无状态变量的状态依赖剪胀方程及其本构模型
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摘要
粗粒土的剪胀行为具有状态依赖特性。为了考虑这一特性,不同的状态依赖变量被唯像地提出,并被经验性地内嵌入已有剑桥、修正剑桥等剪胀方程中。基于分数阶梯度律,用理论推导出了分数阶状态依赖剪胀方程,并阐述了分数阶数的物理意义。所得剪胀比大小受3个因素影响:分数阶求导阶数、当前加载应力以及当前应力到临界状态应力的距离。当分数阶求导阶数从1开始增大时,分数阶剪胀曲线自修正剑桥剪胀曲线向剑桥剪胀曲线移动;而当求导阶数从1开始减小时,分数阶剪胀曲线逐渐远离修正剑桥剪胀曲线;当求导阶数等于1时,分数阶剪胀曲线与修正剑桥剪胀曲线重合。为验证所提出的状态依赖剪胀方程,基于该方程进一步建立了砂土的状态依赖分数阶塑性力学本构模型,并对砂土和堆石料的三轴排水与不排水试验结果进行了模拟。研究表明,基于状态依赖分数阶剪胀方程建立的本构模型,可以合理地描述砂土在不同初始状态及加载条件下的应力-应变行为。与砂土UH模型预测结果对比发现,UH模型预测较好。
It has been recognized that the stress-dilatancy behaviour of granular soil depends on its material state. To consider such state-dependence, a variety of state parameters were suggested phenomenologically and incorporated into existing stress-dilatancy equations, e.g. Cam-clay equation, modified Cam-clay equation, by experience. In this study, a novel state-dependent stress-dilatancy equation is developed by using fractional stress gradient, where physical meaning of the fractional order is provided. The obtained stress-dilatancy ratio is determined by three factors: the fractional order, current load stress, and the distance from current stress to critical state stress. When the fractional order is larger than 1, the stress-dilatancy curve shifts from the modified Cam-clay stress-dilatancy curve towards the Cam-clay stress-dilatancy curve. However, when the fractional order is smaller than 1, the stress-dilatancy curve is located above the modified Cam-clay stress-dilatancy curve. When the fractional order is equal to 1, the proposed stress-dilatancy curve coincides with the modified Cam-clay curve. To validate the proposed approach, a state-dependent fractional plasticity model for sand soils is established by using the proposed stress-dilatancy equation. Then, a series of drained and undrained triaxial compression tests on sand and rockfill with different initial states is simulated and compared, from which a good agreement between the model simulations and the corresponding test results can be observed. Comparison with the model predictions of the UH sand model indicates that the UH model gives better prediction.
It has been recognized that the stress-dilatancy behaviour of granular soil depends on its material state. To consider such state-dependence, a variety of state parameters were suggested phenomenologically and incorporated into existing stress-dilatancy equations, e.g. Cam-clay equation, modified Cam-clay equation, by experience. In this study, a novel state-dependent stress-dilatancy equation is developed by using fractional stress gradient, where physical meaning of the fractional order is provided. The obtained stress-dilatancy ratio is determined by three factors: the fractional order, current load stress, and the distance from current stress to critical state stress. When the fractional order is larger than 1, the stress-dilatancy curve shifts from the modified Cam-clay stress-dilatancy curve towards the Cam-clay stress-dilatancy curve. However, when the fractional order is smaller than 1, the stress-dilatancy curve is located above the modified Cam-clay stress-dilatancy curve. When the fractional order is equal to 1, the proposed stress-dilatancy curve coincides with the modified Cam-clay curve. To validate the proposed approach, a state-dependent fractional plasticity model for sand soils is established by using the proposed stress-dilatancy equation. Then, a series of drained and undrained triaxial compression tests on sand and rockfill with different initial states is simulated and compared, from which a good agreement between the model simulations and the corresponding test results can be observed. Comparison with the model predictions of the UH sand model indicates that the UH model gives better prediction.
引文
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[26]LI X S,WANG Y.Linear representation of steady-state line for sand[J].Journal of Geotechnical and Geoenvironmental Engineering,1998,124(12):1215-1217.
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[28]XIAO Y,LIU H L,ZHU J G,et al.A 3D bounding surface model for rockfill materials[J].Science China Technological Sciences,2011,54(11):2904-2915.
[29]YAO Y P,ZHOU A N.Non-isothermal unified hardening model:a thermo-elasto-plastic model for clays[J].Géotechnique,2013,63(15):1328-1345.
[30]YAO Y P,QU S,YIN Z Y,et al.SSUH model:a small-strain extension of the unified hardening model[J].Science China Technological Sciences,2015,59(2):225-240.
[31]TIAN Y,YAO Y P.A simple method to describe three-dimensional anisotropic failure of soils[J].Computers and Geotechnics,2017,92:210-219.
[2]VERDUGO R,ISHIHARA K.The steady state of sandy soils[J].Soils and Foundations,1996,36(2):81-91.
[3]ISHIHARA K,TATSUOKA F,YASUDA S.Undrained deformation and liquefaction of sand under cyclic stresses[J].Soils and Foundations,1975,15(1):29-44.
