热载荷作用下功能梯度材料板壳的静动态响应
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摘要
功能梯度材料(FGM)是一种新型的功能材料,通过对材料的合理设计可使构件在热环境中具有优于一般均匀材料的力学性能。由于其物理性质的优点,可用作高速航天器、核工业和化工领域的热防护材料。功能梯度材料板壳结构的力学行为特性已成为固体力学研究的重要内容。本论文研究功能梯度材料板壳在热载荷作用下的静动态响应。包括以下几方面:
1)FGM圆板在表面周期动态热载荷作用下的强迫振动
设FGM板材料属性沿板的厚度连续变化,材料属性随组分材料体积分数按幂函数形式分布。首先根据表面热载荷的特点,把板内的温度场设为静态和动态两部分,基于经典热传导理论,采用打靶法数值求解了一维非稳态热传导问题得到了静动态温度场分布和相应的热内力。基于von Karman板理论,建立了FGM薄圆板的轴对称非线性控制方程。通过假设强迫振动的组合谐响应模式,采用Kantorovich时间平均法,把非线性偏微分方程化成了非线性常微分方程组。通过打靶法得到了周边简支FGM圆薄板强迫振动的数值结果。详细讨论了静动态热载荷参数对结构的固有频率的影响,特别分析了共振现象及其特征。
2)FGM薄壳的几何非线性理论
均匀材料壳的简化理论不一定适用FGM壳体。为了研究FGM柱壳的力学行为,我们基于Kirchhoff-Love假设采用矢量方法推导了FGM薄壳的几何非线性理论。几何方程推导时考虑了变形前后曲面的变化所引起的坐标方向单位矢量的变化;精确计算了坐标方向上线段的伸长比及壳体内任一点的非线性应变和位移关系。本构方程中考虑了变形引起的应力分布变化。平衡方程建立在变形以后的微元体上。此基本方程适用于壳体的大变形计算。退化得到的薄壳小挠度问题的方程同已有的结果一致。对于圆柱薄壳的小挠度问题,简化得到了Flügge理论的方程。
3)静态升温场中FGM柱壳的大变形和自由振动
考虑FGM材料属性对温度的依赖特性,求解了表面静态热载荷作用下FGM圆柱壳体内的温度场分布和静态热内力、热弯矩及高阶热内力。由本文导出的FGM薄壳的精确理论,简化得到了圆柱薄壳的非线性控制方程。利用打靶法求得了三种对称支承条件下FGM短圆柱壳的静态大变形响应,并分析了几何参数、载荷参数,材料体积分数指数对壳体挠度的影响。通过假设自由振动位移的时间模式,采用Hamilton原理得到了在热载荷下简支FGM圆柱壳自由振动的位移形式控制方程,通过数值求解,给出了结构小振幅自由振动的自然频率与各控制参数的关系。
4)周期动态热场中FGM柱壳的强迫振动
根据获得的静变形数值结果,利用最小二乘法得到了静弯曲构形的拟合函数。采用三角函数来描述动态响应的空间模态。根据Hamilton原理,得到FGM圆柱壳的非线性振动关于时间的控制方程。对静态热载荷下的小幅自由振动进行分析,其结果与打靶法所得的结果吻合较好。用多尺度法研究了FGM圆柱壳在表面周期热载荷作用下非线性强迫振动的周期解,并分析了振动的动力稳定性。
本文的研究丰富了FGM板壳结构在动态热载荷作用下力学特性的研究成果,对FGM结构的宏观力学特性的研究有借鉴作用,对于此类结构的工程实际应用具有积极意义。
Functionally graded material (FGM) is a kind of new functional materials. Through optimal of design of the material, the FGM structures can have better mechanical properties in thermal environment than homogenous material structures. Due to its virtue of physical characteristics, FGM can be used as heat-shielding material for the high-speed spacecraft, nuclear reactor and chemical engineering. The studies of mechani€al behaviors of FGM plates and shells have being become one of the important direction in solid mechanics. The present study is focus on study the static and dynamic responses of FGM circular plates and cylindrical shells subjected to thermal loads. The main research results and conclusion are summarized as follows:
1) Forced vibration of FGM circular plate with surface periodic thermal loads
The material properties of a FGM plate are graded continuously in the direction of thickness. The variation of the properties followed a simple power-law distribution in the term of the volume fraction of the constituents. In the view of the form of surface thermal loads, the through thickness temperature distribution is assumed to be two parts, static and dynamic. The temperature distributions are solved on the bases of one-dimensional unsteady-state heat conduct equation, the static and dynamic thermal membrane forces and bending moments are obtained numerically by shooting method. On the basis of von Karman's plate theory, the non-linear governing equations of thin FGM circular plate are formulated. Assuming that the vibration is harmonic and using Kantorovich time averaging method, the partial differential equations are converted into a system of nonlinear ordinary differential equations. Numerical results of thin FGM circular plates with simply-supported boundary and subjected to surface periodic thermal loads are obtained by using shooting method. The effects of the static and dynamic thermal load parameters on the natural frequencies are examined in details. Especially, the characteristics of resonance phenomenon are also analyzed.
