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

摘要

Conditions for a maximum or minimum of Poisson's ratio of anisotropic elasticmaterials are derived. For a uniaxial stress in the 1-direction and Poisson'sratio $\nu$ defined by the contraction in the 2-direction, the following threequantities 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 thecompliance tensor. Analogous conditions for stationary values of Young'smodulus and the shear modulus are obtained, along with second derivatives ofthe three engineering moduli at the stationary values. The stationaryconditions and the hessian matrices are presented in forms that are independentof 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 thestretch direction $\bf n$ only. Simplifications for stretch directions in aplane of orthotropic symmetry are also presented, along with numerical examplesfor the extreme values of the three engineering constants in crystals ofmonoclinic symmetry.