Reducible and irreducible forms of stabilised gradient elasticity in dynamics

The continualisation of discrete particle models has been a popular tool to formulate higher-order gradient elasticity models. However, a straightforward continualisation leads to unstable continuum models. Pade approximations can be used to stabilise ´ the model, but the resulting formulation depends on the particular equation that is transformed with the Pade approximation. In this contribution, we study two different stabilised ´ gradient elasticity models; one is an irreducible form with displacement degrees of freedom only, and the other is a reducible form where the primary unknowns are not only displacements but also the Cauchy stresses — this turns out to be Eringen’s theory of gradient elasticity. Although they are derived from the same discrete model, there are significant differences in variationally consistent boundary conditions and resulting finite element implementations, with implications for the capability (or otherwise) to suppress crack tip singularities