【作者】 赵广慧；

【导师】 杨桂通； 张年梅；

【作者基本信息】 太原理工大学 ， 固体力学， 2005， 博士

【摘要】 非线性金属杆件是很多结构物的基本组成部分,当金属杆受力发生变形时就必须克服晶格对变形的抵抗力,其中Peierls-Nabarro(P-N)力是诸多障垒中很基本也很重要的一种。本文系统地研究了计及P-N力效应和材料粘性效应的一维金属杆,在不同扰动作用下的振动以及非线性波的传播问题。通过哈密顿变分原理建立了描述杆中非线性波的运动微分方程,这是受扰的Sine-Gordon(SG)型方程。从数值计算和理论分析两个方面描述和预测系统在不同扰动作用下的动力响应,发现金属杆中不仅存在扭结波,而且在一定条件下,杆的运动呈现出混沌的特征。 1.通过数值计算,系统地研究了计及P-N力和材料粘性效应的一维有限长金属细杆,在应变为零的边界条件和位移沿空间单峰分布的初始条件下,受到沿空间均匀分布、随时间简谐变化的轴向外力扰动时的动力响应。 利用状态变量的时程曲线、时空曲面、功率谱密度函数、最大Lyaounov指数、Poincaré映射、相轨迹图等手段,给出了周期扰动的幅值和频率、初始条件、P-N力的幅值和频率、粘性效应、杆长诸因素对系统动力响应的影响。 发现了当杆的几何条件和材料性质发生变化时,显示出4种典型的动力行为:与空间位置无关的简谐运动、单波的简谐运动、单波的准周期运动和单空间模态的混沌运动。给出系统发生非规则运动的几何条件和材料性质。 2.将石油工程中的钻柱简化为半无限长杆,计入杆中的P-N力效应和材料粘性效应,数值研究了杆端施加随时间简谐变化的外力时,外力的幅值和频率以及P-N力对非线性波的传播特征以及杆中各点振动特征的更多还原

【Abstract】 Nonlinear metallic bar is the basic part of many structures. When the metallic bar deforms after being loaded, resistance of lattice to motion have to be overcome. Peierls-Nabarro(P-N) barrier is not only a basic one but also an important one in many barriers happened in deformed lattices. Considering P-N effect and viscosity of solids, vibration of the bar and propagation of nonlinear wave have been researched under different driving. Governing equation is derived through Hamiltonian variational principle, and dynamic responses of these systems have been described and forecasted through numerical computation and theoretical analysis:1. Under strainless boundary conditions, one-dimensional finite metallic bar subjected to periodic field is simulated systematically with the initial displacement taken as single-hump along the bar.Various diagnostic tools such as time history, power spectrum density function, maximum Lyapunov exponent, Poincare map, phase trajectory portrait have been used to show dynamical responses. Influences of factorssuch as driving amplitude and frequency, initial condition, amplitude and frequency of P-N force, viscous effect and length of the bar have been researched.We find that for different sizes and physical characters, spatial structure will show flat state or single-hump distribution along the bar, and temporal behavior will show periodic, quasi-periodic or chaotic responses respectively.2. Drillstring in oil drilling is simplified as a half-infinite bar. Taking account of P-N force and viscous effect of solid, propagation of nonlinear wave and vibrating of particles are investigated under periodic exciting at the end of the bar. Influences of factors including amplitude and frequency of driving, amplitude of P-N force on the nonlinear responses are researched. We find that all the points of the bar have the same qualitative characters, but vibrating amplitude decreases along the bar. 1-period, 2-period, 3-period, 13-period motion, quasi-periodic motion and chaotic motion would appear under certain driving force and physical properties. Equilibrium point and limit cycle are presented to show transition to periodic and quasi-periodic motion through Poincare mapping section.3. Propagation of soliton in one-dimensional infinite metallic thin bar, which is subjected to axially periodic load and in which P-N effect and viscous effect of metal are taken into account, is investigated qualitatively. Collective coordinate method is employed to reduce the perturbed system into ordinary differential system.Using Melnikov method, thresholds for chaos are given. We find that the critical ratio of the amplitude of external force to damping coefficient (fla)cfor chaos increases along with driving frequency. There is not ultrasubharmonic and even order subharmonic bifurcation orbits in this perturbed system, and a series of odd order harmonic bifurcations will occur before Smale horseshoe mapping appears. By studying the effect of P-N force on the transition to chaos, we find that (fla)c increases with P-N amplitude under small driving frequency. Inversely, (fJa)c decreases with P-N amplitude under large diving frequency.4. In view of P-N force and viscous effect of material, dynamic responses of one-dimensional infinite metallic bar exerted with axial load that is correlative with space and time are investigated. The accurate expressions of librating periodic orbits, rotating periodic orbits and heteroclinic orbits of the perturbation-free system are derived.Melnikov method is utilized to show thresholds for chaos and bifurcations of periodic orbits. It appears that librating periodic orbits reach chaos through odd order subharmonic bifurcation, and rotating periodic orbits reach chaos through even order subharmonic bifurcation. Analysis about effect of P-N force on the onset of chaotic motion denote that chaotic critical value would decrease rapidly and then rise with the parameters values increasing continuously, so as to create a minimum point along every curves.更多还原