【作者】 史姣；

【导师】 王正中；

【作者基本信息】 西北农林科技大学 ， 水利水电工程， 2006， 硕士

【摘要】 文章给出拟膜分析法用于材料设计及分析具有周期性微结构(单胞)的二维非均质材料/结构弹性性质。在非均质材料/结构力学行为研究中,分析对象的单胞的尺度大小差别很大,可以是宏观结构、材料的细观结构,也可以是纳米结构。数值分析时,用传统的非均质连续体理论直接分析这种材料/结构费时费力。为此,人们提出均质化分析方法:先将具有周期性微结构的非均质材料均质化,再用传统的均质连续体理论分析。均质化分析思想最早是以连续体化分析方法出现于土木工程的网架结构分析中。在计算机技术尚不发达的年代里,这种方法成为大型网架结构分析的有效手段之一。即使以当前的计算能力,用有限元分析内部含有数量非常庞大的杆件的大型网架结构时,也会因为总刚度阵阶数太大而很难求解。为此,必须引入连续体化分析方法。二十世纪七十年代末出现的均匀化方法则从数学角度揭示了这种思想的本质。均匀化方法推动复合材料的研究和细观力学的发展。从数学角度来看,均匀化理论是一种极限理论。它通过渐近展开和周期性假定,用具有常数参数或变化很小的参数微分方程替代具有高振荡系数的微分方程,求解原问题的近似解。目前,均匀化方法已经发展成多尺度方法并广泛应用于多种物理和工程领域。在细观力学研究的方法中,除了均匀化分析方法还有代表体元法。后者是在近十年间,人们结合材料力学试验而提出。其基本思想是:对试样中的某点而言,它存在一个邻域,在该邻域内应力和应变平均值之间的关系与荷载无关,这种关系就是该点的宏观弹性本构关系。该点的邻域即为宏观均匀材料的代表体元。近几年来,均质化分析方法也被用于碳纳米管的弹性行为的预测中,并取得了一定成果。综上所述,均质化分析方法起到了利用成熟的连续体理论分析各种尺度的非均质材料/结构的桥梁作用。根据变形能等效原理,通过引入具有周期性微结构的二维网格结构,本文提出一种新的均质化分析方法。由于网格结构中单胞的构造方式多样,无法一一列举,文中仅以一种正交铰接单胞结构作为分析对象,深入地分析了拟膜法及其可靠性。文章工作主要包括如下内容:1.选取正交对角铰接网格结构单胞,提出拟膜的概念并给出其拟膜的弹性更多还原

【Abstract】 Pseudomembrane method is presented to analysis the material design and to predict the elastic properties of two-dimensional heterogeneous materials (or structures) with periodic microstructures (or unit cells).Generally, the scales of the unit cells of the heterogeneous materials may be diverse. The unit cells may be macrostructures, be microstructures or even be the nanostructures. In numerical simulation, it is a big trouble to analyze those materials directly on the theories of traditional continuity mechanics. For this reason, the homogenization analysis method is suggested: firstly, the unit cell of a heterogeneous material is analyzed to obtain the equivalent properties of the initial heterogeneous material, and then the response of the initial material is solved by using those equivalent properties with traditional theories. The pioneer of the homogenization analysis method is the approximation of the lattice structure by a continuum model in civil engineering. It was an effective approach to solve such lattice structure when the computational conditions were poor. Even under the current conditions of computational ability, it is still hard to analyze the lattice structures which contain large number of bars by finite element method directly. The continuum model is necessary to be adopted. The so-called homogenization method, which was presented by applied mathematicians in the end of 1970s, reveals the essential of the idea and promotes the developments both of composites and micromechanical theory. From a mathematical point of view, the theory of homogenization is a limit theory which uses the asymptotic expansion and the assumption of periodicity to substitute the differential equations with rapidly oscillating coefficients are constant or slowly varying in such a way that the solution are close to the initial problem. At present, homogenization method is evaluated to be multiscale method and widely applied in many fields of physics and engineering. Besides homogenization method in micromechanics, representative volume element (RVE) method is another excellent one, which was proposed associating with material experiments almost a decade ago. Its basic idea is summarized as: for any point of a specimen, it has a neighborhood, in which the relationship between the average strain and the average stress is independent of the loading conditions, then the relationship is the elastic constitutive property of this point, and the neighborhood is its representative volume element.更多还原