Homogenisation of periodic lattices with lumped and distributed mass: Beam models, continualisation and stabilisation

Periodic lattice models of beams are homogenised to obtain equivalent continuum models. Various modelling aspects for the discrete model are discussed, such as the selection of a beam theory and the assumption of mass distribution. Straightforward continualisation by means of Taylor series leads, upon truncation, to a hierarchical series of models of increasing accuracy: the lowest order of these belong to the category of couple stress theories, whereas in the higher-order models the equations of motion are extended with strain gradients and acceleration gradients of the displacement and rotational degrees of freedom. Some of the resulting models are affected by instabilities. These instabilities can be eliminated by alternative approaches to continualisation based on Padé approximations, as will be demonstrated. Throughout, a systematic comparison between lumped mass and distributed mass is made.