We work within the context of small deformations elasticity. It can then be shown for a finite solid consisting of a general heterogeneous nonlinearly elastic material, that:
(1) Among all consistent boundary data which yield the same overall average strain, the strain field produced by uniform boundary tractions, renders the elastic strain energy an absolute minimum. (2) Among all consistent boundary data which yield the same overall average stress, the stress field produced by linear boundary displacements, renders the complementary strain energy an absolute minimum.
Similar results are valid when the material of the composite is viscoplastic with convex potentials, such that the stress, (the strain rate), is given by the gradient with respect to the strain rate, (the stress), of a convex potential, φ (a convex potential, Ψ). Based on these general results, computable bounds are developed for the overall stress and strain (strain-rate) potentials of solids of any shape and inhomogeneity, subjected to any set of consistent boundary data. Statistical homogeneity and isotropy are not required. It is shown that the bounds can be improved by incorporating additional material and geometric data specific to a given finite heterogeneous solid. These bounds are not based on an equivalent homogenized reference solid. They remain nonzero and finite even when cavities or rigid inclusions are present. Illustrative examples with explicit results are given.
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