The axisymmetric branching behavior of complete spherical shells
Authors:
Charles G. Lange and Gregory A. Kriegsmann
Journal:
Quart. Appl. Math. 39 (1981), 145-178
MSC:
Primary 73L99; Secondary 35B32, 73H05
DOI:
https://doi.org/10.1090/qam/625467
MathSciNet review:
625467
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: The purpose of this paper is to describe the axisymmetric branching behavior of complete spherical shells subjected to external pressure. By means of an asymptotic integration technique (based on the smallness of the ratio of the shell thickness to the shell radius) applied directly to a differential equation formulation, we are able to continue the solution branches from the immediate vicinity of the bifurcation points, where the solution has the functional form predicted by the linear buckling theory, to the region where the solution consists of either one or two “dimples” with the remainder of the shell remaining nearly spherical. The analysis deals with a novel aspect of bifurcation theory involving “closely spaced” eigenvalues.
J. N. Hutchinson and W. T. Koiter, Postbuckling theory, Appl. Mech. Reviews 23, 1353–1366 (1970)
W. Langford, Bifurcation theory for systems with applications to the buckling of thin spherical shells, Ph. D. Thesis, Calif. Inst. Tech., 1970
W. T. Koiter, The nonlinear buckling problem of a complete spherical shell under external pressure, Proc. Koninklijke Nederland Akademic van Netenschapen, Ser. B, 72, 40–123 (1969)
W. T. Koiter, Aver de stabiliteit van hit elastisch evenwicht (On the stability of elastic equilibrium). Thesis Delft, H. J. Paris, Amsterdam (1945). English translation issued as NASA TT F-10, p. 833 (1967)
- Louis Bauer, Edward L. Reiss, and Herbert B. Keller, Axisymmetric buckling of hollow spheres and hemispheres, Comm. Pure Appl. Math. 23 (1970), 529–568. MR 278605, DOI https://doi.org/10.1002/cpa.3160230402
- Eric Reissner, On axisymmetrical deformations of thin shells of revolution, Proc. Symposia Appl. Math. v. 3, McGraw-Hill Book Co., New York, N. Y., 1950, pp. 27–52. MR 0039489
L. Berke and R. L. Carlson, Experimental studies of the postbuckling behavior of complete spherical shells, Exper. Mech. 8, 548–533 (1968)
- Charles G. Lange, Branching from closely spaced eigenvalues with application to a model biochemical reaction, SIAM J. Appl. Math. 40 (1981), no. 1, 35–51. MR 602498, DOI https://doi.org/10.1137/0140003
- Gregory A. Kriegsmann and Charles G. Lange, On large axisymmetrical deflection states of spherical shells, J. Elasticity 10 (1980), no. 2, 179–192. MR 576166, DOI https://doi.org/10.1007/BF00044502
- E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922
- Julian D. Cole, Perturbation methods in applied mathematics, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0246537
- Ali Hasan Nayfeh, Perturbation methods, John Wiley & Sons, New York-London-Sydney, 1973. Pure and Applied Mathematics. MR 0404788
C. G. Lange and A. C. Newell, The post-buckling problem for thin elastic shells, SIAM J. of Appl. Math. 21, 605–629 (1971)
C. G. Lange and A. C. Newell, Spherical shells like hexagons, cylinders prefer diamonds, Part 1, J. Appl. Mech. 40, 575–581 (1973)
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469
F. von Kármán and S. Tsien, The buckling of spherical shells by external pressure, J. Aero. Sci. 1, 43–50 (1939)
- K. O. Friedrichs, On the minimum buckling load for spherical shells, Theodore von Kármán Anniversary Volume, California Institute of Technology, Pasadena, Calif., 1941, pp. 258–272. MR 0004599
- A. G. Gabril′janc and V. I. Feodos′ev, Axially-symmetric forms of equilibrium of an elastic spherical shell under uniformly distributed pressure, J. Appl. Math. Mech. 25 (1961), 1629–1642. MR 0141279, DOI https://doi.org/10.1016/0021-8928%2862%2990141-7
- U. Ascher, J. Christiansen, and R. D. Russell, A collocation solver for mixed order systems of boundary value problems, Math. Comp. 33 (1979), no. 146, 659–679. MR 521281, DOI https://doi.org/10.1090/S0025-5718-1979-0521281-7
---, Collocation software for boundary value ODE’s, preprint (1979)
C. G. Lange, Axisymmetric buckling of spherical shells, to be submitted to Studies in Appl. Math.
J. N. Hutchinson and W. T. Koiter, Postbuckling theory, Appl. Mech. Reviews 23, 1353–1366 (1970)
W. Langford, Bifurcation theory for systems with applications to the buckling of thin spherical shells, Ph. D. Thesis, Calif. Inst. Tech., 1970
W. T. Koiter, The nonlinear buckling problem of a complete spherical shell under external pressure, Proc. Koninklijke Nederland Akademic van Netenschapen, Ser. B, 72, 40–123 (1969)
W. T. Koiter, Aver de stabiliteit van hit elastisch evenwicht (On the stability of elastic equilibrium). Thesis Delft, H. J. Paris, Amsterdam (1945). English translation issued as NASA TT F-10, p. 833 (1967)
L. Bauer, E. L. Reiss and H. B. Keller, Axisymmetric buckling of hollow spheres and hemispheres, Comm. Pure Appl. Math. 23, 529–568 (1970)
E. Reissner, An axisymmetrical deformation of thin shells of revolution, Proc. Symposia in Appl. Math. 3, 27–52 (1950)
L. Berke and R. L. Carlson, Experimental studies of the postbuckling behavior of complete spherical shells, Exper. Mech. 8, 548–533 (1968)
C. G. Lange, Branching from closely spaced eigenvalues with application to a model biochemical reaction, SIAM J. of Appl. Math. 40, 35–51 (1981)
G. A. Kriegsmann and C. G. Lange, On large axisymmetrical deflection states of spherical shells, J. Elasticity 10, 179–192 (1980)
E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea, New York, 1955
J. D. Cole, Perturbation methods in applied mathematics, Ginn-Blaisdell, Waltham, Mass., 1968
A. H. Nayfeh, Perturbation methods, Wiley—Interscience, New York, 1973
C. G. Lange and A. C. Newell, The post-buckling problem for thin elastic shells, SIAM J. of Appl. Math. 21, 605–629 (1971)
C. G. Lange and A. C. Newell, Spherical shells like hexagons, cylinders prefer diamonds, Part 1, J. Appl. Mech. 40, 575–581 (1973)
E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press, 1927
F. von Kármán and S. Tsien, The buckling of spherical shells by external pressure, J. Aero. Sci. 1, 43–50 (1939)
K. O. Friedrichs, On the minimum buckling load for spherical shells, in von Kármán Anniversary Volume, Calif. Inst. Tech., Pasadena, 1941, pp. 258–272
A. G. Gabriliants and V. J. Feodosèv, Axially-symmetric form of equilibrium of an elastic spherical shell under uniformly distributed pressure, Prikl. Math. Mech. 25, 1091–1101 (1961)
U. Ascher, J. Christinsen and R. D. Russell, A collocation solver for mixed order systems of boundary value problems, Math. Comp. 33, 659–679 (1979)
---, Collocation software for boundary value ODE’s, preprint (1979)
C. G. Lange, Axisymmetric buckling of spherical shells, to be submitted to Studies in Appl. Math.
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73L99,
35B32,
73H05
Retrieve articles in all journals
with MSC:
73L99,
35B32,
73H05
Additional Information
Article copyright:
© Copyright 1981
American Mathematical Society
