Two-Dimensional Polynomial Eigenstrain Formulation of Boundary Integral Equation With Numerical Verification

MA Hang; GUO Zhao; QIN Qing-hua

doi:10.3879/j.issn.1000-0887.2011.05.002

Volume 32 Issue 5

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2011 > 32(5): 522-532

MA Hang, GUO Zhao, QIN Qing-hua. Two-Dimensional Polynomial Eigenstrain Formulation of Boundary Integral Equation With Numerical Verification[J]. Applied Mathematics and Mechanics, 2011, 32(5): 522-532. doi: 10.3879/j.issn.1000-0887.2011.05.002

Citation:

PDF( 963 KB)

Two-Dimensional Polynomial Eigenstrain Formulation of Boundary Integral Equation With Numerical Verification

doi: 10.3879/j.issn.1000-0887.2011.05.002

1.
Department of Mechanics, College of Sciences, Shanghai University, Shanghai 200444, P. R. China;

Received Date: 2011-01-04
Rev Recd Date: 2011-03-19
Publish Date: 2011-05-15

Abstract

Abstract

he low-order polynomial distributed eigenstrain formulation of boundary integral equation (BIE) and the corresponding definition of Eshelby tensors were proposed for elliptical-shaped inhomogeneities in a two-dimensional elastic medium. Taking the results from traditional sub-domain boundary element method (BEM) as the control, effectiveness of the present algorithm was verified for an elastic medium with a single elliptical inhomogeneity. It is shown that, with the present computational model and algorithm, significant improvements are achieved in terms of efficiency as compared with the traditional BEM and in terms of accuracy as compared with the constant eigenstrain formulation of the BIE.
- eigenstrain,
- Eshelby tensor,
- boundary integral equation,
- polynomial,
- inhomogeneity

FullText(HTML)

References(27)

