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The Solitary Wave and Chaos in Nonlinear Solid Structure

Author: ZhangTao
Tutor: ZhangShanYuan
School: Taiyuan University of Technology
Course: Solid Mechanics
Keywords: large-deflection beam fluid-filled pressurized tube axially compressed cylindrical shell solitary wave chaotic motion Jacobi elliptic function expansion method reductive perturbation technique(RPT)
CLC: O322
Type: Master's thesis
Year: 2010
Downloads: 31
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Abstract

In 1960s the research upsurge of the nonlinear problems appeared simultaneously in many branches of natural science. Many new physical phenomena, such as soliton,turbulence,chaos,fractal and complex system etc. were discovered, which indicates that the nonlinear science has been an important symbol in the development of modern science. Owing to the promotion of this upsurge, the investigations on the propagation of nonlinear wave and chaotic motion in solid structure also have made great progress. In this dissertation, on the basis of summarizing the existing research achievement, the propagation property of solitary wave and chaotic behavior in several kinds of typical structural elements are studied. Main works and the important results are as follows:1. On the basis of three classic theories of Bernoulli-Euler,Rayleigh and Timoshenko beam, taking finite-deflection and axial inertia into consideration, the nonlinear partial differential equations governing the propagation of nonlinear flexural wave are derived. The qualitative analyses are carried out, and the exact periodic solutions, the shock wave and solitary wave solution when the modulus m→1 are obtained by means of Jacobi elliptic function expansion method.2. In above three kinds of equilibrium equation, the external load and damping are viewed as small perturbation into the system and the threshold condition of the existence of Smale horseshoe chaos are obtained by Melnikov’s method, which further reveal the interrelationship between the two kinds of nonlinear phenomenon solitary wave and chaos.3. The propagation of nonlinear wave in a fluid-filled elastic thin tube buried inside elastic foundation is studied. In the analysis, the material of the tube is assumed to be linear elastic, the reaction of foundation is calculated by Winkler model, and the fluid is incompressible and inviscid. The solid-liquid coupled equations is obtained by the mass conservation and balance of linear momentum. Employing the reductive perturbation method(RPT) the KdV equation is derived, which indicates the system admits a solitary wave solution.4. The propagation property of nonlinear waves in a viscoelastic thin tube filled with incompressible inviscid fluid is studied. The tube is considered to be made of an incompressible isotropic viscoelastic material described by Kelvin-Voigt model. In the long-wave approximation the nonlinear solid-liquid coupled equations can be derived employing the reductive perturbation technique(RPT). According to the size of viscous effects, the system admits a oscillating solitary wave solution or shock-wave solution and their propagation graph are obtained by numerical calculation.5. Taking the convective term of blood flow and the large deformation of blood vessels into account, the propagation of nonlinear pressure wave in arterial blood vessels is studied by means of strain energy function about soft tissue materials proposed by Hilmi Demiray. Employing the reductive perturbation method(RPT) the KdV equation with soliton solution is derived in the long wave approximate. The effects of system parameters on solution are discussed from a clinical point of view.6. With regard to the nonlinear vibration of an axially compressed cylindrical shell subjected to axial or radial disturbance, two kinds of nonlinear motion equations of cylindrical shell are obtained by adopting Donnell-Kármán large deflection equations or logarithmic circumferential strain definitions, respectively. By means of Bubnov-Galerkin approach two nonlinear partial differential equation can be transformed into an ordinary differential equation containing third-order or second-order nonlinear term. The threshold conditions of the existence of horseshoe-type chaos are presented in the two case of pre-buckling and post-buckling by using of sub-harmonic obit and homoclinic orbit Melnikov function. Lastly, the bifurcation diagram ,the time-history curve, phase portrait and Poincarémap are calculated by means of MATLAB software. The numerical characters of chaotic motion are obtained.

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