分形几何在岩土力学研究中的过去、现在与未来
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摘要
首先简略地介绍了分形几何理论及其方法。从岩土材料结构的定量描述、流体在岩土体中的渗流问题、岩土材料强度的分形模型和分形空间的力学特征四个方面叙述了分形理论在岩土力学研究中取得的成果及应用。指出只有将岩土力学系统在欧氏空间的各种基本变量、原理和方法向分形空间推广和拓展,才能产生对岩土材料非线性力学行为的新认识,这也是分形理论在岩土力学研究中进一步深化的基础。
Brief introduction on fractal theory and its methods are firstly presented.The applications and achievements of fractal theory on rock and soil mechanics are detailedly specified,from four facets: quantitative analysis on geotechnical material structure,liquid flowing in geotechnical materials,fractal models for geotechnical material strength evaluation and mechanics characteristics in fractal space,respectively.The conclusion is drawn that only fundamental variables,principles and methods used in geotechnical mechanics conventionally are extended and generalized from Euclidean space to the fractal space,new understandings on nonlinear mechanics behaviors of geotechnical materials can be brought about,and it is also the base to further and wider application of fractal theory in geotechnical mechanics study.
Brief introduction on fractal theory and its methods are firstly presented.The applications and achievements of fractal theory on rock and soil mechanics are detailedly specified,from four facets: quantitative analysis on geotechnical material structure,liquid flowing in geotechnical materials,fractal models for geotechnical material strength evaluation and mechanics characteristics in fractal space,respectively.The conclusion is drawn that only fundamental variables,principles and methods used in geotechnical mechanics conventionally are extended and generalized from Euclidean space to the fractal space,new understandings on nonlinear mechanics behaviors of geotechnical materials can be brought about,and it is also the base to further and wider application of fractal theory in geotechnical mechanics study.
引文
[1]谢和平.分形几何及其在岩土力学中的应用[J].岩土工程学报,1992,14(1):14-24.
[2]Mandelbrot B B.Fractals:from,chance and dimension[M].San Francisco:W.H.Freeman,1977.
[3]Mandelbrot B B.The Fractal Geometry of Nature[M].San Francisco:W.H.Freeman,1982.
[4]毛灵涛,薛茹,袁则循.软土路基微结构扫描电镜图像的分形[J].长安大学学报(自然科学版),2007,27(2):30-33.
[5]陈颙,陈凌.分形几何学(第2版)[M].北京:地震出版社,2005.
[6]孙博玲.分形维数(Fractal dimension)及其测量方法[J].东北林业大学学报,2004,32(3):116-119.
[7]Fredlund D G,Rahardjo H.Soil Mechanics for Unsaturated Soils[M].New York:John Wiley&Sons,1993.
[8]Friesen W I,Mikula R J.Fractal dimensions of coal particles[J].Colloid Interf.,1987,120:263-271.
[9]XIE He-ping,Chen Zhi-da.Fractal Geometry and Fracture of Rock[J].Acta Mechanica Sinica,1988,4(3):255-264.
[10]谢和平.岩土介质的分形孔隙和分形粒子[J].力学进展,1993,23(2):145-164.
[11]谢和平.岩石节理的分形描述[J].岩土工程学报,1994,17(1):18-23.
[12]XIE He-ping.Fractals in Rock Mechanics[M].Rotterdam:A.A.Balkema Publisher,1993.
[13]Tyler S W,Wheatcraft S W.Fractal scaling of soil particle-size distributions:analysis and limitations[J].Soil Science Society of America,1992,56(1):362-369.
[14]Crawford J W,Sleeman B D,Young I M.On the relation between number-size distributions and the fractal dimensions of aggregates[J].Journal of Soil Science,1993,44:555-565.
[15]XU Yongfu,SUN De’an.A fractal model for soil pores and its application to determination of water permeability[J].Physica A:Statistical Mechanics and its Application,2002,316(1-4):56-64.
[16]Mandelbrot B B.The Fractal Geometry of Nature[M].New York,W.H.Freeman and company,1982.
[17]Turcotte D L.Fractals and fragmentation[J].Journal of Geophysical Research,1986,91(B2):1921-1926.
[18]Crawford J W,Matsui N,Young I M.The relation between the moisture release curve and the structure of soil[J].European journal of soil science,1995,46(3):369-375.
[19]Avnir D,Farin D,Pfeifer P.Surface geometric irregularity of particulate materials:the fractal approach[J].Journal of Colloid Interface Science,1985,103:112-123.
