Asymptotics of the solution of the problem of deformation of an arbitrary locally periodic thin plate

Akimova, E.; Nazarov, S.; Chechkin, G.

doi:10.1090/S0077-1554-04-00142-6

Asymptotics of the solution of the problem of deformation of an arbitrary locally periodic thin plate
HTML articles powered by AMS MathViewer

by E. A. Akimova, S. A. Nazarov and G. A. Chechkin
Translated by: H. H. McFaden

Trans. Moscow Math. Soc. 2004, 1-29

DOI: https://doi.org/10.1090/S0077-1554-04-00142-6

Published electronically: November 2, 2004

PDF | Request permission

Abstract:

This paper is concerned with a problem in the theory of elasticity for a thin composite plate. The principal terms are constructed for the asymptotics of the solution, with only local periodicity of the elastic moduli of the material and the shape of the plate assumed. The asymptotic expression is justified with the help of a weighted anisotropic Korn’s inequality, which is proved by means of the “tetris” procedure for constructing a supporting set.

References

Dietrich Morgenstern, Herleitung der Plattentheorie aus der dreidimensionalen Elastizitätstheorie, Arch. Rational Mech. Anal. 4 (1959), 145–152 (1959) (German). MR 111252, DOI 10.1007/BF00281383
B. A. Shoikhet, On asymptotically exact equations of thin plates of complex structure, Prikl. Mat. Meh. 37 (1973), 914–924 (Russian); English transl., J. Appl. Math. Mech. 37 (1973), 867–877 (1974). MR 0349110, DOI 10.1016/0021-8928(73)90016-6
B. A. Shoikhet, An energy identity in physically nonlinear elasticity, and an error estimate for the equations of plates, Prikl. Mat. Mekh. 40 (1976), 317–326; English transl. in Appl. Math. Mech. 40 (1976), 291–301.
P. G. Ciarlet and P. Destuynder, A justification of the two-dimensional linear plate model, J. Mécanique 18 (1979), no. 2, 315–344 (English, with French summary). MR 533827
Ph. Destuynder, Comparaison entre les modèles tridimensionnels et bidimensionnels de plaques en élasticité, RAIRO Anal. Numér. 15 (1981), no. 4, 331–369 (French, with English summary). MR 642497
S. N. Leora, S. A. Nazarov, and A. V. Proskura, Derivation of limit functions for elliptic problems in thin domains using a computer, Zh. Vychisl. Mat. i Mat. Fiz. 26 (1986), no. 7, 1032–1048, 1119 (Russian). MR 851753
B. Miara, Optimal spectral approximation in linearized plate theory, Appl. Anal. 31 (1989), no. 4, 291–307. MR 1017518, DOI 10.1080/00036818908839832
P. G. Ciarlet, Plates and junctions in elastic multi-structures, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 14, Masson, Paris; Springer-Verlag, Berlin, 1990. An asymptotic analysis. MR 1071376
Alexander Mielke, On the justification of plate theories in linear elasticity theory using exponential decay estimates, J. Elasticity 38 (1995), no. 2, 165–208. MR 1336037, DOI 10.1007/BF00042497
Philippe G. Ciarlet, Mathematical elasticity. Vol. II, Studies in Mathematics and its Applications, vol. 27, North-Holland Publishing Co., Amsterdam, 1997. Theory of plates. MR 1477663
O. V. Motygin and S. A. Nazarov, Justification of the Kirchhoff hypotheses and error estimation for two-dimensional models of anisotropic and inhomogeneous plates, including laminated plates, IMA J. Appl. Math. 65 (2000), no. 1, 1–28. MR 1773872, DOI 10.1093/imamat/65.1.1
S. A. Nazarov, Asymptotic analysis of an arbitrarily anisotropic plate of variable thickness (a shallow shell), Mat. Sb. 191 (2000), no. 7, 129–159 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 7-8, 1075–1106. MR 1809932, DOI 10.1070/SM2000v191n07ABEH000495
S. A. Nazarov, Asymptotic theory of thin plates and rods, Vol. 1, Reduction of dimension and integral estimates, “Nauchnaya Kniga", Novosibirsk, 2002. (Russian)
Denis Caillerie, Plaques élastiques minces à structure périodique de période et d’épaisseur comparables, C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 294 (1982), no. 3, 159–162 (French, with English summary). MR 654232
D. Caillerie, Thin elastic and periodic plates, Math. Methods Appl. Sci. 6 (1984), no. 2, 159–191. MR 751739, DOI 10.1002/mma.1670060112
