A meso–macro finite element modelling of laminate structures: Part II: time-dependent behaviour

A meso–macro modelling is proposed for laminates made of unidirectional layers of a polymer matrix reinforced with long fibres. The time-independent behaviour introduced in the first part of this article is improved herein to account for viscous phenomena through viscoelasticity and viscoplasticity. A spectrum-type viscoelastic model is considered, which is based on the definition of elementary viscoelastic mechanisms. Its mathematical formulation is simplified by using a relaxation times triangular layout. A generalised Norton-type model integrating the elastic domain concept is used to report the plastic strains delay. A zero-valued “dynamic” yield function is incorporated into the traditional viscoplastic format, allowing a same treatment of plastic and viscoplastic problems. The integration of the layer behaviour through the thickness is obtained within a Kirchhoff shell finite element. The constitutive equations are integrated using two families of algorithms that generalise the well-known trapezoidal and mid-point rules, for which accuracy and non-linear stability analysis are carried out. Significant robustness of the local iterative solution is provided by complementing the basic Newton’s scheme with a local line-search strategy. In the case of a fully coupled plastic–viscoplastic behaviour, the local Newton’s iterative scheme is associated with a grid-search method in order to define available initial solutions. A perturbation technique is suggested to evaluate an algorithmic tangent operator since the viscoelasticity renders non-trivial an explicit determination of a consistent tangent operator. The proposed formulation has been implemented in the finite element code CASTEM2000® in order to test its validity. The obtained results are compared with semi-analytical ones in the case of progressive repeated loading tests by applying pure traction and pure pressure. Creep tests are also considered. To avoid ill-posed boundary value problems and to take account of a time-dependent damage process, viscous regularisation of the time-independent damage model is finally introduced with a structure analogous to the Perzyna-type viscoplasticity.