Limits to Poisson's ratio in isotropic materials

TL;DR

This paper investigates the fundamental limits of Poisson's ratio in isotropic materials, revealing that classical elasticity theory only applies within a specific range and explaining why most materials have v > 0.2.

Contribution

The study derives quadratic relations from classical elasticity to identify the valid range of Poisson's ratio, showing that real materials typically have v between 1/5 and 1/2, and classical elasticity is invalid outside this range.

Findings

Poisson's ratio is constrained to 1/5 <= v < 1/2 for real isotropic materials.

Materials with v outside this range are rare and often have complex behaviors.

Classical elasticity cannot accurately describe materials with v < 1/5.

Abstract

A long-standing question is why Poisson's ratio v nearly always exceeds 0.2 for isotropic materials, whereas classical elasticity predicts v to be between -1 to 1/2. We show that the roots of quadratic relations from classical elasticity divide v into three possible ranges: -1 < v <= 0, 0 <= v <= 1/5, and 1/5 <= v < 1/2. Since elastic properties are unique there can be only one valid set of roots, which must be 1/5 <= v < 1/2 for consistency with the behavior of real materials. Materials with Poisson's ratio outside of this range are rare, and tend to be either very hard (e.g., diamond, beryllium) or porous (e.g., auxetic foams); such substances have more complex behavior than can be described by classical elasticity. Thus, classical elasticity is inapplicable whenever v < 1/5, and the use of the equations from classical elasticity for such materials is inappropriate.