Abstract
This paper presents a review which tackles some nonlinear bending problems of plates and shells in a unified way by means of the technique of undetermined small parameters.
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Chien, W. Z., Asymtotic behavior of a thin clamped circular plate under uniform normal pressure at very large deflection, Science Report of Tsinghua University, 5, 1, (1984).
McPherson, A. E., W. Ramburg and S. Levy, Normal pressure tests at circular plates with clamped edges N.A.C.A. Report, 744, (1942).
Chou Huan-wen, The singular perturbation method applied to the problems of large deflection of the circular plates, in “Singular Perturbation Theory and Its Application in Mechanics” (Editor in Chief, Chien Wei-zang), (1981).
Chien Wei-zang and Yeh Kai-yuan, On the large deflections of a circular thin plate, Chinese Journal of Physics, 10, 3 (1954).
Yeh Kai-yuan, Lin Zen-huai, Chang Chuan-dzi and Shue Ih-fan, Nonlinear stability of thin elastic circular shallow sphrical shell under the action of uniform edge moment, Applied Mathematics and Mechanics, 1, 1 (1980), 71–90.
Chou Huan-wen, A perturbation solution in the nonlinear theory of circular plate, Applied Mathematics and Mechanics, 2, 5 (1981).
Bromberg, E., Nonlinear bending of a circular plate under normal pressure, Comm. On Pure and Applied Math., 9, 12 (1956), 597–612.
Srubshchik, L. S. and V. I. Yudovich, Asymptotic treatment of the equations for large bending of a symmetrically loaded circular plate, Dokladi Akad. Nauk SSSR, 139, 2, (1961), 341–344. (in Russian)
Chien Wei-zang, Lin Hong-sun, Hu Hai-chang and Yeh Kai-yuan, The large deflection problem of the circular thin plate, (1954). (in Chinese)
Dickey, R. W., The plane circular elastic surface under normal pressure, Arch. Rational Mech., 26, 3 (1967).
Tucker, J. R., R. Schmidt and D. A. Dadeppa, Moderately large deflections of a loosely clamped circular plate under a uniformly distributed load, J. Industrial Mathematics Society, 25, Part 1, (1975).
Chien Wei-zang, Wang Zhi-zhong, Xu Yin-ge and Chen Shan-lin, The symmetrical deformation of circular membrane under the action of uniformly distributed load in its central portion, Applied Math, and Mech. 2, 6 (1981).
Dickey, R. W., (Chief editor), Nonlinear Elasticity, (1973).
Srubshchik, L. S., Circular plates under the action of discontinuous loadings, Prikl. Nat. Mekh. (PMM), 28, 6, (1964), 1024–1032. (in Russian)
Strubshchik, L. S. and V. I. Yudovich, Asymptotic behavior of the equation of large deflection of a circular-symmetric loaded plate, Sibirsk. Mat. Zh. 4, (1963), 657–672. (in Russian)
Srubshchik, L. S., On the existence of a solution to the problem of the equilibrium of a circular membrane, PMM 30, 3, (1966), 576–579. (in Russian)
Fife, P., Nonlinear deflection of thin elastic plates under tension, Comm. Pure and Appl. Math. 14 (1961), 81–112.
Srubshchik, L. S., On the asymptotic integration of a system of nonlinear equations of plate theory, PMM 28, 2 (1964), 335–349. (in Russian)
Jiang Fu-ru, Unsymmetrical bending problems for the annular and circular thin plates under various supporting conditions (I), Appl. Math. and Mech. 3, 5 (1982), 683–696.
Jiang Fu-ru, On singular perturbations of elliptic equations, Fudan Journal (Edition on Natural Science), 4 (1978), 29–37. (in Chinese).
Jiang Fu-ru, Some applications of perturbation method in thin plate bending problems, Appl. Math. and Mech., 1, 1, (1980), 37–53.
Timoshenko, S. and S. Woinowsky, Krieger, Theory of Plates and Shells, McGraw-Hill, New York, (1959).
Alzheimer, W. E. and R. T. Davis, Unsymmetrical bending of prestressed annular plates, J. Eng. Mech. Div. Proc. ASCE 4 (1968), 905–917.
Mo Jia-ji and Shi, Bing-guo, Perturbation method for thin plate bending problems, Appl. Math. and Mech., 2, 5, (1981), 567–574.
Chien, W. Z., The intrinsic theory of thin shells and plates, Part 1, General theory, Quart. Appl. Math., 1, (1944), 297–327; Part 2, Application to thin plates, ibid. General theory, Quart. Appl. Math., 2(1944), 43–59; Part 3, Application to thin shells, ibid. General theory, Quart. Appl. Math., 2 (1944), 120–135.
