leads to Z Z Z ~ Þ : ru ðr  l ~Þ : ru g dV ¼ ff   u f  u g dV þ fðr  l ft  u t  ug dS; ð53Þ V

V

S

~Þ is the traction vector corresponding to the total stress tensor r ~ ¼ ~s r  l ~ and where t ¼ ^ n  ð~s r  l acting on the boundary S of the body V . In view of Eq. (21b), the above relation is also written in the form (Appendix C(b)) Z Z Z     ~ ~ ff  u f  u g dV þ fð^ n  l Þ : ru ð^ n  lÞ : ru g dS ¼ ft  u t  ug dS: ð54Þ V

S

S

However, as it was mentioned in Section 2, the tensor ru is not independent of u on S. Only its normal component ^ n  ru is independent of u. Thus, in view of Eqs. (9)–(14) the reciprocal integral identity (54) obtains the final form  Z Z Z  ou    ou   ff  u f  u g dV þ fP  u P  u g dS ¼ R dS; ð55Þ R  on on V S S for a smooth boundary S, and  Z Z Z  ou ou ff   u f  u g dV þ fP  u P  u g dS ¼ R dS R  on on V S S I X þ fE  u E  ug dC; ð56Þ Ca

Ca

for a non-smooth boundary S, where the surface tractions P, R and E have the form P¼^ n  ~s ð^ n^ nÞ : ~^ R¼^ nl n; ^ n ^Þ : l ~k: E ¼ kðm

o~ l ~; ~Þ ^ ~ ðrS ^nÞ : l ^ n  ðrS  l n  ½rS  ð~ lÞ213 þ ðrS  ^nÞð^n  ^nÞ : l on

ð57Þ

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D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

From relations (21b), (55)–(57) it is immediately apparent that for g2 ¼ 0, both Eqs. (55) and (56) are reduced to BettiÕs reciprocal identity Z Z ff   u f  u g dV þ ft  u t  u g dS ¼ 0; ð58Þ V

S

with t being the well known surface traction vector defined as t ¼ ^n  ~s.

5. Boundary integral representation of a gradient elastic problem In this section the boundary integral representation of a gradient elastic problem is derived with the aid of the reciprocity identity (56). Since the line integrals involved in the reciprocity identity (56) appear only in 3-D problems, the derived here integral representations concern the most general case of a 3-D body with non-smooth boundary. Consider a finite 3-D gradient elastic body of volume V surrounded by a surface S consisting, for the shake of simplicity, of two smooth surfaces S1 and S2 intersecting across the closed line C. Assume that the displacement field u , appearing in the reciprocal identity (56), is the result of a body force having the form f  ðyÞ ¼ dðx yÞ^e;

ð59Þ

with d being the Dirac d-function and ^e the direction of a unit force acting at point y. Recalling the definition of the fundamental solution derived in Section 3, it is easy to see that the displacement field u can be represented by means of the fundamental displacement tensor ~u ðx; yÞ given by the Eq. (36), according to the relation u ðyÞ ¼ u ðx; yÞ  ^e:

ð60Þ

Inserting the above expression of u in (56) and assuming zero body forces f ¼ 0, one obtains Z Z e  ðx; yÞ  ^e  uðyÞ PðyÞ  ½~u ðx; yÞ  ^e g dSy fdðx yÞ^e  uðyÞg dVy þ f½ P V S " # ) Z (  o~ u ðx; yÞ ouðyÞ  e ðx; yÞ  ^e  ¼ RðyÞ   ^e ½ R dSy ony ony S I n o e  ðx; yÞ  ^e  uðyÞ dCy þ EðyÞ  ½~ u ðx; yÞ  ^e ½ E

ð61Þ

C

or Z

 fdðx yÞuðyÞg dVy

V

 ^e þ

Z n

 o T   e ½ P ðx; yÞ  uðyÞ PðyÞ  ~u ðx; yÞ dSy  ^e

S

9 1 0 8 !T Z < =  o~ u ðx; yÞ ouðyÞ T  e ðx; yÞ  dSy A  ^e  RðyÞ ½ R ¼@ ony ony ; S : þ

I

e  ðx; yÞ  uðyÞg dCy fEðyÞ  u ðx; yÞ ½ E T

  ^e:

ð62Þ

C

Considering that relation (62) is valid for any direction ^e and taking into account the symmetry of the fundamental displacement ~ u , one obtains the boundary integral equation

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

Z n o e  ðx; yÞ T  uðyÞ ~ ~cðxÞ  uðxÞ þ ½P u ðx; yÞ  PðyÞ dSy S 9 8 !T Z < = o~ u ðx; yÞ ouðyÞ e  ðx; yÞ T  dSy ¼  RðyÞ ½ R ony ony ; S : I n o e  ðx; yÞ T  uðyÞ dCy ; þ u ðx; yÞ  EðyÞ ½ E

2855

ð63Þ

C

where ~cðxÞ is the well known jump-tensor of classical boundary integral representations [40]. Utilizing the e , R e , Q e  and E e  instead of ~ e , P e  ÞT , ðo~u =onÞT , ð R e  ÞT and ð E e  ÞT , respectively, Eq. (63) symbols U u , ð P receives the form Z n o e  ðx; yÞ  uðyÞ U e  ðx; yÞ  PðyÞ dSy ~cðxÞ  uðxÞ þ P S  Z  I n o ouðyÞ   e e e  ðx; yÞ  uðyÞ dC : e  ðx; yÞ  EðyÞ E ¼ Q ðx; yÞ  RðyÞ R ðx; yÞ  ð64Þ U dSy þ y ony S C In case the boundary S is smooth or the domain of interest is two dimensional, then the integral equation (64) is reduced to  Z n Z  o 1 e  ðx; yÞ  RðyÞ R e  ðx; yÞ  uðyÞ U e  ðx; yÞ  ouðyÞ dS : e  ðx; yÞ  PðyÞ dSy ¼ uðxÞ þ P Q y 2 ony S S ð65Þ All the kernels appearing in the integral equations (64) and (65) are given explicitly in Appendix D. Observing Eq. (64), one easily realizes that this equation contains three unknown vector fields, uðxÞ, PðxÞ and ouðxÞ=on. For example, for the case of the traction field PðxÞ prescribed on S (classical boundary condition) and the fields RðxÞ and EðxÞ prescribed on S (non-classical boundary condition), the unknown vector fields in (64) are two, uðxÞ and ouðxÞ=on, which as previously stated are independent. Thus, the evaluation of the unknown fields uðxÞ, PðxÞ and ouðxÞ=on requires the existence of one more integral equation. This integral equation is obtained by applying the operator o=onx on (64) and has the form ) Z ( e e  ðx; yÞ ouðxÞ o P ðx; yÞ oU ~cðxÞ  þ  uðyÞ  PðyÞ dSy onx onx onx S ) Z ( e e  ðx; yÞ ouðyÞ o Q ðx; yÞ oR ¼  RðyÞ  dSy onx onx ony S ) I ( e e  ðx; yÞ o U ðx; yÞ oE þ  EðyÞ  uðyÞ dCy : ð66Þ onx onx C For smooth boundaries S the integral equation (66) is reduced to ) ) Z ( e Z ( e e  ðx; yÞ ouðyÞ e  ðx; yÞ 1 ouðxÞ o P ðx; yÞ oU o Q ðx; yÞ oR þ  uðyÞ  PðyÞ dSy ¼  RðyÞ  dSy : 2 onx onx onx onx onx ony S S ð67Þ The kernels appearing in Eqs. (66) and (67) are given explicitly in Appendix E.

