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Asymptotic theory and analysis for displacements and stress distribution in nonlinear elastic straight slender rods. (English) Zbl 0653.73010
From the authors’ abstract: “An asymptotic expansion method is applied to nonlinear three-dimensional elastic straight slender rods. Nonlinear ordinary differential equations for approximate displacements and explicit formulas for approximate stress distributions are obtained. Mathematical properties of these models are studied.”
This is a comprehensive paper written in clear and rigorous style. Examples include: rod clamped at both extremities; rod clamped in mean at both extremities; and rod clamped at one extremity (end). The current theory is discussed in light of strength of materials, showing that the general results are consistent with those of the simpler approach.
This paper should appeal to engineers and scientists concerned with current developments in the theory of ‘elastica’.
Reviewer: R.H.Lance

MSC:
74B20 Nonlinear elasticity
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
34E05 Asymptotic expansions of solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
74G70 Stress concentrations, singularities in solid mechanics
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[1] Adams, R.A.:Sobolev Spaces. Academic Press, New York (1975). · Zbl 0314.46030
[2] Aganovic, I. and Z. Tutek: A justification of the one-dimensional linear model of elastic beam.Math. Meth. Appl. Sci. 8 (1986) 1-14. · Zbl 0617.35136
[3] Antman, S.S.: The Theory of Rods.Handbuch der Physik, Vol. VIa/2, Springer-Verlag, Berlin (1972).
[4] Antman, S.S. and K.B. Jordan: Qualitative aspects of the spatial deformation of nonlinearly elastic rods.Proc. Roy. Soc. Edinburgh 73A (1975) 85-105. · Zbl 0351.73076
[5] Antman, S.S. and C.S. Kenney: Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity.Arch. Rat. Mech. Anal. 76 (1981) 339-354. · Zbl 0472.73036
[6] Berdichevskii, V.I. and L.A. Starosel’skii: On the theory of curvilinear Timoshenko-type rods.PMM U.S.S.R. 47 (1983) 809-817.
[7] Bermudez, A. and J.M. Viaño: Une justification des équations de la thermoélasticité des poutres à section variable par des méthodes asymptotiques.R.A.I.R.O. Analyse Numérique 18, (1984) 347-376. · Zbl 0572.73053
[8] Caillerie, D.: The effect of a thin inclusion of high rigidity in an elastic body.Math. Meth. Appl. Sci. 2 (1980) 251-270. · Zbl 0446.73014
[9] Ciarlet, P.G.:Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity. North Holland (1988). · Zbl 0648.73014
[10] Ciarlet, P.G.: Recent progresses in the two-dimensional approximation of three-dimensional plate models in nonlinear elasticity. InNumerical Approximation of Partial Differential Equations (E. L. Ortiz, editor), pp. 3-19, North-Holland (1987).
[11] Ciarlet, P.G. and P. Destuynder: A justification of a nonlinear model in plate theory.Comp. Meth. Appl. Mech. Engrg. 17/18 (1979) 227-258. · Zbl 0405.73050
[12] Cimetière, A., G. Geymonat, H. Le Dret, A. Raoult and Z. Tutek: Une dérivation d’un modèle non linéaire de poutre à partir de l’élasticité tridimensionnelle.C.R. Acad. Sc. Paris t. 302, série I, (1986) 697-700. · Zbl 0593.73046
[13] Davet, J.L.: Justification de modèles de plaques non linéaires pour des lois de comportement générales.Modélisation Mathématique et Analyse Numérique 20 (1986) 225-249. · Zbl 0634.73048
[14] Destuynder, P.:Une théorie asymptotique des plaques minces en élasticité linéaire. R.M.A. Masson, Paris (1986). · Zbl 0627.73064
[15] Duvaut, G. and J.L. Lions:Inequalities in Mechanics and Physics. Springer-Verlag, Berlin (1976).
[16] Fichera, G.: Remarks on Saint Venant’s principle.Rendiconti di Matematica, serie VI, 12, (1979) 181-200. · Zbl 0443.73002
[17] Geymonat, G., F. Krasucki and J.J. Marigo: Stress distribution in anisotropic elastic composite beams. InApplications of Multiple Scalings in Mechanics (P. G. Ciarlet and E. Sanchez-Palencia, editors) pp. 118-133, R.M.A. Masson, Paris (1987). · Zbl 0645.73029
[18] Gilbert, R.P., G.C. Hsiao and M. Schneider: The two-dimensional linear orthotropic plate.Applicable Analysis. 15 (1983) 147-169. · Zbl 0516.73053
[19] Hay, G.E.: The finite displacement of thin rods.Trans. Am. Math. Soc. 51 (1942) 65-102. · Zbl 0061.42206
[20] Horgan, C. O. and J.K. Knowles: Recent Developments Concerning Saint Venant’s Principle.Advances Appl. Mech. 23 (1983) 179-269. · Zbl 0569.73010
[21] Kato, T.:Perturbation Theory for Linear Operators. Springer-Verlag, Berlin (1966). · Zbl 0148.12601
[22] Landau, L.D. and E.M. Lifchitz:Theory of Elasticity. Pergamon Press, New York (1970). 2nd edn., Elmsford.
[23] Lions, J.L. and E. Magenes:Problèmes aux limites non homogènes et applications. Vol. 1, Dunod, Paris (1968). · Zbl 0165.10801
[24] Love, A.E.:A Treatise on the Mathematical Theory of Elasticity. (1927) 4th edn., Cambridge University Press (Dover reprint, 1944). · JFM 53.0752.01
[25] Pleus, P. and M. Sayir: A second order theory for large deflections of slender beams.Z.A.M.P. 34 (1983) 192-217. · Zbl 0514.73041
[26] Rigolot, A.: Approximation asymptotique d’un cylindre infiniment élancé, élastique, non linéaire, par un milieu curviligne.C.R. Acad. Sc. Paris. t 276, Série A (1973) 559-562. · Zbl 0282.73021
[27] Rigolot, A.: Sur la déformation due à l’effort tranchant dans les poutres droites élastiques.Annales de l’Institut Technique du Bâtiment et des Travaux Publics. 363 (1978) 34-52.
[28] Rigolot, A.: Sur une théorie asymptotique des poutres. Thèse de Doctorat d’Etat. Université Paris 6 (1976).
[29] Sokolnikoff, I.S.:Mathematical Theory of Elasticity. MacGraw Hill (1956). · Zbl 0070.41104
[30] Timoshenko, S. P.:Strength of Materials. Van Nostrand, 2nd edn. (1941). · JFM 67.0794.03
[31] Toupin, R. A.: Saint Venant’s Principle.Arch Rat. Mech. Anal. 18 (1965) 83-96. · Zbl 0203.26803
[32] Truesdell, C. and W. Noll: The Nonlinear Field Theories of Mechanics.Handbuch der Physik Vol. III/3, Springer-Verlag, Berlin (1965). · Zbl 0779.73004
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