Asymptotic solutions for finite deformation of thin shells of revolution with a small circular hole
Authors:
Hubertus J. Weinitschke and Charles G. Lange
Journal:
Quart. Appl. Math. 45 (1987), 401-417
MSC:
Primary 73L99
DOI:
https://doi.org/10.1090/qam/910449
MathSciNet review:
910449
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Abstract: The method of matched asymptotic expansions is used to describe the finite deformation of thin shells of revolution with a small circular hole at the apex. The loading is assumed to be a rotationally symmetric, smoothly varying normal pressure. The mathematical problem is of singular perturbation type characterized by a boundary layer region at the inner edge of the small hole. The analytical results are compared with numerical approximations, and formulas for the stress concentration factors at the hole are presented.
F. Y. M. Wan and H. J. Weinitschke, A boundary layer solution for some nonlinear elastic membrane problems, Report 109, Institute of Applied Mathematics, University of Erlangen (November 1984); revised as Technical Report No. 86–6, Department of Applied Mathematics, University of Washington, Seattle (August 1985) (accepted by Z. Angew. Math. Phys.)
- 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
H. J. Weinitschke, On the stability problem for shallow spherical shells, J. Math. Phys. 38, 209–231 (1960)
- Norman Wagner, Existence theorem for a non-linear boundary value problem in ordinary differential equations, Contributions to Differential Equations 3 (1964), 325–336. MR 162006
- J. Kevorkian and Julian D. Cole, Perturbation methods in applied mathematics, Applied Mathematical Sciences, vol. 34, Springer-Verlag, New York-Berlin, 1981. MR 608029
- U. Ascher, J. Christiansen, and R. D. Russell, A collocation solver for mixed order systems of boundary value problems, Math. Comp. 33 (1979), no. 146, 659–679. MR 521281, DOI https://doi.org/10.1090/S0025-5718-1979-0521281-7
- Hans Grabmüller and Hubertus J. Weinitschke, Finite displacements of annular elastic membranes, J. Elasticity 16 (1986), no. 2, 135–147. MR 849669, DOI https://doi.org/10.1007/BF00043581
- Eleazer Bromberg, Non-linear bending of a circular plate under normal pressure, Comm. Pure Appl. Math. 9 (1956), 633–659. MR 88199, DOI https://doi.org/10.1002/cpa.3160090402
F. Y. M. Wan and H. J. Weinitschke, A boundary layer solution for some nonlinear elastic membrane problems, Report 109, Institute of Applied Mathematics, University of Erlangen (November 1984); revised as Technical Report No. 86–6, Department of Applied Mathematics, University of Washington, Seattle (August 1985) (accepted by Z. Angew. Math. Phys.)
E. Reissner, On axisymmetric deformation of thin shells of revolution, Proc. Sympos. Appl. Math. 3, 27–52 (1950)
H. J. Weinitschke, On the stability problem for shallow spherical shells, J. Math. Phys. 38, 209–231 (1960)
N. Wagner, Existence theorem for a nonlinear boundary value problem in ordinary differential equations, Contrib. Differential Equations 3, 325–336 (1965)
J. Kevorkian and J. D. Cole, Perturbation methods in applied mathematics, Springer-Verlag, New York, 1981
U. Ascher, J. Christiansen, and R. D. Russell, A collocation solver for mixed order systems of boundary value problems, Math. Comput. 33, 659–679 (1979)
H. Grabmuller and H. J. Weinitschke, Finite displacements of annular elastic membranes, Report 115, Institute of Applied Mathematics, University of Erlangen, (December 1984); J. Elasticity 16, 135–147 (1986)
E. Bromberg, Non-linear bending of a circular plate under normal pressure, Comm. Pure Appl. Math. 9. 634–659 (1956)
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Article copyright:
© Copyright 1987
American Mathematical Society
