Journal of Elasticity

, Volume 135, Issue 1–2, pp 73–89

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

• Stuart S. Antman
Article

## 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)
3. 3.
Antman, S.S., Schuricht, F.: Incompressibility in rod and shell theories. Math. Model. Numer. Anal. 33, 289–304 (1999)
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)
5. 5.
Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)
6. 6.
Ball, J.M.: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Q. J. Math. 28, 473–486 (1977)
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)
9. 9.
Calderer, M.C.: Finite time blow-up and stability properties of materials with fading memory. J. Differ. Equ. 63, 289–305 (1986)
10. 10.
Calderer, M.C.: The dynamic behavior of viscoelastic spherical shells. Math. Methods Appl. Sci. 9, 13–34 (1987)
11. 11.
Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3rd edn. Springer, Berlin (2010)
12. 12.
Evans, L.C.: Partial Differential Equations, 2nd edn. AMS, Providence (2010)
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)
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)
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)
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)
18. 18.
Knowles, J.K.: Large amplitude oscillations of a tube of incompressible elastic material. Q. Appl. Math. 18, 71–77 (1960)
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)
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)
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)
22. 22.
Rabier, P., Oden, J.T.: Bifurcation in Rotating Bodies. Masson, Paris (1989)
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)
24. 24.
Truesdell, C., Noll, W.: Non-linear Field Theories of Mechanics, 3rd edn. Springer, Berlin (2004)
25. 25.
Wang, C.-C., Truesdell, C.: Introduction to Rational Elasticity. Noordhoff, Groningen (1973)