MiS Preprint Repository

We present a rigorous derivation of a homogenized, bending-torsion theory for inextensible rods from three-dimensional nonlinear elasticity in the spirit of $\mathrm{\Gamma }$-convergence. We start with the elastic energy functional associated to a nonlinear composite material. In a stress-free reference configuration it occupies a thin cylindrical domain with thickness $h\ll 1$. We consider composite materials that feature a periodic microstructure with period $\epsilon \ll 1$. We study the behavior as $\epsilon$ and $h$ simultaneously converge to zero and prove that the energy (scaled by $h^{-4}$) $\mathrm{\Gamma }$-converges towards a non-convex, singular energy functional. The energy is only finite for configurations that correspond to pure bending and twisting of the rod. In this case, the energy is quadratic in curvature and torsion.

Our derivation leads to a new relaxation formula that uniquely determines the homogenized coefficients. It turns out that their precise structure additionally depends on the ratio $h/\epsilon$ and, in particular, different relaxation formulas arise for $h\ll \epsilon$, $\epsilon \sim h$ and $\epsilon \ll h$. Although, the initial elastic energy functional and the limiting functional are non-convex, our analysis leads to a relaxation formula that is quadratic and involves only relaxation over a single cell. Moreover, we derive an explicit formula for isotropic materials in the cases $h\ll \epsilon$ and $h\gg \epsilon$, and prove that the $\mathrm{\Gamma }$-limits associated to homogenization and dimension reduction in general do not commute.