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

Roberto Paroni (1) , Ali Sili (2, 3)

1 DADU, Università degli Studi di Sassari, Sassari
2 UTLN - Université de Toulon
3 I2M - Institut de Mathématiques de Marseille

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 ε2, 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.

Mots clés: Dimension reduction; Homogenization; Non-local; Rods; Anisotropic

Domaines: Equations aux dérivées partielles [math.AP]

Roberto Paroni, Ali Sili. Non-local effects by homogenization or 3D–1D dimension reduction in elastic materials reinforced by stiff fibers. Journal of Differential Equations, 2016, 260 (3), pp.2026 - 2059. ⟨10.1016/j.jde.2015.09.055⟩. ⟨hal-01256313⟩