Shape Sensitivity Analysis for 2D Mixed Mode Fractures Using Extended FEM (XFEM) and Level Set Method (LSM)#

Abstract

This article presents a shape sensitivity analysis method for calculating gradients of crack growth rate and crack growth direction for 2D structural components under mixed-mode loading using extended finite element method (XFEM) and level set method (LSM). XFEM is a computational technique in which special enrichment functions are used to incorporate the discontinuity of structural responses caused by the crack surfaces and crack tip fields into finite element approximation. The LSM employs level set functions to track the crack during the crack propagation analysis. As a result, this method does not require highly refined mesh around the crack tip nor remesh to conform to the geometric shape of the crack when it propagates, which makes the method extremely attractive for crack propagation analysis. However, shape sensitivity analysis for crack propagation involves calculating derivatives of enrichment functions employed in XFEM that are discontinuous or unsmooth. The proposed sensitivity analysis method in this article overcomes the issues and calculates accurate derivatives of both crack growth rate and direction. The proposed method employs (1) semi-analytical method for the derivatives of stresses and displacements, and (2) material derivatives for the stress intensity factors obtained from domain form of the interaction integral, and therefore, the crack growth rate and direction. A rectangular plate under shear loading and an engine connecting rod example are used to demonstrate the feasibility and accuracy of the proposed method.