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Abstract

A theory of rods 2 or, equivalently, a one-dimensional theory of solids is a characterization of the behavior of slender three-dimensional solid bodies by a set of equations having the parameter of a certain curve and the time as the only independent variables. The formidable mathematical obstacles presented by three-dimensional continuum theories dictate the need for tractable, accurate, and illuminating one-dimensional models.

Keywords

Constitutive Relation Projection Method Reference Configuration Strain Energy Function Entropy Inequality 
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References

  1. Akhiezer, N. I.: Lectures on the calculus of variations [in Russian]. Moscow: Gostekh-teorizdat. (English translation by A. H. Frink, 1962. New York: Blaisdell.) 1955. (22).Google Scholar
  2. Antman, S. S.: General solutions for plane extensible elasticae having nonlinear stress-strain laws. Quart. Appl. Math. 26, 35–47 (1968a). (17, 26).MATHGoogle Scholar
  3. Antman, S. S.: A note on a paper of Tadjbakhsh and Odeh. J. Math. Anal. Appl. 21, 132–135 (1968b). (26).CrossRefMATHMathSciNetGoogle Scholar
  4. Antman, S. S.: Equilibrium states of nonlinearly elastic rods. J.Math. Anal. Appl. 23, 459–470 (1968c). (15, 21).CrossRefMATHMathSciNetGoogle Scholar
  5. Antman, S. S.: The shape of buckled nonlinearly elastic rings. Z. Angew. Math. Phys. 21, 422–438 (1970a). (25, 26).CrossRefMATHMathSciNetGoogle Scholar
  6. Antman, S. S.: Existence of solutions of the equilibrium equations for nonlinearly elastic rings and arches. Indiana Univ. Math. J. (J. Math. Mech.) 20, 281–302 (1970b). (3, 15, 21, 25, 27).CrossRefMATHMathSciNetGoogle Scholar
  7. Antman, S. S.: Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells. Arch. Rational Mech. Anal. 40, 329–372 (1971). (3, 21).MATHMathSciNetGoogle Scholar
  8. Antman, S. S.:, and W. H. Warner: Dynamical theory of hyperelastic rods. Arch. Rational Mech. Anal. 23, 135–162 (1966). (5, 6, 7, 9, 10, 12, 17).MathSciNetGoogle Scholar
  9. Ball, J. M.: Initial-boundary value problems for an extensible beam. J. Math. Anal. Appl. (1972) (to appear). (28).Google Scholar
  10. Bazley, N., and B. Zwahlen: Remarks on the bifurcation of solutions of a nonlinear eigenvalue problem. Arch. Rational Mech. Anal. 28, 51–58 (1967). (25).CrossRefMathSciNetGoogle Scholar
  11. Beck, M.: Die Knicklast des einseitig eingespannten, tangential gedrückten Stabes. Z. Angew. Math. Phys. 3, 225–228, 476-477 (1952). (24).Google Scholar
  12. Bolotin, V. V.: Dynamic stability of elastic systems [in Russian]. Moscow: Gostekh-teorizdat. [English translation (1964). San Francisco: Holden-Day.] 1956. (28).Google Scholar
  13. Bolotin, V. V.: Nonconservative problems of the theory of elastic stability. Moscow: Fizmatgiz. (English translation by T. K. Lusher, 1963. Oxford: Pergamon.) 196I. (24, 28).Google Scholar
  14. Browder, F. E.: Approximation-solvability of nonlinear functional equations in normed linear spaces. Arch. Rational Mech. Anal. 26, 33–42 (1967). (11).MATHMathSciNetGoogle Scholar
  15. Carrier, G. F.: On the buckling of elastic rings. J. Math. Phys. 26, 94–103 (1947). (27).MATHMathSciNetGoogle Scholar
