Lattice theory of three-dimensional cracks

Abstract

The problem of the stability of a three‐dimensional crack will be analyzed within a lattice‐statics approximation. Hsieh and Thomson employed a similar approach for a two‐dimensional crack. In that work the force was taken to be linear up to an arbitrary displacement and set equal to zero for larger displacements. This paper will investigate the consequence of introducing a jog into the crack face as well as the effects of various nonlinear‐force laws. The phenomenon of lattice trapping (upper and lower bounds on the applied stress for an equilibrium crack of given length) is again obtained. By looking at various force laws, it is possible to obtain some physical insight into which aspects of the force law are critical for crack stability. In particular, the inadequacy of a thermodynamic approach (which relates the critical stress to a surface energy corresponding to the area under the cohesive‐force–vs–displacement curve) will be demonstrated. Surface energy is a global property of the cohesive‐force law. Crack stability is sensitive to much more refined aspects of the cohesive‐force law. Crack healing is sensitive to the long‐range portion of the cohesive force. It occurs when an applied load leads to a pair of atoms, whose bond had been broken, acquiring a net displacement such that the cohesive force is non‐negligible (about 5% of peak value) and increasing with decreasing displacment. Crack expansion normally occurs when an increment in a net atomic displacement leads to a change in slope in the cohesive‐force law: a positive spring constant becoming negative. Hence crack expansion is sensitive to the position of the maximum in the cohesive‐force relation.