×

Inverse fracture problems in piezoelectric solids by local integral equation method. (English) Zbl 1244.74123

Summary: The meshless local Petrov-Galerkin (MLPG) method is used to solve the inverse fracture problems in two-dimensional (2D) piezoelectric body. Electrical boundary conditions on the crack surfaces are not specified due to unknown dielectric permittivity of the medium inside the crack. Both stationary and transient dynamic boundary conditions are considered here. The analyzed domain is covered by small circular subdomains surrounding nodes spread randomly over the analyzed domain. A unit step function is chosen as test function in deriving the local integral equations (LIE) on the boundaries of the chosen subdomains. The Laplace-transform technique is applied to eliminate the time variation in the governing equation. The local integral equations are nonsingular and take a very simple form. The spatial variation of the Laplace transforms of displacements and electrical potential are approximated on the local boundary and in the interior of the subdomain by means of the moving least-squares (MLS) method. The singular value decomposition (SVD) is applied to solve the ill-conditioned linear system of algebraic equations obtained from the LIE after MLS approximation. The Stehfest algorithm is applied for the numerical Laplace inversion to retrieve the time-dependent solutions.

MSC:

74R10 Brittle fracture
74F15 Electromagnetic effects in solid mechanics
74H35 Singularities, blow-up, stress concentrations for dynamical problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Sladek, J.; Sladek, V.; Zhang, Ch., A local integral equation method for dynamic analysis in functionally graded piezoelectric materials, (Minutolo, V.; Aliabadi, M. H., Advances in boundary element techniques VIII (2007), Copyright EC, Ltd.: Copyright EC, Ltd. United Kingdom), 141-148
[2] Kuna, M., Finite element analyses of cracks in piezoelectric structures—a survey, Archives of Applied Mechanics, 76, 725-745 (2006) · Zbl 1161.74474
[3] Hadamard, J., Lectures on Cauchy problem in linear partial differential equations (1923), Oxford University Press: Oxford University Press London · JFM 49.0725.04
[4] Kubo, S.; Sakagami, T.; Ojhi, K., Electric-potential CT method based on BEM inverse análisis for measurement of three-dimensional cracks, (Kubo, S., Proceedings of international conference on computational mechanics, vol. 339 (1986), Springer: Springer Berlin)
[5] Tanaka, M.; Masuda, Y., Boundary element method applied to some inverse problems, Engineering Analysis, 3, 138-143 (1986)
[6] Hsieh, C. S.; Mura, T., Nondestructive cavity identification in structures, International Journal of Solids and Structures, 30, 1579-1587 (1993) · Zbl 0774.73075
[7] Kihara, J.; Shen, G.; Yamauchi, T.; Mimura, H.; Makino, L.; Liu, B., Some applications of boundary element method for evaluation of the deformation of tool and calculation of the residual stress distribution in product in the plastic working process, (Brebbia, C. A.; Futagami, T.; Tanaka, M., Boundary element proceedings of the fifth international conference (1983), Springer: Springer Berlin), 393-405
[8] Kubo, S., Inverse problems related to the mechanics and fracture of solids and structures, JSME International Journal, 31, 157-166 (1988)
[9] Bonnet, M.; Constantinescu, A., Inverse problems in elasticity, Inverse Problems, 21, R1-R50 (2005) · Zbl 1070.35118
[10] Bui, H. D., Inverse problems in the mechanics of materials: an introduction (1994), CRC Press: CRC Press Boca Raton FL
[11] Schnur, D.; Zabaras, N., Finite element solution of two-dimensional elastic problems using spatial smoothing, International Journal for Numerical Methods in Engineering, 30, 57-75 (1990) · Zbl 0729.73202
