基于离散元方法的颗粒材料热传导研究
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摘要
颗粒材料的热传导问题是工程科学中面临的重要课题之一,为许多研究者所关注。一般认为,颗粒材料由离散颗粒随机组合而成,其宏观性质由颗粒的形状、尺寸、材料属性以及空间分布等参数决定。为反映其非均质特征,通常将颗粒材料模型化为离散颗粒集合体,采用离散单元法进行数值模拟。离散元法实施的关键在于准确地描述颗粒间的接触行为。因此,修正接触传热模型以提高其反映颗粒物理特征的能力,对于提高模拟颗粒体传热行为的离散元法的准确性大有裨益。由于工程实际中的颗粒体由大量颗粒构成,采用离散元法模拟其热传导问题所需的计算量往往难以承受。采用多尺度计算方法可以在准确描述颗粒材料特性的同时将计算成本控制在可接受的范围之内,因此相关研究具有重要的实际工程意义。
本文从接触传热模型和多尺度计算两方面入手,讨论了颗粒材料热传导问题的数值模拟方法,主要工作包括：通过引入接触面热阻模型,在离散元分析中考虑了颗粒表面性质对颗粒集合热传导能力的影响；发展了周期性颗粒材料各向异性热传导模拟的表征元法；发展了颗粒材料热传导分析的多尺度有限元法。具体内容如下：
简单介绍了颗粒材料以及描述颗粒的形状和表面特性、颗粒体的密度、孔隙比、应力、应变等基本参量。讨论了离散元方法的要点,并以颗粒集合直剪数值试验为例对离散元模拟的实施进行了说明。
在离散元方法中引入了接触面热阻模型,在离散元分析中计入了接触面性质对颗粒集合温度分布和热传导性能的影响,并讨论了数值模拟的主要实施步骤。计算结果显示,颗粒表面性质是影响颗粒集合热传导能力的重要参数,良好的接触面属性有助于提高颗粒集合的整体传热性能。使用所发展的离散元方法对颗粒材料内的热传导问题进行了模拟,分析了颗粒尺寸、颗粒集合体分比、配位数和所受压缩载荷大小对颗粒体传热性能的影响,结果表明上述参数在适当条件下均可影响颗粒体的传热性能。从应用的角度出发,由较大尺寸颗粒组成的密实颗粒材料,在强压缩载荷作用下将表现出相对良好的热传导性能。
针对拓扑结构确定的颗粒材料,发展了其热传导模拟的表征元法。以表征元为研究对象,讨论了获取能够表征颗粒材料各向异性有效热传导系数的均匀化策略：建立了平均热流密度与颗粒位置、颗粒接触对的热流率之间的关系；又以最优拟合的方式描述了颗粒体的平均温度梯度；以离散元模拟获取平均热流密度和平均温度梯度,通过求解傅立叶热传导方程计算有效热传导系数。有效热传导系数矩阵特征值之间的差异显示了颗粒体热传导性能各向异性的程度,反映了颗粒体内部空间结构的复杂性以及单体颗粒属性之间的差异。有效热容系数则根据表征元上的体积平均得到。与离散元分析数据的比较显示,所发展方法可有效地进行颗粒材料的热传导模拟。
发展了颗粒材料热传导分析的多尺度有限元法。对于拓扑结构确定的颗粒材料,其细观热传导信息通过以接触对为基本单元的网络模型反映,热传导模拟则在其宏观粗网格模型上进行。多尺度有限元法的核心是由粗网格单元多尺度基函数建立宏、细观网格模型间的联系,而多尺度基函数则需通过求解粗网格单元上的边值问题数值构造。通过多尺度基函数,颗粒接触对的细观热传导性质被计入粗网格单元的热传导系数阵和热容系数阵中,并在粗网格模型的热传导模拟中得以体现。由于多尺度基函数建立了细观温度场与粗网格单元节点温度之间的联系,颗粒温度可在模拟过程中以降尺度计算直接获得。数值模拟显示,所发展方法可在保证一定计算精度的同时显著提高计算效率。
Thermal conduction within granular materials is one of the challenging research topics in engineering. The thermal properties of granular materials have been discussed in many relevant studies. The macroscopic behavior of granular materials depends on the shape, size, material properties and distribution of the random packed particles, etc. The discrete element method (DEM) is widely used to simulate the behavior of granular materials, of which the key feature is to model the behavior between two particles in contact accurately. Therefore, the thermal DEM simulations of granular materials will benefit from introducing the thermal contact model with a comprehensive consideration of the physical properties of the particles. On the other hand, the computational efforts is far beyond the computer capability, which is required by the quantitatively determination of motions within granular assemblies consisting of the huge number of particles. The advantages of the multi-scale strategy of granular materials lie not only in the reliable description on the material behaviors but also in the acceptable computational efforts of simulations. It is thus necessary to develop the relevant methods.
The present work focuses on the developments of numerical methods to simulating the thermal conduction within granular materials, including:a DEM study on the thermal conduction within granular assemblies with the introduction of the thermal contact resistance, a homogenization technique to model the thermal conduction of periodic granular materials based on the analyses on the granular represent volume element (RVE) and a multi-scale method to simulate the thermal conduction within granular materials consisting of a large number of particles. The major works and findings are summarized as follows:
First, a brief introduction to granular assemblies is provided, including the definitions of the particle shape and surface roughness, the density, solid volume fraction, stress and strain tensor of the granular assemblies, etc. The key features of the DEM are discussed with the numerical direct shear tests performed.
Next, a thermal contact model is introduced into the DEM to study the thermal conduction within granular materials. The numerical examples indicate that the surface properties affect the thermal conductivity (ETC) of the granular assemblies and a smooth particle surface enhances the ETC. The effect on the ETC of the particle size, external load, solid volume fraction and coordination number of granular assemblies have also been investigated via the DEM simulations. The simulation results imply that any of the parameters could be the decisive factor to the ETC of the granular assemblies. From a practice view, a dense granular assembly consisting of relative large particles will exhibit good heat transfer behaviors under heavy compressions.
Then, a homogenization technique is proposed to simulate the thermal conduction of periodic granular materials. The ETC and the effective volumetric heat capacity (EVHC) can be obtained from the granular RVE using homogenization techniques. The average heat flux can be formulated by the positions and heat flows of particles in contact within the RVE as well as the "best fit" average temperature gradient. The ETC of the granular RVE can be computed from solving the heat equation with the average heat flux and temperature gradient obtained from DEM simulations. Moreover, the difference between the three eigenvalues reveals the anisotropy of the ETC, which exhibits the complexity of granular materials. Analogously, the EVHC can be calculated by averaging over all particles in the granular RVE. The ETC and EVHC obtained are then employed to simulate the thermal conduction procedure in periodic granular materials using finite element methods. The similar temperature profiles and heat flux given by both the DEM and DEM simulations verify the proposed techniques.
Finally, a multi-scale method is developed to simulate the thermal conduction within granular materials consisting of a large number of particles. The heat exchange between the two particles in contact is simplified as the heat flow through a truss element so that a network can be used to model these assemblies. Based on multi-scale finite element method, the fundamental idea of the method is to construct the numerical base functions considering both the topology and the material properties of the particles via solving the boundary value problems on the coarse meshes. With the thermal conduction and the heat capacity matrices computed from the contributions of all the contact pairs corresponding to the coarse element, the thermal conduction problem can be solved using the same technique as the finite element method. Moreover, the temperature of every particle can be simultaneously obtained from the downscaling computation during the simulation procedure. The numerical tests indicate that the developed method can provide fairly accurate results with the great reduction of computational efforts for solving the thermal conduction problems within granular materials.
本文从接触传热模型和多尺度计算两方面入手,讨论了颗粒材料热传导问题的数值模拟方法,主要工作包括：通过引入接触面热阻模型,在离散元分析中考虑了颗粒表面性质对颗粒集合热传导能力的影响；发展了周期性颗粒材料各向异性热传导模拟的表征元法；发展了颗粒材料热传导分析的多尺度有限元法。具体内容如下：
简单介绍了颗粒材料以及描述颗粒的形状和表面特性、颗粒体的密度、孔隙比、应力、应变等基本参量。讨论了离散元方法的要点,并以颗粒集合直剪数值试验为例对离散元模拟的实施进行了说明。
在离散元方法中引入了接触面热阻模型,在离散元分析中计入了接触面性质对颗粒集合温度分布和热传导性能的影响,并讨论了数值模拟的主要实施步骤。计算结果显示,颗粒表面性质是影响颗粒集合热传导能力的重要参数,良好的接触面属性有助于提高颗粒集合的整体传热性能。使用所发展的离散元方法对颗粒材料内的热传导问题进行了模拟,分析了颗粒尺寸、颗粒集合体分比、配位数和所受压缩载荷大小对颗粒体传热性能的影响,结果表明上述参数在适当条件下均可影响颗粒体的传热性能。从应用的角度出发,由较大尺寸颗粒组成的密实颗粒材料,在强压缩载荷作用下将表现出相对良好的热传导性能。
针对拓扑结构确定的颗粒材料,发展了其热传导模拟的表征元法。以表征元为研究对象,讨论了获取能够表征颗粒材料各向异性有效热传导系数的均匀化策略：建立了平均热流密度与颗粒位置、颗粒接触对的热流率之间的关系；又以最优拟合的方式描述了颗粒体的平均温度梯度；以离散元模拟获取平均热流密度和平均温度梯度,通过求解傅立叶热传导方程计算有效热传导系数。有效热传导系数矩阵特征值之间的差异显示了颗粒体热传导性能各向异性的程度,反映了颗粒体内部空间结构的复杂性以及单体颗粒属性之间的差异。有效热容系数则根据表征元上的体积平均得到。与离散元分析数据的比较显示,所发展方法可有效地进行颗粒材料的热传导模拟。
发展了颗粒材料热传导分析的多尺度有限元法。对于拓扑结构确定的颗粒材料,其细观热传导信息通过以接触对为基本单元的网络模型反映,热传导模拟则在其宏观粗网格模型上进行。多尺度有限元法的核心是由粗网格单元多尺度基函数建立宏、细观网格模型间的联系,而多尺度基函数则需通过求解粗网格单元上的边值问题数值构造。通过多尺度基函数,颗粒接触对的细观热传导性质被计入粗网格单元的热传导系数阵和热容系数阵中,并在粗网格模型的热传导模拟中得以体现。由于多尺度基函数建立了细观温度场与粗网格单元节点温度之间的联系,颗粒温度可在模拟过程中以降尺度计算直接获得。数值模拟显示,所发展方法可在保证一定计算精度的同时显著提高计算效率。
Thermal conduction within granular materials is one of the challenging research topics in engineering. The thermal properties of granular materials have been discussed in many relevant studies. The macroscopic behavior of granular materials depends on the shape, size, material properties and distribution of the random packed particles, etc. The discrete element method (DEM) is widely used to simulate the behavior of granular materials, of which the key feature is to model the behavior between two particles in contact accurately. Therefore, the thermal DEM simulations of granular materials will benefit from introducing the thermal contact model with a comprehensive consideration of the physical properties of the particles. On the other hand, the computational efforts is far beyond the computer capability, which is required by the quantitatively determination of motions within granular assemblies consisting of the huge number of particles. The advantages of the multi-scale strategy of granular materials lie not only in the reliable description on the material behaviors but also in the acceptable computational efforts of simulations. It is thus necessary to develop the relevant methods.
The present work focuses on the developments of numerical methods to simulating the thermal conduction within granular materials, including:a DEM study on the thermal conduction within granular assemblies with the introduction of the thermal contact resistance, a homogenization technique to model the thermal conduction of periodic granular materials based on the analyses on the granular represent volume element (RVE) and a multi-scale method to simulate the thermal conduction within granular materials consisting of a large number of particles. The major works and findings are summarized as follows:
First, a brief introduction to granular assemblies is provided, including the definitions of the particle shape and surface roughness, the density, solid volume fraction, stress and strain tensor of the granular assemblies, etc. The key features of the DEM are discussed with the numerical direct shear tests performed.
