A norm-preserving mapping of hierarchically symmetric tensors of rank 4 in three dimensions onto 21-dimensional vectors is proposed. It permits the definition of a metric of such tensors, of which the elastic tensors are a subset that satisfies the condition of stability. This metric allows to define the optimum approximation of an arbitrary tensor by one satisfying given symmetry conditions. The deviation of the original tensor from the symmetric approximation is defined as the angle between the two representing vectors. This concept allows to speak of one and the same medium as “deviating from transverse isotropy by a%, from orthorhombic symmetry by b%”(where b ≤ a).
Vectors corresponding to higher symmetry belong to subspaces of the 21-space, and the optimum approximation with given symmetries is the projection of the original vector on the different subspaces. This simple picture is complicated by the equivalence of a tensor to all tensors that can be obtained by a rotation of the 3D reference system. This equivalence is resolved by selecting from the equivalence set the member with the largest projection (and the smallest deviation).
Normalization, i.e., the decomposition of the 21D vectors into magnitude and direction corresponds to a decomposition into overall stiffness and anisotropy. The normalized vectors have their end points on the surface of the unit sphere of the appropriate sub-space. The stability condition constrains the elastic vectors to a fraction 2-k of the surface of the unit sphere, where k is the number of distinct ‘eigenstiffnesses’ of the symmetry system (i.e., two for isotropy, three for cubic symmetry, four for hexagonal and trigonal symmetry, and six for orthorhombic and lower symmetries).
Together with a finite ‘threshold of distinctness’ the finite size of the region of stability leads to an estimate of the number of distinct anisotropic media that might exist. With realistic assumptions for this threshold, the numbers are modest for isotropic and cubic media, barely manageable for hexagonal (transversely isotropic) media, and of the order of 105 for orthorhombic media. With the same threshold, the total number of triclinic media is of the order 1012.
Figures & Tables
“This volume contains a set of papers based on presentations given at the Sixth International Workshop on Seismic Anisotropy.”