On extreme thinning at the notch tip of a neo-Hookean sheet

The deformation and stress fields around the apex of a notch (in an infinite, thin and incompressible sheet of neo-Hookean material) are investigated. General far-field loading and conditions ensuring vanishing tractions at the notch faces are considered. For certain notch angles, the sheet can exhibit a singular asymptotic mechanical behaviour, characterized by unbounded in-plane stresses, accompanied by transversal extreme thinning. The problem, which is formulated within the nonlinear elastostatic plane stress theory, is governed by a quasilinear system of two partial differential equations of the second order with respect to the two in-plane components of the unknown deformation. An asymptotic analysis is performed to extract a solution from such a system. As the notch angle varies, emphasis is placed on the degree of singularity of the Cauchy stress fields as well as on the behaviour of the transverse stretch at the apex.