Next, a thermal contact model is introduced into the DEM to study the thermal conduction within granular materials. The numerical examples indicate that the surface properties affect the thermal conductivity (ETC) of the granular assemblies and a smooth particle surface enhances the ETC. The effect on the ETC of the particle size, external load, solid volume fraction and coordination number of granular assemblies have also been investigated via the DEM simulations. The simulation results imply that any of the parameters could be the decisive factor to the ETC of the granular assemblies. From a practice view, a dense granular assembly consisting of relative large particles will exhibit good heat transfer behaviors under heavy compressions.
Then, a homogenization technique is proposed to simulate the thermal conduction of periodic granular materials. The ETC and the effective volumetric heat capacity (EVHC) can be obtained from the granular RVE using homogenization techniques. The average heat flux can be formulated by the positions and heat flows of particles in contact within the RVE as well as the "best fit" average temperature gradient. The ETC of the granular RVE can be computed from solving the heat equation with the average heat flux and temperature gradient obtained from DEM simulations. Moreover, the difference between the three eigenvalues reveals the anisotropy of the ETC, which exhibits the complexity of granular materials. Analogously, the EVHC can be calculated by averaging over all particles in the granular RVE. The ETC and EVHC obtained are then employed to simulate the thermal conduction procedure in periodic granular materials using finite element methods. The similar temperature profiles and heat flux given by both the DEM and DEM simulations verify the proposed techniques.
Finally, a multi-scale method is developed to simulate the thermal conduction within granular materials consisting of a large number of particles. The heat exchange between the two particles in contact is simplified as the heat flow through a truss element so that a network can be used to model these assemblies. Based on multi-scale finite element method, the fundamental idea of the method is to construct the numerical base functions considering both the topology and the material properties of the particles via solving the boundary value problems on the coarse meshes. With the thermal conduction and the heat capacity matrices computed from the contributions of all the contact pairs corresponding to the coarse element, the thermal conduction problem can be solved using the same technique as the finite element method. Moreover, the temperature of every particle can be simultaneously obtained from the downscaling computation during the simulation procedure. The numerical tests indicate that the developed method can provide fairly accurate results with the great reduction of computational efforts for solving the thermal conduction problems within granular materials.
引文
[1]Vanel L, Claudin P H, Bouchaud J P H, et al. Comparison between theoretical models and new experiments [J]. Physical Review Letters,2000,84:1439-1442
[2]Mobius M E, Lauderdale B E, Nagel S R, et al. Brazil-nut effect:Size separation of granular particles [J]. Nature,2001,414:270
[3]Bak P, Tang C, Wiesenfeld K. Self-organized criticality [J]. Physical Review A,1988: 364-374
[4]王光谦,熊刚,方红卫.颗粒流动的一般本构关系[J].中国科学(E辑),1998,28：282-288
[5]Orpe A V, Khakhar D V. Solid-fluid transition in a granular shear flow [J]. Physical Review Letters,2004,93:068001
[6]Bauer R, Schlunder E U. Effective radial thermal conductivity of packings in gas flow Part Ⅱ Thermal conductivity of the packing fraction without gas flow [J]. International Chemical Engineering 1978,18:189-204
[7]Glatzmaier G C, Ramirez W F. Simultaneous measurement of the thermal conductivity and thermal diffusivity of unconsolidated materials by the transient hot wire method [J]. Review of Scientific Instruments,1985,56:1394-1398
[8]Bala K, Pradhan P R, Saxena N S, Saksena M P. Effective thermal conductivity of copper powders [J]. Journal of Physics D:Applied Physics,1989,22:1068-1072
[9]Woodside W, Messmer J H. Thermal conductivity of porous media I unconsolidated sands [J]. Journal of Applied Physics 1961,32:1688-1698
[10]Hadley G R. Thermal conductivity of packed metal powders [J]. International Journal of Heat and Mass Transfer,1986,29:909-920
[11]Chen K, Cole J, Conger C, Draskovic J, Lohr M, Klein K, Scheidemantel T, Schiffer P. Granular materials:Packing grains by thermal cycling [J]. Nature,2006,442:257
[12]Chen K, Harris A, Draskovic J, Schiffer P. Granular fragility under thermal cycles [J]. Granular Matter,2009,11:237-242
[13]Vargas W L, McCarthy J J. Stress effects on the conductivity of particulate beds [J]. Chemical Engineering Science 2002,57:3119-3131
[14]De Gennes P G. Granular matter:a tentative view [J]. Reviews of Modern Physics,1999, 71:S374-S382
[15]Cundall P A. A computer model for simulating progressive largescale movements in blocky rock systems [C]//Proceedings of Symposium of the International Society for Rock Mechanics,1971, Nancy, France:685-693
[16]徐泳,孙其诚,张凌,黄文彬.颗粒离散元法研究进展[J].力学进展,2003,33：251-260
[17]陆厚根.粉体工程导论[M].上海：同济大学出版社,1993
[18]卢寿慈.粉体技术手册[M].化学工业出版社,2004
[19]魏诗榴.粉体科学与工程[M].广州：华南理工大学出版社,2006
[20]王光谦,傅旭东.快速颗粒流本构关系在固液两相流中的适用条件初探[J].应用基础与工程科学学报,2000,8：416-424
[21]Hou M, Tu H, Liu R, Li Y, Lu K. Temperature oscillations in a compartmentalized bidisperse granular gas[J]. Physical Review Letters,2008,068001
[22]Ji S Y, Shen H H, Internal parameters and regime map for soft polydispersed granular materials [J]. Journal of Rheology,2009,52:87-103
[23]郑晓静,王萍.风沙流中沙粒随机运动的数值模拟研究[J].中国沙漠,2006,26：184-188
[24]郑晓静,黄宁,周又和.风沙运动的沙粒带电机理及其影响的研究进展[J].力学进展,2004,34：77-86
[25]吴爱祥,孙业志,刘湘平.散体动力学理论及其应用[M].冶金工业出版社,2002
[26]李静海,欧阳洁,高士秋,葛蔚,杨宁,宋文立.颗粒流体复杂系统的多尺度模拟[M].北京：科学出版社,2005
[27]Gavrilov D. Discrete Mechanics of Granular Matter [D]. Canada:The University of Calgary,1999
[28]Herrmann H J, Hovi J P, Luding S. Physics of dry granular media [M]. NATO Advanced Study Institute, Kluwer,1998
[29]Terzaghi K. Theoretical soil mechanics [M]. New York:John Wiley & Sons,1943
[30]Fredlund D G, Rahardjo H. Soil mechanics for unsaturated soils [M]. New York:John Wiley & Sons,1943
[31]徐远杰,潘家军,刘祖德.混凝土面板堆石坝的一种坝坡修整算法[J].岩土力学,2009,30(10)：3139-3144
[32]周雷.多孔介质失稳特性研究与化学-水力-力学耦合流形元分析[D].大连：大连理工大学,2007
[33]王鲲鹏.基于物质点方法饱和多孔介质动力学模拟[D].大连：大连理工大学,2009
[34]De Borst R. Simulation of strain localization:a reappraisal of the Cosserat continuum [J]. Engineering Computations,1991,8:317-332
[35]De Borst R. A generalization of the J2-floww theory for polar continua [J]. Computational Methods in Applied Mechanics and Engineering,1993,103:347-362
[36]De Borst R, Sluys L J. Localization in a Cosserat continuum under static and dynamic loading conditions [J]. Computational Methods in Applied Mechanics and Engineering, 1991,90:805-827
[37]Muhlhaus H B, Vardoulakis I. The thickness of shear bands in granular materials [J]. Geotechnique,1987,37:274-283
[38]Muhlhaus H B. Application of Cosserat theory in numerical solutions of limit load problems [J]. Archive of Applied Mechanics,1989,59:124-137
[39]Zhang H W, Wang H, Wriggers P, Schrefler B A. A finite element model for contact analysis of multiple Cosserat bodies [J]. Computational Mechanics,36(6):444-458
[40]Zhang H W, Wang H, Wang J B. Parametric variational principle based elastic-plastic analysis of materials with polygonal and Voronoi cell finite element methods [J]. Finite Elements in Analysis and Design,2007,43:206-217
[41]Zhang H W, Wang H, Chen B S, Xie Z Q. Analysis of Cosserat materials with Voronoi cell finite element and parametric variational principle [J]. Computer Methods in Applied Mechanics and Engineering,2008,197:741-755
[42]Li X K, Tang H X. A consistent return mapping algorithm for pressure-dependent elstoplastic Cosserat continua and modelling of strain localisation [J]. Computers and Structures,2005,83(1):1-10
[43]Wakao N, Kaguei S. Heat and mass transfer in packed beds [M]. New York:Gordon and Breach Sci. Pub,1982
[44]Tsotsas E, Schlunder E U.. Impact of particle size dispersity on thermal conductivity of packed beds:Measurement, numerical simulation, prediction [J]. Chemical Engineering Technology,1991,14:421-427
[45]Kaviany M. Principles of heat transfer in porous media [M]. New York:Springer,1995
[46]Jagota A, Hui C Y. The effective thermal conductivity of a packing of spheres [J]. Journal of Applied Mechanics,1990,57:789-791
[47]Siu W W M, Lee S H K. Transient temperature computation of spheres in three-dimensional random packings [J]. International Journal of Heat and Mass Transfer,2004,47: 887-898
[48]Argento C, Bouvard D. Modeling the effective thermal conductivity of random packing of spheres through densif ication [J]. International Journal of Heat and Mass Transfer, 1996,39:1343-1350
[49]Cheng G J, Yu A B, Zulli P. Evaluation of effective thermal conductivity from the structure of a packed bed [J]. Chemical Engineering Science,1999,54:4199-4209
[50]Feng Y T, Han K, Li C F, Owen D R J. Discrete thermal element modelling of heat conduction in particle systems:Basic formulation [J]. Journal of Computational Physics,2008,227:5072-5089
[51]Feng Y T, Han K, Owen D R J. Discrete thermal element modelling of heat conduction in particle systems:Pipe-network model and transient analysis [J]. Powder Technology, 2009,193:248-256
[52]Kadanof f L P. Built upon sand:Theoretical ideas inspired by granular flows [J]. Reviews of Modern Physics,1999,71:435-444
[53]Herrmann N J, Luding S, Cafiero R. Dynamics of granular systems [J]. Physics A,2001, 295:93-100
[54]Outzourhit, Trefny J U. Simple apparatus for thermal conductivity measurements of unconsolidated powders [J]. Experimental Heat Transfer,1994,7:319-331
[55]Godbee H W, Ziegler W T. Thermal conductivities of MgO, A1203, and ZrO2 powders to 850 C I Experimental [J]. Journal of Applied Physics 1966,37:40-55
[56]Fedina I, Litovsky E, Shapiro M, Shavit A. Thermal conductivity of packed beds of refractory particles:Experimental research [J]. Journal of the American Ceramic Society,1997,80:2100-2107
[57]Lin T H, Watson, Fisher P W. Thermal conductivity of iron-titanium powders [J]. Journal of Chemical & Engineering Data,1985:369-372
[58]Earnshaw J W, Londry F A. The effective thermal conductivity of a bed of 1.2-mm-diam lithium zirconate spherers in helium [J]. Fusion Technology,1998,33:31-37
[59]Tehranian F, Abdou M A. Experimental study of the effect of external pressure on particle bed effective thermal properties [J]. Fusion Technology,1995,27:298-313
[60]Tehranian F, Abdou M A, Tillack M S. Effect of external pressure on particle bed effective thermal properties [J]. Journal of Nuclear Materials,1994,212-215: 885-890
[61]Tavman S, Tavman I H. Measurement of effective thermal conductivity of wheat as a function of moisture contect [J]. International Communications in Heat Mass Transfer, 1998,25:733-741
[62]Weidenfeld G, Weiss Y, Kalman H. The effect of compression and precondsolidation on the effective thermal conductivity (ETC) of particulate beds [J]. Powder Technology, 2003,133:15-22
[63]Weidenfeld G, Weiss Y, Kalman H. A theoretical model for effective thermal conductivity (ETC) of particulate beds under compression [J]. Granular.Matter,2004, 6:121-129
[64]Alder B J, Wainwright T E. Phase transition for a hard sphere system [J]. The Journal of Chemical Physics,1957,27:1208-1209
[65]石根华.数值流形方法与非连续变形方法[M].北京：清华大学出版社,1997
[66]Monaghan J J. An introduction to SPH. Computer Physics Communications,1988,48: 279-287
[67]Chen J K, Beraun J E, Jih C J. An improvement for tensile instability in smoothed particle hydrodynamics [J]. Computers and Mechanics,1999,23:279-287
[68]Cundall P A, Strack 0 D L. A discrete numerical model for granular assemblies [J]. Geotechnique,1979,29:47-65
[69]张雄,刘岩. 《无网格法》[M].清华大学出版社/Springer,2004
[70]Vargas, W L, McCarthy J J. Heat conduction in granular materials [J]. AIChE Journal, 2001,47:1052-1059
[71]Vargas W L, McCarthy J J. Conductivity of granular media with stagnant interstitial fluids via thermal particle dynamics simulation [J]. International Journal of Heat and Mass Transfer,2002,45:4847-4856
[72]Vargas W L, McCarthy J J. Thermal expansion effects and heat conduction in granular materials [J]. Physical Review E,2007,76,041301
[73]Nguyen V D, Cogne C, Guessasma M, Bellenger E, Fortin J. Discrete modeling o granular flow with thermal transfer:Application to the discharge of silos [J]. Applied Thermal Engineering,2009,29:1846-1853
[74]Rognon R, Einav I. Thermal transients and convective particle motion in dense granular materials [J]. Physical Review Letters,2010,105,218301
[75]Strack O D L, Cundall P A. The distinct element method as a tool for research in granular media. Part I. Report to the National Science Foundation [R] Minnesota:University of Minnesota,1978.
