【作者】 李清禄；

【导师】 李世荣；

【作者基本信息】 兰州理工大学 ， 结构工程， 2012， 博士

【摘要】 功能梯度材料(Functionally graded material,简称FGM)通常是由陶瓷和金属复合而成的一种新型先进的非均匀复合材料,兼顾了各组分材料自身的优点,使得FGM材料在航空航天、机械工程、电子工程、核能工程以及土木工程等领域,都有着十分广阔的应用前景。研究材料结构在保守和非保守载荷作用下的静动态力学行为是固体力学研究的重要内容之一。目前,有关FGM结构的总体研究状况为：静态研究多于动态研究,保守系统研究多于非保守系统研究,尤其是在非保守载荷作用下FGM结构的动态特性方面的研究,由于问题本身的强非线性和材料性质在空间位置上的连续变化,在该领域的研究成果不多。本文针对非保守载荷-随动载荷系统,较为全面地研究,尤其是定量研究了FGM直梁、曲梁以及圆板结构的静态和动态力学响应,得到了一些有意义的结果。本文的研究主要包括以下四大方面：(一)本文首先给出了随动载荷作用下FGM梁静动态响应的基本方程。在小振幅和谐振动假设下,将得到的强非线性偏微分方程组被转化为两组相互耦合的常微分方程组,分别是过屈曲问题的控制方程和过屈曲构形附近的振动控制方程。作为特殊情况,这些方程容易退化到保守载荷下的基本方程。在此基础上,利用打靶法,编制了一系列定量分析程序,进一步利用解析延拓法,获得了FGMEuler梁过屈曲平衡路径和平衡构形。考察了材料的梯度指数、边界条件等对梁过屈曲的影响。结果表明：非保守载荷作用下功能梯度梁的平衡路径与保守载荷作用下梁的平衡路径是不同的。在此基础上,本文进一步展开了对切线随动非保守载荷作用下FGM Timoshenko梁过屈曲行为的研究。基于轴向可伸长和一阶横向剪切变形理论,建立了功能梯度Timoshenko梁在非保守载荷作用下的几何非线性控制方程。数据结果表明：Timoshenko梁的过屈曲行为和Euler梁有明显不同。(二)重点研究FGM梁过屈曲附近的自由振动响应。采用打靶法数值求解过屈曲构形附近的线性振动控制方程。获得随动非保守载荷作用下Euler FGM梁前三阶固有频率与载荷之间的特征关系曲线。详细分析和讨论载荷和材料梯度性质变化参数对振动特性的影响。数值结果表明：不论是一端可移简支一端固定梁还是简支梁,在屈曲前,各阶频率都随载荷单调下降,其中一阶频率具有明显非线性性。但过屈曲之后,两种边界条件下,FGM梁的频率随载荷呈现出不同变化。可见边界条件对梁的振动响应有重要影响。另外,材料的体积指数份数对梁的固有频率也有影响。指数份数越大,固有频率越小。保守和非保守载荷下,梁的频率-载荷关系也表现出极大的不同。(三)基于Kirchoff直法线假设,采用可伸长梁的几何非线性理论,建立了FGM弹性曲梁受随动载荷作用下的几何非线性数学模型。精确考虑了轴线伸长和初始曲率对变形的影响,在物理方程中表现出曲梁的轴向变形与弯曲变形之间的相互耦合效应。分别计算了半圆形悬臂FGM曲梁在均匀切线随动载荷和径向随动载荷作用下的弯曲问题。绘出了载荷大范围变化时的平衡路径和平衡构形曲线,分析了梯度指数对曲梁变形的影响。在上述研究基础上,又建立了FGM组合曲梁的静平衡几何非线性大变形控制方程。作为上述模型的特例,分析了fixed-free约束,直梁和四分之一曲梁构成的组合梁的静态大变形静力响应。给出了不同梯度指数下自由端水平和铅垂位移随载荷大范围变化的平衡路径和平衡构形。从计算结果看出：本文建立的基本理论和采用的数值方法可用于分析功能梯度曲梁在任意载荷(保守和非保守)作用下的几何非线性静平衡问题。当梯度指数为零时,就退化为均匀FGM曲梁。当方程中令6=0。=K0=0,则模型又退化为可伸长直梁的静态大变形控制方程,且耦合消失。(四)基于经典非线性板理论,建立了受切向随动分布载荷作用下功能梯度圆板的过屈曲控制方程。假设功能梯度材料性质只沿厚度方向变化,采用打靶法和解析沿拓法获得了相应边值问题的数值解。重点考察了随动载荷、材料的梯度梯度性质参数以及边界条件对板变形的影响。数值结果表明：随动力作用下圆板的临界载荷在周边固支情况下比相应的在周边不可移简支情况下大得多。当梯度指数为零时,两种边界条件下,均匀圆板都会发生通常的分支屈曲。而功能梯度梁则不同,一旦有外部载荷,板就会产生挠度,之后FGM圆板的过屈曲平衡路径介于均匀陶瓷和金属板梁之间。因此,FGM圆板在随动非保守载荷作用下的这种情况类似于通常的屈曲。本文的成果有助于人们进一步认识功能梯度材料结构的静动态响应的特性,同时有助于进一步完善非保守系统的研究成果。更多还原

