The Dirichlet-to-Neumann map for two-dimensional crack problems

A coupled analytic/finite-element method is presented for two-dimensional crack problems. Two classes of problems are studied. The first considers problems where non-linear constitutive processes occur in a region near the crack tip and the remotely applied loading can be characterized by the linear elastic K-field and perhaps the T-stress. In this case, the finite-element method is applied in a circular region around the crack tip where non-linear constitutive response is occurring, and stiffness contributions associated with a numerically implemented Dirichlet-to-Neumann map are imposed on the circular boundary to account for the large surrounding elastic domain and the remote applied loading. The second class of problems considers entirely linear elastic domains with irregular external boundaries and/or complex applied loadings. Here, the discrete Dirichlet-to-Neumann map is used to represent a circular region surrounding the crack tip, and finite-elements are used for the external region. In this case the mixed mode stress intensity factors and the T-stress are retrieved from the map.