[76]Cundall P A, Strack 0 D L. The distinct element method as a tool for research in granular media. Part II. Report to the National Science Foundation, Minnesota:University of Minnesota,1979
[77]Zhu H P, Zhou Z Y, Yang R Y, Yu A B. Discrete particle simulation of particulate systems: Theoretical developments [J]. Chemical Engineering Science,2007,62:3378-3396
[78]Maini T, Cundall P A. Computer modeling of jointed rock mass [R], (AD-AO61658),1978
[79]Cundall P A. UDEC-A Generalized distinct element program for modeling jointed rock [R]. Report PCAR-1-80 Peter Cundall Associates, European Research Office, U. S. Army, 1980
[80]Cundall P A. BALL—A program to model granular media using the distinct element method [R]. Technical Note, Advanced Technology Group, Dames & Moore, London,1978.
[81]Campbell C S, Brennen C E. Computer simulation of granular shear flows [J]. Journal of Fluid Mechanics,1985,151:167-188
[82]Campbell C S, Brennen C E. Chute flows of granular materials:some computer simulations [J]. Journal of Applied Mechanics,1985,52:172-178
[83]Kishino Y. Disc model analysis of the granular media [C]//Cambou B, Jean M, Radjai F. Micromechanics of granular materials,1988:Wiley,143-152
[84]Kishino Y. Computer analysis of dissipation mechanism in granular media [C]//Biarez J, Gourves R. Powders and Grain,1984:Taylor & Francis,323-330
[85]王泳嘉.离散元法—一种适用于节离岩石力学分析的数值方法[C]//第一届全国岩石力学数值计算及模型试验讨论论文集,1986：32-37
[86]剑万禧.离散元法的基本原理及其在岩体工程中的应用[C]//第一届全国岩石力学数值计算及模型试验讨论论文集,1986：43-46
[87]徐泳,黄文彬.离散元法研究进展[J].力学进展,2003,33(2)：251-260
[88]汪远年,李世海.断续节理岩体随机模型三维离散元数值模拟[J].岩石力学与工程学报,2004,23(21)：3652-3658
[89]汪远年,李世海.三维离散元计算参数选取方法研究[J].岩石力学与工程学报,2004,23(21)：3658-3664
[90]俞良群,邢纪波,屠居贤,杨建斌.筒仓装卸料时力场及流场的数值模拟与实验验证[J].烟台大学学报,1999,12(4)：256-262
[91]俞良群,邢纪波.筒仓装卸料时力场及流场的离散元法模拟[J].农业工程学报,2000,16(4)：15-19
[92]刘文白,周健.上拔荷载作用下桩的颗粒流数值模拟[J].岩土工程学报.2004,26(4)：517-521
[93]周健,池泳.砂土力学性质的细观模拟[J].岩土力学,2003,24(6)：901-906
[94]周健,张刚,孔戈.渗流的颗粒流细观模拟[J].水利学报,2006,37(1)：28-32
[95]刘凯欣,高凌天.离散元法在求解三维冲击动力学问题中的应用[J].固体力学学报,2004,25(2)：181-185
[96]刘凯欣,郑文刚,高凌天.脆性材料动态破坏过程的数值模拟[J].计算力学学报,2003,20(2)：127-132
[97]楚锡华.颗粒材料的离散颗粒模型与离散-连续耦合模型及数值方法[D].大连：大连理工大学,2006
[98]Li X K, Chu X H, Feng Y T. A discrete particle model and numerical modeling of the failure modes of granular materials [J]. Engineering Computations,2005(22):894-920
[99]秦建敏.基于离散元模拟的岩土力学性能研究及应变局部化理论分析[D].大连：大连理工大学,2007
[100]张洪武,秦建敏.基于接触价键的颗粒材料微观临界状态[J].岩土力学,2008,29(4);865-870
[101]Campbell CS. Computer simulation of powder flows [C]//Iinoya K, Gotoh K, Higashitani K. Powder technology handbook,1997:777-794
[102]Kruggel-Emden H, Simsek E, Rickelt S, Wirtz S, Scherer V. Review and extension of normal force models for the Discrete Element Method [J]. Powder Technology,2007, 171:157-173
[103]Thornton C. Interparticle sliding in the presence of adhesion [J]. Journal of Physics D:Applied Physics,1991,24:1942-1946
[104]Johnson K L. Contact mechanics [M]. Cambridge, UK:Cambridge University Press,1987
[105]Mindlin R D, Deresiewicz H. Elastic spheres in contact under varying oblique forces [J]. Journal of Applied Mechanics,1953,20:327-344
[106]Johnson K L, Kendall K, Roberts A D. Surface Energy and the Contact of Elastic Solids [J]. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,1971,324:301-313
[107]Savkoor A R, Briggs G A. The Effect of Tangential Force on the Contact of Elastic Solids in Adhesion [J]. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,1977 356:103-114
[108]Oda M, Konishi J, Nemat-Nasser S. Experimental micromechanical evaluation of strength of granular materials:effects of particular rolling [J]. Mechanics of Materials, 1982:269-283
[109]Bardet J P. Observations on the effects of particle rotations on the failure of idealized granular materials [J]. Mechanics of Materials,1994,18:159-182
[110]Oda M, Iwashita K, Kakiuchi T. Importance of particle rotation in the mechanics of granular materials [C]//Behringer R P, Jenkins J T. Powder & Grain 97,1997:207-214
[111]Iwashita K, Oda M. Rolling resistance at contacts in simulation of shear and development by DEM [J]. Journal of engineering mechanics,1998,124:285-292
[112]Jiang M J, Yu H S, Harris D. A novel discrete model for granular material incorporating rolling resistance [J]. Computers and Geotechnics,2005,32:340-357
[113]Yovanovich M M. Thermal contact resistance across elastically deformed spheres [J]. Journal of Spacecraft and Rockets,1967,4:119-122
[114]Holm R. Electric Contacts:Theory and Application [M]. New York:Springer-Verlag, 1967
[115]Batchelor F G K,O' Brien R W. Thermal or electrical conduction through a granular material [J]. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,1977,355:313-333.
[116]Greenwood J A, Williamson J B. Contact of nominally flat surfaces [J]. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,1966, 295:300-319
[117]Kennard E H. Kinetic Theory of Gases [M]. New York:McGraw-Hill,1938
[118]Rose W. Volumes and surface areas of pendular rings [J]. Journal of Applied physics, 1958,4:687-691
[119]Bahrami M, Yovanovich M M, Culham J R. Thermal Joint Resistances of Conforming Rough Surfaces with Gas-Filled Gaps [J]. Journal of Thermophysics and Heat Transfer,2004, 18:318-325
[120]Bahrami M, Yovanovich M M, Culham J R. Thermal Joint Resistances of Non-conforming Rough Surfaces with Gas-Filled Gaps [J]. Journal of Thermophysics and Heat Transfer, 2004,18:326-332
[121]Bahrami M., Culham J R, Yovanovich M M, Schneider GE. Review of thermal joint resistance models for non-conforming rough surfaces [J]. Applied Mechanics Reviews, 2006,59:1-12
[122]Zhang H W, Han W, Chen J T, Duan Q L. A finite element model for numerical simulation of thermo-mechanical frictional contact problems [J]. Acta Mechanica Sinica,2003, 19:551-558
[123]Ting J M, Mahmood K, Larry R M, Rowell J D. An ellipse-based discrete element model for granular materials [J]. International Journal for Numerical and Analytical Methods in Geomechanics,1993,17:603-623.
[124]Ting J M. A robust algorithm for ellipse-based discrete element modeling of granular materials [J].Geotechnical Engineering,1992,13:175-186.
