摘要
稳定性和完整性是许多承受塑性变形的零部件和结构设计中十分重要的问题，它不仅影响机器或结构的工作状态，而且关系到由于结构和部件的失稳而导致的灾难性事故，因此对使用在高温高强度下的功能梯度材料结构的安定分析十分重要。另一方面，因功能梯度材料微观组成和性能沿其厚度方向连续变化，其材料性能的描述方法对研究其力学行为有较大影响。本文在假设功能梯度材料板由线弹性颗粒相材料和各向同性弹塑性基体相材料制备的基础上，分别由一个指数函数分布模型和分段指数函数分布模型描述其材料性能的变化，采用Bree研究核燃料容器的方法分析了受到循环热―机械载荷作用的功能梯度材料板的安定性能，并在此基础上优化了板中颗粒相体积分数的分布。论文主要工作如下：
①基于内时本构方程、静力安定定理和运动安定定理，采用一个指数函数分布模型描述功能梯度材料板的材料性能，借助Bree板的分析方法建立了其安定分析的数学模型，通过数值方法对一个简单的功能梯度材料板进行了详细的安定分析，并与其等效均匀材料板的安定分析结果做了比较。
②借助Eshelby求解的一个椭球形区域点内点外的应变场，推导出颗粒之间的局部相互作用并与基于简单机械模型得到的材料组成相的弹塑性本构关系结合，建立了功能梯度材料的弹塑性本构关系，可用于预测功能梯度材料板中任一点的等效材料性能和其弹塑性响应。
③采用分段指数函数分布模型描述功能梯度材料板的材料性能变化，推导出采用此模型后的安定分析的分段模型。以Al/SiC功能梯度板为例，将其分为20层且每层上下表面处的材料性能采用考虑颗粒之间局部相互作用的细观力学模型预测，通过数值方法对板进行安定分析。
④在以上分析的基础上，以功能梯度材料板处在安定状态时所能承受的最大温度增量为目标，以板中所含颗粒的总体积百分比不变为约束条件，由Matlab语言编写出目标函数和约束函数子程序，借助Matlab软件中的遗传算法工具箱对所提出的板中颗粒体积分数分布函数进行优化，得到所优化参数满足目标函数的最优解。对优化后的板进行安定分析得到相应的安定区域，并与未优化时的板的安定区域比较，并给出各安定边界上典型载荷点对应的板的应力分布进一步分析优化结果。
The shakedown of a FG Bree plate is analyzed with static and kinematicshakedown theorems. The main work in this dissertation is listed as follows:
(1) The mathematical model of shakedown analysis of a functionally graded Breeplate is proposed. The distribution of the material properties in the thickness direction isdescribed with a single exponential function. The shakedown of the plate with thehomogenized material properties is also analyzed. The comparison the distributionadopted is not satisfied and the optimization should be performed foe better shakedowncapability.
(2) A micromechanics method considering the interactions between particles isused to predict thermal and mechanical properties of functionally graded materials, andthe piecewise exponential distribution model is adopted to describe more exactly theactual distribution of material properties in a FG plate. It aims to achieve a morerealistic distribution of the material properties for a more realistic shakedown analysis,because the piecewise exponential distribution model is capable in describing thedistribution of the actual distribution of material properties with sufficient accuracy.
(3) The distribution of the material properties in a FG structure is described withthe piecewise exponential function model. The plate is separated into a number of layers,and the thermal-mechanical properties at the lower and the upper surfaces of eachsegment are obtained with the micromechanics mean-field scheme taking into accountthe interaction between particles. The shakedown of an Al/SiC FG Bree plate isanalyzed. The piecewise concept of the distribution of the material properties is inaccord with the piecewice approach used in the shakedown analysis, therefore, theproposed method is of particularly advantageous for the analysis of the shakedown ofFG structures.
(4) An optimizeztion model for the analysis of the shakedown of the FG Bree plateis proposed, where the Genetics Algorithm is adopted, with which the distribution of theparticles in a FG plate can be optimized for different objective functions. In order toverify this model, two examples are given, in one of which an Al/SiC FG Bree plate isused, and in the other of which a Ti/Si_3N_4FG Bree plate is used. In both examples, theaverage volume fraction of the particles in the plate is fixed as15%, and the distributionof these particles is opitimized for the purpose to enhance the capability of the plate to bear the largest variation of temperature. The results show that the load-bearingcapability od the FG plate is substantially enhanced, demonstrating the capability of theproposed model.
引文
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