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Journal of Elasticity

, Volume 135, Issue 1–2, pp 73–89 | Cite as

Rotationally Symmetric Motions and Their Blowup for Incompressible Nonlinearly Elastic and Viscoelastic Annuli

  • Stuart S. AntmanEmail author
Article
  • 45 Downloads

Abstract

This paper treats rotationally symmetric motions of incompressible, transversely isotropic, nonlinearly elastic and viscoelastic annuli subjected to the live loads of (time-dependent) centrifugal force and hydrostatic pressure. The incompressibility is responsible for some simplification of the analysis of such motions, which is nevertheless more complicated than that for radially symmetric motions. The theory is illustrated with blowup theorems, which have proofs different from those used for compressible media.

Keywords

Rotationally symmetric motion of nonlinearly elastic and viscoelastic annuli Blowup theorems 

Mathematics Subject Classification

34A34 35B44 35Q74 74A05 74B20 74D10 74H35 

Notes

Acknowledgement

I am indebted to a referee who discovered a serious gaffe in the first version of this work.

References

  1. 1.
    Antman, S.S.: Breathing oscillations of rotating nonlinearly elastic and viscoelastic rings. In: Durban, D., Givoli, D., Simmonds, J.G. (eds.) Advances in the Mechanics of Plates and Shells, pp. 1–16. Kluwer Academic, Dordrecht (2001) Google Scholar
  2. 2.
    Antman, S.S.: Nonlinear Problems of Elasticity, 2nd edn. Springer, New York (2005) zbMATHGoogle Scholar
  3. 3.
    Antman, S.S., Schuricht, F.: Incompressibility in rod and shell theories. Math. Model. Numer. Anal. 33, 289–304 (1999) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Antman, S.S., Ulusoy, S.: Blowup of solutions for the planar motions of rotating nonlinearly elastic rods. Int. J. Non-Linear Mech. 94, 28–35 (2017) ADSCrossRefGoogle Scholar
  5. 5.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ball, J.M.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. 28, 473–486 (1977) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ball, J.M.: Finite time blow-up in nonlinear problems. In: Crandall, M.G. (ed.) Nonlinear Evolution Equations, pp. 189–205. Academic Press, San Diego (1978) Google Scholar
  8. 8.
    Calderer, M.C.: The dynamic behavior of nonlinearly elastic spherical shells. J. Elast. 13, 17–47 (1983) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Calderer, M.C.: Finite time blow-up and stability properties of materials with fading memory. J. Differ. Equ. 63, 289–305 (1986) ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Calderer, M.C.: The dynamic behavior of viscoelastic spherical shells. Math. Methods Appl. Sci. 9, 13–34 (1987) ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3rd edn. Springer, Berlin (2010) CrossRefzbMATHGoogle Scholar
  12. 12.
    Evans, L.C.: Partial Differential Equations, 2nd edn. AMS, Providence (2010) zbMATHGoogle Scholar
  13. 13.
    Fosdick, R., Ketema, Y., Yu, J-H.: Dynamics of a viscoelastic spherical shell with a nonconvex strain energy function. Q. Appl. Math. 56, 221–244 (1998) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gradstein, L.S., Ryzhik, M.: Tables of Integrals, Series, and Products. Academic Press, San Diego (1980) Google Scholar
  15. 15.
    Guo, Z.-h., Solecki, R.: Free and forced finite-amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material. Arch. Mech. Stosow. 15, 427–433 (1963) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Knops, R.: Logarithmic convexity and other techniques applied to problems in continuum mechanics. In: Knops, R. (ed.) Symposium on Non-Well-Posed Problems and Logarithmic Convexity. Lect. Notes Math., vol. 316, pp. 31–54. Springer, Berlin (1973) CrossRefGoogle Scholar
  17. 17.
    Knops, R., Levine, H.A., Payne, L.E.: Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics. Arch. Ration. Mech. Anal. 55, 52–72 (1974) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Knowles, J.K.: Large amplitude oscillations of a tube of incompressible elastic material. Q. Appl. Math. 18, 71–77 (1960) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Knowles, J.K., Jakub, M.T.: Finite dynamic deformations of an incompressible elastic medium containing a spherical cavity. Arch. Ration. Mech. Anal. 18, 376–387 (1965) CrossRefzbMATHGoogle Scholar
  20. 20.
    Novozhilova, L., Pence, T.J., Urazhdin, S.: Exact solutions for axially varying three-dimensional twist motion in a neo-Hookean solid. Q. J. Mech. Appl. Math. 56, 123–138 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nowinski, J.L., Wang, S.D.: Finite radial oscillations of a spinning thick-walled cylinder. J. Acoust. Soc. Am. 40, 1548–1553 (1966) ADSCrossRefGoogle Scholar
  22. 22.
    Rabier, P., Oden, J.T.: Bifurcation in Rotating Bodies. Masson, Paris (1989) zbMATHGoogle Scholar
  23. 23.
    Stepanov, A.B., Antman, S.S.: Radially symmetric motions of nonlinearly viscoelastic bodies under live loads. Arch. Ration. Mech. Anal. 226, 1209–1247 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Truesdell, C., Noll, W.: Non-linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004) CrossRefzbMATHGoogle Scholar
  25. 25.
    Wang, C.-C., Truesdell, C.: Introduction to Rational Elasticity. Noordhoff, Groningen (1973) zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Institute for Physical Science and Technology, Institute for Systems ResearchUniversity of MarylandCollege ParkUSA

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