The extension of Beltrami's ellipsoid to anisotropic bodies

Abstract

The spectral decomposition of the compliance, stiffness, and failure tensors for transversely isotropic materials was studied and their characteristic values were calculated using the components of these fourth-rank tensors in a Cartesian frame defining the principal material directions. The spectrally decomposed compliance and stiffness or failure tensors for a transversely isotropic body (fiber-reinforced composite), and the eigenvalues derived from them define in a simple and efficient way the respective elastic eigenstates of the loading of the material. It has been shown that, for the general orthotropic or transversely isotropic body, these eigenstates consist of two double components, σ1 and σ2 which are shears (σ2 being a simple shear and σ1, a superposition of simple and pure shears), and that they are associated with distortional components of energy. The remaining two eigenstates, with stress components σ3, and σ4, are the orthogonal supplements to the shear subspace of σ1 and σ2 and consist of an equilateral stress in the plane of isotropy, on which is superimposed a prescribed tension or compression along the symmetry axis of the material. The relationship between these superimposed loading modes is governed by another eigenquantity, the eigenangle ω.