Thermomagnetoelectroelastic problems for various defects embedded in an infinite matrix are considered in this paper. Using Stroh's formalism, conformal mapping, and perturbation technique, Green's functions are obtained in closed form for a defect in an infinite magnetoelectroelastic solid induced by the thermal analog of a line temperature discontinuity and a line heat source. The defect may be of an elliptic hole or a Griffith crack, a half-plane boundary, a bimaterial interface, or a rigid inclusion. These Green's functions satisfy the relevant boundary or interface conditions. The proposed Green's functions can be used to establish boundary element formulation and to analyzing fracture behaviour due to the defects mentioned above.

2D Green's functions of defective magnetoelectroelastic solids under thermal loading

Recent developments and current understanding on cracks and fracture in piezoelectric ceramic materials are presented. Focus is placed on the description and proper selection of electric boundary conditions along crack surfaces as well as their influence on piezoelectric fracture. Five different types of crack surface boundary conditions are addressed: (i) impermeable cracks with traction-free surfaces; (ii) permeable cracks with traction-free surfaces; (iii) cracks with exact electric boundary and traction-free surfaces; (iv) impermeable cracks with near-tip microstructural features; and (v) impermeable (or permeable) cracks with contacting surfaces. The first three conditions, namely, the impermeable crack (insulating crack), the permeable crack (conducting crack), and the exact electric boundary (which has the impermeable and permeable cracks as its two limiting cases), have been well studied in the literature and are commonly known as the linear models. The last two types, known as the non-linear models, are categorized as the mixed electric boundary condition, and are introduced to treat non-linear effects such as electric-field induced yielding and crack closure.

Five different ways to formulate a fracture criterion for piezoelectric materials are presented and compared: (i) total energy release rate (or the J-integral); (ii) mechanical strain energy release rate (or the mechanical part of the J-integral); (iii) local and global energy release rates due to electrical yielding only; (iv) influence of mode-mixity on fracture toughness; and (v) maximum hoop stress. Contradicting views about the role of applied electric field in piezoelectric fracture are discussed in detail. Whilst most studies found that a positive electric field promotes crack propagation whereas a negative electric field impedes crack propagation, exactly the opposite trends have also been observed. Moreover, some recent investigations with indentation tests reveal that both positive and negative fields can promote crack growth if inelastic deformation is taken into account and/or fracture criterion based on mechanical strain energy release rate is used. In this review article, clarification of the above discrepancies is attempted in terms of different crack surface boundary conditions and different fracture criteria selected. Also discussed are interface crack problems in dissimilar piezoelectric materials and dynamic piezoelectric fracture. Both two- and three-dimensional solutions obtained with either analytical methods or numerical tools such as the method of finite elements are presented.

Cracks and fracture in piezoelectrics

In this paper the dynamic anti-plane problem for a functionally graded piezoelectric strip containing a central crack vertical to the boundary is considered. The crack is assumed to be electrically impermeable or permeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Numerical results show the effects of loading combination parameter, material gradient parameter and crack configuration on the dynamic response. With the permeable assumption, the electric impact has no contribution to the crack tip field singularity. With the impermeable assumption, the direction of applied electric impact loading plays a great role in the behavior of dynamic stress intensity factor, and the existence of electric load always enhances the crack propagation. However, the crack is easier to propagate under the negative electric load than that under the positive electric load.

Wei-Xun Fan

Papers

A general solution for the plane problem in piezoelectric media with collinear cracks