Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, ... eigenstrain distributions were only limited to the homogenously applied ...
Appl. Math. Mech. -Engl. Ed., 32(5), 551–562 (2011) DOI 10.1007/s10483-011-1437-x c Shanghai University and Springer-Verlag Berlin Heidelberg 2011
Applied Mathematics and Mechanics (English Edition)
Two-dimensional polynomial eigenstrain formulation of boundary integral equation with numerical verification∗ Hang MA ( )1 ,
Zhao GUO ( )2 ,
Qing-hua QIN ()3
(1. Department of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, P. R. China; 2. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P. R. China; 3. School of Engineering, Australian National University, Canberra, ACT 0200, Australia)
Abstract The low-order polynomial-distributed eigenstrain formulation of the boundary integral equation (BIE) and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in two-dimensional elastic media. Taking the results of the traditional subdomain boundary element method (BEM) as the control, the effectiveness of the present algorithm is verified for the elastic media with a single elliptical inhomogeneity. With the present computational model and algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE. Key words eigenstrain, Eshelby tensor, boundary integral equation (BIE), polynomial, inhomogeneity Chinese Library Classification O241 2010 Mathematics Subject Classification
1
74S15, 74S05
Introduction
Since the pioneer work of Eshelby[1–2] , inclusion and inhomogeneity problems have been a focus of solid mechanics for several decades. In particular, the determination of elastic states of an embedded inclusion is of considerable importance in a wide variety of physical and engineering applications. By Eshelby’s idea of eigenstrain solutions and equivalent inclusion, a diverse set of research has been reported analytically[3–7] and numerically[8–16] . It should be mentioned that the eigenstrain solution can represent various physical problems, where the eigenstrains can correspond to the thermal-mismatch strains, the strains due to the phase transformation, the plastic strains, as well as the intrinsic strains in the residual stress problems[17] . Based on the concept of eigenstrains and through the substitution of equivalent inclusions[18] , many ∗ Received Jan. 14, 2011 / Revised Mar. 19, 2011 Project supported by the National Natural Science Foundation of China (No. 10972131) and the Graduate Innovation Foundation of Shanghai University (No. SHUCX102351) Corresponding author Hang MA, Professor, Ph. D., E-mail: hangma@staff.shu.edu.cn
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Hang MA, Zhao GUO, and Qing-hua QIN
practical problems can be modeled and solved efficiently. Typical problems are the substituting or interstitial atoms in crystals, the quantum dot/line structures in semiconductors, the effects of voids, and the indigenous phase or the reinforcement in solids. The analytical models available in the literature can be taken as a basis for predicting the stress/strain distributions either within or outside the inhomogeneity. Numerical simulations using the finite element methods (FEM)[10] , the volume integral methods (VIM)[11–13] , or the boundary element methods (BEM)[19] have been conducted for the analysis of engineering problems with the inhomogeneities of various shapes and material properties. The solution scale of the FEM is usually very large since both the matrix and the inhomogeneity should be discretized over their entire solution domains. In contrast with the FEM, the VIM and the BEM seem to be more suitable for the inhomogeneity problems in terms of the solution scale. However, as the interfaces need to be discretized for the unknowns, only small scale problems have been studied for a few inhomogeneities. The situation in the application of the BEM often coupled with the VIM[14–15] is much the same as that of the VIM, in which the problems of simple arrays of inclusions are solved on a small scale because of the similar reason that the unknown appears in the interfaces. For large scale problems with inhomogeneities[16] , special techniques of the fast multipole expansions[20] should be employed. This increases the solution complexity. Following Eshelby’s idea, Ma et al.[21–22] recently proposed the eigenstrain formulation of the boundary integral equation (BIE) for modeling the particle-reinforced materials in the two-dimensional elastic problems, assuming that the eigenstrains in the inhomogeneities were constant, which significantly reduced the solution scale of the problem. However, the constant eigenstrain distributions were only limited to the homogenously applied stress fields and simple geometries of the inhomogeneities. For multiple inhomogeneities and finite domain problems containing even single inhomogeneity, it is more appropriate to employ the assumption of the polynomial distributions of the eigenstrains[23–24] . Aiming at the elliptical inhomogeneity in the finite elastic media, the low-order polynomial-distributed eigenstrain formulation of the BIE and the corresponding definition of the Eshelby tensors are proposed. The elastic media with a single elliptical inhomogeneity are analyzed using the proposed numerical procedure. The effectiveness of the present algorithm is verified in comparison with the results of the traditional subdomain BEM.
