Gradient elasticity and dispersive wave propagation: Model motivation and length scale identification procedures in concrete and composite laminates

Nano-scale experimental findings reveal that wave propagation in heterogeneous materials is dispersive. In order to capture such dispersive behavior, in this paper gradient elasticity theory is resorted to. A popular gradient elasticity model arising from Mindlin's theory incorporates two internal length scale parameters, which correspond to one micro-stiffness and one micro-inertia term. As an extension of Mindlin's model, an expanded three-length-scale gradient elasticity formulation with one additional micro-inertia term is used to improve the description of microstructural effects in dynamics. A non-local lattice model is introduced here to give the above micro-stiffness and micro-inertia terms a physical interpretation based on geometrical and mechanical properties of the microstructure. The purpose of this paper is to assess the effectiveness of such a three-length-scale formulation in predicting wave dispersion against experimental and micro-mechanical data from the literature. The dispersive wave propagation through laminated composites with periodic microstructure is investigated first. Length scale identification is carried out based on higher-order homogenization to link the constitutive coefficients of the gradient theory directly to microstructural properties of the layered composite. Secondly, experimental dispersion curves for phonons propagating in aluminum and bismuth crystals are scrutinized, thus highlighting the motivation for including multiple micro-inertia terms. Finally, ultrasonic wave dispersion experimentally observed in concrete specimens with various sand contents and water/cement ratios is analyzed, along with length scale quantification procedures. It is found that the proposed three-length-scale gradient formulation is versatile and effective in capturing a range of wave dispersion characteristics arising from experiments. Advantages over alternative formulations of gradient elasticity from the literature are discussed throughout the paper.