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Stability in the obstacle problem for a shallow shell

Alain Léger

Laboratoire de Mécanique et d’Acoustique – CNRS, 31, Chemin Joseph Aiguier, 13402 Marseille Cedex 20, France

E-mail Address: leger@lma.cnrs-mrs.fr

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and

Bernadette Miara

Université Paris-Est, Cité Descartes, 93160 Noisy-le-Grand Cedex, France

E-mail Address: bernadette.miara@gmail.com

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https://doi.org/10.1142/S0219530514500225Cited by:1 (Source: Crossref)

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.

Keywords:

AMSC: 35J87, 74K25, 35J86

References

1. S. Alinhac and P. Gérard, Opérateurs Pseudo-différentiels et Théorème de Nash–Moser (InterEditions et CNRS Editions, Paris, 1991). Crossref, Google Scholar
2. A. Bonnet and R. Monneau, Distribution of vortices in a type II superconductor as a free boundary problem: Existence and regularity via Nash–Moser theory, Interfaces Free Bound 2(2) (2000) 181–200. Crossref, Google Scholar
3. L. Caffarelli and A. Friedman, The obstacle problem for the biharmonic operator, Ann. Scuola Norm. Sup. Pisa 6(4) (1979) 151–184. Google Scholar
4. L. Cherfils, A. Léger and B. Miara, On the regularity of the solution in a free boundary problem for a shallow shell, in progress. Google Scholar
5. P. G. Ciarlet and B. Miara, A justification of the two-dimensional equations of a linearly elastic shallow shell, Comm. Pure Appl. Math. 45 (1993) 327–360. Crossref, Web of Science, Google Scholar
6. P. G. Ciarlet and J. C. Paumier, A justification of the Marguerre–von Kármán shell equations, Comput. Mech. 1 (1986) 177–202. Crossref, Google Scholar
7. A. Cimetière and A. Léger, Un résultat de différentiabilité dans le problème d’obstacle pour des poutres en flexion, C. R. Acad. Sci. I 316 (1994) 749–754. Google Scholar
8. R. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. 7(1) (1982) 65–222. Crossref, Web of Science, Google Scholar
9. D. Kinderleherer and L. Nirenberg, Regularity in free boundary problems, Ann. Scuola Norm. Super. Pisa Cl. Sci. (4) 4(2) (1977) 373–391. Google Scholar
10. D. Kinderleherer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications (Academic Press, New York, 1980). Google Scholar
11. A. Léger and B. Miara, Mathematical justification of the obstacle problem in the case of a shallow shell, J. Elasticity 90 (2008) 241–257. Crossref, Web of Science, Google Scholar
12. A. Léger and B. Miara, The obstacle problem for shallow shells: A curvilinear approach, Int. J. Numer. Anal. Model. Ser. B 2(1) (2011) 1–26. Google Scholar
13. A. Léger and C. Pozzolini, Sur la zone de contact entre une plaque élastique et un obstacle rigide, C. R. Mécanique 355 (2007) 144–149. Crossref, Web of Science, Google Scholar
14. A. Léger and C. Pozzolini, A stability result concerning the obstacle problem for a plate, J. Math. Pures Appl. 90(6) (2008) 505–519. Crossref, Web of Science, Google Scholar
15. A. Léger, E. Pratt and Q. J. Cao, A fully nonlinear oscillator with contact and friction, Nonlinear Dynam. 70 (2012) 511–522. Crossref, Web of Science, Google Scholar
16. J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 22(2) (1969) 153–188. Crossref, Web of Science, Google Scholar
17. R. Monneau, On the number of singularities for the obstacle problem in two dimensions, J. Geom. Anal. 13(2) (2003) 359–389. Crossref, Google Scholar
18. E. Sanchez-Palencia and J. Sanchez-Hubert, Coques Élastiques Minces, Propriétés Asymptotiques (Masson, Paris, 1997). Google Scholar
19. D. G. Schaeffer, An example of generic regularity for a nonlinear elliptic equation, Arch. Ration. Mech. Anal. 57 (1974) 134–141. Crossref, Web of Science, Google Scholar
20. D. G. Schaeffer, The capacitor problem, Indiana Univ. Math. J. 24(12) (1975) 1143–1167. Crossref, Web of Science, Google Scholar
21. D. G. Schaeffer, A stability theorem for the obstacle problem, Adv. Math. 16 (1975) 34–47. Crossref, Web of Science, Google Scholar
22. D. G. Schaeffer, Some example of singularities in a free boundary, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(1) (1977) 133–144. Google Scholar
23. R. Schumann, Regularity for Signorini’s problem in linear elasticity, Manuscripta Math. 63 (1989) 255–291. Crossref, Web of Science, Google Scholar
24. J. T. Schwartz, Nonlinear Functional Analysis (Courant Institute of Mathematical Sciences, New York University, 1965). Google Scholar