  We use variational convergence to derive a hierarchy of one-dimensional rod
theories, starting out from three-dimensional models in nonlinear elasticity
subject to local volume-preservation. The densities of the resulting
$\Gamma$-limits are determined by minimization problems with a trace constraint
that arises from the linearization of the determinant condition of
incompressibility. While the proofs of the lower bounds rely on suitable
constraint regularization, the upper bounds require a careful, explicit
construction of locally volume-preserving recovery sequences. After decoupling
the cross-section variables with the help of divergence-free extensions, we
apply an inner perturbation argument to enforce the desired non-convex
determinant constraint. To illustrate our findings, we discuss the special case
of isotropic materials.