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D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

The integral equations (64) and (66) accompanied by the classical and non-classical boundary conditions form the integral representation of any gradient elastic boundary value problem.

6. Integral representation of strains and stresses in the interior of a gradient elastic body According to Eqs. (20b) and (20c), the evaluation of strains ~e and Cauchy stresses ~s, respectively, is accomplished with the aid of the gradient of the displacement field (64), i.e. ~cðxÞ  rx uðxÞ þ ¼

Z  S

þ

Z n

o e  ðx; yÞ  uðyÞ r U  e rx P ðx; yÞ  PðyÞ dSy x

S

e ðx; yÞ  RðyÞ r R e ðx; yÞ  rx Q x 

I n



ouðyÞ ony

 dSy

o e  ðx; yÞ  uðyÞ dC ; e  ðx; yÞ  EðyÞ rx E rx U y

ð68Þ

C

~, where all the kernels of the above integral equation are given in Appendix F. Thus, double stresses l ~ can be obtained through higher order derivatives of Eq. (68) according relative stresses ~s and total stresses r to constitutive relations (21a), (21b) and (20a), respectively, i.e. ~cðxÞ  l ~ðxÞ þ g2 ¼ g2

Z  S

þ g2

Z n

o e ðP Þ ðx; yÞ  uðyÞ rx T e ðU Þ ðx; yÞ  PðyÞ dSy rx T

S 



e ðQ Þ ðx; yÞ  RðyÞ rx T e ðR Þ ðx; yÞ  rx T

ouðyÞ ony

 dSy

I n o e ðU Þ ðx; yÞ  EðyÞ rx T e ðE Þ ðx; yÞ  uðyÞ dCy ; rx T

ð69Þ

C

~cðxÞ  ~sðxÞ g2 Z 

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

 ouðyÞ dSy ony S I n o e ðU Þ ðx; yÞ  EðyÞ r2 T e ðE Þ ðx; yÞ  uðyÞ dCy ; g2 r2x T x

¼ g2





e ðQ Þ ðx; yÞ  RðyÞ r2 T e ðR Þ ðx; yÞ  r2x T x

ð70Þ

C

~cðxÞ  r ~ðxÞ þ ¼

Z 

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

     e ðR Þ ðx; yÞ g2 r2 T e ðQ Þ ðx; yÞ g2 r2 T e ðQ Þ ðx; yÞ  RðyÞ T e ðR Þ ðx; yÞ  ouðyÞ dSy T x x ony S 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 ; þ T x x C

ð71Þ

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

where the kernels h i e  þ ðr P e  Þ213 þ 2lm eI ðr  P e  Þ; e ðP Þ ¼ l rx P T x x 1 2m h i e ðU Þ ¼ l rx U e  Þ; e  þ ðrx U e  Þ213 þ 2lm eI ðrx  U T 1 2m h i e  þ ðr Q e  Þ213 þ 2lm eI ðr  Q e  Þ; e ðQ Þ ¼ l rx Q T x x 1 2m h i e  þ ðr R e  Þ213 þ 2lm eI ðr  R e  Þ; e ðR Þ ¼ l rx R T x x 1 2m h i e  þ ðr E e  Þ213 þ 2lm eI ðr  E e  Þ; e ðE Þ ¼ l rx E T x x 1 2m

2857

ð72Þ

can be easily derived from those given in Appendix F, while their derivatives can be found in [41]. 7. Conclusions A boundary element method for solving 2- and 3-D static, gradient elastic problems has been developed. Microstructural effects on the macroscopic behavior of the considered materials have been taken into account by means of a simple gradient elastic theory proposed by Aifantis. The equation of equilibrium as well as the possible boundary conditions (classical and non-classical) have been determined with the aid of a variational statement of the problem. The fundamental solution and the reciprocity identity of the gradient elastic problem have been explicitly determined. Both have been used to establish the boundary integral equation of the problem consisting of one equation for the displacement and another one for its normal derivative. Finally, the integral forms of the gradient of displacement as well as the Cauchy, double, relative and total stresses in the interior of the body have been derived and presented. 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 second authors also gratefully acknowledge the support of the Karatheodory program for basic research offered by the University of Patras. Appendix A (a) Prove that ~Þ  du ¼ ^ ~  duÞ þ ½ðrS  ^nÞð^n  ^nÞ : l ~  du: rS  ½ð^ nl n  rS ½^ n ð^ nl

ðA:1Þ

~Þ  du can also be written in the form Proof. The vector ð^ nl ~Þ  du ¼ ð^ ~Þ  du ¼ ^ ~  duÞ : ð^ nl n^ nÞð^ nl n  ½^ n  ð^ nl

ðA:2Þ

Thus, utilizing the identity u ¼ rf : ~ u þ f  div ~ uT ; rS  ½f  ~

ðA:3Þ

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D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

with T denoting transposition and f, ~ u denoting vector and tensor, respectively, in three dimensions one obtains ~Þ  du ¼ rS  f^ ~  duÞ g rS  ½ð^ nl n  ½^ n  ð^ nl ~  duÞ þ ^n  frS  ½ð^n  l ~  duÞ  ^n g n : ½^ n  ð^ nl ¼ rS ^

ðA:4Þ

~  duÞ þ ^n  frS  ½ð^n  l ~  duÞ  ^n g: nÞ  ^ n  ð^ nl ¼ ½ðrS ^ However, due to the tangential nature of rS , the inner product ðrS ^nÞ  ^n represents a vector with zero components. Thus (A.4) reduces to ~Þ  du ¼ ^ ~  duÞ  ^ rS  ½ð^ nl n  frS  ½ð^ nl n g:

ðA:5Þ

Making use of the identity rS  ½g  f ¼ r ðf gÞ þ r  ðf  gÞ;

ðA:6Þ

with f, g being vectors, relation (A.5) receives the following form: ~Þ  du ¼ ^ ~  duÞ þ rS  ½^n  ð^n  l ~  duÞ g: rS  ½ð^ nl n  frS ½^ n ð^ nl

ðA:7Þ

The second term of the right hand side of (A.7), with the aid of the identity rS  ðf  gÞ ¼ ðr  fÞg þ f  rg;

ðA:8Þ

can be resolved as follows: ~  duÞ ¼ ðrS  ^ ~  duÞ þ ^n  rS ð^n  l ~  duÞ: rS  ½^ n  ð^ nl nÞð^ nl

ðA:9Þ

Again, due to the tangential nature of rS ^ ~  duÞ ¼ 0: n  rS ð^ nl

ðA:10Þ

Thus, inserting (A.9) into (A.7) and taking into account (A.10) one obtains ~Þ  du ¼ ^ ~  duÞ þ ½ðrS  ^nÞð^n  ^nÞ : l ~  du; nl n  rS ½^ n ð^ nl rS  ½ð^ and the (A.1) has been proven.



