Analysis of a mode-III crack in the presence of surface elasticity and a prescribed non-uniform surface traction

We consider an elastic solid incorporating a mode-III crack in which the crack faces incorporate the effects of surface elasticity and are further subjected to prescribed non-uniform surface tractions. The surface elasticity is modelled using the continuum-based model of Gurtin and Murdoch. Using complex variable techniques, the corresponding problem is reduced to the solution of a first order Cauchy singular integro-differential equation which, in turn, leads to the complete solution of the aforementioned crack problem valid everywhere in the domain of interest (including at the crack tip). Finally, we note that, as a particular case of our analysis, the classical decomposition of a mode-III crack problem in linear elasticity continues to hold even in the presence of surface elasticity.