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A theory of shells with small strain accompanied by moderate rotation

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This paper is concerned with a constrained theory of shells in the presence of small strain accompanied by moderate rotation. The constrained theory accounts for the effect of transverse normal strain and includes, of course, the special case (corresponding to the Kirchhoff-Love theory of shells) in which the effect of transverse normal strain is absent. After precise estimates for (local) moderate rotation and relative displacement gradients in terms of infinitesimal strain have been effected, a complete theory is formulated with the use of linear constitutive equations. The nature of the complete theory is further examined when initially the shell-like body is a plate; and it is shown that our kinematical formulae (strain-displacement relations), as well as the relevant differential equations of the theory in the absence of the effect of transverse normal strain, systematically reduce to those used in the von Kármán plate equations. Also, in the light of the present results, an assessment of kinematical aspects of previously developed theories of shells undergoing small strain and moderate rotation is indicated.

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  1. Casey, J., & P. M. Naghdi, 1981, An invariant infinitesimal theory of motions superposed on a given motion. Arch. Rational Mech. Anal. 76, 335–391.

  2. Eisenhart, L. P., 1947, An introduction to differential geometry. Princeton University Press, Princeton.

  3. Green, A. E., & P. M. Naghdi, 1974, On the derivation of shell theories by direct approach. J. Appl. Mech. 41, 173–176.

  4. Green, A. E., P. M. Naghdi & M. L. Wenner, 1971, Linear theory of Cosserat surface and elastic plates of variable thickness. Proc. Camb. Phil. Soc. 69, 227–254.

  5. Kármán, Th. v., 1910, Festigkeitsprobleme im Maschinenbau. Encyklopädie der Mathematischen Wissenschaften, Vol. 4/4, pp. 311–385 = Collected Works 1, 141–207.

  6. Koiter, W., 1966, On the nonlinear theory of thin elastic shells, Parts I, II and III. Proc. Kon. Ned. Ak. Wet., B 69, 1–54.

  7. Naghdi, P. M., 1972, The theory of shells and plates. In S. Flügge's Handbuch der Physik, Vol. VIa/2 (edited by C. Truesdell), pp. 425–640, Springer-Verlag, Berlin, Heidelberg, New York.

  8. Naghdi, P. M., 1977, Shell theory from the standpoint of finite elasticity. Proc. Symp. on “Finite Elasticity” (edited by R. S. Rivlin), AMD-Vol. 27, Amer. Soc. Mech. Eng., pp. 77–84.

  9. Naghdi, P. M., 1982, Finite deformation of elastic rods and shells. Proc. IUTAM Symp. on “Finite Elasticity” (Bethlehem, PA 1980, edited by D. E. Carlson & R. T. Shield), pp. 47–103, Martinus Nijhoff Publishers, The Hague, Netherlands.

  10. Naghdi, P. M., & R. P. Nordgren, 1963, On the nonlinear theory of elastic shells under the Kirchhoff hypothesis. Quart. Appl. Math. 21, 49–59.

  11. Naghdi, P. M., & L. Vongsarnpigoon, 1982, Small strain accompanied by moderate rotation. Arch. Rational Mech. Anal. 80, 263–294.

  12. Reissner, E., 1950, On axisymmetrical deformation of thin shells of revolution. Proc. Symp. Appl. Math. 3, 27–52.

  13. Sanders, J. L., 1963, Nonlinear theories for thin shells. Quart. Appl. Math. 21, 21–36.

  14. Shield, R. T., 1973, The rotation associated with large strain. SIAM J. Appl. Math. 25, 483–491.

  15. Timoshenko, S., 1940, Theory of plates and shells. McGraw-Hill, New York.

  16. Truesdell, C., & R. A. Toupin, 1960, The classical field theories. In S. Flügge's Handbuch der Physik, Vol. III/1, pp. 226–793, Springer-Verlag, Berlin, Göttingen, Heidelberg.

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Communicated by R. A. Toupin

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Naghdi, P.M., Vongsarnpigoon, L. A theory of shells with small strain accompanied by moderate rotation. Arch. Rational Mech. Anal. 83, 245–283 (1983). https://doi.org/10.1007/BF00251511

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