Energy bounds on the effective moduli of elasticity of anisotropic microheterogeneous media

Most microheterogeneous materials (composites, polycrystals, or rocks) are macroanisotropic elastic bodies. Their properties are anisotropic because of the way they are prepared or treated, and, in the case of rocks, because of the conditions under which they were formed and subsequently evolved. As Hill showed, in the case of a macroisotropic body the Voight approximation yields the upper bound of effective elastic moduli, and the Reuss approximation gives their lower band. Because of an erroneous analogy to these results. There now exists a fairly widespread belief that the elastic properties of macroscopic bodies are bounded on the high end by the Voight approximation and on the lower end by the Reuss approximation. The purpose of this report is to show the baselessness of this generalization of Hill's results and also to derive the inequalities that bound the permissible values of effective elastic moduli of macroanisotropic bodies with different systems of symmetry. The evidence presented here suggests that bounding components of the effective tensor of elasticity of a branch exist only in exceptional cases. The conditions that define the regions of permissible values for most components of an effective tensor take the form of systems of nonlinear inequalities which are difficult to analyze. This shows the error in the assertion that the components in the effective elasticity tensor for an anisotropic solid are bound from is above, by the Voight approximation and from below by the Reuss approximation.