  In this paper we investigate rods made of nonlinearly elastic,
composite--materials that feature a micro-heterogeneous prestrain that
oscillates (locally periodic) on a scale that is small compared to the length
of the rod. As a main result we derive a homogenized bending-torsion theory for
rods as $\Gamma$-limit from 3D nonlinear elasticity by simultaneous
homogenization and dimension reduction under the assumption that the prestrain
is of the order of the diameter of the rod. The limit model features a
spontaneous curvature-torsion tensor that captures the macroscopic effect of
the micro-heterogeneous prestrain. We device a formula that allows to compute
the spontaneous curvature-torsion tensor by means of a weighted average of the
given prestrain. The weight in the average depends on the geometry of the
composite and invokes correctors that are defined with help of boundary value
problems for the system of linear elasticity. The definition of the correctors
depends on a relative scaling parameter $\gamma$, which monitors the ratio
between the diameter of the rod and the period of the composite's
microstructure. We observe an interesting size-effect: For the same prestrain a
transition from flat minimizers to curved minimizers occurs by just changing
the value of $\gamma$. Moreover, in the paper we analytically investigate the
microstructure-properties relation in the case of isotropic, layered
composites, and consider applications to nematic liquid-crystal-elastomer rods
and shape programming.