(b) Prove that Z XI ^ ^  ^nÞ : l ~  duÞ dS ¼ ~k  dug dC: n  rS ½^ n ð^ nl fkðm S

Ca

Proof. The Stokes theorem for an open surface S surrounded by a closed line C reads Z I ^ n  rS f dS ¼ ^s  f dC; S

ðA:11Þ

Ca

ðA:12Þ

C

where ^ n is the normal to S unit vector, ^s is a unit vector tangent to C and f a vector. Consider now a closed surface S formed by two adjacent, across a line C, surface portions S1 and S2 . Then, applying the StokeÕs ~  duÞ one obtains theorem (A.12) for the vector ^ n ð^ nl

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

Z

2859

Z ^ ^n  rS ½^n ð^n  l ~  duÞ dS þ ~  duÞ dS n  rS ½^n ð^n  l S S2 I I1 ^s  ½^ ^s  ½^n ð^n  l ~  duÞ dC ~  duÞ dC n ð^n  l ¼ CðS1 Þ CðS2 Þ I ~  duÞ k dC; ¼ k^s  ½^ n ð^n  l ðA:13Þ

^ ~  duÞ dS ¼ n  rS ½^ n ð^ nl

S

Z

C

where the brackets k  k indicate that the enclosed vector is the difference between the values on the two adjacent surfaces S1 and S2 . Exploiting the vector property u  ðv wÞ ¼ ðu vÞ  w; ^ ¼ ^s ^ and defining the vector m n, the second hand side of (A.13) is written as follows: I I I ^  ð^ ^  ^nÞ : l ~  duÞ k dC ¼ ~  duÞk dC ¼ ~  duk dC: k^s  ½^ n ð^ nl km nl kðm C

C

ðA:14Þ

C

However, the vector du is continuous on S and thus ^ ^ ^ ^ ~  duk ¼ kðm ~k  du: kðm nÞ : l nÞ : l

ðA:15Þ

Finally, inserting (A.15) into (A.14) and then into (A.13) one obtains that Z I ^ ^ ^ ~  duÞ dS ¼ ~k  du dC: n  rS ½^ n ð^ nl kðm nÞ : l S

ðA:16Þ

C

For more than one line C, i.e. Ca (a ¼ 1; 2; . . .) lines, (A.16) is written as Z XI ^ ^  ^nÞ : l ~  duÞ dS ¼ ~k  du dC: n  rS ½^ n ð^ nl kðm  S

a

ðA:17Þ

Ca

(c) Prove that for any two dimensional body with a boundary S Z ^ ~  duÞ dS ¼ 0; n  rS ½^ n ð^ nl

ðA:18Þ

S

~  du belong to the plane where the studied body is Proof. It is easy to see one that both vectors ^ n and ^n  l considered. Thus, Z Z Z ^ dS ¼ ^ dS; ^ ^ ^n  ðrS fÞ k ~  duÞ dS ¼ n  rS ½^ n  rS ½f k n ð^ nl ðA:19Þ S

S

S

^ is the vector being perpendicular to the plane of the body. where k Recalling the definition of surface gradient rS ¼ ðI ^n  ^nÞ  r, one obtains   Z Z Z Z of ^Þ dS ¼ ^ dS ¼ ^Þ dV ^ dS ¼ ^ ^ ^n  ðrf k n  ðrS fÞ k n  rf ^ n r  ðrf k k on S S S V Z ^  r ðrfÞ dV ¼ 0; ¼ k V

and relation (A.18) has been proven. 

ðA:20Þ

2860

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

Appendix B Prove that the scalar function   r=g 1 r g2 C1 2e uðrÞ ¼ þ g þ ; 4pðk þ 2lÞ 2 r r r

ðB:1Þ

is the solution of the differential equation ðk þ 2lÞ½r2 uðrÞ g2 r4 uðrÞ ¼

1 : 4pr

ðB:2Þ

Proof. Eq. (B.2) is also written in the form r2 ð1 g2 r2 ÞuðrÞ ¼

1 1 : 4pðk þ 2lÞ r

ðB:3Þ

Consider two radial functions u1 ðrÞ and u2 ðrÞ which satisfy, respectively, the following differential equations: r2 u 1 ¼

1 1 ; 4pðk þ 2lÞ r

ð1 g2 r2 Þu2 ¼

1 1 : 4pðk þ 2lÞ r

ðB:4Þ

ðB:5Þ

Then it is easy to observe that ð1 g2 r2 Þu ¼ u1

ðB:6Þ

r2 u ¼ u2 :

ðB:7Þ

and Multiplying (B.7) by g2 and adding it to (B.6) one obtains the solution of (B.3) as a function of u1 and u2 , i.e., u ¼ u1 þ g2 u2 :

ðB:8Þ

Taking into account that both u1 and u2 are functions of r only, Eqs. (B.4) and (B.5) are written in the form d2 u1 2 du1 1 1 ; ¼ þ 2 r dr 4pðk þ 2lÞ r dr u2 g 2

d2 u2 2g2 du2 1 1 : ¼ 4pðk þ 2lÞ r dr2 r dr

ðB:9Þ

ðB:10Þ

It is not difficult to see that the functions that satisfy the above equations have the form u1 ¼

c1 1 r ; þ r 4pðk þ 2lÞ 2

u2 ¼

g2 4pðk þ 2lÞ



where c1 is a constant.

 1 e r=g ; r r

ðB:11Þ

ðB:12Þ

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

2861

Finally, inserting (B.11) and (B.12) into (B.8) the radial function of uðrÞ of Eq. (B.1) is obtained. By a similar procedure one can also prove that uðrÞ of Eq. (34) for the 2-D case is the solution of Eq. (30). 

