Simultaneous homogenization and dimension reduction in nonlinear elasticity

A three-dimensional elastic rod is a body that occupies a cylindrical domain with small "thickness h". We consider rods made of composite materials with microstructure on a small "material fine-scale e". We are interested in the effective behavior for small thickness and small material fine-scale. Here, the zero-thickness limit corresponds to dimension reduction, where we expect to obtain a one-dimensional elasticity model. On the other side, in the homogenization limit "e -> 0" we expect that a material law emerges that is homogeneous-in space. In the talk we discuss the simultaneous limit. In a first part, we explore the coupling between relaxation patterns associated to dimension reduction and homogenization by several "mathematical experiments". In a second part, we briefly present a rigorous derivation of a homogenized, nonlinear bending-torsion theory for inextensible rods as Gamma-limit from 3d elasticity.