分数阶导数黏弹性饱和土体一维固结半解析解
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摘要
将分数阶微积分理论引入Kelvin-Voigt本构模型,以描述黏弹性饱和土体的力学行为。对饱和土体一维固结方程和上述分数阶导数Kelvin-Voigt本构方程实施Laplace变换,联立求解得到变换域内有效应力和沉降的解析解。采用Crump方法实现Laplace数值反演,从而获得了物理空间一维固结问题的半解析解,并将其退化到弹性和黏弹性两种经典情形,分析表明,它与经典解析解完全相同,这证明了经典弹性和黏弹性解析解可视为本研究提出分数阶导数黏弹性解的特例。开展了参数研究,即分析了相关各种参数对固结沉降的影响。研究表明,瞬时荷载情形下分数阶导数黏弹性饱和土体一维固结最终沉降量与黏滞系数和分数阶次无关,而不同黏滞系数和分数阶次对固结时间有较大影响。其研究结果有助于深入认识黏弹性饱和土体的固结力学行为。
Theory of fractional calculus is introduced to Kelvin-Voigt constitutive model to describe the mechanical behavior of viscoelastic saturated soil. Applying Laplace transforms upon the one-dimensional consolidation equation of saturated soil and the fractional order derivative Kelvin-Voigt constitutive equation, we derived analytical solutions of the effective stress and the settlement in transformed domains. Then the semi-analytical solution of one-dimensional consolidation problem in physical space was obtained after implementing Laplace numerical inversion by using Crump method. As for two classical cases of elasticity and viscoelasticity, the simplified semi-analytical solutions in this study are the same as those of the two classical cases. It indicates that the analytical solutions of two classical cases can be considered as the special cases of the solutions presented in this paper. Last, parameter studies were conducted to analyze the effects of the various parameters on the consolidation settlement. The results show that, in the case of instantaneous loading, the final settlement is independent on the viscosity coefficient and the fractional order, while the consolidation time is influenced greatly by the viscosity coefficient and the fractional order. The present study contributes to further understand the mechanical behavior of the consolidation of viscoelastic saturated soil.
Theory of fractional calculus is introduced to Kelvin-Voigt constitutive model to describe the mechanical behavior of viscoelastic saturated soil. Applying Laplace transforms upon the one-dimensional consolidation equation of saturated soil and the fractional order derivative Kelvin-Voigt constitutive equation, we derived analytical solutions of the effective stress and the settlement in transformed domains. Then the semi-analytical solution of one-dimensional consolidation problem in physical space was obtained after implementing Laplace numerical inversion by using Crump method. As for two classical cases of elasticity and viscoelasticity, the simplified semi-analytical solutions in this study are the same as those of the two classical cases. It indicates that the analytical solutions of two classical cases can be considered as the special cases of the solutions presented in this paper. Last, parameter studies were conducted to analyze the effects of the various parameters on the consolidation settlement. The results show that, in the case of instantaneous loading, the final settlement is independent on the viscosity coefficient and the fractional order, while the consolidation time is influenced greatly by the viscosity coefficient and the fractional order. The present study contributes to further understand the mechanical behavior of the consolidation of viscoelastic saturated soil.
引文
[1]谢新宇,王龙,刘开富.饱和土一维固结的电阻网络模拟[J].岩土力学,2011,32(11):3455-3460.XIE Xin-yu,WANG Long,LIU Kai-fu.Resistance network analog for one-dimensional consolidation of saturated soil[J].Rock and Soil Mechanics,2011,32(11):3455-3460.
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[11]赵永玲,侯之超.基于分数导数的橡胶材料两种黏弹性本构模型[J].清华大学学报(自然科学版),2013,53(3):378-383.ZHAO Yong-ling,HOU Zhi-chao.Two viscoelastic constitutive models of rubber materials using fractional derivations[J].Journal of Tsinghua University(Science and Technology),2013,53(3):378-383.