  16. Cohen, H.: A non-linear theory of elastic directed curves. Int. J. Engng. Sci. 4, 511–524 (1966). (15).CrossRefGoogle Scholar
  17. Cohen, H.: The elastic fiber strengthened string—first order theory. Rév. Roum. Sci. Techn.—Mec. Appl. 12, 439–449 (1967). (25).Google Scholar
  18. Cosserat, E. et F.: Sur la statique de la ligne déformable. Compt. Rend. 145, 1409–1412 (1907). (14).Google Scholar
  19. Cosserat, E. Théorie des Corps Déformables. Paris: Hermann 1909. (14, 15).Google Scholar
  20. De Silva, C. N., and A. B. Whitman: A thermodynamical theory of directed curves. (To appear 1971). (15, 16).Google Scholar
  21. Dickey, R. W.: Free vibrations and dynamic buckling of the extensible beam. J. Math. Anal. Appl. 29, 443–454 (1970). (28).CrossRefMATHMathSciNetGoogle Scholar
  22. Duhem, P.: Le potentiel thermodynamique et la pression hydrostatique. Ann. École Norm. (3) 10, 187–230 (1893). (14).MathSciNetGoogle Scholar
  23. Dupuis, G.: Stabilité élastique des structures unidimensionelles. Z. Angew. Math. Phys. 20, 94–106 (1969). (28).CrossRefMATHGoogle Scholar
  24. Eisley, J. G.: Nonlinear deformation of elastic beams, rings, and strings. Appl. Mech. Rev. 16, 677–680 (1963). (26).Google Scholar
  25. Ericksen, J. L.: Simpler static problems in nonlinear theories of rods. Int. J. Solids Structures 6, 371–377 (1970). (26, 27).CrossRefMATHGoogle Scholar
  26. Ericksen, J. L., and C. Truesdell: Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal. 1, 295–323 (1958). (14, 15).CrossRefMATHMathSciNetGoogle Scholar
  27. Eringen, A. C.: On the nonlinear vibrations of elastic bars. Quart. Appl. Math. 9, 361–369 (1952). (28).MATHMathSciNetGoogle Scholar
  28. Euler, L.: Additamentum I de curvis elasticis, Methodus inveniendi lineas curvas maximi minimive proprietäte gaudentes. Lausanne = Opera Omnia I 24, 231–297 (1744). (26, 27). Evan-Iwanowski, R. M.: On the parametric response of structures. Appl. Mech. Rev. 18, 699-702 (1965). (28).Google Scholar
  29. Fichera, G.: Semicontinuità ed esistenza del minimo per una classe di integrali multipli. Rév. Roum. Mat. Pures et Appl. 12, 1217–1220 (1967). (22).MATHMathSciNetGoogle Scholar
  30. Friedrichs, K. O.: The edge effect of transverse shear deformation on the boundary of plates. Reissner Anniversary Volume, p. 197–210. Ann Arbor: J. W. Edwards 1949. (13).Google Scholar
  31. Friedrichs, K. O.: Kirchhoff’s boundary condition and the edge effect for elastic plates. Proc. Symp. Appl. Math. 3, p. 117–124. New York: McGraw-Hill 1950. (13).Google Scholar
  32. Frisch-Fay, R.: Flexible bars. London: Butterworths 1962. (26).MATHGoogle Scholar
  33. Funk, P.: Variationsrechnung und ihre Anwendung in Physik und Technik. Berlin-Göttingen-Heidelberg: Springer 1962. (22, 25, 26).CrossRefGoogle Scholar
  34. Goodier, J. N.: On the problem of the beam and the plate in the theory of elasticity. Trans. Roy. Soc. Can., Sect. III, 32, 65–88 (1938). (13).Google Scholar
  35. Green, A. E.: The equilibrium of rods. Arch. Rational Mech. Anal. 3, 417–421 (1959). (5, 6, 12).CrossRefMATHMathSciNetGoogle Scholar