Appendix C (a) Prove that ~s : ru ~s : ru ¼ 0:

ðC:1Þ

Proof. Keeping in mind the relations ~s ¼ 2l~e þ kðtr~eÞeI ; ~e ¼ 12ðru þ urÞ;

ðC:2Þ

ðtr~eÞ ¼ r  u ¼ e; it is easy to observe that ~s : ru ¼ ð2l~e þ keeI Þ : ru ¼ ð2l~e þ keeI Þ : ~e ¼ 2l~e : ~e þ kð~e : eI ÞðeI : ~e Þ; ¼ ~e : 2l~e þ ~e : keI ðeI : ~e Þ ¼ ~e : ð2l~e þ keeI Þ ¼ ~e : ~s ¼ ~s : ~e ¼ ~s : ru:

ðC:3Þ 

~ ¼ g2 r~s prove that (b) If l Z Z   ~ ~ ~ Þ : ru ð^n  l ~Þ : ru g dS: fðr  l Þ : ru ðr  lÞ : ru g dV ¼ fð^n  l V

Proof

Z

ðC:4Þ

S

~ Þ : ru ðr  l ~Þ : ru g dV ¼ fðr  l

Z

V

fg2 r2~s : ru g2 r2~s : ru g dV :

ðC:5Þ

V

With the aid of the second GreenÕs integral identity the right hand side of (C.5) is written as ) Z Z ( s s 2 2  2 2  2 o~ 2 o~  : ru dS : ru g fg r ~s : ru g r ~s : ru g dV ¼ g on on V S Z f^ n  ðg2 r~s Þ : ru ^nðg2 r~sÞ : ru g dS ¼ S Z ~ Þ : ru ð^n  l ~Þ : ru g dS: ¼ fð^ nl 

ðC:6Þ

S

Appendix D In this Appendix the explicit expressions of the kernels appearing in the integral equations (64) and (65) are given as follows: h i 1 e  ðx; yÞ ¼ WeI X^r  ^r ; ðD:1Þ U 16plð1 mÞ

2862

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

e  ðx; yÞ ¼ Q e  ðx; yÞ ¼ R

1 16plð1 mÞ



2X dX r dr

  dW X ð^ny  ^rÞeI ð^ny  ^r þ ^r  ^ny Þ ; ð^ ny  ^rÞ^r  ^r þ dr r

      2 B  g2 dA 3A A dB B  2 ^ny  ^r þ eI ð^ ny  ^rÞ þ ^r  ^r þ dr r r dr r r 16pð1 mÞ      dB B A dC C A BþC ^ny  ^ny ; ny  ^r þ þ ð^ ny  ^rÞ^ þ ð^ny  ^rÞ^r  ^ny þ þ dr r r dr r r r

ðD:2Þ

ðD:3Þ

" #T  ~ ~ ol  e  2134    ~ Þ ^ny  ðr  l ^y  ~s þ ð^ ~ ~~ ~~ ðrS ^ny Þ : l ~~ P ¼ n ny Þ : ^ ny  ðr  l Þ þ ðrS  ^ny Þð^ny  ^ny Þ : l ny  ^ ony ðD:4Þ and e  ðx; yÞ ¼ E

    g2 dA 3A dB B ^ y  ^rÞ^r  ^r þ ^ y  ^rÞeI ð^ny  ^rÞðm ð^ ny  ^rÞðm dr r dr r 16pð1 mÞ      dB B dC C A C B ^ y  ^r þ ^r  m ^ y þ ðm ^ y  ^rÞð^ ^y  ^ ^y þ m m ny  m ny  ^r þ ^r  ^ ny Þ þ ^ ny ; þ dr r dr r r r r ðD:5Þ

where r ^r ¼ ; r ¼ y x; r ¼ jrj; r 8   8g2 r > > > 2 þ 4K 2 < 2 g r  2  X ¼ 2 > 1 6g 6g 6g 2 r=g > > þ þ 2þ e : r r r r3 r3

ðD:6Þ for 2-D; ðD:7Þ for 3-D;

8     4g2 r r > > 2ð3 4mÞ ln r þ 2ð3 4mÞK 2K for 2-D; > 0 2 < g g r2 W¼    2    2  2  > 1 g2 g g r=g g g g 1 r=g > > þ þ þ e e þ 4ð1 mÞ 3 for 3-D; : ð3 4mÞ þ 2ð1 2mÞ 3 þ r r r3 r2 r r3 r 2 r ðD:8Þ  A¼2

2X dX r dr T

ð^ ny  ~s Þ ¼ "

 ;



dW X ; dr r



2m 1 2m



dW dX ða 1ÞX dr dr r



2X ; r

h i 1 Að^ ny  ^rÞ^r  ^r þ Bð^ ny  ^rÞeI þ B^ny  ^r þ C^r  ^ny ; 16pð1 mÞ

e oe l ð^ ny  ^ ny Þ : ony

#T ¼

h i 1 G1^r  ^r þ G2eI þ G3^r  ^ny þ G4 ^ny  ^r þ G5 ^ny  ^ny ; 16pð1 mÞ

ðD:9Þ

ðD:10Þ

ðD:11Þ

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

G1 ¼ g

2

G2 ¼ g

2

G3 ¼ g2

  

    d2 A 7 dA 15A 1 dA 3A 3 ^ ^ þ 2 ð^ 2 ð^ny  rÞ ; ny  rÞ þ 3 dr2 r dr r r dr r

ðD:12Þ

    d2 B 3 dB 3B 1 dB B 3 þ 2 ð^ ny  ^rÞ þ 3 ð^ny  ^rÞ ; dr2 r dr r r dr r2

ðD:13Þ

  d2 C 3 dC 3C 2 dA 6A 2A 1 dC C 2 ^ þ þ  r Þ þ þ ð^ n ; y dr2 r dr r2 r dr r2 r2 r dr r2

ðD:14Þ

  d2 B 3 dB 3B 2 dA 6A 2A 1 dB B 2 ^ þ þ  r Þ þ þ ð^ n ; y dr2 r dr r2 r dr r2 r2 r dr r2   1 dB B 1 dC C A 2 þ 2 ð^ G5 ¼ 2g2 2þ ny  ^rÞ; r dr r r dr r r G4 ¼ g2

2863



~ ~ Þ T ¼ ½^ ny  ðr  l

 2  g2 d A ða 1Þ dA 3ða þ 1ÞA þ ð^ny  ^rÞ^r  ^r dr2 r dr r2 16pð1 mÞ  2  d B ða 1Þ dB ða 1ÞB 2A þ 2 ½ð^ny  ^rÞeI þ ^ny  ^r

þ þ dr2 r dr r2 r  2   d C ða 1Þ dC ða 1ÞC 2A ^ ^ r  n þ þ þ y ; dr2 r dr r2 r2

~~2134 Þ T ¼ ½^ny  ðr  l

ðD:16Þ

ðD:17Þ

 2  g2 d B a dB aB 1 dC C þ ð^ ny  ^rÞeI þ dr2 r dr r2 r dr r2 16pð1 mÞ  2  d A ða 3Þ dA 3ð1 aÞA d2 B 3 dB 3B d2 C 3 dC 3C þ þ 2 þ 2 þ 2 ð^ þ ny  ^rÞ^r  ^r þ 2 þ dr2 r dr r2 dr r dr r dr r dr r    1 dA ða 1ÞA 1 dB B 1 dC C þ ny Þ ; ðD:18Þ þ þ þ ð^ ny  ^r þ ^r  ^ r dr r2 r dr r2 r dr r2 T

