Multiscale Analysis by Γ‐Convergence of a One‐Dimensional Nonlocal Functional Related to a Shell‐Membrane Transition
Nadia Ansini, Andrea Braides, V. Valente · SIAM Journal on Mathematical Analysis · 2006
We study the asymptotic behavior of one‐dimensional functionals associated with the energy of a thin nonlinear elastic spherical shell in the limit of vanishing thickness (proportional to a small parameter) ε and under the assumption of radial deformations. The functionals are characterized by the presence of a nonlocal potential term and defined on suitable weighted functional spaces. The shell‐membrane transition is studied at three different relevant scales. For each we give a compactness result and compute the Γ‐limit. In particular, we show that if the energies on a sequence of configurations scale as $\varepsilon^{3/2}$, then the limit configuration describes a (locally) finite number of transitions between the undeformed and the everted configurations of the shell. We also highlight a kind of “Gibbs phenomenon” by showing that nontrivial optimal sequences restricted between the undeformed and the everted configurations must have energy scaling of at least $\varepsilon^{4/3}$.