ABSTRACT
Here we look at representation of fourth-order tensor moduli E subject to symmetry conditions. Given a symmetry group G we are looking for the general form of E for which
E : A = M[E : (MTAM)]MT (B.1)
for each transformation M in G, and for any second-order tensor A. We only consider orthogonal M. Given E = Eijklei ⊗ ej ⊗ ek ⊗ el, M = Mijei ⊗ ej and A = Aijei ⊗ ej in a base ei of orthogonal unit vectors, we can write this as
EijklAkl = Mim[Emnop(MqoAqrMrp)]Mjn = MimMjnMkoMlpEmnopAkl, (B.2)
which requires that Eijkl = MimMjnMkoMlpEmnop. (B.3)
We assume that E is such that for any second-order tensors A and B we have
A : E : B = AT : E : B = A : E : BT = B : E : A. (B.4)
This indicates that given E = Eijklei ⊗ ej ⊗ ek ⊗ el written in any base ei, the components of E are assumed to have the properties
Eijkl = Ejikl = Eijlk = Eklij , (B.5)
therefore reducing the number of independent components of E to 21. The requirement that A : E : B = AT : E : B indicates that the first two bases of E be those associated with symmetric second-order tensors. There are six base tensors for a symmetric second-order tensor that we can write as
E1 =e1 ⊗ e1, (B.6) E2 =e2 ⊗ e2, (B.7) E3 =e3 ⊗ e3, (B.8) E4 =e2 ⊗ e3 + e3 ⊗ e2, (B.9) E5 =e1 ⊗ e3 + e3 ⊗ e1, (B.10) E6 =e1 ⊗ e2 + e2 ⊗ e1. (B.11)
The requirement that A : E : B = A : E : BT states that base vectors three and four also should be given by the base vectors Ei for a symmetric second-order tensor. These two conditions combined, therefore require that E be expressible by bases of the form Ei ⊗ Ej , which are a total of 36 bases. The final requirement that A : E : B = B : E : A states that E is symmetric in the interchange of the first two bases with bases three and four. This allows us to write the 21 base tensors for fourth-order tensor E as Aij for j ≥ i and define them as
Aij =
Ei ⊗Ej for j = i, Ei ⊗Ej + Ej ⊗Ei for j > i.
