A different catch for Poisson

Abstract

Poisson’s ratio, similar to other material parameters of isotropic elasticity, is determined via experiments corresponding to small strains. Yet at small-strain linear elasticity, Poisson’s ratio has a dual nature; although commonly understood as a geometrical parameter, Poisson’s ratio is also a material parameter. From a geometrical perspective only, the concept of Poisson’s ratio has been extended to large deformations by Beatty and Stalnaker. Here, through a variational analysis, we firstly propose an alternative relationship between the Poisson ratio and stretches at finite deformations such that the nature of Poisson’s ratio as a material parameter is retained. In doing so, we introduce relationships between the Poisson ratio and stretches at large deformations different than those established by Beatty and Stal naker. We show that all the nonlinear definitions of Poisson’s ratio coincide at the reference configuration and thus, material and geometrical descriptions too coincide, at small-strains linear elasticity. Secondly, we employ this variational approach to bring in the notion of nonlinear Poisson’s ratio in peridynamics, for the first time. In particular, we focus on bond-based peridynamics. The nonlinear Poisson’s ratio of bond-based peridynamics coincides with 1/3 for two-dimensional and 1/4 for three-dimensional problems, at the reference configuration.