2
Computational model
2.1 Eigenstrain formulation of BIE In the present work, the inhomogeneity is refered to as a local zone with the material properties different from those of the matrix. Its domain is denoted as ΩI (I=1,2,· · · ,NI , where NI is the number of the inhomogeneities in Ω). If the elastic constants of the local zone ΩI are the same as those of the matrix, ΩI is known as the subdomain, of which the geometry is exactly the same as that of the inhomogeneity. Suppose that the inhomogeneity and the matrix are perfectly bonded, and both of them are isotropic materials. Then, the displacement continuity and the traction equilibrium are remained along their interfaces. The domain of the matrix Ω is finite with the outer boundary Γ. The boundary ΓI (ΓI = ΩI ∩ Ω) of the subdomain ΩI is also taken as the interface. The displacement field and the stress field of the elastic media can be expressed, respectively, by the eigenstrain formulations of the BIE[21–22] as Cui (p) + = Γ
Γ
uj (q)τij∗ (p, q)dΓ(q)
τj (q)u∗ij (p, q)dΓ(q)
+
NI I=1
ΩI
∗ ε0jk (q)σijk (p, q)dΩ(q),
(1)
Two-dimensional polynomial eigenstrain formulation of boundary integral equation
Cσij (p) =
Γ
+
τk (q)u∗ijk (p, q)dΓ(q) −
NI I=1
ΩI −Ωε
Γ
553
∗ uk (q)τijk (p, q)dΓ(q)
∗ ∗ ε0kl (q)σijkl (p, q)dΩ(q) + ε0kl (p)Oijkl ,
(2)
∗ represent the Kelvin’s where p and q are the source and field points, and u∗ij , τij∗ , and σij fundamental solutions for the displacement, the traction, and the stress, respectively. u∗ijk , ∗ ∗ τijk , and σijk are the related derivatives. C is a conventional boundary shape coefficient, and C=1/2 if p is on the smooth boundary Γ[19] . Ωε is a small region of radius ε around p when ∗ p ∈ ΩI , and Oijkl is the corresponding free term induced from the domain integral in (2). If there are the eigenstrains ε0ij in a single subdomain ΩI in the infinite media, using (2) and the constitution relationship of the related material, the constrained strains εC ij in the subdomain can be expressed as follows: −1 C 0 ∗ ∗ (3) εij = εij + Cijkl ε0mn σklmn dΩ + Oklmn ε0mn , ΩI −Ωε
−1 where Cijkl is the elastic compliance tensor of the matrix. By the transformation equations[25] and the assumption that distribution of the eigenstrains in ΩI is constant, the domain integral in (3) can be written in the form of the boundary integral as follows: −1 0 ∗ εC = ε + C ε0mn xn τklm dΓ, (4) ij ij ijkl ΓI
where xk represents the two-point variable or the projection of the distance between the two points p and q, i.e., (5) xk = xk (q) − xk (p). With the transformation of integrals from the domain type to the boundary type, not only the discretization of the internal cells can be avoided, but also the deduction for the free term of the domain integral can be prevented. Thus, the computational process is greatly simplified. As we know, the constrained strains εC ij correlate with the eigenstrains in a subdomain through the Eshelby tensor in the case of constant strains, i.e., 0 εC ij = Sijkl εkl .
