Finite element analysis with staggered gradient elasticity

In this article, staggered gradient elasticity formulations are studied. Firstly, the standard equations of classical elasticity are considered. Afterwards, a set of Helmholtz equations associated with the theory of gradient elasticity is solved to handle the gradient dependence. Due to the two-step nature of the algorithms, image-continuous interpolation functions suffice and finite element discretisations are straightforward and efficient. Different versions of staggered gradient elasticity are treated, whereby the Helmholtz equations operate on the displacements, on the strain tensor, on the stress tensor or on a strain invariant. The governing equations are given with their consistent boundary conditions. The formulations are tested against two criteria: whether all singularities are removed from the strain field, and whether the models are capable of describing size effects.