[125]Lin X, Ng T T. Contact detection algorithms for three-dimensional ellipsoids in discrete element modeling [J]. International Journal for Numerical and Analytical Methods in Geomechanics 1995,19:653-659
[126]Lin X, Ng T T. A three-dimensional discrete element model using arrays of ellipsoids [J]. Geotecnique,1997,47:319-329
[127]Wang C Y, Liang V C. Packing generation scheme for the granular assemblies with planar elliptical particles [J]. International Journal For Numerical And Analytical Methods In Geomechanics,1997,21:347-358
[128]Wang C Y, Wang C F, et al. A packing generation scheme for the granular assemblies with 3D ellipsoidal particles [J]. International Journal For Numerical And Analytical Methods In Geomechanics,1999,23:815-828
[129]Kuhn M R. Smooth convex three-dimensional particle for the discrete element method [J]. Journal of Engineering Mechanics,2003,129:539-547
[130]Vinogradov 0, Sun Y. A Multibody approach in granular dynamics simulations [J]. Computational Mechanics,1997,19:287-296
[131]Moreau J J. Some numerical methods in multibody dynamics:application to granular materials [J]. European Journal of Mechanics A/Solids,1994,13:93-114
[132]Jean M. The non-smooth contact dynamics method [J]. Computer Methods in Applied Mechanics and Engineering,1999,177:235-257
[133]Andrianov I V, Awrejcewica J, Barantsev R G. Asymptotic approaches in mechanis:New parameters and procedures [J]. Applied Mechanics Reviews,2003,56 (1):87-110
[134]Zhang H W, Zhang S, Bi J Y, Schrefler B A. Thermo-mechanical analysis of periodic multiphase materials by a multi-scale asymptotic homogenization approach [J]. International Journal for Numerical Methods in Engineering,2007,69(1):87-113
[135]曹礼群,崔俊芝.整周期复合材料弹性结构的双尺度渐进分析[J].应用数学学报,1999,22(1)：38-46
[136]Wagner G L, Liu W K. Coupling of atomistic and continuum simulations using a bridging scale decomposition [J]. Journal of Computational Physics,2003,190:249-274
[137]Park H, Liu W K. An introduction and tutorial on multi-scale analysis in solids [J]. Computer Methods in Applied Mechanics and Engineering,2004,193:1733-1772
[138]Hou T Y, Wu X H. A multi-scale finite element method for elliptic problems in composite materials and porous media [J]. Journal of Computational Physics,1997,134(1): 169-189
[139]Jenny P, Lee SH, Tchelepi H A. Multi-scale finite-volume method for elliptic problems in subsurface flow simulation [J]. Journal of Computational Physics,2003,187(1): 47-67
[140]Jenny P, Lee SH, Tchelepi H A. Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media [J]. Journal of Computational Physics,2006,217(2):627-641
[141]Weinan E, Engquist B. The heterogeneous multi-scale methods [J]. Communications in Mathematical Science,2003,1(1):87-132
[142]Liu W K, Chen Y, Uras R A, Chang C T. Generalized multiple scale reproducing kernel particle methods [J]. Computer Methods in Applied Mechanics and Engineering,1996, 139(1):91-158
[143]Dorobantu M, Engquist B. Wavelet-based numerical homogenization [J]. SIAM Journal on Numerical Analysis,1998,35(2):540-559
[144]Stfantos G K, Aliabadi M H. Multi-scale boundary element modeling of material degradation and fracture [J]. European Physics Journal E,2002,9:253-259
[145]Hashin Z, Shtrikman S. A variational approach to the theory of the elastic behavior of polycrystals[J]. Journal of the Mechanics and Physics of Solids,1962,10:335-342.
[146]Hashin Z, Shtrikman S. A variational approach to the theory of the elastic behavior of multiphase materials[J]:Journal of the Mechanics and Physics of Solids,1963, 11:127-140
[147]Hill R. Elastic properties of reinforced solids:some theoretical principles[J]. Journal of the Mechanics and Physics of Solids,1963,11:357-372
[148]Hashin Z. Analysis of composite materials-a survey [J]. Journal of Applied Mechanics, 1983,50:481-505
[149]Drugan W J, Willis J R. A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites [J]. Journal of the Mechanics and Physics of Solids,1996,44:497-524
[150]Ren Z Y, Zheng Q S. A quantitative study of minimum sizes of representative volume elements of cubic polycrystals-numerical experiments [J]. Journal of the Mechanics and Physics in Solids,2002,50:881-893
[151]Liu C. Qn the minimum size of representative volume element:an experimental investigation [J]. Experimental Mechanics,2005,45:238-243
[152]Stroeven M, Askes H, Sluys L J. Numerical determination of representative volumes for granular materials [J]. Computer Methods in Applied Mechanic and Engineering, 2004,193:3221-3238
[153]Gitman I M, Askes H, Sluys L J. Representative volume:Existence and size determination [J]. Engineering Fracture Mechanics,2007,74(16):2518-2534
[154]Bianchi G, Sawczuk A. Plasticity today:modeling, methods and applications [M]. London: Elseview Applied Science Publishers,1985
[155]Ghosh S, Lee K, Moorthy S. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method [J]. International Journal of Solids and Structures,1995,32(1):27-62
[156]Terada K, Hori M, Kyoya T, Kikuchi N. Simulation of the multi-scale convergence in computational homogenization approach [J]. International Journal of Solids and Structures,2000,37:2285-2311
[157]Miehe C, Koch A. Computational micro-to-macro transition of discretized microstructures undergoing small strain [J]. Archive of Applied Mechanics,2002,72: 300-317
[158]Kouznetsova V, Geers M G D, Brekelmans W A M. Multi-scale constitutive modeling of heterogeneous materials with a gradient-enhanced computational homogenization scheme [J]. International Journal for Numerical Methods in Engineering,2002,54:1235-1260
[159]Ebinger T, Steeb H, Diebels S. Modeling macroscopic extended continua with the aid of numerical homogenization schemes [J]. Computational Material Science,2005,32: 337-347
[160]Li X K, Liu Q P. A version of Hill's lemma for Cosserat continuum [J]. Acta Mechanica Sinca,2009,25:499-506
[161]Babuska I, Szymczak W G. An error analysis for the finite element method applied to convection-diffusion problems [J]. Computer Methods in Applied Mechanics and Engineering,1982,31:19-42
[162]Babuska I, Osborn E. Generalized finite element methods:Their performance and their relation to mixed methods [J]. SIAM Journal on Numerical Analysis,1983,20:510-536
[163]Hou T Y, Wu X H, Cai Z Q. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients [J]. Mathematics of Computation,1999, 68:913-943
[164]Hou T Y, Wu X H, Zhang Y. Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation [J]. Communications in Mathematical Sciences,2004,2(2):185-205
[165]Efendiev Y, Ginting V, Hou T Y, Ewing R. Accurate multiscale finite element methods for two-phase flow simulations[J]. Journal of Computational Physics,2006,220: 155-174
[166]Hou T Y. Multiscale modelling and computation of fluid flow[J]. International Journal for numerical Methods in Fluids,2005,47:707-719
[167]Efendiev Y, Hou T Y. Multiscale finite element methods for porous media flows and their applications[J]. Applied Numerical Mathematics,2007,57:577-596
[168]Chu J, Efendiev Y, Ginting V, Hou TY. Flow based oversampling technique for multiscale finite element methods[J]. Advances in Water Resources 2008,31:599-608
[169]Zhang H W, Fu Z D, Wu J K. Coupling multiscale finite element method for consolidation analysis of heterogeneous saturated porous media [J]. Advances in Water Resources, 2009,32:268-279
[170]付振东.非均质饱和多孔介质准静态行为分析的耦合多尺度及耦合升尺度有限元法[D].大连：大连理工大学,2010.
[171]Christoffersen J, Mehrabadi M M, Nemat-Nasser S. A Micromechanical description of granular material behavior [J]. Journal of Applied Mechanics,1981,48:339-344
[172]Mehrabadi M M, Nemat-Nasser S. Stress, dilatancy and fabric in granular materials [J]. Mechanics of Materials,1983(2):155-16
[173]Rothenburg L, Bathurst R J. Analytical study of induced anisotropy in idealized granular materials[J]. Geotechnique,1989,39:601-614
[174]Majmudar T S, Behringer R P. Contact force measurements and stress-induced anisotropy in granular materials [J]. Nature,2005,435:1079-1082
[175]Peters J F, Muthuswamy M, Wibowo J, Tordesillas A. Characterization of force chains in granular material [J]. Physical Review E,2005,72,041307
[176]Bouchaud J P, Claudin P, Levine D, Otto M. Force chain splitting in granular materials: A mechanism for large-scale pseudo-elastic behaviour [J]. The European Physical Journal E,2001,4:451-458
[177]Blumenfeld R. Stresses in isostatic granular systems and emergence of force chains [J]. Physical Review Letters,2004,93,108301
[178]Tordesillas A, Walker D M, Lin Q. Force cycles and force chains [J]. Physical Review E,2010,81:011302
[179]Kruyt N P, Rothenburg L. Micromechanical definition of the strain tensor for granular materials [J]. Journal of Applied Mechanics,1996,118:706-711
[180]Bagi K. Stress and strain in granular assemblies [J]. Mechanics of Materials,1996, 22:165-177
[181]Kanatani K. A theory of contact force distribution in granular materials [J]. Powder Technology,1981,28:167-172
[182]Bagi K. Microstructural stress tensor of granular assemblies with volume forces [J]. Journal of Applied Mechanics,1999,66:934-936
[183]Ehlers W, Ramm E, Diebels, S, D'Addetta G A. From particle ensembles to Cosserat continua:homogenization of contact forces towards stresses and couple stresses [J]. International Journal of Solids and Structures,2003,40:6681-6702
[184]Bardet J P. Vardoulakis I. The asymmetry of stress in granular media [J]. International Journal of Solids and Structures,2001,38:353-367
[185]Kruyt NP. Statics and Kinematics of discrete Cosserat-type granular materials[J]. International Journal of Solids and Structures,2003,40:511-534
[186]Chang C S, Kuhn M R. On virtual work and stress in granular media [J]. Internal Journal of Solids and Structures,2005,42:3773-3793
[187]Satake M. Tensorial form definitions of discrete-mechanical quantities for granular assemblies[J]. Internal Journal of Solids and Structures,2004,41:5775-5791
[188]Bagi K. Analysis of microstructural strain tensors for granular assemblies [J]. Internal Journal of Solids and Structures,2006,43:3166-3184
[189]Kuhn MR. Structured deformation in granular materials [J]. Mechanics of Materials, 1999 31:407-429
[190]Cambou B, Chaze M, Dedecker F. Change of scale in granular materials [J]. European Journal of Mechanics A/Solids,2000,19:999-1014
[191]Dedecker F, Chaze M, Dubujet Ph, Cambou B. Specific features of strain in granular materials [J]. Mechanics of Cohesive-Frictional Materials,2000,5:173-193
[192]Liao C L, Chang T P, Young D H, Chang C S. Stress-strain relationship for granular materials based on the hypothesis of best fit [J]. Internal Journal of Solids and Structures,1997,34:4087-4100
[193]Sullivan C O, Bray J D, Li S H. A new approach for calculating strain for particulate media [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2003,27:859-877
[194]Kaneko K, Terada K, kyoya T, Kishino Y. Global-local analysis of granular media in quasi-static equilibrium [J]. International Journal of Solids and structures,2003, 40:4043-4069
[195]Tsutsumi S, Kaneko K. Constitutive response of idealized granular media under the principal axes rotation [J]. International Journal of Plasticity,2008 24:1967-1989
[196]Miehe C, Dettma J. A framework for micro-macro transitions in periodic particle aggregates of granular materials [J]. Computer Methods in Applied Mechanics and Engineering,2004,193:225-256
[197]Li X K, Liu Q P, Zhang J B. A micro-macro homogenization approach for discrete particle assembly-Cosserat continuum modeling of granular materials [J]. International Journal of Solids and Structures,2010,47(2):291-303
[198]刘其鹏.基于平均场理论的颗粒材料离散颗粒集合-Cosserat连续体模型多尺度模拟[D].大连：大连理工大学,2010
[199]Koyama T, Jing L. Effects of model scale and particle size on micro-mechanical properties and failure processes of rocks-a particle mechanics approach [J]. Engineering Analysis with Boundary Elements,2007,31:458-472
[200]Tordesillas A, Walsh S D C. Incorporating rolling resistance and contact anisotropy n micromechanical models of granular media [J]. Powder Technology,2002,124:106-111
[201]Gardiner B S, Tordesillas A. Micromechanics of shear band [J]. International Journal of Solids and Structures,2004,41:5885-5901
[202]Iinoya K, Got oh K, Higashitani K. Powder technology handbook [M]. New York, US:Marcel Dekker Inc,1990
[203]American Society of Mechanical Engineering. Surface Texture:Surface Roughness, Waviness and Lay [S]. ANSI B46.1,1985
[204]Tanner L H, Fahoum M. A study of the surface parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light [J]. Wear,1976,36: 299-316
[205]Antometti V W, White T D, Simons R E. An approximate thermal contact conductance correlation [C]//ASME 28th National Heat Transfer Conference. Minneapolis, MN:1991, HTD-Vol.170,35-42
[206]Lambert M A, Fletcher L S. Thermal contact conductance of spherical rough metals [J]. ASME Journal of Heat Transfer,1997,119(4):684-690
[207]Bahrami M, Culham J R, Yovanovich M M, Schneider G E. Review of thermal joint resistance models for nonconforming rough surfaces [J]. ASME Applied Mechanics Reviews,2006, 59:1-12
[208]Herrmann H J. Granular matter [J]. Physica A,2002,313:188-210
[209]Rothenburg L, Kruyt N P. Critical state and evolution of coordination number in simulated granular materials [J]. International Journal of Solids and Structures, 2004,41:5763-5774
[210]秦建敏,张洪武.颗粒材料的微观临界状态理论模型[J].岩土力学,2010,31(12);3697-3703
[211]季顺迎,Shen H H.颗粒介质在类固-液相变过程中的时空参数特征[J].科学通报,2006,51(3)：255-262
[212]季顺迎.非均匀颗粒材料的类固-液相变行为及本构方程[J].力学学报,2007,39(2)：223-237
[213]Oda M, Iwashita K. Mechanics of Granular Materials:An Introduction [M]. Ballkema, Rotterdam, Netherlands:Taylor & Francis Ltd,1999
[214]Fortin J, Millet O, Saxce G D. Construction of an averaged stress tensor for a granular medium [J]. European Journal of Mechanics A/Solids,2003,2:57-582
[215]Li X, Yu H S, Li X S. Macro-micro relations in granular mechanics [J]. International Journal of Solids and Structures,2009,46:4331-4341
[216]文玉华,朱如曾,周富信,王崇愚.分子动力学模拟的主要技术[J].力学进展,2003,33：65-73
[217]Vemuri B C, Chen L, Vu-Quou L, et al. Efficient and accurate collision detection for granular flow simulation [J]. Graphical Models and Image Processing,1998,60:403-422
[218]Vu-Quoc L, Zhang X, Walton 0 R. A 3-D discrete element method for dry granular flows of ellipsoidal particles [J]. Computational Methods in Applied Mechanics and Engineering,2000,187:483-528
[219]Nezami E G, Hashash M A, Zhao D W, Ghaboussi J. A fast contact detection algorithm for 3-D discrete element method. Computers and Geotechnics,2004,31:575-587.
