Non-local homogenization limits of discrete elastic spring network models with random coefficients

Abstract

This work examines a discrete elastic energy system with local interactions described by a discrete second-order functional in the symmetric gradient and additional non-local random long-range interactions. We analyze the asymptotic behavior of this model as the grid size tends to zero. Assuming that the occurrence of long-range interactions is Bernoulli distributed and depends only on the distance between the considered grid points, we derive -- in an appropriate scaling regime -- a fractional p-Laplace-type term as the long-range interactions' homogenized limit. A specific feature of the presented homogenization process is that the random weights of the p-Laplace-type term are non-stationary, thus making the use of standard ergodic theorems impossible. For the entire discrete energy system, we derive a non-local fractional p-Laplace-type term and a local second-order functional in the symmetric gradient. Our model can be used to describe the elastic energy of standard, homogeneous, materials that are reinforced with long-range stiff fibers.