Analysis of fractures in 3D piezoelectric media by a weakly singular integral equation method

Abstract

A weakly singular, symmetric Galerkin boundary element method (SGBEM) is established to compute stress and electric intensity factors for isolated cracks in threedimensional, generally anisotropic, piezoelectric media. The method is based upon a weakform integral equation, for the surface traction and the surface electric charge, which is established by means of a systematic regularization procedure; the integral equation is in a symmetric form and is completely regularized in the sense that its integrand contains only weakly singular kernels of O(1/r ) (hence allowing continuous interpolations to be employed in the numerical approximation). The weakly singular kernels which appear in the weak-form integral equation are expressed explicitly, for general anisotropy, in terms of a line integral over a unit circle. In the numerical implementation, a special crack-tip element is adopted to discretize the region near the crack front while the remainder of the crack surface is discretized by standard continuous elements. The special crack-tip element allows the relative crack-face displacement and electric potential in the vicinity of the crack front to be captured to high accuracy (even with relatively large elements), and it has the important feature that the mixedmode intensity factors can be directly and independently extracted from the crack front nodal data. To enhance the computational efficiency of the method, special integration quadratures are adopted to treat both singular and nearly singular integrals, and an interpolation strategy is developed to approximate the weakly singular kernels. As demonstrated by various numerical examples for both planar and non-planar fractures, the method gives rise to highly accurate intensity factors with only a weak dependence on mesh refinement.

SGBEM–FEM coupling for analysis of cracks in 3D anisotropic media

“…Finally, we remark that it would be possible to extend the method to allow for treatment of more complex material behavior such as that pertaining to the linear response of certain active materials; such an extension would, of course, require the use of an appropriate set of integral equations as a basis for the SGBEM (e.g. [39,40]). …”

Regularized boundary integral representations for dislocations and cracks in smart media

“…Direct generalization of the existing Somigliana identity for linear elastic media Mear 1998, Rungamornrat andMear 2008a) and linear piezoelectric media (Denda and Lua 1999, Noda 2004, Rungamornrat andMear 2008c) to the case of linear piezoelectromagnetic solids results in a boundary integral relation for the generalized displacement P at any interior point x:…”

“…Chen 2003a, 2003b, Qin et al 2007. Until recently, Rungamornrat and Mear (2008c) derived a weakly singular, weak form integral equation for the generalized traction and it was subsequently used as a basis to implement a weakly singular SGBEM for modeling isolated cracks in threedimensional, linear piezoelectric infinite media. Nevertheless, on the basis of a careful literature search, we can say that the extension of such results to a more general class of smart materials, e.g.…”

Boundary element analysis of 3D shell‐like rigid electrically conducting inclusions in anisotropic thermomagnetoelectroelastic solids

“…A number of works address 3D fracture mechanics under piezoelectric, piezomagnetic and magnetoelectroelastic coupling. Rungamornrat and Mear and Rungamornrat et al . derived a symmetric Galerkin BEM for 3D fracture mechanics analysis of piezoelectric solids.…”

“…A number of works address 3D fracture mechanics under piezoelectric, piezomagnetic and magnetoelectroelastic coupling. Rungamornrat and Mear [3] and Rungamornrat et al [4] derived a symmetric Galerkin BEM for 3D fracture mechanics analysis of piezoelectric solids. Zhao et al [5] presented the extended discontinuity boundary integral equation method for vertical cracks in magnetoelectroelastic medium.…”