【作者】 孙云；

【导师】 李世荣；

【作者基本信息】 兰州理工大学 ， 结构工程， 2014， 博士

【摘要】 功能梯度材料(FGM)由于能够消除传统复合材料中的界面效应和缓解热应力,在现代工程结构中,特别是在高温环境服役的结构中具有重要的应用。因此,FGM结构在热载荷作用下的力学行为研究已成为固体力学新的研究方向。约束屈曲也称为限制失稳,是指构件的屈曲变形由于受到某种限制性约束的作用而不能自由发展的屈曲,这类问题在实际工程中也有大量应用,也是固体力学中的一个经典问题。本论文主要研究FGM梁板在横向约束下的热过屈曲响应,内容包括以下几个方面：1.FGM圆板在横向点间隙约束下的热过屈曲考虑周边不可移夹紧的FGM圆板,其圆心处的横向两侧各有一刚性点约束,这些点约束与圆板间存在着微小间隙。假设圆板的材料性质和升温均只沿厚度方向变化。由升温引起的径向压力超过临界值后,板将产生热过屈曲变形。本文重点研究板与点间隙约束接触前后的热过屈曲响应的变化情况。首先,基于von Karman薄板理论,建立FGM圆板受到点间隙约束作用前后的轴对称热过屈曲控制方程。该方程组为非线性常微分方程组,是以中面位移为基本未知量,且包括温度载荷参数,相应的边界条件在接触前后发生改变。然后,采用打靶法数值求解控制方程,得到了FGM圆板与点约束接触前后的变形和内力的变化情况,分析了材料的非均匀性和温度的非均匀性对热过屈曲响应的影响,给出了有关的平衡构形和平衡路径,并将采用混合律模型和Mori-Tanaka模型计算的结果进行了比较。作为特例,考虑了中心固联有刚性圆盘的圆／环板的情况,分析了外边缘分别为不可移简支和夹紧边界条件下刚性圆盘尺寸对热过屈曲响应的影响。2.非线性弹性地基上FGM Timoshenko梁的热过屈曲研究了置于非线性弹性地基上功能梯度Timoshenko梁的热过屈曲问题。基于精确的几何非线性理论,推导了梁在非均匀升温下的热过屈曲控制方程,分别考虑两端不可移夹紧和简支两种边界条件。从而将研究问题归结为包含七个基本未知函数的非线性一阶常微分方程组的两点边值问题。采用打靶法获得了该边值问题的数值解。临界屈曲温度与地基刚度参数有关,对应的临界屈曲模态会随着地基刚度参数的增加发生跃迁,文中系统分析了临界屈曲模态跃迁问题,给出了临界屈曲模态跃迁点对应的地基刚度跃迁值及其对应的临界温度载荷；给出了热过屈曲的平衡路径和平衡构形,分析了材料和升温的非均匀性、地基刚度参数、边界条件、长细比和剪切变形等对热过屈曲响应的影响。将退化后的均质梁的模态跃迁结果与已有文献进行了比较,结果吻合良好。3.FGM梁在横向点间隙约束下的热过屈曲在前述研究的基础上,基于一维稳态温度场,推导了横向有刚性点间隙约束的功能梯度Euler-Bernoulli梁的热过屈曲控制方程。考虑边界条件为两端不可移夹紧,假设点约束在梁中点的横向两侧并与梁间有微小的间隙,材料性质沿厚度方向按照幂函数连续变化。推导出的控制方程为包含七个基本未知函数的非线性一阶常微分方程组,但梁与点约束接触后相应的边界条件由于点约束反力的出现而与接触前有所不同。采用打靶法数值进行求解,得到了FGM梁与点约束接触前、后的热过屈曲响应,给出了点约束反力随着温度载荷的变化曲线,讨论了材料的梯度变化、梁上下表面升温比、长细比和点间隙位置等对梁的构形、内力和点约束反力的影响。更多还原

