The number of independent invariants of an n-preferred direction anisotropic solid

Sets of classical invariants are used to characterize the mechanical behaviour of elastic solids with preferred directions. In this paper, we prove, for an n-preferred direction anisotropy, that only [Formula: see text] classical invariants are independent. For [Formula: see text], we show that only six strain and [Formula: see text] non-strain invariants are independent.

DIRECTIONAL CONSTITUTIVE LAWS FOR RUBBERS

ABSTRACT Directional laws, also called micro-sphere laws, are based on the rubber elasticity theory and are designed to fit rubber mechanical stress–strain responses at large strain. Because they depend on material directions, directional changes may be introduced accounting for anisotropic damage or residual stretch such as resulting from Mullins softening or accounting for anisotropic strain hardening such as induced by crystallization. Directional laws provide a relevant alternative to strain invariants laws when the material isotropy evolves or when its anisotropy is difficult to guess a priori. In the current contribution, the building process involved when defining directional laws is presented. The major assumptions resulting from this process are reviewed. Finally, recent directional laws from the literature are discussed, highlighting the interest and potential of such a constitutive framework.