On a Crack in a Material with a Nonlinear Stress-Strain Law

A boundary-value problem in longitudinal shear in which the stress-strain law is physically nonlinear, and given by τ =τ ₀γ /(1 + γ ²)^1/2, is considered. The method of solution involves a hodograph transformation, and a further transformation to reduce the governing equations to Laplace's equation for the displacement w in terms of appropriate strain variables. It is shown that near the tip of a crack in this type of material it is not possible to obtain the behaviour of w by a local analysis: the asymptotic behaviour may only be found from a full analysis of the problem. Having found the local crack tip behaviour for our problem, the answer is checked using a path independent integral.