Axisymmetric and nonaxisymmetric buckled states of a shallow spherical cap
Author:
Frank E. Baginski
Journal:
Quart. Appl. Math. 46 (1988), 331-351
MSC:
Primary 73H05; Secondary 35B32, 58E07, 73L99
DOI:
https://doi.org/10.1090/qam/950606
MathSciNet review:
950606
Full-text PDF Free Access
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Additional Information
- Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
- 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
- Melvyn S. Berger, On the existence of equilibrium states of thin elastic shells. I, Indiana Univ. Math. J. 20 (1970/71), 591–602. MR 275739, DOI https://doi.org/10.1512/iumj.1971.20.20048
- Melvyn S. Berger and Paul C. Fife, On von Karman’s equations and the buckling of a thin elastic plate, Bull. Amer. Math. Soc. 72 (1966), 1006–1011. MR 203219, DOI https://doi.org/10.1090/S0002-9904-1966-11620-8
L. Berke and R. L. Carlson, Experimental studies of the postbuckling behavior of complete spherical shells, Experimental Mech. 8, 548–553 (1968)
R. L. Carlson, R. L. Sendelbeck, and N. J. Hoff, Experimental studies of the buckling of complete spherical shells, Experimental Mech. 7, 281–288 (1967)
- Shui Nee Chow and Jack K. Hale, Methods of bifurcation theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 251, Springer-Verlag, New York-Berlin, 1982. MR 660633
- Philippe G. Ciarlet and Patrick Rabier, Les équations de von Kármán, Lecture Notes in Mathematics, vol. 826, Springer, Berlin, 1980 (French). MR 595326
- Fritz John, Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Comm. Pure Appl. Math. 18 (1965), 235–267. MR 175384, DOI https://doi.org/10.1002/cpa.3160180120
- Fritz John, Refined interior equations for thin elastic shells, Comm. Pure Appl. Math. 24 (1971), 583–615. MR 292358, DOI https://doi.org/10.1002/cpa.3160240502
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin/New York, 1963
- George H. Knightly and D. Sather, Nonlinear axisymmetric buckled states of shallow spherical caps, SIAM J. Math. Anal. 6 (1975), no. 6, 913–924. MR 381454, DOI https://doi.org/10.1137/0506080
- George H. Knightly and D. Sather, Buckled states of a spherical shell under uniform external pressure, Arch. Rational Mech. Anal. 72 (1979/80), no. 4, 315–380. MR 549786, DOI https://doi.org/10.1007/BF00248522
- George H. Knightly and D. Sather, Stable subcritical solutions for a class of variational problems, J. Differential Equations 46 (1982), no. 2, 216–229. MR 675908, DOI https://doi.org/10.1016/0022-0396%2882%2990116-4
W. T. Koiter, The nonlinear buckling problem of a complete spherical shell under uniform external pressure, Proc. Kon. Nederl. Akad. Wet. Amsterdam, B72, 40–123 (1969)
W. T. Koiter, On the nonlinear theory of thin elastic shells, Proc. Kon. Nederl. Akad. Wet. Amsterdam, B69, 1–54 (1966)
- J. B. McLeod and D. H. Sattinger, Loss of stability and bifurcation at a double eigenvalue, J. Functional Analysis 14 (1973), 62–84. MR 0353079, DOI https://doi.org/10.1016/0022-1236%2873%2990030-x
- Willard Miller Jr., Symmetry groups and their applications, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 50. MR 0338286
- Paul H. Rabinowitz, A note on topological degree for potential operators, J. Math. Anal. Appl. 51 (1975), no. 2, 483–492. MR 470773, DOI https://doi.org/10.1016/0022-247X%2875%2990134-1
- Edward L. Reiss, Bifurcation buckling of spherical caps, Comm. Pure Appl. Math. 18 (1965), 65–82. MR 177561, DOI https://doi.org/10.1002/cpa.3160180109
- D. Sather, Bifurcation and stability for a class of shells, Arch. Rational Mech. Anal. 63 (1976), no. 3, 295–304 (1977). MR 434079, DOI https://doi.org/10.1007/BF00251585
