A boundary element method for solving 3D static gradient elastic ...

Computational Mechanics 29 (2002) 361–381 Ó Springer-Verlag 2002 DOI 10.1007/s00466-002-0348-5

A boundary element method for solving 3D static gradient elastic problems with surface energy K. G. Tsepoura, S. Papargyri-Beskou, D. Polyzos

361

1 Introduction In linear elastic materials, such as polymers, polycrystals or granular materials, the effects of the microstructure on their mechanical behavior are significant and they have to be taken into account. Classical linear elasticity theory cannot adequately describe their mechanical behavior. These microstructural effects can be successfully modelled in a macroscopic framework by defining the state of stress in a non-local manner with the aid of higher-order strain gradient, micropolar and couple stress theories. Among those who have developed such theories one can mention Mindlin (1964, 1965), Aifantis (1984, 1992) and Vardoulakis and Sulem (1995) in connection with the higher-order strain gradient theories, Eringen (1968) in connection with the micropolar theories and Cosserat (1909), Mindlin and Tiersten (1962), Koiter (1964) and Toupin (1965) in connection with the couple-stress theories. From the above theories, the most general and comprehensive is the one due to Mindlin (1964, 1965), while the simplest is the one due to Aifantis (1984, 1992). During the last fifteen years or so, a variety of boundary value problems of linear elasticity were solved analytically by employing gradient-elasticity theories of rather simple forms and the microstructural effects on the solution were assessed. One can mention here, e.g., static problems Keywords Gradient elasticity, Surface energy, Materials dealing with fracture mechanics, the half-space under varwith microstructure, Boundary Element Method, ious surface loads, a borehole under pressure, a bar under Fundamental solutions, Non-classical boundary tension or a beam in bending (Altan and Aifantis 1992; Ru conditions and Aifantis 1993; Exadaktylos et al. 1996; Exadaktylos and Vardoulakis 1998, 2001; Tsepoura et al. 2002; PapargyriBeskou et al. 2002) and dynamic problems dealing with wave propagation in a bar and the half-space (Altan et al. 1996; Chang and Gao 1997; Georgiadis et al. 2000; Tsepoura Received: 9 November 2001 / Accepted: 20 June 2002 et al. 2002). It was found that use of these non-classical theories may lead to the elimination of singularities or K. G. Tsepoura, D. Polyzos (&) discontinuities of classical elasticity theory and the Department of Mechanical and Aeronautical Engineering, capturing of size effects and wave dispersion in cases where University of Patras, GR-26500 Patras, Greece e-mail: [email protected] this was not possible in the classical elasticity framework. However, for realistic engineering problems characterand Institute of Chemical Engineering and High Temperature ised by complicated geometry and boundary conditions, Chemical processes-FORTH, GR-26500 Patras, Greece analytical methods of solution are inadequate and resort S. Papargyri-Beskou has to be made to numerical methods, such as the finite General Department, School of Technology, element method (FEM) or the boundary element method Aristotle University of Thessaloniki, (BEM). Nakamura et al. (1984) and Huang and Liang GR-54006 Thessaloniki, Greece (1994) for the case of the micropolar elasticity and Shu The first and third authors gratefully acknowledge the support of et al. (1999) and Amanatidou and Aravas (2002) for the case of the more general theories of Mindlin have used the the Karatheodory program for basic research offered by the FEM for solving two-dimensional elastostatic problems. University of Patras. Abstract A boundary element methodology is developed for the static analysis of three-dimensional bodies exhibiting a linear elastic material behavior coupled with microstructural effects. These microstructural effects are taken into account with the aid of a simple strain gradient elastic theory with surface energy. A variational statement is established to determine all possible classical and non-classical (due to gradient with surface energy terms) boundary conditions of the general boundary value problem. The gradient elastic fundamental solution with surface energy is explicitly derived and used to construct the boundary integral equations of the problem with the aid of the reciprocal theorem valid for the case of gradient elasticity with surface energy. It turns out 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. All the kernels in the integral equations are explicitly provided. The numerical implementation and solution procedure are provided. Surface quadratic quadrilateral boundary elements are employed and the discretization is restricted only to the boundary. Advanced algorithms are presented for the accurate and efficient numerical computation of the singular integrals involved. Two numerical examples are presented to illustrate the method and demonstrate its merits.

362

The BEM has also been used for solving two-dimensional strain-gradient elastostatic problems but only in the framework of the micropolar case. One can mention here the works of Dragos (1984), Liang and Huang (1996) and Huang and Liang (1997). In this work the BEM in its direct form is employed for the solution of three-dimensional elastostatic problems in the framework of the strain-gradient theory with surface energy as presented in the book of Vardoulakis and Sulem (1995). The present version of the implementation of the method is restricted to smooth boundaries and computation of boundary displacements and stresses. The paper consists of the following eight sections: Sect. 2 deals with the constitutive equations and the boundary conditions, which are obtained through a variational statement and comprise classical and non-classical ones. Section 3 presents the derivation of the fundamental solution of the problem, while Sect. 4 derives the reciprocal integral identity for gradient elasticity with surface energy. Section 5 presents the boundary integral representation of the problem, while Sect. 6 describes the numerical implementation and solution procedure. Numerical examples are presented in Sect. 7 to illustrate the method and demonstrate its merits. Finally, Sect. 8 consists of the conclusions pertaining to this work.

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 by the gradient elastic theory with surface energy due to Vardoulakis and Sulem (1995) are derived and presented in detail. Since this theory is a special case of Mindlin’s (1964, 1965) strain gradient theory, the derivation of both the equation of equilibrium and boundary conditions is accomplished by taking first the variation of the strain energy defined by Mindlin (1964, 1965) and then inserting the constitutive equation proposed by Vardoulakis and Sulem (1995). Consider a three dimensional (3-D) 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 with surface energy. The geometry of this body is described ^ on S and a Carwith the aid of the unit normal vector n tesian co-ordinate system OX1 X2 X3 with its origin located interior to V . According to Mindlin’s (1964, 1965) strain gradient theory, the stored strain energy depends upon strain and strain gradient, i. e., Z 321 .. ~Þ .r~e dV s~ : ~e þ ðl U¼ V Z ¼ sij eij þ lijk oi ejk dV

ð1Þ

V

where s~ and ~e are the classical second order symmetric elastic stress and strain tensors, r is the gradient operator ~ is a third order tensor, of which the 27 components and l lijk represent double forces per unit area. Furthermore, the first subscript of lijk indicates the normal vector of the

surface on which the double stresses act, while the other two have the same significance with the corresponding ones of the classical stress tensor sij . Finally, the double and triple dots in Eq. (1) indicate dyadic and triadic 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 ðÞ321 means

ða b cÞ321 ¼ c b a ð3Þ 1 Taking into account that ~e ¼ 2 ½ru þ ur , 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 . ð4Þ dU ¼ ½~ s :rdu þ ð~ lÞ321 ..rrdu dV V

Utilizing the identities (Brand 1966)

132 . ~ÞT :ru þ ð~ r ð~ lÞ :ru ¼ ðr l lÞ321 ..rru ~Þ u ¼ ½r ðr l ~Þ u þ ðr l ~ÞT :ru r ½ðr l r ½s~ u ¼ ðr s~Þ u þ s~ :ru

ð5Þ

where T denotes transposition, and the symmetry relation ~Þ321 ¼ l ~, Eq. (4) takes the form lijk ¼ likj or ðl

dU ¼

Z

~Þ du ½r ð~ ~Þ du ðr ½ð~ srl srl

V

þr ½~ l :rðduÞ Þ dV

ð6Þ

Employment of the divergence theorem transforms Eq. (6) into

dU ¼ þ þ

Z

ZV ZS

~Þ du dV ½r ð~ srl ^ ð~ ~Þ du dS n srl ^l ~:rðduÞ dS n

ð7Þ

S

However, as it is mentioned in Mindlin (1964, 1965), the last integral of Eq. (7) contains the function rðduÞ, which is not independent of du on S. Only its normal component ^ rðduÞ is independent of du on S. Splitting the gradient n operator into tangential and normal parts on the surface S, the last integral of Eq. (7) can be written as

Z

Z o ^ ^l ~:rðduÞdS ¼ ^l ~: rS þ n ðduÞ dS n n on

S

S

or

Z S

^l ~:rðduÞdS ¼ n

Z

~n ^ Þ½^ ð^ nl n rðduÞ dS

S

þ

Z S

~Þ:rS ðduÞdS ð^ nl

ð8Þ

where rS is the surface gradient defined as

^n ^Þ r rS ¼ ð~I n with ~I denoting the unit tensor. With the aid of the identities (Brand 1966) i h i h ~ÞT du ¼ rS ð^ ~ÞT du rS ð^ nl nl ~Þ:rS ðduÞ þ ð^ nl ~Þ ¼ ðrS n ^ Þ:~ ^ rS ð~ nl lÞ213 rS ð^ lþn

ð9Þ

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 (Mindlin 1964, 1965)

Z

dW ¼ ð10Þ

V

þ

132

~ , which and recalling the symmetry relation ð~ lÞ ¼ l ~ ÞT ¼ n ^l ~, the last integral of Eq. (8) obtains means ð^ nl the form

Z S

~Þ:rS ðduÞdS ¼ ð^ nl

Z

~Þ du

nl frS ½ð^

Z

Z S

P du dS þ

S

R ½^ n rðduÞ dS XI Ca

fE dugdC

ð11Þ

ð15Þ 363

Ca

In view of the fact that dU ¼ dW, Eqs. (14) and (15) imply that the equation of equilibrium for the gradient elastic body with surface energy is

~Þ þ f ¼ 0 r ð~ srl

S

^Þ:~ ^ rS ð~ ðrS n lÞ213 du dS lþn

f du dV þ

ð16Þ

accompanied by the classical boundary conditions

However, the first term of the integrand of the right hand o~ l side of Eq. (11) can be expressed as ^ ðrS l ^ s~ ð^ ^Þ: n ~Þ n ^ rS ð~ PðxÞ ¼ n nn lÞ213

~Þ du ¼ n ^ rS ½n ^ ð^ ~ duÞ

rS ½ð^ nl nl ^ Þð^ ^Þ:~ þ ½ðrS n nn l du ð12Þ R ^ ð^ ~ duÞ dS vanishes ^ rS ½ n Also, the integral S n nl when the surface S is smooth, while it takes the form Z ^ ð^ ~ duÞ dS ^ r S ½n nl n S

