Stable crack growth along a brittle/ductile interface—I. Near-tip fields

Abstract

In Part I of this work the asymptotic near-tip stress and velocity fields of a crack propagating steadily and quasi-statically along the interface between a ductile and a brittle material are presented. The ductile material is characterized by J2-flow theory with either linear hardening or ideal plasticity. The brittle material is characterized by linear elastic behavior. The cases of antiplane strain and plane strain are considered.The linear-hardening solutions are assumed to be of variable-separable form with a power singularity in the radial distance to the crack tip. Results are given for the strength of the singularity and for the distribution of the stress and velocity fields as functions of the hardening parameter. However, the amplitude of the fields, or plastic stress intensity factor, is left undetermined by this asymptotic analysis. For the case of plane strain, it is found that two distinct solutions exist with slightly different singularity strengths, and very different mixilies on the interfacial line ahead of the crack, For hardening small enough, one of the solutions corresponds to a tensile-like mode, whereas the other solution corresponds to a shear-like mode. These two solutions coalesce at an intermediate value of the hardening, if a certain bimaterial parameter is not zero. In this case, no variable-separable solutions are found for larger values of the hardening parameter. On the other hand, if the bimaterial constant vanishes, the two solutions remain distinct for all values of the hardening parameter up to the perfectly-elastic limit.The ideally plastic solutions are obtained by means of an appropriate assembly of elastic unloading and active plastic sectors, the latter being of either centered-fan or constant-stress type. For simplicity, the substrate material is assumed to be rigid, and the ductile material to be incompressible. The perfectly-plastic results for the stress and velocity fields in this case are continuous and consistent with the small-hardening results showing a tensile- as well as a shear-like solution.In Part II of this work, the corresponding small scale yielding problem will be solved numerically, and the relevance of the asymptotic solutions will be investigated. Where appropriate, the plastic stress intensity factors corresponding to the asymptotic solutions will be determined as functions of the elastic stress intensity factor and the mixity of the applied fields. This information will be useful in determining “resistance curves” for crack growth along brittle/ductile interfaces.