Abstract
In this paper, we consider piezoelectric membrane shells of very small thickness subjected only to mechanical forces and we show that functions which minimize the energy associated with the three-dimensional models converge to the function which minimizes the energy of the two-dimensional model of membrane shells.
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References
Bourquin F, Ciarlet PG, Geymonat G, Raoult A (1992) Gamma-convergence for the asymptotic theory of plates. C R Acad Sci Paris Ser I 315:1017–1024
Ciarlet PG, Destuynder P (1979) A justification of the two dimensional plate model. J Mec 18:315–344
Ciarlet PG, Destuynder P (1979) A justification of the nonlinear model of plate theory. Comp Methods Appl Mech Eng 17(18):227–258
Ciarlet PG, Lods V (1996) Asymptotic analysis of linearly elastic shells. I. Justification of membrane shell equation. Arch Ration Mech Anal 136(2):119–161
Ciarlet PG, Lods V, Miara B (1996) Asymptotic analysis of linearly elastic shells. II. Justification of flexural shell equations. Arch Ration Mech Anal 136:162–190
Ciarlet PG, Miara B (1992) Justification of the two-dimensional equations of a linearly elastic shell. Comm Pure Appl Math 45:327–360
Dal Maso G (1989) An Introduction to gamma convergence, Progress in nonlinear differential equations and its applications. Basel
Frieseke G, James R, Mueller S (2002) A theorem on geometric rigidity and derivation of nonlinear plate theory from three dimensional elasticity. Comm Pure Appl Math 55(11):1461–1506
Genevey K (1995) Justification of the two-dimensional shell models by the use of Gamma-convergence theory. CRM 21:185–197
Ledret H, Raoult A (1995) The nonlinear membrane model as a variational limit of nonlinear three dimensional elasticity. J Math Pures Appl 6(9):549–578
Sabu N (2000) Vibrations of piezoelectric flexural shells: two dimensional approximation. J Elast 68:145–165
Sabu N (2001) Asymptotic analysis of shallow shells with variable thickness. Chin Ann Math 22B(4):405–416
Sabu N (2003) Vibrations of piezoelectric shallow shells: two dimensional approximation. Proc Indian Acad Sci Math Sci 113(3):333–352
Sene A (2001) Modelling of piezoelectric thin plates. Asymptot Anal 25:1–20
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The author would like to thank the referees for their valuable comments.
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Sabu, N. Two-Dimensional Approximation of Thin Piezoelectric Membrane Shells Using Gamma Convergence. Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci. 90, 629–635 (2020). https://doi.org/10.1007/s40010-019-00626-3
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DOI: https://doi.org/10.1007/s40010-019-00626-3