Poisson Ratio Beyond the Limits of the Elasticity Theory

Elastic media can be stable only if the Lame coefficients are positive. This suggests that the Poisson ratio (PR) σ of a regular isotropic elastic media can range from −1 to 0.5. The regular elastic media, described by conventional elasticity theory, means the media which is homogeneous and deforms affinely, that is without formation of hollows or overlapping. In contrast, there are examples of the so-called micropolar media or Cosserat continua, where each microscopic element itself has rotational degrees of freedom. For most of the materials in nature σ lies in the range from 0 to 0.5. For example, cork shows almost zero σ, and σ = 0.5 (constant volume media) is observed for rubber or for the plastic deformation of metals. The materials with a negative σ are quite rare. Usually, the negative σ is attributed to the anisotropy or to the microscopic rotations. The anisotropy is the reason for the negative σ in highly anisotropic crystals like arsenic, antimony and bismuth and also in many single crystals of cubic metals deformed in an oblique direction with respect to the cubic axis. The rotational degrees of freedom cause the negative σ of foams and near the α−β phase transition of quartz and cristobalite. A microscopic model with particles having rotational degrees of freedom has been offered by Ishibashi and Iwata. The model nicely explains the variation of PR from −1 to 0. In the present Short Note we present a microscopic model which demonstrates PR beyond the limits of conventional elasticity. We consider the two-dimensional microscopic model of a crystal shown in Fig. 1. This structure can be found in ref. 11. The model consists of the rigid bars joined to each other by the hinges (open circles in Fig. 1). The bars simulate the rigid clusters of atoms and they have rotational degrees of freedom. The three parameters, the bar lengths a, b and the angle γ (|γ| < π/2), define the geometry of the model. The unit cell has parameters X = 2b cos γ, Y = 2(a− b sin γ). (1)