~ ¼ ~ ½ðrS  ^ ny Þð^ ny  ^ ny Þ : l

~ ~  T ¼ ny Þ : l ½ðrS ^

ðD:15Þ

    g2 ðrS  ^ ny Þ dA 3A A 2 ð^ny  ^rÞ þ ^r  ^r 16pð1 mÞ dr r r      dB B B dB B A þ ð^ ny  ^rÞ2 þ eI þ þ ð^ny  ^rÞ^ny  ^r dr r r dr r r    dC C A BþC ^ny  ^ny ; þ þ ð^ny  ^rÞ^r  ^ny þ dr r r r

   g2 dA 3A A ðrS ^ny Þ : ð^r  ^rÞ þ ðrS  ^ny Þ ^r  ^r dr r r 16pð1 mÞ    dB B B þ ny Þ : ð^r  ^rÞ þ ðrS  ^ny Þ eI ðrS ^ dr r r   A dB B ny Þ þ ^r  ðrS ^ny Þ  ^r þ þ ½ð^r  ^rÞ  ðrS ^ ðrS ^ny Þ  ð^r  ^rÞ r dr r    dC C C B T ^r  ðrS ^ ny Þ  ^r þ ðrS ^ny Þ þ ðrS ^ny Þ ; þ dr r r r

ðD:19Þ

ðD:20Þ

2864

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 a¼

2; 3;

for 2-D case; for 3-D case:

ðD:21Þ

Appendix E In this Appendix the explicit expressions of the kernels appearing in the integral equations (66) and (67) are given as follows:     e  ðx; yÞ oU 1 dW dX 2 X ð^ nx  ^rÞeI þ ¼ X ð^nx  ^rÞ^r  ^r þ ð^nx  ^r þ ^r  ^nx Þ ; onx 16plð1 mÞ dr dr r r    e  ðx; yÞ oQ 1 5 dX 8X d2 X ¼ 2 2 ð^ny  ^rÞð^nx  ^rÞ^r  ^r onx 16plð1 mÞ r dr r dr     2X 1 dX 2X 1 dX ^ ^ ð^ ny  nx Þ^r  r ð^ny  ^rÞð^nx  ^r þ ^r  ^nx Þ r2 r dr r2 r dr    2  d W 1 dW 1 dW 2X 1 dX e e ^ ^ ^ ð^ n  r Þð^ n  r Þ I  n Þ I ð^ n ð^nx  ^rÞ^ny  ^r y x y x dr2 r dr r dr r2 r dr    X 2X 1 dX X ^ ^ ^ ^ þ 2^ ny  ^ n nx  r Þ^ r  n þ  n ð^ n x y x y ; r r2 r dr r2    e  ðx; yÞ oR g2 dA1 4A1 2A1 2 ð^ny  ^rÞð^ny  ^nx Þ^r  ^r ¼ ð^ny  ^rÞ ð^nx  ^rÞ^r  ^r onx 16pð1 mÞ dr r r   A1 dA6 2A6 A6 2 ð^ ny  ^rÞ ð^ nx  ^r þ ^r  ^ nx Þ ð^nx  ^rÞ^r  ^r ð^nx  ^r þ ^r  ^nx Þ r dr r r   dA2 2A2 2A2 dA7 2 ð^ny  ^rÞð^ny  ^nx ÞeI ð^nx  ^rÞeI nx  ^rÞeI ð^ ny  ^rÞ ð^ dr r r dr   dA3 2A3 A3 A3 nx  ^rÞ^ ny  ^r ð^ny  ^nx Þ^ny  ^r ð^ny  ^rÞ^ny  ^nx ð^ ny  ^rÞð^ dr r r r   dA4 2A4 A4 A4 nx  ^rÞ^r  ^ny ð^ny  ^nx Þ^r  ^ny ð^ny  ^rÞ^nx  ^ny ð^ ny  ^rÞð^ dr r r r  dA5 ð^ nx  ^rÞ^ ny  ^ ny ; dr " e  ðx; yÞ ~ ~ oP o ol ~~ Þ ^ny  ðr  l ~~2134 Þ ^y  ~s þ ð^ n ¼ ny Þ : ^ny  ðr  l ny  ^ onx onx ony #T   ~ ~ ~ ðrS ^ny Þ : l ~ ; þ ðrS  ^ ny Þð^ ny  ^ ny Þ : l

ðE:1Þ

ðE:2Þ

ðE:3Þ

ðE:4Þ

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

2865

   e  ðx; yÞ oE g2 dA1 4A1 A ^ y  ^rÞ þ 1 ð^nx  ^ny Þðm ^ y  ^rÞ ¼ ð^ nx  ^rÞð^ny  ^rÞðm onx 16pð1 mÞ dr r r  A1 A1 ^ y Þð^ ^ y  ^rÞð^nx  ^r þ ^r  ^nx Þ nx  m ny  ^rÞðm þ ð^ ny  ^rÞ ^r  ^r ð^ r r    dA2 2A2 A A ^ y  ^rÞ þ 2 ð^nx  ^ny Þðm ^ y  ^rÞ þ 1 ð^nx  m ^ y Þð^ny  ^rÞ eI ny  ^rÞðm ð^ nx  ^rÞð^ r r  r  dr  dA2 A2 A2 dA9 A9 A ^ y  ^nx ^ y  ^r m ^ y 9 ^nx  m ^y ð^ nx  ^rÞm ð^nx  ^rÞ^r  m dr r r dr r r    dA6 2A6 A A ^ y  ^nx Þ ^ny  ^r 6 ðm ^ y  ^rÞ^ny  ^nx ^ y  ^rÞ þ 6 ðm ð^ nx  ^rÞðm r  r r   dr dA6 2A6 A A ^ y  ^nx Þ ^r  ^ny 6 ðm ^ y  ^rÞ^nx  ^ny ^ y  ^rÞ þ 6 ðm ð^ nx  ^rÞðm dr r r r  dA8 dA ^ y 7 ð^ ^ y  ^ny ; ð^ nx  ^rÞ^ nx  ^rÞm ny  m ðE:5Þ dr dr where    oð^ ny  es  ÞT g2 dA 3A A A ¼ ð^ ny  ^rÞð^nx  ^rÞ^r  ^r ð^ny  ^nx Þ^r  ^r ð^ny  ^rÞð^nx  ^r þ ^r  ^nx Þ dr r r r onx 16pð1 mÞ     dB B B dB B B nx  ^rÞeI ð^ny  ^nx ÞeI ð^ ny  ^rÞð^ ð^nx  ^rÞ^ny  ^r ^ny  ^nx dr r r dr r r    dC C C nx  ^ny ; ny ^ ð^ nx  ^rÞ^r  ^ ðE:6Þ dr r r " #T "   e o oe l g2 dB1 5B1 3 ð^ ny  ^ ¼ ny Þ : ð^ny  ^rÞ ð^nx  ^rÞ^r:  ^r onx ony 16pð1 mÞ dr r 3B1 B1 ð^ ny  ^rÞ2 ð^ ny  ^ nx Þ^r  ^r ð^ny  ^rÞ3 ð^nx  ^r þ ^r  ^nx Þ r r  dB2 3B2 B2 ð^ ny  ^rÞð^nx  ^rÞ^r  ^r ð^ny  ^nx Þ^r  ^r dr r r   B2 dB3 3B3 ny  ^rÞð^ nx  ^r þ ^r  ^nx Þ ð^ ð^ny  ^rÞ3 ð^nx  ^rÞeI r dr r   3B3 dB4 B4 B4 2 e ð^ ny  ^rÞ ð^ ny  ^ nx Þ I ð^ny  ^rÞð^nx  ^rÞeI ð^ny  ^nx ÞeI r dr r r   dB5 3B5 2B5 2 ð^ny  ^rÞð^ny  ^nx Þ^r  ^ny ð^ ny  ^rÞ ð^nx  ^rÞ^r  ^ny dr r r   B5 dB6 B6 B6 ny  ^rÞ2 ^ nx  ^ ð^ ny ð^nx  ^rÞ^r  ^ny ^nx  ^ny r dr r r   dB7 3B7 2B7 2 ð^ny  ^rÞð^ny  ^nx Þ^ny  ^r ð^ ny  ^rÞ ð^nx  ^rÞ^ny  ^r dr r r   B7 dB8 B8 B8 2 ny  ^ ny  ^rÞ ^ nx ð^ ð^nx  ^rÞ^ny  ^r ^ny  ^nx r dr r r #   dB9 B9 B9 ð^ ny  ^rÞð^nx  ^rÞ^ny  ^ny ð^ny  ^nx Þ^ny  ^ny ; dr r r

