Second-order homogenisation of crystal plasticity and martensitic transformation

Second-order computational homogenisation is combined with constitutive modelling of crystal plasticity and martensitic transformation to analyse the influence of strain gradients on the multi-scale response of polycrystalline materials and the kind of size effects that can be captured. The impact of the length of the Representative Volume Element (RVE) model and the size of the grains is analysed independently, resorting to parametric studies with several random realisations. It is observed that the captured size effects are due to the RVE length, while the size of the grains, which is related to the number of grains in the RVE, is mainly associated with the representativeness of the homogenised response. Macroscopic deformation states are extracted from a plate subjected to bending to guarantee the physical admissibility of the strain gradients and deformation gradients applied to the RVE. It is also found that the influence of the RVE length on the multi-scale response of polycrystals is also dependent on the relationship between the magnitude of the first and second-order deformations. The TRIP effect is also studied by performing these analyses with and without martensitic transformation. To quantify the differences between the results obtained with first- and second-order homogenisation, analytical expressions that take into consideration the macroscopic deformation measures and the RVE length are identified to approximate numerical observations. Finally, an adaptive framework for multi-scale simulation accounting for strain gradients is proposed, where these expressions can be employed in a criterion for the transition from first- to second-order homogenisation.