Existence theorems for nonlinear elastic plates with periodic boundary conditions. (English) Zbl 0738.73038

The nonlinear problem of three-dimensional elasticity is studied when the elastic body is a rectangular plate. Using the implicit function theorem existence and uniqueness results of a solution are given for different displacement boundary conditions on the lateral surface of the plate.


74K20 Plates
74B20 Nonlinear elasticity
Full Text: DOI


[1] Adams, R.A., Sobolev Spaces, Academic Press, New York (1975).
[2] Brezzi, F., On the existence uniqueness and approximation of saddle point problems-arising from Lagrangian multipliers, R.A.I.R.O., R2 (1974) 129-151. · Zbl 0338.90047
[3] van Buren, M., On the existence and uniqueness of solutions to boundary value problems in finite elasticity, Thesis, Carnegie-Mellon University (1968).
[4] Chillingworth, D.R., Marsden, J.E. and Wan, Y.H., Symmetry and bifurcation in the three dimensional elasticity, Part I, Arch. Rational Mech. Anal. 80 (1982) 295-331. · Zbl 0509.73018 · doi:10.1007/BF00253119
[5] Chillingworth, D.R., Marsden, J.E. and Wan, Y.H., Symmetry and bifurcation in the three dimensional elasticity, Part II, Arch. Rational Mech. Anal. 83 (1983) 363-395. · Zbl 0536.73010 · doi:10.1007/BF00963840
[6] Ciarlet, P.G., A justification of the von K?rm?n equations, Arch. Rational Mech. Anal. 73 (1980) 349-389. · Zbl 0443.73034 · doi:10.1007/BF00247674
[7] Ciarlet, P.G., Mathematical Elasticity, Vol. 1: Three-dimensional Elasticity, Studies in Mathematics and its Applications, North-Holland, Amsterdam.
[8] Ciarlet, P.G. and Destuynder, P., A justification of a nonlinear model in plate theory, Comp. Methods Appl. Mech. Engrg. 17/18 (1979) 227-258. · Zbl 0405.73050 · doi:10.1016/0045-7825(79)90089-6
[9] Ciarlet, P.G. and Paumier, J.-C., A justification of the Marguerre-von K?rm?n equations, Computational Mechanics, 1 (1986) 177-202. · Zbl 0633.73069 · doi:10.1007/BF00272623
[10] Duvaut, G. and Lions, J.L., Les in?quations en m?canique et en physique, Dunod, Paris (1973).
[11] Geymonat, G., Sui problemi ai limiti per i sistemi lineari elliticiti, Ann. Mat. Pura App., 69 (1965) 207-284. · Zbl 0152.11102 · doi:10.1007/BF02414374
[12] Grioli, G., Mathematical theory of elastic equilibrium, Angebnisse der Angew. Math. Vol. 67, Springer, Berlin (1962). · Zbl 0102.17004
[13] Grisvard, P., Singularities in elasticity theory, Applications of Multiple scaling in mechanics, P.G. Ciarlet and E. Sanchez-Palencia (eds), R.M.A. 4, Masson, Paris, 134-150 (1986).
[14] Lanza de, Cristoforis F. and Valent, T., On Neumann’s problem for a quasilinear differential system of the finite elastostatics type, Local theorems of existence and uniquenss, Rend. Sem. Mat. Univ. Padova 68 (1982) 183-206. · Zbl 0532.35014
[15] Marsden, J.E. and Hugues, T.J.R., Topics in the mathematical foundations of elasticity, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. 2, Pitman, London, 30-285 (1978).
[16] Ne?as, J. Les m?thodes directes en th?orie des ?quations elliptiques, Masson, Paris (1967).
[17] Paumier, J.-C., Analyse de certains probl?mes non lin?aires: Mod?les de plaques et de coques, Thesis, Universit? Pierre et Marie Curie (1985).
[18] Rivlin, R.S., Large elastic deformations of isotropic materials. II. Some uniqueness theorems for pure homogeneous deformation, Philos. Trans. Roy. Soc. London. Ser. A 240 (1948) 491-508. · Zbl 0029.32605 · doi:10.1098/rsta.1948.0003
[19] Rivlin, R.S., Stability of pure homogeneous deformations of an elastic cube under dead load, Quat. Applied Math. 32 (1974) 265-271. · Zbl 0324.73039
[20] Stopelli, F., Un theorema di existence e di unicit? relativa alle equazioni dell’elastostatica isoterma per deformazioni finite, Ricerche di Matematica 3 (1954) 247-267. · Zbl 0058.39701
[21] Stopelli, F., Sulla svilluppabilit? in serie di potenze di un parametro delle soluzioni delle equazioni dell’elastostatica isoterma, Ricerche di Matematica 4 (1955) 58-73. · Zbl 0067.41503
[22] Truesdell, C. and Noll, W., The non-linear field theories of Mechanics, Handbuch der Physik, Vol. III/3, Springer Berlin (1965). · Zbl 0779.73004
[23] Valent, T., Teoremi di existenza e unicit? in elastostatica finita, Rend. Sem. Mat. Univ. Padova 60 (1978) 165-181. · Zbl 0425.73011
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