An anisotropic viscoelastic fibre–matrix model at finite strains: Continuum formulation and computational aspects

Abstract

This paper presents a fully three-dimensional constitutive model for anisotropic viscoelasticity suitable for the macroscopic description of fibre reinforced composites that experience finite strains. An essential feature of the model is that the matrix and the fibres are treated separately allowing then as many bundles of fibres as desired. Moreover, the relaxation and/or creep response is based on the multiplicative viscoelastic split of the deformation gradient combined with the assumption of viscoelastic potentials for each compound. Here the composite is thought to be the superposition of an isotropic matrix material and further one-dimensional continua, each of them representing one family of fibres. The deformation gradient and its multiplicative decomposition apply to all the continua linking them implicitly. The global anisotropic response is obtained by an assembly of all the contributions. Constitutive models for orthotropic and transversely isotropic materials are included as special cases. It is shown how the continuum thermodynamics is crucial in setting the correct forms for the constitutive and evolution equations. For the algorithmic design within the context of the finite element method, the numerical effort is of the order of that devoted for isotropic computations. In fact, only a single scalar-valued resolution procedure is added for each fibre bundle. The algorithmic tangent moduli are derived for each compound and their assembly leads to consistent viscoelastic tangent modulus which is suitable for a quadratic rate of convergence when the Newton–Raphson iterative scheme is employed. The numerical efficiency of the model is illustrated through a set of representative simulations.