Ellipsoidal anisotropies in linear elasticity Extension of Saint Venant's work to phenomenological modelling of materials

TL;DR

This paper explores ellipsoidal anisotropies in linear elasticity, extending Saint Venant's work to understand when ellipsoidal indicator surfaces fully determine elastic constants and their implications for material modeling.

Contribution

It systematically analyzes classes of ellipsoidal anisotropy and conditions under which indicator surfaces determine elastic parameters in phenomenological models.

Findings

Ellipsoidal anisotropies include several basic classes.

Conditions for determining elastic constants from indicator surfaces are identified.

Ellipsoids of certain indicator surfaces must share principal axes, implying orthotropy.

Abstract

Several families of elastic anisotropies were introduced by Saint Venant (1863) for which the polar diagram of elastic parameters in different directions of the material (indicator surface) are ellipsoidal. These families recover a large variety of models introduced in recent years for damaged materials or as effective modulus of heterogeneous materials. On the other hand, ellipsoidal anisotropy has been used as a guideline in phenomenological modeling of materials. A question that then naturally arises is to know in which conditions the assumption that some indicator surfaces are ellipsoidal allows one to entirely determine the elastic constants. This question has not been rigorously studied in the literature. In this paper, first several basic classes of ellipsoidal anisotropy are presented. Then the problem of determination of the elastic parameters from indicator surfaces is discussed in several basic cases that can occur in phenomenological modelling. Finally the compatibility between the assumption of ellipsoidal form for different indicator surfaces is discussed and in particular, it is shown that if the indicator surfaces of the fourth root of E(n) and of 1/c(n) where E(n) and c(n) are respectively the Young's modulus and the elastic coefficient in the direction n) are ellipsoidal, then the two ellipsoids have necessarily the same principal axes and the material in this case is orthotropic.