- D. H. Sattinger, Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rational Mech. Anal. 43 (1971), 154–166. MR 336485, DOI https://doi.org/10.1007/BF00252776
- C. Truesdell, Invariant and complete stress functions for general continua, Arch. Rational Mech. Anal. 4 (1959), 1–29 (1959). MR 122083, DOI https://doi.org/10.1007/BF00281376
G. N. Watson, Theory of Bessel Functions, Cambridge University Press, Cambridge, 1966
- H. J. Weinitschke, On asymmetric buckling of shallow spherical shells, J. Math. and Phys. 44 (1965), 141–163. MR 181162
S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, 1965
L. Bauer, H. B. Keller, and E. L. Reiss, Axisymmetric buckling of hollow spheres and hemispheres, Comm. Pure Appl. Math. 23, 529–568 (1970)
M. S. Berger, The existence of equilibrium states of thin elastic shells (I), Indiana Univ. Math. J. 20, 591–602 (1971)
M. S. Berger and P. C. Fife, On Von Kármán’s equations and the buckling of a thin elastic plate, Bull. Amer. Math. Soc. 72, 1006–1011 (1966)
L. Berke and R. L. Carlson, Experimental studies of the postbuckling behavior of complete spherical shells, Experimental Mech. 8, 548–553 (1968)
R. L. Carlson, R. L. Sendelbeck, and N. J. Hoff, Experimental studies of the buckling of complete spherical shells, Experimental Mech. 7, 281–288 (1967)
S. Chow and J. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982
P. Ciarlet and P. Rabier, Les équations de Von Kármán, Springer Mathematics Lecture Notes, Vol. 826, Springer-Verlag, Berlin, 1980
F. John, Estimates for the derivatives of the stresses in a thin shell and interior shell equations, Comm. Pure Appl. Math. 18, 235–267 (1965)
F. John, Refined interior equations for thin elastic shells, Comm. Pure Appl. Math. 24, 583–615 (1971)
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin/New York, 1963
G. H. Knightly and D. Sather, Nonlinear axisymmetric buckled states of shallow spherical caps, SIAM J. Math. Anal. 6, 913–924 (1975)
G. H. Knightly and D. Sather, Buckled states of a spherical shell under uniform external pressure, Arch. Rat. Mech. Anal. 72, 315–380 (1980)
G. H. Knightly and D. Sather, Stable subcritical solutions for a class of variational problems, J. Differential Equations 46, 216–229 (1982)
W. T. Koiter, The nonlinear buckling problem of a complete spherical shell under uniform external pressure, Proc. Kon. Nederl. Akad. Wet. Amsterdam, B72, 40–123 (1969)
W. T. Koiter, On the nonlinear theory of thin elastic shells, Proc. Kon. Nederl. Akad. Wet. Amsterdam, B69, 1–54 (1966)
J. B. McLeod and D. H. Sattinger, Loss of stability and bifurcation at a double eigenvalue, J. Functional Analysis 14, 62–84 (1973)
W. Miller, Symmetry Groups and Their Applications, Academic Press, New York, 1972
P. H. Rabinowitz, A note on topological degree for potential operators, J. Math. Anal. Appl. 51, 483–492 (1975)
E. L. Reiss, Bifurcation buckling of spherical caps, Comm. Pure Appl. Math. 18, 65–82 (1965)
D. Sather, Bifurcation and stability for a class of shells, Arch. Rat. Mech. Anal. 63, 295–304 (1977)
D. H. Sattinger, Stability of bifurcating solutions by Leray-Schauder degree, Arch. Rat. Mech. Anal. 43, 154–166 (1971)
C. Truesdell, Invariant and complete stress functions for general continua, Arch. Rat. Mech. Anal. 4, 1–29 (1959/60)
G. N. Watson, Theory of Bessel Functions, Cambridge University Press, Cambridge, 1966
H. J. Weinitschke, On asymmetric buckling of shallow spherical shells, J. Math. and Phys. 44, 141–163 (1965)
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Article copyright:
© Copyright 1988
American Mathematical Society
