On slender shells and related problems suggested by Torroja's structures

We study the rigidification phenomenon for several thin slender bodies or shells, with a small curvature in the transversal direction to the main length, for which the propagation of singularities through the characteristics is of parabolic type. The asymptotic behavior is obtained starting with the two-dimensional Love–Kirchoff theory of plates. We consider, in a progressive study, a starting basic geometry, we pass then to consider the “V-shaped” structure formed by two slender plates pasted together along two long edges forming a small angle between their planes and, finally, we analyze the periodic extension to a infinite slab. We introduce a scalar potential φ and prove that the equation and constrains satisfied by the limit displacements are equivalent to a parabolic higher-order equation for φ. We get some global informations on φ, some on them easely associated to the different momenta and others of a different nature. Finally, we study the associate obstacle problem and obtain a global comparison result between the third component of the displacements with and without obstacle.