Bounds on the effective anisotropic elastic constants

Hill [12] showed that it was possible to construct bounds on the effective isotropic elastic coefficients of a material with triclinic or greater symmetry. Hill noted that the triclinic symmetry coefficients appearing in the bounds could be specialized to those of a greater symmetry, yielding the effective isotropic elastic coefficients for a material with any elastic symmetry. It is shown here that it is possible to construct bounds on the effective elastic constants of a material with any anisotropic elastic symmetry in terms of triclinic symmetry elastic coefficients. Similarly, it is then possible to specialize the triclinic symmetry coefficients appearing in the bounds to those of a greater symmetry. Specific bounds are given for the effective elastic coefficients of cubic, hexagonal, tetragonal and trigonal symmetries in terms of the elastic coefficients of triclinic symmetry. These results are obtained by combining the approach of Hill [12] with a representation of the stress-strain relations due, in principle, to Kelvin [25,26] but recast in the structure of contemporary linear algebra.