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

Wenzheng Zhang, Xuegang Yuan and Hongwu Zhang

Journal of Applied Mathematics, 2012, vol. 2012, issue 1

Abstract: 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.

Date: 2012
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https://doi.org/10.1155/2012/872161

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2012:y:2012:i:1:n:872161

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