岩土类颗粒物质宏-细观力学研究进展
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摘要
岩土类颗粒物质在自然界、工程建设以及日常生活中普遍存在,其运动特性的研究在力学界已经开展了几十年.在近20年开展的一系列小尺寸物理实验中,颗粒物质表现出许多新奇现象,人们从物理角度开展了系统研究,在统计力学中,颗粒固体的流体动力学等理论研究以及实验检测技术等方面都取得突破性进展,深刻地揭示了颗粒材料的物理机制,促使力学界对颗粒物质研究中原有的概念、假设、模型和理论进行重新认识.本文主要介绍颗粒物质的细观结构特点,以Rowe应力剪胀关系为例的宏细观研究思路,非线性、非弹性本构关系的细观推导,宏观强度与颗粒摩擦系数的关系,尺寸效应和相似关系等基础问题,介绍了岩土颗粒力学的新进展,介绍了常用的实验量测手段.基于我们的经验,提出了今后需要加强的几个方面:如分析颗粒离散元理论和计算细节;开展理想实验来验证理论和计算结果;结合迅速发展的高性能计算能力,开展高科学置信度大规模计算.目前,岩土类颗粒物质的宏细观研究尚处于起步阶段,到达实际工程应用还有很长的路要走;相关的工程技术要取得实质进展,还需依赖颗粒物质等基础研究的突破.
A granular material is a conglomeration of discrete solid,macroscopic particles characterized by a loss of energy whenever the particles interact.Granular materials are omnipresent in nature,industrial practice and daily life,with size spanning from 1μm to a few meters.The mechanical properties have long been studied. In recent two decades,many laboratory scale tests have been conducted and novel phenomena were observed. Both theoretical studies,such as granular matter statistical mechanics,granular solid hydrodynamics,and measurement techniques,such as X-ray CT,photoelastics,have witnessed great breakthrough.The revealed physical mechanism prompt ones to examine the assumptions,concepts,models and theories that previously proposed and being used.In this review paper,major fundamental problems are presented introduced and discussed,including the characteristics of microscale structure,procedure of micro-macro study with Rowe stress dilatancy relation as an example,derivation of stress and strain from microscale structure,the relation of macroscale strength with particle friction,size effect and similarity.In addition,granular soil as a typical granular material,its new developments are summarized.The frequently used techniques are introduced as well. Finally,we present a few aspects in which research should be reinforced,such as examining the assumptions in the discrete element method and checking the computing details,conducting the ideal physical and numerical experiments;by lying on the fast development of high performance computing,carrying on high scientific confidence degree large scale computation of granular materials with high scientific confidence degree;with the help of fast development of high performence computers.The mechanics of granular materials is still at its infancy stage,and there is a long way to directly solve the engineering problems.Essential development of related engineering technologies depends certainly on progress or break-through in the of fundamental studies of granular material mechanics.
A granular material is a conglomeration of discrete solid,macroscopic particles characterized by a loss of energy whenever the particles interact.Granular materials are omnipresent in nature,industrial practice and daily life,with size spanning from 1μm to a few meters.The mechanical properties have long been studied. In recent two decades,many laboratory scale tests have been conducted and novel phenomena were observed. Both theoretical studies,such as granular matter statistical mechanics,granular solid hydrodynamics,and measurement techniques,such as X-ray CT,photoelastics,have witnessed great breakthrough.The revealed physical mechanism prompt ones to examine the assumptions,concepts,models and theories that previously proposed and being used.In this review paper,major fundamental problems are presented introduced and discussed,including the characteristics of microscale structure,procedure of micro-macro study with Rowe stress dilatancy relation as an example,derivation of stress and strain from microscale structure,the relation of macroscale strength with particle friction,size effect and similarity.In addition,granular soil as a typical granular material,its new developments are summarized.The frequently used techniques are introduced as well. Finally,we present a few aspects in which research should be reinforced,such as examining the assumptions in the discrete element method and checking the computing details,conducting the ideal physical and numerical experiments;by lying on the fast development of high performance computing,carrying on high scientific confidence degree large scale computation of granular materials with high scientific confidence degree;with the help of fast development of high performence computers.The mechanics of granular materials is still at its infancy stage,and there is a long way to directly solve the engineering problems.Essential development of related engineering technologies depends certainly on progress or break-through in the of fundamental studies of granular material mechanics.
