On the conventional boundary integral equation formulation for piezoelectric solids with defects or of thin shapes. (English) Zbl 1114.74497
Summary: In this paper, the conventional boundary integral equation (BIE) formulation for piezoelectric solids is revisited and the related issues are examined. The key relations employed in deriving the piezoelectric BIE, such as the generalized Green’s identity (reciprocal work theorem) and integral identities for the piezoelectric fundamental solution, are established rigorously. A weakly singular form of the piezoelectric BIE is derived for the first time using the identities for the fundamental solution, which eliminates the calculation of any singular integrals in the piezoelectric boundary element method (BEM). The crucial question of whether or not the piezoelectric BIE will degenerate when applied to crack and thin shell-like problems is addressed. It is shown analytically that the conventional BIE for piezoelectricity does degenerate for crack problems, but does not degenerate for thin piezoelectric shells. The latter has significant implications in applications of the piezoelectric BIE to the analysis of thin piezoelectric films used widely as sensors and actuators. Numerical tests to show the degeneracy of the piezoelectric BIE for crack problems are presented and one remedy to this degeneracy by using the multi-domain BEM is also demonstrated.
MSC:
| 74S15 | Boundary element methods applied to problems in solid mechanics |
| 74F15 | Electromagnetic effects in solid mechanics |
References:
| [1] | Tiersten, H. F., Linear piezoelectric plate vibrations (1969), Plenum Press: Plenum Press New York |
| [2] | Tzou, H. S., Piezoelectric shells: distributed sensing and control of continua (1993), Kluwer Academic: Kluwer Academic Dordrecht |
| [3] | Banks, H. T.; Smith, R. C.; Wang, Y., Smart material structures — modeling, estimation and control. Research in applied mathematics (1996), Wiley: Wiley Chichester · Zbl 0882.93001 |
| [4] | Rizzo, F. J., An integral equation approach to boundary value problems of classical elastostatics, Q Appl Math, 25, 83-95 (1967) · Zbl 0158.43406 |
| [5] | Mukherjee, S., Boundary element methods in creep and fracture (1982), Applied Science: Applied Science New York · Zbl 0534.73070 |
| [6] | Cruse, T. A., Boundary element analysis in computational fracture mechanics (1988), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0648.73039 |
| [7] | Brebbia, C. A.; Dominguez, J., Boundary elements — an introductory course (1989) · Zbl 0691.73033 |
| [8] | Banerjee, P. K., The boundary element methods in engineering (1994), McGraw-Hill: McGraw-Hill New York |
| [9] | Krishnasamy, G.; Rizzo, F. J.; Liu, Y. J., Boundary integral equations for thin bodies, Int J Numer Methods Engng, 37, 107-121 (1994) · Zbl 0795.73076 |
| [10] | Liu, Y. J.; Rizzo, F. J., Scattering of elastic waves from thin shapes in three dimensions using the composite BIE formulation, J Acoust Soc Am, 102, 2, 926-932 (1997), (Part 1, August) |
| [11] | Liu, Y. J., Analysis of shell-like structures by the boundary element method based on 3D elasticity: formulation and verification, Int J Numer Methods Engng, 41, 541-558 (1998) · Zbl 0910.73068 |
| [12] | Luo, J. F.; Liu, Y. J.; Berger, E. J., Analysis of two-dimensional thin structures (from micro- to nano-scales) using the boundary element method, Comput Mech, 22, 404-412 (1998) · Zbl 0938.74075 |
| [13] | Liu, Y. J.; Xu, N.; Luo, J. F., Modeling of interphases in fiber-reinforced composites under transverse loading using the boundary element method, J Appl Mech, 67, 1 (March), 41-49 (2000) · Zbl 1110.74568 |
| [14] | Luo, J. F.; Liu, Y. J.; Berger, E. J., Interfacial stress analysis for multi-coating systems using an advanced boundary element method, Comput Mech, 24, 6, 448-455 (2000) · Zbl 0961.74069 |
| [15] | Liu, Y. J.; Xu, N., Modeling of interface cracks in fiber-reinforced composites with the presence of interphases using the boundary element method, Mech Mater, 32, 12, 769-783 (2000) |
| [16] | Chen, S. H.; Liu, Y. J., A unified boundary element method for the analysis of sound and shell-like structure interactions. I. Formulation and verification, J Acoust Soc Am, 103, 3, 1247-1254 (1999) |
| [17] | Liu, Y. J.; Zhang, D.; Rizzo, F. J., Nearly singular and hypersingular integrals in the boundary element method. Nearly singular and hypersingular integrals in the boundary element method, Boundary elements XV (1993), Computational Mechanics Publications: Computational Mechanics Publications Worcester, MA |
| [18] | Barnett, D. M.; Lothe, J., Dislocations and line charges in anisotropic piezoelectric insulators, Phys Status Solidi B, 67, 105-111 (1975) |
| [19] | Meric, R. A.; Saigal, S., Shape sensitivity analysis of piezoelectric structures by the adjoint variable method, AIAA J, 29, 8, 1313-1318 (1991) |
