Extreme values of Poisson's ratio and other engineering moduli in anisotropic materials

TL;DR

This paper derives conditions for the extrema of Poisson's ratio and other elastic moduli in anisotropic materials, providing algorithms to identify global extreme values based on compliance tensor properties.

Contribution

It introduces a systematic method to find maximum and minimum values of Poisson's ratio, Young's modulus, and shear modulus in anisotropic elastic materials.

Findings

Derived stationary conditions for elastic moduli extrema.

Presented coordinate-independent search algorithms.

Provided numerical examples for monoclinic crystals.

Abstract

Conditions for a maximum or minimum of Poisson's ratio of anisotropic elastic materials are derived. For a uniaxial stress in the 1-direction and Poisson's ratio $\nu$ defined by the contraction in the 2-direction, the following three quantities vanish at a stationary value: $s_{14}$, $[2\nu s_{15} + s_{25}]$ and $[(2 \nu -1)s_{16} + s_{26}]$, where $s_{IJ}$ are the components of the compliance tensor. Analogous conditions for stationary values of Young's modulus and the shear modulus are obtained, along with second derivatives of the three engineering moduli at the stationary values. The stationary conditions and the hessian matrices are presented in forms that are independent of the coordinates, which lead to simple search algorithms for extreme values. In each case the global extremes can be found by a simple search over the stretch direction $\bf n$ only. Simplifications for stretch directions in a plane of orthotropic symmetry are also presented, along with numerical examples for the extreme values of the three engineering constants in crystals of monoclinic symmetry.