Symmetry analysis of extreme areal Poisson's ratio in anisotropic crystals

Poisson's ratio is defined as the negative of the ratio of the transverse strain to the longitudinal strain in response to a longitudinal uniaxial stress. In the presence of anisotropy, this means that the ratio depends on two directions. With a view to assessing crystals that exhibit directions for which the ratio is negative, we resort to a transverse average to eliminate one directional variable and at the same time to arrive at a measure that poses a challenge to achieving significant negative values. The areal Poisson ratio coincides with the Poisson ratio for an isotropic material. We determine the stationary directions of the areal Poisson ratio for all crystal symmetry classes. The directions represented by invariant stationary points-those that hold independently of the material-we identify and explain class-by-class in terms of the axes of symmetry for the class. It is shown that for cubic crystals, positive definiteness of the strain energy requires that the areal Poisson ratio lie between -1 and 1/2, as it does for isotropy. We conclude that the areal Poisson ratio for the classes of lower symmetry are not restricted.