On the Scale of the Nonlinear Effect in a Crack Problem

Abstract

When crack problems are analyzed on the basis of nonlinear theories, such as finite elasticity or deformation theory of plasticity, it is inevitable that nonlinear effects will predominate near a crack-tip, even if the loads are small. The most favorable circumstance in this regard occurs when the loads are so small that the zone of significant nonlinearity lies within the region of validity of the near-tip approximation to the global solution of the associated linearized crack problem. This situation - called small-scale yielding for crack problems in plasticity - permits simplifications in analysis which are often decisive; see, e.g., Knowles (1977) and Rice (1968). Insofar as we know, there are no analytical estimates available of the level of load below which nonlinear effects are guaranteed to be small-scale in the above sense. Indeed, even a precise version of the question seems to be lacking. In the present note we formulate and answer such a question for an especially cooperative crack problem; that corresponding to finite anti-plane shear of an infinite medium containing a crack of finite length for an elastic material of Neo-Hookean type. The associated boundary value problem is a linear one for Laplace's equation and thus can be solved globally. Nevertheless, there is a significant nonlinear effect of Kelvin type in the stress field. We give a condition under which this nonlinear response occurs on a small scale near the crack tips.