(6)
The boundary integral representation of the Eshelby tensor in the plane strain condition can be written as 1 1 ∗ ∗ ∗ Sijkl = (δik δjl + δil δjk ) + xl (τijk + τjik − 2νδij τmmk )dΓ, (7) 2 4μ ΓI where μ and ν stand for the shear modulus and Poisson’s ratio of the matrix, respectively. Using the boundary integral (7), the Eshelby tensor Sijkl can be conveniently computed. Moreover, the machine precision can be easily achieved. 2.2 Polynomial eigenstrain and Eshelby tensor In practical problems, since the size of the subdomain is relatively small compared with that of the matrix, it is reasonable to describe the eigenstrains with the low-order polynomials. In the present work, the second-order polynomial with respect to the field point q is used and expressed with the local variables ξ1 and ξ2 in the subdomain ΩI as follows: ε0ij =
m+n=2 m=0,n=0
(0)mn m ξ1 (q)ξ2n (q),
εij
(8)
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Hang MA, Zhao GUO, and Qing-hua QIN (0)mn
where εij is the coefficient of the polynomial, and m and n are integers. In Fig. 1, the subdomain ΩI is mapped into a unit circle. By the conservation theorem of polynomials[23–24] , the constrained strains εC ij in the elliptical subdomain distribute in a form similar to that of the second-order polynomials, and the even term and the odd term correspond to each other. In 0 this case, εC ij and εij are no longer correlated with the conventional Eshelby tensor. To express it conveniently, the following notation is employed in the present work: 0 εC IJ = SIJKL εKL ,
(9)
where SIJKL is still named as the Eshelby tensor, of which the subscripts are listed in Table 1. For example, let I = 1, K = 3, J = 4, and L = 6. Then, (9) becomes (0)02
2 ξ12 (q)εC20 11 = S1436 ξ2 (q)ε22
.
(10)
The Eshelby tensor S1436 in (10) means the component of the constrained strains εC 11 that are distributed in the form of ξ12 in the elliptical subdomain and caused by the component of the eigenstrains ε022 distributed in the form of ξ22 .
Fig. 1
Table 1
Local coordinates and fitting points in ΩI
Definition of subscripts in the Eshelby tensor SIJ KL ‚
I, K
Subscript Number Definition
1 ε11
2 ε12
3 ε22
‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚
J, L
1 1
2 ξ1
3 ξ2
4 ξ12
5 ξ1 ξ2
6 ξ22
In the two-dimensional elasticity problem, there are 78 nonzero terms of the Eshelby tensor, where five terms correspond to the constant strains that can be determined using (7). The numbers of the terms corresponding to the linear strain and the quadratic strain are 18 and 41, respectively. There are also 14 terms of constant constrained strains caused by the quadratic eigenstrains. The domain integrals in (1) and (2) with the polynomial-distributed eigenstrains can also be transformed into the boundary integrals. First, the domain integrals in (1) and (2), with any term of the polynomial eigenstrains in the global coordinates xk (q) after the variable mapping in (8), can be written in the form of two-point variables in (5) as follows: n ∗ xm 1 (q)x2 (q)σijk dΩ ΩI
=
m n s=0 t=0
m!n! (x1 (p))m−s (x2 (p))n−t (m − s)!s!(n − t)!t!
ΩI
∗ xs1 xt2 σijk dΩ,
(11)
Two-dimensional polynomial eigenstrain formulation of boundary integral equation
=
ΩI −Ωε m n s=0 t=0
·
555
n ∗ m n ∗ xm 1 (q)x2 (q)σijkl dΩ + x1 (p)x2 (p)Oijkl
m!n! (x1 (p))m−s (x2 (p))n−t (m − s)!s!(n − t)!t!
ΩI −Ωε
∗ n ∗ xs1 xt2 σijkl dΩ + xm 1 (p)x2 (p)Oijkl ,
(12)
where s and t are integers. Then, the domain integrals at the right-hand side of (1) and (2) with the eigenstrains in the form of two-point polynomials can be transformed term by term into the boundary integrals[25] using the formulae listed in Appendices A and B, respectively. In this way, the favorable features of the boundary-only discretization are preserved. The numerical computation of SIJKL is carried out term by term for the polynomial eigenstrains using (3). Take a typical term of the polynomial as an example. The local coordinate ξk (q) is first mapped into the global coordinate xk (q) with the origin placed at the center of the subdomain ΩI . Then, transform the field-point polynomial in the domain integral in (3) into the two-point polynomial using (12). With the transformation equations in Appendix B and the definition (9), the corresponding SIJKL can be numerically determined with the boundaryonly discretization. It should be pointed out that the free term in (12) exists only when the eigenstrains are constant. In this case, only (7) needs to be used for computing the Eshelby tensor. 2.3 Eigenstrains in equivalent inclusion Based on the concept of the equivalent inclusion[1–2] and under the applied stress or strain εij , the inhomogeneous zone can be replaced by the equivalent inclusion or the subdomain with the eigenstrains without unaltering the stress state according to the theory of elasticity, i.e., I C 0 (εC Cijkl kl + εkl ) = Cijkl (εkl + εkl − εkl ),
