On Variationally Consistent Homogenisation for Composite Structural Elements

Paper in proceeding, 2017

In the current contribution, a new method for computational multiscale, or so-called FE2, modelling of composite structures is proposed. This method allows the inclusion of detailed microor mesoscale composite features e.g. manufacturing defects, reinforcement architectures, matrix cracks etc. and their impact on the mechanical response. However it does not require that these features be fully resolved in the macro-scale analysis. In FE2, at least two geometrical scales, the macro scale and the subscale(s), are linked in a nested FE procedure: on the macroscale, the homogenised material response (normally the stress) is given by a volume averaged lower scale quantity (the locally varying stress) obtained from a coupled subscale FE analysis driven by imposed macroscopic strains or tractions. Of special concern here is to devise a computationally efficient procedure in which the macroscopic problem can be modelled by structural finite elements (beams, plates and shells) while the nested subscale problem is modelled with continuum elements. Several authors have indeed previously contributed to this field, cf. e.g. Refs [1], [2], [3]. Therein, a mixture of in-plane displacement (periodic or Dirichlet) and out-of-plane traction (free) boundary conditions (BCs) have been proposed for the subscale problem. Commonly, they are all based on predefined assumptions on the kinematics of the macroscale problem as well as the assumption that the macroscale in-plane bending response is well represented. However, sufficient consideration has not yet been given to the appropriate handling of the outof- plane shear response. In fact, we have previously shown that using a mixture of displacement and traction based BCs for the subscale problem is not appropriate to capture a consistent outof- plane shear response with increasing geometrical size of the subscale problem [4]. Alternatively, we propose herein a new approach based on Variationally Consistent Homogenisation (VCH) [5]. The advantage of this approach is that the macroscopic problem, i.e. the complete shell theory, does not have to be specified beforehand. Instead, the appropriate problems on the macroscale and the subscale are derived from the fully resolved problem. We will in the current contribution present the modelling framework and discuss the results of using alternative prolongation (macro-to-subscale) and homogenisation (sub-to-macroscale) strategies and how these influence the accuracy in predictions of e.g. the transverse shear response of a thin-walled composite with various subscale features.