¼

XI Ca

^ n ^ Þ:~ fkðm lk dugdC

ð13Þ

for non-smooth boundaries, where Ca are the edge lines formed by the intersection of two surface portions, S1 and ^ ¼ ^s n ^ with ^s being the tangential vector to Ca , S2 , of S, m and the brackets kk indicate that the enclosed quantity is the difference between the values on the surface portions S1 and S2 . Inserting Eqs. (12) and (13) into Eq. (11) and the resulting integral into Eq. (8), the variation of the strain energy given by Eq. (7) takes eventually the form

dU ¼ þ

ZV

~n ^ Þ ½^ ð^ nl n rðduÞ dS

S

Z o~ l ^ ðrS l ^ s~ ð^ ^ Þ: n ~Þ n nn þ on S 213 ^ rS ð~ dudS lÞ n Z ^ Þð^ ^ Þ:l ðrS n ^ Þ:lÞ du dS þ ððrS n nn

Ca

and the non-classical ones

Ca

^ n ^ Þ:~ lk dugdC fkðm

ouðxÞ ¼ q0 on

ð18Þ

^ n ^ Þ:~ EðxÞ ¼ kðm lk ¼ E0 where P0 , u0 , R0 , q0 and E0 denote prescribed values. Comparing the equilibrium equation (16) with that of classical elasticity theory, one can see that in the gradient elastic theory with surface energy the total macroscopic stress tensor r~ is the sum of the classical stress tensor s~ and the relative stress tensor ~s , as illustrated by the relations

r~ ¼ s~ þ ~s ~s ¼ r l ~

s~ ¼ 2l~e þ ktr~e~I 1 ~e ¼ ½ru þ ur

2 tr~e ¼ r u

S

þ

uðxÞ ¼ u0

ð19Þ

The stress tensor s~ , called by Mindlin (1964, 1965) the Cauchy stress tensor, is dual in energy to the strain tensor ~e , while the double stress tensor l ~ is dual in energy to the strain gradient r~e . Mindlin (1964, 1965), considering isotropic materials and a special case of his theory where the macroscopic strain coincides with the micro-deformation, relates the two tensors s~ and ~e via the classical Hooke’s law, which for a linear isotropic elastic medium is written in the form

~Þ du dV ½r ð~ srl

XI

ð17Þ

and/or

^l ~n ^ ¼ R0 and/or RðxÞ ¼ n

Ca

Z

on ^ Þð^ ^Þ:~ ^ Þ:~ þ ðrS n nn l ðrS n l ¼ P0

ð14Þ

ð20Þ

where k and l are the Lame’ constants and ~I the unit tensor in three dimensions. In the same theory the third ~ and r~e are related to each other through a order tensors l

constitutive law which employ’s five material constants plus the two classical Lame’ constants. A simpler and mathematically more tractable constitutive equation which correlates the double stress tensor with the space derivatives of strains is that proposed by Vardoulakis and Sulem (1995), i. e.,

~ ¼ ‘ ~s þ g 2 r~s ¼ 2l ‘ ~e þ ktr~e ‘ ~I l

2

2

þ 2lg r~e þ kg ðrr uÞ~I 364

ð21Þ

where g 2 is the volumetric energy strain gradient with surface energy coefficient, the only constant which relates the microstructure with the macrostructure, and ‘ is the surface energy strain gradient coefficient. Adopting the above strain gradient theory with surface energy and inserting the constitutive Eqs. (20) and (21) into (16) one obtains the equation of equilibrium of a gradient elastic continuum with surface energy in terms of the displacement field u in the form

lr2 u þ ðk þ lÞrr u g 2 r2 lr2 u þ ðk þ lÞrr u þ f ¼ 0

ð22Þ

3 Gradient elastic fundamental solution with surface energy In this section the static fundamental solution of an infinitely extended gradient elastic material with surface energy is explicitly derived. This fundamental solution is defined as the solution of the partial differential equation =~ u ðrÞ ¼ dðx yÞ~I

ð23Þ

rr ðk þ 2lÞ r2 uðrÞ g 2 r4 uðrÞ þ rr ðk þ 2lÞ r2 AðrÞ g 2 r4 AðrÞ ~ ðrÞ g 2 r4 G ~ ðrÞ þ r r l r2 G 1 1 ~ r r ¼ rr I 4pr 4pr

ð28Þ

Due to the irrotational and solenoidal nature of uðrÞ and ~ ðrÞ, respectively, Eq. (26) is satisfied identically G ~ ðrÞ are solutions of the equations if AðrÞ ¼ 0 and uðrÞ, G

1 ðk þ 2lÞ r2 uðrÞ g 2 r4 uðrÞ ¼ ð29Þ 4pr ~ ðrÞ g 2 r4 G ~ ðrÞ ¼ 1 ~I l r2 G ð30Þ 4pr ~ ðrÞ functions, which satisfy The scalar uðrÞ and tensor G Eqs. (29) and (30), respectively, have the form 1 r g2 er=g c1 þ uðrÞ ¼ ð31Þ þ g2 r r 4pðk þ 2lÞ 2 r 2 r=g ~ ðrÞ ¼ 1 r þ g g 2 e ~I c2 ~I G ð32Þ r r 4pl 2 r where c1 and c2 are constants, both equal to zero, since for ~ should give, via Eq. (25), the classical g 2 ¼ 0, u and G elastic fundamental solution. Inserting Eqs. (31) and (32) into Eq. (25) and considering A ¼ 0, the fundamental solution of Eq. (22) takes the final form

~ ðr; l; m; g Þ ¼ u

1 Wðr; m; g Þ~I Xðr; gÞ^r ^r 16plð1 mÞ ð33Þ

where d is the Dirac d-function, x is the point where the where m is the Poisson ratio, ^r the unit vector in the di~ due to a unit force applied at a point rection r ¼ y x and X, W are scalar functions given by displacement field u y should be obtained, r ¼ jx yj and = is the linear op- the relations erator 2 2

1 6g 6g 6g 2 Xðr; gÞ ¼ þ 3 þ 2 þ er=g ð34Þ 3 r r r r r ð24Þ 1 Wðr; m; gÞ ¼ ð3 4mÞ þ 2ð1 2mÞ r According to the Helmholtz decomposition applied to 2 2 g g g r=g dyadic fields (Dassios and Lindell 2001), the fundamental

3þ 3þ 2 e ~ ðrÞ can be decomposed into irrotational and solution u r r r solenoidal parts as 2 2 g g g 1 r=g ð35Þ þ 4ð1 mÞ 3 3 þ 2 þ e ~ ðrÞ ~ ðr Þ ¼ rruðrÞ þ rr Aðr Þ þ r r G u r r r r ð25Þ For the gradient coefficient g being equal to zero, one can where uðrÞ is a scalar function, AðrÞ a vector function and easily prove that ~ ðrÞ a dyadic function. Substituting Eq. (25) into Eq. (23) G 1 ð3 4mÞ and taking into account the relation ð36Þ XðrÞ ¼ ; Wðr; mÞ ¼ r r 1 ¼ dðrÞ ð26Þ which are the expressions of the 3-D classical elastostatic r2 4pr fundamental solution (Brebbia and Dominguez 1992). Utilising the Taylor expansion and the identity = lr2 þ ðk þ lÞrr g 2 r2 lr2 þ ðk þ lÞrr

r2 ¼ rdiv r rot Eq. (23) takes the form

ð27Þ

r r2 r3 r4 er=g ¼ 1 þ 2 3 þ 4 3!g 4!g g 2!g

ð37Þ

it is easy to prove that both functions X and W given by forces f and f , respectively, the integral relation (46) Eqs. (34) and (35), respectively, are regular with respect to leads to Z the distance r ! 0 according to the asymptotic relations Z 2 2 2

XðrÞ ¼ Oð1Þ;

Wðr; mÞ ¼ OðrÞ

ff u f u gdV þ g

ð38Þ

V

4 Reciprocal integral identity for gradient elasticity with surface energy It is well known that in the framework of classical linear elastic theory, Betti’s reciprocal identity (Brebbia and Dominguez 1992) 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

¼

r w ¼ r ½ð~ s þ ~sÞ u ð~ s þ ~s Þ u

Z

¼

Z

ð41Þ

2

s~:ru s~ :ru ¼ 0

þ

Z S

¼

ð43Þ

Thus, Eq. (42) takes the form

r w ¼ ½r ð~ s þ ~sÞ u ½r ð~ s þ ~s Þ u g 2 r2 s~:ru þ g 2 r2 s~ :ru

ð44Þ

The Gauss divergence theorem reads

r wdV ¼

V

Z

^ wdV n

Z

ð45Þ

g

¼

Z V Z

ft u t ugdS

ð49Þ

ff u f u gdV þ

Z

fp u p u gdS

S

Z ou ou dS R R ¼ on on

ð50Þ

S

Z V

r s~:ru r s~ :ru dV 2

Z

for a smooth boundary S, and

s þ ~sÞ u ½r ð~ s þ ~s Þ ugdV f½r ð~ 2

~ Þ:ru ð^ ~Þ:ru gdS nl nl fð^

However, as it was mentioned in Sect. 2, the tensor ru is not independent of u on S. Only its normal ^ ru is independent of u. Thus, in view of component n Eqs. (8)–(13) the reciprocal integral identity (49) obtains the final form

S

V

ð48Þ

S

V

and in view of Eqs. (39) and (44), takes the form

Z

ft u t ugdS

2

However,

Z

S

ff u f u gdV

s g r s~ Þ:ru ð42Þ þ ð~ s g r s~Þ:ru ð~

2

In view of Eq. (21), the above relation is also written in the form

r w ¼ ½r ð~ s þ ~sÞ u ½r ð~ s þ ~s Þ u 2

Z

S

and inserting Eqs. (19)2 and (21) into Eq. (40) one obtains 2

Z o~ s o~ s :ru :ru dV ff u f u gdV þ g on on

V

r ½s~ u ¼ ðr s~Þ u þ s~:ru

ð47Þ

^ ð~ where t ¼ n s þ ~sÞ is the traction vector corresponding to the total stress tensor r~ ¼ s~ þ ~s and acting on the boundary S of the body V. With the aid of the second Green’s integral identity, Eq. (47) is transformed to

ð40Þ

Recalling the identity

ft u t ugdS

S

ð39Þ

where r~ and u are the total stress tensor and the displacement field of a gradient elastic continuum body of 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 Eq. (19)1 , the divergence of w can be written as

