连续排水边界下分数阶黏弹性饱和土体一维固结分析
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摘要
采用分数阶Kelvin模型来描述饱和土体的流变特性,在连续排水边界条件下,利用Laplace变换推导出变换域内的饱和土体一维固结解析解,再通过Crump法对有效应力和固结沉降进行数值反演,得到了分数阶导数黏弹性饱和土体的一维固结半解析解。将连续排水边界退化到Terzaghi排水边界,退化后的结果与已有文献一致,验证了所得半解析解的可靠性。最后,基于所得解,分析了相关参数对土体固结的影响。结果表明,界面参数反映排水边界的透水性,从而影响土体中超孔隙水压力的消散速率;黏滞系数在固结后期对沉降发展影响较大,其值越大,沉降发展越慢;当阶次a不为零时,分数阶次越小,土体表现出的黏滞性越强,整体固结沉降发展越缓慢,次固结沉降发展也越缓慢。
The fractional order derivative Kelvin constitutive model is used to describe the rheological characteristics of saturated soils. With continuous drainage boundary,the Laplace transform is used to derive the analytical solutions of the one-dimensional consolidation of soil in the transformed domain. Then the corresponding semi-analytical solutions of the effective stress and the settlement are obtained through inverse Laplace transform based on Crump's method. The rationality of the proposed solutions is validated by reducing the presented solutions based on continuous drainage boundary to those based on the Terzaghi drainage boundary. Finally,the effects of relevant parameters on consolidation behavior of soil are investigated by the obtained solutions. The results show that the interface parameter reflects the permeability of the drainage boundary,which has an obvious effect on the dissipation rate of the excess pore water pressure in soil. The viscosity coefficient greatly affects the settlement development in the later stage of consolidation,and the larger the viscosity coefficient is,the slower the settlement development is. When a is not zero,a smaller fractional order α indicates a stronger viscosity,a longer consolidation process and a slower development of secondary consolidation.
The fractional order derivative Kelvin constitutive model is used to describe the rheological characteristics of saturated soils. With continuous drainage boundary,the Laplace transform is used to derive the analytical solutions of the one-dimensional consolidation of soil in the transformed domain. Then the corresponding semi-analytical solutions of the effective stress and the settlement are obtained through inverse Laplace transform based on Crump's method. The rationality of the proposed solutions is validated by reducing the presented solutions based on continuous drainage boundary to those based on the Terzaghi drainage boundary. Finally,the effects of relevant parameters on consolidation behavior of soil are investigated by the obtained solutions. The results show that the interface parameter reflects the permeability of the drainage boundary,which has an obvious effect on the dissipation rate of the excess pore water pressure in soil. The viscosity coefficient greatly affects the settlement development in the later stage of consolidation,and the larger the viscosity coefficient is,the slower the settlement development is. When a is not zero,a smaller fractional order α indicates a stronger viscosity,a longer consolidation process and a slower development of secondary consolidation.
引文
[1]黄文熙.土的工程性质[M].北京:水利电力出版社,1983.HUANG Wen-xi.The engineering properties of soil[M].Beijing:Water Resources and Electric Power Press,1983.
[2]CHRISTIE I F.A reappraisal of Merchant's contribution to the theory of consolidation[J].Heat&Fluid Flow,1964,14(4):309-320.
[3]LO K Y.Secondary compression of clays[C]//Proceedings of the 14th Southeast Asian Geotechnical Conference.[S.l.]:[s.n.],2012.
[4]CHEN Z J.Secondary time effects and consolidation of clays[J].Science in China Series A,1958,1(11):1060-1075.
[5]赵维炳.广义Voigt模型模拟的饱水土体一维固结理论及其应用[J].岩土工程学报,1989,11(5):78-85.ZHAO Wei-bing.Theory of one-dimensional consolidation of saturated clays with generalized Voigt model and its applications[J].Chinese Journal of Geotechnical Engineering,1989,11(5):78-85.
[6]刘加才,赵维炳,宰金珉,等.双层黏弹性地基一维固结分析[J].岩土力学,2007,28(4):743-752.LIU Jia-cai,ZHAO Wei-bing,ZAI Jin-min,et al.Analysis of one-dimensional consolidation of double-layered viscoelastic ground[J].Rock and Soil Mechanics,2007,28(4):743-752.
[7]XIE K H,XIE X Y,LI X B.Analytical theory for one-dimensional consolidation of clayey soils exhibiting rheological characteristics under time-dependent loading[J].International Journal for Numerical and Analytical Methods in Geomechanics,2010,32(14):1833-1855.