ðE:7Þ

2866

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   # $& o % g2 dF1 3F1 F1 ~ T ¼ ^ ~ ny  r  l ð^ny  ^rÞð^nx  ^rÞ^r  ^r ð^ny  ^nx Þ^r  ^r onx 16pð1 mÞ dr r r   F1 dF2 F2 F2 ð^ ny  ^rÞð^ nx  ^r þ ^r  ^nx Þ ð^ny  ^rÞð^nx  ^rÞeI ð^ny  ^nx ÞeI r dr r r      dF2 F2 F2 dF3 F3 F3 ny  ^r ^ny  ^nx ð^ nx  ^rÞ^ ð^nx  ^rÞ^r  ^ny ^nx  ^ny ; dr r r dr r r ðE:8Þ    # $& o % g2 dD1 3D1 2134 T ~ ^ ~ ny  r  l ¼ ð^ny  ^rÞð^nx  ^rÞ^r  ^r onx 16pð1 mÞ dr r D1 D1 ð^ ny  ^ nx Þ^r  ^r ð^ny  ^rÞð^nx  ^r þ ^r  ^nx Þ r r   dD2 D2 D2 ð^ ny  ^rÞð^nx  ^rÞeI ð^ny  ^nx ÞeI dr r r    # $ D3 # $ dD3 D3 ^ny  ^nx þ ^nx  ^ny ; ð^ nx  ^rÞ ^ny  ^r þ ^r  ^ny dr r r

ðE:9Þ

   & o % g2 ðrS  ^ ny Þ dA1 4A1 2  T ~ ~ ny Þ : l ðrS  ^ ny Þð^ ny  ^ ¼ ð^ny  ^rÞ ð^nx  ^rÞ^r  ^r onx 16pð1 mÞ dr r 2A1 A1 ð^ ny  ^rÞð^ny  ^nx Þ^r  ^r ð^ny  ^rÞ2 ð^nx  ^r þ ^r  ^nx Þ r r   dA6 2A6 A6 ð^nx  ^rÞ^r  ^r ð^nx  ^r þ ^r  ^nx Þ dr r r   dA2 2A2 2A2 ð^ny  ^rÞð^ny  ^nx ÞeI ð^ny  ^rÞ2 ð^nx  ^rÞeI dr r r   dA7 dA3 2A3 e ^ ð^ nx  r Þ I ð^ny  ^rÞð^nx  ^rÞ^ny  ^r dr dr r

A3 A3 ð^ ny  ^ nx Þ^ny  ^r ð^ny  ^rÞ^ny  ^nx r r   dA4 2A4 ð^ny  ^rÞð^nx  ^rÞ^r  ^ny dr r



 A4 A4 dA5 ð^ ny  ^ ð^nx  ^rÞ^ny  ^ny ; nx Þ^r  ^ny ð^ny  ^rÞ^nx  ^ny r r dr

ðE:10Þ

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

2867

  &T o % g2 dA1 4A1 ~~ ¼ ðrS ^ny Þ : l ð^ nx  ^rÞðrS ^ ny Þ : ^r  ^r 16pð1 mÞ dr r onx    A1 dA6 2A6 ð^ nx  ^rÞðrS  ^ ny Þ^r  ^r ^r  ^r ny Þ þ þ ð^nx  ^r þ ^r  ^nx Þ : ðrS ^ r dr r    dA2 2A2 A2 dA7 nx Þ : ðrS ^ ny Þ eI ð^nx  ^rÞðrS ^ nx  ^r þ ^r  ^ ðrS  ^ þ ny Þ : ^r  ^r þ ð^ ny Þ þ dr r r dr   A6 A1 ðrS  ^ny Þ þ ðrS ^ny Þ : ^r  ^r ð^ nx  ^r þ ^r  ^ nx Þ þ r r   dA6 2A6 ð^nx  ^rÞ½ð^r  ^rÞ  ðrS ^ þ ny Þ þ ^r  ðrS ^ ny Þ  ^r

dr r A6 nx þ ð^ nx  ^r þ ^r  ^ nx Þ  ðrS ^ þ ½^nx  ðrS ^ny Þ  ^r þ ^r  ðrS ^ ny Þ  ^ ny Þ

r  dA2 2A2 A2 nx  ^r þ ^r  ^ nx Þ ð^nx  ^rÞðrS ^ þ ny Þ  ð^r  ^rÞ þ ðrS ^ ny Þ  ð^ dr r r   dA9 2A9 A9 þ nx Þ  ðrS ^ ð^nx  ^rÞð^r  ^rÞ  ðrS ^ nx  ^r þ ^r  ^ ny ÞT þ ð^ n y ÞT dr r r  dA8 dA7 T þ ðE:11Þ ð^nx  ^rÞðrS ^ny Þ þ ð^ nx  ^rÞðrS ^ ny Þ ; dr dr

dA 3A dB B dB B A dC C A ; A2 ¼ ; A3 ¼ þ ; A4 ¼ þ ; dr r dr r dr r r dr r r ðB þ CÞ A B C dC C A5 ¼ ; A 6 ¼ ; A7 ¼ ; A8 ¼ ; A9 ¼ ; r r r r dr r