[12] Yeih, W. C.; Koya, T.; Mura, T., An inverse problem in elasticity with partially overspecified boundary conditions, I. Theoretical approach, ASME Journal of Applied Mechanics, 60, 595-600 (1993) · Zbl 0795.73015
[13] Koya, T.; Yeih, W. C.; Mura, T., An inverse problem in elasticity with partially overspecified boundary conditions, II. Numerical details, ASME Journal of Applied Mechanics, 60, 601-606 (1993) · Zbl 0795.73015
[14] Marin, L.; Elliot, L.; Ingham, D. B.; Lesnic, D., Boundary element method for the Cauchy problem in linear elasticity, Engineering Analysis with Boundary Elements, 25, 783-793 (2001) · Zbl 1048.74049
[15] Marin, L.; Lesnic, D., Boundary element solution for Cauchy problem in linear elasticity using singular value decomposition, Computer Methods in Applied Mechanics and Engineering, 191, 3257-3270 (2002) · Zbl 1045.74050
[16] Marin, L.; Lesnic, D., The method of fundamental solution for the Cauchy problem in two-dimensional linear elasticity, International Journal of Solids and Structures, 41, 3425-3438 (2004) · Zbl 1071.74055
[17] Galybin, A. N., A method for determination of stress distributions in the process zone ahead of a 2D crack, (Sarler, B.; Brebbia, C. A., Moving Boundaries VI (2001), WIT Press: WIT Press Southampton), 243-252
[18] Pak, Y. E., Crack extension force in a piezoelectric material, ASME Journal of Applied Mechanics, 57, 647-653 (1990) · Zbl 0724.73191
[19] Park, S. B.; Sun, C. T., Effect of electric field on fracture of piezoelectric ceramics, International Journal of Fracture, 70, 203-216 (1995)
[20] Shindo, Y.; Narita, F.; Tanaka, K., Electroelastic intensification near anti-plane shear crack in orthotropic piezoelectric ceramic strip, Theoretical and Applied Fracture Mechanics, 25, 65-71 (1996)
[21] Shindo, Y.; Tanaka, K.; Narita, F., Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear, Acta Mechanica, 120, 31-45 (1997) · Zbl 0889.73062
[22] Yang, J. H.; Lee, K. Y., Penny-shaped crack in a three-dimensional piezoelectric strip under in-plane normal loadings, Acta Mechanica, 148, 187-197 (2001) · Zbl 0987.74057
[23] Gruebner, O.; Kamlah, M.; Munz, D., Finite element analysis of cracks in piezoelectric materials taking into account the permittivity of the crack medium, Engineering Fracture Mechanics, 70, 1399-1413 (2003)
[24] Govorukha, V.; Kamlah, M., Asymptotic fields in the finite element analysis of electrically permeable interfacial cracks in piezoelectric bimaterials, Archives of Applied Mechanics, 74, 92-101 (2004) · Zbl 1158.74450
[25] Enderlein, M.; Ricoeur, A.; Kuna, M., Finite element techniques for dynamic crack analysis in piezoelectrics, International Journal of Fracture, 134, 191-208 (2005) · Zbl 1196.74264
[26] Kuna, M., Finite element analyses of crack problems in piezoelectric structures, Computational Materials Science, 13, 67-80 (1998)
[27] Pan, E., A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids, Engineering Analysis with Boundary Elements, 23, 67-76 (1999) · Zbl 1062.74639
[28] Davi, G.; Milazzo, A., Multidomain boundary integral formulation for piezoelectric materials fracture mechanics, International Journal of Solids and Structures, 38, 2557-2574 (2001) · Zbl 1058.74084
[29] Gross, D.; Rangelov, T.; Dineva, P., 2D wave scattering by a crack in a piezoelectric plane using traction BIEM, SID: Structural Integrity & Durability, 1, 35-47 (2005)
[30] Garcia-Sanchez, F.; Saez, A.; Dominguez, J., Anisotropic and piezoelectric materials fracture analysis by BEM, Computers and Structures, 83, 804-820 (2005)