[20]涂新斌,王思敬,岳中琦.风化岩石的破碎分形及其工程地质意义[J].岩石力学与工程学报,2005,24(4):587-595.
[21]刘晓明,赵明华,苏永华.沉积岩土粒度分布分形模型改进及应用[J].岩石力学与工程学报,2006,25(8):1691-1697.
[22]XIE H-ping.Fractal Characteristic in Geological Material Mechanics[J].In:Applied Mechanics,Int.Academic Publishers,Pergamon Press,1989:1255-1260.
[23]谢和平,于广明,杨伦,等.采动岩体分形裂隙网络研究[J].岩石力学与工程学报,1999,18(2):147-151.
[24]王利,高谦.基于损伤能量耗散的岩体块度分布预测[J].岩石力学与工程学报,2007,26(6):1202-1211.
[25]WANG Jin-an,XIE He-ping.Fractal properties of rock fracture surfaces[J].Journal of Coal Science and Engineering,1996,2(1):16-23.
[26]王金安,谢和平.岩石断裂面的各向异性分形和多重分形研究[J].岩土工程学报,1998,20(6):16-21.
[27]孙洪泉,谢和平.岩石断裂表面的分形模拟[J].岩土力学,2008,29(2):347-352.
[28]王金安,谢和平,田晓燕,等.岩石断裂表面分形测量的尺度效应[J].岩石力学与工程学报,2000,19(1):11-17.
[29]谢卫红,谢和平,赵鹏.分形节理粗糙度对应力状态影响的研究[J].岩石力学与工程学报,1998,17(3):253-258.
[30]谢和平,王金安.岩石节理(断裂)表面的多重分形性质[J].力学学报,1998,30(3):314-320.
[31]Gefen Y,Aharony A,Alexander S.Anomalous diffusion on percolating clusters[J].Physical Review Letters,1983,50(1):77-80.
[32]Yortsos Y C.Heterogeneity description using fractal concepts[J].Modeling and applications of transport phenomena in porous media.Rhode Saits Genese,Von Karman Institute for Fluid Dynamics,1990,1:1-35.
[33]Guerrini I A,Swartzendruber D.Soil water diffusivity as explicitly dependent on both time and water content[J].Soil Science Society of America,1992,56(1):335-340.
[34]Guerrini I A,Swartzendruber D.Fractal characteristics of the horizontal movement of water in soils[J].Fractals,1994,2(3):465-468.
[35]Yakov P,Dennis T.Water transport in soils as in fractal media[J].Journal of Hydrology,1998,204:98-107.
[36]Thevanayagam S,Nesarajah S.Fractal Model for Flow Through Saturated Soils[J].Journal of Geotechnical and Geoenvironmental Engineering,1998,124(1):53-66.
[37]周宏伟,谢和平.岩体中渗流形貌演化的随机理论描述[J].岩土工程学报,2001,23(2):183-186.
[38]LI Shou-ju,LIU Ying-xi.Application of Fractal Models to Water and Solute Transport in Unsaturated Soils[J].Advances in Unsaturated Soil,Seepage,and Environmental Geotechnics,2006,(148):195-202.
[39]褚卫江,徐卫亚.分形介质饱和渗流应力耦合数值模拟研究[J].岩石力学与工程学报,2007,26(S1):2641-2647.
[40]任强,徐卫亚.裂隙岩体非饱和渗流的分形模型[J].岩土力学,2008,29(10):2735-3740.
[41]Jarvis N J,Messing I.Near-saturated hydraulic conductivity in soils of contrasting texture as measured by tension infiltrometers[J].Soil Science Society of America Journal,1995,59(1):27-34.
[42]徐永福,黄寅春.分形理论在研究非饱和土力学性质中的应用[J].岩土工程学报,2006,28(5):635-638.
[43]徐永福,董平.非饱和土的水分特征曲线的分形模型[J].岩土力学,2002,23(4):400-405.
[44]孙大松,刘鹏,夏小和.非饱和土的渗透系数[J].水利学报,2004,(3):71-75.
[45]HUANG Guan-hua,ZHANG Ren-duo.Evaluation of Soil Water Retention Curve with the Pore-Solid Fractal Model[J].Geoderma,2005,(127):52-61.
[46]HUANG Guan-hua,ZHANG Ren-duo,HUANG Quan-zhong.Modeling Soil Water Retention Curve with a Fractal Method[J].Soil Science Society of China,2006,16(2):137-146.
