Non-local effects by homogenization or 3D-1D dimension reduction in elastic materials reinforced by stiff fibers

TL;DR

This paper investigates the nonlocal effects in elastic materials reinforced with stiff fibers, demonstrating that homogenization and 3D-1D reduction lead to nonlocal models in thin heterogeneous cylinders and periodic rod arrays.

Contribution

It establishes the connection between homogenization and 3D-1D dimension reduction, revealing nonlocal models in elastic composites with stiff fibers.

Findings

One-dimensional model is a nonlocal system.

Homogenized model is a second order nonlocal problem.

Direct link between homogenization and 3D-1D reduction.

Abstract

We first consider an elastic thin heterogeneous cylinder of radius of order epsilon: the interior of the cylinder is occupied by a stiff material (fiber) that is surrounded by a soft material (matrix). By assuming that the elasticity tensor of the fiber does not scale with epsilon and that of the matrix scales with epsilon square, we prove that the one dimensional model is a nonlocal system. We then consider a reference configuration domain filled out by periodically distributed rods similar to those described above. We prove that the homogenized model is a second order nonlocal problem. In particular, we show that the homogenization problem is directly connected to the 3D-1D dimensional reduction problem.