Poisson’s ratio of anisotropic systems

The Poisson’s ratio of anisotropic materials depends, in general, both on a “longitudinal” direction along which the stress is changed and on a “transverse” direction in which the transverse deformation is measured. For cubic media there exist “longitudinal” directions, parallel to the 4-fold and 3-fold axes, for which the Poisson’s ratio does not depend on the “transverse” direction. Depending on the tensor of elastic compliances (or elastic constants), crystals of cubic symmetry can exhibit negative Poisson’s ratio in both these directions (they are called strongly auxetics), in one of them (i.e. either along the 4-fold axis or along the 3-fold one; they are called partially auxetic) or in none of them. For crystals exhibiting 3-fold symmetry axis the Poisson’s ratio along this axis does not depend on the “transverse” direction. For other “longitudinal” directions the Poisson’s ratio depends, in general, on the “transverse” direction. The Poisson’s ratio averaged with respect to the “transverse” direction depends only on the “longitudinal” direction and can be conveniently presented graphically. As an example the f.c.c. hard sphere crystal is considered. It is shown that the average (with respect to “transverse” direction) Poisson’s ratio of the hard sphere crystal is positive for all “longitudinal” directions. One should add, however, that there exist directions for which the (not averaged) Poisson’s ratio of hard spheres is negative.