Poisson's ratio and the incompressibility relation for various strain measures with the example of a silica-filled SBR rubber in uniaxial tension tests

The controversy in the definition of Poisson's ratio (PR) as a material constant is discussed in this study. PR of an isotropic material is usually defined as the ratio, taken with the opposite sign, between its lateral and longitudinal strains under the action of longitudinal stresses. However, if deformations of the material are large, the value of PR depends on the strain measure used. Five different measures of strain are considered, and a unified relation in terms of stretch ratios is obtained for calculating the PR. It is demonstrated that only for Hencky strains is the value of PR of an incompressible material constant and equal to 0.5 over its entire extension range. Other measures lead to stretch-dependent PRs. A generalized relation for the volume strain is found in terms of strain invariants. With the example of uniaxial tension of a silica-filled rubber, the Cauchy and Hencky strains are used to demonstrate different ways for checking the incompressibility of a material and the evaluation of PR. The level of incompressibility of the rubber and its practical importance are evaluated.