Nonlinear multiple-scale solution of a cylindrical shell

Authors:
James R. Stafford and Adolf T. Hsu

Journal:
Quart. Appl. Math. **30** (1973), 491-499

DOI:
https://doi.org/10.1090/qam/99718

MathSciNet review:
QAM99718

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Abstract | References | Additional Information

Abstract: A multiple-scale perturbation technique is used with a two-parameter expansion to study the asymptotic solution of Reissner's axisymmetric finite-deformation equations for a circular cylindrical shell with an edge-bending moment load. Beyond the assumptions of Reissner's differential equations, it is assumed that (1) the rotations of a shell element are finite but not excessively large, (2) thickness variations in the differential equations are of order one and (3) the boundary-layer behavior is of the linear bending type to a first approximation. An asymptotic solution is then found which is uniformly valid in that it contains boundary-layer effects and corrections for extending the analysis into the shell's interior. Upon considering certain limits, it is observed that the solution contains well-established linear and nonlinear approximations to the solution.

**[1]**M. Van Dyke,*Perturbation methods in fluid dynamics*, 1st edition, Academic Press, New York, 1969**[2]**James Alan Cochran,*PROBLEMS IN SINGULAR PERTURBATION THEORY*, ProQuest LLC, Ann Arbor, MI, 1962. Thesis (Ph.D.)–Stanford University. MR**2613484****[3]**J. R. Stafford,*A multiple scale solution for circular cylindrical shells*, Int. J. Solids Structures**5**, 855-861 (1969)**[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. D. Cole and J. Kevorkian,*Uniformly valid asymptotic approximations for certain non-linear differential equations*, Internat. Sympos. Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 113–120. MR**0147701****[6]**F. B. Hildebrand,*On asymptotic integration in shell theory*, Proc. Symposia Appl. Math. v. 3, McGraw-Hill Book Co., New York, N. Y., 1950, pp. 53–66. MR**0039490****[7]**T. J. Lardner,*Symmetric deformation of a circular cylindrical shell of variable thickness*, Zeit. Ange. Math. Phys.**19**, 270-277 (1968)**[8]**Eric Reissner and H. J. Weinitschke,*Finite pure bending of circular cylindrical tubes*, Quart. Appl. Math.**20**(1962/1963), 305–319. MR**0148283**, https://doi.org/10.1090/S0033-569X-1963-0148283-6**[9]**Eric Reissner,*On influence coefficients and non-linearity for thin shells of revolution*, J. Appl. Mech.**26**(1959), 69–72. MR**0101674**

Additional Information

DOI:
https://doi.org/10.1090/qam/99718

Article copyright:
© Copyright 1973
American Mathematical Society