饱和分数导数型粘弹性土层竖向振动放大效应
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摘要
将土骨架视为具有分数阶导数本构关系的粘弹性体,采用Biot动力固结方程,在频率域内研究了饱和分数导数粘弹性土层的土骨架粘性、土层厚度等对竖向振动放大系数的影响。通过动力控制方程解耦和边界条件束缚,给出了经典弹性饱和土、分数导数型粘弹性饱和土和经典粘弹性饱和土三种情况下饱和土层的位移、应力和孔压解析表达式。考察了饱和土各物理和几何参数对竖向振动放大系数的影响,结果表明:在不同土层厚度时,经典弹性饱和土、分数导数型粘弹性饱和土及经典粘弹性饱和土的竖向振动放大系数各不相同;分数导数模型的材料参数对振动放大系数有较大影响。
The soil skeleton is treated as a viscoelastic medium with a fractional derivative constitutive relation.By utilizing the Biot’s theory,the influences of the viscosity of soil skeleton and the thickness of soil layer for a saturated soil layer on the vertical vibration amplification coefficient are investigated in the frequency domain.By decoupling dynamic control equations and bounding the boundary conditions,the analytical expressions of the displacement,stress and pore water pressure for the saturated classic elastic,saturated fractional derivative viscoelastic soil and saturated classic viscoelastic soil are obtained.On the basis,the vertical vibration amplification of the soil layer is analyzed for different physical and geometrical parameters of the soil.It is shown that the vertical vibration amplification of the saturated classic elastic,fractional derivative viscoelastic saturated soil and saturated classic viscoelastic soil are different for the different thickness of soil layers;the material parameters of the fractional derivative model have great influences on the vertical vibration amplification coefficient.
The soil skeleton is treated as a viscoelastic medium with a fractional derivative constitutive relation.By utilizing the Biot’s theory,the influences of the viscosity of soil skeleton and the thickness of soil layer for a saturated soil layer on the vertical vibration amplification coefficient are investigated in the frequency domain.By decoupling dynamic control equations and bounding the boundary conditions,the analytical expressions of the displacement,stress and pore water pressure for the saturated classic elastic,saturated fractional derivative viscoelastic soil and saturated classic viscoelastic soil are obtained.On the basis,the vertical vibration amplification of the soil layer is analyzed for different physical and geometrical parameters of the soil.It is shown that the vertical vibration amplification of the saturated classic elastic,fractional derivative viscoelastic saturated soil and saturated classic viscoelastic soil are different for the different thickness of soil layers;the material parameters of the fractional derivative model have great influences on the vertical vibration amplification coefficient.
引文
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[12]朱正佑,李根国,程昌钧.具有分数导数本构关系的粘弹性Timoshenko梁的静动力学行为分析[J].应用数学和力学,2002,23(1):1―10.Zhu Zhengyou,Li Genguo,Cheng Changjun.Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation[J].Applied Mathematics and Mechanics,2002,23(1):1―10.(in Chinese)
[13]刘林超,杨骁.分数导数模型描述的饱和土桩纵向振动分析[J].岩土力学,2011,32(2):526―532.Liu Linchao,Yang Xiao.Analysis of vertical vibrations of a pile in saturated soil described by fractional derivative model[J].Rock and Soil Mechanics,2011,32(2):526―532.(in Chinese)
[14]杨骁,闻敏杰.饱和分数导数型粘弹性土-深埋圆形隧洞衬砌系统的动力特性[J].工程力学,2012,29(12):248―255.Yang Xiao,Wen Minjie.Dynamic characteristics of saturated fractional derivative type viscoelastic soil and lining system with a deeply embedded circular tunnel[J].Engineering Mechanics,2012,29(12):248―255.(in Chinese)
[15]Biot M A.Theory of propagation of elastic waves in a fluid-saturated porous media.I.low frequency range[J].The Journal of the Acoustical Society of America,1956,28(2):168―178.
