On the effective strain tensor in heterogeneous materials

Abstract

This paper considers effective strain tensors within the context of linear elastic equilibrium theory. The elastic properties of structured materials are often averaged over subvolumes of various scales inside the material. For subvolumes smaller than a representative volume element, simple volume-averaging of the stress and strain may not preserve the elastic energy. We introduce an averaging process which preserves the energy for all boundary conditions. This averaging process emphasizes the parts of the material which carry the most stress. Here the effective strain is weighted by the local stress, and can be interpreted as an average strain over all paths taken by loads and forces through the volume. This alternative effective strain may be especially appropriate for materials with voids, such as foams and granular matter, as the averaging only involves the material itself. For uniform boundary conditions the weighted strain matches the volume-averaged strain. This paper investigates the properties of this weighted strain tensor. First, for each path taken by loads and forces through the volume we can measure a net length as well as a net extension due to the linear deformation. The weighted effective strain equals the ratio of average length to average extension, where the averaging is over all possible force paths. Thus this method provides a connection to load path analysis. Secondly, even when the average rotation within the subvolume is zero, there may be local fluctuations in the rotation field. These rotations can act like a mechanism, transferring elastic energy between boundaries or degrees of freedom. The effective strain defined here highlights this mechanism effect.