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Published in: Meccanica 9/2023

17-07-2023

Exact solution to axisymmetric interface crack problem in piezo-electric transversely isotropic materials

Author: V. I. Fabrikant

Published in: Meccanica | Issue 9/2023

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Abstract

This seems to be the first exact closed form solution to the problem of a penny-shaped interface crack, subjected to an axisymmetric normal and tangential loading. The crack is located at the boundary between two bonded piezo-electric transversely isotropic half-spaces, made of different materials. We use the combination of Green’s functions for 2 different half-spaces and Fourier transform. We derive first the governing equations, which are valid for a crack of arbitrary shape. The usual approach leads to 4 hypersingular integral equations, we arrive at 3 integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications or not noticed at all. In the particular case of axial symmetry, the problem is reduced to just one singular equation, for which exact closed form solution is known. We are not aware of any other publication with which our results can be compared.

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Appendix
Available only for authorised users
Literature
1.
go back to reference Fabrikant VI (2010) Contact and crack problems in linear elasticity. fabrikant-research.webs.com Fabrikant VI (2010) Contact and crack problems in linear elasticity. fabrikant-research.webs.com
2.
go back to reference Fabrikant VI (2021) New approach to interface crack problems in transversely isotropic materials. ZAMP, pp. 72-86 Fabrikant VI (2021) New approach to interface crack problems in transversely isotropic materials. ZAMP, pp. 72-86
3.
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go back to reference Willis JR (1972) The penny-shaped crack on an interface. Quart J Mech Appl Math 25:367–385CrossRefMATH Willis JR (1972) The penny-shaped crack on an interface. Quart J Mech Appl Math 25:367–385CrossRefMATH
5.
go back to reference Zhao MH, Fang PZ, Shen YP (2004) Boundary integral-differential equations and boundary element method for interfacial cracks in three-dimensional piezoelectric media. Eng Anal Bound Elem 28:753–762CrossRefMATH Zhao MH, Fang PZ, Shen YP (2004) Boundary integral-differential equations and boundary element method for interfacial cracks in three-dimensional piezoelectric media. Eng Anal Bound Elem 28:753–762CrossRefMATH
Metadata
Title
Exact solution to axisymmetric interface crack problem in piezo-electric transversely isotropic materials
Author
V. I. Fabrikant
Publication date
17-07-2023
Publisher
Springer Netherlands
Published in
Meccanica / Issue 9/2023
Print ISSN: 0025-6455
Electronic ISSN: 1572-9648
DOI
https://doi.org/10.1007/s11012-023-01687-w

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