[16]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2):239―244.He Lijun,Kong Lingwei,Wu Wenjun,et al.A description of creep model for soft soil with fractional derivative[J].Rock and Soil Mechanics,2011,32(Suppl 2):239―244.(in Chinese)
[2]Idriss I M,Seed H B.Seismic response of horizontal soil layers[J].Journal of the Soil Mechanics and Foundations Division,1968,94(4):1003―1031.
[3]Wolf J P.土-结构动力相互作用[M].吴世明,译.北京:地震出版社,1989:146―151.Wolf J P.Dynamic soil-structure interaction[M].Translated by Wu Shiming.Beijing:Seismic Press,1989:146―151.(in Chinese)
[4]Davis R.Effects of weathering on site response[J].Earthquake Engineering and Structure Dynamics,1995,24(2):301―309.
[5]Zhao J X.Modal analysis of soft-soil sites including radiation damping[J].Earthquake Engineering and Structure Dynamics,1997,26(1):93―113.
[6]尚守平,李刚,任慧.剪切模量沿深度按指数规律增大的场地土的地震放大效应[J].工程力学,2005,22(5):153―157,35.Shang Shouping,Li Gang,Ren Hui.Seismic amplification of sites with exponentially increasing shear modulus with depth[J].Engineering Mechanics,2005,22(5):153―157,35.(in Chinese)
[7]Yang J,Sato T.Characterization of a reclaimed site and its seismic vertical amplification[C]//Proceedings of Sessions of Geo-Denver2000――Use of Geophysical Methods in Construction.Denver,Colorado,United States:American Society of Civil Engineers,2000,296:187―200.
[8]Yang J,Sato T.Analytical study of saturation effects on seismic vertical amplification of a soil layer[J].Geotechnique,2001,51(2):161―165.
[9]Eringen A C.Mechanics of continua[M].New York:Huntington Press,1980:385―406.
[10]马晓华,徐长节,蔡袁强.粘弹性中竖向振动放大系数的研究[J].岩石力学与工程学报,2005,24(24):4534―4539.Ma Xiaohua,Xu Changjie,Cai Yuanqiang.Analytical study on vertical vibration amplification of a viscoelastic soil layer[J].Chinese Journal of Rock Mechanics and Engineering,2005,24(24):4534―4539.(in Chinese)
[11]孙海忠,张卫.服从分数导数Kelvin本构模型的粘弹性阻尼器的阻尼性能分析及试验研究[J].振动工程学报,2008,21(1):48―53.Sun Haizhong,Zhang Wei.Analysis and experiment on damping properties of visco-elastic damper modeled byfractional Kelvin method[J].Journal of Vibration Engineering,2008,21(1):48―53.(in Chinese)
[12]朱正佑,李根国,程昌钧.具有分数导数本构关系的粘弹性Timoshenko梁的静动力学行为分析[J].应用数学和力学,2002,23(1):1―10.Zhu Zhengyou,Li Genguo,Cheng Changjun.Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation[J].Applied Mathematics and Mechanics,2002,23(1):1―10.(in Chinese)
[13]刘林超,杨骁.分数导数模型描述的饱和土桩纵向振动分析[J].岩土力学,2011,32(2):526―532.Liu Linchao,Yang Xiao.Analysis of vertical vibrations of a pile in saturated soil described by fractional derivative model[J].Rock and Soil Mechanics,2011,32(2):526―532.(in Chinese)
[14]杨骁,闻敏杰.饱和分数导数型粘弹性土-深埋圆形隧洞衬砌系统的动力特性[J].工程力学,2012,29(12):248―255.Yang Xiao,Wen Minjie.Dynamic characteristics of saturated fractional derivative type viscoelastic soil and lining system with a deeply embedded circular tunnel[J].Engineering Mechanics,2012,29(12):248―255.(in Chinese)
[15]Biot M A.Theory of propagation of elastic waves in a fluid-saturated porous media.I.low frequency range[J].The Journal of the Acoustical Society of America,1956,28(2):168―178.
[16]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2):239―244.He Lijun,Kong Lingwei,Wu Wenjun,et al.A description of creep model for soft soil with fractional derivative[J].Rock and Soil Mechanics,2011,32(Suppl 2):239―244.(in Chinese)
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