[47]CHEN Jing-yu,GONG Xiao-nan,WANG Ming-yuan.A Fractal-Based Soil-Water Characteristic Curve Model for Unsaturated Soils[J].Advances in Unsaturated Soil,Seepage,and Environmental Geotechnics,2006,(148):55-61.
[48]Soto M A,Vilar O M.Evaluation of a Pore Fractal Model for the Prediction of Soil Water Retention Curve[C].International Conference on Unsaturated Soils,Carefree,AZ(US),2006:2441-2452.
[49]谢卫红,谢和平,李世平,等.分形节理力学性能试验研究[J].工程力学,1997,14(4):128-138.
[50]谢卫红,谢和平,李世平,等.分形节理的强度和变形研究[J].长春地质学院学报,1997,27(3):284-288.
[51]王金安,谢和平,Kwasniewski M A.剪切过程中岩石节理粗糙度分形演化及力学特征[J].岩土工程学报,1997,19(4):2-9.
[52]王金安,谢和平,Kwasniewski M A.岩石节理面在剪切中表面损伤的分形演化[J].力学与实践,1997,19(4):56-58.
[53]高峰,钟卫平,黎立云,等.节理岩体强度的分形统计分析[J].岩石力学与工程学报,2004,23(21):3608-3612.
[54]高峰,谢和平,巫静波.岩石损伤和破碎相关性的分形分析[J].岩石力学与工程学报,1999,18(5):503-506.
[55]徐志斌,谢和平.断裂尺度的分形分布与其损伤演化的关系[J].地质力学学报,2004,10(3):268-275.
[56]XU Y F.Fractal Approach to Unsaturated Shear Strength[J].Journal of Geotechnical and Geoenvironmental Engineering,2004,130(3):264-273.
[57]谢学斌,潘长良.排土场散体岩石粒度分布与剪切强度的分形特征[J].岩土力学,2004,25(2):287-291.
[58]舒志乐,刘新荣,刘保县,等.基于分形理论的土石混合体强度特征研究[J].岩石力学与工程学报,2009,28(S1):2651-2656.
[59]Bonala M V S,Reddi L N.Fractal Representation of Soil Cohesion[J].Journal of Geotechnical and Geoenvironmental Engineering,1999,125(10):901-904.
[60]Panagiotopoulos P D,Panagouli O K,Mistakidis E S.Fractal geometry and fractal material behavior in solids and structure[J].Archive of Applied Mechanics,1993,63(1):1-24.
[61]谢和平.分形力学的数学基础[J].力学进展,1995,25(2):174-185.
[62]谢和平.分形力学研究进展[J].力学与实践,1996,18(2):10-18.
[63]谢和平,鞠杨.分数维空间中的损伤力学研究初探[J].力学学报,1999,31(3):300-310.
[64]苏维宜.Fractal空间[C].全国第二次分形学术讨论会论文集.武汉,1991.
[65]苏维宜.Fractal与导数[R].全国第二次分形学术讨论会特邀报告.武汉,1991.
[66]谢和平.分形-岩石力学导论[M].北京:科学出版社,1997.
[67]谢和平,薛秀谦.分形应用中的数学基础与方法[M].北京:科学出版社,1997.
[68]XIE He-ping,Parisean W G.Fractal estimation of joint roughness coefficients[J].Science in China(Series B),1994,37(12):1516-1524.
[2]Mandelbrot B B.Fractals:from,chance and dimension[M].San Francisco:W.H.Freeman,1977.
[3]Mandelbrot B B.The Fractal Geometry of Nature[M].San Francisco:W.H.Freeman,1982.
[4]毛灵涛,薛茹,袁则循.软土路基微结构扫描电镜图像的分形[J].长安大学学报(自然科学版),2007,27(2):30-33.
[5]陈颙,陈凌.分形几何学(第2版)[M].北京:地震出版社,2005.
[6]孙博玲.分形维数(Fractal dimension)及其测量方法[J].东北林业大学学报,2004,32(3):116-119.
[7]Fredlund D G,Rahardjo H.Soil Mechanics for Unsaturated Soils[M].New York:John Wiley&Sons,1993.
[8]Friesen W I,Mikula R J.Fractal dimensions of coal particles[J].Colloid Interf.,1987,120:263-271.
[9]XIE He-ping,Chen Zhi-da.Fractal Geometry and Fracture of Rock[J].Acta Mechanica Sinica,1988,4(3):255-264.
