The nonlinear bending–torsion theory for curved rods as $\varGamma $ -limit of three-dimensional elasticity

The problem of the rigorous derivation of one-dimensional models for nonlinearly elastic curved beams is studied in a variational setting. Considering different scalings of the three-dimensional energy and passing to the limit as the diameter of the beam goes to zero, a nonlinear model for strings and a bending–torsion theory for rods are deduced.