Uniform asymptotic solutions for lamellar inhomogeneities in piezoelectric solids
Authors:
Cristian Dascalu and Dorel Homentcovschi
Journal:
Quart. Appl. Math. 61 (2003), 657-682
MSC:
Primary 74F15; Secondary 74B05, 74G10, 74G70, 78A30
DOI:
https://doi.org/10.1090/qam/2019617
MathSciNet review:
MR2019617
Full-text PDF Free Access
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Abstract: We study the problem of a lamellar inhomogeneity of arbitrary shape embedded in a piezoelectric matrix of infinite extent. Uniform asymptotic solutions for the equations of elastostatics and electrostatics on this configuration are obtained. The first order terms, in the inhomogeneity thickness, are explicitly determined for piezoelectric inclusions, rigid inclusions of electric conductor, impermeable cracks, and cracks with inside electric field. We give real-form expressions of mechanical and electric fields at the interface and on the inhomogeneity axis. Detailed first order solutions are obtained for elliptic and lemon-shaped inhomogeneities. It is found that, while for elliptic piezoelectric inclusions the perturbation stresses and electric displacements at the inclusion ends have the same order as those given at infinity, for a lemon-shaped inclusion they are an order-of-magnitude smaller. Intensity factors are calculated for lemon-shaped cavities. It is shown that, when inside electric fields are considered, the stress intensity coefficients are influenced by the material anisotropy.
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Barnett, D. M. and Lothe, J., 1975, Dislocations and line charges in anisotropic piezoelectric insulators, Phys. Status Solidi B 67, 105β111
Chung, M. Y. and Ting, T. C. T., 1996, Piezoelectric solid with an elliptic inclusion or hole, Int. J. Solids Structures 33 (23), 3343β3361
Dascalu, C., 1997, Electroelasticity equations and energy approaches to fracture, Int. J. Engng. Sci. 35, 1185β1196
Dascalu, C. and Maugin, G. A., 1995, On the dynamic fracture of piezoelectric materials, Q. J. Mech. Appl. Mat. 48, 237β251
Dascalu, C. and Homentcovschi, D., 1999, Uniform asymptotic solutions for lamellar inhomogeneities in anisotropic elastic solids, SIAM J. Appl. Math. 60, 18β42
Geer, J. F. and Keller, J. B., 1968, Uniform asymptotic solutions for potential flow around a thin airfoil and the electrostatic potential about a thin conductor, SIAM J. Appl. Math. 16, 75β101
Homentcovschi, D., 1979, Conformal mapping of the domain exterior to a thin region, SIAM J. Math. Anal. 10, 1246β1257
Homentcovschi, D., 1982, Uniform asymptotic solutions for the two-dimensional potential field problem with joining relations on the surface of a slender body, Int. J. Engng. Sci. 20, 753β767
Homentcovschi, D., 1984, Uniform asymptotic solutions of two-dimensional problems of elasticity for the domain exterior to a thin region, SIAM J. Appl. Math. 44, 1β10
Homentcovschi, D. and Dascalu, C., 2000, Uniform asymptotic solutions for lamellar inhomogeneities in plane elasticity, J. Mech. Phys. Solids 48, 153β173
Liang, J., Han, J., Wang, B., and Du, S., 1995, Electroelastic modelling of anisotropic piezoelectric materials with an elliptic inclusion, Int. J. Solids Structures 32 (20), 2989β3000
Lothe, J. and Barnett, D. M., 1976, Integral formalism for surface waves in piezoelectric crystals. Existence considerations, J. Appl. Phys. 47, 1799β1807
Lu, P., Tan, M. J., and Liew, K. M., 2000, A further investigation of Green functions for a piezoelectric material with a cavity or a crack, Int. J. Solids Structures 37, 1065β1078
Maugin, G. A., 1988, Continuum mechanics of electromagnetic solids, North-Holland, Amsterdam
Pak, Y. E., 1990, Crack extension force in a piezoelectric material, J. Appl. Mech. 57, 647β653
Pak, Y. E., 1992, Linear electroelastic fracture mechanics of piezoelectric materials, Int. J. Fracture 54, 79β100
Park, S. B. and Sun C. T., 1995, Effect of electric field on fracture of piezoelectric ceramics, Int. J. Fracture 70, 203β216
Pisarenko, G. G., Chushko, V. M., and Kovalev, S. P., 1985, Anisotropy of fracture toughness of piezoelectric ceramics, J. Am. Ceram. Soc. 68 (5), 259β265
Sosa, H., 1991, Plane problems in piezoelectric media with defects, Int. J. Solids Structures 28, 491β505
Sosa, H. and Khutoryansky, N., 1996, New developments concerning piezoelectric materials with defects, Int. J. Solids Structures 33, 3399β3414
Suo, Z., Kuo, C.-M., Barnett, D. M., and Willis, J. R., 1992, Fracture mechanics for piezoelectric ceramics, J. Mech. Phys. Solids 40, 739β765
Tiersten, H. F., 1969, Linear piezoelectric plate vibrations, Plenum Press, New York
Ting, T. C. T., 1988, Some identities and the structure of N$_{i}$ in the Stroh formalism of anisotropic elasticity, Q. Appl. Math. 46, 109β120
Ting, T. C. T., 1996, Anisotropic elasticity: Theory and applications, Oxford University Press, New York
Wu, C. H., 1994, Regularly and singularly perturbed crack, Q. Appl. Math. 52, 529β543
Zhang, T.-Y., Qian, C.-F., and Tong, P., 1998, Linear electro-elastic analysis of a cavity or a crack in a piezoelectric material, Int. J. Solids Structures 35 (17), 2121β2149
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© Copyright 2003
American Mathematical Society