[8]夏开宗,陈从新,刘秀敏,等.石膏矿矿柱-护顶层支撑体系的流变力学模型分析[J].岩土力学,2017,38(10):2924-2930.XIA Kai-zong,CHEN Cong-xin,LIU Xiu-min,et al.Rheological mechanical modeling of pillar-protective roof supporting system in gypsum mines[J].Rock and Soil Mechanics,2017,38(10):2924-2930.
[9]谷任国,邹育,房营光,等.引入非线性瞬时弹性模量的软土流变模型[J].岩土力学,2018,39(1):237-241.GU Ren-guo,ZOU Yu,FANG Ying-guang,et al.Rheological model of soft soils using nonlinear instantaneous elastic modulus[J].Rock and Soil Mechanics,2018,39(1):237-241.
[10]时刚,刘忠玉,李永辉.循环荷载作用下考虑非达西渗流的软黏土一维流变固结分析[J].岩土力学,2018,39(增刊1):525-528.SHI Gang,LIU Zhong-yu,LIU Yong-hui.Onedimensional rheological consolidation of soft clay under cyclic loadings considering non-Darcy flow[J].Rock and Soil Mechanics,2018,39(Suppl.1):525-528.
[11]GEMANT A.A method of analyzing experimental results obtained from elasto-viscous bodies[J].Physics,1936,7(8):311-317.
[12]SMIT W,VRIES H D.Rheological models containing fractional derivatives[J].Rheologica Acta,1970,9(4):525-534.
[13]BAGLEY R L,TORVIK P J.On the fractional calculus model of viscoelastic behavior[J].Journal of Rheology,1986,30(1):133-155.
[14]BAGLEY R L,TORVIK P J.Fractional calculus in the transient analysis of viscoelastically damped structures[J].AIAA Journal,2012,23(6):918-925.
[15]KOELLER R C.Applications of fractional calculus to the theory of viscoelasticity[J].Journal of Applied Mechanics,1984,51(2):299-307.
[16]刘林超,闫启方,孙海忠.软土流变特性的模型研究[J].岩土力学,2006,27(增刊1):214-217.LIU Lin-chao,YAN Qi-fang,SUN Hai-zhong.Study on model of rheological property of soft clay[J].Rock and Soil Mechanics,2006,27(Suppl.1):214-217.
[17]何利军,孔令伟,吴文军,等.采用分数阶导数描述软黏土蠕变的模型[J].岩土力学,2011,32(增刊2):239-243.HE Li-jun,KONG Ling-wei,WU Wen-jun,et al.Adescription of creep model for soft soil with fractional derivative[J].Rock and Soil Mechanics,2011,32(Suppl.2):239-243.
[18]YIN D S,LI Y Q,WU H,et al.Fractional description of mechanical property evolution of soft soils during creep[J].Water Science and Engineering,2013,6(4):446-455.
[19]何志磊,朱珍德,朱明礼,等.基于分数阶导数的非定常蠕变本构模型研究[J].岩土力学,2016,37(3):737-744.HE Zhi-lei,ZHU Zhen-de,ZHU Ming-li,et al.An unsteady creep constitutive model based on fractional order derivatives[J].Rock and Soil Mechanics,2016,37(3):737-744.
[20]ZHU H H,ZHANG C C,MEI G X,et al.Prediction of one-dimensional compression behavior of Nansha clay using fractional derivatives[J].Marine Georesources&Geotechnology,2016,35(5):688-697.
[21]WANG L,SUN D A,LI P,et al.Semi-analytical solution for one-dimensional consolidation of fractional derivative viscoelastic saturated soils[J].Computers&Geotechnics,2017,83:30-39.
[22]解益,李培超,汪磊,等.分数阶导数黏弹性饱和土体一维固结半解析解[J].岩土力学,2017,38(11):3240-3246.XIE Yi,LI Pei-chao,WANG Lei,et al.Semi-analytical solution for one-dimensional consolidation of viscoelastic saturated soil with fractional order derivative[J].Rock and Soil Mechanics,2017,38(11):3240-3246.
[23]刘忠玉,杨强.基于分数阶Kelvin模型的饱和黏土一维流变固结分析[J].岩土力学,2017,38(12):1001-1009.LIU Zhong-yu,YANG Qiang.One-dimensional rheological consolidation analysis of saturated clay based on fractional order Kelvin’s model[J].Rock and Soil Mechanics,2017,38(12):1001-1009.
