Capturing wave dispersion in heterogeneous and microstructured materials through a three-length-scale gradient elasticity formulation

Long-range interactions occurring in heterogeneous materials are responsible for the dispersive character of wave propagation. To capture these experimental phenomena without resorting to molecular and/or atomistic models, generalized continuum theories can be conveniently used. In this framework, this paper presents a three-length-scale gradient elasticity formulation whereby the standard equations of elasticity are enhanced with one additional strain gradient and two additional inertia gradients to describe wave dispersion in microstructured materials. It is well known that continualization of lattice systems with distributed microstructure leads to gradient models. Building on these insights, the proposed gradient formulation is derived by continualization of the response of a non-local lattice model with two-neighbor interactions. A similar model was previously proposed in the literature for a two-length-scale gradient formulation, but it did not include all the terms of the expansions that contributed to the response at the same order. By correcting these inconsistencies, the three-length-scale parameters can be linked to geometrical and mechanical properties of the material microstructure. Finally, the ability of the gradient formulation to simulate wave dispersion in a broad range of materials (aluminum, bismuth, nickel, concrete, mortar) is scrutinized against experimental observations.