[10]谢和平.岩土介质的分形孔隙和分形粒子[J].力学进展,1993,23(2):145-164.
[11]谢和平.岩石节理的分形描述[J].岩土工程学报,1994,17(1):18-23.
[12]XIE He-ping.Fractals in Rock Mechanics[M].Rotterdam:A.A.Balkema Publisher,1993.
[13]Tyler S W,Wheatcraft S W.Fractal scaling of soil particle-size distributions:analysis and limitations[J].Soil Science Society of America,1992,56(1):362-369.
[14]Crawford J W,Sleeman B D,Young I M.On the relation between number-size distributions and the fractal dimensions of aggregates[J].Journal of Soil Science,1993,44:555-565.
[15]XU Yongfu,SUN De’an.A fractal model for soil pores and its application to determination of water permeability[J].Physica A:Statistical Mechanics and its Application,2002,316(1-4):56-64.
[16]Mandelbrot B B.The Fractal Geometry of Nature[M].New York,W.H.Freeman and company,1982.
[17]Turcotte D L.Fractals and fragmentation[J].Journal of Geophysical Research,1986,91(B2):1921-1926.
[18]Crawford J W,Matsui N,Young I M.The relation between the moisture release curve and the structure of soil[J].European journal of soil science,1995,46(3):369-375.
[19]Avnir D,Farin D,Pfeifer P.Surface geometric irregularity of particulate materials:the fractal approach[J].Journal of Colloid Interface Science,1985,103:112-123.
[20]涂新斌,王思敬,岳中琦.风化岩石的破碎分形及其工程地质意义[J].岩石力学与工程学报,2005,24(4):587-595.
[21]刘晓明,赵明华,苏永华.沉积岩土粒度分布分形模型改进及应用[J].岩石力学与工程学报,2006,25(8):1691-1697.
[22]XIE H-ping.Fractal Characteristic in Geological Material Mechanics[J].In:Applied Mechanics,Int.Academic Publishers,Pergamon Press,1989:1255-1260.
[23]谢和平,于广明,杨伦,等.采动岩体分形裂隙网络研究[J].岩石力学与工程学报,1999,18(2):147-151.
[24]王利,高谦.基于损伤能量耗散的岩体块度分布预测[J].岩石力学与工程学报,2007,26(6):1202-1211.
[25]WANG Jin-an,XIE He-ping.Fractal properties of rock fracture surfaces[J].Journal of Coal Science and Engineering,1996,2(1):16-23.
[26]王金安,谢和平.岩石断裂面的各向异性分形和多重分形研究[J].岩土工程学报,1998,20(6):16-21.
[27]孙洪泉,谢和平.岩石断裂表面的分形模拟[J].岩土力学,2008,29(2):347-352.
[28]王金安,谢和平,田晓燕,等.岩石断裂表面分形测量的尺度效应[J].岩石力学与工程学报,2000,19(1):11-17.
[29]谢卫红,谢和平,赵鹏.分形节理粗糙度对应力状态影响的研究[J].岩石力学与工程学报,1998,17(3):253-258.
[30]谢和平,王金安.岩石节理(断裂)表面的多重分形性质[J].力学学报,1998,30(3):314-320.
[31]Gefen Y,Aharony A,Alexander S.Anomalous diffusion on percolating clusters[J].Physical Review Letters,1983,50(1):77-80.
[32]Yortsos Y C.Heterogeneity description using fractal concepts[J].Modeling and applications of transport phenomena in porous media.Rhode Saits Genese,Von Karman Institute for Fluid Dynamics,1990,1:1-35.
[33]Guerrini I A,Swartzendruber D.Soil water diffusivity as explicitly dependent on both time and water content[J].Soil Science Society of America,1992,56(1):335-340.
[34]Guerrini I A,Swartzendruber D.Fractal characteristics of the horizontal movement of water in soils[J].Fractals,1994,2(3):465-468.
[35]Yakov P,Dennis T.Water transport in soils as in fractal media[J].Journal of Hydrology,1998,204:98-107.
[36]Thevanayagam S,Nesarajah S.Fractal Model for Flow Through Saturated Soils[J].Journal of Geotechnical and Geoenvironmental Engineering,1998,124(1):53-66.
[37]周宏伟,谢和平.岩体中渗流形貌演化的随机理论描述[J].岩土工程学报,2001,23(2):183-186.
