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Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity. (English) Zbl 0472.73036

MSC:
74G60 Bifurcation and buckling
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74B20 Nonlinear elasticity
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[1] 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. · Zbl 0479.58005
[2] S. S. Antman (1972), The Theory of Rods, Handbuch der Physik VIa/2, edited by C. Truesdell, Springer-Verlag, 641–703.
[3] S. S. Antman (1974), Kirchhoff’s problem for nonlinearly elastic rods, Quart. Appl. Math. 32, 221–240. · Zbl 0302.73031
[4] 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. · Zbl 0354.73046
[5] S. S. Antman & E. R. Carbone (1977), Shear and necking instabilities in nonlinear elasticity, J. Elasticity 7, 125–151. · Zbl 0356.73048
[6] S. S. Antman & K. B. Jordan (1975), Qualitative aspects of the spatial deformation of nonlinearly elastic rods, Proc. Roy. Soc. Edinburgh 73A, 85–105. · Zbl 0351.73076
[7] S. S. Antman & T.-P. Liu (1979), Travelling waves in hyperelastic rods, Quart. Appl. Math. 36, 377–399. · Zbl 0408.73043
[8] S. S. Antman & A. Nachman (1980), Large buckled states of rotating rods, Nonlinear Analysis, 4, 303–327. · Zbl 0437.73028
[9] S. S. Antman & G. Rosenfeld (1978), Global behavior of buckled states of nonlinearly elastic rods, SIAM Rev. 20, 513–566. Correcti · Zbl 0395.73039
[10] M. Beck (1952), Die Knicklast des einseitig eingespannten tangential gedrückten Stabes, Zeitschr. Angew. Math. Phys. 3, 225–228. · Zbl 0046.17703
[11] V. V. Bolotin (1961), Nonconservative Problems of the Theory of Elastic Stability (in Russian) GIFML; English translation (1963), MacMillan.
[12] M. Born (1906), Untersuchungen über die Stabilität der elastischen Linie in Ebene und Raum, unter verschiedenen Grenzbedingungen, Dieterich Univ.-Buchdruckerei, Göttingen. · JFM 38.0984.03
[13] R. C. Browne (1979), Dynamic stability of one-dimensional viscoelastic bodies, Arch. Rational Mech. Anal. 68, 231–262.
[14] J. Carr & M. Z. M. Malhardeen (1981), Beck’s problem, to appear.
[15] A. Clebsch (1862), Theorie der Elasticität fester Körper, Leipzig, Teubner.
[16] E. A. Coddington & N. Levinson (1955), Theory of Ordinary Differential Equations, McGraw-Hill. · Zbl 0064.33002
[17] H. Cohen (1966), A nonlinear theory of elastic directed curves, Int. J. Eng. Sci. 4, 511–524.
[18] E. Cosserat & F. Cosserat (1907), Sur la statique de la ligne déformable, C. R. Acad. Sci. Paris 145, 1409–1412. · JFM 38.0693.02
[19] E.& F. Cosserat (1909), Théorie des Corps Déformables, Hermann.
[20] C. N. Desilva & A. B. Whitman (1971), A thermodynamic theory of directed curves, J. Math. Phys. 12, 1603–1609. · Zbl 0242.73004
[21] J. L. Ericksen & C. Truesdell (1958), Exact theory of stress and strain in rods and shells, Arch. Rational Mech. Anal. 1, 295–323. · Zbl 0081.39303
[22] 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.
[23] 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.
[24] 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.
[25] R. Grammel (1923), Das kritische Drillungsmoment von Wellen, Z. angew. Math. Mech. 3, 262–271. · JFM 49.0593.02
[26] A. E. Green & N. Laws (1966), A general theory of rods, Proc. Roy. Soc. London A 293, 145–155.
[27] 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. · JFM 13.0741.03
[28] 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.
[29] J. A. Haringx (1942), On the buckling and the lateral rigidity of helical compression springs, Proc. Nederl. Akad. Wet. 45, 533–539, 650–654. · JFM 68.0548.03
[30] 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.
