Ellipsoidal Anisotropies in Linear Elasticity: Extension of Saint Venant's Work to Phenomenological Modeling of Materials

Several families of elastic anisotropies have been introduced by Saint Venant (Saint Venent, B. (de) (1863). Sur la distribution des élasticitiés autour de chaque point d'un solide ou d'un milieu de contexture quelconque, particulièrement lorsqu'il est amorphe sams être isotrope, Journal de Math. Pures et Appliquées, Tome VIII (2 ^éme série) pp. 257–430) for which the polar diagram of elastic parameters in different directions of the material (indicator surface) is ellipsoidal. These families cover a large variety of models introduced in recent years for damaged materials or as effective moduli of heterogeneous materials. Ellipsoidal anisotropy has also been used as a guideline in phenomenological modeling of materials. Then a question that 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 study, first, several basic classes of ellipsoidal anisotropy are presented. Then the problem of the determination of elastic parameters from indicator surfaces is discussed in several basic cases that can occur in phenomenological modeling. Finally, the compatibility between the assumption of ellipsoidal form for different indicator surfaces is discussed. In particular, it is shown that if the indicator surfaces of ⁴ √ E(n) and of ^-4 √ 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.