Calculus of Variations and Geometric Measure Theory

N. Ansini - A. Braides - V. Valente

Multi-scale analysis by $\Gamma$-convergence of a one-dimensional non-local functional related to a shell-membrane transition

created by ansini on 10 May 2005
modified on 18 Dec 2006


Published Paper

Inserted: 10 may 2005
Last Updated: 18 dec 2006

Journal: SIAM J. Math. Anal.
Volume: 38
Number: 3
Pages: 944-976
Year: 2006


We study the asymptotic behavior of one-dimensional functionals associated to the energy of a thin nonlinear elastic spherical shell in the limit of vanishing thickness (proportional to a small parameter) $\varepsilon$ 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 transition shell-membrane is studied at three relevant different scales. For each of them we give a compactness result and compute the $\Gamma$-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 non-trivial optimal sequences restricted between the undeformed and the everted configurations must have energy scaling at least as $\varepsilon^{4/3}$.