V

Z

V

w ¼ r~ u r~ u

r s~ :ru r s~ :ru dV

2

ff u f u gdV þ

S

þ

S

ð46Þ Taking into account that both fields ð~ r; uÞ and ð~ r ; u Þ satisfies the equilibrium equation (16) with body

fp u p u gdS

Z ou ou R dS ¼ R on on S

^ ð~ ^ ð~ s þ ~s Þ ugdS s þ ~sÞ u ½n f½n

Z

XI Ca

fE u E ugdC

ð51Þ

Ca

for a non-smooth boundary S, where the surface tractions p, R and E have the form

365

o~ l ^ ðrS l ~Þ n ^ rS ð~ n lÞ213 on ^ Þð^ ^ Þ:~ ^ Þ:~ þ ðrS n nn l ðrS n l

^ s~ ð^ ^ Þ: P¼n nn

^l ~n ^ R¼n

0 1 Z @ fdðx yÞuðyÞgdVy A ê V

! Z n o T ~ ðx; yÞ uðyÞ PðyÞ u ~ ðx; yÞ dSy ê ½P

þ

^ n ^Þ:~ E ¼ kðm lk

S

ð52Þ 366

¼

From relations (21), (50), (51) and (52) it is immediately apparent that for g 2 ¼ 0, both Eqs. (50) and (51) are reduced to Betti’s reciprocal identity

Z

ff u f u gdV þ

V

Z

ft u t u gdS

S

ð53Þ þ

S

where t is the well known surface traction vector defined as ^ s~. t¼n

5 Boundary integral representation of a gradient elastic problem with surface energy In this section the boundary integral representation of a gradient elastic problem, for the most general case of a non-smooth boundary, is derived with the aid of the reciprocity identity (51). 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 (51), is the result of a body force having the form f ðyÞ ¼ dðx yÞê

ð54Þ

Z ( o~ u ðx; yÞ T RðyÞ ony ! ouðyÞ ~ ðx; yÞ T dSy ê ½R ony I

~ ðx; yÞ fEðyÞ u

C

o ~ ðx; yÞ uðyÞ dCy ½E T

! ê

ð56Þ

Considering that relation (56) is valid for any direction ê and taking into account the symmetry of the fundamental ~ , one obtains the boundary integral displacement u equation

Z n o ~ ðx; yÞ T uðyÞ u ~cðxÞ uðxÞ þ ~ ðx; yÞ PðyÞ dSy ½P S ) Z ( o~ u ðx; yÞ T ouðyÞ T ~ ðx; yÞ dSy ¼ RðyÞ ½R ony ony S I n o ~ ðx; yÞ T uðyÞ dCy ~ ðx; yÞ EðyÞ ½E u þ

C where d is the Dirac d-function and ê the direction of a ð57Þ unit force acting at point y. Recalling the definition of the fundamental solution derived in Sect. 3, it is easy to see where ~cðxÞ is the well known jump-tensor of classical that the displacement field u can be represented by means boundary integral representations (Brebbia and Dom ~ ðx; yÞ given by of the fundamental displacement tensor u ~ and E ~ ~ , R ~ , P ~, Q inguez 1992). Utilizing the symbols U Eq. (33) according to the relation T T T T ~ , o~u , R ~ and E ~ respectively, ~ , P instead of u on ~ ðx; yÞ ê u ðyÞ ¼ u ð55Þ Eq. (57) receives the form Z Inserting the above expression of u in Eq. (51) and as ~ ðx; yÞ PðyÞ dSy ~ ðx; yÞ uðyÞ U ~ cðxÞ uðxÞ þ P suming zero body forces f ¼ 0, one obtains

Z

fdðx yÞê uðyÞgdVy

V

þ

Z

~ ðx; yÞ ê uðyÞ PðyÞ ½~ ½P u ðx; yÞ ê dSy

Z S ouðyÞ ~ ~ Q ðx; yÞ RðyÞ R ðx; yÞ dSy ¼ ony S I n o ~ ðx; yÞ uðyÞ dC ð58Þ ~ ðx; yÞ EðyÞ E þ U y

S

Z o~ u ðx; yÞ ¼ RðyÞ ê ony S ouðyÞ ~ ðx; yÞ ê dSy ½R ony I þ fEðyÞ ½~ u ðx; yÞ ê

C

or

~ ðx; yÞ ê uðyÞ dCy ½E

C

In case the boundary S is smooth, then the integral Eq. (57) is reduced to

Z 1 ~ ðx; yÞ PðyÞ dSy ~ ðx; yÞ uðyÞ U uðxÞ þ P 2 S Z ouðyÞ ~ ~ Q ðx; yÞ RðyÞ R ðx; yÞ dSy ð59Þ ¼ ony S

All the kernels appearing in the integral Eqs. (58) and (59) are given explicity in Appendix I.

Observing Eq. (58), one easily realizes that this equation contains two unknown vector fields, uðxÞ and ouðxÞ on . For example, for the case of the traction field PðxÞ prescribed on S (classical boundary condition) as well as the fields RðxÞ and EðxÞ prescribed on S (non-classical boundary conditions), the unknown vector fields in Eq. (58) are two, uðxÞ and ouðxÞ=on . Thus, the evaluation of the unknown fields uðxÞ and ouðxÞ=on requires the existance of one more integral equation. This integral equation is obtained by applying the operator o=onx on Eq. (58) and has the form

Z ~ ~ ðx; yÞ ouðxÞ oP ðx; yÞ oU ~cðxÞ þ uðyÞ PðyÞ dSy onx onx onx S ) Z ( ~ ~ ðx; yÞ ouðyÞ oQ ðx; yÞ oR dSy ¼ RðyÞ onx onx ony S ) I ( ~ ~ ðx; yÞ oU ðx; yÞ oE þ EðyÞ uðyÞ dCy onx onx

The above described boundary value problem for a 3-D gradient elastic body with smooth surface S admits an integral representation of the form

Z Z 1 ~ ðx;yÞ qðyÞdSy ~ uðxÞ þ P ðx;yÞ uðyÞdSy þ R 2 S S Z Z ~ ðx;yÞ RðyÞdSy ~ ðx;yÞ pðyÞdSy þ Q ¼ U S

S

1 qðxÞ þ 2 ¼

Z S

Z S

~ ðx;yÞ oP uðyÞdSy þ onx

Z S

~ ðx;yÞ oR qðyÞdSy onx

Z ~ ~ ðx;yÞ oU oQ ðx;yÞ pðyÞdSy þ RðyÞdSy onx onx S

ð65Þ

where x is a field point and y a source point laying on the surface S. The gradient fundamental solution of the C ~ , R ~ as well as the remaining kernels P ~, Q ~ problem U ð60Þ appeared in the integral equation (65)1 , as well as the ~ ðx; yÞ=onx ~ ðx; yÞ=onx , oP ~ ðx; yÞ=onx , oQ The kernels appearing in Eq. (60) are given explicitly in kernels oU ~ Appendix II. and oR ðx; yÞ=onx appeared in the integral equation The integral Eqs. (58) and (60) accompanied by the (65)2 , are given in Appendix I and Appendix II, classical and non-classical boundary conditions form the respectively. integral representation of any gradient elastic boundary Next, the boundary element formulation and solution value problem. procedure of a three-dimensional static gradient elastic problem is presented in detail. The goal of the Boundary Element methodology is to solve numerically the 6 well-posed boundary value problem constituted by the Numerical implementation and solution procedure Consider a finite three dimensional (3-D) gradient elastic system of two integral equations (65) and the boundary body of volume V surrounded by a surface S. According to conditions (Eqs. (62)). To this end the smooth surface S is discretised into E eight-noded quadrilateral and/or Mindlin’s (1964, 1965) theory and assuming zero body six-noded triangular quadratic continuous forces acting on the body the static equation of equilibrium and the boundary conditions for the gradient elastic isoparametric boundary elements. For a nodal point k the discretized integral equations (65) have the following body with surface energy read form

~Þ ¼ 0 r ðs~ r l PðxÞ ¼ P0

ð61Þ

and/or u ¼ u0

ð62Þ ou0 ¼ q0 on ^ is the normal unit vector, s~ the classical elastic where n ~ the third order tensor of double forces per stress tensor, l unit, p the external surface traction and R the surface double stresses. Boundary conditions (62)1 and (62)2 are known as classical and non-classical boundary conditions, respectively. Considering the special case where s~ ¼ lðru þ urÞ þ kðr uÞ~I ð63Þ ~ ¼ ‘ ~ s þ g 2 r~ s l ^l ~n ^ ¼ R0 R¼n

and/or q ¼

The governing equation (16) in terms of displacements takes the form

lr2 u þ ðk þ lÞrr u g 2 lr2 u þ ðk þ lÞrr u ¼ 0 ð64Þ

AðeÞ Z 1 Z 1 E X 1 k X ~ xk ;ye ðn1 ; n2 Þ N a ðn1 ; n2 Þ P u x þ 2 e¼1 a¼1 1 1

J ðn1 ;n2 Þdn1 dn2 uea þ

AðeÞ Z 1 Z 1 E X X e¼1 a¼1

~ xk ; ye ðn1 ; n2 Þ N a ðn1 ;n2 Þ R

1 1

J ðn1 ; n2 Þdn1 dn2 qea ¼

AðeÞ Z 1 Z 1 E X X e¼1 a¼1

~ xk ; ye ðn1 ; n2 Þ N a ðn1 ;n2 Þ U

1 1

J ðn1 ; n2 Þdn1 dn2 pea AðeÞ Z 1 Z 1 E X X ~ xk ; ye ðn1 ; n2 Þ N a ðn1 ;n2 Þ Q þ e¼1 a¼1

1 1

J ðn1 ; n2 Þdn1 dn2 Rea

367

AðeÞ

E X 1 k X q x þ 2 e¼1 a¼1

Z1 Z1 ~ k e oP x ; y ðn1 ; n2 Þ a N ðn1 ; n2 Þ onx

~ kb G

¼

1 1

Z1 Z1 1 1

ðn1 ; n2 ÞJ ðn1 ; n2 Þdn1 dn2

uea

J ðn1 ; n2 Þdn1 dn2 AðeÞ Z 1 Z 1 ~ k e E X X oR x ; y ðn1 ; n2 Þ a þ N ðn1 ; n2 Þ onx e¼1 a¼1 1 1

qea

J ðn1 ; n2 Þdn1 dn2 AðeÞ Z 1 Z 1 ~ k e E X X oU x ; y ðn1 ; n2 Þ a ¼ N ðn1 ; n2 Þ onx e¼1 a¼1

