Abstract
This work deals with the variation of the solution to an obstacle problem with respect to the variation of its parameters. More precisely, a mechanical structure is pushed by some external forces against an obstacle in such a way that the equilibrium solution involves a part of the domain in which the structure is strictly in contact with the obstacle. It is known from the theory of variational inequalities that studying the variation of the solution as the external forces vary amounts to studying the variation of the boundary of this contact zone. This problem has been studied in previous works in the scalar case, and it was open in the general case where the unknown is a vector field, due to the coupling between the components. As a first step, the present work considers the case of a linearly elastic shallow membrane shell where the coupling between the in-plane and normal components of the displacement arises from the curvature.
Highlights
We are studying a coupled obstacle problem
The harmonic operator is the simplest model for the out-of-plane component of the displacement of a linearly elastic flat membrane subjected to a distribution of forces normal to its plane in the reference configuration
The stability result for the obstacle problem in the case of the scalar harmonic operator has been given in the pioneer paper by Schaeffer [21]
Summary
We are studying a coupled obstacle problem. A mechanical structure is bent over an obstacle by some external distribution of forces so that the natural unknown is made of the three components of the displacement field. The harmonic operator is the simplest model for the out-of-plane component of the displacement of a linearly elastic flat membrane subjected to a distribution of forces normal to its plane in the reference configuration. The obstacle problem in presence of coupling was known as intricate, so that we tried to restrict our attention to the simplest coupled problem we could find: that of a shallow linear membrane shell over a flat obstacle, in which the coupling is only due to the curvature This means that the present paper could be seen as a first step in the stability analysis of more general coupled problems. An implicit function argument, which is Nash–Moser Theorem, states that if the external force varies in a set of smooth functions, the boundary of the contact zone will vary smoothly in such a way that one have a smooth diffeomorphism between the force and the solution
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