Derivations of two one-dimensional models for transversely curved shallow shells: one leads to relaxation

We study the $\Gamma$-limit of sequences of variational problems for straight, transversely curved shallow shells, as the width of the planform $\varepsilon$ goes to zero. The energy is of von K\'arm\'an type for shallow shells under suitable boundary conditions. What distinguishes the various regimes is the scaling of the stretching energy $\sim \varepsilon^{2\beta}$, with $\beta$ a positive number. We derive two one-dimensional models as $\beta$ ranges in $(0, 2]$. Remarkably, boundary conditions are essential to get compactness. We show that for $\beta \in (0, 2)$ the $\Gamma$-limit leads to relaxation: the limit membrane energy vanishes on compression. For $\beta=2$ there is no relaxation, and the limit model is a nonlinear energy coupling four kinematical descriptors in a nontrivial way. As special cases of the latter limit model, a nonlinear Vlasov torsion theory and a nonlinear Euler-Bernoulli beam theory can be deduced.