引文
1孙其诚,王光谦.颗粒物质力学导论.北京:科学出版社, 2009
2黄文熙.土的工程性质北京:水利电力出版社,1983
3蒋亦民,刘佑.砂土的流体动力学方程与本构模型的比较.岩土力学,2010,31(6):1729-1738
4 Kolyrnbas D.Introduction to Hypoplasticity.In:Adavances in Geotechnical Engineering.Rotterdam:Balkema A A,2000
5 Huang W X,Bauer E.Numerical Investigations of shear localization in a micro-polar hypoplastic material.Int J Numer Anal Method Geomechan,2003,27:325-352
6 Jiang Y M,Liu M.Granular solid hydrodynamics.Granular Matter,2009,11:139-156
7蒋亦民,刘佑.基于流体动力学理论的颗粒物质本构关系.科学通报,2009,54:1504-1510
8 Edwards S F,Oakeshott R B S.Theory of powders.Phys- ica A,1989,157:1080-1090
9 Edwards S F.The full canonical ensemble of a granular system.Physica A,2005,353:114-118
10 Coniglio A,de Candia A,Fierro A,et al.On Edwards's theory of powders.Physica A,2004,339:1-6
11陆坤权,刘寄星.软物质物理学导论.北京:北京大学出版社,2006
12孙其诚,厚美瑛,金峰.颗粒物质物理与力学.北京:科学出版社,2011
13殷跃平,潘桂堂,刘宇平,等.汶川地震地质与滑坡灾害概论北京:地质出版社,2009
14 Pica Ciamarra M,Lippiello E,Godano C,et al.Unjamming dynamics:the micromechanics of a seismic fault model.Phys Rev Lett,2010,104:238001
15 Cheng X.Experimental study of the jamming transition at zero temperature.Phys Rev E,2010,81:031301
16 Snoeijer J H,Vlugt T J H,van Hecke M,et al.Force network ensemble:a new Approach to static granular matter. Phys Rev Lett,2004,92:054302
17 Rowe P W.The stress-dilatancy relation for static equilibrium of an assembly of particles in contact.Proc R Soc A,1962,269(1339):500-527
18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:[博士论文].大连:大连理工大学,2007.71-74
19 Skinner A E.A note on the influence of interparticle friction on the shearing strength of a random assembly of spherical particles.Geotechnique,1969,19(1):150-157
20 Oda M,Nemat-Nasser S,Mehrabadi M M.A statistical study of fabric in a random assembly of spherical granules. Int J Num.Anal Meth Geomech,1982,6:77-94
21 Davy C A,Fleck N A,Bolton M D.The collapse behaviour of a sugar aggregate.Acta Materialia.,2004,52:3587- 3601
22 Suiker A S J,Fleck N A.Frictional collapse of granular assemblies.J Appl Mech,2004,71(3):350-358
23 Kruyt N P,Rothenburg L.Shear strength,dilatancy,energy and dissipation in quasi-static deformation of granular materials.J Stat Mech,2006,P07021
24 Thornton C.Numerical simulations of deviatoric shear deformation of granular media.Geotechnique,2000,50(1): 43-53
25 Iwashita K,Oda M.Rolling resistance at contacts in simulation of shear band development by DEM.J Enq Mech ASCE,1998,124:285-292
26 Jiang M J,Yu H S,Harris D.A novel discrete model for granular material incorporating rolling resistance.Comput Geotech,2005,32:340-357
27常在,杨军,程晓辉.砂土强度和剪胀性的颗粒力学分析.工程力学,2010,27(4):095-104
28刘海涛.无黏性颗粒材料剪切试验和贯入试验的离散元分析:[硕士论文].北京:清华大学,2010
29刘海涛,程晓辉.粗粒土尺寸效应的离散元研究岩土力学, 2009,30(增刊):287-292
30 Li X.Micro-scale investigation on the quasi-static behavior of granular material:[PhD Thesis].Hong Kong:The Hong Kong University of Science and Technology,2006
31 Li X,Li X S.Micro-macro quantification of the internal structure of granular materials.J Eng Mech,2009,135 (7):641-656
32 Bagi K.Microstructural stress tensor of granular assemblies with volume forces.J Appl Mech,1999,66(4),934- 936
33 Love A E H.A Treatise of Mathematical Theory of Elasticity. Cambridge:Cambridge University Press,1927
34 Goddard J.An elastohydrodynamics theory for the rheology of concentrated suspensions of deformable particles. J Non-Newtonian Fluid Mech,1977,2:169-189
35 Rothenburg L,Selvadurai,A P S,A micromechanical definition of the Cauchy stress tensor for particular media. In:Selvadurai,A.ed.Mechanics of Structured Media. Amsterdam:Elsevier,1981.469-486