  36. Green, A. E., R. J. Knops, and N. Laws: Large deformations, superposed small deformations and stability of elastic rods. Int. J. Solids Structures 4, 555–577 (1968). (16, 28).CrossRefMATHGoogle Scholar
  37. Green, A. E., and N. Laws: A general theory of rods. Proc. Roy. Soc. (London) 293, 145–155 (1966). (15, 16, 17).CrossRefGoogle Scholar
  38. Green, A. E., and P. M. Naghdi: A linear theory of straight elastic rods. Arch. Rational Mech. Anal. 25, 285–298 (1967). (11, 12, 16).MATHMathSciNetGoogle Scholar
  39. Green, A. E. Rods, plates and shells. Proc. Cambridge Phil. Soc. 64, 895–913 (1968). (5, 6, 7, 9, 10, 12).CrossRefMATHGoogle Scholar
  40. Green, A. E. Rods, and P.M. Naghdi: Non-isothermal theory of rods, plates, and shells. Int. J. Solids Structures 6, 209–244 (1970). (5, 6, 7, 9, 10, 12, 15).CrossRefMATHGoogle Scholar
  41. Green, A. E., and W. Zerna: Theoretical elasticity, eds. 1, 2. Oxford: Clarendon Press 1954, 1968. (9).MATHGoogle Scholar
  42. Greenberg, J.M.: On the equilibrium configurations of compressible slender bars. Arch. Rational Mech. Anal. 27, 181–194 (1967). (24, 25).CrossRefMATHMathSciNetGoogle Scholar
  43. Hay, G. E.: The finite displacement of thin rods. Trans. Am. Math. Soc. 51, 65–102 (1942). (12, 13).MathSciNetGoogle Scholar
  44. Herrmann, G.: Stability of equilibrium of elastic systems subject to nonconservative forces. Appl. Mech. Rev. 20, 103–108 (1967). (24, 28).Google Scholar
  45. Kantorovich, L. V., and G. P. Akilov: Functional analysis in normed spaces [in Russian]. Moscow: Fizmatgiz. (English translation by D. E. Brown, 1964. Oxford: Pergamon.) 1959. (11).Google Scholar
  46. Kantorovich, L. V., and Krylov: Approximate methods of higher analysis [in Russian], eds. 1-5. Moscow: Fizmatgiz. (English translation of 3rd ed. by C. D. Benster, 1958. New York: Interscience.) 1952-1962. (11).Google Scholar
  47. Keller, J. B.: The shape of the strongest column. Arch. Rational Mech. Anal. 5, 275–285 (1960). (27).CrossRefMathSciNetGoogle Scholar
  48. Keller, J. B.: Bifurcation theory for ordinaxy differential equations. In: Bifurcation theory and nonlinear eigenvalue problems, ed. by J. B. Keller and S. Antman, p. 17–48. New York: Benjamin 1969. (25).Google Scholar
  49. Keller, J. B., and S. Antman (editors): Bifurcation theory and nonlinear eigenvalue problems. New-York: Benjamin 1969. (25).MATHGoogle Scholar
  50. Keller, J. B., and F. I. Niordsen: The tallest column. J. Math. Mech. 16, 433–446 (1966). (27).MATHMathSciNetGoogle Scholar
  51. Keller, J. B., and L. Ting: Periodic vibrations of systems governed by nonlinear partial differential equations. Comm. Pure Appl. Math. 19, 371–420 (1966). (28).CrossRefMATHMathSciNetGoogle Scholar
  52. Kolodner, I. I.: Heavy rotating string—a nonlinear eigenvalue problem. Comm. Pure Appl. Math. 8, 395–408 (1955). (25).CrossRefMATHMathSciNetGoogle Scholar
  53. Kovari, K.: Räumliche Verzweigungsprobleme des dünnen elastischen Stabes mit endlichen Verformungen. Ing.-Arch. 37, 393–416 (1969). (26).CrossRefMATHGoogle Scholar
  54. Krasnosel’ska, M. A.: Topological methods in the theory of nonlinear integral equations [in Russian]. Moscow: Gostekhteorizdat. (English transi, by A. H. Armstrong, 1964. Oxford: Pergamon.) 1956. (20, 25).Google Scholar
