Anisotropic nonlocality for crack tip fields in microstructured periodic beam lattices

Taking a discrete beam lattice as a starting point, continuum models are derived using continualisation and asymptotic series equivalence. The models contain higher-order spatial gradients of the displacements and, therefore, belong to the class of so-called generalised continua. Furthermore, the continuum models are anisotropic, not only regarding the lower-order terms (i.e. the classical elasticity terms) but also the higher-order terms (i.e. the gradient-enrichment terms). We show that the resulting continuum models can be interpreted as particular cases of the Theory of Critical Distances, which itself is a special case of nonlocal elasticity. Two minor simplifications are suggested in order to facilitate straightforward finite element implementation. Taking the compact tension test as a numerical example, the resulting models are shown to avoid singularities in the stress fields around sharp crack tips. Finally, a comparison is carried out with the results of the associated discrete beam lattice.