Extreme Lamé Compliance in Anisotropic Crystals

Abstract

For a crystalline material, Poisson's ratio depends upon two orthogonal directions, one corresponding to the applied uniaxial stress and another for the resulting transverse strain. Interest in directions that yield a negative value leads to a consideration of extreme values and associated directions. Another indicator of a negative Poisson's ratio is the Lamé compliance, defined as the transverse strain response to a unit uniaxial stress. Where this quantity is positive, Poisson's ratio is negative and it is natural to associate a near minimum value of the ratio with a maximum of the Lamé compliance. Moreover, the stationary directions associated with the compliance bear a clearer relationship to the crystallographic directions than those of the ratio. Indeed, many of these directions are not dependent upon the elastic constants within a given crystal symmetry class. In the case of 1-cristobalite, the maximum value of the Lamé compliance is associated with such invariant stationary points. In the present work, we describe the invariant stationary directions and touch on a few of the simplest material-dependent stationary points.