  55. Krasnosel’ska, M. A., G. M. Va’inikko, P. P. Zabre’iko, Ya. B. Rutitskh, and V. Ya. Stetsenko: Approximate solutions of operator equations [in Russian]. Moscow: Izdat. Nauka 1969. (11).Google Scholar
  56. Laws, N.: A simple dipolar curve. Int. J. Engng. Sci. 5, 653–661 (1967a). (17).CrossRefGoogle Scholar
  57. Laws, N.: A theory of elastic-plastic rods. Quart. J. Mech. Appl. Math. 20, 167–181 (1967b). (16).CrossRefMATHGoogle Scholar
  58. Leipholz, H.: Stabilitätstheorie. Stuttgart: Teubner. (English translation by Scientific Translation Service, 1970: Stability theory. New York: Academic Press.) 1968. (24, 28).Google Scholar
  59. Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Paris: Dunod, Gauthier-Villars 1969. (11).MATHGoogle Scholar
  60. Love, A. E. H.: A treatise on the mathematical theory of elasticity, eds. 1-4. Cambridge: University Press. [Reprinted (1944). New York: Dover Publications.] 1892-1927. (11, 12, 13, 15, 24, 26).Google Scholar
  61. MacCamy, R. C, and V. J. Mizel: Existence and nonexistence in the large of solutions of quasilinear wave equations. Arch. Rational Mech. Anal. 25, 299–320 (1967). (28).MATHMathSciNetGoogle Scholar
  62. Medick, M. A.: One dimensional theories of wave propagation and vibrations in elastic bars of rectangular cross-section. J. Appl. Mech. 33, 489–495 (1966). (11, 12).CrossRefMathSciNetGoogle Scholar
  63. Meissonnier, B.: Étude de quelques figures d’équilibre d’une ligne hyperélastique et considérée comme un milieu orienté. J. Mécanique 4, 423–437 (1965). (15, 24).Google Scholar
  64. Mettler, E.: Combination resonances in mechanical systems under harmonic excitation. Proc. Fourth Conf. on Nonlinear Oscillations, Prague, 1967, p. 51-70. Prague: Academia 1968. (28).Google Scholar
  65. Mindlin, R. D., and G. Herrmann: A one-dimensional theory of compressional waves in an elastic rod. Proc. First U.S. Natl. Cong. Appl. Mech. Chicago, 1951, p. 187–191. New York: A.S.M.E. 1952. (11, 12).Google Scholar
  66. Mindlin, R. D., and H. D. McNiven: Axially symmetric waves in elastic rods. J. Appl. Mech. 27, 145–151 (1960). (11,12).CrossRefMATHMathSciNetGoogle Scholar
  67. Morrey, C. B.: Multiple integrals in the calculus of variations. Berlin-Heidelberg-New York: Springer 1966. (19, 22).MATHGoogle Scholar
  68. Naghdi, P. M., and R. P. Nordgren: On the nonlinear theory of elastic shells under the Kirchhoff hypothesis. Quart. Appl. Math. 21, 49–59 (1963). (9).MATHMathSciNetGoogle Scholar
  69. Nariboli, G. A.: Asymptotic theory of wave motion in rods (longitudinal wave-motion). Z. Angew. Math. Mech. 49, 525–531 (1969). (13).CrossRefMATHGoogle Scholar
  70. Nemat-Nasser, S.: Thermoelastic stability under general loads. Appl. Mech. Revs. 23, 615–624 (1970). (24, 28).Google Scholar
  71. Niordsen, F. I.: On the optimal design of a vibrating beam. Quart. Appl. Math. 23, 47–53 (1965). (27).MATHMathSciNetGoogle Scholar
  72. Novozhilov, V. V.: Foundations of the nonlinear theory of elasticity [in Russian]. Moscow: Gostekhizdat. (English translation by F. Bagemihl, H. Komm, and W. Seidel, 1953. Rochester: Graylock.) 1948. (11, 12).Google Scholar
