Minimal Rotationally Invariant Bases for Hyperelasticity

Rotationally invariant polynomial bases of the hyperelastic strain energy function are rederived using methods of group theory, invariant theory, and computational algebra. A set of minimal basis functions is given for each of the 11 Laue groups, with a complete set of rewriting syzygies. The ideal generated from this minimal basis agrees with the classic work of Smith and Rivlin [Trans. Amer. Math. Soc., 88 (1958), pp. 175–193]. However, the structure of the invariant algebra described here calls for fewer terms, beginning with the fourth degree in strain, for most groups.