  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.