[4]刘洋,李爽.散粒介质临界状态细观力学结构特征的数值模拟与分析[J].岩土力学,2018,39(6):2237-2248.LIU Yang,LI Shuang.Numerical simulation and analysis of meso-mechanical structure characteristic at critical state for granular media[J].Rock and Soil Mechanics,2018,39(6):2237-2248.
[5]BEEN K,JEFFERIES M G.A state parameter for sands[J].Géotechnique,1985,22(6):99-112.
[6]姚仰平,刘林,罗汀.砂土的UH模型[J].岩土工程学报,2016,38(12):2147-2153.YAO Yang-ping,LUO Lin,LUO Ting.UH model for sands[J].Chinese Journal of Geotechnical Engineering,2016,38(12):2147-2153.
[7]ISHIHARA K.Liquefaction and flow failure during earthquakes[J].Geotechnique,1993,43(3):351-451.
[8]WAN R,GUO P.A simple constitutive model for granular soils:modified stress-dilatancy approach[J].Computers and Geotechnics,1998,22(2):109-133.
[9]WANG Z L,DAFALIAS Y F,LI X S,et al.State pressure index for modeling sand behavior[J].Journal of Geotechnical and Geoenvironmental Engineering,2002,128(6):511-519.
[10]LI X S,DAFALIAS Y F.Dilatancy for cohesionless soils[J].Géotechnique,2000,50(4):449-460.
[11]RUSSELL A R,KHALILI N.A bounding surface plasticity model for sands exhibiting particle crushing[J].Canadian Geotechnical Journal,2004,41(6):1179-1192.
[12]XIAO Y,LIU H,CHEN Y,et al.State-dependent constitutive model for rockfill materials[J].Interntional Journal of Geomechanics,2014,15(5):04014075.
[13]XIAO Y,LIU H,CHEN Y,et al.Bounding surface model for rockfill materials dependent on density and pressure under triaxial stress conditions[J].Journal of Engineering Mechanics,2014,140(4):04014002.
[14]YANG J,LI X.State-dependent strength of sands from the perspective of unified modeling[J].Journal of Geotechnical and Geoenvironmental Engineering,2004,130(2):186-198.
[15]SUN Y F,XIAO Y.Fractional order plasticity model for granular soils subjected to monotonic triaxial compression[J].International Journal of Solids and Structures,2017,118-119:224-234.
[16]DAFALIAS Y F,TAIEBAT M,SANISAND Z.Zero elastic range sand plasticity model[J].Géotechnique,2016,66(12):999-1013.
[17]PASTOR M,ZIENKIEWICZ O C,CHAN A H C.Generalized plasticity and the modelling of soil behaviour[J].International Journal of Numerical and Analytical Methods in Geomechanics,1990,14(3):151-190.
[18]DAFALIAS Y F.Bounding surface plasticity,I:mathematical foundation and hypoplasticity[J].Journal of Engineering Mechanics,1986,112(9):966-987.
[19]YAO Y P,KONG Y X.Extended UH model:three-dimensional unified hardening model for anisotropic clays[J].Journal of Engineering Mechanics,2012,138(7):853-866.
[20]YAO Y P,HOU W,ZHOU A N.UH model:threedimensional unified hardening model for overconsolidated clays[J].Géotechnique,2009,59(5):451-469.
[21]YAO Y P,KONG L,ZHOU A N,et al.Time-dependent unified hardening model:three-dimensional elastoviscoplastic constitutive model for clays[J].Journal of Engineering Mechanics,2014,141(6):04014162.
[22]SUN Y F,SHEN Y.Constitutive model of granular soils using fractional order plastic flow rule[J].Interntional Journal of Geomechanics,2017,17(8):04017025.
[23]PODLUBNY I.Fractional differential equations:an introduction to fractional derivatives,fractional differential equations,to methods of their solution and some of their applications[M].San Diego:Academic Press,1998.
[24]SCHOFIELD A,WROTH P.Critical state soil mechanics[M].New York:McGraw-Hill,1968.
[25]MATSUOKA H,YAO Y P,SUN D A.The cam-clay models revised by the SMP criterion[J].Soils and Foundations,1999,39(1):81-95.
[26]LI X S,WANG Y.Linear representation of steady-state line for sand[J].Journal of Geotechnical and Geoenvironmental Engineering,1998,124(12):1215-1217.
[27]RICHART JR FE.Vibration of soils and foundations[M].Englewood Cliffs:Prentice-Hall,1970.
[28]XIAO Y,LIU H L,ZHU J G,et al.A 3D bounding surface model for rockfill materials[J].Science China Technological Sciences,2011,54(11):2904-2915.
[29]YAO Y P,ZHOU A N.Non-isothermal unified hardening model:a thermo-elasto-plastic model for clays[J].Géotechnique,2013,63(15):1328-1345.
[30]YAO Y P,QU S,YIN Z Y,et al.SSUH model:a small-strain extension of the unified hardening model[J].Science China Technological Sciences,2015,59(2):225-240.
[31]TIAN Y,YAO Y P.A simple method to describe three-dimensional anisotropic failure of soils[J].Computers and Geotechnics,2017,92:210-219.