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 symm etry. Hill noted that the triclinic symmetry coefficients appearing in the bounds could be specialized to those of a greater symmetry, yielding the ef fective isotropic elastic coefficients for a material with any elastic symm etry. It is shown here that it is possible to construct bounds on the effec tive 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 e ffective elastic coefficients of cubic, hexagonal, tetragonal and trigonal symmetries in terms of the elastic coefficients of triclinic symmetry. Thes e results are obtained by combining the approach of Hill [12] with a repres entation of the stress-strain relations due, in principle, to Kelvin [25,26 ] but recast in the structure of contemporary linear algebra.