Integral representations for deformation (velocity) gradients in elastic or elastic-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 velocity gradients, 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 the 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 present results are also useful in directly establishing traction boundary-integral equations in linear and non-linear solid mechanics.
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