[12]孙雅珍,朱传江,王立保.基于分数阶导数沥青路面松弛应力强度因子分析[J].中外公路,2015,25(2):65-68.SUN Ya-zhen,ZHU Chuan-jiang,WANG Li-bao.Analysis based on fractional derivative asphalt pavement relaxed stress intensity factor[J].Journal of China and Foreign Highway,2015,25(2):65-68.
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[15]殷德顺,任俊娟,和成亮,等.一种新的岩土流变模型元件[J].岩石力学与工程学报,2007,26(9):1899-1903.YIN De-shun,REN Jun-juan,HE Cheng-liang,et al.Anew rheological model element for geomaterials[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(9):1899-1903.
[16]王智超,罗迎社,罗文波,等.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,2011,30(1):208-216.WANG Zhic-hao,LUO Ying-she,LUO Wen-bo,et al.Mechanical characterization and parameter identification of rheological deformation of subgrade compacted soil[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(1):208-216.
[17]NONNENMACHER T F,METZLER R.On the Riemann-Liouville fractional calculus and some recent applications[J].Fractals,1995,3(3):557-566.
[18]MANDELBORT B B.The fractal geometry of nature[M].San Francisco:Freeman,1982.
[19]尹检务,旷杜敏,王智超.压实土固结蠕变特征及分数阶流变模型参数分析[J].湖南科技大学学报(自然科学版),2015,30(3):46-51.YIN Jian-wu,KUANG Du-min,WANG Zhi-chao.Consolidation creep characteristics of compacted clay and its parameters analysis of rheological constitutive model based on fractional calculus[J].Journal of Hunan University of Science and Technology(Natural Science Edition),2015,30(3):46-51.
[20]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):96-103.HE Guang-yu,WANG Wei-hong.Accurate numerical Laplace inversion method and its application[J].Acta Petroleisinaca,1995,16(1):96-103.
[21]STEHFEST H.Algorithm 368:Numerical inversion of Laplace transforms[J].Communications of the ACM,1970,13(1):47-49.
[22]CRUMP K S.Numerical inversion of Laplace transforms using a Fourier series approximation[J].Journal of the ACM,1976,23(1):89-96.
[23]张先伟,王常明.结构性软土的黏滞系数[J].岩土力学,2011,32(11):3276-3282.ZHANG Xian-wei,WANG Chang-ming.Viscosity coefficient of structural soft clay[J].Rock and Soil Mechanics,2011,32(11):3276-3282.
[2]TERZAGHI K.Theoretical soil mechanics[M].New York:John Wiley and Sons Inc.,1943.
[3]刘林超,闫启方.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,2011,28(12):139-145.LIU Lin-chao,YAN Qi-fang.Lateral vibration of single pile in viscoelastic soil described by fractional derivative model[J].Engineering Mechanics,2011,28(12):139-145.
[4]陈宗基.固结及次时间效应的单向问题[J].土木工程学报,1958,5(1):1-10.CHEN Zong-ji.Unidirectional issue several time consolidation effect[J].China Civil Engineering Journal,1958,5(1):1-10.
[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.Generalized Voigt model to simulate the saturated soil and water body one-dimensional consolidation theory and its application[J].Chinese Journal of Geotechnical Engineering,1989,11(5):78-85.
[6]蔡袁强,徐长节,袁海明.任意荷载下成层黏弹性地基的一维固结[J].应用力学和数学,2001,22(3):307-313.CAI Yuan-qiang,XU Chang-jie,YUAN Hai-ming.Under arbitrary loading viscoelastic foundation layer of one-dimensional consolidation[J].Applied Mathematics and Mechanics,2001,22(3):307-313.
[7]蔡袁强,梁旭,郑灶锋,等.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,2003,36(8):86-90.CAI Yuan-qiang,LIANG Xu,ZHENG Zao-feng,et al.One-dimensional consolidation of viscoelastic soil layer with semi-permeable boundaries under cyclic loading[J].China Civil Engineering Journal,2003,36(8):86-90.
