The finite element over-deterministic method to calculate the coefficients of crack tip asymptotic fields in anisotropic planes

This article applies the finite element over-deterministic (FEOD) method to determine the coefficients of the crack tip asymptotic fields in an anisotropic plane. This method relies on finding the best fit of the crack tip displacement equations to the displacements taken from a large number of nodes around the crack tip. We first revisit the asymptotic solution of crack tip fields in an anisotropic plane, followed by giving the exact solution of the crack tip parameters for an infinite anisotropic plane with a central crack. Using the derived displacement field, we then formulate the FEOD method to determine the crack tip parameters from a set of finite element nodal displacements. Numerical examples of cracks in finite and infinite plates are then investigated, where the FEOD results manifest a good agreement with exact or alternative numerical solutions. A convergence study is performed to examine the robustness of the methodology and the reliability of the results. By implementing a least squares error function, we also calculate the corresponding error in approximating the displacement field. The low values of errors demonstrate the high precision of the FEOD method.