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Existence theorems for some boundary value problems in the nonlinear theory of annular elastic membranes

Published online by Cambridge University Press:  16 July 2009

Hans Grabmüller  and
Robert Pirner
Hans Grabmüller
Affiliation:
Institut für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-8520 Erlangen, Germany
Robert Pirner
Affiliation:
Institut für Angewandte Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-8520 Erlangen, Germany
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Abstract

A simplified version of Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations reduces, when specialized to axi-symmetric deformations of annular membranes under a vertical surface load, to a nonlinear ordinary differential equation which is free of Poisson'ratio. Within this framework the questions of existence and non-existence of non-negative solutions of the associated stress and displacement boundary value problem are brought to a final answer. Progress in this direction was made in an earlier study (Grabmüller & Pirner 1987). In this paper a continuous monotone curve is constructed which effects a subdivision of the respective ranges of boundary data into complementary domains of existence and non-existence of strictly positive solutions

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 3 , September 1992 , pp. 299 - 317
Copyright
Copyright © Cambridge University Press 1992

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References

Arango, J. 1989 Existenz und Eindeutigkeitsaussagen über Lösungen von Randwertproblemen der nichtlinearen Membrantheorie. Thesis, Universität Erlangen-Nürnberg, 07, 121 pp.Google Scholar
Baxley, J. V. 1988 A singular nonlinear boundary value problem: membrane response of a spherical cap. SIAM J. Appl. Math. 48, 497–505.CrossRefGoogle Scholar
Beck, A. 1989 Die Reissnersche Kreismembran: Existenz und Eindeutigkeit von positiven Lösungen. Diploma-Thesis, Fachbereich Mathematik, Universität Erlangen-Nürnberg, 04, 63 pp.Google Scholar
Beck, A. & Grabmüller, H. 1992 Wrinkle-free solutions of circular membrane problems. J. Appl. Math. Phys. 43Google Scholar
Clark, R. A. & Narayanaswamy, O. S. 1967 Nonlinear membrane problems for elastic shells of revolution. Proceedings-Symposium on the Theory of Shells (ed. Muster, D.). Houston: University of Houston, 81–110.Google Scholar
Dickey, R. W. 1987 Membrane caps. Quart. Appl. Math. 45, 697–712.CrossRefGoogle Scholar
Dickey, R. W. 1989 Rotationally symmetric solutions for shallow membrane caps. Quart. Appl. Math. 47, 571–581.CrossRefGoogle Scholar
Dörfler, W. 1987 Private communication.Google Scholar
Erwe, F. 1962 Differential- und Integralrechnung 1. Mannheim: Bibliograhisches Institut.Google Scholar
Föppl, A. 1907 Vorlesungen über Technische Mechanik, Bd. 3. Leipzig: B. O. Teubner.Google Scholar
Grabmüller, H. 1988 Monotone method for second order differential equations with decreasing nonlinearity and nonlinear boundary conditions. SFB 123, Universität Heidelberg, Report 467, June.Google Scholar
Grabmüller, H. & Novak, E. 1987 Nonlinear boundary value problems for the annular membrane: A note on uniqueness of positive solutions. J. Elasticity 17, 279–284.Google Scholar
Grabmoller, H. & Novak, E. 1988 Nonlinear boundary value problems for the annular membrane: New results on existence of positive solutions. Math. Meth. Appl. Sci. 10, 37–49.CrossRefGoogle Scholar
Grabmuller, H. & Pirner, R. 1987 Positive solutions of annular elastic membrane problems with finite rotations. Stud. Appl. Math. 77, 223–252.CrossRefGoogle Scholar
Grabmüller, H. 1991 Wrinkle-free solutions in the theory of annular elastic membranes. J. Appl. Math., Phys. 42, 783–805.Google Scholar
Hencky, H. 1915 Über den Spannungszustand in kreisrunden Platten. Z. Math. Phys. 63, 311–317.Google Scholar
Libai, A. & Simmonds, J. G. 1988 The Nonlinear Theory of Elastic Shells of one Spatial Dimension. Boston: Academic Press.Google Scholar
Heuser, H. 1980 Lehrbuch der Analysis, Teil I. Stuttgart: B. O. Teubner.Google Scholar
Pirner, R. 1987 Randwertprobleme der nichtlinearen Membranentheorie: Über Existenz und Nichtexistenz positiver Lösungen. Diploma-Thesis, Fachbereich Mathematik, Universität Erlangen-Nurnberg, 04, 82 pp.Google Scholar
Reissner, E. 1950 On axisymmetric deformation of thin shells of revolution. Proc. Symp. Appl. Math. 3, 27–52.CrossRefGoogle Scholar
Schwerin, E. 1929 Über Spannungen und Formanderungen kreisringformiger Membranen. Z. Techn. Phj's. 12, 651–659.Google Scholar
Stuart, C. A. 1975 Integral equations with decreasing nonlinearities and applications. J. Differential Equations 18, 202–217.Google Scholar
Weinitschke, H. J. 1976 Some mathematical problems in the nonlinear theory of elastic membranes, plates and shells. Trends in Applications of Pure Mathematics to Mechanics (ed. Fichera, G.), London: Pitman, 409–424.Google Scholar
Weinitschke, H. J. 1980 On axisymmetric deformation of nonlinear elastic membranes. Mechanics Today 5 (ed.Nemat-Nasser, S.), Oxford: Pergamon Press, 523–542.CrossRefGoogle Scholar
Weinitschke, H. J. 1987 On finite displacements of circular elastic membranes. Math. Meth. Appl. Sci. 9, 76–98.Google Scholar
Weinitschke, H. J. 1989 Stable and unstable axisymmetric solutions for membranes of revolution. Appl. Mech. Rev. 42 (11) Part 2, S289–S294.CrossRefGoogle Scholar
Weinitschke, H. J. & Grabmuller, H. 1992 Recent mathematical results in the nonlinear theory of flat and curved elastic membranes of revolution. J. Eng. Math. 26.Google Scholar