基于耦合扩展多尺度有限元方法的功能梯度材料热应力分析
Thermal stress analysis of functionally graded material based on coupling extended multiscale finite element method

以高效模拟功能梯度材料(FGM)微观非均质性对整体热力学性能的影响为研究目的, 通过随机形态描述函数(RMDF)法和体积分数的指数分布建立FGM二维微结构, 在此基础上, 发展了FGM热应力分析的耦合扩展多尺度有限元方法(CEMsFEM)。该方法基于扩展多尺度有限元方法(EMsFEM)的基本思想, 对温度场和位移场构造数值基函数, 以把微观非均质材料性质带到宏观响应中。同时为了考虑泊松效应导致的不同方向间的耦合作用, 在位移场数值基函数中增加了耦合附加项。通过数值基函数建立宏微观单元信息的映射关系, 在宏观尺度求解有效方程, 节约计算量。为了更好地考虑微观载荷的影响, 把结构的真实响应分解为宏观响应和微观扰动, 进一步推导出修正的宏观载荷向量。通过不同体积分数分布的FGM在不同载荷工况下的热应力分析算例验证了本文中方法的正确性和有效性, 最后讨论了微结构的尺寸效应对结构热力学响应的影响。 This paper aims at effectively simulating the influence of microscopic heterogeneous properties of functionally graded material (FGM) to the overall thermomechanics performances. The two-dimensional microstructures of FGM were generated based on the random morphology description functions (RMDF) method and exponential function distribution of volume fraction, and then the coupling extended multiscale finite element method (CEMsFEM) was developed for the thermal stress analysis of FGM. Based on the basic idea of extended multiscale finite element method (EMsFEM), two sets of numerical base functions of temperature and displacement were constructed to bring the microscopic heterogeneous properties to the macroscopic response. The additional coupling items for base functions of displacement were added to consider the coupling effects caused by the Poisson's ratio. The mapping relationship between the element information on macroscale and microscale was then constructed by the base functions, thus the equivalent equations were solved on macroscale and the computational complexity can be greatly reduced. To better consider the influence of microscopic load, the actual response of the structure was decomposed into macroscopic response and microscopic perturbation and then the modified macroscopic load vectors were derived. Finally the thermal stress analysis of FGM examples in different load cases was presented and demonstrates the accuracy and efficiency of the proposed method. The size effect of microscopic structure to the structural thermo-mechanical response was also discussed. 国家自然科学基金(11232003;91315302);国家"973"计划(2010CB832704);教育部博士点基金(20130041110050)