Directional Constitutive Laws for Rubbers

Industrial rubbers show large strain non-linear viscohyperelastic behaviour with possible softening and residual stretch. Discarding their viscoelasticity and focusing on modelling the material hyperelasticity with possible damage, one may write constitutive equations based on strain invariants or on directional laws inspired by the representation of the material by macromolecular chains. Macromolecular network constitutive equations are determined by the choices of a network theory, affine or phantom, a chains statistics, Gaussian or Langevin, the assumption of a perfect network and the thermodynamics principles stating that a change of strain energy is due to a change of entropy. Due to the assumptions made, the parameters of network laws are often mechanical parameters difficult to relate to the physics of the materials. Stemming from the physics of rubber elasticity, directional laws have evolved into versatile mechanical constitutive equations that may account for initial or induced anisotropy or/and residual stretch. At the end, they offer an interesting alternative to strain invariant behaviour laws when non-isotropic behaviour is considered like in the case of filled rubbers undergoing Mullins softening.