层状非饱和土地基的轴对称稳态动力响应
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摘要
考虑土颗粒、孔隙流体的压缩性及各相物质间的黏性、惯性耦合作用,采用理论上更加严谨的饱和度本构关系式,建立了非饱和土的动力控制方程。该组方程与饱和土的经典Biot波动方程完全兼容,因此,具有更广泛的适用性。通过引入一组状态向量,在圆柱坐标系下将非饱和土满足的波动方程转化为状态方程组,利用Hankel变换,求解状态方程组,得到了传递矩阵。结合边界条件及层间接触连续条件,求解了层状非饱和地基的稳态动力响应问题。数值算例表明:土层对地表动位移的影响主要集中在临界深度范围内;软硬土层的相对次序对地表动位移幅值有显著影响;饱和度增大会引起土的物理参数发生相应的改变,尤其是动剪切模量通常降幅较大,而动剪切模量是决定位移幅值的关键因素,最终的结果是导致地表动位移幅值明显增大。
Considering compression of solid grain and pore fluids,viscous-coupling interactions and inertial force of fluids,the dynamic governing equations for unsaturated soils are established by adopting an exact constitutive formula of saturation.These equations are highly versatile and completely compatible with Biot's wave equations for the special case of fully saturated soils.The governing equations in cylindrical coordinates are firstly transformed into a group of state differential equations by introducing the state vector.Then the transfer matrix for layered media is derived by means of a Hankel transform.Using the transfer matrix followed by boundary and continuity conditions,solutions of steady state dynamic response for multilayer unsaturated soils are obtained.Numerical examples show that the displacement of the ground surface is mainly affected by stratum within critical depth.The relative position of soft and hard strata has a significant influence on displacement.The displacement of the ground surface increases with saturation degree since dynamic shear modulus will decrease distinctly for most soils,which is the key factor to determine displacement amplitude.
Considering compression of solid grain and pore fluids,viscous-coupling interactions and inertial force of fluids,the dynamic governing equations for unsaturated soils are established by adopting an exact constitutive formula of saturation.These equations are highly versatile and completely compatible with Biot's wave equations for the special case of fully saturated soils.The governing equations in cylindrical coordinates are firstly transformed into a group of state differential equations by introducing the state vector.Then the transfer matrix for layered media is derived by means of a Hankel transform.Using the transfer matrix followed by boundary and continuity conditions,solutions of steady state dynamic response for multilayer unsaturated soils are obtained.Numerical examples show that the displacement of the ground surface is mainly affected by stratum within critical depth.The relative position of soft and hard strata has a significant influence on displacement.The displacement of the ground surface increases with saturation degree since dynamic shear modulus will decrease distinctly for most soils,which is the key factor to determine displacement amplitude.
引文
[1]BIOT M A.Theory of propagation of elastic waves in a fluid-saturated porous solid(I.low-frequency range)[J].Journal of the Acoustical Society of America,1956,28(2):168-178.
[2]BIOT M A.Theory of propagation of elastic waves in a fluid-saturated porous solid(Ⅱ.higher-frequency range)[J].Journal of the Acoustical Society of America,1956,28(2):179-191.
[3]PHILIPPACOPOULOS A J.Lamb's problem for fluid-saturated porous media[J].Bulletin of the Seismological Society of America,1988,78(2):908-923.
[4]SENJUNTICHAI T,RAJAPAKSE R K N D.Dynamic Green's functions of homogeneous poroelastic half-space[J].Journal of Engineering Mechanics,ASCE,1994,120(11):2381-2404.
[5]RAJAPAKSE R K N D,SENJUNTICHAI T.Dynamic response of a multi-layered poroelastic medium[J].Earthquake Engineering and Structural Dynamics,1995,24(5):703-722.
[6]蔡袁强,占宏,郑灶锋,等.横观各向同性饱和土体振动分析[J].岩土力学,2005,26(12):1917-1920.CAI Yuan-qiang,ZHAN Hong,ZHENG Zao-feng,et al.Vibration analysis of transversely isotropic saturated soils[J].Rock and Soil Mechanics,2005,26(12):1917-1920.
[7]DOMENICO S N.Effect of water saturation on seismic reflectivity of sand reservoirs encased in shale[J].Geophysics,1974,39(6):759.
[8]VARDOULAKIS I,BESKOS D E.Dynamic behavior of nearly saturated porous media[J].Mechanics of Materials,1986,5(1):87-108.
