Solution to line loading of a semi-infinite solid in gradient elasticity

Abstract

A semi-infinite elastic solid subjected to line loading is considered with higher order boundary conditions is utilized to obtain a general solution in terms of Fourier integral transforms for a symmetrical line loading with prescribed normal tractions. The boundary conditions are obtained by variational method and shown that they differ from the boundary conditions reported in the literature for the simple theory. Closed form solutions are then obtained for a concentrated normal force (the Flamant problem), for constant normal traction, and a typical Hertzian normal traction distribution from classical elasticity. It is verified that undesirable displacement singularity predicted by classical elasticity in the Flamant problem is eliminated by the gradient elasticity solution. The solution for constant normal tractions also illustrate the capability of gradient elasticity to predict size effect by taking into account the effect of micro-structure, which classical elasticity does not adequately describe.