Boundary element‐free method for fracture analysis of 2‐D anisotropic piezoelectric solids

Abstract

SUMMARYThis paper considers a 2-D fracture analysis of anisotropic piezoelectric solids by a boundary elementfree method. A traction boundary integral equation (BIE) that only involves the singular terms of order 1/r is first derived using integration by parts. New variables, namely, the tangential derivative of the extended displacement (the extended displacement density) for the general boundary and the tangential derivative of the extended crack opening displacement (the extended displacement dislocation density), are introduced to the equation so that solution to curved crack problems is possible. This resulted equation can be directly applied to general boundary and crack surface, and no separate treatments are necessary for the upper and lower surfaces of the crack. The extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions, and the unknown weight functions are modelled using the moving least-squares (MLS) approximation. The numerical scheme of the boundary element-free method is established, and an effective numerical procedure is adopted to evaluate the singular integrals. The extended 'stress intensity factors' (SIFs) are computed for some selected example problems that contain straight or curved cracks, and good numerical results are obtained.

An interpolating boundary element-free method (IBEFM) for elasticity problems

“…There are also fewer coefficients in the improved moving least-squares approximation than there are in the MLS approximation, and hence the computing speed and efficiency have increased. Combining the boundary integral equation method with the improved moving least-squares approximation, Cheng and Liew et al [9][10][11][12][13][14][15][16][17][18][19] come up with a direct meshless boundary integral equation method, called boundary element-free method (BEFM), to solve the problems, such as potential problems, elasticity, elastodynamics, and fracture. And the improved element-free Galerkin method based on the improved moving least-squares approximation was discussed by Zhang, Liew and Cheng [21][22][23].…”

“…The meshless method based on the MLS approximation can generate a solution possessing great precision. The meshless boundary integral equation methods are developed by combining the MLS approximation with boundary integral equation methods [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The boundary node method (BNM) is one of the meshless boundary integral equation methods, and Mukherjee et al [3][4][5] used it to solve potential problems and linear elasticity problems.…”

“…Combining different approximation functions in the meshless method with the boundary integral equation method, researchers have developed meshless boundary integral equation methods, such as the boundary node method (BNM) [3][4][5], the local boundary integral equation (LBIE) method [6][7][8], and the boundary element-free method (BEFM) [9][10][11][12][13][14][15][16][17][18][19].…”

Element Free Gelerkin Method for 2-D Potential Problems

“…Consider the problem stated in Equation (18), a penalty factor is applied to penalize the difference between the potential of the MLS approximation and the prescribed potential on the essential boundary [2]. The constrained Galerkin weak form uses the penalty method and with substituting the expression of MLS approximation of Equation (13) can then be posed as…”

“…In the past few decades, a variety of new meshless methods have been developed, including the smoothed particle hydrodynamics (SPH) method [4], the finite point method (FPM) [5], the diffuse element method (DEM) [6], the element free Galerkin (EFG) method [7], the point interpolation method (PIM) [8], the hp clouds method [9], the partition of unity method (PUM) [10], the meshless local Petrov-Galerkin (MLPG) method [11], the local point interpolation method (LPIM) [12], the discrete least squares meshless (DLSM) method [13], the boundary point interpolation method (BPIM) [14], and the meshless method with boundary integral equations [15]- [18].…”