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Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity

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References

  • J. C. Alexander & S. S. Antman (1981), Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Rational Mech. Anal. 76, 339–354.

    Google Scholar 

  • S. S. Antman (1972), The Theory of Rods, Handbuch der Physik VIa/2, edited by C. Truesdell, Springer-Verlag, 641–703.

  • S. S. Antman (1974), Kirchhoff's problem for nonlinearly elastic rods, Quart. Appl. Math. 32, 221–240.

    Google Scholar 

  • S. S. Antman (1976), Ordinary differential equations of one-dimensional nonlinear elasticity I: Foundations of the theories of nonlinearly elastic rods and shells, Arch. Rational Mech. Anal., 61, 307–351.

    Google Scholar 

  • S. S. Antman & E. R. Carbone (1977), Shear and necking instabilities in nonlinear elasticity, J. Elasticity 7, 125–151.

    Google Scholar 

  • S. S. Antman & K. B. Jordan (1975), Qualitative aspects of the spatial deformation of nonlinearly elastic rods, Proc. Roy. Soc. Edinburgh 73A, 85–105.

    Google Scholar 

  • S. S. Antman & T.-P. Liu (1979), Travelling waves in hyperelastic rods, Quart. Appl. Math. 36, 377–399.

    Google Scholar 

  • S. S. Antman & A. Nachman (1980), Large buckled states of rotating rods, Nonlinear Analysis, 4, 303–327.

    Google Scholar 

  • S. S. Antman & G. Rosenfeld (1978), Global behavior of buckled states of nonlinearly elastic rods, SIAM Rev. 20, 513–566. Corrections and additions, Ibid. 22 (1980), 186–187.

    Google Scholar 

  • M. Beck (1952), Die Knicklast des einseitig eingespannten tangential gedrückten Stabes, Zeitschr. Angew. Math. Phys. 3, 225–228.

    Google Scholar 

  • V. V. Bolotin (1961), Nonconservative Problems of the Theory of Elastic Stability (in Russian) GIFML; English translation (1963), MacMillan.

  • M. Born (1906), Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen, Dieterich Univ.-Buchdruckerei, Göttingen.

    Google Scholar 

  • R. C. Browne (1979), Dynamic stability of one-dimensional viscoelastic bodies, Arch. Rational Mech. Anal. 68, 231–262.

    Google Scholar 

  • J. Carr & M. Z. M. Malhardeen (1981), Beck's problem, to appear.

  • A. Clebsch (1862), Theorie der Elasticität fester Körper, Leipzig, Teubner.

    Google Scholar 

  • E. A. Coddington & N. Levinson (1955), Theory of Ordinary Differential Equations, McGraw-Hill.

  • H. Cohen (1966), A nonlinear theory of elastic directed curves, Int. J. Eng. Sci. 4, 511–524.

    Google Scholar 

  • E. Cosserat & F. Cosserat (1907), Sur la statique de la ligne déformable, C. R. Acad. Sci. Paris 145, 1409–1412.

    Google Scholar 

  • E.& F. Cosserat (1909), Théorie des Corps Déformables, Hermann.

  • C. N. Desilva & A. B. Whitman (1971), A thermodynamic theory of directed curves, J. Math. Phys. 12, 1603–1609.

    Google Scholar 

  • J. L. Ericksen & C. Truesdell (1958), Exact theory of stress and strain in rods and shells, Arch. Rational Mech. Anal. 1, 295–323.

    Google Scholar 

  • L. Euler (1744), Additamentum I de curvis elasticis, in Methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes = Opera Omnia Ser. I Vol. 24, Füssli, 1960, 231–297.

  • L. Euler (1775), De motu turbinatorio chordarum musicarum ubi simul universa theoria tam aequilibrii quam motus corporum flexibilium simulque etiam elasticorum breviter explicatur, Novi Comm. Acad. Sci. Petrop. 19, 340–370 and Opera Omnia, Ser. II, Vol. 11, Füssli, Zürich, 158–179.

    Google Scholar 

  • L. Euler (1780), Determinatio onerum, quae columnae gestare valent, Acta Acad. Sci. Petrop. 2, 121–145; Examen insignis paradoxi in theoria columnarum occurentis, loc. cit., 146–162; De altitudine columnarum sub proprio pondere corruentium, loc. cit., 163–193, in Opera Omnia, Ser. II, Vol. 17, Füssli, Zürich.

    Google Scholar 

  • R. Grammel (1923), Das kritische Drillungsmoment von Wellen, Z. angew. Math. Mech. 3, 262–271.

    Google Scholar 

  • A. E. Green & N. Laws (1966), A general theory of rods, Proc. Roy. Soc. London A 293, 145–155.

    Google Scholar 

  • A. G. Greenhill (1881), Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow. Proc. Camb. Phil. Soc. 4, 65–75.

    Google Scholar 

  • A. G. Greenhill (1883), On the strength of shafting when exposed both to torsion and to end thrust. Institution of Mechanical Engineers, Proc., 182–209.

  • J. A. Haringx (1942), On the buckling and the lateral rigidity of helical compression springs, Proc. Nederl. Akad. Wet. 45, 533–539, 650–654.

    Google Scholar 

  • J. A. Haringx (1948–1949), On highly compressible helical springs and rubber rods, and their application for vibration-free mountings, Philips Res. Reports 3, 401–449; 4, 49–80, 206–220, 261–290, 375–400, 407–448.

    Google Scholar 

  • G. Herrmann (1967), Stability of equilibrium of elastic systems subjected to nonconservative forces, Appl. Mech. Rev. 20, 103–108.