368

~ xk ; ye ðn1 ; n2 Þ N a U

~kb L

¼

Z1 Z1 1 1

~ xk ; ye ðn1 ; n2 Þ N a Q


ðe;aÞ!b

1 1

Z1

pea

J ðn1 ; n2 Þdn1 dn2 S~kb ¼ 1 Z1 Z AðeÞ E ~ xk ; ye ðn1 ; n2 Þ XX oQ 1 þ N a ðn1 ; n2 Þ on x e¼1 a¼1 1 1

J ðn1 ; n2 Þdn1 dn2 Rea ð66Þ

ðe;aÞ!b

~k ¼ T

Z1 1

~ xk ; ye ðn1 ; n2 Þ oP Na onx


Z1 Z1

ðe;aÞ!b

oR xk ; ye ðn1 ; n2 Þ a N onx ’

b where A(e) is the number of nodes of the current element e 1 1 (A = 8 or 6 for quadrilateral or triangular elements, re spectively), N a (a ¼ 1; 2; . . . ; A) the shape functions of a

ðn1 ; n2 ÞJ ðn1 ; n2 Þdn1 dn2 typical quadrilateral or triangular quadratic element, J the corresponding Jacobian magnitude of the transformation ðe;aÞ!b from the global to the local co-ordinate system n1 , n2 and 1 1 Z Z ~ k e oU x ; y ðn1 ; n2 Þ a uea , qea , pea and Rea are the nodal values of the corresponding k ~ N field functions. Adopting now a global numbering for the Vb ¼ onx nodes, each pair (e, a) is associated to a number b and the 1 1 integral equations (66) are written as

1 k u þ 2

L X

~ kb ub þ H

b¼1

L X b¼1

~ kb qb ¼ K

L X

~ kb pb þ G

b¼1

L X


~kb Rb L

b¼1

L L L L X X X 1 k X k b ~ k qb ¼ ~Skb ub þ ~ ~ kb Rb q þ p þ T V W b b 2 b¼1 b¼1 b¼1 b¼1

~ kb ¼ W

Z1

1 1

~ kb ¼ H

Z1

Z1

~ kb K

¼

Z1 Z1

ðe;aÞ!b

’ R xk ; ye ðn1 ; n2 Þ N a

1 1


Na

ðe;aÞ!b

~ ~ L ~ K p u G ð69Þ 1~ ~ q ¼ V ~ W ~ R I þ T 2 ~, G ~, L ~, K ~ , ~S, T ~, V ~ and W ~ contain all the where matrices H submatrices given by Eqs. (68), respectively. Applying the boundary conditions (Eqs. (62)) and rearranging Eq. (69) one produces the final linear system of algebraic equations of the form ~ X¼B A ð70Þ ~þ H ~ S~

2I

1 1


onx

ðe;aÞ!b

Collocating Eqs. (67) at all nodal points L, one obtains the following linear system of algebraic equations

1

~ xk ; ye ðn1 ; n2 Þ N a P

~ xk ; ye ðn1 ; n2 Þ oQ


ð67Þ where L is the total number of nodes and

Z1

ð68Þ

ðe;aÞ!b

where the vectors X and B contain all the unknown and known nodal components of the boundary fields. When b 6¼ k, integrals of Eqs. (68)1–8 are non-singular and can be easily computed by Gauss quadrature, utilizing

in the present work 6 6 integration points. In the case of b ¼ k, the integrals of Eqs. (68)3 , (68)4 and (68)7 are also non-singular, while the remaining integrals of Eqs. (68)2 , (68)8 , (68)1 , (68)6 and (68)5 become singular with the first two being weakly singular integrals, the next two strongly singular (CPV) integrals and the last one a hypersingular integral. In the present work the singular integrals are evaluated with high accuracy applying a methodology for direct treatment in unified manner of CPV and hypersingular integrals proposed by Guiggiani et al. (1990) and Huber et al. (1993), respectively. According to this methodology a local polar co-ordinate (q, h) system, centered at the singularity, is introduced and the aforementioned singular integrals, those of Eqs. (68)2 , (68)8 , (68)1 , (68)6 and (68)5 , are transformed as follows

~ kk ¼ W

Zh2 Zq^

Zh2 Zq^

Zh2

! ~ðhÞ h ~ ðq; hÞ dq dh H q

c2 ¼

ð71Þ

^ðhÞ ~ðhÞ lnq h bðhÞdh

0

h1

Zh2 Zq^ 2 1 ~Sðq; hÞ ~sðhÞ ~sðhÞ dq dh ¼ q2 q h1

þ

where ur is the radial displacement, R is the double surface traction 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

~ ðq; hÞdq dh W

Zh2 Zh2 Zq^ q^ðhÞ ~tðhÞ dh ~ ðq; hÞ dq dh þ ~tðhÞ ln ¼ T q bðhÞ h1

~Skk

ð73Þ

c1 ¼

h1

~k T k

Rjr¼a ¼ 0

0

Zh2 ( h1

ð74Þ

where

0

h1

þ

ð72Þ

1 ‘F ðaÞ 2 a d FðrÞ d F ðr Þ 2 ag a ‘F ðaÞ a‘ dr r¼a dr2 r¼a

0

h1

~ kk ¼ H

~ ðq; hÞdq dh K

0

h1

ur ðr ¼ aÞ ¼ u0

ur ðrÞ ¼ c1 r þ c2 F ðr Þ

Zh2 Zq^

~ kk ¼ K

7.1 Radial deformation of a sphere Consider a gradient elastic with surface energy solid sphere with radius a subjected to an external uniform radial deformation and assume that the double surface traction vanishes at the boundary, i.e.,

" #) ^ c ð h Þ 1 q ð h Þ 1 2~sðhÞ ~sðhÞ ln dh þ ^ðhÞ bðhÞ ðbðhÞÞ2 q

~ ðq; hÞ and S~ðq; hÞ are the ~ ðq; hÞ; W ~ ðq; hÞ; H ~ ðq; hÞ; T where K integrands of the integrals in Eqs. (68)2 , (68)8 , (68)1 , (68)6 and (68)5 respectively, expressed in polar co-ordinates, while all the remaining parameters are given in the Appendix IV. The obtained integrals (Eqs. (71)) are now regular and can be easily computed by Gauss quadrature, involving in the present work 6 6 integration points for the first four integrals and 8 8 integration points for the last one.

‘F ðaÞ a‘dFðrÞ dr

‘ 2

ag 2 d Fð2rÞ dr r¼a

ð75Þ

ð76Þ r¼a

sinhðr=g Þ coshðr=g Þ ð77Þ þg 2 r r The problem has also been solved numerically by the BEM presented in the previous section for a ¼ 1 and u0 ¼ 1. Due to the symmetry of the problem, only one octant of the sphere needs to be discretized. In the present work, a mesh of thirty-eight quadrilateral quadratic elements was used. The radial displacement and its first derivative (radial strain) as functions of the distance r for a value of the material characteristic length g 2 and different values of the surface energy parameter ‘ have been evaluated. The results, as it is evident in Fig. 1a and b, are in a very good agreement with those obtained analytically by using Eq. (74). In the same figures the classical elasticity solution is also displayed for reasons of comparison. F ðrÞ ¼ g 2

7.2 Tension of a bar The second numerical problem deals with a gradient elastic with surface energy bar of diameter D and length 2L subjected to a constant axial stress. The classical boundary conditions of the problem read rðþLÞ ¼ T0 duðxÞ ¼0 dx

ð78Þ ð79Þ

x¼þL

7 and the non-classical ones Numerical Examples In this section two characteristic problems with known RðLÞ ¼ 0 analytical solutions are presented to illustrate the accuracy uðxÞjx¼L ¼ 0 of the proposed 3-D BEM.

ð80Þ ð81Þ

369

where u and r are the axial displacement and stress, reT0 spectively, R denotes the surface double traction and T0 is c ¼ ; the applied axial load at one end of the bar. The analytical 1 E solution of this one-dimensional (1-D) problem has the form (Tsepoura et al. 2002)

uðxÞ ¼ c1 jxj þ c2 þ c3 g 2 ejxj=g þ c4 g 2 ejxj=g ;

where 370

jx jL ð82Þ

w 2 T0 g‘ 1 þ e g

; c2 ¼ 2w 2w E g þ ge g þ ‘ ‘e g w w T0 ‘ þ ge g ‘e g ; c3 ¼ 2w 2w Eg g þ ge g þ ‘ ‘e g w w T0 e g g ‘ þ ‘e g c4 ¼ 2w 2w Eg g þ ge g þ ‘ ‘e g

ð83Þ

with E being the Young modulus and w ¼ 2L. In order to demonstrate the accuracy of the proposed 3D BEM, the above-described 1-D problem has been solved utilizing a 3-D model. According to this model, the axial bar in tension is modelled by a thick solid axisymmetric cylinder with height 2L ¼ 2:4a and diameter D ¼ 8:4a, as shown in Fig. 2. 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 of 0:2a. The discretization was restricted to one quarter of the cylinder because of symmetry and involved 268 quadratic quadrilateral boundary elements. The problem has been solved for a ¼ 0:5 and T0 =E ¼ 1 and the distribution of the axial displacement and its first derivative (axial strain) along the axis x are displayed in Figs. 3 and 4 for different values of the material characteristic length g 2 and the surface energy parameter ‘. The numerical results are in a very good agreement with those obtained by using Eq. (82). In the same figures the classical elasticity solution is also displayed for reasons of comparison.

8 Conclusions On the basis of the material presented in the previous sections, one can draw the following conclusions: 1) A boundary element method has been developed for the static analysis of three-dimensional bodies characterized by a linear elastic material behavior taking into account microstructural effects with the aid of a simple strain gradient theory with surface energy.

Fig. 1. a Radial displacement versus radial distance for the solid sphere for g 2 ¼ 0:09 and various values of ‘. b Radial strain versus radial distance for the solid sphere for g 2 ¼ 0:09 and Fig. 2. Geometry of the thick solid axisymmetric cylinder in axial various values of ‘ tension

371

Fig. 3. a Axial displacement versus lateral distance for g 2 ¼ 0:01 Fig. 4. a Axial displacement versus lateral distance for g 2 ¼ 0:09 and various values of ‘. b Axial strain versus lateral distance for and various values of ‘. b Axial strain versus lateral distance for g 2 ¼ 0:01 and various values of ‘ g 2 ¼ 0:09 and various values of ‘

2) All possible boundary conditions (classical and nonimplementation is accomplished with the aid of quaclassical ones) have been determined with the aid of a dratic quadrilateral elements and advanced integration variational statement of the problem. algorithms for the highly accurate evaluation of singular 3) The fundamental solution of the gradient elastic with integrals. surface energy problem has been explicitly determined 5) Two numerical examples have been used to illustrate and used to establish the boundary integral representhe application of the method and demonstrate its adtation of the solution of the problem with the aid of a vantages, which are the surface-only discretization reciprocal identity, specifically constructed for this character of the method and its high accuracy. The gradient elastic with surface energy case. present version of the implementation is restricted to 4) The boundary integral equations of the problem smooth boundaries and computation of boundary consist of one equation for the displacement and tractions and stresses. Work is under way to remove another one for its normal derivative. Their numerical both of these restrictions.

