热机载荷作用下功能梯度曲梁和圆柱壳的静态响应

发布时间：2018-12-10 23:54

【摘要】：功能梯度梁板壳结构的静动态响应是非均匀固体力学研究的重要内容。本文以功能梯度变曲率曲梁和圆柱壳为研究对象,采用理论分析与数值计算相结合的方法,研究其在热机载荷共同作用下的静态力学响应,内容主要由两部分组成。1.基于弹性曲梁平面问题的精确几何非线性理论和一阶剪切变形理论,在几何方程中精确计入了轴线伸长和横向剪切变形,分别建立了变曲率的功能梯度Euler曲梁和Timoshenko曲梁在机械和热载荷共同作用下的弹性大变形微分控制方程。因考虑曲率可变化,与曲率有关的刚度系数是轴线弧长坐标的函数,所以控制微分方程是变系数的。其中基本未知量均被表示为变形前的轴线弧长坐标的函数,弧长用曲梁轴线参数方程的参变量来表示。采用打靶法数值求解上述多未知量的强非线性常微分方程两点边值问题,定量分析了不同边界条件下功能梯度椭圆弧Euler曲梁的大变形弯曲问题,讨论了材料梯度指数、温度载荷参数、结构几何参数等对曲梁内力和变形的影响。利用同样的几何非线性数学模型,还分别研究了两端固定的椭圆弧Euler曲梁在不同机械载荷作用下的非线性稳定性问题,给出了曲梁的过屈曲平衡路径特性曲线。随后,数值求解了功能梯度摆线和椭圆弧Timoshenko曲梁在均匀升温和横向非均匀升温下的热弹性大变形问题,通过比较Timoshenko曲梁和相应Euler曲梁的解答,分析了横向剪切变形对曲梁内力和变形的影响。2.假设圆柱壳的材料性质和升温场均只沿厚度方向变化,研究了功能梯度圆柱壳在热载荷下的屈曲行为。基于经典的线性薄壳理论,推导了用几何中面位移表示的无量纲热屈曲控制方程。采用分离变量法将控制方程从复杂的偏微分方程组转化为未知函数相互耦合的常微分方程组,考虑边界条件为两端简支和两端固定的情形,利用打靶法求解了所得两点边值问题,获得了临界屈曲温度载荷。讨论了功能梯度圆柱壳在均匀升温和非均匀升温时的临界屈曲温度随着材料梯度指数n、厚径比δ=h/R和长径比λ=l/R、非均匀升温参数fT(壳的外表面和内表面升温之比)等的变化关系。数值结果表明:功能梯度圆柱壳的临界屈曲温度随着材料梯度指数的增加,即陶瓷组分的增加而增加;在均匀和非均匀升温场下,无量纲临界屈曲温度随着厚径比的增大而减小,但对长径比的变化不敏感;升温参数的取值反映了温度场的非均匀程度,升温参数越大,无量纲临界屈曲温度越小;边界约束的增强会引起临界屈曲温度的提高,但随着厚径比的增大,边界约束效应减弱。
[Abstract]:The static and dynamic response of a functionally graded beam shell structure is an important part of the research on non-uniform solid mechanics. In this paper, the static mechanical responses of curved beams and cylindrical shells with functionally gradient variable curvature are studied by means of theoretical analysis and numerical calculation, which are mainly composed of two parts. 1. Based on the exact geometric nonlinear theory and the first order shear deformation theory of elastic curved beam plane problem, the axis elongation and transverse shear deformation are accurately considered in the geometric equation. The differential governing equations of large elastic deformation of functionally gradient Euler curved beams with variable curvature and Timoshenko curved beams under mechanical and thermal loads are established respectively. The governing differential equation is variable because the curvature can be taken into account and the stiffness coefficient related to curvature is a function of the arc length coordinate of the axis. The basic unknown is expressed as the function of the axis arc length coordinate before deformation, and the arc length is represented by the parameter variable of the curved beam axis parameter equation. The two-point boundary value problem of strongly nonlinear ordinary differential equations with multiple unknowns is solved numerically by shooting method. The problem of large deformation and bending of functionally gradient elliptic arc Euler curved beams under different boundary conditions is quantitatively analyzed, and the material gradient exponents are discussed. The influence of temperature load parameter and structure geometry parameter on the internal force and deformation of curved beam. Using the same geometric nonlinear mathematical model, the nonlinear stability of elliptic arc Euler curved beams with fixed ends under different mechanical loads is studied, and the equilibrium path characteristic curves of the curved beams are given. Then, the problem of large thermoelastic deformation of functionally gradient cycloid and elliptic arc Timoshenko curved beams under uniform and transverse non-uniform heating is numerically solved. The solutions of Timoshenko curved beams and corresponding Euler curved beams are compared. The influence of transverse shear deformation on the internal force and deformation of curved beam is analyzed. The buckling behavior of functionally gradient cylindrical shells under thermal loading is studied on the assumption that the material properties and the temperature field of cylindrical shells change only along the thickness direction. Based on the classical linear thin shell theory, a dimensionless governing equation of thermal buckling expressed by geometric midplane displacement is derived. By using the method of separating variables, the governing equations are transformed from complex partial differential equations to ordinary differential equations with unknown functions coupled with each other. The boundary conditions are considered as simple support at both ends and fixed at both ends. The two point boundary value problem is solved by shooting method, and the critical buckling temperature load is obtained. The critical buckling temperature of functionally graded cylindrical shells under uniform and non-uniform heating conditions is discussed. The critical buckling temperature of functionally graded cylindrical shells with the material gradient exponent n, thickness to diameter ratio 未 = h / R and aspect diameter ratio 位 = l / R are discussed. The variation of fT (the ratio of the outer surface to the inner surface of the shell), etc. The numerical results show that the critical buckling temperature of functionally graded cylindrical shells increases with the increase of material gradient index, that is, the increase of ceramic composition. The dimensionless critical buckling temperature decreases with the increase of the ratio of thickness to diameter, but is not sensitive to the change of the ratio of length to diameter. The values of the heating parameters reflect the inhomogeneity of the temperature field. The larger the heating parameters are, the smaller the dimensionless critical buckling temperature is. The enhancement of boundary constraints will lead to the increase of critical buckling temperature, but with the increase of thick-diameter ratio, the effect of boundary constraints will weaken.
【学位授予单位】：扬州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O342

【相似文献】