| Abstract: |
| We consider a nonlinear elastic composite material with $\varepsilon$-periodic microstructure that occupies a thin cylindrical domain (in $\mathbb R^3$) with small thickness $h$. We are interested in the situation when the different phases of the composite are prestrained (i.e.~the reference configuration is not a stress-free state for the individual phases). As a consequence the rod will show a non-flat equilibrium shape that depends in a non-linear and non-local way on the heterogeneity of the material and the distribution of the prestrain. By combining homogenization and dimension reduction, we derive a one-dimensional nonlinear bending-torsion theory for rods that invokes a spontaneous curvature/torsion tensor that captures the macroscopic effect of the microstructural prestrain. The spontaneous curvature/torsion tensor characterizes the equilibrium shape in the asymptotic limit $(h,\varepsilon)\downarrow 0$. It can be computed by solving linear elliptic systems. |
|