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Stability theory for an extensible beam. (English) Zbl 0247.73054


MSC:

74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
45K05 Integro-partial differential equations
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References:

[1] Ball, J. M., Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42, 61-90 (1973) · Zbl 0254.73042
[2] Ball, J. M., Topological methods in the nonlinear analysis of beams, (Ph.D. thesis (1973), University of Sussex)
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[10] Hsu, C. S., The effects of various parameters on the dynamic stability of a shallow arch, Trans. ASME Ser. E Appl. Mech., 34, 349-358 (1967)
[11] Hsu, C. S., Equilibrium configurations of a shallow arch of arbitrary shape and their dynamic stability character, Internat. J. Non-Linear Mech., 3, 113-136 (1968) · Zbl 0162.28401
[12] Hsu, C. S., Stability of shallow arches against snap-through under timewise step loads, Trans. ASME Ser. E Appl. Mech., 35, 618-620 (1968), and see discussion
[13] Huang, N. C.; Nachbar, W., Dynamic snap through of imperfect visco-elastic shallow arches, Trans. ASME Ser. E Appl. Mech., 35, 289-297 (1968)
[14] Lefschetz, S., Differential Equations: Geometric Theory (1962), Wiley-Interscience: Wiley-Interscience New York · Zbl 0107.07101
[15] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod Gauthier-Villars: Dunod Gauthier-Villars Paris · Zbl 0189.40603
[16] Lions, J. L.; Magenes, E., (Problèmes aux limites non homogènes et applications, Vol. 1 (1968), Dunod: Dunod Paris) · Zbl 0165.10801
[17] Mettler, E., Dynamic buckling, (Flügge, S., Handbook of Engineering Mechanics (1962), McGraw-Hill: McGraw-Hill New York), Chapter 62
[18] Mikhlin, S. G., Variational Methods in Mathematical Physics (1964), Pergamon Press: Pergamon Press London, (T. Boddington, Transl.) · Zbl 0119.19002
[19] Reiss, E. L., Column buckling—An elementary example of bifurcation, (Keller, J. B.; Antman, S., Bifurcation Theory and Nonlinear Eigenvalue Problems (1969), Benjamin: Benjamin New York) · Zbl 0185.53002
[20] Reiss, E. L.; Matkowsky, B. J., Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math., 29, 245-260 (1971) · Zbl 0224.73064
[21] Slemrod, M., Asymptotic behaviour of a class of abstract dynamical systems, J. Differential Equations, 7, 584-600 (1970) · Zbl 0275.93023
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