Mathematical justification of the obstacle problem for elastic elliptic membrane shells

Abstract

We consider a family of linearly elastic shells with thickness (where is a small parameter). The shells are clamped along a portion of their lateral face, all having the same middle surface S, and may enter in contact with a rigid foundation along the bottom face. We are interested in studying the limit behavior of both the three-dimensional problems, given in curvilinear coordinates, and their solutions (displacements of covariant components ) when tends to zero. To do that, we use asymptotic analysis methods. Guided by the insight of a previous work of the author where the formal asymptotical analysis was developed, in this paper it is shown that if the density of applied body forces is O(1) with respect to and the density of surface tractions is , the solution of the three-dimensional problem converges to the solution of the two-dimensional variational inequality which can be identified as the variational formulation of the obstacle problem of an elastic elliptic membrane shell.

Keywords:

• Asymptotic analysis
• elasticity
• shells
• elliptic membrane
• contact
• rigid foundation
• Signorini

AMS Subject Classifications:

• 41A60
• 35Q74
• 74K25
• 74K15
• 74B05
• 74M15

Acknowledgements

I am grateful to the reviewers of this paper for their valuable remarks and suggestions, which contributed to improve the original manuscript.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was supported by the project MTM2016-78718-P financed by the Government of Spain with the participation of FEDER.