Many materials are anisotropic and inhomogeneous due to the varying composition of their constituents. The analysis of the elastic constant measurements for these materials is difficult because the problem of the identification of the type of elastic symmetry is complicated by the varying composition of the material and vice versa. These two factors are intertwined and each complicates the other. A solution to this problem in which these two aspects are separated and analyzed independently is presented here.

This solution is illustrated here by application to human cancellous bone, hardwoods and softwoods. The solid volume fraction (or apparent density) is the compositional variable for the elastic constants of these natural materials. The solid volume fraction-dependent anisotropic Hooke's law for cancellous bone and a density-dependent one for hardwoods and softwoods are established. The analysis leading to this form of Hooke’s law shows that human cancellous bone has orthotropic elastic symmetry at the 95% confidence level and it provides expressions for all elastic constants as functions of the bone solid volume fraction (or apparent density) and orientation only;​ no measures of trabecular structural architecture are involved. The squared correlation coefficients between model and data are in a range that is about one-third higher than those obtained with previous models based upon volume fraction only, and are approximately the same or higher than those obtained with models involving volume fraction plus the measures of architecture.

The identification of the best elastic symmetry representation for these natural materials is accomplished by measuring the closeness of two symmetries. Bounds on the effective elastic constants of a material with any anisotropic elastic symmetry in terms of lower symmetry elastic coefficients are constructed to measure the closeness. An interesting result obtained from the applications of this bounding method shows that cancellous bone, generally considered to have orthotropic symmetry, may be considered to have transversely isotropic symmetry with a small error. A similar conclusion applies for hardwoods and softwoods. It is also shown that human cancellous bone has lesser degrees of anisotropy than hardwoods and softwoods.