A crack with surface elasticity in finite plane elastostatics

Abstract

We consider the effect of surface elasticity on a finite crack in a particular class of compressible hyperelastic materials of harmonic type subjected to uniform remote Piola stresses. The surface mechanics is incorporated into the model of finite deformation by employing a version of the continuum-based surface/interface theory of Gurtin and Murdoch. A complete solution valid throughout the entire domain of interest is obtained by reducing the problem to two series of coupled Cauchy singular integrodifferential equations that are solved numerically using a collocation method. Our model predicts that, in general, the size-dependent Piola stresses exhibit a weak logarithmic singularity at the crack tips. For a crack in a special class of materials subjected to mode II loading, the stresses are bounded whereas the deformation gradients exhibit a logarithmic-type singularity at the crack tips.