[38]LI Shou-ju,LIU Ying-xi.Application of Fractal Models to Water and Solute Transport in Unsaturated Soils[J].Advances in Unsaturated Soil,Seepage,and Environmental Geotechnics,2006,(148):195-202.
[39]褚卫江,徐卫亚.分形介质饱和渗流应力耦合数值模拟研究[J].岩石力学与工程学报,2007,26(S1):2641-2647.
[40]任强,徐卫亚.裂隙岩体非饱和渗流的分形模型[J].岩土力学,2008,29(10):2735-3740.
[41]Jarvis N J,Messing I.Near-saturated hydraulic conductivity in soils of contrasting texture as measured by tension infiltrometers[J].Soil Science Society of America Journal,1995,59(1):27-34.
[42]徐永福,黄寅春.分形理论在研究非饱和土力学性质中的应用[J].岩土工程学报,2006,28(5):635-638.
[43]徐永福,董平.非饱和土的水分特征曲线的分形模型[J].岩土力学,2002,23(4):400-405.
[44]孙大松,刘鹏,夏小和.非饱和土的渗透系数[J].水利学报,2004,(3):71-75.
[45]HUANG Guan-hua,ZHANG Ren-duo.Evaluation of Soil Water Retention Curve with the Pore-Solid Fractal Model[J].Geoderma,2005,(127):52-61.
[46]HUANG Guan-hua,ZHANG Ren-duo,HUANG Quan-zhong.Modeling Soil Water Retention Curve with a Fractal Method[J].Soil Science Society of China,2006,16(2):137-146.
[47]CHEN Jing-yu,GONG Xiao-nan,WANG Ming-yuan.A Fractal-Based Soil-Water Characteristic Curve Model for Unsaturated Soils[J].Advances in Unsaturated Soil,Seepage,and Environmental Geotechnics,2006,(148):55-61.
[48]Soto M A,Vilar O M.Evaluation of a Pore Fractal Model for the Prediction of Soil Water Retention Curve[C].International Conference on Unsaturated Soils,Carefree,AZ(US),2006:2441-2452.
[49]谢卫红,谢和平,李世平,等.分形节理力学性能试验研究[J].工程力学,1997,14(4):128-138.
[50]谢卫红,谢和平,李世平,等.分形节理的强度和变形研究[J].长春地质学院学报,1997,27(3):284-288.
[51]王金安,谢和平,Kwasniewski M A.剪切过程中岩石节理粗糙度分形演化及力学特征[J].岩土工程学报,1997,19(4):2-9.
[52]王金安,谢和平,Kwasniewski M A.岩石节理面在剪切中表面损伤的分形演化[J].力学与实践,1997,19(4):56-58.
[53]高峰,钟卫平,黎立云,等.节理岩体强度的分形统计分析[J].岩石力学与工程学报,2004,23(21):3608-3612.
[54]高峰,谢和平,巫静波.岩石损伤和破碎相关性的分形分析[J].岩石力学与工程学报,1999,18(5):503-506.
[55]徐志斌,谢和平.断裂尺度的分形分布与其损伤演化的关系[J].地质力学学报,2004,10(3):268-275.
[56]XU Y F.Fractal Approach to Unsaturated Shear Strength[J].Journal of Geotechnical and Geoenvironmental Engineering,2004,130(3):264-273.
[57]谢学斌,潘长良.排土场散体岩石粒度分布与剪切强度的分形特征[J].岩土力学,2004,25(2):287-291.
[58]舒志乐,刘新荣,刘保县,等.基于分形理论的土石混合体强度特征研究[J].岩石力学与工程学报,2009,28(S1):2651-2656.
[59]Bonala M V S,Reddi L N.Fractal Representation of Soil Cohesion[J].Journal of Geotechnical and Geoenvironmental Engineering,1999,125(10):901-904.
[60]Panagiotopoulos P D,Panagouli O K,Mistakidis E S.Fractal geometry and fractal material behavior in solids and structure[J].Archive of Applied Mechanics,1993,63(1):1-24.
[61]谢和平.分形力学的数学基础[J].力学进展,1995,25(2):174-185.
[62]谢和平.分形力学研究进展[J].力学与实践,1996,18(2):10-18.
[63]谢和平,鞠杨.分数维空间中的损伤力学研究初探[J].力学学报,1999,31(3):300-310.
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[65]苏维宜.Fractal与导数[R].全国第二次分形学术讨论会特邀报告.武汉,1991.
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