Reconsidering the Boundary Conditions for a Dynamic, Transient Mode I Crack Problem

A careful examination of a dynamic mode I crack problem leads to the conclusion that the commonly used boundary conditions do not always hold in the case of an applied crack face loading, so that a modification is required to satisfy the equations. In particular, a transient compressive stress wave travels along the crack faces, moving outward from the loading region on the crack face. This does not occur in the quasi-static or steady state problems, and is a special feature of the transient dynamic problem that is important during the time interval immediately following the application of crack face loading. We demonstrate why the usual boundary conditions lead to a prediction of crack face interpenetration, and then examine how to modify the boundary condition for a semi-infinite crack with a cohesive zone. Numerical simulations illustrate the resulting approach. 1. Introductory remarks The subject of the present contribution is unsteady, dynamic crack propagation in brittle polymers. The subject has received considerable attention in the literature, mostly focused upon experimental and numerical studies with comparatively few results obtained via analytical methods. Analytical solutions to select canonical fracture boundary value problems have an important role to play in gaining a deep understanding of the physical processes involved in dynamic fracture of polymeric materials and their numerical simulation. However, constructing analytical solutions to such boundary values, even subject to various simplifying idealizations, presents many technical obstacles. There is a growing literature devoted to constructing analytical solutions to dynamic fracture boundary value problems in the context of either linear elasticity or viscoelasticity. The treatises by Broberg [1] and Freund [4] give extensive accounts of these developments prior to 2000. Analytical solutions for dynamic steady (constant crack speed) crack growth in linear viscoelastic material have been constructed for both mode III (anti-plane shear) [5, 13] and mode I (planar opening) [15, 6] fracture conditions. For dynamic, unsteady crack growth, the catalog of analytical solutions in the literature is much smaller, and almost entirely confined to mode III cracks in elastic material, two exceptions being a paper by Sarakin and Slepyan [11], which points the way to solving dynamically accelerating mode I cracks in elastic material but does not explicitly exhibit a full solution for general loading, and [9, 10] which consider a mode III accelerating crack in a linear viscoelastic material.