372

Appendix I In this Appendix the explicit expressions of the kernels appearing in the integral equations (58) are given as follows: 1 ~ ðx; yÞ ¼ W~I X^r ^r ðI:1Þ U 16plð1 mÞ T ~ ~ ðx;yÞ ¼ oU ðx; yÞ Q ony 1 2X dX ^ y ^r ^r ^r n ¼ 16plð1 mÞ r dr dW X ^y ^ y ^r ~I n ^y ^r þ ^r n þ n dr r ðI:2Þ ’ g dA 3A A R ðx; yÞ ¼ ð^ ny ^rÞ2 þ ^r ^r 16pð1 mÞ dr r r 2 B dB B dB B A ^ y ^r þ ~I þ n þ þ dr r r dr r r dC C A ^ y ^r ^r n ^y ^r þ ^y ^y ^r n þ n

n dr r r BþC 1 ^y ‘ ^y þ ^y n A n þ n r 16pð1 mÞ ^y ^r þ B n ^y ‘ ^ y ^r ^r ^r þ B n ^y ‘ n

n

~ are given as follows The terms of kernel P

1 ^y ^r ^r ^r þ B n ^ y ^r ~I A n 16pð1 mÞ 1 dA 3A ^y þ þ B^ ny ^r þ C^r n 16pð1 mÞ dr r A ^ y ^r ^r ^r þ ^ y ‘ ^r ^r

‘ ^r n n r dB B ^ y ^r ‘ ^r þ ^ y ^r ^r ‘ þ n þ n dr r dC C ^y ^r ~I þ n ^ y ^r þ n

‘ ^r ‘ ^r dr r B B C ^y ^y þ ^y ‘ ~I þ n ^y ‘ þ ‘ n

^r n n r r r

T ^ y s~ ¼ n

2

~ ^y ^ y ^r I þ C n ^ y ‘ ^r n

n

ðI:3Þ

"

o l ^y s~ þ n ^y n ^y : ^ y r l n P ¼ n ony ^y n ^ y : l ^ y r l2134 þ rS n ^y n n #T ^y : l rS n ðI:4Þ

~

’

g2 dA 3A ^ ^rÞðm ^ ^rÞ^r ^r ðn 16pð1 mÞ dr r dB B dB B ~ ^ ^ ^ ^ ^ ^r ðn rÞðm rÞI þ m þ dr r dr r dC C A ^r m ^ þ ðm ^ ^rÞðn ^ ^r þ ^r n ^Þ þ dr r r C B 1 ^m ^þ m ^ n ^ þ þ n r r 16pð1 mÞ ^ ^rÞ^r ^r þ B n ^‘ m ^ ^r ^ ‘ ðm

A n

E ðx; yÞ ¼

~ ^ ^rÞI þ C n ^ ‘ ^r m ^ ^ ‘ ðm þB n

ðI:5Þ

ðI:6Þ "

#T

o l 1 ^y : ^y n ¼ n ony 16pð1 mÞ i h ^y ^ y þ G4 n ^y ^r þ G5 n ^y n

G1^r ^r þ G2~I þ G3^r n

ðI:7Þ 2 g d A 2 dA 12A 2 þ 16pð1 mÞ dr2 r dr r 2 d B 2 dB 2B 2A ^ y ^r ^r ^r þ þ þ 2

n dr2 r dr r2 r 2 d C 2 dC 2C 2A ^ y ^r þ ^y ^r ~I þ n 2 þ 2 þ

n dr 2 r dr r r 1 A ^y þ ^ y ^r ‘ ^r þ ^r ‘

^r n n 16pð1 mÞ r A dA 3A ^ y ^r þ ^y ‘ ^r ^r þ n ‘ ^r n dr r r B dB B ^ y ^r þ ^y ‘ ~I þ n ‘ ^r n dr r r dB B dC C ^ y ^r þ þ ‘ ^r n ‘ ^r dr r dr r B C ^ y ‘ þ ‘ ^ ^y þ n ðI:8Þ ny

^r n r r 2 h iT g2 d B 3 dB 3B 2134 ^y r l ¼ þ n 16pð1 mÞ dr2 r dr r2 1 dC C 1 ^ y ^r ~I þ 2 n þ r dr r 16pð1 mÞ dB 3B C ^ y ‘ ~I þ þ n

ðI:9Þ dr r r h iT ^ y r l ¼ n

2

h iT g 2 rS n ^y ^y n ^y : l ¼ ^y n rS n 16pð1 mÞ 2 A dA 3A ^ y ^r þ ^r ^r n

dr r r 2 B dB B dB B A ~ ^y ^r þ I þ n þ þ dr r r dr r r dC C A ^y ^r ^r n ^ y ^r þ ^y ^ y ^r n þ n

n dr r r ^y rS n BþC ^y ‘ ^y þ ^y n A n þ n 16pð1 mÞ r ^y ^r þ B n ^y ‘ n ^y ‘ ^y ^r ^r ^r þ B n

n ^y ^y ‘ ^r n ^ y ^r ~I þ C n ðI:10Þ

n

h iT g2 ^ y : l ¼ rS n 16pð1 mÞ dA 3A A ^ y ^r ^r ^y : ð^r ^rÞ þ rS n rS n

dr r r dB B B ^ y ~I ^ y : ð^r ^rÞ þ rS n þ rS n dr r r A ^ y þ ^r rS n ^y ^r þ ð^r ^rÞ rS n r dB B ^ y ð^r ^rÞ rS n þ dr r dC C ^y Þ ^r ^r ðrS n þ dr r C B T ^y Þ þ ðrS n ^y Þ þ ðrS n r r 1 ^ y Þ:ð‘ ^rÞ½A^r ^r þ B~I

ðrS n þ 16pð1 mÞ T ^y ‘ ^r þ C rS n ^ y ‘ ^r þ B rS n

ðI:11Þ where

2 3 d A 7 dA 15A ^ y ^r G1 ¼ g n þ 2 2 dr r dr r 1 dA 3A ^ y ^r 2 n þ3 r dr r 2 A dA 3A ^y ‘ n ^y ^r þ ^y ‘ n þ n dr r r 2

ðI:16Þ 2 3 d B 3 dB 3B ^ y ^r þ n G2 ¼ g 2 2 2 dr r dr r 1 dB B ^ y ^r 2 n þ3 r dr r 2 B dB B ^y ‘ n ^ y ^r þ ^y ‘ þ n n dr r r ðI:17Þ 2 2 d C 3 dC 3C 2 dA 6A ^y ^r þ 2 þ 2 n G3 ¼ g 2 2 dr r dr r r dr r 2A 1 dC C þ 2 þ r r dr r2 A dC C ^ y ^r n ^y ‘ þ n ðI:18Þ þ r dr r 2 2 d B 3 dB 3B 2 dA 6A 2 ^ y ^r G4 ¼ g þ 2 þ 2 n 2 dr r dr r r dr r 2A 1 dB B þ 2 þ r r dr r 2 A dB B ^ y ^r n ^y ‘ þ n ðI:19Þ þ r dr r 1 dB B 1 dC C A 2 ^ y ^r þ þ n G5 ¼ g 2 r dr r2 r dr r2 r2 B C ^y ‘ þ þ n ðI:20Þ r r

r ^r ¼ ; r

r ¼ y x; r ¼ jrj ðI:12Þ 1 6g 2 6g 2 6g 2 r=g X¼ þ 3 þ 2þ e ðI:13Þ r r3 r r r 2 2 1 g g g r=g W ¼ ð3 4mÞ þ 2ð1 2mÞ 3 þ 3 þ 2 e r r r r 2 2 g g g 1 ðI:14Þ þ 4ð1 mÞ 3 3 þ 2 þ er=g r r r r 2X dX dW X A¼2 ; B¼ ; r dr dr r ðI:15Þ 2m dW dX 2X 2X C¼ 1 2m dr dr r r

Appendix II In this Appendix the explicit expressions of the kernels appearing in the integral equation (60) are given as follows: ~ ðx; yÞ o U 1 dW ^x ^rÞ~I ¼ ðn onx 16plð1 mÞ dr dX 2 ^x ^rÞ^r ^r X ðn þ dr r X ^x Þ ^x ^r þ ^r n þ ðn ðII:1Þ r

373

374

~ ðx; yÞ o Q 1 5 dX 8 d2 X X 2 ¼ onx 16plð1 mÞ r dr r2 dr 2 1 dX ^ x ^r ^r ^y n

ð^ ny ^rÞð^ nx ^rÞ^r ^r 2 X n r r dr 2 1 dX ^y ^r ðn ^ x ^r þ ^r n ^x Þ n 2X r r dr 2 d W 1 dW 1 dW ^ y ^r ðn ^ x ^rÞ~I ^ x ~I ^y n n n 2 dr r dr r dr 2 1 dX 1 ^ x ^rÞ^ ^x ðn ny ^r þ 2 X^ ny n 2X r r dr r 2 1 dX 1 ^ ^ ^ ^ 2X ðnx rÞ^r ny þ 2 X^ nx ny ðII:2Þ r r dr r h’ i o R ðx; yÞ

dB B ^ y ^r ðn ^ x ^rÞ n ^y ‘ n dr r B ^x n ^y ‘ ~I ^y n þ n r dC C ^ x ^rÞ n ^y ^ y ‘ ^r n ðn dr r C ^x n ^y ^y ‘ n ðII:3Þ n r ~ ðx; yÞ o l o P o h ^ y ~s þ n ^y n ^y : : n ¼ onx ony onx ^ y r l2134 ^y r l n n iT ^y n ^ y : l rS n ^ y : l ^y n þ rS n