[8]余湘娟,殷宗泽,高磊.软土的一维次固结双曲线流变模型研究[J].岩土力学,2015(2):320-324.YU Xiang-juan,YIN Zong-ze,GAO Lei.A hyperbolic rheological model for one-dimensional secondary consolidation of soft soils[J].Rock and Soil Mechanics,2015(2):320-324.
[9]GEMANT A.A method of analyzing experimental results obtained from elasto-viscous bodies[J].Journal of Applied Physics,1936,7(1):311-317.
[10]段玮玮,闻敏杰,李强.饱和分数导数型黏弹性土层竖向振动放大效应[J].工程力学,2013,30(4):235-240.DUAN Wei-wei,WEN Min-jie,LI Qiang.Vertical vibration amplification of a saturated fractional derivative type viscoelastic soil layer[J].Engineering Mechanics,2013,30(4):235-240.
[11]赵永玲,侯之超.基于分数导数的橡胶材料两种黏弹性本构模型[J].清华大学学报(自然科学版),2013,53(3):378-383.ZHAO Yong-ling,HOU Zhi-chao.Two viscoelastic constitutive models of rubber materials using fractional derivations[J].Journal of Tsinghua University(Science and Technology),2013,53(3):378-383.
[12]孙雅珍,朱传江,王立保.基于分数阶导数沥青路面松弛应力强度因子分析[J].中外公路,2015,25(2):65-68.SUN Ya-zhen,ZHU Chuan-jiang,WANG Li-bao.Analysis based on fractional derivative asphalt pavement relaxed stress intensity factor[J].Journal of China and Foreign Highway,2015,25(2):65-68.
[13]DI PAOLA M,HEUER R,PIRROTTA A.Fractional visco-elastic Euler-Bernoulli beam[J].International Journal of Solids and Structures,2013,50(22):3505-3510.
[14]YIN De-shun,LI Yan-qing,WU Hao,et al.Fractional description of mechanical property evolution of soft soils during creep[J].Water Science and Engineering,2013,6(4):446-455.
[15]殷德顺,任俊娟,和成亮,等.一种新的岩土流变模型元件[J].岩石力学与工程学报,2007,26(9):1899-1903.YIN De-shun,REN Jun-juan,HE Cheng-liang,et al.Anew rheological model element for geomaterials[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(9):1899-1903.
[16]王智超,罗迎社,罗文波,等.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,2011,30(1):208-216.WANG Zhic-hao,LUO Ying-she,LUO Wen-bo,et al.Mechanical characterization and parameter identification of rheological deformation of subgrade compacted soil[J].Chinese Journal of Rock Mechanics and Engineering,2011,30(1):208-216.
[17]NONNENMACHER T F,METZLER R.On the Riemann-Liouville fractional calculus and some recent applications[J].Fractals,1995,3(3):557-566.
[18]MANDELBORT B B.The fractal geometry of nature[M].San Francisco:Freeman,1982.
[19]尹检务,旷杜敏,王智超.压实土固结蠕变特征及分数阶流变模型参数分析[J].湖南科技大学学报(自然科学版),2015,30(3):46-51.YIN Jian-wu,KUANG Du-min,WANG Zhi-chao.Consolidation creep characteristics of compacted clay and its parameters analysis of rheological constitutive model based on fractional calculus[J].Journal of Hunan University of Science and Technology(Natural Science Edition),2015,30(3):46-51.
[20]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):96-103.HE Guang-yu,WANG Wei-hong.Accurate numerical Laplace inversion method and its application[J].Acta Petroleisinaca,1995,16(1):96-103.
[21]STEHFEST H.Algorithm 368:Numerical inversion of Laplace transforms[J].Communications of the ACM,1970,13(1):47-49.
[22]CRUMP K S.Numerical inversion of Laplace transforms using a Fourier series approximation[J].Journal of the ACM,1976,23(1):89-96.
[23]张先伟,王常明.结构性软土的黏滞系数[J].岩土力学,2011,32(11):3276-3282.ZHANG Xian-wei,WANG Chang-ming.Viscosity coefficient of structural soft clay[J].Rock and Soil Mechanics,2011,32(11):3276-3282.