多物理现象耦合层状复合材料均匀化研究
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摘要
随着材料科学的快速发展,诸如层压材料之类的多层复合材料在工程实际中得到了广泛应用,而在这些实际应用中所关注的主要是材料的整体宏观性能。而且,这些功能材料在实际工程应用中,通常伴随着多物理场耦合现象,比如压电材料、磁致伸缩材料等。所以多层复合材料的多物理场耦合现象的研究具有很重要的意义,本文对该类问题进行了一些研究。
本文以伴随有多物理现象的平面多层材料为研究对象,假定材料具有周期性微观结构,而且各层材料间粘结完好,基于平均场的均匀化方法,主要包含如下三个方面：
首先,对多物理场耦合现象进行了分析,将耦合多物理场处理为多个单独场和相关耦合场的组合,并通过一个无散场和无旋场用统一的控制方程将其完整表述出来；然后,基于平均场的均匀化方法和Hadamard界面连续性条件,对该类材料进行了均匀化研究,推导出了一般情况下该类材料宏观有效模量的精确表达,它能够描述材料的宏观有效性能；最后,以热传导问题和弹性问题为例处理了非耦合现象,并以压电现象和压-电-磁现象为例处理了耦合多物理现象的均匀化,结果验证了本文方法的合理性和结论的有效性。
鉴于结论的是一系列简洁紧凑、与坐标无关的张量表达式,它们可以应用于一般性情况,甚至处理更加复杂的多物理场耦合现象,这为相关问题的理论研究和实际应用提供了新的思路和可借鉴的方法。
With the rapid development of material science, layered materials like laminates are widely used in practice, and the overall properties or macroscopic effective properties of the materials are mainly concerned about in application. What's more, in application environment or through some function materials such as piezoelectric materials and magnetostrictive materials, there are often multi-physical coupling phenomena. It is significant to focus on these problems, and some investigations have been done in the paper.
Plane layered composites coupled with multi-physical phenomena are taken as study objects. The investigations are based on mean-filed homogenization theory and the assumptions that the micro-structure is periodic and each layer is well bonded. The paper includes the following aspects:
First, multi-physical phenomena are analyzed and treated as a combination of several single fields and corresponding coupled fields, and then the multi-physical phenomena are governed by a unifying constitutive equation with the help of a divergence-free vector field and a curl-free vector field. Second, based on the mean-filed homogenization and Hadamard interface continuous conditions, material homogenization process of this category is given and subsequently, the exacted general notations of macroscopic effective modulus that can fully characterize the effective properties of the material are derived. At last, thermal conductivity and elasticity are treated as uncoupled illustrative examples, then, piezoelectricity and piezomagnetoelectricity are shown as coupled examples to verify the rationality of the method and the validity of the general notations.
Since the derived solutions are tensorial notations, which are brief, coordinate-free, thus, they are suitable for a general case, even a more complicated n-coupled multi-physical case. This provides a good way for the corresponding problem both theoretically and practically.
本文以伴随有多物理现象的平面多层材料为研究对象,假定材料具有周期性微观结构,而且各层材料间粘结完好,基于平均场的均匀化方法,主要包含如下三个方面：
首先,对多物理场耦合现象进行了分析,将耦合多物理场处理为多个单独场和相关耦合场的组合,并通过一个无散场和无旋场用统一的控制方程将其完整表述出来；然后,基于平均场的均匀化方法和Hadamard界面连续性条件,对该类材料进行了均匀化研究,推导出了一般情况下该类材料宏观有效模量的精确表达,它能够描述材料的宏观有效性能；最后,以热传导问题和弹性问题为例处理了非耦合现象,并以压电现象和压-电-磁现象为例处理了耦合多物理现象的均匀化,结果验证了本文方法的合理性和结论的有效性。
鉴于结论的是一系列简洁紧凑、与坐标无关的张量表达式,它们可以应用于一般性情况,甚至处理更加复杂的多物理场耦合现象,这为相关问题的理论研究和实际应用提供了新的思路和可借鉴的方法。
With the rapid development of material science, layered materials like laminates are widely used in practice, and the overall properties or macroscopic effective properties of the materials are mainly concerned about in application. What's more, in application environment or through some function materials such as piezoelectric materials and magnetostrictive materials, there are often multi-physical coupling phenomena. It is significant to focus on these problems, and some investigations have been done in the paper.
Plane layered composites coupled with multi-physical phenomena are taken as study objects. The investigations are based on mean-filed homogenization theory and the assumptions that the micro-structure is periodic and each layer is well bonded. The paper includes the following aspects:
First, multi-physical phenomena are analyzed and treated as a combination of several single fields and corresponding coupled fields, and then the multi-physical phenomena are governed by a unifying constitutive equation with the help of a divergence-free vector field and a curl-free vector field. Second, based on the mean-filed homogenization and Hadamard interface continuous conditions, material homogenization process of this category is given and subsequently, the exacted general notations of macroscopic effective modulus that can fully characterize the effective properties of the material are derived. At last, thermal conductivity and elasticity are treated as uncoupled illustrative examples, then, piezoelectricity and piezomagnetoelectricity are shown as coupled examples to verify the rationality of the method and the validity of the general notations.
