Non-Hyper-Singular Integral-Representations for Velocity (Displacement) Gradients in Elastic/Plastic Solids Undergoing Small or Finite Deformations

Summary

New integral representations for deformation (velocity) gradients in elastic or elasto-plastic solids undergoing small or large deformations are presented. Compared to the cases wherein direct differentiation of the integral representations for displacements (or velocities) were carried out to obtain displacement (or velocity) gradients (which gave rise to hyper-singularity when the source point was taken to the boundary), the present integral representations have lower order singularities which are quite tractable from a numerical point of view. Moreover, the present representations, allow the source point to be taken in the limit, to the boundary, without any difficulties. This obviates a need for a two tier system of evaluation of deformation gradients in the interior of the domain on one hand and at the boundary of the domain on the other. It is expected that the present formulations would yield more accurate and stable deformation gradients in problems dominated by geometric and material nonlinearities.

The velocity gradients for four different cases of practical interest are considered here. They are (i) infinitestimal deformation of a linear elastic solid; (ii) small-strain elasto-plastic behavior; (iii) finite-strain elasto plastic behavior; and (iv) large deformation behavior of a semi-linear elastic solid. All singular integral representations are derived in the sense of Cauchy principal values. Hence residual or jump terms arise out of these singular integrals.