36 Bagi K.Stress and strain in granular assemblies.Mech Mater,1996,22:165-177
37 Kruyt N P,Rothenburg L.Kinematic and static assumptions for homogenization in micromechanics of granular materials.Mech Mater,2004,36:1157-1173
38 Li X,Yu H S,Li X S.Macro-micro relations in granular mechanics.Int J Solids Struct,2009,46:4331-4341
39 De Saxce G,Fortin J,Millet O.About the numerical simulation of the dynamics of granular media and the definition of the mean stress tensor.Mech Mater,2004,36: 1175-1184
40 Bardet J P,Vardoulakis I.The asymmetry of stress in granular media.Int J Solids Struct,2001:38:353-367
41 Bardet J P,Vardoulakis I.Reply to Dr.Kuhn's discussion. Int J Solids Struct,2003,40:1809
42 Kruyt N P.Statics and Kinematics of discrete Cosserattype granular materials.Int J Solids Struct,2003,40: 511-534
43 Kuhn M R.Discussion of "The asymmetry of stress in granular media".Int J Solids Struct,2003,40:1805-1807
44 Chang C S,Kuhn M R.On virtual work and stress in granular media.Int J Solids Struct,2005,42:2373-3779
45 Bagi K.Analysis of microstructural strain tensor for granular assemblies.Int J Solids Struct,2006,43(1):3166- 3184
46 Varadarajan A,Sharma K G,Venkatachalam K,et al. Testing and modeling two rockfill materials.J Geotech Geoenviron Eng ASCE,2003,129(3):206-218
47 Aqil U,Tatsuoka F,Uchimura T,et al.Strength and deformation characteristics of recycled concrete aggregate as a backfill material.Soils Found,2005,45(5):53-72
48 Huang H Y.Discrete element modeling of tool-rock interaction: [PhD Thesis].Minnesota:University of Minnesota, US,1999
49 Yang B,Jiao Y,Lei S.A study on the effects of microparameters on macroproperties for specimens created by bonded particles.Int J Computer-Aided Eng Software, 2006,23:607-631
50 Nardin A,Schrefler B A.Modelling of cutting tool-soil interaction-Part II:macromechanical model and upscaling. Computer Mech,2005,36:343-359
51 Potyondy D O,Cundall P A.A bonded particle model for rock.Int J Rock Mech Mining Sci,2004,41:1329-1364
52 Dantu P.Contribution a l'etude mecanique et geometrique des milieux pulverulents.In:Proceedings of the 4th International Conference on Soil Mechanics and Foundation Engineering,1957,1:144-148
53 Wakabayashi T.Photoelastic method for determination of stress in powdered mass.In:Proc 7th Jpn Nat Cong Appl Mech,1957.153-158
54 De Josselin de Jong G,Verruijt A.Etude photo-elastique d'un empilement de disques.Cahier Groupe Francais Rheologie,1969,2:73-86
55 Drecher A,De Josselin de Jong G.Photoelastic verification of a mechanical model for the flow of a granular material.J Mech Phys Solids,1972,20:337-351
56 Oda M,Konishi J,Nemat Nasser S.Some experimentally based fundamental results on the mechanical behavior of granular material:effects of particle rolling.Geotechnique, 1980,30(4):479-495
57 Drescher A.An experimental investigation of flow rules for granular materials using optically sensitive glass particles. Geotechnique,1976,26(4):591-901
58 Allersma HGB.Optical analysis of stress and strain in photoelastic particle assemblies:[PhD Thesis].TUDelft, Netherlands,1987
59 Cundall P A,Strack O 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
60 Cundall P A,Strack O D L.A discrete numerical model for granular assembles.Geotechnique,1979,29(1):47-65
61 Shukla A,Damania C.Experimental investigation of wave velocity and dynamic contact stresses in an assembly of disks.Exp Mech,1987,27(3):268-281
