The effects of arbitrary loading in nonlinear fracture mechanics

The effects of arbitrary loading on the asymptotic stress and strain fields in nonlinear fracture for a stationary crack tip are investigated. The material behavior is modeled as pure power law hardening within the theory of J₂-deformation plasticity. Of particular interest is the role of two-parameter higher order solutions in “mixed mode” fracture, where the first two terms in the expansion of the displacement field are variable-separable with real, but unequal eigenvalues. A displacement-based finite element method coupled with singular value decomposition is developed to solve the nonlinear higher order eigenvalue problem. The versatility of this method to account for arbitrary loading and geometry is used to investigate asymptotic crack solutions by studying the limit as a wedge becomes a crack. By using this approach it has been determined that in homogeneous 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 to as mode I dominant and mode II dominant. Each of the three fracture cases has at least one higher order solution, where the dominant term in the series is identical to the pure mode dominant term. In these solutions two asymptotic terms are necessary to account for the mixed nature of the far-field loading.; For plane strain homogeneous fracture the asymptotic mixed mode solution of Shih (1974), where the effects of arbitrary loading combine to enter the leading term of the asymptotic series, does not apply close to mode I. Instead, a mode I dominant higher order asymptotic form applies, where the leading term is the pure mode I symmetric HRR solution and the second term is antisymmetric. The significant difference between the two asymptotic solutions is the singular nature of the antisymmetric part of the stresses and strains. Therefore, near mode I the antisymmetric part of the local fields does not contribute to the J-integral.; For plane stress plane stress homogeneous fracture, it is shown that two higher order asymptotic solutions are necessary to represent arbitrary loading. The mode II dominant asymptotic solution consists of the pure mode II HRR term and a second symmetric term with an eigenvalue slightly weaker than the HRR eigenvalue. The mode I dominant asymptotic solution also consists of a symmetric and an antisymmetric term with different eigenvalues. The pure mode I HRR term is the symmetric term. Contrary to expected behavior based on energy considerations and experience with higher order solutions, the antisymmetric term has an eigenvalue that is stronger than the HRR eigenvalue. It is demonstrated that this term does not contribute to the J-integral, so there is no physical violation of unbounded strain energy.; For both plane stress and plane strain the asymptotic results are confirmed with full-field finite element analysis by using the J₂ deformation theory of plasticity. This validates the asymptotic solutions and shows that a two-parameter fracture theory can be used near mode I and near mode II. The full-field analysis also reveals that the transition from one asymptotic solution to the other gives rise to a complete loss of dominance of these two-term asymptotic solutions.