On the second gradient nonlinear spectral constitutive modelling of viscoelastic composites reinforced with stiff fibers.

General novel nonlinear second-gradient spectral constitutive models for rate dependent fiber-reinforced viscoelastic solids that consider bending stiffness are developed. The constitutive models are characterized by spectral invariants, each with a clearer physical meaning compared to the classical invariants. Hence, they are experimentally useful if a rigorous experimental curve fitting exercise is used to obtain a specific form of free energy function. The number of complete-irreducible-minimal spectral invariants is significantly less than the number of 'classical' complete-irreducible invariants given in the literature, and hence modelling complexity is drastically reduced when a spectral technique is used. Our spectral approach in this paper is different from the classical invariant approach that have been done in the last decades regarding nonlinear solid mechanics. A detailed proof to show that the spherical part of the couple stress is just a Lagrange multiplier, is given. Results for pure bending and, the extension and inflation of a solid cylinder, that could be useful for experiments and numerical validations, are given.