Explicit Expression of the Stationary Values of Young's Modulus and the Shear Modulus for Anisotropic Elastic Materials

Abstract

AbstractExplicit expressions of the directionnand the stationary values (maximum, minimum and saddle point) of Young's modulusE(n) for orthotropic, tetragonal, trigonal, hexagonal and cubic materials are presented. For the shear modulusG(n, m), explicit expressions of the extrema (maximum and minimum) and the two mutually orthogonal unit vectorsn, mare given for cubic and hexagonal materials. We also present a general procedure for computing the extrema ofG(n, m) for more general anisotropic elastic materials. It is shown that Young's modulusE(n) can be made as large as we wish for certainnwithout assuming that the elastic compliance s11, s22or s33is very small. As to the shear modulusG(n,m), it can be made as large as we wish for certainnandmwithout assuming that any one of the elastic compliance sαβis very small. We also show that Young's modulusE(n) can be independent ofnfor orthotropic and hexagonal materials while the shear modulusG(n, m) can be independent ofnandmfor hexagonal materials.