  Starting from three-dimensional nonlinear elasticity under the restriction of
incompressibility, we derive reduced models to capture the behavior of strings
in response to external forces. Our $\Gamma$-convergence analysis of the
constrained energy functionals in the limit of shrinking cross sections gives
rise to explicit one-dimensional limit energies. The latter depend on the
scaling of the applied forces. The effect of local volume preservation is
reflected either in their energy densities through a constrained minimization
over the cross-section variables or in the class of admissible deformations.
Interestingly, all scaling regimes allow for compression and/or stretching of
the string. The main difficulty in the proof of the $\Gamma$-limit is to
establish recovery sequences that accommodate the nonlinear differential
constraint imposed by the incompressibility. To this end, we modify classical
constructions in the unconstrained case with the help of an inner perturbation
argument tailored for $3$d-$1$d dimension reduction problems.

Analysis of PDEs