Thermoelectroelastic analysis of cracks in piezoelectric half-plane by BEM

The Green's function and the boundary element method for analysing fracture behaviour of cracks in piezoelectric half-plane are presented in this paper. By combining Stroh formalism and the concept of perturbation, a general thermoelectroelastic solution for half-plane solid subjected to point heat source and/or temperature discontinuity has been derived. Using the proposed solution and the potential variational principle, a boundary element model (BEM) for 2-D half-plane solid with multiple cracks has been developed and used to calculate the stress intensity factors of the multiple crack problem. The method is available for multiple crack problems in both ®nite and in®nite solids. Numerical results for a two-crack system are presented and compared with those from ®nite element method (FEM). 1 Introduction The thermoelectroelastic analysis of multiple cracks inside a piezoelectric solid is of considerable importance in the ®eld of fracture mechanics, as the piezoelectric materials often contains many internal microcracks before in use. Stress analysis of multiple crack problems in isotropic materials has been reported by many researchers such as Chen (1984), Horri and Nemat-Nasser (1985, 1987), Wang and Atluri (1995, 1996). For anisotropic materials, Hwu (1991) obtained a solution for collinear cracks in an in®nite plate. Chen and Hasebe (1994) treated the elastic interaction between a main-crack and a parallel micro-crack in an orthotropic plate. Atluri and his colleagues (Chow, Beom and Atluri, 1995; Chow and Atluri, 1995) analysed the mixed-mode stress intensity factors of an interface crack using the mutual integral techniques. They found that the virtual crack closure integral method could lead to very accurate results with a relatively coarse ®nite element mesh. Later, Beom and Atluri (1996) extended their results to the case of anisotropic piezoelectric materials. Using the strain energy density criterion, Ma et al. (1996) studied the direction of initial crack growth of two interacting cracks in an anisotropic solid. Most of the developments in the ®eld can also be found in (Aliabadi and Brebbia, 1993; Kachanov, 1993; Atluri, 1997). Relatively little work has been, however, found in the literature for the thermal analysis of multiple crack problems in piezoelectric solid. In this paper, a general thermoelectroelastic solution for a half-plane solid subjected to point heat source and/or temperature discontinuity is ®rst derived by using Stroh formalism and the concept of perturbation. The solution is, then, implemented with BEM to study fracture behaviour of multiple cracks in a half-plane solid. Based on the thermoelectroelastic solution for half-plane solid, a boundary element model for temperature discontinuity as well as dislocation of elastic displacement and electric potential (EDEP) is developed and used to calculate stress and electric displacement (SED) intensity factors. Numerical results of SED intensity factors for a two-crack system are presented to illustrate applications of the proposed formulation, and comparison is made with those obtained from ®nite element method. 2 Basic formulations 2.1 General solution for linear thermopiezoelectricity Consider a linear piezoelectric solid in which all ®elds are assumed to depend only on in-plane coordinates x1 and x2. Boldfaced symbols stand for either column vectors or matrices, depending on whether lower case or upper case is used. The SED vector Pi, The EDEP vector u, temperature T and heat ̄ux hi in the solid subjected to loading can be expressed in terms of complex analytic functions as follows (Qin, 1998): T ˆ g0…zt† ‡ g0…zt†; # ˆ ÿikg0…zt† ‡ ikg0…zt†; h1 ˆ ÿ#;2; h2 ˆ #;1; u ˆ AF…z†q‡ cg…zt† ‡ AF…z†q‡ cg…zt†; / ˆ BF…z†q‡ dg…zt† ‡ BF…z†q‡ dg…zt†; P1 ˆ ÿ/;2; P2 ˆ /;1 …1† with F…z† ˆ diag f …z1†f …z2†f …z3†f …z4† ‰ Š zt ˆ x1 ‡ sx2; zi ˆ x1 ‡ pix2 …2† Computational Mechanics 23 (1999) 353±360 Ó Springer-Verlag 1999