    Google Scholar 

  • W. Hess (1884), Über die Biegung und Drillung eines unendlich dünnen elastischen Stabes, dessen eines Ende von einem Kräftepaar angegriffen wird, Math. Annalen 23, 181–212.

    Google Scholar 

  • W. Hess (1885), Über die Biegung und Drillung eines unendlich dünnen elastischen Stabes mit zwei gleichen Widerständen, auf dessen freies Ende eine Kraft und ein um die Hauptaxe ungleichen Widerstandes drehendes Kräftepaar einwirkt. Math. Annalen 25, 1–38.

    Google Scholar 

  • J. B. Keller (1960), The shape of the strongest column, Arch. Rational Mech. Anal. 5, 275–285.

    Google Scholar 

  • C. S. Kenney (1979), Greenhill's problem for nonlinearly elastic rods, dissertation, Univ. Maryland.

  • G. Kirchhoff (1859), Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. reine angew. Math. (Crelle) 56, 285–313.

    Google Scholar 

  • K. Kovári (1969), Räumliche Verzweigungsprobleme des dünnen elastischen Stabes mit endlichen Verformungen, Ing. Arch. 37, 393–416. H. H. E. Leipholz, ed. (1978), Stability of Elastic Structures, Springer.

    Google Scholar 

  • J. L. Lions (1969), Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris.

    Google Scholar 

  • A. E. H. Love (1892–1927), A Treatise on the Mathematical Theory of Elasticity, First edn., Cambridge Univ. Press; Fourth edn., 1927.

  • R. Magnus (1976), A generalization of multiplicity and the problem of bifurcation, Proc. London Math. Soc. (3) 32, 251–278.

    Google Scholar 

  • J. E. Marsden & M. McCracken (1976), The Hopf Bifurcation and its Applications, Springer-Verlag.

  • J. H. Michell (1890), On the stability of a bent and twisted wire, Mess. of Math. 19, 181.

    Google Scholar 

  • M. A. Naimark (1952), Linear Differential Operators (in Russian), English transl., 1967, Ungar.

  • E. L. Nikolai (1916), On the problem of elastic lines of double curvature (in Russian), dissertation, Petrograd, reprinted in E. L. Nikolai (1955), 45–277.

  • E. L. Nikolai (1928), On the stability of the straight equilibrium form of a compressed and twisted rod (in Russian), Izv. Leningr. Politekh. Inst. 31, reprinted in E. L. NikoLai (1955), 357–387.

    Google Scholar 

  • E. L. Nikolai (1929), On the question of the stability of a twisted rod, (in Russian), Vestnik. Prikl. Mat. i. Mekh., 1, reprinted in E. L. Nikolai (1955), 388–406.

    Google Scholar 

  • E. L. Nikolai (1955), Works on Mechanics, (in Russian), G.I.T.T.L.

  • F. Odeh & I. Tadjbakhsh (1965), A nonlinear eigenvalue problem for rotating rods, Arch. Rational Mech. Anal. 20, 81–94.

    Google Scholar 

  • J. Pierce & A. P. Whitman (1980), Topological properties of the manifolds of configurations for several simple deformable bodies, Arch. Rational Mech. Anal. 74, 101–113.

    Google Scholar 

  • M. Potier-Ferry (1981), On the mathematical foundations of elastic stability theory, Arch. Rational Mech. Anal., to appear.

  • I. Tadjbakhsh & J. B. Keller (1962), Strongest columns and isoperimetric inequalities for eigenvalues, ASME Trans. 84E (J. Appl. Mech.), 159–164.

    Google Scholar 

  • W. Thomson (Lord Kelvin) & P. G. Tait (1867), Treatise on Natural Philosophy, Part I; Cambridge Univ. Press; second edition, 1879.

  • A. Trösch (1952), Stabilitätsprobleme bei tordierte Stäben und Wellen, Ing. Arch. 20, 258–277.

    Google Scholar 

  • C. Truesdell (1960), The Rational Mechanics of Flexible or Elastic Bodies 1638–1788, L. Euleri Opera Omnia II 112, Füssli, Zürich.

  • C. Truesdell & W. Noll (1965), The Non-linear Field Theories of Mechanics, Handbuch der Physik III/3, Springer-Verlag.

  • C. Truesdell & R. A. Toupin (1960), The Classical Field Theories, Handbuch der Physik, III/1, Springer-Verlag.

  • M. M. Vainberg (1956), Variational Methods for the Study of Nonlinear Operators (in Russian), Gostekhteorizdat, English transl. (1964), Holden-Day.

  • H. Weyl (1949), Philosophy of Mathematics and Natural Science, Princeton Univ. Press.

  • A. B. Whitman & C. N. Desilva (1974), An exact solution in a nonlinear theory of rods, J. Elast. 4, 265–280.

    Google Scholar 

  • D. W. Zachmann (1979), Nonlinear analysis of a twisted axially loaded elastic rod, Quart. Appl. Math. 37, 67–72.

    Google Scholar 

  • H. Ziegler (1951), Stabilitätsprobleme bei geraden Stäben und Wellen, Zeitschr. Angew. Math. Phys. 2, 265–289.

    Google Scholar 

  • H. Ziegler (1968), Principles of Structural Stability, First ed., Ginn; Second ed. (1977), Birkhäuser.

  • M. Życzkowski (1978), Part III of Leipholz (1978).

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Antman, S.S., Kenney, C.S. Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity. Arch. Rational Mech. Anal. 76, 289–338 (1981). https://doi.org/10.1007/BF00249969

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