Remarks on explicit strong ellipticity conditions for anisotropic or pre-stressed incompressible solids

Abstract

We present a set of explicit conditions, involving the components of the elastic stiffness tensor, which are necessary and sufficient to ensure the strong ellipticity of an orthorhombic incompressible medium. The derivation is based on the procedure developed by Zee & Sternberg (Arch. Rat. Mech. Anal., 83, 53-90 (1983)) and, consequently, is also applicable to the case of the homogeneously pre-stressed incompressible isotropic solids. This allows us to reformulate the results by Zee & Sternberg in terms of components of the incremental stiffness tensor. In addition, the resulting conditions are specialized to higher symmetry classes and compared with strong ellipticity conditions for plane strain, commonly used in the literature.