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

Abstract

We first consider an elastic thin heterogeneous cylinder of radius of order ε: 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 ε and that of the matrix scales with ${\epsilon }^2$, 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.