任意荷载下双面半透水边界分数阶导数黏弹性饱和土层一维固结
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摘要
基于Terzaghi一维固结理论,分析了考虑半透水边界条件的分数阶导数黏弹性饱和土层在随时间变化的任意荷载作用下一维固结问题。首先,应用Laplace变换联立求解饱和土层一维固结微分方程和分数阶Kelvin-Voigt黏弹性本构方程,推导出有效应力和沉降在Laplace变换域内的解析解,采用Crump方法进行Laplace逆变换,得到了时间域内的半解析解。然后将本文得到的半解析解分别退化为半透水边界条件下基于黏弹性假设的一维固结半解析解和双面透水边界条件下基于分数阶黏弹性假设的一维固结半解析解,结果与已有文献的半解析解相同,验证了本研究所提出解的可靠性。最后通过算例分别考察了半透水边界参数、分数阶黏弹性模型参数和荷载参数对饱和土层固结沉降的影响。研究表明,半透水边界条件参数、分数阶次与黏滞系数主要影响饱和土层固结的发展快慢,而饱和土层的最终沉降量主要受到土层压缩模量的影响;另外,饱和土层的固结规律与外荷载变化规律一致。
Based on one-dimensional consolidation theory of Terzaghi,this paper studies the one-dimensional consolidation problem of saturated soil layer with fractional viscoelastic model under symmetric semi-permeable boundaries subjected to arbitrary loading. By using the Laplace transform upon the one-dimensional consolidation equation of saturated soils and the fractional Kelvin-Voigt viscoelastic constitutive equation,the analytical solutions of effective stress and settlement in the Laplace transform domain are obtained. Crump's method is adopted toperform the inverse Laplace transform in order to obtain semi-analytical solutions in the time domain. It is shown that the present solution is reliable and in a good agreement with the existing solutions from literatures by reducing the proposed solution. Last,several numerical examples are provided to investigate the consolidation behavior of saturated soils with the fractional viscoelastic model under symmetric semi-permeable boundaries subjected to arbitrary loading. The results illustrate that,in the case of arbitrary loading,the consolidation rate is affected by the semi-permeable boundary parameters, fractional order, viscosity coefficient and load parameters. The final settlement of saturated soil layer is affected by the modulus of compressibility. In addition,the consolidation behavior of soil layer is consistent with the characteristics of loadings.
Based on one-dimensional consolidation theory of Terzaghi,this paper studies the one-dimensional consolidation problem of saturated soil layer with fractional viscoelastic model under symmetric semi-permeable boundaries subjected to arbitrary loading. By using the Laplace transform upon the one-dimensional consolidation equation of saturated soils and the fractional Kelvin-Voigt viscoelastic constitutive equation,the analytical solutions of effective stress and settlement in the Laplace transform domain are obtained. Crump's method is adopted toperform the inverse Laplace transform in order to obtain semi-analytical solutions in the time domain. It is shown that the present solution is reliable and in a good agreement with the existing solutions from literatures by reducing the proposed solution. Last,several numerical examples are provided to investigate the consolidation behavior of saturated soils with the fractional viscoelastic model under symmetric semi-permeable boundaries subjected to arbitrary loading. The results illustrate that,in the case of arbitrary loading,the consolidation rate is affected by the semi-permeable boundary parameters, fractional order, viscosity coefficient and load parameters. The final settlement of saturated soil layer is affected by the modulus of compressibility. In addition,the consolidation behavior of soil layer is consistent with the characteristics of loadings.
引文
Cai Y Q,Liang X,Zheng Z F,et al.2003.One-dimensional consolidation of viscoelastic soil layer with semi-permeable boundaries under cyclic loadings[J].China Civil Engineering Journal,36(8):86-90.
Cai Y Q,Xu C J,Yuan H M.2001.One dimensional consolidation o layered and visco-elastic solids under arbitrary loading[J].Applied Mathematics and Mechanics,22(3):307-313.
Crump K S.1976.Numerical inversion of Laplace transforms using a fourier series approximation[J].Journal of the Acm,23(1):89-96.
Guo B,Gong X N,Li Y J.2016.Analytical solution for consolidation o stone columnreinforced foundations taking into account consolidation deformation of pile under progressive loading[J].Journal o Engineering Geology,24(3):409-417.
He J,Xiao S F.2003.Some influence of bound water on rheologica properties of marine soft soils[J].Journal of Jilin University(Earth Science Edition),33(2):204-207.
Lei H Y,Wang X C,Ding X D,et al.2014.Exploration on division of primary-secondary consolidation and influential factors with soft soil[J].Journal of Engineering Geology,22(3):489-497.
