Multiscale modelling of heterogeneous beams

Abstract

Material heterogeneities, such as pores, inclusion or manufacturing defects can have a detrimental impact on the performance of structural components, such as beams, plates and shells. These heterogeneities are typically defined on a much finer scale than that of the structural component, meaning that fully resolving the substructure in numerical analyses is computationally expensive. A method known as FE2 is therefore considered. As the name suggests, it links at least two finite element (FE) analyses, one defining the macroscale, the other the subscale, in a nested solution procedure. Of particular interest however, are the prolongation (macro-subscale) and homogenisation (subscale-macroscale) techniques used to link a macroscale beam to a statistical volume element (SVE), used to characterise the subscale. Multiple prolongation and homogenisation methods are presented. Although capturing an accurate elongation and bending response is straightforward, the same cannot be said for the shear response. The standard use of Dirichlet, Neumann, and periodic boundary conditions is insufficient. As the length of a statistical volume element (SVE) increases, there is a deterioration in geometric behaviour. More specifically, the SVE begins to bend in an unphysically manner, leading to overly soft results. Variationally Consistent Homogenisation (VCH), provides a systematic way to formulate the macroscale and subscale problem, as well as the link between them. Through the introduction of VCH, an additional volumetric constraints, which imposes an internal rotation, is formulated. The additional constraint provides a drastic improvement. The degradation in shear behaviour is no longer apparent and an accurate shear response is captured. It is important to note however, that this is not an ideal solution, as adding the volumetric constraint perturbs the physicality of the subscale problem. Keywords: Beams, heterogeneities, homogenisation, prolongation