[24]DURBIN F.Numerical inversion of Laplace transforms:an efficient improvement to Dubner and Abate’s method[J].Computer Journal,1974,17(4):371-376.
[25]何光渝,王卫红.精确的拉普拉斯数值反演方法及其应用[J].石油学报,1995,16(1):96-104.HE Guang-yu,WANG Wei-hong.An application of an accurate and efficient numerical inversion of Laplace transform in model well-testing analysis[J].Acta Petrolei Sinica,1995,16(1):96-104.
[26]梅国雄,夏君,梅岭.基于不对称连续排水边界的太沙基一维固结方程及其解答[J].岩土工程学报,2011,33(1):28-31.MEI Guo-xiong,XIA Jun,MEI Ling.Terzaghi’s one-dimensional consolidation equation and its solution based on asymmetric continuous drainage boundary[J].Chinese Journal of Geotechnical Engineering,2011,33(1):28-31.
[27]蔡烽,何利军,周小鹏,等.连续排水边界下一维固结不排水对称面的有限元分析[J].岩土工程学报,2012,34(11):2141-2147.CAI Feng,HE Li-jun,ZHOU Xiao-peng,et al.Finite element analysis of one-dimensional consolidation of undrained symmetry plane under continuous drainage boundary[J].Chinese Journal of Geotechnical Engineering,2012,34(11):2141-2147.
[28]李西斌.软土流变固结理论与试验研究[D].杭州:浙江大学,2005.LI Xi-bin.Theoretical and experimental studies on rheological consolidation of soft soil[D].Hangzhou:Zhejiang University,2005.
[2]CHRISTIE I F.A reappraisal of Merchant's contribution to the theory of consolidation[J].Heat&Fluid Flow,1964,14(4):309-320.
[3]LO K Y.Secondary compression of clays[C]//Proceedings of the 14th Southeast Asian Geotechnical Conference.[S.l.]:[s.n.],2012.
[4]CHEN Z J.Secondary time effects and consolidation of clays[J].Science in China Series A,1958,1(11):1060-1075.
[5]赵维炳.广义Voigt模型模拟的饱水土体一维固结理论及其应用[J].岩土工程学报,1989,11(5):78-85.ZHAO Wei-bing.Theory of one-dimensional consolidation of saturated clays with generalized Voigt model and its applications[J].Chinese Journal of Geotechnical Engineering,1989,11(5):78-85.
[6]刘加才,赵维炳,宰金珉,等.双层黏弹性地基一维固结分析[J].岩土力学,2007,28(4):743-752.LIU Jia-cai,ZHAO Wei-bing,ZAI Jin-min,et al.Analysis of one-dimensional consolidation of double-layered viscoelastic ground[J].Rock and Soil Mechanics,2007,28(4):743-752.
[7]XIE K H,XIE X Y,LI X B.Analytical theory for one-dimensional consolidation of clayey soils exhibiting rheological characteristics under time-dependent loading[J].International Journal for Numerical and Analytical Methods in Geomechanics,2010,32(14):1833-1855.
[8]夏开宗,陈从新,刘秀敏,等.石膏矿矿柱-护顶层支撑体系的流变力学模型分析[J].岩土力学,2017,38(10):2924-2930.XIA Kai-zong,CHEN Cong-xin,LIU Xiu-min,et al.Rheological mechanical modeling of pillar-protective roof supporting system in gypsum mines[J].Rock and Soil Mechanics,2017,38(10):2924-2930.
[9]谷任国,邹育,房营光,等.引入非线性瞬时弹性模量的软土流变模型[J].岩土力学,2018,39(1):237-241.GU Ren-guo,ZOU Yu,FANG Ying-guang,et al.Rheological model of soft soils using nonlinear instantaneous elastic modulus[J].Rock and Soil Mechanics,2018,39(1):237-241.
[10]时刚,刘忠玉,李永辉.循环荷载作用下考虑非达西渗流的软黏土一维流变固结分析[J].岩土力学,2018,39(增刊1):525-528.SHI Gang,LIU Zhong-yu,LIU Yong-hui.Onedimensional rheological consolidation of soft clay under cyclic loadings considering non-Darcy flow[J].Rock and Soil Mechanics,2018,39(Suppl.1):525-528.
[11]GEMANT A.A method of analyzing experimental results obtained from elasto-viscous bodies[J].Physics,1936,7(8):311-317.
[12]SMIT W,VRIES H D.Rheological models containing fractional derivatives[J].Rheologica Acta,1970,9(4):525-534.