(13)
where Cijkl is the elastic tensor of the matrix. Define the ratio of Young’s modulus β = EI /E and denote 2ν βνI β(1 + ν) γ γ= , β1 = , β2 = − , (14) 1 − 2ν 1 + νI 2 1 − 2νI where EI and E are Young’s moduli of the inhomogeneity and the matrix, respectively. The subscript or the superscript I represents the inhomogeneity. Then, (13) can be explicitly written as C 0 0 (1 − β1 )εC (15) ij + β2 δij εkk − εij − 0.5γδij εkk = −(1 − β1 )εij − β2 δij εkk . Substitute (9) into (15). The eigenstrains in the equivalent inclusion can be computed through the applied strains in the subdomain. In the present work, 18 coefficients of the secondorder polynomials for the eigenstrains are determined via the least square fitting method with 21 components of the applied strains over the subdomain ΩI shown in Fig. 1. 2.4 Solution procedure In addition to the applied load on the outer boundary, the eigenstrain-induced stresses from an inhomogeneity should superimpose on other inhomogeneities, since the eigenstrains in each subdomain produce an internal stress field self-balanced over the whole domain Ω, disturbing not only the stress state of other inhomogeneities but also the tractions on the displacementprescribed outer boundary and the displacements on the traction-prescribed outer boundary. As a result, the applied strains and the eigenstrains on the equivalent inclusion should be computed in an iterative way in the solution procedure. After the discretization, incorporated with the boundary conditions, (1) can be written in the matrix form as Ax = b + Bα,
(16)
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Hang MA, Zhao GUO, and Qing-hua QIN
where A is the system matrix, B is the coefficient matrix related to the kernels in the domain integrals in (1), b is the right vector related to the known quantities on the outer boundary and the corresponding kernels, and x is the unknown vector on the outer boundary. α is the coefficient vector of the polynomial eigenstrains in all the subdomains, which needs to be corrected in the iteration. It should be pointed out that all the coefficients in A, B, and b are constants so that they need to be computed only once during the solution process. In the process of calculation, the eigenstrains are assigned first with the initial values through the applied strains in the analysis via (2) at each position of the inhomogeneities. Then, the unknown vector x can be computed by the following simple iterative formula: x(k+1) = A−1 (b + Bα(k) ),
(17)
where k is the iteration count. The flowchart of the algorithm is shown in Fig. 2, where the iteration procedure of the polynomial eigenstrain formulation is similar to that of the constant eigenstrain formulation. The details of the procedure can be found in [21−22], which are omitted here for conciseness.
Fig. 2
3
Flowchart of algorithm for eigenstrain formulation of BIE
Numerical verification
3.1 Numerical model It can be shown from Fig. 3 that the computing domain in the present work is a square zone with the side length l containing a single elliptical inhomogeneity (NI = 1). The long axis and the ratio of the axes of the ellipse are set as 2a = 0.4l and b/a = 0.6, respectively. The center of the ellipse is placed at (l/3, l/2). The stresses in the matrix Ω and the inhomogeneity ΩI are computed under different loading manners along the vertical dashed line x1 = 0.553l and the horizontal dot and dashed line x2 = 0.5l (see Fig. 3). The computed results from the proposed algorithm are compared with those from the constant eigenstrain formulation. Here, the results from the domain decomposition or the subdomain method of the BEM[21–22] are taken as the control. In the subdomain method, the domains Ω and ΩI are described by two separate BIEs that are coupled on the interface ΓI with each other under the conditions of the displacement continuity and the traction equilibrium. In the BEM algorithm, both the outer boundary Γ and the interface ΓI are discretized
Two-dimensional polynomial eigenstrain formulation of boundary integral equation
557
with 32 quadratic boundary elements. In the two eigenstrain algorithms, the outer boundary Γ is still discretized with 32 quadratic boundary elements. However, the interface ΓI is discretized with 120 nodes by the boundary point methed (BPM)[26–27] . The purpose of discretizing the interface is for computing the domain integrals in (1)–(3), which have been transformed into the boundary integrals[25] before the computation.
