On finite displacements of curved annular elastic membranes without wrinkling
Authors:
Armin Beck and Hans Grabmüller
Journal:
Quart. Appl. Math. 53 (1995), 527-550
MSC:
Primary 73G05; Secondary 73K10
DOI:
https://doi.org/10.1090/qam/1343465
MathSciNet review:
MR1343465
Full-text PDF Free Access
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Abstract: Axisymmetric deformations of curved annular elastic membranes subjected to vertical surface loads and radial edge loads or displacements are considered within Reissner’s finite-rotation theory of thin shells of revolution, assuming linear stress-strain relations. The principal stresses in the membrane are determined by the solutions of a nonlinear second-order ODE, dependent on a geodesic variable. Analytical methods are used in order to determine the range of those boundary data for which the solutions of the differential equation are wrinkle-free in the sense that both the radial and the circumferential stress components are nonnegative everywhere.
J. Arango, Existenz und Eindeutigkeit über Lösungen von Randwertproblemen der nichtlinearen Membrantheorie, Dissertation, Faculty of Science I, Univ. Erlangen-Nürnberg, July, 1989, 121 pp
A. Beck, Nichtflache Kreis- und Kreisringmembranen unter axialsymmetrischer Vertikallast: Existenz und Eindeutigkeitsbereiche positiver Hauptspannungen und deren numerische Berechnung, Dissertation, Faculty of Science I, Univ. Erlangen-Nürnberg, July, 1992, 79 pp
A. Beck and H. Grabmüller, Wrinkle-free solutions in the theory of curved circular membrane problems, J. Engineering Math. 27, 389–409 (1993)
A. J. Callegari and E. L. Reiss, Nonlinear boundarv-value problems for the circular membrane, Arch. Rat. Mech. Anal. 31, 390–400 (1968)
R. A. Clark and O. S. Narayanaswamy, Nonlinear membrane problems for elastic shells of revolution, Proc. Sympos. Theory of Shells (L. H. Donnell Anniversary Volume; D. Muster, ed.), Univ. of Houston Press, Houston, 1967, pp. 81–110
F. Erwe, Differential- und Integralrechnung I, Mannheim, Bibliographisches Institut, 1962
H. Grabmüller, Wrinkle-free solutions in the theory of annular elastic membranes, J. Appl. Math. and Phys. (ZAMP) 42, 783–805 (1991)
H. Grabmüller and E. Novak, Nonlinear boundary value problems for the annular membrane: A note on uniqueness of positive solutions, J. Elasticity 17, 279–284 (1987)
H. Grabmüller and E. Novak, Nonlinear boundary value problems for the annular membrane: New results on existence of positive solutions, Math. Methods Appl. Sci. 10, 37–49 (1988)
H. Grabmüller and R. Pirner, Positive solutions of annular elastic membrane problems with finite rotations, Studies in Applied Math. 77, 223–252 (1987)
H. Grabmüller and R. Pirner, Existence theorems for some boundary value problems in the nonlinear theory of elastic membranes, European J. Appl. Math. 3, 299–317 (1992)
H. Grabmüller and H. J. Weinitschke, Finite displacements of annular elastic membranes, J. Elasticity 16, 135–147 (1986)
A. Libai and J. G. Simmonds, The nonlinear theory of elastic shells, One spatial dimension, Academic Press, Boston, 1988
R. Pirner, Randwertprobleme der nichtlinearen Membrantheorie: Über Existenz und Nicht-Existenz postiver Lösungen, Diploma-Thesis, Faculty of Science I, Univ. Erlangen-Nürnberg, April, 1987, 82 pp
M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice Hall, Inc., Englewood Cliffs, N.J., 1967
E. Reissner, On axisymmetrical deformation of thin shells of revolution, Proc. Sympos. Appl. Math., Vol. III, Amer. Math. Soc., Providence, RI, 1950, pp. 27–52
E. Schwerin, Über Spannungen und Formänderungen kreisringförmiger Membranen, Z. Tech. Phys. 12, 651–659 (1929)
J. G. Simmonds and A. Libai, A simplified version of Reissner’s nonlinear equations for a first-approximation theory of shells of revolution, Computational Mechanics 2, 99–103 (1987)
D. J. Steigmann, Proof of a conjecture in elastic membrane theory, ASME J. Appl. Mech. 53, 955–956 (1986)