A1 ¼

d2 A 7 dA 15A 3 dA 9A d2 B 3 dB 3B þ 2 ; B2 ¼ 2 ; B3 ¼ 2 þ 2; 2 dr r dr r r dr r dr r dr r 2 3 dB 3B d C 3 dC 3C 2 dA 6A 2A 1 dC C 2 ; B5 ¼ 2 þ 2 þ 2 ; B6 ¼ 2 þ ; B4 ¼ r dr r dr r dr r r dr r r r dr r2 d2 B 3 dB 3B 2 dA 6A 2A 1 dB B þ 2 þ 2 ; B8 ¼ 2 þ ; B7 ¼ 2 dr r dr r r dr r r r dr r2 2 dB 2B 2 dC 2C 2A 2 þ 2 þ 2; B9 ¼ r dr r r dr r r

ðE:12Þ

B1 ¼

d2 A ða 1Þ dA 3ða þ 1ÞA d2 B ða 1Þ dB ða 1ÞB 2A þ ; F ¼ þ þ 2; 2 dr2 r dr r2 dr2 r dr r2 r 2 d C ða 1Þ dC ða 1ÞC 2A F3 ¼ 2 þ þ 2; dr r dr r2 r

ðE:13Þ

F1 ¼

d2 A ða 3Þ dA 3ð1 aÞA d2 B 3 dB 3B d2 C 3 dC 3C þ þ 2 þ 2 þ 2 þ þ 2 dr2 r dr r2 dr r dr r dr r dr r d2 B a dB aB 1 dC C 1 dA ða 1ÞA 1 dB B 1 dC C 2 þ þ þ ; D2 ¼ 2 þ D3 ¼ þ dr r dr r r dr r2 r dr r2 r dr r2 r dr r2  2 for 2-D case; a¼ 3 for 3-D case:

ðE:14Þ

D1 ¼

ðE:15Þ ðE:16Þ

2868

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Appendix F In this Appendix the explicit expressions of the kernels appearing in the integral equation (68) are given as follows:    1 dX 2X X X dW e ¼ e  ¼ r U ^r  ^r  ^r þ eI  ^r þ ð^r  eI Þ213 ^r  eI ; ðF:1Þ rx U 16plð1 mÞ dr r r r dr  2  1 d X 5 dX 8X þ 2 ð^ny  ^rÞ^r  ^r  ^r 16plð1 mÞ dr2 r dr r       2  213 1 dX 2X d W 1 dW þ 2 ð^ ð^ny  ^rÞ^r  eI ny  ^rÞ eI  ^r þ ð^r  eI r dr r dr2 r dr   1 dX 2X þ 2 ð^ ny  ^r  ^r þ ^r  ^ny  ^r þ ^r  ^r  ^ny Þ r dr r   213  1 dW # X ^ny  eI ; þ 2 eI  ^ ny  eI ny þ ^ r r dr

e ¼ e  ¼ r Q rx Q

ðF:2Þ

 g2 e  ¼ r R e ¼ Kð^ ny  ^rÞ2^r  ^r  ^r þ F ^r  ^r  ^r þ G^r  eI rx R 16pð1 mÞ    213  A þ F ð^ ny  ^rÞ2 þ 2 eI  ^r þ ^r  eI Lð^ny  ^rÞ2^r  eI r   A e ^ þ 2Gð^ ny  rÞ^ ny  I þ G þ 2 ð^ ny  ^rÞð^ny  eI Þ213 r   A þ H þ 2 ð^ ny þ 2F ð^ ny  ^rÞ^ny  ^r  ^r ny  ^rÞeI  ^ r ny  ^r þ ðM þ F Þð^ny  ^rÞ^r  ^r  ^ny þ ðL þ F Þð^ ny  ^rÞ^r  ^     A A ny  ^ þ Gþ 2 ^ ny  ^r þ H þ 2 ^ny  ^r  ^ny r r  þ ðG þ H Þ^r  ^ ny  ^ ny ;  ~ % ~ ol   ~~ Þ ^ny  ðr  l ~~2134 Þ þ ðrS  ^ny Þð^ny  ^ny Þ : l ~~ e ny Þ : ^ ny  ðr  l rx P ¼ rx n^y  ~s þ ð^ ny  ^ ony &T ~ ~ ; ny Þ : l ðrS ^

ðF:3Þ

ðF:4Þ

where T

rx ð^ ny  ~s Þ ¼ rð^ ny  ~s Þ

T

 h i 1 A ðFrÞð^ ny  ^rÞ^r  ^r  ^r þ ð^ny  ^rÞ eI  ^r þ ð^r  eI Þ213 þ ðGrÞð^ny  ^rÞ^r 16pð1 mÞ r i A Bh ny  ^r  ^r þ ðGrÞ^r  ^ny  ^r þ ðHrÞ^r  ^r  ^ny þ ^ny  eI þ ð^ny  eI Þ213  eI þ ^ r r  Ce þ I^ ny ; r

¼

ðF:5Þ

D. Polyzos et al. / Comput. Methods Appl. Mech. Engrg. 192 (2003) 2845–2873

"

#  T

~ ~ ol rx ð^ ny  ^ ny Þ : ony

2869

"

#  T ~ ~ ol ¼ r ð^ ny  ^ ny Þ : ony        g2 dK 5K dF 3F 3 ¼ ð^ny ^rÞ þ 3 ð^ny ^rÞ ^r ^r ^r dr r dr r 16pð1 mÞ h h i 3F i K 3 213 213 þ ð^ ny ^rÞ eI ^r þ ð^r  eI Þ þ ð^ny ^rÞ eI ^r þ ð^r  eI Þ r r      # $ dL 3L dG G 3 ^ny ^r ^r  eI þ ð^ ny ^rÞ þ 3 dr r dr r      L 3G ðM þ 2F Þ H 2A e 2 2 ^ þ 3 ð^ ny ^rÞ þ ny  eI þ ð^ny ^rÞ þ þ 3 I  ^ny r r r r r      ðL þ 2F Þ H 2A 3K 3F 2 213 2 e ^ny ^r ^r ð^ ny ^rÞ þ ð^ny ^rÞ þ þ þ ð^ny  I Þ þ r r r3 r r     dðM þ 2F Þ 3ðM þ 2F Þ dðH þ 2A=r2 Þ ðH þ 2A=r2 Þ ^r ^r  ^ny þ ð^ny ^rÞ2 þ dr r dr r     dðL þ 2F Þ 3ðL þ 2F Þ dðG þ 2A=r2 Þ ðG þ 2A=r2 Þ 2 ^r  ^ny ^r þ ð^ny ^rÞ þ dr r dr r ðM þ 2F Þ ðL þ 2F Þ ð^ ny ^rÞ^ ð^ny ^rÞ^ny  ^ny ^r ny ^r  ^ny þ 2 r r   dðG þ H þ A=r2 Þ ðG þ H þ A=r2 Þ þ2 ð^ny ^rÞ^r  ^ny  ^ny dr r  ðG þ H þ A=r2 Þ ^ ny  ^ny  ^ny ; þ2 r þ2