62 Baxter G W.Stress distributions in a two dimensional granular material.In:Behringer R P,Jenkins J T. Balkema.ed Powders and Grains 97.1997.345-348
63 Majmudar T S,Behringer R P.Contact force measurements and stress-induced anisotropy in granular materials. Nature,2005,435(2):1079-1082
64 Majmudar T S.Experimental studies of two-dimensional granular material systems using grain-scale contact force measurements:[PhD Thesis].Durham:Duke University, 2006★65杨荣伟,程晓辉.光弹颗粒材料的直剪实验研究.岩土力学, 2009.30:103-109
66 Liu J,Sun Q,Jin F.Visualization of force networks in 2D dense granular materials.Front Archit Civ Eng China, 2010,4:109-115
67 Desrues J,Chambon R,Mokni M,et al.Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography.Geotechnique,1996,46(3): 529-546
68 Oda M,Takemura T,Takahashi M.Microstructure in shear band observed by microfocus X-ray computed tomography. Geotechnique,2004,54(8):539-542
69 Matsushima T,Katagiri J,Uesugi K,et al.Micro X-ray CT at Spring-8 for granular mechanics.In:Soil StressStrain Behavior:Measurement,Modeling and Analysis, 2007.225-234
70 Terzaghi K.Old earth pressure theories and new test results. Engineering News-Record,1920,85(14):632-637
71孙广忠.岩体结构力学.北京:科学出版社,1988
72 Roscoe K H,Schofield A N,Wroth C P.On the yielding of soils.Geotechnique,1958,8(1):22-53
73 Schofield A N,Wroth C P.Critical State Soil Mechanics. London:McGraw-Hill,1968
74 Skempton A W.Long-term stability of clay slopes. Geotechnique,1964,14(2):77-101
75 Itasca Consulting Group Inc.PFC2D Theory and Background. Minneapolis,Minnesota,1999
76莫则尧,张爱清.主编.JASMIN框架用户指南(1.8.0版),技术报告T09-JMJL-01,北京应用物理与计算数学研究,2010
77夏蒙棼,韩闻生,柯孚久,等.统计细观损伤力学和损伤演化诱致突变.力学进展,1995,25(1):1-23
2黄文熙.土的工程性质北京:水利电力出版社,1983
3蒋亦民,刘佑.砂土的流体动力学方程与本构模型的比较.岩土力学,2010,31(6):1729-1738
4 Kolyrnbas D.Introduction to Hypoplasticity.In:Adavances in Geotechnical Engineering.Rotterdam:Balkema A A,2000
5 Huang W X,Bauer E.Numerical Investigations of shear localization in a micro-polar hypoplastic material.Int J Numer Anal Method Geomechan,2003,27:325-352
6 Jiang Y M,Liu M.Granular solid hydrodynamics.Granular Matter,2009,11:139-156
7蒋亦民,刘佑.基于流体动力学理论的颗粒物质本构关系.科学通报,2009,54:1504-1510
8 Edwards S F,Oakeshott R B S.Theory of powders.Phys- ica A,1989,157:1080-1090
9 Edwards S F.The full canonical ensemble of a granular system.Physica A,2005,353:114-118
10 Coniglio A,de Candia A,Fierro A,et al.On Edwards's theory of powders.Physica A,2004,339:1-6
11陆坤权,刘寄星.软物质物理学导论.北京:北京大学出版社,2006
12孙其诚,厚美瑛,金峰.颗粒物质物理与力学.北京:科学出版社,2011
13殷跃平,潘桂堂,刘宇平,等.汶川地震地质与滑坡灾害概论北京:地质出版社,2009
14 Pica Ciamarra M,Lippiello E,Godano C,et al.Unjamming dynamics:the micromechanics of a seismic fault model.Phys Rev Lett,2010,104:238001
15 Cheng X.Experimental study of the jamming transition at zero temperature.Phys Rev E,2010,81:031301
16 Snoeijer J H,Vlugt T J H,van Hecke M,et al.Force network ensemble:a new Approach to static granular matter. Phys Rev Lett,2004,92:054302
17 Rowe P W.The stress-dilatancy relation for static equilibrium of an assembly of particles in contact.Proc R Soc A,1962,269(1339):500-527
18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:[博士论文].大连:大连理工大学,2007.71-74
19 Skinner A E.A note on the influence of interparticle friction on the shearing strength of a random assembly of spherical particles.Geotechnique,1969,19(1):150-157
20 Oda M,Nemat-Nasser S,Mehrabadi M M.A statistical study of fabric in a random assembly of spherical granules. Int J Num.Anal Meth Geomech,1982,6:77-94