[13]BAGLEY R L,TORVIK P J.On the fractional calculus model of viscoelastic behavior[J].Journal of Rheology,1986,30(1):133-155.
[14]BAGLEY R L,TORVIK P J.Fractional calculus in the transient analysis of viscoelastically damped structures[J].AIAA Journal,2012,23(6):918-925.
[15]KOELLER R C.Applications of fractional calculus to the theory of viscoelasticity[J].Journal of Applied Mechanics,1984,51(2):299-307.
[16]刘林超,闫启方,孙海忠.软土流变特性的模型研究[J].岩土力学,2006,27(增刊1):214-217.LIU Lin-chao,YAN Qi-fang,SUN Hai-zhong.Study on model of rheological property of soft clay[J].Rock and Soil Mechanics,2006,27(Suppl.1):214-217.
[17]何利军,孔令伟,吴文军,等.采用分数阶导数描述软黏土蠕变的模型[J].岩土力学,2011,32(增刊2):239-243.HE Li-jun,KONG Ling-wei,WU Wen-jun,et al.Adescription of creep model for soft soil with fractional derivative[J].Rock and Soil Mechanics,2011,32(Suppl.2):239-243.
[18]YIN D S,LI Y Q,WU H,et al.Fractional description of mechanical property evolution of soft soils during creep[J].Water Science and Engineering,2013,6(4):446-455.
[19]何志磊,朱珍德,朱明礼,等.基于分数阶导数的非定常蠕变本构模型研究[J].岩土力学,2016,37(3):737-744.HE Zhi-lei,ZHU Zhen-de,ZHU Ming-li,et al.An unsteady creep constitutive model based on fractional order derivatives[J].Rock and Soil Mechanics,2016,37(3):737-744.
[20]ZHU H H,ZHANG C C,MEI G X,et al.Prediction of one-dimensional compression behavior of Nansha clay using fractional derivatives[J].Marine Georesources&Geotechnology,2016,35(5):688-697.
[21]WANG L,SUN D A,LI P,et al.Semi-analytical solution for one-dimensional consolidation of fractional derivative viscoelastic saturated soils[J].Computers&Geotechnics,2017,83:30-39.
[22]解益,李培超,汪磊,等.分数阶导数黏弹性饱和土体一维固结半解析解[J].岩土力学,2017,38(11):3240-3246.XIE Yi,LI Pei-chao,WANG Lei,et al.Semi-analytical solution for one-dimensional consolidation of viscoelastic saturated soil with fractional order derivative[J].Rock and Soil Mechanics,2017,38(11):3240-3246.
[23]刘忠玉,杨强.基于分数阶Kelvin模型的饱和黏土一维流变固结分析[J].岩土力学,2017,38(12):1001-1009.LIU Zhong-yu,YANG Qiang.One-dimensional rheological consolidation analysis of saturated clay based on fractional order Kelvin’s model[J].Rock and Soil Mechanics,2017,38(12):1001-1009.
[24]DURBIN F.Numerical inversion of Laplace transforms:an efficient improvement to Dubner and Abate’s method[J].Computer Journal,1974,17(4):371-376.
[25]何光渝,王卫红.精确的拉普拉斯数值反演方法及其应用[J].石油学报,1995,16(1):96-104.HE Guang-yu,WANG Wei-hong.An application of an accurate and efficient numerical inversion of Laplace transform in model well-testing analysis[J].Acta Petrolei Sinica,1995,16(1):96-104.
[26]梅国雄,夏君,梅岭.基于不对称连续排水边界的太沙基一维固结方程及其解答[J].岩土工程学报,2011,33(1):28-31.MEI Guo-xiong,XIA Jun,MEI Ling.Terzaghi’s one-dimensional consolidation equation and its solution based on asymmetric continuous drainage boundary[J].Chinese Journal of Geotechnical Engineering,2011,33(1):28-31.
[27]蔡烽,何利军,周小鹏,等.连续排水边界下一维固结不排水对称面的有限元分析[J].岩土工程学报,2012,34(11):2141-2147.CAI Feng,HE Li-jun,ZHOU Xiao-peng,et al.Finite element analysis of one-dimensional consolidation of undrained symmetry plane under continuous drainage boundary[J].Chinese Journal of Geotechnical Engineering,2012,34(11):2141-2147.
[28]李西斌.软土流变固结理论与试验研究[D].杭州:浙江大学,2005.LI Xi-bin.Theoretical and experimental studies on rheological consolidation of soft soil[D].Hangzhou:Zhejiang University,2005.