On the polynomial invariants of the elasticity tensor

We consider the old problem of finding a basis of polynomial invariants of the fourth rank tensor C of elastic moduli of an anisotropic material. Decomposing C into its irreducible components we reduce this problem to finding joint invariants of a triplet (a, b, D), where a and b are traceless symmetric second rank tensors, and D is completely symmetric and traceless fourth rank tensor (D ∈ T₄^ss).We obtain by reinterpreting the results of classical invariant theory a polynomial basis of invariants for D which consists of 9 invariants of degrees 2 to 10 in components of D. Finally we use this result together with a well-known descriptin of joint invariants of a number of second-rank symmetric tensors to obtain joint invariants of the triplet (a, b, D) for a generic D.