References

[1]	Eshelby J D. The determination of the elastic field of an ellipsoidal inclusion and related problems[J]. Proceedings of the Royal Society of London A, 1957, 241(1226): 376-396. doi: 10.1098/rspa.1957.0133
[2]	Eshelby J D. The elastic field outside an ellipsoidal inclusion[J]. Proceedings of the Royal Society of London A, 1959, 252(1271): 561-569. doi: 10.1098/rspa.1959.0173
[3]	Mura T, Shodja H M, Hirose Y. Inclusion problems (part 3)[J]. Applied Mechanics Review, 1996, 49(10S): S118-S127.
[4]	Federico S, Grilloc A, Herzog W. A transversely isotropic composite with a statistical distribution of spheroidal inclusions: a geometrical approach to overall properties[J]. Journal of the Mechanics and Physics of Solids, 2004, 52(10): 2309-2327. doi: 10.1016/j.jmps.2004.03.010
[5]	Cohen I. Simple algebraic approximations for the effective elastic moduli of cubic arrays of spheres[J]. Journal of the Mechanics and Physics of Solids, 2004, 52(9): 2167-2183. doi: 10.1016/j.jmps.2004.02.008
[6]	Franciosi P, Lormand G. Using the radon transform to solve inclusion problems in elasticity[J]. International Journal of Solids and Structures, 2004, 41(3/4): 585-606. doi: 10.1016/j.ijsolstr.2003.10.011
[7]	Feng X Q, Mai Y W, Qin Q H. A micromechanical model for interpenetrating multiphase composites[J]. Computational Material Science, 2003, 28(3/4): 486-493. doi: 10.1016/j.commatsci.2003.06.005
[8]	Kompis V, Kompis M, Kaukic M. Method of continuous dipoles for modeling of materials reinforced by short micro-fibers[J]. Engineering Analysis With Boundary Elements, 2007, 31(5): 416-424. doi: 10.1016/j.enganabound.2006.10.008
[9]	Doghri I, Tinel L. Micromechanics of inelastic composites with misaligned inclusions: numerical treatment of orientation[J]. Computer Methods in Applied Mechanics and Engineering, 2006, 195(13/16): 1387-1406. doi: 10.1016/j.cma.2005.05.041
[10]	Kakavas P A, Kontoni D N. Numerical investigation of the stress field of particulate reinforced polymeric composites subjected to tension[J]. International Journal for Numerical Methods in Engineering, 2006, 65(7): 1145-1164. doi: 10.1002/nme.1483
[11]	Kanaun S K, Kochekseraii S B. A numerical method for the solution of thermo- and electro-static problems for a medium with isolated inclusions[J]. Journal of Computational Physics, 2003, 192(2): 471-493. doi: 10.1016/j.jcp.2003.07.010
[12]	Lee J, Choi S, Mal A. Stress analysis of an unbounded elastic solid with orthotropic inclusions and voids using a new integral equation technique[J]. International Journal of Solids and Structures, 2001, 38(16): 2789-2802. doi: 10.1016/S0020-7683(00)00182-7
[13]	Dong C Y, Cheung Y K, Lo S H. A regularized domain integral formulation for inclusion problems of various shapes by equivalent inclusion method[J]. Computer Methods in Applied Mechanics and Engineering, 2002, 191(31): 3411-3421. doi: 10.1016/S0045-7825(02)00261-X
[14]	Dong C Y, Lee K Y. Boundary element analysis of infinite anisotropic elastic medium containing inclusions and cracks[J]. Engineering Analysis With Boundary Elements, 2005, 29(6): 562-569. doi: 10.1016/j.enganabound.2004.12.011
[15]	Dong C Y, Lee K Y. Effective elastic properties of doubly periodic array of inclusions of various shapes by the boundary element method[J]. International Journal of Solids and Structures, 2006, 43(25/26): 7919-7938. doi: 10.1016/j.ijsolstr.2006.04.009
[16]	Liu Y J, Nishimura N, Tanahashi T, Chen X L, Munakata H. A fast boundary element method for the analysis of fiber-reinforced composites based on a rigid-inclusion model[J]. ASME Journal of Applied Mechanics, 2005, 72(1): 115-128. doi: 10.1115/1.1825436
[17]	Ma H, Deng H L. Nondestructive determination of welding residual stresses by boundary element method[J]. Advances in Engineering Software, 1998, 29(2): 89-95. doi: 10.1016/S0965-9978(98)00051-9
[18]	Nakasone Y, Nishiyama H, Nojiri T. Numerical equivalent inclusion method: a new computational method for analyzing stress fields in and around inclusions of various shapes[J]. Materials Science and Engineering A, 2000, 285(1/2): 229-238. doi: 10.1016/S0921-5093(00)00637-7
[19]	Qin Q H. Nonlinear analysis of Reissner plates on an elastic foundation by the BEM[J]. International Journal of Solids and Structures, 1993, 30(22): 3101-3111. doi: 10.1016/0020-7683(93)90141-S
[20]	Greengard L F, Rokhlin V. A fast algorithm for particle simulations[J]. Journal of Computational Physics, 1997, 73(2): 325-348.
[21]	Ma H, Yan C, Qin Q H. Eigenstrain formulation of boundary integral equations for modeling particle-reinforced composites[J]. Engineering Analysis with Boundary Elements, 2009, 33(3): 410-419. doi: 10.1016/j.enganabound.2008.06.002
[22]	马杭，夏利伟，秦庆华. 短纤维复合材料的本征应变边界积分方程计算模型[J]. 应用数学和力学, 29(6): 687-695.(MA Hang, XIA Li-wei, QIN Qing-hua. Computational model for short-fiber composites with eigen-strain formulation of boundary integral equations[J]. Applied Mathematics and Mechanics(English Edition), 2008, 29(6): 757-767.) doi: 10.1007/s10483-008-0607-4
[23]	Rahman M. The isotropic ellipsoidal inclusion with a polynomial distribution of eigenstrain[J]. ASME Journal of Applied Mechanics, 2002, 69(5): 593-601. doi: 10.1115/1.1491270
[24]	Nie G H, Guo L, Chan C K, Shin F G. Non-uniform eigenstrain induced stress field in an elliptic inhomogeneity embedded in orthotropic media with complex roots[J]. International Journal of Solids and Structures, 2007, 44(10): 3575-3593. doi: 10.1016/j.ijsolstr.2006.10.005
[25]	Ma H, Kamiya N, Xu S Q. Complete polynomial expansion of domain variables at boundary for two-dimensional elasto-plastic problems[J]. Engineering Analysis With Boundary Elements, 1998, 21(3): 271-275. doi: 10.1016/S0955-7997(98)00017-4
[26]	Ma H, Qin Q H. Solving potential problems by a boundary-type meshless method—the boundary point method based on BIE[J]. Engineering Analysis With Boundary Elements, 2007, 31(9): 749-761. doi: 10.1016/j.enganabound.2007.03.001
[27]	Ma H, Zhou J, Qin Q H. Boundary point method for linear elasticity using constant and quadratic moving elements[J]. Advances in Engineering Software, 2010, 41(3): 480-488. doi: 10.1016/j.advengsoft.2009.10.006