A Shape Design Sensitivity Approach for Two-Dimensional Mixed-Mode Fracture Analysis Under General Loading

Under general loadings including body forces and crack-face traction, the energy release rate equation for a two-dimensional cracked body is derived by a shape design sensitivity approach. Defining the virtual crack extension (VCE) as the variation of the geometry, the virtual work principle and the material derivative concept are used to obtain the final analytical equation for the energy release rate. In contrast to the results of other researchers, the functionals which appear in the derived energy release rate equation do not involve the derivative of the displacement field on the crack surface, thereby improving the numerical accuracy in the computation of the energy release rate. Although the finite element method (FEM) is applied to crack problems in this paper, any numerical analysis method can be applied to the resulting equation. In addition, if body forces and crack-face traction are constant with respect to VCE, i.e., their material derivatives are identically zero, then the energy release rate equation is domain independent for domains which exclude the crack-tip region. Three example problems are treated which demonstrate the generality, accuracy, and domain-independent nature of the derived energy release rate equation.