zbMATH — the first resource for mathematics

Stability of nonlinearly elastic rods. (English) Zbl 0545.73039
The author studies in a quasistatic theory buckling of nonlinearly elastic rods that can deform in space under endloading. The governing rod equations are formulated in Euler angles using director theory and are derived from variational principles. Nonlinearity follows as well as from geometry and the constitutive relations. From the formulation as variational principle the stability and instability of planar equilibrium positions can be determined from the corresponding second variation. However, due to isoperimetric constrains the standard tests concerning the second variation are not applicable and a novel devise is used.
A wealth of interesting, previously unknown, results which are physically well interpreted, are presented which makes the paper very worthwhile to study also for the physicist and engineer.
Reviewer: H.Troger

74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74B20 Nonlinear elasticity
PDF BibTeX Cite
Full Text: DOI
[1] J. C. Alexander & S. S. Antman (1982), The ambiguous twist of Love, Q. Appl. Math., XL, p. 83. · Zbl 0486.73032
[2] S. S. Antman & E. R. Carbone (1977), Shear and Necking Instabilities in Nonlinear Elasticity, J. of Elasticity, 7, pp. 125–151. · Zbl 0356.73048
[3] S. S. Antman & C. S. Kenney (1981), Large, buckled states of nonlinearly elastic rods under torsion, thrust, and gravity, Arch. Rational Mech. and Anal., 76, pp. 289–338. · Zbl 0472.73036
[4] S. S. Antman & G. Rosenfeld (1978), Global behavior of buckled states of nonlinearly elastic rods, SIAM Review 20, p. 513, Corrigenda (1980), 22, p. 186. · Zbl 0395.73039
[5] M. Born (1906), Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen, Dieterichsche Univ.-Buchdruckerei, Göttingen. · JFM 38.0984.03
[6] R. C. Browne (1979), Dynamic Stability of One-dimensional Viscoelastic Bodies, Arch. Rational Mech. Anal., 68, pp. 257–282. · Zbl 0401.73064
[7] R. E. Caflisch & J. H. Maddocks (1984), Nonlinear dynamical theory of the elastica, submitted to Proc. Roy. Soc. Edinburgh A.
[8] C. V. Coffman (1976), The nonhomogeneous classical elastica, Technical Report, Department of Mathematics, Carnegie-Mellon University.
[9] L. Euler (1744), Methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes, Opera Omnia I, Vol. 24, Füssli, Zurich 1960, pp. 231–297.
[10] R. L. Fosdick & R. D. James (1981), The elastica and the problem of the pure bending for a non-convex stored energy function, J. Elasticity 11, pp. 165–186. · Zbl 0481.73018
[11] I. M. Gelfand & S. V. Fomin (1963), Calculus of Variations, Prentice-Hall, Inc., New Jersey. · Zbl 0127.05402
[12] H. Goldstein (1980), Classical Mechanics, Addison-Wesley, Reading, Mass., Second Edition. · Zbl 0491.70001
[13] J. Gregory (1980), Quadratic Form Theory and Differential Equations, Academic Press, New York. · Zbl 0468.15015
[14] M. R. Hestenes (1951), Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific Journal of Mathematics, 1, pp. 525–581. · Zbl 0045.20806
[15] M. R. Hestenes (1966), Calculus of Variations and Optimal Control Theory, John Wiley, New York. · Zbl 0173.35703
[16] E. L. Ince (1927), Ordinary Differential Equations, Longmans, Green & Co; London.
[17] R. D. James (1981), The equilibrium and post-buckling behaviour of an elastic curve governed by a non-convex energy, J. Elasticity 11, pp. 239–269. · Zbl 0514.73029
[18] G. Kirchhoff (1859), Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. reine angew. Math. (Crelle), 56. · ERAM 056.1494cj
[19] K. Kovári (1969), Räumliche Verzweigungsprobleme des dünnen elastischen Stabes mit endlichen Verformungen, Ing.-Arch., 37, pp. 393–416. · Zbl 0167.23602
[20] A. E. H. Love (1927), A Treatise on the Mathematical Theory of Elasticity, Fourth Edition, Cambridge University Press.
[21] J. H. Maddocks (1981), Analysis of nonlinear differential equations governing the equilibria of an elastic rod and their stability, Thesis, University of Oxford.
[22] J. H. Maddocks (1984), Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles, SIAM J. of Math. Anal., in press. · Zbl 0581.47049
[23] J. Pierce & A. P. Whitman (1980), Topological properties of the Manifolds of configurations of Several Simple Deformable Bodies, Arch. for Rational Mech. & Anal., 74, p. 101. · Zbl 0439.58012
[24] E. L. Reiss (1969), Column buckling–An elementary example of bifurcation, pp. 1–16 of Bifurcation Theory and Nonlinear Eigenvalue Problems, Eds. J. B. Keller & S. S. Antman, Benjamin, New York. · Zbl 0185.53002
[25] H. F. Weinberger (1973), Variational Methods for Eigenvalue Approximation, C. B. M. S. Conference Series # 15, SIAM.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.