功能梯度材料梁、壳结构的静态力学响应
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摘要
本文研究了功能梯度Timoshenko梁和扁薄锥壳结构在变温场作用下的几何
非线性静态响应问题。主要内容包括以下几个方面：
1． 介绍了功能梯度材料(FGMs)的性质、特点及其在现代科技和工程中的应
用。在查阅大量文献的基础上，总结了近几年国内外研究人员对这种材料结构力
学行为的研究成果，特别是对FGM梁、板(壳)结构在机械和热载荷作用下的
宏观力学响应研究现状和最新进展进行了详细说明。
2． 基于轴线可伸长和横向可剪切的几何非线性理论，采用打靶法研究了功
能梯度材料Timoshenko梁在横向非均匀升温下的静态热屈曲和热过屈曲响应。
在精确考虑轴线伸长和一阶横向剪切变形的基础上，建立了功能梯度材料
Timoshenko梁在热-机载荷同时作用下的大变形控制方程。其中，功能梯度材
料梁的材料性质采用了沿厚度方向按照幂函数形式连续变化的形式。采用打靶法
数值求解所得强非线性边值问题，获得了横向非均匀升温载荷作用下两边固支
Timoshenko梁的静态非线性屈曲和过屈曲数值解。绘出了梁的挠度随温度载荷
及材料梯度参数变化的特性曲线，分析和讨论了温度载荷及材料的梯度性质参数
对梁变形的影响。研究结果表明，由于材料在横向的非均匀性，变形过程中存在
拉-弯耦合效应。
3． 研究了陶瓷／金属FGM扁薄锥壳在横向非均匀温度场中的几何非线性大
变形问题。在假设材料的物性参数只沿厚度按幂函数均匀变化的前提下，基于
Kirchhoff直法线假设和von Karman几何非线性理论推导出了以中面位移为基本
未知量的功能梯度材料扁薄锥壳在横向非均匀热载荷作用下的轴对称大挠度控
制方程。采用打靶法数值求解所得非线性常微分方程边值问题，得到了扁锥壳在
静态温度载荷作用下大挠度弯曲变形数值解。给出了壳体变形随壳体的形状参
数、载荷和材料参数变化的特征关系曲线，重点分析和讨论了温度参数和材料梯
度参数对变形的影响。
In this thesis, geometrically nonlinear static responses of a functionally graded Timoshenko beam and a shallow conical shell, subjected to thermal loadings, are studied. The main contents of it are as follows:
1. Firstly, the characteristics of functionally graded materials (FGMs) and the applications of FGMs in modern engineering, science and technology are briefly introduced. And then, according to my survey of the corresponding literatures, an outline about the research results and advances in the structural mechanical behaviors of FGM structures is summarized, especially about the macroscopical mechanical responses of FGM beams and plates (shells), subjected to thermal and mechanical loadings, are also presented in details.
2. Based on the geometrically nonlinear theory for axially extensible and transversely shearable beams, the thermal buckling and post-buckling of the FGM Timoshenko beams, subjected to transversely non-uniform temperature rise, are investigated. Accurately considering the axial extension and transverse shearing of the beams, the governing equations for FGM Timoshenko beams, subjected to thermal and mechanical loadings with large elastic deformations, were formulated. In the analysis, the material properties of the functionally graded Timoshenko beam are assumed to vary continuously through the thickness of the beam, according to a power law distribution of the volume fraction of the constituents. By using shooting method to numerically solve the above mentioned strongly nonlinear boundary value problems of ordinary differential equations. The results of thermal buckling and post-buckling of transversely non-uniformly heated Timoshenko beams with pinned-pinned edges are obtained. The deformation param
eters of the FGM beams versus thermal loading and material gradient constant are plotted. The effects of material gradient properties and thermal loading on deformations of beam are discussed in details. The results show that there exist tension-bend coupling in the deformation because of the transversely non-uniform characteristic of materials.
3. Geometrically nonlinear large deformation problem of ceramic/metal FGM thin shallow conical shell is examined. Firstly, by assuming the mechanical properties of the functionally graded conical shells varying continuously through the thickness of the shells and obeying a power law distribution of the fraction of the graded shells and on the basis of the Kirchhoff straight normal assumption together with von Karman's geometrically nonlinear theory, the governing equations for axi-symmetrical large deformation of functionally graded shallow conical shell subjected to non-uniform
temperature rise are derived. Numerical results of bending of the statically thermal loaded shell with large deflection are obtained through solving the boundary value problem for nonlinear ordinary differential equations by using shooting method. The characteristic curves of the deformation parameter versus the shape parameter, thermal loadings and material gradient constant of the FGM shells are illustrated and also the effects of material gradient property, shape parameters and temperature parameters on deformations of shells are discussed in details.
