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.