【Abstract】 Functionally graded materials (FGM) are widely applied in modern engineering structures, especially in high temperature environment, because they can eliminate the interfaces exiting in composite materials and reduce the thermal stress concentration in the high temterature gradient enviromemt. The researches on the mechanical behaviors of FGM structures under thermal loadings have been a new research area in solid mechanics. Constrained buckling, which is also called confined buckling, means that the bucking configurations of structures can not evolve freely due to the existence of the constraint. This type of problems also has many applications in the practical engineering and is a classic problem of solid mechanics, In this thesis, thermal post-buckling response of functionally graded beams and plates subjected to transverse constraints is studied including the following parts:1. Thermal post-buckling of FGM circular plates subjected to transverse point-constraint.The FGM circular plates with immovably clamped edges are bilaterally constrained by rigid point-constraints at the center with clearance. It is assumed that material properties and non-uniform temperature rise change only through the thickness direction. The plates will go into thermal post-buckling when the radial pressure caused by temperature rise exceeds the critical value. The focus of this thesis is the changes of thermal post-bucking responses during the contact. Firstly, based on von Karman’s nonlinear thin plate theory, governing equations of axisymmetric thermal post-buckling of constrained FGM circular plates are derived, which including parameter of thermal loading are nonlinear ordinary differential equations in terms of the displacements of the middle plane, with boundary conditions changing when the plates contact the point-constraints. Then, by using the shooting method, thermal post-buckling responses of the FGM circular plates are obtained before and after it contact the point-space constraint. The changes in the characteristics of the deformation and the internal forces of FGM plates are discussed. The effects of gradients of material properties and non-uniform temperature rise parameters on the thermal post-buckling behaviors of the plates are also examined. The equilibrium configurations and equilibrium paths are given. The results according to the linear mixture rule and Mori-Tanaka model are compared. As an example, the behaviors of the circular/annular plates with a centric rigid mass are analyzed. The effect of the size of the centric rigid circular plate on the constrained post-buckling response is studied for the plates with both immovably clamped and simply supported edges. 2. Thermal post-buckling of FGM Timoshenko beams on nonlinear elastic foundations.Based on the accurate geometrically nonlinear theory for Timoshenko beams, thermal post-buckling of FGM Timoshenko beams on nonlinear elastic foundations is studied. Governing equations of this problem consist of a two-point boundary value problem of nonlinear ordinary differential equations including seven basic unknown functions with two types of boundary conditions, namely; immovably clamped and simply supported ends. The numerical solutions are arrived at by the shooting method. The critical buckling temperatures vary with the foundation stiffness and the corresponding critical buckling modes transit with the increasing foundation stiffness. Stability boundaries, or critical load curves, are presented for different values of the power exponent and boundary conditions. Post-buckling equilibrium configurations and equilibrium paths are demonstrated for different values of nonlinear elastic foundation parameters, the power exponent, boundary conditions, non-uniform temperature rise, and slenderness ratio. The results of Timoshenko beams are compared with Euler-Bernoulli beams to examine the effect of the shearing deformation. The results of uniform beams are compared with those in the literature, which show a good agreement.3. Thermal post-buckling of FGM beams subjected to transverse point-constraint.On the basis of the aforementioned study, governing equations for large post-buckling deformation of FGM beams with point-constraint are formulated. One dimensional steady state temperature field is considered. It is assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate and the point-constraint is at the side of the middle point of the beam and the space value is in the range of thermal post-buckling deformation. The derived governing equations are nonlinear ordinary differential equations including seven basic unknown functions, with boundary conditions changing due to the appearance of point constraint force. By using shooting method to solve the nonlinear bounadary value promlem numerically, thermal post-buckling responses of the FGM beam are obtained when it contacts the point constraint. Curves of the point constraint force changing with the thermal load are plotted. The effects of parameters of the gradients of material properties, non-uniform temperature rise, slenderness ratio and the space value on the thermal post-buckling behaviors of FGM beams are also examined.更多还原