非线性静态响应问题。主要内容包括以下几个方面：
1． 介绍了功能梯度材料(FGMs)的性质、特点及其在现代科技和工程中的应
用。在查阅大量文献的基础上，总结了近几年国内外研究人员对这种材料结构力
学行为的研究成果，特别是对FGM梁、板(壳)结构在机械和热载荷作用下的
宏观力学响应研究现状和最新进展进行了详细说明。
2． 基于轴线可伸长和横向可剪切的几何非线性理论，采用打靶法研究了功
能梯度材料Timoshenko梁在横向非均匀升温下的静态热屈曲和热过屈曲响应。
在精确考虑轴线伸长和一阶横向剪切变形的基础上，建立了功能梯度材料
Timoshenko梁在热-机载荷同时作用下的大变形控制方程。其中，功能梯度材
料梁的材料性质采用了沿厚度方向按照幂函数形式连续变化的形式。采用打靶法
数值求解所得强非线性边值问题，获得了横向非均匀升温载荷作用下两边固支
Timoshenko梁的静态非线性屈曲和过屈曲数值解。绘出了梁的挠度随温度载荷
及材料梯度参数变化的特性曲线，分析和讨论了温度载荷及材料的梯度性质参数
对梁变形的影响。研究结果表明，由于材料在横向的非均匀性，变形过程中存在
拉-弯耦合效应。
3． 研究了陶瓷／金属FGM扁薄锥壳在横向非均匀温度场中的几何非线性大
变形问题。在假设材料的物性参数只沿厚度按幂函数均匀变化的前提下，基于
Kirchhoff直法线假设和von Karman几何非线性理论推导出了以中面位移为基本
未知量的功能梯度材料扁薄锥壳在横向非均匀热载荷作用下的轴对称大挠度控
制方程。采用打靶法数值求解所得非线性常微分方程边值问题，得到了扁锥壳在
静态温度载荷作用下大挠度弯曲变形数值解。给出了壳体变形随壳体的形状参
数、载荷和材料参数变化的特征关系曲线，重点分析和讨论了温度参数和材料梯
度参数对变形的影响。
In this thesis, geometrically nonlinear static responses of a functionally graded Timoshenko beam and a shallow conical shell, subjected to thermal loadings, are studied. The main contents of it are as follows:
1. Firstly, the characteristics of functionally graded materials (FGMs) and the applications of FGMs in modern engineering, science and technology are briefly introduced. And then, according to my survey of the corresponding literatures, an outline about the research results and advances in the structural mechanical behaviors of FGM structures is summarized, especially about the macroscopical mechanical responses of FGM beams and plates (shells), subjected to thermal and mechanical loadings, are also presented in details.
2. Based on the geometrically nonlinear theory for axially extensible and transversely shearable beams, the thermal buckling and post-buckling of the FGM Timoshenko beams, subjected to transversely non-uniform temperature rise, are investigated. Accurately considering the axial extension and transverse shearing of the beams, the governing equations for FGM Timoshenko beams, subjected to thermal and mechanical loadings with large elastic deformations, were formulated. In the analysis, the material properties of the functionally graded Timoshenko beam are assumed to vary continuously through the thickness of the beam, according to a power law distribution of the volume fraction of the constituents. By using shooting method to numerically solve the above mentioned strongly nonlinear boundary value problems of ordinary differential equations. The results of thermal buckling and post-buckling of transversely non-uniformly heated Timoshenko beams with pinned-pinned edges are obtained. The deformation param
eters of the FGM beams versus thermal loading and material gradient constant are plotted. The effects of material gradient properties and thermal loading on deformations of beam are discussed in details. The results show that there exist tension-bend coupling in the deformation because of the transversely non-uniform characteristic of materials.
3. Geometrically nonlinear large deformation problem of ceramic/metal FGM thin shallow conical shell is examined. Firstly, by assuming the mechanical properties of the functionally graded conical shells varying continuously through the thickness of the shells and obeying a power law distribution of the fraction of the graded shells and on the basis of the Kirchhoff straight normal assumption together with von Karman's geometrically nonlinear theory, the governing equations for axi-symmetrical large deformation of functionally graded shallow conical shell subjected to non-uniform
temperature rise are derived. Numerical results of bending of the statically thermal loaded shell with large deflection are obtained through solving the boundary value problem for nonlinear ordinary differential equations by using shooting method. The characteristic curves of the deformation parameter versus the shape parameter, thermal loadings and material gradient constant of the FGM shells are illustrated and also the effects of material gradient property, shape parameters and temperature parameters on deformations of shells are discussed in details.
引文
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10． 沈惠申．功能梯度复合材料板壳结构的弯曲、屈曲和振动．力学进展，2004， 34(1) ：53-60．
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2． M Koizumi and M Niino. Overview of FGM research in Japan. MRS Bull, 1995, 20: 19-21．
3． W.A.Kaysser and B.Iisechner. FGM research in Europe. MRS Bull, 1995, 20: 22-26．
4． Z.H.Jin, R.C.Batra. Some basie fracture mechanics concepts in functionally graded materials. Journal of the Mechanics and Physics of Solids, 1996, 44: 1221-1235．
5． J.Rdel and A.Neubrand: Proc.Conf. Functionally Graded Materials 1996, Tsukuba, Japan, 1996,I. Shiota and Y.Miyamoto, eds, Elsevier, Amsterdam, pp. 9-14．
6． 郑慧雯等．功能梯度材料的研究进展．西南师范大学学报，2002，27(5) ： 788-793．
7． 王引真等．功能梯度材料的研究动态．机械工程材料，1997，21(4) ：35-37．
8． 张伟．功能梯度材料研究的现状和前景．新材料·新技术，2003，17(4) ：216-29．
9． 陈东等．功能梯度材料的进展．青岛建筑工程学院学报，2001，22(4) ：92-95．
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