| [20] | Lee, J. S.; Jiang, L. Z., Boundary element formulation for electro-elastic interaction in piezoelectric materials. Boundary element formulation for electro-elastic interaction in piezoelectric materials, Boundary elements XV (1993), Computational Mechanics Publications: Computational Mechanics Publications Worcester, MA |
| [21] | Lee, J. S.; Jiang, L. Z., A boundary integral formulation and 2D fundamental solution for piezoelastic media, Mech Res Commun, 21, 1, 47-54 (1994) · Zbl 0801.73063 |
| [22] | Lee, J. S., Boundary element method for electroelastic interaction in piezoceramics, Engng Anal Bound Elem, 15, 4, 321-328 (1995) |
| [23] | Lu, P.; Mahrenholtz, O., A variational boundary element formulation for piezoelectricity, Mech Res Commun, 21, 605-611 (1994) · Zbl 0843.73078 |
| [24] | Chen, T.; Lin, F. Z., Boundary integral formulations for three-dimensional anisotropic piezoelectric solids, Comput Mech, 15, 485-496 (1995) · Zbl 0826.73066 |
| [25] | Chen, T., Green’s functions and the non-uniform transformation problem in a piezoelectric medium, Mech Res Commun, 20, 271-278 (1993) · Zbl 0773.73077 |
| [26] | Chen, T.; Lin, F. Z., Numerical evaluation of derivatives of the anisotropic piezoelectric Green’s functions, Mech Res Commun, 20, 501-506 (1993) · Zbl 0925.73737 |
| [27] | Wang, C.-Y., Green’s functions and general formalism for 2D piezoelectricity, Appl Math Lett, 9, 4, 1-7 (1996) · Zbl 0864.73060 |
| [28] | Dunn, M. L.; Wienecke, H. A., Green’s functions for transversely isotropic piezoelectric solids, Int J Solids Struct, 33, 30, 4571-4581 (1996) · Zbl 0919.73293 |
| [29] | Hill, L. R.; Farris, T. N., Three-dimensional piezoelectric boundary element method, AIAA J, 36, 1, 102-108 (1998) · Zbl 0908.73086 |
| [30] | Ding, H.; Wang, G.; Chen, W., A boundary integral formulation and 2D fundamental solutions for piezoelectric media, Comput Methods Appl Mech Engng, 158, 65-80 (1998) · Zbl 0954.74077 |
| [31] | Ding, H.; Liang, J., The fundamental solutions for transversely isotropic piezoelectricity and boundary element method, Comput Struct, 71, 447-455 (1999) |
| [32] | Jiang, L. Z., Integral representation and Green’s functions for 3-D time dependent thermo-piezoelectricity, Int J Solids Struct, 37, 42, 6155-6171 (2000) · Zbl 0963.74021 |
| [33] | Xu, X.-L.; Rajapakse, R. K.N. D., Boundary element analysis of piezoelectric solids with defects, Compos Part B — Engng, 29, 655-669 (1998) |
| [34] | Zhao, M. H.; Shen, Y. P.; Liu, G. N.; Liu, Y. J., Fundamental solutions for infinite transversely isotropic piezoelectric media, (Santini, P.; Marchetti, M.; Brebbia, C. A., Computational methods for smart structures and materials (1998), WIT Press/Computational Mechanics Publications: WIT Press/Computational Mechanics Publications Boston), 45-54 |
| [35] | Zhao, M. H.; Shen, Y. P.; Liu, G. N.; Liu, Y. J., Dugdale model solutions for a penny-shaped crack in three-dimensional transversely isotropic piezoelectric media by boundary-integral equation method, Engng Anal Bound Elem, 23, 573-576 (1999) · Zbl 0960.74075 |
| [36] | Pan, E., A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids, Engng Anal Bound Elem, 23, 67-76 (1999) · Zbl 1062.74639 |
| [37] | Qin, Q., Thermoelectroelastic analysis of cracks in piezoelectric half-space by BEM, Comput Mech, 23, 353-360 (1999) · Zbl 0951.74077 |
| [38] | Rudolphi, T. J., The use of simple solutions in the regularization of hypersingular BIEs, Math Comput Modell, 15, 269-278 (1991) · Zbl 0728.73081 |
| [39] | Liu, Y. J.; Rudolphi, T. J., Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations, Engng Anal Bound Elem, 8, 6, 301-311 (1991) |
| [40] | Liu, Y. J.; Rudolphi, T. J., New identities for fundamental solutions and their applications to non-singular boundary element formulations, Comput Mech, 24, 4, 286-292 (1999) · Zbl 0969.74073 |
| [41] | Zemanian, A. H., Distribution theory and transform analysis — an introduction to generalized functions, with applications (1987), Dover: Dover New York · Zbl 0643.46028 |
| [42] | Liu, Y. J., On the simple-solution method and non-singular nature of the BIE/BEM — a review and some new results, Engng Anal Bound Elem, 24, 10, 787-793 (2000) |
| [43] | Tanaka, M.; Sladek, V.; Sladek, J., Regularization techniques applied to boundary element methods, Appl Mech Rev, 47, 10, 457-499 (1994) |
| [44] | Sladek, V.; Sladek, J., (Brebbia, C. A.; Aliabadi, M. H., Singular integrals in boundary element methods. Singular integrals in boundary element methods, Advances in boundary element series (1998), Computational Mechanics Publications: Computational Mechanics Publications Boston) · Zbl 0961.74072 |
| [45] | Mukherjee, S., On boundary integral equations for cracked and for thin bodies, Math Mech Solids (2001), (in press) · Zbl 1042.74055 |
| [46] | Sosa, H. A., Plane problems in piezoelectric media with defects, Int J Solids Struct, 28, 491-505 (1991) · Zbl 0749.73070 |
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.