ðII:4Þ h’ i 2 g2 dA1 4A1 o E ðx; yÞ ^ y ^r ðn ^x ^rÞ^r ^r n ¼ 16pð1 mÞ dr r onx 2 2A1 A1 ^x Þ ^ y ^r ð^ ny ^ nx Þ^r ^r nx ^r þ ^r n ð^ ny ^rÞð^ n g2 dA1 4A1 ^ x ^rÞ n ^ y ^r m ^ y ^r r r ¼ ðn 16pð1 mÞ dr r dA6 2A6 A6 ^ x ^rÞ^r ^r ðn ^x Þ ^ x ^r þ ^r n ðn A1 A1 dr r r ^y m ^y n ^ y ^r þ ^ y ^r ^r ^r ^x n ^x m þ n n r r 2 dA2 2A2 ^ y ^r ðn ^x ^rÞ~I n A1 dr r ^ y ^r ðn ^x ^r þ ^r n ^x Þ ^y ^r m n r dA7 2A2 ^y n ^x ~I ^ y ^r n ^ x ^rÞ~I ðn n dA2 2A2 r dr ^ x ^rÞ n ^y ^r m ^ y ^r ðn dr r dA3 2A3 ^ y ^r ðn ^ x ^rÞ^ n ny ^r A1 A2 dr r ^y m ^y n ^ y ^r þ ^y ^r ~I ^x n ^x m þ n n r r A3 A3 ^x n ^y n ^x ^y ^r ^y n ^ y ^r n n n dA2 A2 A r r ^ x ^rÞm ^ y ^r 2 m ^x ^y n ðn dA4 2A4 dr r r ^ y ^r ðn ^ x ^rÞ^r n ^y n dr r dA9 A9 A9 ^ x ^rÞ^r m ^y n ^y ^x m ðn A4 A4 dr r r ^x ^r n ^x n ^y ^y ^y n ^ y ^r n n n r r dA6 2A6 ^ x ^rÞ m ^ y ^r ðn dA5 1 dr r ^y þ ^ x ^rÞ^ ðn ny n dr 16pð1 mÞ A6 A ^ y ^r 6 m ^ ^y n ^x ^ ^ y ^r n þ my nx n dA 3A r r ^y ^r ðn ^x ^rÞ n ^y ‘ n

dr r dA6 2A6 ^ x ^rÞ m ^ y ^r ðn A ^x n dr r ^y ‘ ^r ^r ^y n þ n r A6 A6 ^y ^ x ^r n ^x n ^y A ^y n ^ y ^r n þ m m ^ y ‘ ðn ^x Þ ^ x ^r þ ^r n ^y ^r n n r r r dA8 dA7 dB B ^y ^y n ^y ^x ^rÞ^ ^ x ^rÞm ðn ny m ðn ^ y ^r ^ x ^rÞ n ^y ‘ n ðn dr dr dr r 1 B þ ^x ^y n ^y ‘ n n 16pð1 mÞ r onx

dA 3A dA6 2A6 ^y ^r ðn ^ y ^r ðn ^x ^rÞ n ^y ‘ ^ x ^rÞ n m

dr r dr r A A6 A6 ^y n ^x ^r ^r ^ x ‘ ^r ^y n ^ y ^r ‘ ^ ^y ‘ m nx n n þ n þ r r r A dA2 2A2 ^y ^r ðn ^ x ^r þ ^r n ^ y ‘ ðn ^x Þ ^ x ^rÞ ‘ ^r ^ y ^r n m n dr r r dB B A2 A2 ^ y ‘ ðn ^ x ^rÞm ^ y ^r ^x ‘ þ ^x ‘ ^r ~I ^y ^r n ^y n n n n þ r r dr r B dA2 2A2 ^ x ^rÞ ‘ ^r ^y n ^x ^y ‘ m ðn n dr r r dB B A2 A ^y ^r 2 ‘ ^r n ^ y ^r ðn ^ x ^rÞ n ^y ‘ ^y n ^x ^x ‘ n m þ n r r dr r B dA9 2A9 ~ ^ x ^rÞ ‘ ^r ^y n ^x I ^y ‘ m ðn þ n dr r r dC C A9 A ^ y ‘ ðn ^ x ^rÞ^r m ^ y 9 ‘ ^r n ^y ^x n ^y ^ x ‘ ^r n n n þ r r dr r C dA7 dA ^x m ^ x ^rÞ~I 7 ðn ^y ^ y ‘ ðn ^ x ^rÞ^ ^y ‘ n ðII:5Þ ny ‘ n n dr dr r ~ ðx;yÞ

o½ P dA8 The terms of kernel onx are given as follows ^ x ^rÞ‘ ^ ðII:6Þ ðn ny T dr ^ y ~s o n " # T o l onx o ^y : ^y n n g2 dA 3A ony onx ^y ^r ðn ^x ^rÞ^r ^r n ¼ 16pð1 mÞ dr r 3 g2 dB1 5B1 ^y ^r ðn ^x ^rÞ^r ^r n ¼ A A 16pð1 mÞ dr r ^ x ^r ^r n ^ x ^r þ ^r n ^x Þ ^y n ^ y ^r ðn n r r 2 3B1 ^ x ^r ^r ^y n ^ y ^r n n dB B B ~ ~ r ^ y ^r ðn ^ x ^rÞI n ^x I ^y n n dr r r 3 B1 ^x Þ ^ x ^r þ ^r n ^ y ^r ðn n dB B B ^ x ^rÞ^ ^x ^y n ðn ny ^r n r dr r r dB2 3B2 ^ y ^r ðn ^x ^rÞ^r ^r n dC C C dr r ^ x ^rÞ^r n ^y n ^y ^x n ðn B2 B2 dr r r ^ x ^r ^r ^ x ^r þ ^r n ^x Þ ^y n ^ y ^r ðn n n r r 1 dA1 4A1 ^ y ^r ðn ^x ^rÞ ‘ ^r n þ 3 dB3 3B3 dr r 16pð1 mÞ ^ y ^r ðn ^ x ^rÞ~I n dr r A1 A1 ^x ‘ þ ^x ‘ ^r ^r ^r ^y ^r n ^y n n n þ 2 3B3 r r ^ x ~I ^y n ^ y ^r n n r A1 ^ x ^r þ ^r n ^x Þ ^y ^r ‘ ^r ðn n dB4 B4 B4 ^y ^r ðn ^x ^rÞ~I ^ x ~I ^y n r n n dr r r dA6 2A6 2 ^ x ^rÞ n ^y ‘ ^r ^r ðn dB5 3B5 ^ y ^r ðn ^y ^ x ^rÞ^r n dr r n dr r A6 2B5 ^ x ^r þ ^r n ^x Þ ^ y ‘ ðn n ^ x ^r n ^ y ^r n ^y ^y n n r r dA6 2A6 2 B5 dB6 B6 ^y ^r ðn ^ x ^rÞ n ^ x ^rÞ^r n ^y ^y ^x n ^ y ^r n ðn n dr r r dr r A6 A 2 B6 dB7 3B7 ^ x ^r ‘ 6 n ^y n ^ y ^r n ^x ‘ þ n ^ y ^r ðn ^ ^ x ^rÞ^ ^ n n ny ^r n x y r r r dr r

375

2B7 ^y n ^x n ^ y ^r n ^ y ^r n r 2 B7 ^x ^y n ^ y ^r n n r dB8 B8 B8 ^ x ^rÞ^ ^x ^y n ðn ny ^r n dr r r dB9 B9 ^y ^r ðn ^x ^rÞ^ ^y n ny n dr r B9 ^x n ^y ^y n ^y n n r 1 dA1 4A1 ^ y ^r ðn ^ x ^rÞ n þ dr r 16pð1 mÞ 2A1 ^ ^ ^ ^ ^ þ ny nx ny r ny ‘ ^r ^r r 2 A1 ^ x ^r þ ^r n ^x Þ ^ y ‘ ðn ^y ^r n n r dA6 2A6 ^ y ‘ ðn ^x ^rÞ^r ^r n dr r A6 ^ x ^r þ ^r n ^x Þ ^ y ‘ ðn n r 2 dA7 d B 2 dB ~ ^ ^ ^ x ^rÞ ^ n ðnx ^rÞ ny ‘ I ^ r ðn y dr dr2 r dr 2 dB ^ y ^r n ^x n ^ y ‘ ~I ^y n þ n r dr dA7 2A7 ^y ^r ðn ^ x ^rÞ n þ dr r 2A7 ^y ^r n ^x n ^ y ‘ ~I ^y n n þ r dA4 2A4 ^y ^r ðn ^ x ^rÞ n dr r A4 ^y ‘ ^r n ^y ^x ^y n n n þ r A4 ^x n ^y ‘ n ^y ^y ^r n n r dA5 ^y n ^y ‘ n ^y ^x ^rÞ n ðn dr dA3 2A3 ^y ^r ðn ^ x ^rÞ n dr r A3 ^y ‘ n ^ y ^r ^x ^y n n n þ r A3 ^y n ^y ‘ n ^x ^y ^r n ðII:7Þ n r

376

iT o h ^ y r l n onx g2 dF1 3F1 ^y ^r ðn ^x ^rÞ^r ^r n ¼ 16pð1 mÞ dr r F1 F1 ^ x ^r ^r ^x ^r þ ^r n ^x Þ ^y n ^y ^r ðn n n r r dF2 F2 F2 ^ y ^r ðn ^ x ^rÞ~I ^x ~I ^y n n n dr r r dF2 F2 F2 ^ x ^rÞ^ ^x ^y n ðn ny ^r n dr r r dF3 F3 F3 ^ x ^rÞ^r n ^y n ^y ^x n ðn dr r r 1 dA6 2A6 ^ y ^r ðn ^ x ^rÞ n þ dr r 16pð1 mÞ A6 A6 ^ x ‘ ^r ^y n ^ y ^r ‘ ^ þ nx n n r r dA6 2A6 ^ y ^r ðn ^ x ^rÞ n dr r A6 A6 ^ x ^r ‘ ^x ‘ ^y n ^ y ^r n þ n n r r A1 ^ x ^r þ ^r n ^x Þ ^ y ^r ‘ ^r ðn n r dA1 4A1 ^ y ^r ðn ^ x ^rÞ ‘ ^r n dr r A1 A1 ^x ‘ þ ^x ‘ ^r ^r ^r ^y ^r n ^y n n n þ r r dA6 2A6 ^ x ^rÞ n ^ y ‘ ^r ^r ðn dr r A6 ^x ^r þ ^r n ^x Þ ^ y ‘ ðn n r dA2 2A2 ^ y ^r ðn ^ x ^rÞ ‘ ^r n dr r A2 A2 ^x ‘ þ ^x ‘ ^r ~I ^y ^r n ^y n þ n n r r dA7 ^ x ^rÞ~I ^y ‘ ðn n dr dA2 2A2 ^x ^rÞ ‘ ^r ðn dr r A2 A2 ^ y ^r ^y n ^x ^x ‘ n n ‘ ^r n þ r r dA9 2A9 ^x ^rÞ ‘ ^r ðn dr r A9 A9 ^y ^x n ^y ^x ‘ ^r n ‘ ^r n þ n r r dA7 dA8 ^ x ^rÞ^ ^ x ^rÞ ‘ ^ ðII.8Þ ðn ny ‘ ðn ny dr dr

iT o h ^ y r l2134 n onx g2 dD1 D1 ^ y ^r ðn ^ x ^rÞ~I n ¼ 16pð1 mÞ dr r D1 ~ ^x I ^y n n r 1 dD2 ^ x ^rÞ~I ^ y ‘ ðn ðII:9Þ n 16pð1 mÞ dr iT o h ^y n ^ y : l ^y n rS n onx 2 ^y g 2 rS n dA1 4A1 ^ y ^r ðn ^x ^rÞ^r ^r n ¼ 16pð1 mÞ dr r 2A1 ^y n ^ x ^r ^r ^ y ^r n n r 2 A1 ^x Þ ^x ^r þ ^r n ^ y ^r ðn n r dA6 2A6 ^ x ^rÞ^r ^r ðn dr r

