Discrete and continuum modelling of micro-lattices in dynamics

Materials with a sparse, periodic lattice microstructure exhibit excellent mechanical performance compared to their weight. Particularly, their engineered microstructure allows optimised mechanical behaviour under dynamic loading conditions. For instance, the material's microstructure can be manipulated such that wave filters emerge so that only certain frequencies can propagate through the material.

Various techniques can be used to model such materials with lattice microstructures. For instance, a discrete model can be deployed whereby every strut of the lattice structure is modelled as a beam element. However, a more efficient approach is to replace such a detailed microscopic material model with an enriched continuum model for certain dynamic problems. Compared to the classical continuum, the new model allows the microstructural effects to be captured efficiently and effectively by equipping the continuum equations of elasticity with an appropriate set of higher-order spatial derivatives; hence, a gradient elasticity formulation is obtained.

In order to link the additional constitutive coefficients of gradient elasticity to the geometric and mechanical properties of the lattice, in this thesis we use continualisation techniques whereby a representative volume element of a discrete square lattice model is translated into a homogeneous continuum formulation. Taylor series expansions and Pad\'{e} approximations are usually required to ensure stability of the gradient elasticity model. The resulting continuum formulation is equipped with a range of strain gradient and inertia gradient terms. The dispersive properties of the model are then tested to check for the occurrence of wave filters. Applications of the Ru-Aifantis theorem are considered in detail. Finally, implementability with finite elements of the new continuum is examined. The research first reviews the one-dimensional case and subsequently applies the procedure to two-dimensional lattices of square, trapezium and hexagonal geometry arrangements.