Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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.


References [Enhancements On Off] (What's this?)

  • [1] J. N. Hutchinson and W. T. Koiter, Postbuckling theory, Appl. Mech. Reviews 23, 1353-1366 (1970)
  • [2] W. Langford, Bifurcation theory for systems with applications to the buckling of thin spherical shells, Ph. D. Thesis, Calif. Inst. Tech., 1970
  • [3] 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)
  • [4] 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)
  • [5] L. Bauer, E. L. Reiss and H. B. Keller, Axisymmetric buckling of hollow spheres and hemispheres, Comm. Pure Appl. Math. 23, 529-568 (1970) MR 0278605
  • [6] E. Reissner, An axisymmetrical deformation of thin shells of revolution, Proc. Symposia in Appl. Math. 3, 27-52 (1950) MR 0039489
  • [7] L. Berke and R. L. Carlson, Experimental studies of the postbuckling behavior of complete spherical shells, Exper. Mech. 8, 548-533 (1968)
  • [8] C. G. Lange, Branching from closely spaced eigenvalues with application to a model biochemical reaction, SIAM J. of Appl. Math. 40, 35-51 (1981) MR 602498
  • [9] G. A. Kriegsmann and C. G. Lange, On large axisymmetrical deflection states of spherical shells, J. Elasticity 10, 179-192 (1980) MR 576166
  • [10] E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea, New York, 1955 MR 0064922
  • [11] J. D. Cole, Perturbation methods in applied mathematics, Ginn-Blaisdell, Waltham, Mass., 1968 MR 0246537
  • [12] A. H. Nayfeh, Perturbation methods, Wiley--Interscience, New York, 1973 MR 0404788
  • [13] C. G. Lange and A. C. Newell, The post-buckling problem for thin elastic shells, SIAM J. of Appl. Math. 21, 605-629 (1971)
  • [14] C. G. Lange and A. C. Newell, Spherical shells like hexagons, cylinders prefer diamonds, Part 1, J. Appl. Mech. 40, 575-581 (1973)
  • [15] E. T. Whittaker and G. N. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press, 1927 MR 1424469
  • [16] F. von Kármán and S. Tsien, The buckling of spherical shells by external pressure, J. Aero. Sci. 1, 43-50 (1939)
  • [17] K. O. Friedrichs, On the minimum buckling load for spherical shells, in von Kármán Anniversary Volume, Calif. Inst. Tech., Pasadena, 1941, pp. 258-272 MR 0004599
  • [18] 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) MR 0141279
  • [19] 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) MR 521281
  • [20] -, Collocation software for boundary value ODE's, preprint (1979)
  • [21] 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

DOI: https://doi.org/10.1090/qam/625467
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society