W. Walter, Gewöhnliche Differentialgleichungen, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986
H. J. Weinitschke, Stable and unstable axisymmetric solutions for membranes of revolution, Appl. Mech. Rev. 42, No. 11, Part 2, S289–S294 (1989)
H. J. Weinitschke and H. Grabmüller, Recent mathematical results in the nonlinear theory of flat and curved elastic membranes of revolution, J. Engineering Math. 26, 159–194 (1992)
J. Arango, Existenz und Eindeutigkeit über Lösungen von Randwertproblemen der nichtlinearen Membrantheorie, Dissertation, Faculty of Science I, Univ. Erlangen-Nürnberg, July, 1989, 121 pp
A. Beck, Nichtflache Kreis- und Kreisringmembranen unter axialsymmetrischer Vertikallast: Existenz und Eindeutigkeitsbereiche positiver Hauptspannungen und deren numerische Berechnung, Dissertation, Faculty of Science I, Univ. Erlangen-Nürnberg, July, 1992, 79 pp
A. Beck and H. Grabmüller, Wrinkle-free solutions in the theory of curved circular membrane problems, J. Engineering Math. 27, 389–409 (1993)
A. J. Callegari and E. L. Reiss, Nonlinear boundarv-value problems for the circular membrane, Arch. Rat. Mech. Anal. 31, 390–400 (1968)
R. A. Clark and O. S. Narayanaswamy, Nonlinear membrane problems for elastic shells of revolution, Proc. Sympos. Theory of Shells (L. H. Donnell Anniversary Volume; D. Muster, ed.), Univ. of Houston Press, Houston, 1967, pp. 81–110
F. Erwe, Differential- und Integralrechnung I, Mannheim, Bibliographisches Institut, 1962
H. Grabmüller, Wrinkle-free solutions in the theory of annular elastic membranes, J. Appl. Math. and Phys. (ZAMP) 42, 783–805 (1991)
H. Grabmüller and E. Novak, Nonlinear boundary value problems for the annular membrane: A note on uniqueness of positive solutions, J. Elasticity 17, 279–284 (1987)
H. Grabmüller and E. Novak, Nonlinear boundary value problems for the annular membrane: New results on existence of positive solutions, Math. Methods Appl. Sci. 10, 37–49 (1988)
H. Grabmüller and R. Pirner, Positive solutions of annular elastic membrane problems with finite rotations, Studies in Applied Math. 77, 223–252 (1987)
H. Grabmüller and R. Pirner, Existence theorems for some boundary value problems in the nonlinear theory of elastic membranes, European J. Appl. Math. 3, 299–317 (1992)
H. Grabmüller and H. J. Weinitschke, Finite displacements of annular elastic membranes, J. Elasticity 16, 135–147 (1986)
A. Libai and J. G. Simmonds, The nonlinear theory of elastic shells, One spatial dimension, Academic Press, Boston, 1988
R. Pirner, Randwertprobleme der nichtlinearen Membrantheorie: Über Existenz und Nicht-Existenz postiver Lösungen, Diploma-Thesis, Faculty of Science I, Univ. Erlangen-Nürnberg, April, 1987, 82 pp
M. H. Protter and H. F. Weinberger, Maximum principles in differential equations, Prentice Hall, Inc., Englewood Cliffs, N.J., 1967
E. Reissner, On axisymmetrical deformation of thin shells of revolution, Proc. Sympos. Appl. Math., Vol. III, Amer. Math. Soc., Providence, RI, 1950, pp. 27–52
E. Schwerin, Über Spannungen und Formänderungen kreisringförmiger Membranen, Z. Tech. Phys. 12, 651–659 (1929)
J. G. Simmonds and A. Libai, A simplified version of Reissner’s nonlinear equations for a first-approximation theory of shells of revolution, Computational Mechanics 2, 99–103 (1987)
D. J. Steigmann, Proof of a conjecture in elastic membrane theory, ASME J. Appl. Mech. 53, 955–956 (1986)
W. Walter, Gewöhnliche Differentialgleichungen, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1986
H. J. Weinitschke, Stable and unstable axisymmetric solutions for membranes of revolution, Appl. Mech. Rev. 42, No. 11, Part 2, S289–S294 (1989)
H. J. Weinitschke and H. Grabmüller, Recent mathematical results in the nonlinear theory of flat and curved elastic membranes of revolution, J. Engineering Math. 26, 159–194 (1992)
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Article copyright:
© Copyright 1995
American Mathematical Society
