Nonlinear theories for thin shells

Author:
J. Lyell Sanders Jr.

Journal:
Quart. Appl. Math. **21** (1963), 21-36

MSC:
Primary 73.99

DOI:
https://doi.org/10.1090/qam/147023

MathSciNet review:
147023

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Abstract: Strain-displacement relations for thin shells valid for large displacements are derived. With these as a starting point approximate strain-displacement relations and equilibrium equations are derived by making certain simplifying assumptions. In particular the middle surface strains are assumed small and the rotations are assumed moderately small. The resulting equations are suitable as a basis for stability investigations or other problems in which the effects of deformation on equilibrium cannot be ignored, but in which the rotations are not too large.

**[1]**K. Marguerre,*Zur Theorie der gekrümmten Platte grosser Formänderung*, Proc. of Fifth International Congress of Appl. Mech., Wiley and Sons, 1938, pp. 93-101**[2]**L. H. Donnell,*Stability of thin-walled tubes under torsion*, N. A. C. A. TR 479, 1934**[3]**Kh. M. Mushtari and K. Z. Galimov,*Non-linear theory of thin elastic shells*, Translated by J. Morgenstern, J. J. Schorr-Kon and PST Staff, Published for The National Science Foundation, Washington, D.C. and The National Aeronautics and Space Administration by The Israel Program for Scientific Translations, Jerusalem, 1961. MR**0134005****[4]**Eric Reissner,*On axisymmetrical deformations of thin shells of revolution*, Proc. Symposia Appl. Math. v. 3, McGraw-Hill Book Co., New York, N. Y., 1950, pp. 27–52. MR**0039489****[5]**J. L. Synge and W. Z. Chien,*The intrinsic theory of elastic shells and plates*, Theodore von Kármán Anniversary Volume, University of California Press, Berkeley, Calif., 1941, pp. 103–120. MR**0004596****[6]**Wei-Zang Chien,*The intrinsic theory of thin shells and plates. I. General theory*, Quart. Appl. Math.**1**(1944), 297–327. MR**0009744**, https://doi.org/10.1090/S0033-569X-1944-09744-3**[7]**Wei-Zang Chien,*The intrinsic theory of thin shells and plates. III. Application to thin shells*, Quart. Appl. Math.**2**(1944), 120–135. MR**0010867**, https://doi.org/10.1090/qam/10867**[8]**J. L. Ericksen and C. Truesdell,*Exact theory of stress and strain in rods and shells*, Arch. Rational Mech. Anal.**1**(1958), 295–323. MR**0099135**, https://doi.org/10.1007/BF00298012**[9]**A. E. Green and W. Zerna,*The equilibrium of thin elastic shells*, Quart. J. Mech. Appl. Math.**3**(1950), 9–22. MR**0035175**, https://doi.org/10.1093/qjmam/3.1.9**[10]**F. B. Hildebrand, E. Reissner, and G. B. Thomas,*Notes on the foundations of the theory of small displacements of orthotropic shells*, Tech. Notes Nat. Adv. Comm. Aeronaut.,**1949**(1949), no. 1833, 59. MR**0030886****[11]**A. E. Green and W. Zerna,*Theoretical elasticity*, Oxford, at the Clarendon Press, 1954. MR**0064598****[12]**V. V. Novozhilov,*Foundations of the nonlinear theory of elasticity*, Graylock Press, Rochester, N. Y., 1953. MR**0054465****[13]**R. W. Leonard,*Nonlinear first approximation thin shell and membrane theory*, Thesis, Virginia Polytechnic Inst., June 1961**[14]**W. T. Koiter,*A consistent first approximation in the general theory of thin elastic shells*, Proc. Sympos. Thin Elastic Shells (Delft, 1959) North-Holland, Amsterdam, 1960, pp. 12–33. MR**0142241****[15]**A. E. H. Love,*A treatise on the Mathematical Theory of Elasticity*, Dover Publications, New York, 1944. Fourth Ed. MR**0010851****[16]**V. V. Novožilov,*The theory of thin shells*, Translated by P. G. Lowe. Edited by J. R. M. Radok, P. Noordhoff Ltd., Groningen, 1959. MR**0107406****[17]**J. L. Sanders,*An improved first approximation theory for thin shells*, N. A. S. A. Report 24, June 1959**[18]**P. M. Naghdi and J. G. Berry,*On the equations of motion of cylindrical shells*, J. Appl. Mech.**21**(1954), 160–166. MR**0063246****[19]**James K. Knowles and Eric Reissner,*Note on the stress-strain relations for thin elastic shells*, J. Math. and Phys.**37**(1958), 269–282. MR**0134930****[20]**J. W. Cohen,*The inadequacy of the classical stress-strain relations for the right helicoidal shell*, Proc. of the I. U. T. A. M. Symposium on the Theory of Thin Elastic Shells, North-Holland Pub. Co., 1959, pp. 415-433

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DOI:
https://doi.org/10.1090/qam/147023

Article copyright:
© Copyright 1963
American Mathematical Society