[220]孙其诚,王光谦.颗粒物质力学导论[M].北京：科学出版社,2009
[221]Silbert L E, Ertas D, Grest G S, Halsey T C, Levine D, Plimpton S J. Granular flow down an inclined plane:Bagnold scaling and rheology [J]. Physics Review E,2001, 64:051302
[222]Kuwabara G, Kono K. Restitution coefficient in a collision between two spheres [J]. Japanese Journal of Applied Physics,1987,26:1230-1233
[223]Brilliantov N V, Spahn F,. Hertzsch J, Poschel T. Model for collisions in granular gases [J]. Physical Review E,1996,53:5382
[224]Poschel T, Schwager T. Computational Granular Dynamics:Models and Algorithms [M]. Berlin:Springer,2005
[225]Ramirez R, Roschel T, Brilliantov N V, Schwager T. Coefficient of restitutation of colliding viscoelastic spheres [J]. Physical Review E,1999,60(4):4465-4472
[226]Kremmer M, Favier J F. A method for representing boundaries in discrete element modelling-Part Ⅱ:Kinematics [J]. International Journal for Numerical Methods in Engineering,2001,51:1423-1436
[227]Parsons J D. Progress report on an investigation of the shearing resistance of cohesionless soils [C]. Proceedings of the 1st International Conference on Soil Mechanics and Foundation Engineering, Cambridge, USA,1936:133-138
[228]Cerato A B, Lutenegger A J. Specimen size and scale effects of direct shear box tests of sands [J]. Geotechnical Testing Journal,2006,29:507-516
[229]Zhang L, Thornton C. A numerical examination of the direct shear test [J]. Geotechnique, 2007,57:343-354
[230]Murakmi A, Sakaguchi H, Hasegawa T. Dislocation, vortex and couple stress in the formation of shear bands under trap-door problems [J]. Soils and foundations,1997, 37:123-135
[231]Incropera F P, DeWitt D P, Bergman T L, Lavine A S. Fundamentals of Heat and Mass Transfer (sixth ed.) [M]. New York:John Wiley & Sons,2007
[232]ABAQUS theory manual, version 6.4 [CP]. Hibbit, Karlsson and Sorensen Inc.,2003.
[233]Clausing A M, Chao B T. Thermal contact resistance in a vacuum environment [J]. ASME Journal of Heat Transfer,1965,87:243-251
[234]Bahrami M, Yovanovich M M, Culham J R A compact model for spherical rough contacts [J]. Journal of Tribology,2005,127:884-889
[235]Yovanovich M M, Hegazy A. An accurate universal contact conductance correlation for conforming rough surfaces with different micro-hardness profiles, AIAA, Paper No.83-1434,1983
[236]Sridhar M R, Yovanovich M M. Empirical methods to predict Vickers microhardness [J]. Wear,1996,193:91-98
[237]D'Addetta. Discrete models for Cohesive Frictional Materials [D].Germany: University of Stuttgart,2004
[238]Wellmann C, Lillie C, Wriggers. Homogenization of granular material modeled by a three-dimensional discrete element method [J]. Computers and Geotechnics,2008,35: 394-405
[239]Zwillinger D. Standard Mathematical Tables and Formulae 30th edition[M].CRC-press, 1996
[240]Wen X H, Durlofsky L J, Edwards M G. Use of border regions for improved permeability upscaling [J].Mathematical Geology,2003,35:521-547
[241]Zhang H W, Fu Z D. Coupling upscaling finite element method for consolidation analysis of heterogeneous saturated porous media [J]. Advances in Water Resources,2010,33: 34-47
[242]Kanit T, Forest S, Galliet I, et. al. Determination of the size of the representative volume element for random composites:statistical and numerical approach [J]. International Journal of Solids and Structures,2003,40:3647-3679
[243]Auriault J L. Effective macroscopic description for heat conduction in periodic composites [J]. International Journal of Heat and Mass Transfer,1983,26:861-869
[244]Bahrami M, Yovanovich M M, Culham J R. Effective thermal conductivity of rough spherical packed beds [J], International Journal of Heat and Mass Transfer,2006, 49:3691-3701
[245]Zhang H W, Wu J K, Fu Z D. Extended multiscale finite element method for elasto-plastic analysis of 2D periodic lattice truss materials [J]. Computational Mechanics,2010, 45:623-635
[246]Zhang H W, Wu J K, Fu Z D. Extended multiscale finite element method for mechanical analysis of periodic lattice truss materials [J]. International Journal for Multiscale Computational Engineering,2010,8(6):597-613
[247]Efendiev Y R, Hou T Y, Wu X H. Convergence of a nonconforming multiscale finite element method [J]. SIAM Journal on Numerical Analysis,2000,37:888-910
[248]Guo Z Y, Cheng X G, Xia Z Z., Least dissipation principle of heat transport potential capacity and its application in heat conduction optimization [J]. Chinese Science Bulletin,2003,48:406-410
[249]Wood W L, Lewis R H. A comparison of time marching schemes for the transient heat conduction equation [J]. International Journal for Numerical Methods in Engineering,
[2]Mobius M E, Lauderdale B E, Nagel S R, et al. Brazil-nut effect:Size separation of granular particles [J]. Nature,2001,414:270
[3]Bak P, Tang C, Wiesenfeld K. Self-organized criticality [J]. Physical Review A,1988: 364-374
[4]王光谦,熊刚,方红卫.颗粒流动的一般本构关系[J].中国科学(E辑),1998,28：282-288
[5]Orpe A V, Khakhar D V. Solid-fluid transition in a granular shear flow [J]. Physical Review Letters,2004,93:068001
[6]Bauer R, Schlunder E U. Effective radial thermal conductivity of packings in gas flow Part Ⅱ Thermal conductivity of the packing fraction without gas flow [J]. International Chemical Engineering 1978,18:189-204
[7]Glatzmaier G C, Ramirez W F. Simultaneous measurement of the thermal conductivity and thermal diffusivity of unconsolidated materials by the transient hot wire method [J]. Review of Scientific Instruments,1985,56:1394-1398
[8]Bala K, Pradhan P R, Saxena N S, Saksena M P. Effective thermal conductivity of copper powders [J]. Journal of Physics D:Applied Physics,1989,22:1068-1072
[9]Woodside W, Messmer J H. Thermal conductivity of porous media I unconsolidated sands [J]. Journal of Applied Physics 1961,32:1688-1698
[10]Hadley G R. Thermal conductivity of packed metal powders [J]. International Journal of Heat and Mass Transfer,1986,29:909-920
[11]Chen K, Cole J, Conger C, Draskovic J, Lohr M, Klein K, Scheidemantel T, Schiffer P. Granular materials:Packing grains by thermal cycling [J]. Nature,2006,442:257
[12]Chen K, Harris A, Draskovic J, Schiffer P. Granular fragility under thermal cycles [J]. Granular Matter,2009,11:237-242
[13]Vargas W L, McCarthy J J. Stress effects on the conductivity of particulate beds [J]. Chemical Engineering Science 2002,57:3119-3131
[14]De Gennes P G. Granular matter:a tentative view [J]. Reviews of Modern Physics,1999, 71:S374-S382
[15]Cundall P A. A computer model for simulating progressive largescale movements in blocky rock systems [C]//Proceedings of Symposium of the International Society for Rock Mechanics,1971, Nancy, France:685-693
[16]徐泳,孙其诚,张凌,黄文彬.颗粒离散元法研究进展[J].力学进展,2003,33：251-260
[17]陆厚根.粉体工程导论[M].上海：同济大学出版社,1993
[18]卢寿慈.粉体技术手册[M].化学工业出版社,2004
[19]魏诗榴.粉体科学与工程[M].广州：华南理工大学出版社,2006
[20]王光谦,傅旭东.快速颗粒流本构关系在固液两相流中的适用条件初探[J].应用基础与工程科学学报,2000,8：416-424
[21]Hou M, Tu H, Liu R, Li Y, Lu K. Temperature oscillations in a compartmentalized bidisperse granular gas[J]. Physical Review Letters,2008,068001
[22]Ji S Y, Shen H H, Internal parameters and regime map for soft polydispersed granular materials [J]. Journal of Rheology,2009,52:87-103
[23]郑晓静,王萍.风沙流中沙粒随机运动的数值模拟研究[J].中国沙漠,2006,26：184-188
[24]郑晓静,黄宁,周又和.风沙运动的沙粒带电机理及其影响的研究进展[J].力学进展,2004,34：77-86
[25]吴爱祥,孙业志,刘湘平.散体动力学理论及其应用[M].冶金工业出版社,2002
[26]李静海,欧阳洁,高士秋,葛蔚,杨宁,宋文立.颗粒流体复杂系统的多尺度模拟[M].北京：科学出版社,2005
[27]Gavrilov D. Discrete Mechanics of Granular Matter [D]. Canada:The University of Calgary,1999
[28]Herrmann H J, Hovi J P, Luding S. Physics of dry granular media [M]. NATO Advanced Study Institute, Kluwer,1998
[29]Terzaghi K. Theoretical soil mechanics [M]. New York:John Wiley & Sons,1943
[30]Fredlund D G, Rahardjo H. Soil mechanics for unsaturated soils [M]. New York:John Wiley & Sons,1943
[31]徐远杰,潘家军,刘祖德.混凝土面板堆石坝的一种坝坡修整算法[J].岩土力学,2009,30(10)：3139-3144
[32]周雷.多孔介质失稳特性研究与化学-水力-力学耦合流形元分析[D].大连：大连理工大学,2007
[33]王鲲鹏.基于物质点方法饱和多孔介质动力学模拟[D].大连：大连理工大学,2009
[34]De Borst R. Simulation of strain localization:a reappraisal of the Cosserat continuum [J]. Engineering Computations,1991,8:317-332
[35]De Borst R. A generalization of the J2-floww theory for polar continua [J]. Computational Methods in Applied Mechanics and Engineering,1993,103:347-362
[36]De Borst R, Sluys L J. Localization in a Cosserat continuum under static and dynamic loading conditions [J]. Computational Methods in Applied Mechanics and Engineering, 1991,90:805-827
[37]Muhlhaus H B, Vardoulakis I. The thickness of shear bands in granular materials [J]. Geotechnique,1987,37:274-283
[38]Muhlhaus H B. Application of Cosserat theory in numerical solutions of limit load problems [J]. Archive of Applied Mechanics,1989,59:124-137
[39]Zhang H W, Wang H, Wriggers P, Schrefler B A. A finite element model for contact analysis of multiple Cosserat bodies [J]. Computational Mechanics,36(6):444-458
[40]Zhang H W, Wang H, Wang J B. Parametric variational principle based elastic-plastic analysis of materials with polygonal and Voronoi cell finite element methods [J]. Finite Elements in Analysis and Design,2007,43:206-217
[41]Zhang H W, Wang H, Chen B S, Xie Z Q. Analysis of Cosserat materials with Voronoi cell finite element and parametric variational principle [J]. Computer Methods in Applied Mechanics and Engineering,2008,197:741-755
[42]Li X K, Tang H X. A consistent return mapping algorithm for pressure-dependent elstoplastic Cosserat continua and modelling of strain localisation [J]. Computers and Structures,2005,83(1):1-10
[43]Wakao N, Kaguei S. Heat and mass transfer in packed beds [M]. New York:Gordon and Breach Sci. Pub,1982