21 Davy C A,Fleck N A,Bolton M D.The collapse behaviour of a sugar aggregate.Acta Materialia.,2004,52:3587- 3601
22 Suiker A S J,Fleck N A.Frictional collapse of granular assemblies.J Appl Mech,2004,71(3):350-358
23 Kruyt N P,Rothenburg L.Shear strength,dilatancy,energy and dissipation in quasi-static deformation of granular materials.J Stat Mech,2006,P07021
24 Thornton C.Numerical simulations of deviatoric shear deformation of granular media.Geotechnique,2000,50(1): 43-53
25 Iwashita K,Oda M.Rolling resistance at contacts in simulation of shear band development by DEM.J Enq Mech ASCE,1998,124:285-292
26 Jiang M J,Yu H S,Harris D.A novel discrete model for granular material incorporating rolling resistance.Comput Geotech,2005,32:340-357
27常在,杨军,程晓辉.砂土强度和剪胀性的颗粒力学分析.工程力学,2010,27(4):095-104
28刘海涛.无黏性颗粒材料剪切试验和贯入试验的离散元分析:[硕士论文].北京:清华大学,2010
29刘海涛,程晓辉.粗粒土尺寸效应的离散元研究岩土力学, 2009,30(增刊):287-292
30 Li X.Micro-scale investigation on the quasi-static behavior of granular material:[PhD Thesis].Hong Kong:The Hong Kong University of Science and Technology,2006
31 Li X,Li X S.Micro-macro quantification of the internal structure of granular materials.J Eng Mech,2009,135 (7):641-656
32 Bagi K.Microstructural stress tensor of granular assemblies with volume forces.J Appl Mech,1999,66(4),934- 936
33 Love A E H.A Treatise of Mathematical Theory of Elasticity. Cambridge:Cambridge University Press,1927
34 Goddard J.An elastohydrodynamics theory for the rheology of concentrated suspensions of deformable particles. J Non-Newtonian Fluid Mech,1977,2:169-189
35 Rothenburg L,Selvadurai,A P S,A micromechanical definition of the Cauchy stress tensor for particular media. In:Selvadurai,A.ed.Mechanics of Structured Media. Amsterdam:Elsevier,1981.469-486
36 Bagi K.Stress and strain in granular assemblies.Mech Mater,1996,22:165-177
37 Kruyt N P,Rothenburg L.Kinematic and static assumptions for homogenization in micromechanics of granular materials.Mech Mater,2004,36:1157-1173
38 Li X,Yu H S,Li X S.Macro-micro relations in granular mechanics.Int J Solids Struct,2009,46:4331-4341
39 De Saxce G,Fortin J,Millet O.About the numerical simulation of the dynamics of granular media and the definition of the mean stress tensor.Mech Mater,2004,36: 1175-1184
40 Bardet J P,Vardoulakis I.The asymmetry of stress in granular media.Int J Solids Struct,2001:38:353-367
41 Bardet J P,Vardoulakis I.Reply to Dr.Kuhn's discussion. Int J Solids Struct,2003,40:1809
42 Kruyt N P.Statics and Kinematics of discrete Cosserattype granular materials.Int J Solids Struct,2003,40: 511-534
43 Kuhn M R.Discussion of "The asymmetry of stress in granular media".Int J Solids Struct,2003,40:1805-1807
44 Chang C S,Kuhn M R.On virtual work and stress in granular media.Int J Solids Struct,2005,42:2373-3779
45 Bagi K.Analysis of microstructural strain tensor for granular assemblies.Int J Solids Struct,2006,43(1):3166- 3184
46 Varadarajan A,Sharma K G,Venkatachalam K,et al. Testing and modeling two rockfill materials.J Geotech Geoenviron Eng ASCE,2003,129(3):206-218
47 Aqil U,Tatsuoka F,Uchimura T,et al.Strength and deformation characteristics of recycled concrete aggregate as a backfill material.Soils Found,2005,45(5):53-72
48 Huang H Y.Discrete element modeling of tool-rock interaction: [PhD Thesis].Minnesota:University of Minnesota, US,1999
49 Yang B,Jiao Y,Lei S.A study on the effects of microparameters on macroproperties for specimens created by bonded particles.Int J Computer-Aided Eng Software, 2006,23:607-631
