Finite strain coordinates and the stability of solid phases

Abstract

It is shown that finite strain coordinates may be obtained from the unique factorization of a non-singular matrix into the product of a triangular matrix, with positive diagonal elements, and an orthogonal matrix. These finite strain coordinates are the six elements of the triangular matrix. They are the usual strain elements of infinitesimal strain theory, if a suitable geometrical restriction is applied. The use of this restriction and of these coordinates facilitates a discussion of the stability of a phase, which differs from previous treatments. It is shown that it is necessary to ensure that the mechanical equilibrium condition of vanishing total moments be maintained throughout any variation, by the use of geometrical constraints. The well known difficulty of the so-called trivial instability of a compressed strut does not occur in this treatment, and the thermodynamic stability of a phase can be tested against any variation of intrinsic elastic strain whatsoever. A fundamental error in a recent criticism by the author of the Coleman-Noll inequality is acknowledged, and this inequality is briefly discussed.