【Abstract】 Functionally graded material (FGM) has been regarded as one of the advanced inhomogeneous composite materials, usually made from metal and ceramic, taking advantage of the merits of constituent materials adequately which has found wider application foreground in many fields such as aerospace, mechanical, electronics, nuclear engineering, civil engineering, and so on. It is of great important to analyze the static and dynamic mechanical behavior of non-conservative and conservative FGM structures and the problem of static deformation and dynamic response of FGM structures have been extensively investigated and become one of the most important issues in solid mechanics. Now, in the study of FGM structures, the static researches are more than the dynamic ones, and also the researchers on conservative systems are more than these on non-conservative ones. Especially, because of strong nonlinearity and continuous change in position of material properties, the investigations on the dynamics of FGM structures under non-conservative follower force are very few. In this paper, for a non-conservative system of FGM beams and circular plates subjected to distributed follower force, a more comprehensive study, especially quantitative research on static and dynamic characteristics are carried out, and then some useful results are obtained.The main contents of this dissertation are involved as the following four parts.(I) Firstly, the basic equations of static and dynamic response under distributed follower force of FGM beams, are presented. By assuming that the amplitude of the beam vibration is small and its response is harmonic, the above non-linear partial differential equations are reduced to two sets of coupled ordinary differential equations, the one is for postbuckling and the other is for the dynamic response. As a special case, these equations mentioned above are easily degraded to basic equations subjected to conservative load. On this basis, the numerical methods which can be used to deal with the static and dynamic response of FGM beams under non-conservative distributed follower force are established. The equilibrium paths as well as the post-buckling configurations of the deformed beam are presented. The effects of material gradient property and boundary conditions on the buckling deformation of the FGM beam are discussed in details. The numerical results show that features of the equilibrium paths of FGM beams subjected to a non-conservative load are different from those to a conservative one. Next, we further study the post-buckling behavior of FGM Timoshenko beams subjected to a distributed tangential follower force along the central axis. By accurately considering the axial extension and transverse shear deformation, geometrically nonlinear governing equations for FGM beams under non-conservative force are formulated. The numerical results indicate that the post-buckling behavior of FGM Timoshenko and Euler beam are significantly different.(Ⅱ) The free vibrations in the vicinity of post-buckling of FGM Euler beam subject to axially distributed follow force are investigated. By employing the numerical shooting technique to solve the ordinary differential equations obtained in chapter2, Characteristic curves of the first three natural frequencies of the FGM beam versus the dimensionless load parameter are illustrated. The effect of the parameters of material gradients and load on the responses of vibration of the beam was analyzed and discussed. The numerical results show that the three lowest frequencies of the pre-buckled beam decrease with an increment of load parameter for both hinged-fixed beam and hinged-pinned beam. The first natural frequency has a strong nonlinearity. However, when the beam is in post-buckled state, the frequency versus the load parameter is showing a different variation. The frequency-load relationship is between the pure ceramics and metal.(Ⅲ) Based on the assumption of straight normal line and by employing geometrically nonlinear theory for the extensible beams, governing equations of large static deformation of curved elastic FGM beam subjected to a distributed tangential follower force along the central axis are established. In the mathematical model, not only the effects of the axial elongation and the initial curvature of the curved FGM beam on the deformation are accurately taken into account but also the tension-bending coupling is considered. By using shooting method, the nonlinear plane bending of a semicircle curved FGM beam made of metal and ceramic subjected to both tangentially distributed follower force along the axial line and distributed follower force along the radial direction are analyzed. The equilibrium paths and configurations of the deformed FGM beams with different material gradient properties, varying with the load parameter in a large range, are presented, and compared with those of the curved beams made of pute metal and ceramic. Then governing equations of large static deformation of FGM elastic composite curved beams subjected a non-conservative distributed tangential load along axis are established. As a numerical example, the nonlinear plane bending of large static deformations of a cantilever composite beam composed of straight beam and one fourth circle part is presented. The results indicate that the basic theory and numerical method we built can be used to analyze the problem of FGM curved beams subjected to arbitrarily distributed load(conservative and non-conservative).(Ⅳ) Based on the classical plate theory, governing equations of post-buckling of a functionally graded circular plate subjected to uniformly distributed follower compressive force are established. It is assumed that the material properties of the beam vary continuously in the thickness direction. By using shooting method to solve the ordinary differential equations with clamped and simply supported boundary in the middle plane, the response of post-buckling of the FGM circular plate is obtained. Effects of material gradient parameter on the critical buckling, post-buckling behaviors of the FGM plate are discussed in details.The research results will be helpful to us to further understand the characteristics of static and dynamic responses of FGM structures, and improve the research achievement of non-conservative system.更多还原