Some remarks on the range of Poisson's ratio in isotropic linear elasticity

Abstract

We study the range of Poisson's ratio ν in the theory of isotropic linear elasticity. Using the ratio of the Lamé constants χ = λ/μ, we defined the parametrized Poisson's ratio as ν = ν(χ) and found that it represents a standard hyperbola with center (χ, ν) = (−1/3, 1/2) and eccentricity . One of the focal points of the hyperbola represents a Born instability criterion (or the equivalent strong ellipticity condition) and the other determines the sign of ν. The hyperbola has a vertex point in the admissible range of ν, and, hence, it naturally divides the range into the two subranges: and . We also found that there are three equivalent formulas among the nine elastic-constant ratios χ i considered in this study. The geometric conjugate of Poisson's ratio, which is defined as ν* ≔ 1 − ν, has also been proposed.