2) Geometrical nonlinear theory of thin FGM shells
In genral, the reduced shell theories of homogenous material structures are not fit for FGM shells. In order to study mechanical behaviors of FGM cylindrical shell, the geometrical nonlinear theory for thin FGM shells is derived based on Kirchhoff-Love assumption by using vector tools. In derivation of the geometrical equations, the variations of deformed curve surface resulting in the direction change of basis vectors are considered, and the elongation ratio of line segment in basis vectors and the nonlinear strain and displacement relations in any point of the shell are obtained. The changes of stress distribution due to deformation are taken into account in constitutive equations. The equilibrium equations are formulated on deformed element. Above mentioned basic equations are applicable to the large deformation problems of shells. The results of degenerative forms of the equations for the thin shell with small deflection are consistent with those in the literature. For the small deflection problem of cylindrical shells, the equations of Flugge's shell theory can be obtain.
3) Large deformation and free vibration of FGM cylindrical shell under static temperature rise field
Considering the temperature-dependent properties of FGM, the temperature distribution, the thermal membrane forces, thermal bending moments and high-order thermal forces of the FGM cylindrical shell are evaluated under static thermal loads on the surface. Nonlinear governing equations for thin FGM circular cylindrical shell are reduced from the general ones derived in this dissertation. By using the numerical method, response of the static large deformation of a short FGM cylindrical shell with three kinds of symmetry boundary conditions are investigated. and the effects of the geometric and load parameters, the volume fraction of material on the deflection of the cylindrical shell. By using the time-assumed mode of the free vibration and on the basis of Hamilton principle, the governing equations in terms of displacements for free vibration of FGM cylindrical shell subjected to thermal loads are obtained. By numerical method, the relations between the linear frequencies and the geometrical and material parameters of the structure are presented.
4) Forced vibration of cylindrical shell under periodic thermal loads
According to the numerical results of the static deformation obtained in the former analysis, the fitting functions of the static bending configurations can be obtained by least square method, the space mode shape of dynamic responses are assumed by trigonometric functions. By using Hamilton principle, nonlinear ordinary differential equations in the time domain governing the large amplitude vibration of FGM cylindrical shell are derived. By analyzing the free vibration with small amplitude under static thermal loads, we arrived at the results coincided with those by shooting method, which shows that the method is feasible. The periodic solution of nonlinear forced vibration of FGM cylindrical shell under periodic surface thermal loads is investigated by multi-scale method. Further more, the dynamic stability of nonlinear vibration is examined.
The investigation of this dissertation will enrich the research results of mechanical behavior of the functionally graded structures under dynamic thermal loads, which may be helpful and beneficial in better understanding for such advanced composite material structures in the engineering application.
1)FGM圆板在表面周期动态热载荷作用下的强迫振动
设FGM板材料属性沿板的厚度连续变化,材料属性随组分材料体积分数按幂函数形式分布。首先根据表面热载荷的特点,把板内的温度场设为静态和动态两部分,基于经典热传导理论,采用打靶法数值求解了一维非稳态热传导问题得到了静动态温度场分布和相应的热内力。基于von Karman板理论,建立了FGM薄圆板的轴对称非线性控制方程。通过假设强迫振动的组合谐响应模式,采用Kantorovich时间平均法,把非线性偏微分方程化成了非线性常微分方程组。通过打靶法得到了周边简支FGM圆薄板强迫振动的数值结果。详细讨论了静动态热载荷参数对结构的固有频率的影响,特别分析了共振现象及其特征。
2)FGM薄壳的几何非线性理论
均匀材料壳的简化理论不一定适用FGM壳体。为了研究FGM柱壳的力学行为,我们基于Kirchhoff-Love假设采用矢量方法推导了FGM薄壳的几何非线性理论。几何方程推导时考虑了变形前后曲面的变化所引起的坐标方向单位矢量的变化;精确计算了坐标方向上线段的伸长比及壳体内任一点的非线性应变和位移关系。本构方程中考虑了变形引起的应力分布变化。平衡方程建立在变形以后的微元体上。此基本方程适用于壳体的大变形计算。退化得到的薄壳小挠度问题的方程同已有的结果一致。对于圆柱薄壳的小挠度问题,简化得到了Flügge理论的方程。
3)静态升温场中FGM柱壳的大变形和自由振动
考虑FGM材料属性对温度的依赖特性,求解了表面静态热载荷作用下FGM圆柱壳体内的温度场分布和静态热内力、热弯矩及高阶热内力。由本文导出的FGM薄壳的精确理论,简化得到了圆柱薄壳的非线性控制方程。利用打靶法求得了三种对称支承条件下FGM短圆柱壳的静态大变形响应,并分析了几何参数、载荷参数,材料体积分数指数对壳体挠度的影响。通过假设自由振动位移的时间模式,采用Hamilton原理得到了在热载荷下简支FGM圆柱壳自由振动的位移形式控制方程,通过数值求解,给出了结构小振幅自由振动的自然频率与各控制参数的关系。
4)周期动态热场中FGM柱壳的强迫振动
根据获得的静变形数值结果,利用最小二乘法得到了静弯曲构形的拟合函数。采用三角函数来描述动态响应的空间模态。根据Hamilton原理,得到FGM圆柱壳的非线性振动关于时间的控制方程。对静态热载荷下的小幅自由振动进行分析,其结果与打靶法所得的结果吻合较好。用多尺度法研究了FGM圆柱壳在表面周期热载荷作用下非线性强迫振动的周期解,并分析了振动的动力稳定性。
本文的研究丰富了FGM板壳结构在动态热载荷作用下力学特性的研究成果,对FGM结构的宏观力学特性的研究有借鉴作用,对于此类结构的工程实际应用具有积极意义。
Functionally graded material (FGM) is a kind of new functional materials. Through optimal of design of the material, the FGM structures can have better mechanical properties in thermal environment than homogenous material structures. Due to its virtue of physical characteristics, FGM can be used as heat-shielding material for the high-speed spacecraft, nuclear reactor and chemical engineering. The studies of mechani€al behaviors of FGM plates and shells have being become one of the important direction in solid mechanics. The present study is focus on study the static and dynamic responses of FGM circular plates and cylindrical shells subjected to thermal loads. The main research results and conclusion are summarized as follows:
1) Forced vibration of FGM circular plate with surface periodic thermal loads
The material properties of a FGM plate are graded continuously in the direction of thickness. The variation of the properties followed a simple power-law distribution in the term of the volume fraction of the constituents. In the view of the form of surface thermal loads, the through thickness temperature distribution is assumed to be two parts, static and dynamic. The temperature distributions are solved on the bases of one-dimensional unsteady-state heat conduct equation, the static and dynamic thermal membrane forces and bending moments are obtained numerically by shooting method. On the basis of von Karman's plate theory, the non-linear governing equations of thin FGM circular plate are formulated. Assuming that the vibration is harmonic and using Kantorovich time averaging method, the partial differential equations are converted into a system of nonlinear ordinary differential equations. Numerical results of thin FGM circular plates with simply-supported boundary and subjected to surface periodic thermal loads are obtained by using shooting method. The effects of the static and dynamic thermal load parameters on the natural frequencies are examined in details. Especially, the characteristics of resonance phenomenon are also analyzed.
2) Geometrical nonlinear theory of thin FGM shells
In genral, the reduced shell theories of homogenous material structures are not fit for FGM shells. In order to study mechanical behaviors of FGM cylindrical shell, the geometrical nonlinear theory for thin FGM shells is derived based on Kirchhoff-Love assumption by using vector tools. In derivation of the geometrical equations, the variations of deformed curve surface resulting in the direction change of basis vectors are considered, and the elongation ratio of line segment in basis vectors and the nonlinear strain and displacement relations in any point of the shell are obtained. The changes of stress distribution due to deformation are taken into account in constitutive equations. The equilibrium equations are formulated on deformed element. Above mentioned basic equations are applicable to the large deformation problems of shells. The results of degenerative forms of the equations for the thin shell with small deflection are consistent with those in the literature. For the small deflection problem of cylindrical shells, the equations of Flugge's shell theory can be obtain.
3) Large deformation and free vibration of FGM cylindrical shell under static temperature rise field
Considering the temperature-dependent properties of FGM, the temperature distribution, the thermal membrane forces, thermal bending moments and high-order thermal forces of the FGM cylindrical shell are evaluated under static thermal loads on the surface. Nonlinear governing equations for thin FGM circular cylindrical shell are reduced from the general ones derived in this dissertation. By using the numerical method, response of the static large deformation of a short FGM cylindrical shell with three kinds of symmetry boundary conditions are investigated. and the effects of the geometric and load parameters, the volume fraction of material on the deflection of the cylindrical shell. By using the time-assumed mode of the free vibration and on the basis of Hamilton principle, the governing equations in terms of displacements for free vibration of FGM cylindrical shell subjected to thermal loads are obtained. By numerical method, the relations between the linear frequencies and the geometrical and material parameters of the structure are presented.
4) Forced vibration of cylindrical shell under periodic thermal loads
According to the numerical results of the static deformation obtained in the former analysis, the fitting functions of the static bending configurations can be obtained by least square method, the space mode shape of dynamic responses are assumed by trigonometric functions. By using Hamilton principle, nonlinear ordinary differential equations in the time domain governing the large amplitude vibration of FGM cylindrical shell are derived. By analyzing the free vibration with small amplitude under static thermal loads, we arrived at the results coincided with those by shooting method, which shows that the method is feasible. The periodic solution of nonlinear forced vibration of FGM cylindrical shell under periodic surface thermal loads is investigated by multi-scale method. Further more, the dynamic stability of nonlinear vibration is examined.
The investigation of this dissertation will enrich the research results of mechanical behavior of the functionally graded structures under dynamic thermal loads, which may be helpful and beneficial in better understanding for such advanced composite material structures in the engineering application.
引文
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