Universal Bounds for Overall Properties of Linear and Nonlinear Heterogeneous Solids

For a sample of a general heterogeneous nonlinearly elastic material, it is shown that, among all consistent boundary data which yield the same overall average strain (stress), the strain (stress) field produced by uniform boundary tractions (linear boundary displacements), renders the elastic strain (complementary strain) energy an absolute minimum. Similar results are obtained when the material of the composite is viscoplastic. Based on these results, universal bounds are presented for the overall elastic parameters of a general, possibly finite-sized, sample of heterogeneous materials with arbitrary microstructures, subjected to any consistent boundary data with a common prescribed average strain or stress. Statistical homogeneity and isotropy are neither required nor excluded. 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. The bounds can be improved by incorporating additional material and geometric data specific to the given finite heterogeneous solid. Any numerical (finite-element or boundary-element) or analytical solution method can be used to analyze any subregion under uniform boundary tractions or linear boundary displacements, and the results can be incorporated into the procedure outlined here, leading to exact bounds. These bounds are not based on the equivalent homogenized reference solid (discussed in Sections 3 and 4). They may remain finite even when cavities or rigid inclusions are present. Complementary to the above-mentioned results, for linear cases, eigenstrains and eigenstresses are used to homogenize the solid, and general exact bounds are developed. In the absence of statistical homogeneity, the only requirement is that the overall shape of the sample be either parallelepipedic (rectangular or oblique) or ellipsoidal, though the size and relative dimensions of the sample are arbitrary. Then, exact analytically computable, improvable bounds are developed for the overall moduli and compliances, without any further assumptions or approximations. Bounds for two elastic parameters are shown to be independent of the number of inhomogeneity phases, and their sizes, shapes, or distribution. These bounds are the same for both parallelepipedic and ellipsoidal overall sample geometries, as well as for the statistically homogeneous and isotropic distribution of inhomogeneities. These bounds are therefore universal. The same formalism is used to develop universal bounds for the overall non-mechanical (such as thermal, diffusional, or electrostatic) properties of heterogeneous materials.