Computational challenges in using strain-gradient theories in three dimensions

Strain-gradient models are useful for modelling scale-dependent phenomena such as boundary-layer formation and localisation of deformation. Most problems involving straingradient models in the literature are studied in two dimensions, since this allows for easier theoretical and numerical treatment. In order to determine the applicability of strain-gradient models to real-world problems it is however necessary to consider these models in threedimensional boundary value problems. Additionally, some types of material behaviour (e.g. behaviour in torsion) are only observable in three dimensions. We therefore consider here the computational challenges arising when implementing straingradient models in three dimensions, in the common case where the finite element method is used. Using both theoretical arguments and practical examples, we present issues related to the computational cost of the available finite elements and the different possible plasticity models and integration algorithms. Moreover, we consider in more detail the modes of localisation of deformation in three-dimensional problems, showing the dependence of the results on the way localisation is triggered.