非局部压电纳米板的振动和稳定性研究

发布时间：2018-03-27 11:00

本文选题：压电纳米板　切入点：非局部理论　出处：《北京交通大学》2016年博士论文

【摘要】：压电材料由于其特有的机-电耦合性能,在传感技术、信息技术、新兴的智能材料系统等工程领域中有着广泛的应用。而随着材料制备技术的发展,低维压电纳米结构在微/纳米级机电系统中应用前景广阔,因而受到了国内外学者的普遍关注。大量研究表明,在纳米尺度下,压电纳米材料的特性会受到尺度效应的影响,与宏观尺寸下的特性相比会表现出明显的差异,并导致新特性出现。为准确描述压电纳米结构的尺度效应,本文将采用Eringen提出的非局部理论建立非局部压电纳米板模型,研究其振动和稳定性问题,包括线性振动、非线性振动、屈曲和后屈曲等。在此基础上,我们将进一步讨论磁电弹纳米板的非线性振动问题。与压电纳米板相比,磁电弹纳米板的机-电-磁耦合特性使其非局部本构关系更为复杂。研究以理论分析为重点,通过解析和数值求解,分析非局部系数、热-电-机械荷载、边界条件等不同因素对压电纳米板的线性振动、非线性振动、屈曲和后屈曲等特性的影响,以及对磁电弹纳米板非线性振动特性的影响。具体来说,本文的讨论分为以下几个部分:1.文章首先讨论的是压电纳米板的自由振动问题。基于非局部本构关系,分别建立Kirchhoff压电纳米板和Mindlin压电纳米板模型,分析其在热-电-机械荷载作用下的自由振动特性。对于Kirchhoff压电纳米板,仅考虑四周简支边界条件的影响;对于Mindlin压电纳米板,考虑不同边界条件的影响。通过Hamilton原理推导出问题的控制方程和相应的边界条件,再利用解析和数值方法求解得出压电纳米板的固有频率和振动模态。2.在压电纳米板自由振动问题的研究基础上,进一步研究压电纳米板在热-电-机械荷载作用下的非线性振动特性。先针对Kirchhoff压电纳米板,仅考虑四周简支的边界条件,通过Navier法求解得到非线性频率的解析解。之后讨论在不同边界条件下,Mindlin压电纳米板的非线性振动问题;利用微分求积法(DQ法)对控制方程进行离散,再利用迭代法求解出非线性频率和振动模态的解析解。3.考查在不同边界条件下Mindlin压电纳米板受到面内轴向荷载时的屈曲和后屈曲行为。通过最小势能原理,可推导出问题的控制方程和相应的边界条件。由于压电纳米板的屈曲问题仅考虑了线性几何关系,因而可以直接通过DQ法对控制方程进行离散求解,得出板的临界屈曲荷载;而对于后屈曲问题,由于考虑了 vonKarman几何非线性,因而需要通过迭代法求出压电纳米板的后屈曲荷载。4.在前面的研究基础之上,进一步研究非局部磁电弹纳米板的非线性振动问题。与压电纳米材料相比,磁电弹纳米材料的非局部本构关系更为复杂。基于Mindlin板理论、非局部本构关系和von Karman非线性几何关系,考虑不同边界条件的影响,分析磁电弹纳米板在热-电-磁-机械荷载作用下的非线性振动特性。本文通过建立了非局部压电纳米板和磁电弹纳米板模型,分析了尺度效应、多场耦合效应等因素对压电纳米板和磁电弹纳米板振动和稳定特性的影响,为微/纳米电子机械器件的设计提供理论依据。
[Abstract]:The piezoelectric material due to its unique electro-mechanical coupling performance information technology in sensor technology, has been widely used new intelligent material system in the engineering field. With the development of material preparation technology, low dimensional piezoelectric nanostructures in micro / nano electromechanical system application in Jing Guangkuo, which has been the attention of scholars at home and abroad. A large number of studies show that, in the nanometer scale, the characteristics of piezoelectric nano materials will be affected by the scale effect, compared with the characteristics of the macro dimensions will show obvious differences, and lead to new features. In order to accurately describe the scale effect of piezoelectric nano structure, this paper will the use of Eringen's nonlocal theory to establish the non local piezoelectric nano plate model, study the problems of vibration and stability, including linear vibration, nonlinear vibration, buckling and post buckling. On this basis, we will further discuss The nonlinear vibration problem of magnetoelectroelastic nano plate. Compared with the piezoelectric nano plate, magneto electro elastic nano board machine - electric - magnetic coupling characteristics of the non local constitutive relationship is more complicated. The research on the theoretical analysis for the key, through the analytical and numerical analysis, non local coefficient, thermal electrical mechanical load. Linear vibration, different boundary conditions on piezoelectric nano plate nonlinear vibration, impact buckling characteristics such as buckling and post, and the influence on the nonlinear vibration characteristics of magneto electro elastic nano plate. Specifically, this paper is divided into the following parts: 1. firstly discussed is the free vibration of piezoelectric nano plate the non local constitutive relation. Based on the established Kirchhoff piezoelectric nano nano plate and Mindlin plate model electric pressure, analysis of its role in the thermo electro mechanical loading. The free vibration characteristics of Kirchhoff piezoelectric nano plate, only consider around Effect of simply supported boundary conditions; for Mindlin piezoelectric nano plate, considering the influence of different boundary conditions. The Hamilton is derived from the principle of the control equations and the corresponding boundary conditions, using the analytical and numerical methods for solving the piezoelectric nano plate natural frequency and vibration mode of.2. based on the piezoelectric nano plate free vibration on the problem of further study on nonlinear vibration characteristics of piezoelectric nano plate in thermal electrical mechanical loads. For Kirchhoff piezoelectric nano plate, only considering the boundary conditions of four simply supported, through the Navier method to obtain analytic solutions of nonlinear frequency. After discussing the various boundary conditions, the nonlinear vibration problem of Mindlin the piezoelectric nano plate; by using differential quadrature method (DQ method) was used to discrete the governing equations, and then solved by the iterative method of nonlinear analytical solution of frequency and vibration modal test in.3. With the boundary conditions of Mindlin piezoelectric nano plates under in-plane buckling and buckling behavior of axial load and by the principle of minimum potential energy can be deduced the governing equations and corresponding boundary conditions. The buckling problems of piezoelectric nano plate only considers the linear relation of geometry, which can direct the discrete control equations are solved by by DQ method, the critical buckling load of the plate; and for the post buckling problems, due to the vonKarman geometric nonlinearity, thus need to through the iterative method to calculate the post buckling load of.4. piezoelectric nano plate based on the above study, further study on the nonlinear vibration problems of non local magnetoelectroelastic nano plate compared with piezoelectric. Nano materials, nano material magnetoelectroelastic non local constitutive relationship is more complicated. Based on Mindlin plate theory, the non local constitutive relation and von Karman nonlinear geometric relationship, considering the different The influence of boundary conditions and nonlinear vibration analysis of effect of magneto electro elastic nano plate in thermal - electric - magnetic - mechanical loads. This paper established the non local piezoelectric nano plate and nano magneto electro elastic plate model, analyzes the influence of scale effect, multi field coupling effect of piezoelectric and magneto electro elastic nano nano plate plate vibration characteristics and stability, to provide the theoretical basis for design of micro / nano electronic and mechanical devices.

【学位授予单位】：北京交通大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O327;TB34

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