[44]Tsotsas E, Schlunder E U.. Impact of particle size dispersity on thermal conductivity of packed beds:Measurement, numerical simulation, prediction [J]. Chemical Engineering Technology,1991,14:421-427
[45]Kaviany M. Principles of heat transfer in porous media [M]. New York:Springer,1995
[46]Jagota A, Hui C Y. The effective thermal conductivity of a packing of spheres [J]. Journal of Applied Mechanics,1990,57:789-791
[47]Siu W W M, Lee S H K. Transient temperature computation of spheres in three-dimensional random packings [J]. International Journal of Heat and Mass Transfer,2004,47: 887-898
[48]Argento C, Bouvard D. Modeling the effective thermal conductivity of random packing of spheres through densif ication [J]. International Journal of Heat and Mass Transfer, 1996,39:1343-1350
[49]Cheng G J, Yu A B, Zulli P. Evaluation of effective thermal conductivity from the structure of a packed bed [J]. Chemical Engineering Science,1999,54:4199-4209
[50]Feng Y T, Han K, Li C F, Owen D R J. Discrete thermal element modelling of heat conduction in particle systems:Basic formulation [J]. Journal of Computational Physics,2008,227:5072-5089
[51]Feng Y T, Han K, Owen D R J. Discrete thermal element modelling of heat conduction in particle systems:Pipe-network model and transient analysis [J]. Powder Technology, 2009,193:248-256
[52]Kadanof f L P. Built upon sand:Theoretical ideas inspired by granular flows [J]. Reviews of Modern Physics,1999,71:435-444
[53]Herrmann N J, Luding S, Cafiero R. Dynamics of granular systems [J]. Physics A,2001, 295:93-100
[54]Outzourhit, Trefny J U. Simple apparatus for thermal conductivity measurements of unconsolidated powders [J]. Experimental Heat Transfer,1994,7:319-331
[55]Godbee H W, Ziegler W T. Thermal conductivities of MgO, A1203, and ZrO2 powders to 850 C I Experimental [J]. Journal of Applied Physics 1966,37:40-55
[56]Fedina I, Litovsky E, Shapiro M, Shavit A. Thermal conductivity of packed beds of refractory particles:Experimental research [J]. Journal of the American Ceramic Society,1997,80:2100-2107
[57]Lin T H, Watson, Fisher P W. Thermal conductivity of iron-titanium powders [J]. Journal of Chemical & Engineering Data,1985:369-372
[58]Earnshaw J W, Londry F A. The effective thermal conductivity of a bed of 1.2-mm-diam lithium zirconate spherers in helium [J]. Fusion Technology,1998,33:31-37
[59]Tehranian F, Abdou M A. Experimental study of the effect of external pressure on particle bed effective thermal properties [J]. Fusion Technology,1995,27:298-313
[60]Tehranian F, Abdou M A, Tillack M S. Effect of external pressure on particle bed effective thermal properties [J]. Journal of Nuclear Materials,1994,212-215: 885-890
[61]Tavman S, Tavman I H. Measurement of effective thermal conductivity of wheat as a function of moisture contect [J]. International Communications in Heat Mass Transfer, 1998,25:733-741
[62]Weidenfeld G, Weiss Y, Kalman H. The effect of compression and precondsolidation on the effective thermal conductivity (ETC) of particulate beds [J]. Powder Technology, 2003,133:15-22
[63]Weidenfeld G, Weiss Y, Kalman H. A theoretical model for effective thermal conductivity (ETC) of particulate beds under compression [J]. Granular.Matter,2004, 6:121-129
[64]Alder B J, Wainwright T E. Phase transition for a hard sphere system [J]. The Journal of Chemical Physics,1957,27:1208-1209
[65]石根华.数值流形方法与非连续变形方法[M].北京：清华大学出版社,1997
[66]Monaghan J J. An introduction to SPH. Computer Physics Communications,1988,48: 279-287
[67]Chen J K, Beraun J E, Jih C J. An improvement for tensile instability in smoothed particle hydrodynamics [J]. Computers and Mechanics,1999,23:279-287
[68]Cundall P A, Strack 0 D L. A discrete numerical model for granular assemblies [J]. Geotechnique,1979,29:47-65
[69]张雄,刘岩. 《无网格法》[M].清华大学出版社/Springer,2004
[70]Vargas, W L, McCarthy J J. Heat conduction in granular materials [J]. AIChE Journal, 2001,47:1052-1059
[71]Vargas W L, McCarthy J J. Conductivity of granular media with stagnant interstitial fluids via thermal particle dynamics simulation [J]. International Journal of Heat and Mass Transfer,2002,45:4847-4856
[72]Vargas W L, McCarthy J J. Thermal expansion effects and heat conduction in granular materials [J]. Physical Review E,2007,76,041301
[73]Nguyen V D, Cogne C, Guessasma M, Bellenger E, Fortin J. Discrete modeling o granular flow with thermal transfer:Application to the discharge of silos [J]. Applied Thermal Engineering,2009,29:1846-1853
[74]Rognon R, Einav I. Thermal transients and convective particle motion in dense granular materials [J]. Physical Review Letters,2010,105,218301
[75]Strack O D L, Cundall P A. The distinct element method as a tool for research in granular media. Part I. Report to the National Science Foundation [R] Minnesota:University of Minnesota,1978.
[76]Cundall P A, Strack 0 D L. The distinct element method as a tool for research in granular media. Part II. Report to the National Science Foundation, Minnesota:University of Minnesota,1979
[77]Zhu H P, Zhou Z Y, Yang R Y, Yu A B. Discrete particle simulation of particulate systems: Theoretical developments [J]. Chemical Engineering Science,2007,62:3378-3396
[78]Maini T, Cundall P A. Computer modeling of jointed rock mass [R], (AD-AO61658),1978
[79]Cundall P A. UDEC-A Generalized distinct element program for modeling jointed rock [R]. Report PCAR-1-80 Peter Cundall Associates, European Research Office, U. S. Army, 1980
[80]Cundall P A. BALL—A program to model granular media using the distinct element method [R]. Technical Note, Advanced Technology Group, Dames & Moore, London,1978.
[81]Campbell C S, Brennen C E. Computer simulation of granular shear flows [J]. Journal of Fluid Mechanics,1985,151:167-188
[82]Campbell C S, Brennen C E. Chute flows of granular materials:some computer simulations [J]. Journal of Applied Mechanics,1985,52:172-178
[83]Kishino Y. Disc model analysis of the granular media [C]//Cambou B, Jean M, Radjai F. Micromechanics of granular materials,1988:Wiley,143-152
[84]Kishino Y. Computer analysis of dissipation mechanism in granular media [C]//Biarez J, Gourves R. Powders and Grain,1984:Taylor & Francis,323-330
[85]王泳嘉.离散元法—一种适用于节离岩石力学分析的数值方法[C]//第一届全国岩石力学数值计算及模型试验讨论论文集,1986：32-37
[86]剑万禧.离散元法的基本原理及其在岩体工程中的应用[C]//第一届全国岩石力学数值计算及模型试验讨论论文集,1986：43-46
[87]徐泳,黄文彬.离散元法研究进展[J].力学进展,2003,33(2)：251-260
[88]汪远年,李世海.断续节理岩体随机模型三维离散元数值模拟[J].岩石力学与工程学报,2004,23(21)：3652-3658
[89]汪远年,李世海.三维离散元计算参数选取方法研究[J].岩石力学与工程学报,2004,23(21)：3658-3664
[90]俞良群,邢纪波,屠居贤,杨建斌.筒仓装卸料时力场及流场的数值模拟与实验验证[J].烟台大学学报,1999,12(4)：256-262
[91]俞良群,邢纪波.筒仓装卸料时力场及流场的离散元法模拟[J].农业工程学报,2000,16(4)：15-19
[92]刘文白,周健.上拔荷载作用下桩的颗粒流数值模拟[J].岩土工程学报.2004,26(4)：517-521
[93]周健,池泳.砂土力学性质的细观模拟[J].岩土力学,2003,24(6)：901-906
[94]周健,张刚,孔戈.渗流的颗粒流细观模拟[J].水利学报,2006,37(1)：28-32
[95]刘凯欣,高凌天.离散元法在求解三维冲击动力学问题中的应用[J].固体力学学报,2004,25(2)：181-185
[96]刘凯欣,郑文刚,高凌天.脆性材料动态破坏过程的数值模拟[J].计算力学学报,2003,20(2)：127-132
[97]楚锡华.颗粒材料的离散颗粒模型与离散-连续耦合模型及数值方法[D].大连：大连理工大学,2006
[98]Li X K, Chu X H, Feng Y T. A discrete particle model and numerical modeling of the failure modes of granular materials [J]. Engineering Computations,2005(22):894-920
[99]秦建敏.基于离散元模拟的岩土力学性能研究及应变局部化理论分析[D].大连：大连理工大学,2007
[100]张洪武,秦建敏.基于接触价键的颗粒材料微观临界状态[J].岩土力学,2008,29(4);865-870
[101]Campbell CS. Computer simulation of powder flows [C]//Iinoya K, Gotoh K, Higashitani K. Powder technology handbook,1997:777-794
[102]Kruggel-Emden H, Simsek E, Rickelt S, Wirtz S, Scherer V. Review and extension of normal force models for the Discrete Element Method [J]. Powder Technology,2007, 171:157-173
[103]Thornton C. Interparticle sliding in the presence of adhesion [J]. Journal of Physics D:Applied Physics,1991,24:1942-1946
[104]Johnson K L. Contact mechanics [M]. Cambridge, UK:Cambridge University Press,1987
[105]Mindlin R D, Deresiewicz H. Elastic spheres in contact under varying oblique forces [J]. Journal of Applied Mechanics,1953,20:327-344
[106]Johnson K L, Kendall K, Roberts A D. Surface Energy and the Contact of Elastic Solids [J]. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,1971,324:301-313
[107]Savkoor A R, Briggs G A. The Effect of Tangential Force on the Contact of Elastic Solids in Adhesion [J]. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,1977 356:103-114
[108]Oda M, Konishi J, Nemat-Nasser S. Experimental micromechanical evaluation of strength of granular materials:effects of particular rolling [J]. Mechanics of Materials, 1982:269-283
[109]Bardet J P. Observations on the effects of particle rotations on the failure of idealized granular materials [J]. Mechanics of Materials,1994,18:159-182
[110]Oda M, Iwashita K, Kakiuchi T. Importance of particle rotation in the mechanics of granular materials [C]//Behringer R P, Jenkins J T. Powder & Grain 97,1997:207-214
[111]Iwashita K, Oda M. Rolling resistance at contacts in simulation of shear and development by DEM [J]. Journal of engineering mechanics,1998,124:285-292
[112]Jiang M J, Yu H S, Harris D. A novel discrete model for granular material incorporating rolling resistance [J]. Computers and Geotechnics,2005,32:340-357
[113]Yovanovich M M. Thermal contact resistance across elastically deformed spheres [J]. Journal of Spacecraft and Rockets,1967,4:119-122
[114]Holm R. Electric Contacts:Theory and Application [M]. New York:Springer-Verlag, 1967
[115]Batchelor F G K,O' Brien R W. Thermal or electrical conduction through a granular material [J]. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,1977,355:313-333.
