【作者】 李超；

【导师】 许佳；

【作者基本信息】 天津大学 ， 一般力学与力学基础， 2019， 硕士

【摘要】 自从碳纳米管被首次发现开始,由于其卓越的电、磁、热以及力学属性使得碳纳米管材料被广泛应用于生物化学、医疗工程、电子信息、航空航天等领域。伴随着纳米材料技术的飞速发展,人们可以根据实际需求精准制造出各种类型的纳米材料以满足科研以及工程领域的需求。目前对纳米材料的力学、电学和热学等问题进行研究时,广泛采用连续介质模型来进行理论分析。但是在纳米尺度上,小尺寸效应对纳米结构单元产生的影响不可忽略。同时,在生产、生活等实际应用场景中,纳米结构或设备常常处于复杂的外部环境状态中,随机因素的干扰也不可忽略。本文基于非局部弹性理论和随机动力学理论研究了碳纳米管在随机磁场作用下的强非线性动力学特性,得到如下研究成果:1)基于Eringen的非局部弹性理论来考虑碳纳米结构的小尺度效应,提出用非局部弹性理论的本构关系来推导单壁碳纳米管的动力学方程,随后考虑纳米管被嵌入到粘弹性基体中的作用效果,利用Winkler模型将其等效为施加线性刚度与阻尼,其中阻尼包括线性阻尼和强非线性阻尼。考虑到纳米管处于随机磁场作用下,因此分析了洛伦兹力的作用效果,从而推导出在随机磁场作用下嵌入到粘性基体中的碳纳米管动力学方程;2)研究了碳纳米管在随机磁场作用下的强非线性动力学特性,利用能量函数方法获得了强非线性动力学系统的固有频率,并求解了漂移系数和扩散系数,从而推导出系统动力学响应的平稳概率密度函数以及稳态概率密度函数,随后分析了系统的分岔特性并利用Mathematica软件进行数值仿真。研究结果表明:随着磁场强度的增大,系统的稳定性先减小后增强,这种现象称为随机共振现象;验证了系统安全域的分形边界,随着磁导率的增加,安全域的面积逐步减小,系统可靠性显著降低。本文所做工作对碳纳米管的应用有一定的理论指导意义与工程实用价值。更多还原

【Abstract】 Since the discovery of carbon nanotubes,carbon nanotubes have been widely applied in biochemistry,medical engineering,electronic information,aerospace and some other fields due to their excellent electrical,magnetic,thermal and mechanical properties.With the development of nanomaterials technology,people can accurately manufacture various types of nanomaterials to meet the needs of scientific research and Engineering.Nowadays,the continuum model is widely used to analyse the mechanical,electrical and thermal problems.However,at the nanoscale,the effect of small scale on the nanostructure is significant.Besides,the interference with stochastic factors can not be ignored when applied to complex environments.Based on the theory of nonlocal elasticity and stochastic dynamics,this paper focus the research on the strongly nonlinear dynamic response of carbon nanotubes under stochastic magnetic field,and the main conclusions are as follows: 1)Based on Eringen’s nonlocal elasticity theory,the constitutive relationship of nonlocal elasticity theory is derived,thus we can propose the dynamic equation of single-walled carbon nanotubes,and then we considered the effect of nanotube which embedded in viscoelastic matrix.At this step,Winkler model is introduced which includes linear damping and strong nonlinear damping.Beacause of the stochastic magnetic field,the we should consider the effect of Lorentz force.Therefore,we obtained the dynamic equation of carbon nanotubes which embedded in the Viscoelastic matrix under stochastic magnetic field;2)The strongly nonlinear dynamic characteristics of carbon nanotubes under stochastic magnetic field are studied,the natural frequency of the strong nonlinear dynamic system is obtained by using the energy function method,and the drift coefficient and diffusion coefficient are solved.Besides,the stationary probability density function of the dynamic response of the system is given.The bifurcation characteristics of the system are simulated by the software of Mathematica.The results show that,with the increase of magnetic field strength,the stability of the system decreases firstly and then increases,which is called stochastic resonance phenomenon.The fractal boundary of the system’s safety basin decreases gradually with the increase of magnetic permeability,leading to lower reliability.更多还原