50 Nardin A,Schrefler B A.Modelling of cutting tool-soil interaction-Part II:macromechanical model and upscaling. Computer Mech,2005,36:343-359
51 Potyondy D O,Cundall P A.A bonded particle model for rock.Int J Rock Mech Mining Sci,2004,41:1329-1364
52 Dantu P.Contribution a l'etude mecanique et geometrique des milieux pulverulents.In:Proceedings of the 4th International Conference on Soil Mechanics and Foundation Engineering,1957,1:144-148
53 Wakabayashi T.Photoelastic method for determination of stress in powdered mass.In:Proc 7th Jpn Nat Cong Appl Mech,1957.153-158
54 De Josselin de Jong G,Verruijt A.Etude photo-elastique d'un empilement de disques.Cahier Groupe Francais Rheologie,1969,2:73-86
55 Drecher A,De Josselin de Jong G.Photoelastic verification of a mechanical model for the flow of a granular material.J Mech Phys Solids,1972,20:337-351
56 Oda M,Konishi J,Nemat Nasser S.Some experimentally based fundamental results on the mechanical behavior of granular material:effects of particle rolling.Geotechnique, 1980,30(4):479-495
57 Drescher A.An experimental investigation of flow rules for granular materials using optically sensitive glass particles. Geotechnique,1976,26(4):591-901
58 Allersma HGB.Optical analysis of stress and strain in photoelastic particle assemblies:[PhD Thesis].TUDelft, Netherlands,1987
59 Cundall P A,Strack O 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
60 Cundall P A,Strack O D L.A discrete numerical model for granular assembles.Geotechnique,1979,29(1):47-65
61 Shukla A,Damania C.Experimental investigation of wave velocity and dynamic contact stresses in an assembly of disks.Exp Mech,1987,27(3):268-281
62 Baxter G W.Stress distributions in a two dimensional granular material.In:Behringer R P,Jenkins J T. Balkema.ed Powders and Grains 97.1997.345-348
63 Majmudar T S,Behringer R P.Contact force measurements and stress-induced anisotropy in granular materials. Nature,2005,435(2):1079-1082
64 Majmudar T S.Experimental studies of two-dimensional granular material systems using grain-scale contact force measurements:[PhD Thesis].Durham:Duke University, 2006★65杨荣伟,程晓辉.光弹颗粒材料的直剪实验研究.岩土力学, 2009.30:103-109
66 Liu J,Sun Q,Jin F.Visualization of force networks in 2D dense granular materials.Front Archit Civ Eng China, 2010,4:109-115
67 Desrues J,Chambon R,Mokni M,et al.Void ratio evolution inside shear bands in triaxial sand specimens studied by computed tomography.Geotechnique,1996,46(3): 529-546
68 Oda M,Takemura T,Takahashi M.Microstructure in shear band observed by microfocus X-ray computed tomography. Geotechnique,2004,54(8):539-542
69 Matsushima T,Katagiri J,Uesugi K,et al.Micro X-ray CT at Spring-8 for granular mechanics.In:Soil StressStrain Behavior:Measurement,Modeling and Analysis, 2007.225-234
70 Terzaghi K.Old earth pressure theories and new test results. Engineering News-Record,1920,85(14):632-637
71孙广忠.岩体结构力学.北京:科学出版社,1988
72 Roscoe K H,Schofield A N,Wroth C P.On the yielding of soils.Geotechnique,1958,8(1):22-53
73 Schofield A N,Wroth C P.Critical State Soil Mechanics. London:McGraw-Hill,1968
74 Skempton A W.Long-term stability of clay slopes. Geotechnique,1964,14(2):77-101
75 Itasca Consulting Group Inc.PFC2D Theory and Background. Minneapolis,Minnesota,1999
76莫则尧,张爱清.主编.JASMIN框架用户指南(1.8.0版),技术报告T09-JMJL-01,北京应用物理与计算数学研究,2010
77夏蒙棼,韩闻生,柯孚久,等.统计细观损伤力学和损伤演化诱致突变.力学进展,1995,25(1):1-23
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