10 - Matter tensors of rank 6

Summary

The only matter tensor having a rank as high as 6 that appears in Chapters 1 and 2 is the so-called ‘third-order elastic constant’ tensor of type T_s(6). (See Section 1–6 and Table 1–1.) This tensor couples a quantity Y, which is a thermodynamic tension, t_i, of type T_s(2) to a quantity X, which is the symmetric product of Lagrangian strains, η_iη_i, of type T_s(4). (Here, both t_i and η_j are second-rank symmetric tensors whose six components are written in single-index hypervector notation.) The resulting matter tensor K is then a T_s(6) tensor, symmetric in the interchange of all the indices. The first objective of this chapter will be to obtain the independent components of such a T_s(6) tensor. This will require us to go beyond the material contained in the S-C-T tables (Appendix E).

Relation betweenT_s(2) andT_s(4)

We wish to consider the usual relation: Y = KX, in which Y is a T_s(2) tensor that transforms as a six-vector, and X is a T_s(4) tensor which transforms as the 21 symmetric products of six-vectors, α_iα_j. (Here we use the notation α for a T_s(2) tensor as in the S-C-T tables and Eq. (4–4).) In terms of the single-index quantities Y and X, K then become a two-index (6 × 21) matrix.