Li X B,Xie K H,Wang K H,et al.2004.Analytical Solution of 1-Dvisco-elastic consolidation of soils with impeded boundaries under cyclic loadings[J].Engineering Mechanics,21(5):103-108.
Li X B,Xie K H,Wu S M,et al.2005.Theory of 1-D consolidation of soft clayey soil with impeded boundaries under cyclic loading[J].Rock and Soil Mechanics,26(1):155-159.
Liang X,Cai Y Q.2002.1-D consolidation analysis of saturated soil with semi-pervious boundaries under cyclie loads[J].Journal of Hydraulic Engineering,33(7):31-36.
Liu L C,Yan Q F,Sun H Z.2006.Study on model of rheological property of soft clay[J].Rock and Soil Mechanics,27(S1):214-217.
Liu L C,Yan Q F.2011.Lateral vibration of single pile in viscoelastic soil described by fractional derivative model[J].Engineering Mechanics,28(12):139-145.
Liu Z Y,Yan F Y,Wang X J.2013.One-dimensional rheological consolidation analysis of saturated clay considering non-Darcy flow[J].Chinese Journal of Rock Mechanics and Engineering,32(9):1937-1944.
Sun H Z,Zhang W.2007.Analysis of soft soil with viscoelastic fractional derivative Kelvin model[J].Rock and Soil Mechanics,28(9):1983-1986.
Sun J,Xie X Y,Xie K H.2007.Analytical theory for consolidation of double-layered composite ground under impeded boundaries[J].Journal of Zhejiang University(Engineering Science),41(9):1467-1471.
Sun M Q,Wang Q,Niu C C,et al.2015.Rheological consolidation theory of soft soils with secondary consolidation effect[J].Journal of Engineering Geology,23(4):675-680.
Tan T K.1958.One dimensional problems of consolidation and secondary time effects[J].China Civil Engineering Journal,5(1):1-10.
Terzaghi K.1943.Theoretical soil mechanics[M].New York:John Wiley and Sons Inc.
Wang K H,Xie K H,Zeng G X.1998.A study on 1-D consolidation of soils exhibiting rheological characteristics with impeded boundaries[J].Chinese Journal of Geotechnical Engineering,20(2):34-36.
Wang L,Sun D A,Li P C,et al.2017.Semi-analytical solution for onedimensional consolidation of fractional derivative viscoelastic saturated soils[J].Computers and Geotechnics,83:30-39.
Wang Z C,Luo Y S,Luo W B,et al.2011.Mechanical characterization and parameter identification of rheological deformation of subgrade compacted soil[J].Chinese Journal of Rock Mechanics and Engineering,30(1):208-216.
Xie K H,Xie X Y,Gao X.1999.Theory of one dimensional consolidation of two-layered soil with partially drained boundaries[J].Computers and Geotechnics,24(4):265-278.
Xu S,Chen Y L,Zhao C X.2008.One-dimensional consolidation tests of creep deformation and secondary consolidation characteristics of soft soils in Shanghai area[J].Journal of Engineering Geology,16(4):495-510.
Yin D S,Li Y Q,Wu H,et al.2013.Fractional description of mechanica property evolution ofsoft soils during creep[J].Water Science and Engineering,6(4):446-455.
Yin D S,Ren J J,He C L,et al.2007.A new rheological model elemen for geomaterials[J].Chinese Journal of Rock Mechanics and Engineering,26(9):1899-1903.
Zhao W B.1989.Generalized Voigt model to simulate the saturated soi and water body one-dimensional consolidation theory and its application[J].Chinese Journal of Geotechnical Engineering,11(5):78-85.
蔡袁强,梁旭,郑灶锋,等.2003.半透水边界的黏弹性土层在循环荷载下的一维固结[J].土木工程学报,36(8):86-90.
蔡袁强,徐长节,袁海明.2001.任意荷载下成层黏弹性地基的一维固结[J].应用数学和力学,22(3):307-313.
陈宗基.1958.固结及次时间效应的单向问题[J].土木工程学报,5(1):1-10.
郭彪,龚晓南,李亚军.2016.考虑加载过程及桩体固结变形的碎石桩复合地基固结解析解[J].工程地质学报,24(3):409-417.
何俊,肖树芳.2003.结合水对海积软土流变性质的影响[J].吉林大学学报(地球科学版),33(2):204-207.
雷华阳,王学超,丁小冬,等.2014.软土主次固结划分及影响因素探讨[J].工程地质学报,22(3):489-497.
李西斌,谢康和,王奎华,等.2004.双面半透水边界饱和土层在循环荷载作用下的一维黏弹性固结解析解[J].工程力学,21(5):103-108.
