Material frame representation of equivalent stress tensor for discrete solids

Abstract

In this paper, we derive expressions for equivalent Cauchy and Piola stress tensors that can be applied to discrete solids and are exact for the case of homogeneous deformation. The main principles used for this derivation are material frame formulation, long wave approximation and decomposition of particle motion into continuum and thermal parts. Equivalent Cauchy and Piola stress tensors for discrete solids are expressed in terms of averaged interparticle distances and forces. No assumptions about interparticle forces are used in the derivation, thereby ensuring our expressions are valid irrespective of the choice of interatomic potential used to model the discrete solid. The derived expressions are used for calculation of the local Cauchy stress in several test problems. The results are compared with prediction of the classical continuum definition (force per unit area) as well as existing discrete formulations (Hardy, Lucy, and Heinz-Paul-Binder stress tensors). It is shown that in the case of homogeneous deformations and finite temperatures the proposed expression leads to the same values of stresses as classical continuum definition. Hardy and Lucy stress tensors give the same result only if the stress is averaged over a sufficiently large volume. Thus, given the lack of sensitivity to averaging volume size, the derived expressions can be used as benchmarks for calculation of stresses in discrete solids.