% # $&T % # $& T ~ ~ ^y  r  l ~ ~ ¼ r ^ ny  r  l rx n   h i g2 dA 3A A ¼ ð^ny  ^rÞ^r  ^r  ^r þ ð^ny  ^rÞ eI  ^r þ ð^r  eI Þ213 dr r r 16pð1 mÞ     dB B A dB B e ^r  ^ny  ^r þ ð^ ny  ^rÞ^r  I þ ^ny  ^r  ^r þ dr r r dr r    i C dC C Bh 213 e e e ^ ^ ^ ^ ^ r  r  ny þ ny  I þ ð^ny  I Þ þ þ I  ny ; dr r r r

ðF:6Þ

ðF:7Þ

% # $&T % # $&T ~ ~ ^y  r  l ~2134 ~2134 ¼ r ^ ny  r  l rx n   h i g2 dF 3F F 213 ¼ ð^ny  ^rÞ^r  ^r  ^r þ ð^ny  ^rÞ eI  ^r þ ð^r  eI Þ dr r r 16pð1 mÞ     dG G F dH H e ^ ^r  ^ny  ^r ^ ^ ^ þ ð^ ny  rÞ^r  I þ ny  r  r þ dr r r dr r    213 H dH H G H# ^r  ^r  ^ny þ ^ny  eI þ ^ny  eI þ þ eI  ^ny ; ðF:8Þ dr r r r r

2870

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%# $ &T % # $  &T ~ ~ ~~ ¼ r ðrS  ^ny Þ ^ ny : l rx rS  ^ny ð^ny  ^ny Þ : l ny  ^  g2 ðrS  ^ny Þ ¼ ½Kð^ ny  ^rÞ2 þ F ^r  ^r  ^r 16pð1 mÞ "

# i A he þ F ð^ny  ^rÞ þ 2 I  ^r þ ð^r  eI Þ213 þ ½Lð^ ny  ^rÞ2 þ G ^r  eI r 2

ny  ^rÞ^r  ^ ny  ^r þ 2F ð^ny  ^rÞ^ny  ^r  ^r þ ðL þ F Þð^    A A ^y  ^ ny  ^r  ^ ny þ G þ 2 n ny  ^r þ H þ 2 ^ ny þ ðM þ F Þð^ny  ^rÞ^r  ^r  ^ r r 

þ ðG þ H Þ^r  ^ ny  ^ ny þ 2Gð^ ny  ^rÞ^ ny  eI      213  A A e e ny  ^rÞ I  ^ ny  I þ H þ 2 ð^ ny ; þ G þ 2 ð^ny  ^rÞ ^ r r

ðF:9Þ

% &T % &T ~ ~ ~ ¼ r ðrS ^ ~ ny Þ : l ny Þ : l rx ðrS ^ ( h i g2 ny Þ : ^r  ^r þ F ðrS ^ny Þ : eI ^r  ^r  ^r KðrS ^ ¼ 16pð1 mÞ 

i h i Ah eI eI  ^r þ ð^r  eI Þ213 ^ n ðr Þ : S y r2   e ny Þ : ^r  ^r þ GðrS ^ny Þ : I ^r  eI þ LðrS ^

þ

F ðrS ^ ny Þ : ^r  ^r þ

T

þ F ½ðrS ^ ny Þ  ^r  ^r  ^r þ ðrS ^ny Þ  ^r  ^r  ^r þ ^r  ^r  ^r  ðrS ^ny Þ

ny ÞT þ L½^r  ^r  ^r  ðrS ^ny ÞT 132 þ M½^r  ^r  ^r  ðrS ^ ny Þ 132 þ F ½^r  ^r  ^r  ðrS ^ A 132 T ny  ^rÞ þ rS ^ny  ^r þ G½ðrS ^ny Þ  ^r þ ^r  ðrS ^ny Þ

½ðrS ^ r2  1324 h i T 321 þ H ½^r  ðrS ^ ny Þ þ ðrS ^ny  ^rÞ þ H eI  ðrS ^ny Þ  ^r þ G eI  ðrS ^ny Þ  ^r

þ

  i Ah T T e e ny Þ  I þ ^r  ðrS ^ny Þ  I þ 2 eI  ðrS ^ny Þ  ^r þ G ^r  ðrS ^ r ) i1423 Ah þ 2 eI  ðrS ^ ny Þ  ^r ; r

ðF:10Þ

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2871

 g2 ^  ^r  ^r þ ðm ^  ^rÞ^ny  ^r  ^r

F ½ð^ ny  ^rÞm 16pð1 mÞ h i ^  ^rÞ^r  ^r  ^r þ F ð^ ^  ^rÞ eI  ^r þ ð^r  eI Þ213 þ Kð^ ny  ^rÞðm ny  ^rÞðm h i ^  eI þ ðm ^  ^rÞ^ ^  ^rÞ^r  eI þ L½^r  m ^  ^r þ ^r  ^r  m ^

þ G ð^ ny  ^rÞm ny  eI þ Lð^ny  ^rÞðm

e ¼ e  ¼ r E rx E

h i A ^ þ ðm ^  eI Þ213 þ 2 ½m ^ ^ ^  ^r  ^ny

þ G eI  m ny  ^r þ m r h i A ^  ^rÞ eI  ^ ^  ^rÞ½^r  ^ny  ^r þ ^r  ^r  ^ny

þ 2 ðm ny þ ð^ ny  eI Þ213 þ F ðm r  ^ þ G^r  m ^ ^ ny ; þ H^r  ^ ny  m



d2 A 7 dA 15A þ 2 ; dr2 r dr r

F ¼

1 dA 3A 2; r dr r



d2 B 3 dB 3B þ 2; dr2 r dr r



1 dB B ; r dr r2



d2 C 3 dC 3C þ 2 ; dr2 r dr r



1 dC C ; r dr r2

d2 A ða 1Þ dA 3ða þ 1ÞA A¼ 2 þ ; dr r dr r2 B¼

d2 B ða 1Þ dB ða 1ÞB 2A þ þ 2; dr2 r dr r2 r



d2 C ða 1Þ dC ða 1ÞC 2A þ þ 2; dr2 r dr r2 r

F ¼

d2 A ða 3Þ dA 3ð1 aÞA d2 B 3 dB 3B d2 C 3 dC 3C þ þ 2 þ 2 þ 2 ; þ þ 2 dr2 r dr r2 dr r dr r dr r dr r



d2 B a dB aB 1 dC C 2 þ ; þ dr2 r dr r r dr r2

1 dA ða 1ÞA 1 dB B 1 dC C þ þ ; þ r dr r2 r dr r2 r dr r2  2 for 2-D case; a¼ 3 for 3-D case:

ðF:11Þ

ðF:12Þ



ðF:13Þ

2872

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