Distillation of nonlocality in porous solids

Abstract Nonlocal interactions are intrinsic to multiscale heterogeneous solids. The strength of the interactions exhibits a position-dependent (heterogeneous) character whose spatial distribution strongly correlates with the underlying microstructure. In this work, a variable-order fractional calculus-based framework to distill the position-dependent nonlocal effects is developed. By considering the example of porous plates, theoretical and numerical analyses will illustrate the ability of variable-order mechanics to enable parsimonious and causal models via a synthetic variable-order map that naturally measures the non-classical role of the microstructure in determining the macroscopic response. Parsimony and causality ultimately enable the variable-order model to meaningfully measure the heterogeneous nonlocality and embed it in the variable-order map. The characteristic measure for nonlocality, different from the entanglement noted in the local porosity map, indicates a distillation of the macroscopic nonlocality, analogous to quantum nonlocality distillation. The macroscopic distillation also enables the additivity of the order map, that is the ability of assembled variable-order maps to capture the response of combined porous plates.