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We use variational convergence to derive a hierarchy of one-dimensional rodtheories, starting out from three-dimensional models in nonlinear elasticitysubject to local volume-preservation. The densities of the resulting$\Gamma$-limits are determined by minimization problems with a trace constraintthat arises from the linearization of the determinant condition ofincompressibility. While the proofs of the lower bounds rely on suitableconstraint regularization, the upper bounds require a careful, explicitconstruction of locally volume-preserving recovery sequences. After decouplingthe cross-section variables with the help of divergence-free extensions, weapply an inner perturbation argument to enforce the desired non-convexdeterminant constraint. To illustrate our findings, we discuss the special caseof isotropic materials.

Theories for incompressible rods: a rigorous derivation via $\Gamma$-convergence