[31] G. Herrmann (1967), Stability of equilibrium of elastic systems subjected to nonconservative forces, Appl. Mech. Rev. 20, 103–108.
[32] 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. · JFM 16.0875.01
[33] 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. · JFM 17.0953.01
[34] J. B. Keller (1960), The shape of the strongest column, Arch. Rational Mech. Anal. 5, 275–285.
[35] C. S. Kenney (1979), Greenhill’s problem for nonlinearly elastic rods, dissertation, Univ. Maryland.
[36] G. Kirchhoff (1859), Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. reine angew. Math. (Crelle) 56, 285–313. · ERAM 056.1494cj
[37] 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. · Zbl 0167.23602
[38] J. L. Lions (1969), Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris.
[39] A. E. H. Love (1892–1927), A Treatise on the Mathematical Theory of Elasticity, First edn., Cambridge Univ. Press; Fourth edn., 1927.
[40] R. Magnus (1976), A generalization of multiplicity and the problem of bifurcation, Proc. London Math. Soc. (3) 32, 251–278. · Zbl 0316.47042
[41] J. E. Marsden & M. McCracken (1976), The Hopf Bifurcation and its Applications, Springer-Verlag. · Zbl 0346.58007
[42] J. H. Michell (1890), On the stability of a bent and twisted wire, Mess. of Math. 19, 181.
[43] M. A. Naimark (1952), Linear Differential Operators (in Russian), English transl., 1967, Ungar.
[44] 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.
[45] 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.
[46] 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.
[47] E. L. Nikolai (1955), Works on Mechanics, (in Russian), G.I.T.T.L.
[48] F. Odeh & I. Tadjbakhsh (1965), A nonlinear eigenvalue problem for rotating rods, Arch. Rational Mech. Anal. 20, 81–94. · Zbl 0136.22304
[49] 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. · Zbl 0439.58012
[50] M. Potier-Ferry (1981), On the mathematical foundations of elastic stability theory, Arch. Rational Mech. Anal., to appear. · Zbl 0497.35006
[51] I. Tadjbakhsh & J. B. Keller (1962), Strongest columns and isoperimetric inequalities for eigenvalues, ASME Trans. 84E (J. Appl. Mech.), 159–164. · Zbl 0106.38301
[52] W. Thomson (Lord Kelvin) & P. G. Tait (1867), Treatise on Natural Philosophy, Part I; Cambridge Univ. Press; second edition, 1879.
[53] A. Trösch (1952), Stabilitätsprobleme bei tordierte Stäben und Wellen, Ing. Arch. 20, 258–277. · Zbl 0046.41404
[54] C. Truesdell (1960), The Rational Mechanics of Flexible or Elastic Bodies 1638–1788, L. Euleri Opera Omnia II 112, Füssli, Zürich.
[55] C. Truesdell & W. Noll (1965), The Non-linear Field Theories of Mechanics, Handbuch der Physik III/3, Springer-Verlag.
[56] C. Truesdell & R. A. Toupin (1960), The Classical Field Theories, Handbuch der Physik, III/1, Springer-Verlag.
[57] M. M. Vainberg (1956), Variational Methods for the Study of Nonlinear Operators (in Russian), Gostekhteorizdat, English transl. (1964), Holden-Day.
[58] H. Weyl (1949), Philosophy of Mathematics and Natural Science, Princeton Univ. Press.
[59] A. B. Whitman & C. N. Desilva (1974), An exact solution in a nonlinear theory of rods, J. Elast. 4, 265–280. · Zbl 0295.73052
[60] D. W. Zachmann (1979), Nonlinear analysis of a twisted axially loaded elastic rod, Quart. Appl. Math. 37, 67–72. · Zbl 0403.73062
[61] H. Ziegler (1951), Stabilitätsprobleme bei geraden Stäben und Wellen, Zeitschr. Angew. Math. Phys. 2, 265–289. · Zbl 0043.39302
[62] H. Ziegler (1968), Principles of Structural Stability, First ed., Ginn; Second ed. (1977), Birkhäuser.
[63] M. \.Zyczkowski (1978), Part III of Leipholz (1978).
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