Higher-order gradient continuum modelling of periodic lattice materials

Dispersion of elastic waves in periodic materials is investigated. Gradient elasticity models approximating discrete materials lead to physically consistent results. A continuum enriched with higher-order inertia terms is proposed to capture dispersion. The model can be used to solve boundary value problems via classical finite element implementation. The dynamic behaviour of periodic lattice materials is investigated using an equivalent higher-order continuum model obtained by homogenisation of the equations of motion. A gradient continuum enriched with higher-order inertia terms is developed using a combination of finite element discretisation of the unit cell and the continualisation approach. The analysis of the dispersion relations shows that the proposed model is able to capture correctly the physical phenomenon of wave dispersion in lattice structures which is overlooked by classical continuum theories.