李西斌,谢康和.2005.循环荷载下半透水边界土层一维固结解析解[J].岩土力学,26(1):155-159.
梁旭,蔡袁强,吴世明,等.2002.半透水边界饱和土层在循环荷载作用下的一维固结分析[J].水利学报,33(7):31-36.
刘林超,闫启方,孙海忠.2006.软土流变特性的模型研究[J].岩土力学,27(S1):214-217.
刘林超,闫启方.2011.分数导数模型描述的黏弹性土层中桩基水平振动研究[J].工程力学,28(12):139-145.
刘忠玉,闫富有,王喜军.2013.基于非达西渗流的饱和黏土一维流变固结分析[J].岩石力学与工程学报,32(9):1937-1944.
孙海忠,张卫.2007.一种分析软土黏弹性的分数导数开尔文模型[J].岩土力学,28(9):1983-1986.
孙举,谢新宇,谢康和.2007.半透水边界双层复合地基固结解析理论[J].浙江大学学报(工学版),41(9):1467-1471.
孙明乾,王清,牛岑岑,等.2015.考虑次固结效应的流变固结理论研究[J].工程地质学报,23(4):675-680.
王奎华,谢康和,曾国熙.1998.双面半透水边界的一维黏弹性固结理论[J].岩土工程学报,20(2):34-36.
王智超,罗迎社,罗文波,等.2011.路基压实土流变变形的力学表征及参数辨识[J].岩石力学与工程学报,30(1):208-216.
徐珊,陈有亮,赵重兴.2008.单向压缩状态下上海地区软土的蠕变变形与次固结特性研究[J].工程地质学报,16(4):495-510.
殷德顺,任俊娟,和成亮,等.2007.一种新的岩土流变模型元件[J].岩石力学与工程学报,26(9):1899-1903.
赵维炳.1989.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,11(5):78-85.
Cai Y Q,Xu C J,Yuan H M.2001.One dimensional consolidation o layered and visco-elastic solids under arbitrary loading[J].Applied Mathematics and Mechanics,22(3):307-313.
Crump K S.1976.Numerical inversion of Laplace transforms using a fourier series approximation[J].Journal of the Acm,23(1):89-96.
Guo B,Gong X N,Li Y J.2016.Analytical solution for consolidation o stone columnreinforced foundations taking into account consolidation deformation of pile under progressive loading[J].Journal o Engineering Geology,24(3):409-417.
He J,Xiao S F.2003.Some influence of bound water on rheologica properties of marine soft soils[J].Journal of Jilin University(Earth Science Edition),33(2):204-207.
Lei H Y,Wang X C,Ding X D,et al.2014.Exploration on division of primary-secondary consolidation and influential factors with soft soil[J].Journal of Engineering Geology,22(3):489-497.
Li X B,Xie K H,Wang K H,et al.2004.Analytical Solution of 1-Dvisco-elastic consolidation of soils with impeded boundaries under cyclic loadings[J].Engineering Mechanics,21(5):103-108.
Li X B,Xie K H,Wu S M,et al.2005.Theory of 1-D consolidation of soft clayey soil with impeded boundaries under cyclic loading[J].Rock and Soil Mechanics,26(1):155-159.
Liang X,Cai Y Q.2002.1-D consolidation analysis of saturated soil with semi-pervious boundaries under cyclie loads[J].Journal of Hydraulic Engineering,33(7):31-36.
Liu L C,Yan Q F,Sun H Z.2006.Study on model of rheological property of soft clay[J].Rock and Soil Mechanics,27(S1):214-217.
Liu L C,Yan Q F.2011.Lateral vibration of single pile in viscoelastic soil described by fractional derivative model[J].Engineering Mechanics,28(12):139-145.
Liu Z Y,Yan F Y,Wang X J.2013.One-dimensional rheological consolidation analysis of saturated clay considering non-Darcy flow[J].Chinese Journal of Rock Mechanics and Engineering,32(9):1937-1944.
Sun H Z,Zhang W.2007.Analysis of soft soil with viscoelastic fractional derivative Kelvin model[J].Rock and Soil Mechanics,28(9):1983-1986.
Sun J,Xie X Y,Xie K H.2007.Analytical theory for consolidation of double-layered composite ground under impeded boundaries[J].Journal of Zhejiang University(Engineering Science),41(9):1467-1471.
Sun M Q,Wang Q,Niu C C,et al.2015.Rheological consolidation theory of soft soils with secondary consolidation effect[J].Journal of Engineering Geology,23(4):675-680.
Tan T K.1958.One dimensional problems of consolidation and secondary time effects[J].China Civil Engineering Journal,5(1):1-10.
