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Journal of Applied Mathematics
Volume 2012, Article ID 872161, 9 pages
Research Article

Nonlinear Periodic Oscillation of a Cylindrical Microvoid Centered at an Isotropic Incompressible Ogden Cylinder

1State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116024, China
2School of Science, Dalian Nationalities University, Dalian 116600, China

Received 31 August 2012; Accepted 5 December 2012

Academic Editor: Kale Oyedeji

Copyright © 2012 Wenzheng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study the dynamic mathematical model for an infinitely long cylinder composed of an isotropic incompressible Ogden material with a microvoid at its center, where the outer surface of the cylinder is subjected to a uniform radial tensile load. Using the incompressibility condition and the boundary conditions, we obtain a second-order nonlinear ordinary differential equation that describes the motion of the microvoid with time. Qualitatively, we find that this equation has two types of solutions. One is a classical nonlinear periodic solution which describes that the motion of the microvoid is a nonlinear periodic oscillation; the other is a blow-up solution. Significantly, for the isotropic incompressible Ogden material, there exist some special values of material parameters, the phase diagrams of the motion equation have homoclinic orbits, which means that the amplitude of a nonlinear periodic oscillation increases discontinuously with the increasing load.