Fig. 3
Computing domain
In the computation, two ratios of the moduli EI /E = 0.01 and EI /E = 10 are used, and Poisson’s ratios ν = νI = 0.3 are assumed to be the same for the inhomogeneity and the matrix. In Fig. 4, four kinds of loading manners are applied, i.e., the single tension (see Fig. 4(a)), the pure shear (see Fig. 4(b)), the quadratic distributed single tension (see Fig. 4(c)), and the simple shear (see Fig. 4(d)). In all the loading cases, the traction p is taken as 1 (see Fig. 4).
Fig. 4
Loading manners
3.2 Computed results The computed stresses in the matrix along the line x1 = 0.553l and in the inhomogeneity along the line x2 = 0.5l (see Fig. 3) are shown in Figs. 5–10 by three algorithms, i.e., the subdomain algorithm, the constant eigenstrain, and the quadratic eigenstrain algorithm, where the results obtained from the subdomain algorithm are taken as the control. It is seen from Figs. 5 and 6 that, for the loading cases of the single tension (see Fig. 4(a)) and the pure shear (see Fig. 4(b)), there is no obvious difference between the constant and the quadratic eigenstrains except that some improvements are observed in the normal stress σ22 in the inhomogeneity under the single tension for EI /E = 0.01 (see Fig. 5) and the shear stress σ12 in the matrix under the pure shear for EI /E = 0.01 (see Fig. 6) by the quadratic eigenstrain algorithm. The reason is that the strain distribution is almost constant in the inhomogeneity under these two loading cases, although the computing zone is finite. Hence, the strains in the inhomogeneity can be described properly with the constant eigenstrain model. Under the quadratic tension (see Fig. 4(c)) and the simple shear (see Fig. 4(d)), the computed stresses in the matrix are shown in Figs. 7 and 8, respectively, while the stress distributions in
558
Hang MA, Zhao GUO, and Qing-hua QIN
Fig. 5
Normal stresses in inhomogeneity under single tension
Fig. 6
Shear stresses in matrix under pure shear
Fig. 7
Normal stresses in matrix under quadratic tension
Fig. 8
Shear stresses in matrix under simple shear
the inhomogeneity are shown in Figs. 9 and 10, respectively. It can be seen from Figs. 7–10 that the results from the quadratic eigenstrain algorithm are in well agreement with those from the subdomain approach but obviously different from those obtained by the constant eigenstrain algorithm. This indicates that the accuracy of the results is greatly improved by the quadratic eigenstrain algorithm, showing that it is not enough to describe the strain distributions in the inhomogeneities with the constant eigenstrain model under the two loading cases, i.e., the quadratic tension and the simple shear. Therefore, the quadratic eigenstrain algorithm is capable for modeling complex loading cases for the problems with inhomogeneities. Moreover, the correctness of the algorithm is verified. The maximum error and the root mean square error εMAX and εRMS of the computed stresses in the matrix are listed in Table 2 for comparison, showing that the accuracy achieved by the quadratic eigenstrain algorithm is generally higher than that obtained by the constant eigenstrain algorithm under each loading manner. Although there seems to be no distinguishable difference in the shear stress σ12 in the matrix under the single tension shown in Fig. 6 for EI /E = 10, the errors listed in Table 2 clearly show that the results obtained by the quadratic eigenstrain algorithm are better than those obtained by the constant eigenstrain algorithm. It is noted that the convergence is achieved after 3 or 4 iterations by the eigenstrain algo-
