A multi-step crack closure technique for the energetically orthogonal partitioning of the energy release rate

The linear elastic solution for a bimaterial interface crack is characterised by an oscillating singularity of the near crack-tip stresses. Consequently, the mode mixity (i.e. the relative amount of fracture modes I and II) cannot be defined based on the ratio of the shear and normal stresses at the crack tip. Moreover, the linear solution predicts oscillating displacements with overlap of the crack faces. Despite such inconsistencies, the total energy release rate, G, associated with crack growth, can be correctly calculated based on the linear elastic solution. However, the contributions, GI and GII, related to the basic fracture modes do not converge to well-defined limits, if the expressions derived for homogeneous materials are used. In a finite element formulation, G can be evaluated by using the virtual crack closure technique (VCCT). Nevertheless, the standard VCCT is unable to evaluate GI and GII, which are dependent on the crack length increment, Δa, also equal to the element size in the crack-tip region. The standard VCCT may also be inappropriate to evaluate GI and GII for homogeneous bodies with asymmetric cracks, for which physically unacceptable negative values may be obtained. This shortcoming is due to the lack of energetic orthogonality between the force components used to calculate GI and GII. As a remedy, a physically consistent VCCT has been proposed, in which the crack-tip nodal force is decomposed into the sum of two energetically orthogonal components. In this work, the energetically orthogonal partitioning of the energy release rate is extended to bimaterial interface cracks. To this aim, a multi-step crack closure technique is introduced, in which the convergence of the finite element solution is kept separate from the numerical evaluation of the limits for Δa → 0. The extended crack faces are subdivided into a number, n, of elements with size lx = Δa/n. The extended crack is progressively closed by introducing displacement constraints in the tangential and normal directions with respect to the crack plane. The works of closure done by the corresponding reaction forces are suitably attributed to the mode I and mode II contributions to the energy release rate.