Abstract
Three problems are solved for a nonlinear model of elastic plates with transverse shear deformations. The material of the plate may be anisotropic. An existence theorem is formulated and proved for a class of boundary conditions. A uniqueness theorem is given for small loadings. The dual problem is derived and the minimax or Lagrangian approach is discussed.
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J.M. Ball, J.C. Curie and P.J. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal., 41 (1981) 135–174.
W.R. Bielski and J.J. Telega, A contribution to contact problems for a class of solids and structures, Arch. Mech. 37 (1985) 303–320.
W.R. Bielski and J.J. Telega, The complementary energy principle in finite elastostatics as a dual problem. In: Lecture Notes in Engineering, vol. 19, pp. 62–81, Springer-Verlag, Berlin (1986).
W.R. Bielski and J.J. Telega, On existence of solutions for geometrically nonlinear shells and plates, ZAMM 68 (1988) T155-T157.
W.R. Bielski and J.J. Telega, On existence of solutions and duality for a model of non-linear elastic plates with transverse shear deformations, IFTR Reports 35/1992.
J. Cea, Optimisation: Theórie et Algorithme, Herrmann, Paris (1971).
C.-Y. Chia, Nonlinear Analysis of Plates, McGraw-Hill, New York (1980).
P.G. Ciarlet, Recent progress in the two dimensional approximation of three-dimensional plate models in nonlinear elasticity. In: E.L. Ortiz (ed.), Numerical Approximation of Partial Differential Equations. North-Holland, Amsterdam (1987) pp. 3–19.
P.G. Ciarlet, Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis, Masson, Paris, Springer-Verlag, Berlin (1990).
P.G. Ciarlet and P. Rabier, Les Equations de von Kármán, Springer-Verlag, Berlin (1980).
A. Curnier, Q.-C. He and J.J. Telega, Formulation of unilateral contact between two elastic bodies undergoing finite deformation, C.R. Acad. Sci. Paris, Série II 314 (1992) 1–6.
B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin (1989).
G. Duvaut and J.-L. Lions, Problèmes unilatéraux dans la théorie de la flexion forte des plaques, Part I. Le cas stationnaire. J. Méc. 13: 51–74; II. Le cas d'évolution, ibid. (1974) 245–266.
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).
Y.C. Fung, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey (1965).
A. Gaŀka and J.J. Telega, The complementary energy principle as a dual problem for a specific model of geometrically non-linear elastic shells with an independent rotation vector: general results, European J. Mech. 11 (1992) 1–26.
A. Gaŀka and J.J. Telega, Duality and the complementary energy principle for a class of geometrically nonlinear structures. Part I. Five parameter shell model; Part II. Anomalous dual variational principles for compressed elastic beams, Arch. Mech. 47 (1995) 677–698, 699–724.
N.F. Hanna and A.W. Leissa, Higher order shear deformation theory for the vibration of thick plates, J. Vib. Acoust. 170 (1994) 545–555.
G. Jemielita, On the windings paths of the theory of plates, Pol. Warszawska, Prace Naukowe, Budownictwo, z.117, Warszawa, (1991) (in Polish).
J.L. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia (1989).
J.L. Lagnese and J.-L. Lions, Modelling Analysis and Control of Thin Plates, RMA6, Masson, Paris (1988).
T. Lewiński, On refined plate models based on kinematical assumptions, Ing.-Arch. (1987) 133–146.
C.B. Morrey, Multiple Integrals in the Calculus of Variations. Berlin-Heidelberg-New York; Springer (1966).
J. Nečas, Les Methodes Directes en Théorie des Equations Elliptiques, Masson, Paris, (1967).
J. Nečas and I. Hlavaček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier, Amsterdam (1981).
P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Verlag, Boston-Basel (1985).
J.N. Reddy, A refined nonlinear theory of plates with transverse shear deformation, Int. J. Solids Structures 20 (1984) 881–896.
J.N. Reddy, A general non-linear third-order theory of plates with moderate thickness, Int. J. Non-Linear Mech. 25 (1990) 677–686.
E. Reissner, Reflections on the theory of elastic plates, Appl. Mech. Reviews 38 (1985) 1453–1464.
J.J. Telega, Variational methods in contact problems of mechanics, Uspekhi Mekhaniki (Adv. in Mech.) 10 (1987) 3–95, (in Russian).
J.J. Telega, On the complementary energy principle in non-linear elasticity. Part I: Von Kármán plates and three-dimensional solids, C.R. Acad. Sci. Paris, Série II, 308, 1193–1198; Part II: Linear elastic solid and non-convex boundary condition. Minimax approach, ibid, (1989) pp. 1313–1317.
I.I. Vorovich, Mathematical Problems of Nonlinear Theory of Shallow Shells, Nauka, Moskva 1989, in Russian.
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Dedicated to the memory of Paweŀek Telega, son of the second author
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Bielski, W.R., Telega, J.J. A non-linear elastic plate model of moderate thickness: Existence, uniquenness and duality. J Elasticity 42, 243–273 (1996). https://doi.org/10.1007/BF00041792
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DOI: https://doi.org/10.1007/BF00041792