An eigenvalue problem for tensors used in mechanics and the number of independent Saint-Venant strain compatibility conditions

Abstract

A number of questions concerning the eigenvalue problem for a tensor \(\mathop A\limits_ \approx\) ∈ ℝ₄(Ω) with special symmetries are considered; here Ω is a domain of a four-dimensional (three-dimensional) Riemannian space. It is proved that a nonsingular fourth-rank tensor has no more than six (three) independent components in the case of a four-dimensional (three-dimensional) Riemannian space. It is shown that the number of independent Saint-Venant strain compatibility conditions is less than six.