A mechanism-based spatiotemporal non-local constitutive formulation for elastodynamics of composites

Non-locality in elasticity of continua with microstructures has been described via integral formulations or by high-order gradient theories with respect to spatial variables. Although various non-local constitutive relations have been proposed, the origins of spatial and temporal non-localities and their physical mechanisms are still elusive. In this work, we develop an explicit spatiotemporal non-local dynamic constitutive formulation for composites composed of conventional local elastic constituents. All the involved parameters are quantitatively related to the microstructure and properties of the composite. Then, we show that this formulation can be made to correspond to the Mindlin equation that contains high-order spatial and temporal derivatives, to the Willis formalism, and to the spatial non-local Eringen constitutive relation, and the peridynamic formulation. The most salient feature of the dispersion relations of the developed formulation is that it yields both an optical branch and an acoustic branch, whereas the previous spatial non-local formulations only give the latter. These correlations shed light on physical mechanisms of the relevant theories in the context of composites. The present approach can be readily applied to other multi-field coupled mechanical problems of heterogeneous media involving heat conduction, mass diffusion, electrical and other conductive phenomena.