Since the derived solutions are tensorial notations, which are brief, coordinate-free, thus, they are suitable for a general case, even a more complicated n-coupled multi-physical case. This provides a good way for the corresponding problem both theoretically and practically.
引文
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[6]Tauchert,T.R. Piezothermoelastic Behavior of a Laminated Plate[J]. Journal of Thermal Stresses,1992,15(1):25-37.
[7]Tzou, H.S. and Gadre, M. Theoretical Analysis of a Multi-Layered Thin Shell Coupled with Piezoelectric Shell Actuators for Distributed Vibration Controls[J]. Journal of Sound and Vibration,1989,132(3):433-450.
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[9]曹礼群.材料物性的多尺度关联与数值模拟[J].世界科技研究与发展,24(6)：23-30.
[10]王崇愚.多尺度模型及相关分析方法[J].复杂系统与复杂性科学,2004,1(1)：9-19.
[11]何国威,夏蒙葬,柯孚久,白以龙.多尺度耦合现象：挑战和机遇[J].自然科学进展,2004,14(2)：121-124.
[12]Hashin Z, Rosen BW. The elastic moduli of heterogeneous materials[J]. ASME Journal of Applied Mechanics,1964,31(2):223-232.
[13]Chou, P.C., Carlleone, J., and Hsu, C.M. Elastic constants of layered media[J]. Journal of Composite Materials,1972,6:80-93.
[14]Frankfort, G.A., and Murat, F. Homogenization and optimal bounds in linear elasticity.Arch[J]. Rational Mechanics and Analysis.1986,94:307-334.
[15]Naoki Takano, Masaru Zako, Manabu ishizono. Multi-scale computational method for elastic bodies with globle and local heterogeneity [J]. Journal of Computer-Aided Materials Design, 2000,7:111-132.
[16]Boutin, Henri Wong. Study of thermosensitive heterogeneous media via space-time homogenization[J]. European Journal of Mechanics-A/Solids,1998,17(6):939-968.
[17]Pagano, N.J. Exact moduli of anisotropic laminates[J]. Composite Materials,1974,2:23-45.
[18]曹礼群,崔俊芝.整周期复合材料弹性结构的双尺度渐近分析[J].应用数学学报,1999,22(1)：38-46.
[19]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,1999,22(1)：331-344.
[20]崔俊芝,曹礼群.一类具有小周期系数的椭圆型边值问题的双尺度渐近分析方法[J].计算数学,1999,21(1)：19-28.
[21]潘燕环,秘醒薛,松涛.复合材料中的渐近均匀化方法[J].上海力学,1997,18(4)：290-297.
[22]戴兰宏,黄祝平.N-层涂层夹杂体复合材料有效模量显示表达[J].高分子材料科学与工程,2005,16(3)：17-19.
[23]宋士仓,崔俊芝,刘红生.复合材料稳态热传导问题对尺度计算的一个数学模型[J].应用数学,2005,18(4)：560-566.
[24]罗剑兰,曹礼群.复相介质平稳随机场热传导方程的多尺度模型与计算[J].工程热物理学报,2005,26(6)：983-985.
[25]罗剑兰,曹礼群.随机多孔复相介质稳态温度场的多尺度数值模拟[J].工程热物理学报,2001,22(5)：593-596.
[26]刘涛,邓子辰.材料弹塑性性能数值模拟的多尺度方法[J].固体力学学报,2007,28(1)：97-101
[27]刘更,刘天祥.宏观-微观多尺度数值计算方法研究进展[J].中国机械工程,2005,16(16)：1493-1498.
[28]张洪武.弹性接触颗粒状周期性结构材料力学分析的均匀化方法(一)局部RVE分析[J].复合材料学报.2001,18(4)：93-97.
[29]张洪武.弹性接触颗粒状周期性结构材料力学分析的均匀化方法(二)宏观均匀化分析[J]复合材料学报,2001,18(4)：9-102.
[30]Q. Yu, Fish. Temporal homogenization of viscodlastic and viscoplastic solids subjected to locally periodic loading[J]. Computational Mechanics,2002,29(3):199-211.
[31]Benveniste, Milton. New exact results for the effective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages[J]. Journal of the Mechanics and Physics of Solids,2003,51:1773-1813.
[32]Chen, T.Y. Exact moduli of layered piezoelectric media[J]. International Journal of Solids Structures,1997,34:847-858.
[33]Fish, Qing Yu. Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales:a coupled thermo-viscoelastic example problem[J]. Journal of Solids and Structures,2002,39:6429-6452.
[34]He. Q.C. and Bary. B. Exact relations for the effective properties of nonlinearly elastic inhomogeneous materials[J]. International journal of Multiscale Computational Engineering 2004,2:19-31.
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