周期性复合材料有效性能的均匀化计算
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摘要
本论文应用双尺度渐近展开均匀化方法,考虑了组分材料性能、纤维的形状、材料与几何随机性,对周期复合材料的有效性能进行了研究。从小参数位移展开形式出发,应用了线弹性问题的控制方程,采用摄动展开得到了一系列的摄动方程,引入了特征函数和均匀化系数得到了均匀化方法的基本理论公式。
结合均匀化方程中特征函数的控制方程,利用最小位能原理推导了求解特征函数的有限元列式。通过对特征函数的进一步推导,得到作用在单胞内部材料边界上分布力的具体形式,提出了求解特征函数的边界力方法。本文中提出的边界力方法,解释了均匀化系数构成的物理意义,增进了对均匀化过程的理解。在通用有限元软件ABAQUS中便捷的实现了求解特征函数的边界力方法,并与热应力方法计算结果进行了比较,得到了一致的结果。
在对单向纤维增强复合材料有效性能研究中,讨论了边界条件、单胞选取、纤维体积分数、纤维排列、纤维形状对宏观性能的影响。对于三维四向编织复合材料,采用两次均匀化预报了宏观有效性能,并考虑了纤维体积分数和编织角对宏观有效性能的影响。
从灵敏度分析的一般理论出发,结合均匀化系数的有限元计算公式,推导了均匀化系数的灵敏度计算公式。针对单向纤维增强复合材料,计算了均匀化系数对于弹性模量、泊松比、纤维体积分数、纤维形状的灵敏性系数。考虑了设计参数、网格尺寸对灵敏度计算精度的影响规律。通过引入了标准化的灵敏度系数,比较了同一设计参数对不同均匀化系数的影响程度。
针对复合材料细观微结构的随机性,将地质统计学中的空间变异理论,包括变异函数(变差函数),克里金估计方法引入到应用均匀化方法对复合材料有效性能的预报中。提出了复合材料性能的随机分析方法,其中包括密度函数的克里金近似,基于克里金的积分近似。考虑了单向纤维增强复合材料中弹性模量、泊松比、纤维形状的随机性对宏观性能的影响,计算了宏观性能的均值与方差,并同Monte Carlo计算结果进行了比较。
The effective properties of composites with periodicity were studied by the two-scaleasymptotic homogenization method. In the process, the performance of the componentmaterials, the geometry of the fibres and the stochastic effects were considered. Basedon the displacement expansion of the small parameter epsilon, a series of perturbationequations were obtained by applying the perturbation method to the control equationsof linear elasticity. And the basic theoretical formulas were achieved by introducing theeigendisplacement and the homogenization coeffcient.
The finite element solution of the characteristic function was derived from the controlformulas of the homogenization equations with the principle of the minimum potentialenergy. Through the further derivation, the distribution force between the interface ofthe hetergenous materials was obtained, and the boundary force method was proposed tosolve the characteristic function. To evaluate the invalidity of the method, simulationswere carried out by ABAQUS, of which the results were coincident with the ones ofthe thermal-stress method. Besides, the boundary force method is helpful to the greatercomprehension of the homogenization coeffcient and the process of homogenization.
The effective properties of fibre reinforced composites, which were divided into uni-directional fiber reinforced composites and three dimension four direction braided com-posites, were discussed in the paper. And the inffuences of the boundary conditions, thechoice of unit cell, the fibre volume fraction, the arrangement of fibres and the shape offibres on the effective properties were investigated. For the three dimension four directionbraided composites, the homogenization process was applied twice to calculate the ef-fective properties, considering the inffuences of the fibre volume fraction and the braidedangle.
The sensitivity formula of the homogenization coeffcient was derived from thegeneral homogenization-oriented sensitivity analysis with the aid of the finite elementmethod. For the unidirectional fibre reinforced composites, the sensitivity of effectiveproperties to the component modulus, the Poisson’s ratio, the fibre volume fraction andthe shape of fibres was calculated. The inffuences of the design parameters and the meshsize on the accuracy of the sensitivity were also discussed. Furthermore, the effects of the same design parameters on the different effective properties were compared by introduc-ing the scaled sensitivity coeffcient.
In the last section, the spatial variability theory in geostatistics, such as the vari-ogram and the kriging method, was introduced to the homogenization method to forecastthe effective properties of the composites in consideration of the stochastic microstruc-ture of the composites. Then the stochastic analysis method was proposed, including thekriging-based approximation of density function and kriging-based integral approxima-tion. Based on the method, the inffuences of the random variation of the elastic modulus,the Poisson’s ratio and the shape of fibres on the effective properties of the unidirec-tional fibre reinforced composites were discussed. The average and the variance of theeffective properties were also calculated, which were compared with the results of MonteCarlo simulations.
