粗粒土的分数阶应变率及其与分形维度的关系
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摘要
在波浪荷载、潮汐作用下砂土等粗粒土常常经受长期动力变形。运用分数阶微积分理论,分析了5种不同粗粒土在不同加载条件下的累积变形特性及粗粒土的分数阶应变率,传统的整数阶应变率随着加载次数的变化而变化,而粗粒土的分数阶应变率在同一加载条件下保持为常数。通过粗粒土颗粒破碎的分形理论,尝试建立分数阶应变率与土颗粒分布的分形维度之间的关系,分析土体分形维度对分数阶应变率大小的影响,发现随着分形维度的增加,分数阶应变率的数值降低。
Due to the wave-load and tidal load, granular soils, such as sand, often suffers from long-term cyclic deformation. Cumulative strains of five different granular soils under different loading conditions are analyzed by using the fractional calculus, and its there exists a fractional strain rate for granular soils subjected to repeated loads. Unlike the traditional integral strain rate which is varying with the load cycles, the fractional strain rate remains constant for a given loading condition. To investigate the physical origin of the fractional approach, fractal breakage theory of granular soils is used. It is found that the fractional strain rate has a strong connection with the corresponding fractal dimension of a given granular soil. It decreases with the increase of the fractal dimension.
Due to the wave-load and tidal load, granular soils, such as sand, often suffers from long-term cyclic deformation. Cumulative strains of five different granular soils under different loading conditions are analyzed by using the fractional calculus, and its there exists a fractional strain rate for granular soils subjected to repeated loads. Unlike the traditional integral strain rate which is varying with the load cycles, the fractional strain rate remains constant for a given loading condition. To investigate the physical origin of the fractional approach, fractal breakage theory of granular soils is used. It is found that the fractional strain rate has a strong connection with the corresponding fractal dimension of a given granular soil. It decreases with the increase of the fractal dimension.
引文
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[2]WANG Z,YANG Y,YU H S.Effects of principal stress rotation on the wave-seabed interactions[J].Acta Geotechnica,2016,1-10.
[3]沈扬,闫俊,刘汉龙,等.主应力轴循环旋转下高密实粉土稳定性影响研究[J].岩土力学,2011,32(10):2957-2964.SHEN Yang,YAN Jun,LIU Han-long,et al.Stability of high relative density silt under cyclic principal stress axis rotation[J].Rock and Soil Mechanics,2011,32(10):2957-2964.
[4]陈生水,彭成,傅中志.基于广义塑性理论的堆石料动力本构模型研究[J].岩土工程学报,2012,34(11):1961-1968.CHEN Sheng-shui,PENG Cheng,FU Zhong-zhi.Dynamic constitutive model for rockfill materials based on generalized plasticity theory[J].Chinese Journal of Geotechnical Engineering,2012,34(11):1961-1968.
[5]王磊,朱斌,来向华.砂土循环累积变形规律与显式计算模型研究[J].岩土工程学报,2015,37(11):2024-2029.WANG Lei,ZHU Bing,LAI Xiang-hua Cyclic accumulative deformation of sand and its explicit model[J].Chinese Journal of Geotechnical Engineering,2015,37(11):2024-2029.
[6]SUN Y,XIAO Y,HANIF K F.Fractional order modelling of the cumulative deformation of granular soils under cyclic loading[J].Acta Mechanica Solida Sinica,2015,28(6):647-658.
[7]INDRARATNA B,SUN Y,NIMBALKAR S.Laboratory assessment of the role of particle size distribution on the deformation and degradation of ballast under cyclic loading[J].Journal of Geotechnical and Geoenvironmental Engineering.2016,142(7):04016016.
[8]张季如,胡泳,张弼文,等.石英砂砾破碎过程中粒径分布的分形行为研究[J].岩土工程学报,2015,37(5):784-791.ZHANG Ji-ru,HU Yong,ZHANG Bi-wen,et al.Fractal behavior of particle-size distribution during particle crushing of quartz sand and gravel[J].Chinese Journal of Geotechnical Engineering,2015,37(5):784-791.
[9]徐永福.分形介质土力学理论[J].岩土工程学报,2015,37(增刊1):16-20.XU Yong-fu.Fractals in soil mechanics[J].Chinese Journal of Geotechnical Engineering,2015,37(Supp.1):16-20.
[10]YIN D,WU H,CHENG C,et al.Fractional order constitutive model of geomaterials under the condition of triaxial test[J].International Journal for Numerical and Analytical Methods in Geomechanics,2013,37(8):961-972.
[11]何明明,李宁,陈蕴生,等.基于分数阶微积分岩石的动态变形行为研究[J].岩土工程学报.2015,37(增刊1):178-184.HE Ming-ming,LI Ning,CHEN Yun-sheng,et al.Dynamic deformation behavior of rock based on fractional order calculus[J].Chinese Journal of Geotechnical Engineering,2015,37(Supp.1):178-184.
[12]SUN Y,XIAO Y,ZHENG C,et al.Modelling long-term deformation of granular soils incorporating the concept of fractional calculus[J].Acta Mechanica Sinica,2016,32(1):112-124.
[13]Mc Dowell G,de Bono J P.On the micro mechanics of one-dimensional normal compression[J].Géotechnique,2013,63(11):895-908.
[14]PODLUBNY I.Fractional differential equations[M].San Diego,California:Academic Press,1998.
[15]WICHTMANN T,NIEMUNIS A,TRIANTAFYLLIDIST H.Validation and calibration of a high-cycle accumulation model based on cyclic triaxial tests on eight sands[J].Soils and Foundations,2009,49(5):711-728.
[16]LACKENBY J,INDRARATNA B,Mc Dowell G,et al.Effect of confining pressure on ballast degradation and deformation under cyclic triaxial loading[J].Géotechnique,2007,57(6):527-536.
[17]SUIKER A S J,de BORST R.A numerical model for the cyclic deterioration of railway tracks[J].International Journal for Numerical Methods in Engineering,2003,57(4):441-470.
[18]SEVI A,GE L.Cyclic behaviors of railroad ballast within the parallel gradation scaling framework[J].Journal of Materials in Civil Engineering,2012,24(7):797-804.
[19]COOP M,SORENSEN K,FREITAS T B,et al.Particle breakage during shearing of a carbonate sand[J].Géotechnique,2004,54(3):157-163.
[20]UENG T S,CHEN T J.Energy aspects of particle breakage in drained shear of sands[J].Géotechnique.2000,50(1):65-72.
[21]CHO G,DODDS J,SANTAMARINA J.Particle shape effects on packing density,stiffness,and strength:natural and crushed sands[J].Journal of Geotechnical and Geoenvironmental Engineering.2006,132(5):591-602.
[22]XIAO Y,SUN Y,LIU H,et al.Critical state behaviors of a coarse granular soil under generalized stress conditions[J].Granular Matter,2016,18(2):1-13.INDRARATNA B,KHABBAZ H,SALIM W,et al.Geotechnical properties of ballast and the role of geosynthetics in rail track stabilisation[J].Proceedings of the ICE-Ground Improvement.2006,10(3):91-101.