On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials

In this paper, we show that the well known decomposition of a matrix into an orthogonal matrix (Q) and a upper triangular matrix (R), can be used to develop a remarkably simple representation for constitutive equations for anisotropic Green elastic materials. Using this decomposition (in which the upper triangular matrix is related to the Cholesky factorization of the Cauchy Green stretch tensor C), in lieu of the traditional polar decomposition, provides significant advantages: (1) the decomposition is very simple, fast and widely implemented in every major linear algebra package and does not require knowledge of eigenvalues. (2) The factorization allows us to decompose the total deformation into a sequence of simple, physically visualizable, meaningful elementary deformations. (3) It provides a remarkably simple representation of the stress response in terms of the derivatives of the strain energy (see Eqs. (26), (27) and (28)) especially for anisotropic materials. (4) It provides a unified and simple way of considering triclinic, monoclinic, orthorhombic, and transversely isotropic materials and satisfies the “orthogonality” criterion described by Criscione (2005) in order to minimize experimental error propagation. The results, which were inspired by and are closely related to the developments by Criscione et al., 2001 and Criscione and Hunter, 2003 and follows many of their ideas, are in a much simpler and computationally tractable form than that given by Criscione and coworkers while retaining every desirable feature of the decomposition presented by them.