Some identities and the structure of 𝑁ᵢ in the Stroh formalism of anisotropic elasticity

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Abstract
The Stroh formalism of anisotropic elasticity leads to a 6 × 6 6 \times 6 real matrix N that can be composed from three 3 × 3 3 \times 3 real matrices N i ( i = 1 , 2 , 3 ) {N_i} \left ( {i = 1, 2, 3} \right ) . The eigenvalues and eigenvectors of N are all complex. New identities are derived that express certain combinations of the eigenvalues and eigenvectors in terms of the real matrices N i {N_i} and the three real matrices H, S, L introduced by Barnett and Lothe. It is shown that the elements of N 1 {N_1} and N 3 {N_3} have simple expressions in terms of the reduced elastic compliances. We prove that − N 3 - {N_3} is positive semidefinite and, with this property, we present a direct proof that L is positive definite.