[116]Greenwood J A, Williamson J B. Contact of nominally flat surfaces [J]. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences,1966, 295:300-319
[117]Kennard E H. Kinetic Theory of Gases [M]. New York:McGraw-Hill,1938
[118]Rose W. Volumes and surface areas of pendular rings [J]. Journal of Applied physics, 1958,4:687-691
[119]Bahrami M, Yovanovich M M, Culham J R. Thermal Joint Resistances of Conforming Rough Surfaces with Gas-Filled Gaps [J]. Journal of Thermophysics and Heat Transfer,2004, 18:318-325
[120]Bahrami M, Yovanovich M M, Culham J R. Thermal Joint Resistances of Non-conforming Rough Surfaces with Gas-Filled Gaps [J]. Journal of Thermophysics and Heat Transfer, 2004,18:326-332
[121]Bahrami M., Culham J R, Yovanovich M M, Schneider GE. Review of thermal joint resistance models for non-conforming rough surfaces [J]. Applied Mechanics Reviews, 2006,59:1-12
[122]Zhang H W, Han W, Chen J T, Duan Q L. A finite element model for numerical simulation of thermo-mechanical frictional contact problems [J]. Acta Mechanica Sinica,2003, 19:551-558
[123]Ting J M, Mahmood K, Larry R M, Rowell J D. An ellipse-based discrete element model for granular materials [J]. International Journal for Numerical and Analytical Methods in Geomechanics,1993,17:603-623.
[124]Ting J M. A robust algorithm for ellipse-based discrete element modeling of granular materials [J].Geotechnical Engineering,1992,13:175-186.
[125]Lin X, Ng T T. Contact detection algorithms for three-dimensional ellipsoids in discrete element modeling [J]. International Journal for Numerical and Analytical Methods in Geomechanics 1995,19:653-659
[126]Lin X, Ng T T. A three-dimensional discrete element model using arrays of ellipsoids [J]. Geotecnique,1997,47:319-329
[127]Wang C Y, Liang V C. Packing generation scheme for the granular assemblies with planar elliptical particles [J]. International Journal For Numerical And Analytical Methods In Geomechanics,1997,21:347-358
[128]Wang C Y, Wang C F, et al. A packing generation scheme for the granular assemblies with 3D ellipsoidal particles [J]. International Journal For Numerical And Analytical Methods In Geomechanics,1999,23:815-828
[129]Kuhn M R. Smooth convex three-dimensional particle for the discrete element method [J]. Journal of Engineering Mechanics,2003,129:539-547
[130]Vinogradov 0, Sun Y. A Multibody approach in granular dynamics simulations [J]. Computational Mechanics,1997,19:287-296
[131]Moreau J J. Some numerical methods in multibody dynamics:application to granular materials [J]. European Journal of Mechanics A/Solids,1994,13:93-114
[132]Jean M. The non-smooth contact dynamics method [J]. Computer Methods in Applied Mechanics and Engineering,1999,177:235-257
[133]Andrianov I V, Awrejcewica J, Barantsev R G. Asymptotic approaches in mechanis:New parameters and procedures [J]. Applied Mechanics Reviews,2003,56 (1):87-110
[134]Zhang H W, Zhang S, Bi J Y, Schrefler B A. Thermo-mechanical analysis of periodic multiphase materials by a multi-scale asymptotic homogenization approach [J]. International Journal for Numerical Methods in Engineering,2007,69(1):87-113
[135]曹礼群,崔俊芝.整周期复合材料弹性结构的双尺度渐进分析[J].应用数学学报,1999,22(1)：38-46
[136]Wagner G L, Liu W K. Coupling of atomistic and continuum simulations using a bridging scale decomposition [J]. Journal of Computational Physics,2003,190:249-274
[137]Park H, Liu W K. An introduction and tutorial on multi-scale analysis in solids [J]. Computer Methods in Applied Mechanics and Engineering,2004,193:1733-1772
[138]Hou T Y, Wu X H. A multi-scale finite element method for elliptic problems in composite materials and porous media [J]. Journal of Computational Physics,1997,134(1): 169-189
[139]Jenny P, Lee SH, Tchelepi H A. Multi-scale finite-volume method for elliptic problems in subsurface flow simulation [J]. Journal of Computational Physics,2003,187(1): 47-67
[140]Jenny P, Lee SH, Tchelepi H A. Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media [J]. Journal of Computational Physics,2006,217(2):627-641
[141]Weinan E, Engquist B. The heterogeneous multi-scale methods [J]. Communications in Mathematical Science,2003,1(1):87-132
[142]Liu W K, Chen Y, Uras R A, Chang C T. Generalized multiple scale reproducing kernel particle methods [J]. Computer Methods in Applied Mechanics and Engineering,1996, 139(1):91-158
[143]Dorobantu M, Engquist B. Wavelet-based numerical homogenization [J]. SIAM Journal on Numerical Analysis,1998,35(2):540-559
[144]Stfantos G K, Aliabadi M H. Multi-scale boundary element modeling of material degradation and fracture [J]. European Physics Journal E,2002,9:253-259
[145]Hashin Z, Shtrikman S. A variational approach to the theory of the elastic behavior of polycrystals[J]. Journal of the Mechanics and Physics of Solids,1962,10:335-342.
[146]Hashin Z, Shtrikman S. A variational approach to the theory of the elastic behavior of multiphase materials[J]:Journal of the Mechanics and Physics of Solids,1963, 11:127-140
[147]Hill R. Elastic properties of reinforced solids:some theoretical principles[J]. Journal of the Mechanics and Physics of Solids,1963,11:357-372
[148]Hashin Z. Analysis of composite materials-a survey [J]. Journal of Applied Mechanics, 1983,50:481-505
[149]Drugan W J, Willis J R. A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites [J]. Journal of the Mechanics and Physics of Solids,1996,44:497-524
[150]Ren Z Y, Zheng Q S. A quantitative study of minimum sizes of representative volume elements of cubic polycrystals-numerical experiments [J]. Journal of the Mechanics and Physics in Solids,2002,50:881-893
[151]Liu C. Qn the minimum size of representative volume element:an experimental investigation [J]. Experimental Mechanics,2005,45:238-243
[152]Stroeven M, Askes H, Sluys L J. Numerical determination of representative volumes for granular materials [J]. Computer Methods in Applied Mechanic and Engineering, 2004,193:3221-3238
[153]Gitman I M, Askes H, Sluys L J. Representative volume:Existence and size determination [J]. Engineering Fracture Mechanics,2007,74(16):2518-2534
[154]Bianchi G, Sawczuk A. Plasticity today:modeling, methods and applications [M]. London: Elseview Applied Science Publishers,1985
[155]Ghosh S, Lee K, Moorthy S. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method [J]. International Journal of Solids and Structures,1995,32(1):27-62
[156]Terada K, Hori M, Kyoya T, Kikuchi N. Simulation of the multi-scale convergence in computational homogenization approach [J]. International Journal of Solids and Structures,2000,37:2285-2311
[157]Miehe C, Koch A. Computational micro-to-macro transition of discretized microstructures undergoing small strain [J]. Archive of Applied Mechanics,2002,72: 300-317
[158]Kouznetsova V, Geers M G D, Brekelmans W A M. Multi-scale constitutive modeling of heterogeneous materials with a gradient-enhanced computational homogenization scheme [J]. International Journal for Numerical Methods in Engineering,2002,54:1235-1260
[159]Ebinger T, Steeb H, Diebels S. Modeling macroscopic extended continua with the aid of numerical homogenization schemes [J]. Computational Material Science,2005,32: 337-347
[160]Li X K, Liu Q P. A version of Hill's lemma for Cosserat continuum [J]. Acta Mechanica Sinca,2009,25:499-506
[161]Babuska I, Szymczak W G. An error analysis for the finite element method applied to convection-diffusion problems [J]. Computer Methods in Applied Mechanics and Engineering,1982,31:19-42
[162]Babuska I, Osborn E. Generalized finite element methods:Their performance and their relation to mixed methods [J]. SIAM Journal on Numerical Analysis,1983,20:510-536
[163]Hou T Y, Wu X H, Cai Z Q. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients [J]. Mathematics of Computation,1999, 68:913-943
[164]Hou T Y, Wu X H, Zhang Y. Removing the cell resonance error in the multiscale finite element method via a Petrov-Galerkin formulation [J]. Communications in Mathematical Sciences,2004,2(2):185-205
[165]Efendiev Y, Ginting V, Hou T Y, Ewing R. Accurate multiscale finite element methods for two-phase flow simulations[J]. Journal of Computational Physics,2006,220: 155-174
[166]Hou T Y. Multiscale modelling and computation of fluid flow[J]. International Journal for numerical Methods in Fluids,2005,47:707-719
[167]Efendiev Y, Hou T Y. Multiscale finite element methods for porous media flows and their applications[J]. Applied Numerical Mathematics,2007,57:577-596
[168]Chu J, Efendiev Y, Ginting V, Hou TY. Flow based oversampling technique for multiscale finite element methods[J]. Advances in Water Resources 2008,31:599-608
[169]Zhang H W, Fu Z D, Wu J K. Coupling multiscale finite element method for consolidation analysis of heterogeneous saturated porous media [J]. Advances in Water Resources, 2009,32:268-279
[170]付振东.非均质饱和多孔介质准静态行为分析的耦合多尺度及耦合升尺度有限元法[D].大连：大连理工大学,2010.