Terzaghi K.1943.Theoretical soil mechanics[M].New York:John Wiley and Sons Inc.
Wang K H,Xie K H,Zeng G X.1998.A study on 1-D consolidation of soils exhibiting rheological characteristics with impeded boundaries[J].Chinese Journal of Geotechnical Engineering,20(2):34-36.
Wang L,Sun D A,Li P C,et al.2017.Semi-analytical solution for onedimensional consolidation of fractional derivative viscoelastic saturated soils[J].Computers and Geotechnics,83:30-39.
Wang Z C,Luo Y S,Luo W B,et al.2011.Mechanical characterization and parameter identification of rheological deformation of subgrade compacted soil[J].Chinese Journal of Rock Mechanics and Engineering,30(1):208-216.
Xie K H,Xie X Y,Gao X.1999.Theory of one dimensional consolidation of two-layered soil with partially drained boundaries[J].Computers and Geotechnics,24(4):265-278.
Xu S,Chen Y L,Zhao C X.2008.One-dimensional consolidation tests of creep deformation and secondary consolidation characteristics of soft soils in Shanghai area[J].Journal of Engineering Geology,16(4):495-510.
Yin D S,Li Y Q,Wu H,et al.2013.Fractional description of mechanica property evolution ofsoft soils during creep[J].Water Science and Engineering,6(4):446-455.
Yin D S,Ren J J,He C L,et al.2007.A new rheological model elemen for geomaterials[J].Chinese Journal of Rock Mechanics and Engineering,26(9):1899-1903.
Zhao W B.1989.Generalized Voigt model to simulate the saturated soi and water body one-dimensional consolidation theory and its application[J].Chinese Journal of Geotechnical Engineering,11(5):78-85.
蔡袁强,梁旭,郑灶锋,等.2003.半透水边界的黏弹性土层在循环荷载下的一维固结[J].土木工程学报,36(8):86-90.
蔡袁强,徐长节,袁海明.2001.任意荷载下成层黏弹性地基的一维固结[J].应用数学和力学,22(3):307-313.
陈宗基.1958.固结及次时间效应的单向问题[J].土木工程学报,5(1):1-10.
郭彪,龚晓南,李亚军.2016.考虑加载过程及桩体固结变形的碎石桩复合地基固结解析解[J].工程地质学报,24(3):409-417.
何俊,肖树芳.2003.结合水对海积软土流变性质的影响[J].吉林大学学报(地球科学版),33(2):204-207.
雷华阳,王学超,丁小冬,等.2014.软土主次固结划分及影响因素探讨[J].工程地质学报,22(3):489-497.
李西斌,谢康和,王奎华,等.2004.双面半透水边界饱和土层在循环荷载作用下的一维黏弹性固结解析解[J].工程力学,21(5):103-108.
李西斌,谢康和.2005.循环荷载下半透水边界土层一维固结解析解[J].岩土力学,26(1):155-159.
梁旭,蔡袁强,吴世明,等.2002.半透水边界饱和土层在循环荷载作用下的一维固结分析[J].水利学报,33(7):31-36.
刘林超,闫启方,孙海忠.2006.软土流变特性的模型研究[J].岩土力学,27(S1):214-217.
刘林超,闫启方.2011.分数导数模型描述的黏弹性土层中桩基水平振动研究[J].工程力学,28(12):139-145.
刘忠玉,闫富有,王喜军.2013.基于非达西渗流的饱和黏土一维流变固结分析[J].岩石力学与工程学报,32(9):1937-1944.
孙海忠,张卫.2007.一种分析软土黏弹性的分数导数开尔文模型[J].岩土力学,28(9):1983-1986.
孙举,谢新宇,谢康和.2007.半透水边界双层复合地基固结解析理论[J].浙江大学学报(工学版),41(9):1467-1471.
孙明乾,王清,牛岑岑,等.2015.考虑次固结效应的流变固结理论研究[J].工程地质学报,23(4):675-680.
王奎华,谢康和,曾国熙.1998.双面半透水边界的一维黏弹性固结理论[J].岩土工程学报,20(2):34-36.
王智超,罗迎社,罗文波,等.2011.路基压实土流变变形的力学表征及参数辨识[J].岩石力学与工程学报,30(1):208-216.
徐珊,陈有亮,赵重兴.2008.单向压缩状态下上海地区软土的蠕变变形与次固结特性研究[J].工程地质学报,16(4):495-510.
殷德顺,任俊娟,和成亮,等.2007.一种新的岩土流变模型元件[J].岩石力学与工程学报,26(9):1899-1903.
赵维炳.1989.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,11(5):78-85.