Variationally consistent homogenisation of plates

Journal article, 2023

Advanced fibre composite materials are often used for weight-efficient thin-walled designs, making a plate-based modelling approach suitable for their structural assessment. However, as the sub-structural geometrical features of these materials govern much of their behaviour, a multi-scale approach is necessary. A related challenge, however, is that the in-plane variation of these sub-structural features may be much larger than the total thickness of the material, whereby tailored homogenisation techniques for shell elements are needed. Existing frameworks for plate- and shell-based homogenisation are typically developed using second-order homogenisation in combination with the Hill–Mandel (macro-homogeneity) condition. However, it has been reported in the literature that this approach can lead to kinematic inconsistencies in the macro- to micro-scale transition. One inconsistency that is commonly reported, is the inability to correctly account for the macro-scale transverse shear behaviour on the sub-scale level. In this contribution, we show how the method of Variationally Consistent Homogenisation (VCH) can be used to develop a homogenisation framework for Reissner-Mindlin plate elements, which guarantees kinematically consistent prolongation and homogenisation operations. The homogenisation approach is demonstrated in four numerical examples, where it is shown that the method accurately homogenise the effective sectional plate stiffnesses of homogeneous and heterogeneous sub-structures.