Isogeometric shape design sensitivity analysis of stress intensity factors for curved crack problems

Abstract

An isogeometric shape design sensitivity analysis method is developed for the stress intensity factors (SIFs) in curved crack problems. In order to obtain an enhanced shape sensitivity of SIFs, exact higher-order geometric information of curvature and normal vector is embedded in the isogeometric shape sensitivity expressions. A direct differentiation method is employed for the design sensitivity analysis and the size and orientation of the crack are selected as design variables. In the isogeometric approach, the NURBS basis functions in CAD system are directly utilized in the response analysis, which enables a seamless incorporation of exact geometry and higher continuity into the computational framework. The precise evaluation of the crack-face integral as well as the interaction integral is essential for the computation of the stress intensity factors for the curved crack problems in a mixed-mode. The CAD-based exact representation of tangential and normal vectors enables us to exactly define a local coordinate system at the crack-tip, whose shape dependency naturally leads to configuration design variations that include the change of crack orientation. Compared with the conventional finite element approach, a higher continuity of stress and strain fields are expected in the interaction integral domain. The design dependencies of auxiliary field and q-function are considered in the design sensitivity formulation, reflecting the convection effects due to the non-constant design velocity in the interaction integral domain. Various numerical examples of curved crack problems are presented to verify the developed isogeometric sensitivity analysis method through the comparison with exact and finite element solutions.