R. Kohn and M. Vogelius, A new model for thin plates with rapidly varying thickness, Internat. J. Solids Structures 20 (1984), 333–350.
Robert V. Kohn and Michael Vogelius, A new model for thin plates with rapidly varying thickness. II. A convergence proof, Quart. Appl. Math. 43 (1985), no. 1, 1–22. MR 782253, DOI 10.1090/S0033-569X-1985-0782253-6
Robert V. Kohn and Michael Vogelius, A new model for thin plates with rapidly varying thickness. III. Comparison of different scalings, Quart. Appl. Math. 44 (1986), no. 1, 35–48. MR 840441, DOI 10.1090/S0033-569X-1986-0840441-7
N. S. Bakhvalov and G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, “Nauka”, Moscow, 1984 (Russian). Matematicheskie zadachi mekhaniki kompozitsionnykh materialov. [Mathematical problems of the mechanics of composite materials]. MR 797571
G. P. Panasenko and M. V. Reztsov, Averaging of a three-dimensional problem of elasticity theory in an inhomogeneous plate, Dokl. Akad. Nauk SSSR 294 (1987), no. 5, 1061–1065 (Russian). MR 898314
Tomasz Lewiński, Effective models of composite periodic plates. I. Asymptotic solution, Internat. J. Solids Structures 27 (1991), no. 9, 1155–1172. MR 1087930, DOI 10.1016/0020-7683(91)90116-W
S. A. Nazarov, A general scheme for averaging selfadjoint elliptic systems in multidimensional domains, including thin domains, Algebra i Analiz 7 (1995), no. 5, 1–92 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 5, 681–748. MR 1365812
I. Aganović, E. Marušić-Paloka, and Z. Tutek, Slightly wrinkled plate, Asymptotic Anal. 13 (1996), no. 1, 1–29. MR 1406165
I. Aganović, M. Jurak, E. Marušić-Paloka, and Z. Tutek, Moderately wrinkled plate, Asymptot. Anal. 16 (1998), no. 3-4, 273–297. MR 1612817
T. Lewiński and J. J. Telega, Plates, laminates and shells, Series on Advances in Mathematics for Applied Sciences, vol. 52, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. Asymptotic analysis and homogenization. MR 1758600, DOI 10.1142/9789812813695
Doina Cioranescu (ed.), Nonlinear partial differential equations and their applications. Collège de France Seminar. Vol. XIV, Studies in Mathematics and its Applications, vol. 31, North-Holland Publishing Co., Amsterdam, 2002. To the memory of Jacques-Louis Lions; Papers from the Seminar on Applied Mathematics held at the Collège de France, Paris, 1997–1998. MR 1933963
S. A. Nazarov, Korn inequalities that are asymptotically exact for thin domains, Vestnik S.-Peterburg. Univ. Mat. Mekh. Astronom. vyp. 2 (1992), 19–24, 113–114 (Russian, with English and Russian summaries); English transl., Vestnik St. Petersburg Univ. Math. 25 (1992), no. 2, 18–22. MR 1280920
G. A. Chechkin and E. A. Pichugina, Weighted Korn’s inequality for a thin plate with a rough surface, Russ. J. Math. Phys. 7 (2000), no. 3, 279–287. MR 1832718
E. A. Akimova, S. A. Nazarov, and G. A. Chechkin, The weighted Korn inequality: the “tetris” procedure that serves an arbitrary periodic plate, Dokl. Akad. Nauk 380 (2001), no. 4, 439–442 (Russian). MR 1875497
O. A. Oleĭnik, G. A. Iosif′yan, and A. S. Shamaev, Matematicheskie zadachi teorii sil′no neodnorodnykh uprugikh sred, Moskov. Gos. Univ., Moscow, 1990 (Russian). MR 1115306
K. O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn’s inequality, Ann. of Math. (2) 48 (1947), 441–471. MR 22750, DOI 10.2307/1969180
Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
G. Duvaut and J.-L. Lions, Les inéquations en mécanique et en physique, Travaux et Recherches Mathématiques, No. 21, Dunod, Paris, 1972 (French). MR 0464857
V. A. Kondrat′ev and O. A. Oleĭnik, Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities, Uspekhi Mat. Nauk 43 (1988), no. 5(263), 55–98, 239 (Russian); English transl., Russian Math. Surveys 43 (1988), no. 5, 65–119. MR 971465, DOI 10.1070/RM1988v043n05ABEH001945
Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
Yakov Roitberg, Elliptic boundary value problems in the spaces of distributions, Mathematics and its Applications, vol. 384, Kluwer Academic Publishers Group, Dordrecht, 1996. Translated from the Russian by Peter Malyshev and Dmitry Malyshev. MR 1423135, DOI 10.1007/978-94-011-5410-9