A6 ^x Þ ^x ^r þ ^r n ðn r 2 dA2 2A2 ^ y ^r ðn ^ x ^rÞ~I n dr r

dA7 2A2 ^y n ^ x ~I ^ y ^r n ^ x ^rÞ~I ðn n r dr dA3 2A3 ^x ^rÞ^ ^ y ^r ðn ny ^r n dr r

A3 A3 ^x n ^y n ^x ^y n ^ y ^r n ^ y ^r n n r r dA4 2A4 ^ y ^r ðn ^x ^rÞ^r n ^y n dr r

A4 ^ x ^r n ^y n ^y n r A4 dA5 ^x n ^y ^y ^ y ^r n ^ x ^rÞ^ ðn ny n n r dr ^y rS n dA 3 ^ y ^r ðn ^ x ^rÞ A n þ 16pð1 mÞ dr r 1 ^ y ‘ ^r ^r ^y n ^x n þ A n r 1 ^ y ^r n ^ x ^r þ ^r n ^x Þ ^ y ‘ ðn A n r dB 1 ^ ^ y ^r ^ ^ B ðnx rÞ ny ‘ n dr r 1 ^y ‘ n ^y n ^x B n r

dB 1 ^ y ^r ðn ^ x ^rÞ B n dr r 1 ^ y ‘ ~I ^y n ^x n þ B n r dC 1 ^ y ‘ ^r n ^y ^ x ^rÞ n C ðn dr r 1 ^y ‘ n ^x n ^y ðII:10Þ C n r iT o h ^ y : l rS n onx g2 dA1 4A1 ^ x ^rÞ rS n ^ y : ^r ^r ¼ ðn 16pð1 mÞ dr r A1 ^ x Þ : rS n ^y ^x ^r þ ^r n þ ðn r dA6 2A6 ^ x ^rÞ rS n ^ y ^r ^r ^r ^r þ ðn dr r dA2 2A2 ^ x ^rÞ rS n ^ y : ^r ^r ðn þ dr r dA7 A2 ^ x Þ : rS n ^ y ~I ^y þ ^x ^r þ ^r n rS n þ ðn r dr A1 A6 ^x Þ ^ y : ^r ^r ðn ^x ^r þ ^r n ny þ rS ^ rS n þ r r dA6 2A6 ^ x ^rÞ ðn þ dr r ^ y þ ^r rS n ^ y ^r

ð^r ^rÞ rS n A6 ^ y ^r þ ^r rS n ^y n ^x ^ x rS n þ n r ^ x Þ rS n ^y ^ x ^r þ ^r n þðn dA2 2A2 ^ x ^rÞ rS n ^ y ð^r ^rÞ ðn þ dr r A2 ^x Þ ^ y ðn ^ x ^r þ ^r n rS n þ r T dA9 2A9 ^ x ^rÞð^r ^rÞ rS n ^y ðn þ dr r T A9 ^x Þ rS n ^y ^x ^r þ ^r n þ ðn r T dA7 dA8 ^ ^ ^ ^ ^ ^ þ ðnx rÞ rS ny þ ðnx rÞ rS ny dr dr 1 dA 3A ^ x ^rÞ ‘ ^r : rS n ^y ðn 16pð1 mÞ dr r A ^ y ^r ^r nx : r S n þ ‘ ^ r dB B ^ x ^rÞ ‘ ^r : rS n ^y ðn þ dr r B ^ y ~I þ nx : rS n ‘ ^ r

377

A ^x Þ ^ y ðn ^ x ^r þ ^r n ^ r : rS n ‘ r dB B ^ x ^rÞ rS n ^ y ‘ ^r þ ðn dr r dC C ^ x ^rÞ^r rS n ^y ‘ ðn þ dr r B C ^ y ‘ ^ ^y ‘ ^ x rS n þ rS n nx þ n r r þ

378

2X dX dW X A¼2 ; B¼ ; r dr dr r 2m dW dX 2X 2X C¼ 1 2m dr dr r r dA 3A dB B ; A2 ¼ ; A1 ¼ dr r dr r dB B A dC C A þ ; A4 ¼ þ ; A3 ¼ dr r r dr r r ðB þ C Þ A ; A6 ¼ ; A5 ¼ r r B C dC C A7 ¼ ; A8 ¼ ; A9 ¼ r r dr r d2 A 7 dA 15A 3 dA 9A þ 2 ; B2 ¼ 2; B1 ¼ 2 dr r dr r r dr r 2 d B 3 dB 3B þ ; B3 ¼ 2 dr r dr r 2 3 dB 3B ; B4 ¼ r dr r2 d2 C 3 dC 3C 2 dA 6A B5 ¼ 2 þ 2 þ 2; dr r dr r r dr r 2A 1 dC C B6 ¼ 2 þ ; r r dr r2 d2 B 3 dB 3B 2 dA 6A þ 2; B7 ¼ 2 þ dr r dr r 2 r dr r 2A 1 dB B B8 ¼ 2 þ ; r r dr r2 2 dB 2B 2 dC 2C 2A 2 þ 2 þ B9 ¼ r dr r2 r dr r r 2 d A 2 dA 12A 2 ; F1 ¼ 2 þ dr r dr r d2 B 2 dB 2B 2A F2 ¼ 2 þ þ 2; dr r dr r2 r 2 d C 2 dC 2C 2A 2 þ 2 F3 ¼ 2 þ dr r dr r r 2 d B 3 dB 3B 1 dC C ; þ D1 ¼ 2 þ dr r dr r2 r dr r2 dB 3B C D2 ¼ þ þ dr r r

ðII:11Þ

ðII:12Þ

ðII:13Þ

Appendix III Let n the local coordinates of the singular point. After the transformation into polar coordinates (q; h), the Taylor expansion of all the involved functions are given as follows:

n1 ¼ n1 þ q cos h n2 ¼ n2 þ q sin h r ¼ yðn1 ; n2 Þ xðn1 ; n2 Þ # ox ox cos h þ sin h ¼q on1 n¼n on2 n¼n " 2 cos2 h o2 x 2 o x þ cos h sin h þq on1 on2 n¼n on21 n¼n 2 # o2 x sin2 h þ Oðq3 Þ þ 2 2 on2 n¼n ¼ qAðhÞ þ q2 BðhÞ þ Oðq3 Þ ðA BÞ þ O qnþ2 ; rn ¼ jrjn ¼ qn A 1 þ nq 2 A n ¼ 1; 2; 3; . . . A ¼ jAj

2 r A B ðA BÞ ^r ¼ ¼ þ q A þ O q r A A A3 2 ¼ d0 þ qd1 þ O q 1 1 1 ¼ þO r2 q2 A2 q 1 1 3ðA BÞ 1 ¼ þ O r3 q3 A3 q2 A5 q ðII:14Þ

ðIII:1Þ

ðIII:2Þ

ðIII:3Þ ðIII:4Þ

ðIII:5Þ ðIII:6Þ ðIII:7Þ

Expansion of the interpolation functions

a oN oN a N ðnÞ ¼ N ðnÞ þ q cos h þ sin h on1 n¼n on2 n¼n þ O q2 ¼ N0a þ qN1a þ O q2 ðIII:8Þ a

a

Expansion of the Jacobian

"

# oJ oJ JðnÞ ¼ JðnÞ þ q cos h þ sin h on1 n¼n on2 n¼n þ O q2 ¼ J0 þ qJ1 þ O q2

ðIII:9Þ

Expanded form of the Jacobian magnitude

ðII:15Þ

ðJ J Þ J ¼ jJj ¼ jJ0 j þ q 0 1 þ O q2 jJ j 2 0 ¼ J0 þ qJ1 þ O q

ðIII:10Þ

Expanded form of the normal vector

ðII:16Þ

J J0 J0 J1 ^ y ¼ ¼ þ q J 0 2 þ O q2 n J0 J J J0 2 0 0 1 ^ y þ q^ ny þ O q ¼n

ðIII:11Þ

Expanded form of the dot products

^ x d0 Þ þ qðn ^ x d 1 Þ þ O q2 ^ x ^rÞ ¼ ðn ðn ^x ^rÞ1 þO q2 ^ x ^rÞ0 þqðn ¼ ðn h i ^ 1y d0 þ O q2 ^ 0y d1 þ n ^ y ^r ¼ q n n ^y ^r 1 þO q2 ¼q n ^x n ^y ¼ n ^ 0y þ q n ^ 1y þ O q2 ^x n ^x n n ^y 0 þq n ^x n ^ y 1 þO q2 ^x n ¼ n ^ 0y ‘ þ þq n ^1y ‘ þ O q2 ^y ‘ ¼ n n