Srubshchik, L. S., Asymptotic method of determining the critical buckling loads of shallow strictly convex shells of revolution, PMM 36, 4, (1972), 705–716. (in Russian)
Chou Huan-wen, An application of the method of composite expansions in the large deflection problems of the spherical shells, Symposium on Appl. Math. and Mech. (to be published). (in Chinese)
Srubshchik, L. S. and V. I. Yudovich, Asymptotic integration of a system of the large deflection equations of symmetrically loaded shells of revolution, PMM 26, 5, (1962), 913–922. (in Russian)
Zhukow, M. Iu., and L. S. Srubshchik, Post-buckling behavior of a closed spherical shell, PMM 35, 5, (1971), 840–847. (in Russian)
Srubshchik, L. S., Asymptotic behavior of the Reissner's equations for nonlinear theory of symmetrically loaded shells of revolution, Dokladi Akad. Nauk SSSR 182, 3, (1968).
Srubshchik, L. S., On the question of nonrigidity in the nonlinear theory of shallow shells, Izv. Akad. Nauk SSSR Ser. Mat. 36(1972), 890–909. (in Russian)
Chou Huan-wen, An iterative method on the basis of the perturbation method (to be published).
Chou Huan-wen, The method of composite expansion applied to boundary layer problems in symmetric bending of the spherical shells, Appl. Math. and Mech. 4, 6(1983), 855–864.
O'Malley, R. W., Jr, Two-Parameter Singular Perturbation Problems, Doctoral Dissertation, Stanford University, U.S.A. (1965).
O'Malley Jr, R. E., Arch. Rational Mech. Anal., 26(1967).
O'Malley Jr, R. E., Arch. Rational Mech. Anal., 40(1971).
O'Malley Jr, R. E., J. Math. Mech., 16(1967).
O'Malley Jr, R. E., SIAM J. Appl. Math., 26, 4(1974).
Lin Zong-chi, Singular perturbation of general boundary value problem for higher order elliptic equation containing two parameters, Appl. Math. and Mech., 3, 5(1982), 697–708.
Srubshchik, L. S., Influence of initial imperfections on the buckling of elastic shells under multiple critical loads, PMM 44, 5, (1980), 629–637. (in Russian)
Reissner, E., The edge effect in symmetric bending of shallow shells of revolution, Pure and Appl. Math., 12, 2, (1959), 385.
Srubshchik, L. S., Precritical equilibrium of a thin shallow shell of revolution and its stability, PMM 44, 2(1980), 229–235. (in Russian)
Chang, C. H., J. Appl. Mech. Trans. ASME, E43, 1, 168–169.
Srubshchik, L. S., Influence of initial imperfections on the convexity of elastic shells under multiple critical loads, PMM 44, 5, (1980), 892–904. (in Russian)
Srubshchik, L. S., On the convexity of flexible plates, PMM 32, 4, (1968), 721–727. (in Russian)
Srubshchik, L. S., On the convexity of elastic shells with initial imperfections by many proper forms, Dokladi Akad. Nauk SSSR, 249, 4 (1979). (in Russian)
Bauer, L., H. Keller and E. Reiss, Multiple eigenvalues lead to secondary bifurcation, SIAM Rev., 17, 1, (1975), 101–122.
Mallet-Paret, J., Buckling of cylindrical shells with small curvature, Quart. Appl. Math., 35, 3, (1977), 383–400.
Srubshchik, L. S., On the loss of stability of nonsymmetric strictly convex thin shallow shells, PMM 37, 1, (1973), 118–131. (in Russian)
Srubshchik, L. S., Nonstiffness of spherical shells, PMM 31, 4, (1967), 723–729. (in Russian)
Vorovich, I. I., Certain problems of shell stability in the large, Dokl. Akad. Nauk SSSR 122 (1958), 37–40. (in Russian)
Srubshchik, L. S., On the asymptotic integration of the equilibrium equation for surface tensioned fluid in gravitative field, Zh. Vychisl. Mat. i Ma.. Fiz. 6, 6, (1966). (in Russian)
Srubshchik, L. S., and V. I. Iudovich, Note on the stability of membrane solutions in the nonlinear theory of plates shells, PMM 30, 1, (1966), 116–123. (in Russian)
Srubshchik, L. S., On solvability of nonlinear equations of Reissner for nonshallow symmetrically loaded shells of revolution, PMM 32, 3, (1968), 328–332. (in Russian)
Zhukov, M. Iu. and L. S. Srubshchik, Stability of thin nonsymmetric piecewiseconvex elastic shells, PMM 41, 3, (1977), 530–542. (in Russian)
Reiss, E. L., Singular perturbation of bifurcations, SIAM J. Appl. Math., 33, 2, (1977).
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Huan-wen, C. Some applications of the singular perturbation method to the bending problems of thin plates and shells. Appl Math Mech 5, 1449–1457 (1984). https://doi.org/10.1007/BF01910435
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DOI: https://doi.org/10.1007/BF01910435