Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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SINGULAR PERIODIC SOLUTIONS OF A CLASS OF ELASTODYNAMICS EQUATIONS

  • Yuan, Xuegang (Department of Engineering Mechanics, Dalian University of Technology) ;
  • Zhang, Yabo (Department of Mathematics and Information Science, Tianjin University of Technology)
  • Published : 2009.05.31
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Abstract

A second order nonlinear ordinary differential equation is obtained by solving the initial-boundary value problem of a class of elas-todynamics equations, which models the radially symmetric motion of a incompressible hyper-elastic solid sphere under a suddenly applied surface tensile load. Some new conclusions are presented. All existence conditions of nonzero solutions of the ordinary differential equation, which describes cavity formation and motion in the interior of the sphere, are presented. It is proved that the differential equation has singular periodic solutions only when the surface tensile load exceeds a critical value, in this case, a cavity would form in the interior of the sphere and the motion of the cavity with time would present a class of singular periodic oscillations, otherwise, the sphere remains a solid one. To better understand the results obtained in this paper, the modified Varga material is considered simultaneously as an example, and numerical simulations are given.

Keywords

References

  1. J.M. Ball, Discontinuous equilibrium solutions and cavitations in nonlinear elasticity, Philos Trans Roy Soc Lond Ser A306 (1982), 557-611.
  2. M. -S Chou-Wang and C.O. Horgan, Void nucleation and growth for a class of incom- pressible nonlinearly elastic materials, Int. J. Solid Structures 25 (1989), 1239-1254. https://doi.org/10.1016/0020-7683(89)90088-7
  3. C.O. Horgan and D.A. Polignone , Cavitation in nonlinear elastic solids: A review, Ap-plied Mechanics Review 48 (1995) 471-485. https://doi.org/10.1115/1.3005108
  4. X. C. Shang and C. J. Cheng, Exact Solution for cavitated bifurcation for compressible hyper-elastic materials, Int. J. Engin. Sci. 39 (2001) 1101-1117. https://doi.org/10.1016/S0020-7225(00)00090-2
  5. J. S. Ren and C. J. Cheng, Cavity bifurcation for incompressible hyper-elastic material. Applied Mathematics and Mechanics (English Edition) 23 (2002) 881-888. https://doi.org/10.1007/BF02437792
  6. J. S. Ren and C. J. Cheng, Bifurcation of cavitation solutions for incompressible trans- versely isotropic hyper-elastic materials, J. Engin. Math. 44 (2002) 245-257. https://doi.org/10.1023/A:1020910210880
  7. X. G. Yuan, Z. Y. Zhu and C. J. Cheng, Qualitative study of cavity bifurcation for a class of incompressible generalized neo-Hookean spheres, Applied Mathematics and Mechanics (English Edition) 26 (2005) 185-194. https://doi.org/10.1007/BF02438240
  8. X. G. Yuan, Z. Y. Zhu and C. J. Cheng, Study on cavitated bifurcation problem for spheres composed of hyper-elastic materials, J. Engin. Math. 51 (2005) 15-34. https://doi.org/10.1007/s10665-004-1242-2
  9. J. K. Knowles, Large amplitude oscillations of a tube of incompressible elastic material, Q. Appl. Math. 18(1960), 71-77.
  10. J. K. Knowles, On a class of oscillations in the finite deformation theory of elasticity, J. Appl. Math. 29(1962), 283-286.
  11. C. Calderer, The dynamical behavior of nonlinear elastic spherical shells. J. lasticity 13 (1983) 17-47. https://doi.org/10.1007/BF00041312
  12. E. Verron, R. E. Khayat, A. Derdouri and B. Peseux, Dynamic inflation of hyperelastic spherical membranes, J. Rheo. 43 (1999), 1083-1097. https://doi.org/10.1122/1.551017
  13. N. Roussos and D. P. Mason, Radial oscillations of this cylindrical and spherical shell, Commu. Non. Sci. Num. Simu. 10 (2005), 139-150. https://doi.org/10.1016/S1007-5704(03)00112-6
  14. X. G. Yuan, Z. Y. Zhu and C. J. Cheng, Qualitative analysis of dynamical behavior for an imperfect incompressible neo-Hookean spherical shell, Applied Mathematics and Mechanics (English Edition) 26 (2005), 973-981. https://doi.org/10.1007/BF02466409
  15. X. G. Yuan, Z. Y. Zhu and C. J. Cheng, Dynamical analysis of cavitation for a transversely isotropic incompre-ssible hyper-elastic medium: periodic motion of a pre-existing microvoid, Int. J. Nonlinear Mech. 42(2007), 442-449. https://doi.org/10.1016/j.ijnonlinmec.2007.01.006
  16. M. -S Chou-Wang and C.O. Horgan, Cavitation in nonlinearly elastodynamics for neo- Hookean materials, Int. J. Engin. Sci. 27 (1989) 967-973. https://doi.org/10.1016/0020-7225(89)90037-2