具有周期性单胞的二维非均质材料/结构的拟膜分析法
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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
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.
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
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.
引文
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[7] Noor AK, Russell WC. Anisotropic continuum models for beamlike lattice trusses [J] . Computer Methods in Applied Mechanics and Engineering, 1986, 57: 257-277.
[8] Noor AK, Continuum modeling for repetitive lattice structures [J] . Applied Mechanics Review, 1988, 41(7): 285-296.
[9] Nayfeh AH, Hefzy MS. Continuum modeling of the mechanical and thermal behavior of discrete large structures [J] . AIAA J, 1981, 19: 766-773.
[10] 董石麟,夏亨熹. 正交正放类网架结构的拟板(夹层板)分析法(上,下)[J] . 建筑结构学报,1982,3(2): 14-25; 3(3)14-25.
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[23] 崔俊芝 曹礼群. 基于双尺度渐近分析的有限元算法 [J] . 计算数学, 1998, 20(1): 89-102.
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[2] 蓝天.空间结构的十年—从中国看世界. 第六届空间结构学术会议论文集[A] .北京:地震出版社,1992,1-8.
[3] 董石麟,姚谏. 网壳结构的未来与发展[J] . 空间结构,1994,1(1): 3-10.
[4] 沈世钊. 大跨空间结构的发展——回顾与展望[J] . 土木工程学报,1998,31(3): 5-14.
[5] Noor AK, Anderson MS, Greene WH. Continuum models for beamlike and platelike lattice structures [J] . AIAA J, 1978, 16: 1219-1228.
[6] Noor AK, Anderson CM, Analysis of beam-like lattice trusses [J] . Computer Methods in Applied Mechanics and Engineering, 1979, 20: 53-70.
[7] Noor AK, Russell WC. Anisotropic continuum models for beamlike lattice trusses [J] . Computer Methods in Applied Mechanics and Engineering, 1986, 57: 257-277.
[8] Noor AK, Continuum modeling for repetitive lattice structures [J] . Applied Mechanics Review, 1988, 41(7): 285-296.
[9] Nayfeh AH, Hefzy MS. Continuum modeling of the mechanical and thermal behavior of discrete large structures [J] . AIAA J, 1981, 19: 766-773.
[10] 董石麟,夏亨熹. 正交正放类网架结构的拟板(夹层板)分析法(上,下)[J] . 建筑结构学报,1982,3(2): 14-25; 3(3)14-25.
[11] Abrate S. Continuum modeling of lattice structures [J] . Shock Vibr. Dig, 1985, 18(1): 16-21.
[12] Abrate S. Continuum modeling of lattice structures for dynamic analysis [J] . Shock Vibr. Dig, 1988, 21(10): 3-8.
[13] Abrate S. Continuum modeling of lattice structures 3 [J] . Shock Vibr. Dig., 1991, 23(3): 16-21.
[14] Tollenaere H, Caillerie D. Continuous modeling of lattice structures by homogenization [J] . Advances in Engineering Software, 1998, 29(7-9): 699-705.
[15] 钟万勰,李锡夔. JIGFEX 系统及其在大型结构分析中的应用[J] . 土木工程学报, 1982,15(3): 20-28.
[16] Vasiliev VV, Barynin VA, Rasin AF. Anisogrid lattice structures – survey of development and application [J] . Composite Structures. 2001,54(2-3): 361-370.
[17] Peng HX, Fan Z, Evanz JRG. Factors affecting the microstructure of a fine ceramic foam [J] . Ceramics International, 2000, 26(8): 887-895.
[18] Banhart J. Manufacture, characterisation and application of cellular metals and metal foams [J] . Progress in Materials Science, 2001, 46(6): 559-632.
[19] Wadley HNG, Fleck NA, Evans AG. Fabrication and structural performance of periodic cellular metal sandwich structures [J] . Composites Science and Technology, 2003, 63(16):2331-2343.
[20] Zhou J, Shrotriya P, Soboyejo WO. On the deformation of aluminum lattice block structures: from struts to structures [J] . Mechanics of Materials, 2004, 36(8): 723-737.
[21] Benssousan A, Lions JL, Papanicoulau G . Asymptotic analysis for periodic structures. Amsterdam, North-Holland, 1978.
[22] Cioranescu D, Paulin JSJ. Homogenization in open sets with holes [J] . Journal of Mathematical Analysis and Applications, 1979, 71: 590-607.
[23] 崔俊芝 曹礼群. 基于双尺度渐近分析的有限元算法 [J] . 计算数学, 1998, 20(1): 89-102.
[24] 曹礼群 崔俊芝. 复合材料拟周期结构的均匀化方法 [J] . 计算数学, 1999, 21(3): 331-344.
[25] Swan CC, Kosaka I, Homogenization-based analysis and design of composites [J] . Computers and structures, 1997, 64(1-4): 603-621.
[26] Hassani B, Henton E, A review of homogenization and topology optimization I—homogenization theory for media with periodic structure [J] . Computers and structures, 1998, 69(6): 707-717.
[27] Okada H, Fukui Y, Kumazawa N. Homogenization method for heterogeneous material based on boundary element method [J] . Computers and structures, 2001, 79(20-21): 1987-2007.
[28] Drugan WJ, Willis JR. A miciromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites [J] . Journal of the Mechanics and Physics of Solids, 1996, 44(4): 497-524.
[29] Gusev AA. Representative volume elements size for elastic composites: a numerical study [J]. Journal of the Mechanics and Physics of Solids, 1997, 45(9): 1449-1459.
[30] 胡更开,郑泉水,黄筑平. 复合材料有效弹性性质分析方法[J] . 力学进展,2001,31(3): 361-393.
[31] Ren ZY, Zheng QS. A quantitative study on minimum sizes of representative volume elements of cubic polycrystals-numerical experiments [J] . Journal of the Mechanics and Physics of Solids, 2002, 50(4): 881-893.
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