Article
MSC:
34B15,
35B32,
35J65,
37G99,
58E07,
58F14,
73C50,
73H05,
73K10,
74G60,
74K20 | MR 0990299 | Zbl 0682.73036 | DOI: 10.21136/AM.1989.104340
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Keywords:
Fredholm operator; static equilibrium; plate of constant thickness; Fredholm map of index zero; singular point; rotationally symmetric buckled states; von Kármán plate equations; operator equation; proper Sobolev space; local bifurcation behavior; nodal properties
Fredholm operator; static equilibrium; plate of constant thickness; Fredholm map of index zero; singular point; rotationally symmetric buckled states; von Kármán plate equations; operator equation; proper Sobolev space; local bifurcation behavior; nodal properties
Summary:
This paper deals with the exact number of solutions of von Kármán equations for a rotationally symmetric buckling of a thin elastic plate. The plate of constant thickness is in static equilibrium under a uniform compressive thrust applied along its edge in the plane of the plate. The theory of M. G. Crandall, P. H. Rabinowitz [4], is used and the theory of M. S. Berger [1], [3] and M. S. Berger and P. C. Fife [2] is adapted. This work is a part of [6].
References:
[1] M. S. Berger: On von Kármán's Equations and the Buckling of a Thin Elastic Plate, I The Clamped Plate. Comm. on Pure and Appl. Math. vol. XX, 1967, 687-719. DOI 10.1002/cpa.3160200405 | MR 0221808 | Zbl 0162.56405
[2] M. S. Berger P. C. Fife: Von Kármán's Equations and the Buckling of a Thin Elastic Plate, II Plate with General Edge Conditions. Comm. on Pure and Appl. Math., vol. XXI, 1968, 227-241. MR 0229978
[3] M. S. Berger: Nonlinear Problems with Exactly Three Solutions. Indiana University Math. Journal, vol. 28, No 4 (1979), 689-698. DOI 10.1512/iumj.1979.28.28047 | MR 0542952 | Zbl 0431.47037
[4] M. G. Crandall P. H. Rabinowitz: Bifurcation from Simple Eigenvalue. J. Funct. Anal. 8 (1971), 321-340. DOI 10.1016/0022-1236(71)90015-2 | MR 0288640
[5] A. Kufner: Weighted Sobolev Spaces. Teubner, Leipzig, 1980. MR 0664599 | Zbl 0455.46034
[6] Ľ. Marko: Buckled States of Circular Plates. thesis, 1985 (Slovak).
[7] L. Nirenberg: Topics in Nonlinear Functional Analysis. Russian translation, Mir, Moscow 1977. MR 0488104 | Zbl 0426.47034
[8] A. S. Voľmir: Elastic Plates and Shells. (Russian). GITTL, Moscow 1956.
[9] J. H. Wolkowisky: Existence of Buckled States of Circular Plates. Comm. on Pure and Appl. Math. vol. XX, 1967, 549-560. MR 0213087 | Zbl 0168.45206
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