Obstacle Problems for Shells

Shells are ubiquitous structures, whether they be cooling towers, sails of a sailboat, or inner tubes. Since the validity of numerical simulations of such shells depends first and foremost on the validity of the mathematical equations that model them, the reliability of these equations is a crucial issue.Shells are three-dimensional structures and as such could be in principle modeled by means of three- dimensional equations. But, because they are often very “thin” (for instance, the ratio between the thickness and the height of a cooling tower is of the order of 1/1000), shells are most often modeled by two-dimensional equations, i.e., equations that are “posed on their middle surface”.More specifically, consider a linearly elastic shell subjected to body forces and surface forces on its “upper” and “lower” faces, and to boundary conditions of clamping or simple support on a portion of its lateral surface. Then the mathematical modeling of such a shell, in both the static and time-dependent cases, is now on firm grounds: a rigorous asymptotic analysis of the three-dimensional equations, when the thickness (considered as a parameter) approaches zero, shows that there are three types of shells, the two-dimensional equations for each type being determined by the “geometry” of the middle surface and the boundary conditions: the “flexural shells” (such as a cylindrical hangar clamped on the ground), the “membrane shells” (such as a shell with a portion of an ellipsoid as its middle surface and clamped on its entire lateral face), and the most common “generalized membrane shells” (such as a cooling tower resting on its lower rim). In addition, one can also rigorously justify, by means of another asymptotic analysis, a different type of two-dimensional shell equations, due to W.T. Koiter, which have the definite advantage that they apply to any shell, regardless of its type. For this reason, this highly convenient “all-purpose” model is most often preferred.An “obstacle problem”, also known as a “contact problem”, for an elastic structure is one in which the structure is in addition “geometrically constrained” to stay in a given spatial domain (for instance, a half-space), with or without friction when contact occurs. While the mathematical modeling, by means of variational inequalities, of the obstacle problem without friction for three-dimensional bodies, or for specific thin bodies such as plates or shallow shells, is on firm grounds at least in the static case, the obstacle problem for a general shell does not seem to have been tackled in the existing literature. The objective of this Research Proposal is to remedy this situation.More specifically, our research plan consists first in identifying, and justifying by means of a rigorous asymptotic analysis, three classes of variational inequalities that model in the static case the contact problem without friction for a shell constrained to stay in a given half-space, according to each type of shell (flexural, membrane, or generalized membrane): this will constitute our first three objectives, while our fourth objective, again in the static case, will be to introduce and justify a natural type of “all-purpose” variational inequalities “of Koiter type” that model the contact problem, regardless of the type of shell. Our fifth objective will be to venture into a seemingly virgin territory, viz., that of time- dependent obstacle problems for shells, where the existence of solutions to time-dependent variational inequalities of hyperbolic type with three unknowns (such as those that can be expected to model such problems) seems to be essentially an open problem.There are two long-term impacts of this research: First, since the various two-dimensional models that will have been identified will be amenable to numerical simulations, a natural (and certainly challenging especially in the time-dependent case) future objective will consist in devising convergent numerical schemes for approximating their solutions. Second, the present research should pave the ground for studying the even more challenging mathematical modeling of self-contact and non-interpenetration for shells.