Diffusion of incremental loads in prestrained bars

Abstract

Axial diffusion of perturbed fields generated by concentrated traction rates applied to the surface of a prestrained bar are examined within the framework of finite strain theory. The analysis centres on a plane-strain compatibility equation for a stress rate potential. This formulation covers a broad class of elastic and elastoplastic solids at large strains. A family of elemental rate boundary-value problems for concentrated incremental loads are solved exactly in terms of Fourier integrals. This is done in the spirit of earlier studies by Filon and von Kármán for linear elastic materials. Numerical examples for the Blatz–Ko constitutive relation reveal that the rate of axial diffusion has a strong sensitivity to the level of prestrain. Of special interest here is the surprising redistribution of the perturbed transversely symmetric stress rates for axial strains near necking. This behaviour is accompanied by considerable reduction in the rate of axial diffusion. A similar pattern is displayed by antisymmetric disturbances near the stress free configuration. These findings are supported by an asymptotic expansion based on the residue integration method. At a distance from the applied loads, axial diffusion is dominated by the exponential decay of the first self-equilibrating eigenfunction of the associated end problem for a semi-infinite strip.