The existence of riemann invariants in the one-dimensional equations of the non-linear theory of elasticity

The conditions for the existence of Riemann invariants of a one-dimensional system of equations of the non-linear theory of elasticity are investigated. Haantjes' diagonalization criterion is used to determine the form of the elastic potential for which the system has six Riemann invariants or three Riemann invariants (for waves which propagate in one direction). In particular, it is shown that the Haantjes criterion is satisfied and there are three Riemann invariants in the case of the elastic potential for slightly-non-linear weakly-anisotropic elastic media [1–3]. A procedure for computing Riemann invariants is described. The Riemann invariants are computed approximately for a form of elastic potential which satisfies the Haantjes criterion.