Two-dimensional polynomial eigenstrain formulation of boundary integral equation
Fig. 9
Normal stresses in inhomogeneity under quadratic tension
Fig. 10
559
Stresses in inhomogeneity under simple shear
rithms in the present work. The degrees of freedom and the efficiencies of the three algorithms are listed in Table 3 for comparison. The results in Table 3 show that the efficiency of the eigenstrain algorithms is better than that of the subdomain algorithm, because the unknowns have to be solved on both the outer boundary and the interfaces in the subdomain method. With the subdomain algorithm, the solution scale is proportional to the number of the inhomogeneities. Thus, the CPU time increases in the geometric series. The program even cannot run on an ordinary desk-top computer when the number of the inhomogeneities is relatively large. In contrast, the solution scale of the eigenstrain algorithms is independent of the number of the inhomogeneities. The CPU time is largely spent on the iteration, which is in proportion to the number of the inhomogeneities. It is considered to be the major difference from the subdomain algorithm. Table 2
Errors of stresses in matrix along x1 = 0.553l for EI /E = 10
Loading
Algorithm
εMAX (σ11 )
εMAX (σ12 )
εMAX (σ22 )
εRMS (σ11 )
εRMS (σ12 )
εRMS (σ22 )
Single tension
Constant Quadratic
0.010 8 0.002 8
0.008 4 0.008 6
0.007 8 0.001 7
0.011 6 0.005 7
0.009 4 0.006 0
0.011 9 0.003 9
Pure shear
Constant Quadratic
0.040 1 0.031 3
0.009 3 0.002 4
0.014 4 0.025 0
0.021 2 0.012 6
0.012 6 0.005 7
0.016 2 0.012 0
Quadratic tension
Constant Quadratic
0.126 7 0.057 8
0.120 5 0.011 4
0.322 2 0.064 5
0.043 8 0.029 9
0.050 3 0.013 9
0.065 6 0.027 0
Simple shear
Constant Quadratic
0.128 4 0.024 7
0.040 0 0.011 6
0.174 3 0.021 4
0.039 9 0.016 4
0.031 7 0.009 8
0.054 7 0.017 5
εMAX is the maximum error; εRMS is the root mean square error
Table 3
Degrees of freedom and efficiencies of three algorithms
Algorithm
Subdomain
Constant
Degree of freedom
256
128
Quadratic 128
CPU time/ms
562
172
235
It can also be seen from Table 3 that the CPU time required by the quadratic eigenstrain algorithm is longer than that of the constant eigenstrain algorithm, since there are more coefficients to be determined through the least square fitting in the quadratic eigenstrain algorithm. However, the efficiency of the quadratic eigenstrain algorithm is much better than that of the subdomain algorithm, and the quadratic eigenstrain algorithm can handle the problems with nonuniform eigenstrains.
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Conclusions
In the paper, a low-order polynomial-distributed eigenstrain algorithm and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in the twodimensional elastic media. Taking the traditional subdomain BEM as the control, the elastic media with a single elliptical inhomogeneity are analyzed numerically, verifying the effectiveness and the feasibility of the present algorithm with the polynomial eigenstrain formulation of the BIE. With the proposed algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE.
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[16] Liu, Y. J., Nishimura, N., Tanahashi, T., Chen, X. L., and Munakata, H. A fast boundary element method for the analysis of fiber-reinforced composites based on a rigid-inclusion model. ASME Journal of Applied Mechanics, 72, 115–128 (2005) [17] Ma, H. and Deng, H. L. Nondestructive determination of welding residual stresses by boundary element method. Advances in Engineering Software, 29, 89–95 (1998) [18] Nakasone, Y., Nishiyama, H., and Nojiri, T. Numerical equivalent inclusion method: a new computational method for analyzing stress fields in and around inclusions of various shapes. Materials Science and Engineering, A285, 229–238 (2000) [19] Qin, Q. H. Nonlinear analysis of Reissner plates on an elastic foundation by the BEM. International Journal of Solids and Structures, 30, 3101–3111 (1993) [20] Greengard, L. F. and Rokhlin, V. A fast algorithm for particle simulations. Journal of Computational Physics, 73, 325–348 (1987) [21] Ma, H., Yan, C., and Qin, Q. H. Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites. Engineering Analysis with Boundary Elements, 33, 410– 419 (2009) [22] Ma, H., Xia, L. W., and Qin, Q. H. Computational model for short-fiber composites with eigenstrain formulation of boundary integral equations. Applied Mathematics and Mechanics (English Edition), 29, 757–767 (2008) DOI 10.1007/s10483-008-0607-4 [23] Rahman, M. The isotropic ellipsoidal inclusion with a polynomial distribution of eigenstrain. ASME Journal of Applied Mechanics, 69, 593–601 (2002) [24] Nie, G. H., Guo, L., Chan, C. K., and Shin, F. G. Non-uniform eigenstrain induced stress field in an elliptic inhomogeneity embedded in orthotropic media with complex roots. International Journal of Solids and Structures, 44, 3575–3593 (2007) [25] Ma, H., Kamiya, N., and Xu, S. Q. Complete polynomial expansion of domain variables at boundary for two-dimensional elasto-plastic problems. Engineering Analysis with Boundary Elements, 21, 271–275 (1998) [26] Ma, H. and Qin, Q. H. Solving potential problems by a boundary-type meshless method — the boundary point method based on BIE. Engineering Analysis with Boundary Elements, 31, 749–761 (2007) [27] Ma, H., Zhou, J., and Qin, Q. H. Boundary point method for linear elasticity using constant and quadratic moving elements. Advances in Engineering Software, 41, 480–488 (2010) Appendix A The transformation equations from the domain type to the boundary type with two-point polynomials in (1) are as follows[25] : Z Z 1 m n ∗ ∗ x1 x2 σ111 dΩ = xm+1 xn (A1) 2 τ11 dΓ, 1 m + n + 1 Ω Γ Z Z 1 n ∗ n+1 ∗ xm xm τ22 dΓ, (A2) 1 x2 σ222 dΩ = 1 x2 m+n+1 Γ Ω Z Z 1 n ∗ n+1 ∗ xm xm τ11 dΓ, (A3) 1 x2 σ112 dΩ = 1 x2 m + n + 1 Ω Γ Z Z 1 n ∗ ∗ xm xm+1 xn (A4) 1 x2 σ221 dΩ = 2 τ22 dΓ, m+n+1 Γ 1 Ω Z Z Z 1 m n ∗ n+1 ∗ ∗ xm xm τ12 dΓ − xm−1 xn+2 τ11 dΓ, (A5) 1 x2 σ122 dΩ = 1 x2 2 n+1 Γ (n + 1)(m + n + 1) Γ 1 Ω Z Z Z 1 n n ∗ m+1 n ∗ ∗ xm x σ dΩ = x x τ dΓ − xm+2 xn−1 τ22 dΓ, (A6) 1 2 211 2 21 2 m+1 Γ 1 (n + 1)(m + n + 1) Γ 1 Ω where m 0 and n 0.
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Appendix B Denoting γ=2ν/(1 − 2ν), the transformation equations from the domain type to the boundary type with two-point polynomials in (2) are as follows[25] Z −1 n ∗ μ xm 1 x2 σ1111 dΩ Ω
=
(γ + 2)m m+n
Z Γ
n ∗ xm 1 x2 τ11 dΓ −
Z Γ
n ∗ ∗ xm 1 x2 ((γ + 2)σ111 n1 + γσ211 n2 )dΓ
Z Z γn(n − 1) γn ∗ ∗ xm+1 xn−1 τ21 dΓ − xm+2 xn−2 τ22 dΓ, 1 2 2 m+1 Γ (m + 1)(m + n) Γ 1 Z n ∗ μ−1 xm 1 x2 σ2222 dΩ +
Ω
=
(γ + 2)n m+n
Z Γ
n ∗ xm 1 x2 τ22 dΓ −
Z Γ
(B1)
n ∗ ∗ xm 1 x2 ((γ + 2)σ222 n2 + γσ122 n1 )dΓ
Z Z γm(m − 1) γm n+1 ∗ ∗ xm−1 x τ dΓ − xm−2 xn+2 τ11 dΓ, 12 2 2 n+1 Γ 1 (n + 1)(m + n) Γ 1 Z n ∗ xm μ−1 1 x2 σ1122 dΩ +
(B2)
Ω
=
Z Z n m n ∗ n ∗ xm xm 1 x2 τ11 dΓ + 1 x2 τ22 dΓ m+n Γ m+n Γ Z n ∗ ∗ − xm 1 x2 (σ221 n1 + σ112 n2 )dΓ,
(B3)
Γ
μ−1
Z Ω
n ∗ xm 1 x2 σ1112 dΩ Z Z ∗ ∗ n ∗ ∗ xm+1 xn−1 (τ11 − τ22 )dΓ + xm 1 x2 (σ221 − σ111 )n2 dΓ, 1 2
n m+n Γ Γ Z n ∗ μ−1 xm 1 x2 σ2221 dΩ Ω Z Z m n+1 ∗ ∗ n ∗ ∗ = xm−1 x (τ − τ )dΓ + xm 22 11 1 x2 (σ112 − σ222 )n1 dΓ, 2 m+n Γ 1 Γ =
(B4)
(B5)
where m 0, n 0, and m + n 0. Moreover, if m = n = 0, there is Z Ω−ΩI
∗ ∗ σijkl dΩ + Oijkl =
Z Γ
∗ xl τijk dΓ.
(B6)
