A semi-analytical method based on the mixed formulation for the elastic analysis of a crack in an anisotropic homogeneous medium

In this study, a semi-analytical method based on the Reisner's mixed formulation is presented for the elastic analysis of anisotropic homogeneous solids with an edge or interior crack. In this method, the displacement and the stress fields are represented as the sum of a known function and a finite series of functions with unknown coefficients. The functions are constructed in such a way that the displacement discontinuity across the crack faces and the exact singular behavior of the stress field at the crack tip are captured, moreover all essential and natural homogeneous and inhomogeneous boundary conditions are satisfied exactly. The equilibrium and the compatibility equations are also applied with the desired accuracy using the Reissner's variational principle. Solution of the variational equation leads to a set of linear algebraic equations in terms of the unknown coefficients. After computing of the unknown coefficients, the displacement and the stress fields are obtained and subsequently the stress intensity factors (SIFs) and crack opening displacement (COD) are calculated. The results show that computing of SIFs has high convergence rate and the results of the proposed approach are in good agreement with those of the analytical solutions reported in the literature.