[171]Christoffersen J, Mehrabadi M M, Nemat-Nasser S. A Micromechanical description of granular material behavior [J]. Journal of Applied Mechanics,1981,48:339-344
[172]Mehrabadi M M, Nemat-Nasser S. Stress, dilatancy and fabric in granular materials [J]. Mechanics of Materials,1983(2):155-16
[173]Rothenburg L, Bathurst R J. Analytical study of induced anisotropy in idealized granular materials[J]. Geotechnique,1989,39:601-614
[174]Majmudar T S, Behringer R P. Contact force measurements and stress-induced anisotropy in granular materials [J]. Nature,2005,435:1079-1082
[175]Peters J F, Muthuswamy M, Wibowo J, Tordesillas A. Characterization of force chains in granular material [J]. Physical Review E,2005,72,041307
[176]Bouchaud J P, Claudin P, Levine D, Otto M. Force chain splitting in granular materials: A mechanism for large-scale pseudo-elastic behaviour [J]. The European Physical Journal E,2001,4:451-458
[177]Blumenfeld R. Stresses in isostatic granular systems and emergence of force chains [J]. Physical Review Letters,2004,93,108301
[178]Tordesillas A, Walker D M, Lin Q. Force cycles and force chains [J]. Physical Review E,2010,81:011302
[179]Kruyt N P, Rothenburg L. Micromechanical definition of the strain tensor for granular materials [J]. Journal of Applied Mechanics,1996,118:706-711
[180]Bagi K. Stress and strain in granular assemblies [J]. Mechanics of Materials,1996, 22:165-177
[181]Kanatani K. A theory of contact force distribution in granular materials [J]. Powder Technology,1981,28:167-172
[182]Bagi K. Microstructural stress tensor of granular assemblies with volume forces [J]. Journal of Applied Mechanics,1999,66:934-936
[183]Ehlers W, Ramm E, Diebels, S, D'Addetta G A. From particle ensembles to Cosserat continua:homogenization of contact forces towards stresses and couple stresses [J]. International Journal of Solids and Structures,2003,40:6681-6702
[184]Bardet J P. Vardoulakis I. The asymmetry of stress in granular media [J]. International Journal of Solids and Structures,2001,38:353-367
[185]Kruyt NP. Statics and Kinematics of discrete Cosserat-type granular materials[J]. International Journal of Solids and Structures,2003,40:511-534
[186]Chang C S, Kuhn M R. On virtual work and stress in granular media [J]. Internal Journal of Solids and Structures,2005,42:3773-3793
[187]Satake M. Tensorial form definitions of discrete-mechanical quantities for granular assemblies[J]. Internal Journal of Solids and Structures,2004,41:5775-5791
[188]Bagi K. Analysis of microstructural strain tensors for granular assemblies [J]. Internal Journal of Solids and Structures,2006,43:3166-3184
[189]Kuhn MR. Structured deformation in granular materials [J]. Mechanics of Materials, 1999 31:407-429
[190]Cambou B, Chaze M, Dedecker F. Change of scale in granular materials [J]. European Journal of Mechanics A/Solids,2000,19:999-1014
[191]Dedecker F, Chaze M, Dubujet Ph, Cambou B. Specific features of strain in granular materials [J]. Mechanics of Cohesive-Frictional Materials,2000,5:173-193
[192]Liao C L, Chang T P, Young D H, Chang C S. Stress-strain relationship for granular materials based on the hypothesis of best fit [J]. Internal Journal of Solids and Structures,1997,34:4087-4100
[193]Sullivan C O, Bray J D, Li S H. A new approach for calculating strain for particulate media [J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2003,27:859-877
[194]Kaneko K, Terada K, kyoya T, Kishino Y. Global-local analysis of granular media in quasi-static equilibrium [J]. International Journal of Solids and structures,2003, 40:4043-4069
[195]Tsutsumi S, Kaneko K. Constitutive response of idealized granular media under the principal axes rotation [J]. International Journal of Plasticity,2008 24:1967-1989
[196]Miehe C, Dettma J. A framework for micro-macro transitions in periodic particle aggregates of granular materials [J]. Computer Methods in Applied Mechanics and Engineering,2004,193:225-256
[197]Li X K, Liu Q P, Zhang J B. A micro-macro homogenization approach for discrete particle assembly-Cosserat continuum modeling of granular materials [J]. International Journal of Solids and Structures,2010,47(2):291-303
[198]刘其鹏.基于平均场理论的颗粒材料离散颗粒集合-Cosserat连续体模型多尺度模拟[D].大连：大连理工大学,2010
[199]Koyama T, Jing L. Effects of model scale and particle size on micro-mechanical properties and failure processes of rocks-a particle mechanics approach [J]. Engineering Analysis with Boundary Elements,2007,31:458-472
[200]Tordesillas A, Walsh S D C. Incorporating rolling resistance and contact anisotropy n micromechanical models of granular media [J]. Powder Technology,2002,124:106-111
[201]Gardiner B S, Tordesillas A. Micromechanics of shear band [J]. International Journal of Solids and Structures,2004,41:5885-5901
[202]Iinoya K, Got oh K, Higashitani K. Powder technology handbook [M]. New York, US:Marcel Dekker Inc,1990
[203]American Society of Mechanical Engineering. Surface Texture:Surface Roughness, Waviness and Lay [S]. ANSI B46.1,1985
[204]Tanner L H, Fahoum M. A study of the surface parameters of ground and lapped metal surfaces, using specular and diffuse reflection of laser light [J]. Wear,1976,36: 299-316
[205]Antometti V W, White T D, Simons R E. An approximate thermal contact conductance correlation [C]//ASME 28th National Heat Transfer Conference. Minneapolis, MN:1991, HTD-Vol.170,35-42
[206]Lambert M A, Fletcher L S. Thermal contact conductance of spherical rough metals [J]. ASME Journal of Heat Transfer,1997,119(4):684-690
[207]Bahrami M, Culham J R, Yovanovich M M, Schneider G E. Review of thermal joint resistance models for nonconforming rough surfaces [J]. ASME Applied Mechanics Reviews,2006, 59:1-12
[208]Herrmann H J. Granular matter [J]. Physica A,2002,313:188-210
[209]Rothenburg L, Kruyt N P. Critical state and evolution of coordination number in simulated granular materials [J]. International Journal of Solids and Structures, 2004,41:5763-5774
[210]秦建敏,张洪武.颗粒材料的微观临界状态理论模型[J].岩土力学,2010,31(12);3697-3703
[211]季顺迎,Shen H H.颗粒介质在类固-液相变过程中的时空参数特征[J].科学通报,2006,51(3)：255-262
[212]季顺迎.非均匀颗粒材料的类固-液相变行为及本构方程[J].力学学报,2007,39(2)：223-237
[213]Oda M, Iwashita K. Mechanics of Granular Materials:An Introduction [M]. Ballkema, Rotterdam, Netherlands:Taylor & Francis Ltd,1999
[214]Fortin J, Millet O, Saxce G D. Construction of an averaged stress tensor for a granular medium [J]. European Journal of Mechanics A/Solids,2003,2:57-582
[215]Li X, Yu H S, Li X S. Macro-micro relations in granular mechanics [J]. International Journal of Solids and Structures,2009,46:4331-4341
[216]文玉华,朱如曾,周富信,王崇愚.分子动力学模拟的主要技术[J].力学进展,2003,33：65-73
[217]Vemuri B C, Chen L, Vu-Quou L, et al. Efficient and accurate collision detection for granular flow simulation [J]. Graphical Models and Image Processing,1998,60:403-422
[218]Vu-Quoc L, Zhang X, Walton 0 R. A 3-D discrete element method for dry granular flows of ellipsoidal particles [J]. Computational Methods in Applied Mechanics and Engineering,2000,187:483-528
[219]Nezami E G, Hashash M A, Zhao D W, Ghaboussi J. A fast contact detection algorithm for 3-D discrete element method. Computers and Geotechnics,2004,31:575-587.
[220]孙其诚,王光谦.颗粒物质力学导论[M].北京：科学出版社,2009
[221]Silbert L E, Ertas D, Grest G S, Halsey T C, Levine D, Plimpton S J. Granular flow down an inclined plane:Bagnold scaling and rheology [J]. Physics Review E,2001, 64:051302
[222]Kuwabara G, Kono K. Restitution coefficient in a collision between two spheres [J]. Japanese Journal of Applied Physics,1987,26:1230-1233
[223]Brilliantov N V, Spahn F,. Hertzsch J, Poschel T. Model for collisions in granular gases [J]. Physical Review E,1996,53:5382
[224]Poschel T, Schwager T. Computational Granular Dynamics:Models and Algorithms [M]. Berlin:Springer,2005
[225]Ramirez R, Roschel T, Brilliantov N V, Schwager T. Coefficient of restitutation of colliding viscoelastic spheres [J]. Physical Review E,1999,60(4):4465-4472
[226]Kremmer M, Favier J F. A method for representing boundaries in discrete element modelling-Part Ⅱ:Kinematics [J]. International Journal for Numerical Methods in Engineering,2001,51:1423-1436
[227]Parsons J D. Progress report on an investigation of the shearing resistance of cohesionless soils [C]. Proceedings of the 1st International Conference on Soil Mechanics and Foundation Engineering, Cambridge, USA,1936:133-138
[228]Cerato A B, Lutenegger A J. Specimen size and scale effects of direct shear box tests of sands [J]. Geotechnical Testing Journal,2006,29:507-516
[229]Zhang L, Thornton C. A numerical examination of the direct shear test [J]. Geotechnique, 2007,57:343-354
[230]Murakmi A, Sakaguchi H, Hasegawa T. Dislocation, vortex and couple stress in the formation of shear bands under trap-door problems [J]. Soils and foundations,1997, 37:123-135
[231]Incropera F P, DeWitt D P, Bergman T L, Lavine A S. Fundamentals of Heat and Mass Transfer (sixth ed.) [M]. New York:John Wiley & Sons,2007
[232]ABAQUS theory manual, version 6.4 [CP]. Hibbit, Karlsson and Sorensen Inc.,2003.
[233]Clausing A M, Chao B T. Thermal contact resistance in a vacuum environment [J]. ASME Journal of Heat Transfer,1965,87:243-251
[234]Bahrami M, Yovanovich M M, Culham J R A compact model for spherical rough contacts [J]. Journal of Tribology,2005,127:884-889
[235]Yovanovich M M, Hegazy A. An accurate universal contact conductance correlation for conforming rough surfaces with different micro-hardness profiles, AIAA, Paper No.83-1434,1983
[236]Sridhar M R, Yovanovich M M. Empirical methods to predict Vickers microhardness [J]. Wear,1996,193:91-98
[237]D'Addetta. Discrete models for Cohesive Frictional Materials [D].Germany: University of Stuttgart,2004
[238]Wellmann C, Lillie C, Wriggers. Homogenization of granular material modeled by a three-dimensional discrete element method [J]. Computers and Geotechnics,2008,35: 394-405
[239]Zwillinger D. Standard Mathematical Tables and Formulae 30th edition[M].CRC-press, 1996
[240]Wen X H, Durlofsky L J, Edwards M G. Use of border regions for improved permeability upscaling [J].Mathematical Geology,2003,35:521-547
[241]Zhang H W, Fu Z D. Coupling upscaling finite element method for consolidation analysis of heterogeneous saturated porous media [J]. Advances in Water Resources,2010,33: 34-47
[242]Kanit T, Forest S, Galliet I, et. al. Determination of the size of the representative volume element for random composites:statistical and numerical approach [J]. International Journal of Solids and Structures,2003,40:3647-3679
[243]Auriault J L. Effective macroscopic description for heat conduction in periodic composites [J]. International Journal of Heat and Mass Transfer,1983,26:861-869
[244]Bahrami M, Yovanovich M M, Culham J R. Effective thermal conductivity of rough spherical packed beds [J], International Journal of Heat and Mass Transfer,2006, 49:3691-3701
[245]Zhang H W, Wu J K, Fu Z D. Extended multiscale finite element method for elasto-plastic analysis of 2D periodic lattice truss materials [J]. Computational Mechanics,2010, 45:623-635
[246]Zhang H W, Wu J K, Fu Z D. Extended multiscale finite element method for mechanical analysis of periodic lattice truss materials [J]. International Journal for Multiscale Computational Engineering,2010,8(6):597-613
[247]Efendiev Y R, Hou T Y, Wu X H. Convergence of a nonconforming multiscale finite element method [J]. SIAM Journal on Numerical Analysis,2000,37:888-910
[248]Guo Z Y, Cheng X G, Xia Z Z., Least dissipation principle of heat transport potential capacity and its application in heat conduction optimization [J]. Chinese Science Bulletin,2003,48:406-410
[249]Wood W L, Lewis R H. A comparison of time marching schemes for the transient heat conduction equation [J]. International Journal for Numerical Methods in Engineering,