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

We study the $Γ$-limit of sequences of variational problems for straight, transversely curved shallow shells, as the width of the planform ε goes to zero. The energy is of von Kármán type for shallow shells under suitable boundary conditions. What distinguishes the various regimes is the scaling of the stretching energy ~ ε2β, with $β$ a positive number. We derive two one-dimensional models as $β$ ranges in $(0, 2]$. Remarkably, boundary conditions are essential to get compactness. We show that for β∈ (0, 2) the $Γ$-limit leads to relaxation: the limit membrane energy vanishes on compression. For $β=2$ 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.