[31] Garcia-Sanchez, F.; Zhang, Ch.; Sladek, J.; Sladek, V., 2-D transient dynamic crack analysis in piezoelectric solids by BEM, Computational Materials Science, 39, 179-186 (2007)
[32] Sheng, N.; Sze, K. Y., Multi-region Trefftz boundary element method for fracture analysis in plane piezoelectricity, Computational Mechanics, 37, 381-393 (2006) · Zbl 1103.74055
[33] Ohs, R. R.; Aluru, N. R., Meshless analysis of piezoelectric devices, Computational Mechanics, 27, 23-36 (2001) · Zbl 1005.74078
[34] Liu, G. R.; Dai, K. Y.; Kim, K. M.; Gu, Y. T., A point interpolation mesh free method for static and frequency analysis of two-dimensional piezoelectric structures, Computational Mechanics, 29, 510-519 (2002) · Zbl 1146.74371
[35] Atluri, S. N.; Han, Z. D.; Shen, S., Meshless local Petrov-Galerkin (MLPG) approaches for solving the weakly-singular traction & displacement boundary integral equations, CMES: Computer Modeling in Engineering & Sciences, 4, 507-516 (2003) · Zbl 1108.74385
[36] Atluri, S. N., The meshless method, (MLPG) for domain & BIE discretizations (2004), Tech Science Press · Zbl 1105.65107
[37] Sladek, J.; Sladek, V.; Atluri, S. N., Meshless local Petrov-Galerkin method in anisotropic elasticity, CMES: Computer Modeling in Engineering & Sciences, 6, 477-489 (2004) · Zbl 1082.74059
[38] Sladek, J.; Sladek, V.; Zhang, Ch.; Garcia-Sanchez, F.; Wunsche, M., Meshless local Petrov-Galerkin method for plane piezoelectricity, CMC: Computers, Materials & Continua, 4, 109-118 (2006)
[39] Sladek, J.; Sladek, V.; Zhang, Ch.; Solek, P.; Starek, L., Fracture analyses in continuously nonhomogeneous piezoelectric solids by the MLPG, CMES: Computer Modeling in Engineering & Sciences, 19, 247-262 (2007) · Zbl 1184.74054
[40] Belytschko, T.; Krogauz, Y.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods; an overview and recent developments, Computer Methods in Applied Mechanics and Engineering, 139, 3-47 (1996) · Zbl 0891.73075
[41] Stehfest, H., Algorithm 368: numerical inversion of Laplace transform, Communications of the Association for Computing Machinery, 13, 47-49 (1970)
[42] Parton, V. Z.; Kudryavtsev, B. A., Electromagnetoelasticity, piezoelectrics and electrically conductive solids (1988), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York
[43] Sladek, J.; Sladek, V.; Zhang, Ch.; Solek, P.; Pan, E., Evaluation of fracture parameters in continuously nonhomogeneous piezoelectric solids, International Journal of Fracture, 145, 313-326 (2007) · Zbl 1198.74024
[44] Sosa, H., Plane problems in piezoelectric media with defects, International Journal of Solids and Structures, 28, 491-505 (1991) · Zbl 0749.73070
[45] Pak, Y. E., Linear electro-elastic fracture mechanics of piezoelectric materials, International Journal of Fracture, 54, 79-100 (1992)
[46] Park, S. B.; Sun, C. T., Effect of electric field on fracture of piezoelectric ceramics, International Journal of Fracture, 70, 203-216 (1995)
[47] Hansen, P. C., Rank-deficient and discrete Ill-posed problems: numerical aspects of linear inversion (1998), SIAM: SIAM Philadelpia
[48] Golub, G. H.; Van Loan, V. F., Matrix computations (1996), The John Hopkins University Press: The John Hopkins University Press Baltimore · Zbl 0865.65009
[49] Lesnic, D.; Elliot, L.; Ingham, D. B., The boundary element solution of the Laplace and biharmonic equations subjected to noisy boundary data, International Journal for Numerical Methods in Engineering, 43, 479-492 (1998) · Zbl 0941.76056
[50] Jin, B.; Marin, L., The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction, International Journal for Numerical Methods in Engineering, 69, 1570-1589 (2007) · Zbl 1194.80101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.