On the independence of strain invariants of two preferred direction nonlinear elasticity

It is often assumed in the literature that the nine classical strain invariants, which are used to characterize the strain energy of a compressible anisotropic elastic solid with two preferred non-orthogonal directions are independent. In this paper, it is shown that only six of the classical strain invariants are independent, and syzygies exist between the classical invariants. Alternatively, using principal axis techniques, it is simply proven that, only six of the classical strain invariants are independent and syzygies exist between the principal axis strain invariants. Consequently, all other sets of strain invariants, proposed in the literature, which are uniquely related to the set of principal axis strain invariants, have only six independent invariants. Due to syzygies, it is shown that the number of ground state constants required to fully describe the quadratic linear strain energy function of two-fibre solids is fourteen, not thirteen, as assumed in the literature.