A meso–macro finite element modelling of laminate structures: Part I: time-independent behaviour

A meso–macro modelling is proposed for laminates made of unidirectional layers of a polymer matrix reinforced with long fibres. The behaviour of a layer is expressed through elasticity coupled to damage and plasticity. The anisotropic damage is completely described by a single scalar variable and its evolution law is specified from the principle of maximum dissipation. The effective undamaged space concept associated with the strain equivalence principle is used for the plastic analysis. The flow rules are defined within the context of the generalised standard materials assuming the existence of a dissipation pseudo-potential and a plastic criterion expressed as scalar valued functions of the effective stresses and the thermodynamic force associated with a non-linear kinematic hardening. The integration of the layer behaviour through the thickness is obtained within a Kirchhoff shell element. The constitutive equations of the layer are numerically integrated using an unconditionally stable algorithm consisting of a multi-level iterative scheme. A consistent integration of the equilibrium equations is performed to assure a quadratic convergence of the global solution scheme. A methodology is also proposed for the evaluation of the inter-laminar shear stresses in a consistent manner with the stresses and internal variables computation algorithm. The proposed formulation has been implemented in the finite element code CASTEM2000 in order to test its validity. The obtained results have been compared with the available experimental ones in the case of progressive repeated loading tests by applying pure traction, pure internal pressure and internal pressure with end effect on a tube.