Asymptotic solutions for mixed mode loading of cracks and wedges in power law hardening materials

The effects of arbitrary loading on the asymptotic stress and strain fields in non-linear fracture for a stationary crack tip are investigated. The ma terial behavior is modeled as pure power law hardening within the theory of J(2)-deformation plasticity. Of particular interest is the role of two-par ameter higher order solutions in "mixed mode" fracture, where the first two terms in the expansion of the displacement field are variable-separable wi th real, but unequal eigenvalues. A displacement-based finite element metho d coupled with singular value decomposition, which is presented herein, is used to solve the non-linear higher order eigenvalue problem. The versatili ty of this method to account for arbitrary loading and geometry is used to investigate asymptotic crack solutions by studying the limit as a wedge bec omes a crack. By using this approach it has been determined that in homogen eous plane strain and plane stress fracture, and also plane strain fracture of an interface crack between a hardening material and a rigid substrate, there are two distinct asymptotic solutions. These solutions are referred t o as mode I dominant and mode II dominant. Each of the three fracture cases has one higher order solution, where the dominant term in the series is id entical to the pure mode dominant term. In these solutions two asymptotic t erms are necessary to account for the mixed nature of the far-field loading . In both plane strain fracture problems this higher order solution is mode I dominant, while for plane stress it is mode II dominant. The mode I domi nant interface crack higher order solution was first presented by Sharma an d Aravas [Int J Solids Struct 30 (1993) 695]. The "mode II dominant" soluti on for homogeneous plane strain fracture, first presented by Shih [Fract An al, ASTM STP 560 (1974) 187], is a true mixed mode asymptotic solution gove rned by a single real eigenvalue, where the mixed nature of the loading ent ers into the leading term. Contrary to this mixed mode asymptotic solution, there is strong evidence that the mode II dominant plane strain interface crack solution is oscillatory, i.e., non-separable with a form similar to t hat of the compressible linear elastic case that results from a complex eig envalue. The mode I dominant plane stress asymptotic solution could not be determined by the current approach. (C) 2001 Elsevier Science Ltd. All righ ts reserved.