BIEM analysis of dynamically loaded anti-plane cracks in graded piezoelectric finite solids

Abstract

a b s t r a c tAnti-plane cracks in finite functionally graded piezoelectric solids under time-harmonic loading are studied via a non-hypersingular traction based boundary integral equation method (BIEM). The formulation allows for a quadratic variation of the material properties in two directions. The boundary integral equation (BIE) system is treated by using the frequency dependent fundamental solution based on Radon transforms. Its numerical solution provides the displacements and tractions on the external boundary as well as the crack opening displacements from which the mechanical stress intensity factor (SIF) and the electrical displacement intensity factor (EDIF) are determined. Several examples for single and multiple straight and curved cracks demonstrate the applicability of the method and show the influence of the different system parameters.

Anti-plane dynamic hole–crack interaction in a functionally graded piezoelectric media

“…(6), (7), and (11) can be formulated by a system of traction boundary integral equations on using the representation formulas for the piezoelectric continua, see [25,26], and the non-hypersingular traction BIEM proposed by Zhang and Gross [27] for the pure elastic solid. For the considered problem, it is a combination of the BIEM numerical schemes given in Dineva et al [20,21]:…”

“…Applying the shifted point scheme, the singular integrals converge in Cauchy principal value (CPV) sense, since the smoothness requirements u J ∈ C 1+α ( cr ), u J ∈ C 1+α ( h ), t J ∈ C α ( ) of the approximation are fulfilled, see Rangelov et al [29]. Due to the form of the fundamental solution as an integral over the unit circle, see Appendix A in [20], and all integrals in Eq. (14) are two dimensional.…”

“…(14) are two dimensional. After discretization procedure of the BIE, solution of all type of integrals, see Appendix B in [20], and satisfying the boundary conditions, an algebraic system of equations for the crack opening displacement u J along the crack cr and displacement u J along the hole boundary h is obtained and solved. Following this procedure, a program code based on Mathematica and FORTRAN has been created.…”

“…There is a lack of results for dynamic hole-crack interaction problems even for homogeneous piezoelectrics, and to the authors' knowledge, there are no results for this problem in the case of functionally graded piezoelectric solids. In Dineva et al [20], the authors solved a dynamic anti-plane crack problem for a functionally graded material, while in Dineva et al [21], the similar problem was solved for a single hole. The present paper is based on the results obtained in [20,21] and investigates the dynamic interaction between hole and crack.…”

A boundary element and radial basis function approximation method for a second order elliptic partial differential equation with general variable coefficients

“…In this paper, we present a boundary element procedure for a more general steady‐state anisotropic diffusion equation by including a linear source term and taking the coefficients k ij to be any general smoothly varying functions of space. No restrictive form (such as with constant γ ij , as assumed in, for example, the works of Ang et al Dineva et al, and Rangelov et al) is imposed on k 11 , k 12 , and k 22 here. The coefficients k 11 , k 12 , and k 22 may be individually given by any smoothly varying functions as long as they satisfy the positive definiteness condition (for elliptic partial differential equations) in the solution domain.…”