  73. Novozhilov, V. V., and L. I. Slepian: On St. Venant’s principle in the dynamics of beams. Prikl. Mat. Mekh. 29, 261-281. English translation in J. Appl. Math. Mech. 29, 293–315 (1965). (11).MATHMathSciNetGoogle Scholar
  74. Odeh, F., and I. Tadjbakhsh: A nonlinear eigenvalue problem for rotating rods. Arch. Rational Mech. Anal. 20, 81–94 (1965). (25).MATHMathSciNetGoogle Scholar
  75. Olmstead, W. E.: On the preclusion of buckling of a nonlinearly elastic rod. To appear (1971). (24).Google Scholar
  76. Panovko, Ya. G., and I.I. Gubanova: Stability and vibrations of elastic systems, eds. 1, 2 [in Russian]. Moscow: Izdat. Nauka. (English translation by C. V. Larrick, 1965. New York: Consultants Bureau.) 1964-1967. (24).Google Scholar
  77. Petryshyn, W. V.: Remarks on approximation-solvability of nonlinear functional equations. Arch. Rational Mech. Anal. 26, 43–49 (1967). (11).MATHMathSciNetGoogle Scholar
  78. Pimbley, G. H.: Eigenfunction branches of nonlinear operators and their bifurcations. Berlin-Heidelberg-New York: Springer 1969. (25).CrossRefMATHGoogle Scholar
  79. Popov, E. P.: Nonlinear problems of the statics of thin rods [in Russian]. Leningrad: Gostekh-teorizdat 1948. (26).Google Scholar
  80. Prodi, G.: Problemi di diramazione per equazioni funzionali. Boll. Un. Mat. Ital. (3) 22, 413–433 (1967). (25).MATHMathSciNetGoogle Scholar
  81. Reiss, E. L.: Column buckling—an elementary example of bifurcation. In: Bifurcation theory and nonlinear eigenvalue problems, ed. by J. B. Keller and S. Antman, p. 1–16. New York: Benjamin 1969. (24).Google Scholar
  82. Reiss, E. L.: To appear (1972). (13).Google Scholar
  83. Reiss, E. L., and B. J. Matkowsky: Nonlinear dynamic buckling of a compressed elastic column. Quart. Appl. Math. (1971) (to appear). (28).Google Scholar
  84. Rothe, E.: Weak topology and the calculus of variations. In: C.I.M.E. course, Calculus of variations, classical and modern, 1966. Rome: Cremonese 1968. (22).Google Scholar
  85. Shack, W. J.: On linear viscoelastic rods. Int. J. Solids Structures 6, 1–20 (1970). (16).CrossRefMATHGoogle Scholar
  86. Sheu, C. Y., and W. Prager: Recent developments in optimal structural design. Appl. Mech. Rev. 21, 985–992 (1968). (27).Google Scholar
  87. Stakgold, I.: Branching of solutions of nonlinear equations. S.I.A.M. Rev. (1971) (to appear). (25, 26).Google Scholar
  88. Tadjbakhsh, I.: The variational theory of the plane motion of the extensible elastica. Int. J. Engng. Sci. 4, 433–450 (1966). (15, 17, 28).CrossRefMATHGoogle Scholar
  89. Tadjbakhsh, I.: An optimal design problem for the nonlinear elastica. S.I.A.M. J. Appl. Math. 16, 964–972 (1968). (27).CrossRefMATHGoogle Scholar
  90. Tadjbakhsh, I., and J. B. Keller: Strongest columns and isoperimetric inequalities for eigenvalues. J. Appl. Mech. 29, 157–164 (1962). (27).MathSciNetGoogle Scholar
  91. Tadjbakhsh, I., and F. Odeh: Equilibrium states of elastic rings. J. Math. Anal. Appl. 18, 59–74 (1967). (15, 21, 25, 26).CrossRefMATHMathSciNetGoogle Scholar
  92. Timoshenko, S. P.: On the correction for shear of the differential equations for transverse vibrations of prismatic bars. Phil. Mag. (6) 41, 744–746 (1921). (11).CrossRefGoogle Scholar