结合均匀化方程中特征函数的控制方程,利用最小位能原理推导了求解特征函数的有限元列式。通过对特征函数的进一步推导,得到作用在单胞内部材料边界上分布力的具体形式,提出了求解特征函数的边界力方法。本文中提出的边界力方法,解释了均匀化系数构成的物理意义,增进了对均匀化过程的理解。在通用有限元软件ABAQUS中便捷的实现了求解特征函数的边界力方法,并与热应力方法计算结果进行了比较,得到了一致的结果。
在对单向纤维增强复合材料有效性能研究中,讨论了边界条件、单胞选取、纤维体积分数、纤维排列、纤维形状对宏观性能的影响。对于三维四向编织复合材料,采用两次均匀化预报了宏观有效性能,并考虑了纤维体积分数和编织角对宏观有效性能的影响。
从灵敏度分析的一般理论出发,结合均匀化系数的有限元计算公式,推导了均匀化系数的灵敏度计算公式。针对单向纤维增强复合材料,计算了均匀化系数对于弹性模量、泊松比、纤维体积分数、纤维形状的灵敏性系数。考虑了设计参数、网格尺寸对灵敏度计算精度的影响规律。通过引入了标准化的灵敏度系数,比较了同一设计参数对不同均匀化系数的影响程度。
针对复合材料细观微结构的随机性,将地质统计学中的空间变异理论,包括变异函数(变差函数),克里金估计方法引入到应用均匀化方法对复合材料有效性能的预报中。提出了复合材料性能的随机分析方法,其中包括密度函数的克里金近似,基于克里金的积分近似。考虑了单向纤维增强复合材料中弹性模量、泊松比、纤维形状的随机性对宏观性能的影响,计算了宏观性能的均值与方差,并同Monte Carlo计算结果进行了比较。
The effective properties of composites with periodicity were studied by the two-scaleasymptotic homogenization method. In the process, the performance of the componentmaterials, the geometry of the fibres and the stochastic effects were considered. Basedon the displacement expansion of the small parameter epsilon, a series of perturbationequations were obtained by applying the perturbation method to the control equationsof linear elasticity. And the basic theoretical formulas were achieved by introducing theeigendisplacement and the homogenization coeffcient.
The finite element solution of the characteristic function was derived from the controlformulas of the homogenization equations with the principle of the minimum potentialenergy. Through the further derivation, the distribution force between the interface ofthe hetergenous materials was obtained, and the boundary force method was proposed tosolve the characteristic function. To evaluate the invalidity of the method, simulationswere carried out by ABAQUS, of which the results were coincident with the ones ofthe thermal-stress method. Besides, the boundary force method is helpful to the greatercomprehension of the homogenization coeffcient and the process of homogenization.
The effective properties of fibre reinforced composites, which were divided into uni-directional fiber reinforced composites and three dimension four direction braided com-posites, were discussed in the paper. And the inffuences of the boundary conditions, thechoice of unit cell, the fibre volume fraction, the arrangement of fibres and the shape offibres on the effective properties were investigated. For the three dimension four directionbraided composites, the homogenization process was applied twice to calculate the ef-fective properties, considering the inffuences of the fibre volume fraction and the braidedangle.
The sensitivity formula of the homogenization coeffcient was derived from thegeneral homogenization-oriented sensitivity analysis with the aid of the finite elementmethod. For the unidirectional fibre reinforced composites, the sensitivity of effectiveproperties to the component modulus, the Poisson’s ratio, the fibre volume fraction andthe shape of fibres was calculated. The inffuences of the design parameters and the meshsize on the accuracy of the sensitivity were also discussed. Furthermore, the effects of the same design parameters on the different effective properties were compared by introduc-ing the scaled sensitivity coeffcient.
In the last section, the spatial variability theory in geostatistics, such as the vari-ogram and the kriging method, was introduced to the homogenization method to forecastthe effective properties of the composites in consideration of the stochastic microstruc-ture of the composites. Then the stochastic analysis method was proposed, including thekriging-based approximation of density function and kriging-based integral approxima-tion. Based on the method, the inffuences of the random variation of the elastic modulus,the Poisson’s ratio and the shape of fibres on the effective properties of the unidirec-tional fibre reinforced composites were discussed. The average and the variance of theeffective properties were also calculated, which were compared with the results of MonteCarlo simulations.
引文
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