0

ðIII:13Þ

ðIII:15Þ

aðn1 ; n2 Þ ¼ aðn1 ; n2 Þ þ OðqÞ ¼ a0 þ OðqÞ

ðIII:17Þ

a ðn1 ; n2 Þ ¼ a ðn1 ; n2 Þ þ OðqÞ ¼ a 0 þ OðqÞ

ðIII:18Þ

bðn1 ; n2 Þ ¼ bðn1 ; n2 Þ þ OðqÞ ¼ b0 þ OðqÞ

ðIII:19Þ

b0 ðn1 ; n2 Þ ¼ b0 ðn1 ; n2 Þ þ OðqÞ ¼ b0 0 þ OðqÞ

ðIII:20Þ

0

~ kk ¼ H

m¼1

þ

ðIII:14Þ

1

0

^y

n

; a ¼ oy oy on1 on2 ^y n

oy on1

; b ¼ oy oy on1 on2 ~ kk ¼ T

Z1 Z1 1 1 a

a0 ¼

o^ ny on1

~Skk

N ðn1 ; n2 ÞJ ðn1 ; n2 Þdn1 dn2

m¼1

hm 1

ðIII:30Þ

qm

0

hm 2

o^ ny b0 ¼ on2

m q^ ðhÞ ~tðhÞ ln m dh b ðhÞ

ðIII:27Þ

l Z Z 2 1 X ~Sðq; hÞ ~sðhÞ ~sðhÞ dq dh q2 q m¼1

ðIII:21Þ þ

l Z X m¼1

(

m q^ ðhÞ ~sðhÞ ln m b ðhÞ

1

hm 1

"

#) cm ðhÞ 1 ~sðhÞ dh þ ðbm ðhÞÞ2 q^m ðhÞ 2

ðIII:22Þ

0

hm 2

þ

hm 2

hm 1

h qm l Z2 Z X ~tðhÞ ~ dqdh ¼ Tðq; hÞ q m¼1 l Z X

hm 1

1 1 a

~ xk ; ye ðn1 ; n2 Þ oR onx

hm 1

m q^ ðhÞ ~ hðhÞ ln m dh b ðhÞ

l Z X

Z1 Z1 ~ k e oP x ; y ðn1 ; n2 Þ ¼ onx

~Skk ¼

m

~ kk T

0

~ ~ ðq; hÞ ¼ hðhÞ þ Oð1Þ ðIII:28Þ H q N0a J0 ~ðhÞ ¼ h 32pð1 mÞA2 h i ^ 0y ð3 4mÞ^

ð1 4mÞd0 n n0y d0 ðIII:29Þ

0

N ðn1 ; n2 ÞJ ðn1 ; n2 Þdn1 dn2

hm 1

m¼1

where the vectors a; a ; b; b are defined as follows oy on2

! ~ðhÞ h ~ ðq; hÞ dq dh H q

h qm l Z2 Z X

hm 2

ðIII:16Þ

0

ðIII:26Þ m

^ y ¼ rS n ^0y þ OðqÞ ¼ rS n ^ y 0 þOðqÞ rS n 0

~ xk ; ye ðn1 ; n2 Þ N a ðn1 ; n2 ÞJ ðn1 ; n2 Þdn1 dn2 P

1 1

ðIII:12Þ

^ ^ ¼ ny ‘ þq ny ‘ þO q2

~ kk ¼ H

Z1 Z1

2 1 ~Sðq; hÞ ¼ ~sðhÞ þ ~sðhÞ þ Oð1Þ ðIII:32Þ q2 q h N0a J0 2 ^x n ^ y 0 d0 d0 ~sðhÞ ¼ 3 n 3 32pð1 mÞA ^ y 0~I þ 3ð1 4mÞðn ^0y ^x n ^ x ^rÞ0 d0 n þ ð3 4mÞ n

^ 0y þ 3ð4m 3Þðn ^x ^rÞ0 n ^ 0y d0 þ ð4m 1Þ^ nx n i 0 ^x 2 n ^x n ^y 0 n ^ 0y ^y n þ ð3 4mÞ^ n0y n

ðIII:23Þ

~ ~ ðq; hÞ ¼ tðhÞ þ Oð1Þ ðIII:24Þ T q N0a J0 ~tðhÞ ¼ ^ x ^rÞ0 d0 d0 3ðn 32pð1 mÞA2 ^x d0 þ d0 n ^ x Þ ð3 4mÞðn ^x ^rÞ0~I þ 2ðn ^ y 0 d0 n ^0y n ^ 0y d0 ^x n þ 2ð1 2mÞ n 0 0 ^y ^y n ^x ^rÞ0 n ðIII:25Þ 2ðn

ðIII:31Þ

ðIII:33Þ 1

1

1

1

~sðhÞ ¼ ~s1 þ ~s2 þ ~s3 ðIII:34Þ ( 1 1 ^ y ^r 1 ðn ~s1 ¼ ^ x ^rÞ0 N0a J0 d0 d0 15 n 3 32pð1 mÞA ^ y ^r 1 N0a J0 ½n ^x

^ x d0 þ d0 n þ3 n 9ðA BÞ ^ y 0 N0a J0 d0 d0 ^x n n 2 A

379

h ^x n ^ y 0 N0a J1 d0 d0 þ3 n ^ y 0 N1a J0 d0 d0 þ n ^x n ^ y 1 N0a J0 d0 d0 ^x n þ n i ^x n ^ y 0 N0a J0 ðd0 d1 þ d1 d0 Þ þ n ^ y ^r 1 ðn ^ x ^rÞ0 N0a J0~I þ 3ð4m 3Þ n

380

a 3ðA BÞð3 4mÞ ^ ^ n N J ~I n x y 0 0 0 A2 h ^ y 0 N0a J1 ^x n þ ð3 4mÞ n i ^x n ^ y 0 N1a J0 þ n ^x n ^ y 1 N0a J0 ~I þ n ^ y 0 N0a J0 d0 n ^0y ^x n ^ y ^r 1 n 6ð3 4mÞ n 9ðA BÞð1 4mÞ ^0y ^ x ^rÞ0 N0a J0 d0 n ðn A2 h ^ x ^rÞ0 N0a J1 d0 n ^0y þ 3ð1 4mÞ ðn

^0y þ ðn ^ x ^rÞ1 N0a J0 d0 n ^0y ^x ^rÞ0 N1a J0 d0 n þ ðn i ^x ^rÞ0 N0a J0 d1 n ^ 0y þ d0 n ^ 1y þ ðn 3ðA BÞð1 4mÞ a ^ 0y ^x n N0 J0 n A2 h ^ 0y þ N1a J0 n ^ 0y ^x n ^x n ð1 4mÞ N0a J1 n i ^1y ^x n þN0a J0 n ^ y 0 N0a J0 n ^x n ^ 0y d0 ^ y ^r 1 n þ 6ð1 4mÞ n

þ

9ðA BÞð3 4mÞ ^ x ^rÞ0 N0a J0 n ^ 0y d0 ðn A2 h ^ x ^rÞ0 N0a J1 n ^ 0y d0 3ð3 4mÞ ðn þ

^ x ^rÞ1 N0a J0 n ^x ^rÞ0 N1a J0 n ^ 0y d0 þ ðn ^ 0y d0 þ ðn i ^ x ^rÞ0 N0a J0 n ^ 1y d0 ^ 0y d1 þ n þðn 3ðA BÞð3 4mÞ a 0 ^x ^y n N0 J0 n A2 h ^ x þ N1a J0 n ^x ^ 0y n ^ 0y n þ ð3 4mÞ N0a J1 n i ^x ^1y n þN0a J0 n ^ 0y ^ x ^rÞ0 N0a J0 n ^0y n ^ y ^r 1 ðn þ6 n

a 0 6ðA BÞ ^ ^0y ^ ^ n n N Jn n x y 0 0 0 y A2 h ^x n ^ y 0 N0a J1 n ^0y ^ 0y n 2 n þ

^x n ^ y 0 N1a J0 n ^0y þ n ^x n ^ y 1 N0a J0 n ^ 0y ^ 0y n ^ 0y n þ n ) i a 1 0 0 1 ^x n ^ y 0 N 0 J0 n ^y þ n ^y n ^y ^y n þ n ðIII:35Þ 1

~s2 ¼

N0a J0

32pð1 mÞA2 ^ y 0 2ð1 2mÞ n ^x n ^ y 0 d0 n ^ 0y n ^ 0y d0

r n ^ 0y ^ y 0 ðn ^ x ^rÞ0 n ^ 0y n 2 r n ^ x ^rÞ0 ða0 d0 Þ a00 d0 15ðn þðb0 d0 Þ b00 d0 d0 d0 ^x a0 Þ a00 d0 þ3 ðn ^ x b0 Þ b00 d0 ^ x a00 ða0 d0 Þþ ðn þ n ^x b00 ðb0 d0 Þ d0 d0 þ n þ3 ða0 d0 Þ a00 d0 ^x

^ x d0 þ d0 n þðb0 d0 Þ b00 d0 ½n 0 ^ x ^rÞ0 ða0 d0 Þ a0 d0 3ð3 4mÞðn þðb0 d0 Þ b00 d0 ~I ^ x a00 ða0 d0 Þ ^ x a0 Þ a00 d0 þ n þð3 4mÞ ðn ^ x b0 Þ b00 d0 þ n ^x b00 ðb0 d0 Þ ~I þðn ^ x ^rÞ0 ða0 d0 Þ d0 a00 þa00 d0 þ3ðn ^ x ^rÞ0 ðb0 d0 Þ d0 b00 þ b00 d0 þ3ðn ^x a0 Þ d0 a00 þ a00 d0 ðn ^x b0 Þ d0 b00 þb00 d0 ðn ^x a00 þ a00 n ^x ða0 d0 Þ n ^ x b00 þb00 n ^x ðb0 d0 Þ n ^ x ^rÞ0 a00 d0 a0 d0 3ð3 4mÞðn þ b00 d0 b0 d0 ^x a00 a0 d0 þ n ^ x b00 b0 d0 þð3 4mÞ n ^ x þ b00 d0 b0 n ^x þ ð3 4mÞ a00 d0 a0 n ^ x ^rÞ0 a00 d0 d0 a0 þ b00 d0 d0 b0 þ3ð1 4mÞðn ^ x a00 d0 a0 þ n ^ x b00 d0 b0 ð1 4mÞ n ^ x a0 þ b00 d0 n ^ x b0 ð1 4mÞ a00 d0 n ^x ^rÞ0 a00 a0 þb00 b0 ð1 4mÞðn 0 0 ^x ^rÞ0 a0 a0 þb0 b0 þ ð3 4mÞðn ðIII:36Þ

1

~s3 ¼

N0a J0 ^x Þ ^ ^x d0 þ d0 n ðn n y ‘ 32pð1 mÞg 2 A2 0 ^x ^rÞ0 d0 d0 ^ y ‘ ðn 3 n 0 ^x n ^y 0 n ^y ‘ þ 2ð1 2mÞ n 0 ^ 0y n ^0y d0

d0 n ^ 0y ^x ^rÞ0 n ^ 0y n ^ y ‘ ðn 2 n 0 ^ y ‘ ðn ^x ^rÞ0~I þ ð5 4mÞ n ðIII:37Þ

0

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