  93. Timoshenko, S. P.: History of the strength of materials with a brief account of the history of the theory of elasticity and the theory of structures. New York: McGraw-Hill 1953. (12).Google Scholar
  94. Todhunter, I., and K. Pearson: A history of the theory of elasticity and of the strength of materials. Cambridge: University Press. 1886, 1893. [Reprinted (1960). New York: Dover.] (12).Google Scholar
  95. Truesdell, C.: A new chapter in the theory of the elastica. Proc. First Midwestern Conf. on Solid Mech. 1953, 52–55 (1954). (24).MathSciNetGoogle Scholar
  96. Truesdell, C.: The rational mechanics of materials—past, present, future. Appl. Mech. Rev. 12, 75–80 (1959). Corrected reprint in: Applied Mechanics Surveys. Washington: Spartan Books 1966. (12).MathSciNetGoogle Scholar
  97. Truesdell, C.: The rational mechanics of flexible or elastic bodies, 1638–1788. L. Euleri Opera Omnia (2) 112. Zürich: Füssli 1960. (12, 15).Google Scholar
  98. Vainberg, M. M.: Variational methods for the study of nonlinear operators [in Russian]. Moscow: Gostekhteorizdat. (English translation by A. Feinstein, 1964. San Francisco: Holden-Day.) 1956. (20, 22).Google Scholar
  99. Vainberg, M. M., and V. A. Trenogin: The methods of Lyapunov and Schmidt in the theory of nonlinear equations and their further development. Usp. Mat. Nauk 17 (2), 13-75. [English transi. (1962) in Russian Math. Surveys. 17, 1-60.] (1962). (25).Google Scholar
  100. Vainberg, M. M.: The theory of branching of solutions of nonlinear equations [in Russian]. Moscow: Izdat. Nauka 1969. (25).Google Scholar
  101. Vol’mir, A. S.: Stability of deformable systems, eds. 1, 2 [in Russian]. Moscow: Izdat. Nauka 1963-1967. (24, 28).Google Scholar
  102. Volterra, E.: Equations of motion for curved elastic bars by the use of the “method of internal constraints”. Ing.-Arch. 23, 402–409 (1955). (5, 11, 12).CrossRefMATHMathSciNetGoogle Scholar
  103. Volterra, E.: Equations of motion for curved and twisted elastic bars deduced by the use of the “method of internal constraints”. Ing.-Arch. 24, 392–400 (1956). (5, 11, 12).CrossRefMATHMathSciNetGoogle Scholar
  104. Volterra, E.: Second approximation of the method of internal constraints and its applications. Int. J. Mech. Sci. 3, 47–67 (1961). (5, 11, 12).CrossRefGoogle Scholar
  105. Wasiutynski, Z., and A. Brandt: The present state of knowledge in the field of optimal design of structures. Appl. Mech. Rev. 16, 341–350 (1963). (27).Google Scholar
  106. Woodall, S. R.: On the large amplitude oscillations of a thin elastic beam. Int. J. Nonlin. Mech. 1, 217–238 (1966). (28).CrossRefMATHGoogle Scholar
  107. Whitman, A. B., and C. N. De Silva: A dynamical theory of elastic directed curves. Z. Angew. Math. Phys. 20, 200–212 (1969). (15).CrossRefMATHGoogle Scholar
  108. Whitman, A. B., and C. N. De Silva: Dynamics and stability of elastic Cosserat curves. Int. J. Solids Structures 6, 411–422 (1970). (28).CrossRefMATHGoogle Scholar
  109. Ziegler, H.: Principles of structural stability. Waltham: Ginn/Blaisdell 1968. (24).Google Scholar

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© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • Stuart S. Antman
    • 1
  1. 1.New York UniversityNew YorkUSA

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