[9]黄义,张引科.非饱和土本构关系的混合物理论(Ⅱ)——线性本构方程和场方程[J].应用数学和力学,2003,24(2):124-137.HUANG Yi,ZHANG Yin-ke.Constitutive relation of unsaturated soil by use of the mixture theory(Ⅱ)—linear constitutive equations and field equations[J].Applied Mathematics and Mechanics,2003,24(2):124-137.
[10]蔡袁强,李保忠,徐长节.两种不混溶流体饱和岩石中弹性波的传播[J].岩石力学与工程学报,2006,25(10):2009-2016.CAI Yuan-qiang,LI Bao-zhong,XU Chang-jie.Analysis of elastic wave propagation in sandstone saturated by two immiscible fluids[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(10):2009-2016.
[11]徐长节,史焱永.非饱和土中波的传播特性[J].岩土力学,2004,25(3):354-358.XU Chang-jie,SHI Yan-yong.Characteristics of wave propagation in unsaturated soils[J].Rock and Soil Mechanics,2004,25(3):354-358.
[12]弗雷德隆德D G,拉哈尔佐H.非饱和土土力学[M].陈仲颐,张在明,陈愈炯,等译.北京:中国建筑工业出版社,1997.
[13]BROOKS R H,COREY A T.Hydraulic propertied of porous media[C]//Hydranlic Paper 3.Fort Collins:Colorado State University,1964:3-27.
[14]陈龙珠,陈胜立,梁发云.饱和地基竖向振动的衰减特性[J].上海交通大学学报,2002,36(3):376-381.CHEN Long-zhu,CHEN Sheng-li,LIANG Fa-yun.Vibration attenuation of saturated strata by vertical surface loads[J].Journal of Shanghai Jiaotong University,2002,36(3):376-381.
[15]BOLZON G,SCHREFLER B A,ZIENKIEWICZ O C.Elastoplastic soil constitutive laws generalized to partially saturated states[J].Geotechnique,1996,46(2):279-289.
[16]陈国兴,谢君斐,张克绪.土的动模量和阻尼比的经验估计[J].地震工程与工程振动,1995,15(1):73-84.CHEN Guo-xing,XIE Jun-fei,ZHANG Ke-xu.The empirical evaluation of soil moduli and damping ratio for dynamic analysis[J].Earthquake Engineering and Engineering Vibration,1995,15(1):73-84.
[17]尚守平,熊伟,杜运兴,等.饱和场地土动力特性试验研究[J].岩土力学,2008,29(1):23-27.SHANG Shou-ping,XIONG Wei,DU Yun-xing,et al.Experimental study of dynamic characteristics of saturated soils[J].Rock and Soil Mechanics,2008,29(1):23-27.
[18]蔡袁强,孟楷,徐长节.成层地基在轴对称荷载下的稳态振动分析[J].岩土力学,2004,25(12):1994-1998.CAI Yuan-qiang,MENG Kai,XU Chang-jie.Steady-state vibration analysis of layered saturated soils under axisymmetric cyclic loading[J].Rock and Soil Mechanics,2004,25(12):1994-1998.3.02.52.00-5m)地表排水、排气地表不排水、不排气1(/)0,r(zu1.51.00.500 2 4 6 8 10r/m
[2]BIOT M A.Theory of propagation of elastic waves in a fluid-saturated porous solid(Ⅱ.higher-frequency range)[J].Journal of the Acoustical Society of America,1956,28(2):179-191.
[3]PHILIPPACOPOULOS A J.Lamb's problem for fluid-saturated porous media[J].Bulletin of the Seismological Society of America,1988,78(2):908-923.
[4]SENJUNTICHAI T,RAJAPAKSE R K N D.Dynamic Green's functions of homogeneous poroelastic half-space[J].Journal of Engineering Mechanics,ASCE,1994,120(11):2381-2404.
[5]RAJAPAKSE R K N D,SENJUNTICHAI T.Dynamic response of a multi-layered poroelastic medium[J].Earthquake Engineering and Structural Dynamics,1995,24(5):703-722.
[6]蔡袁强,占宏,郑灶锋,等.横观各向同性饱和土体振动分析[J].岩土力学,2005,26(12):1917-1920.CAI Yuan-qiang,ZHAN Hong,ZHENG Zao-feng,et al.Vibration analysis of transversely isotropic saturated soils[J].Rock and Soil Mechanics,2005,26(12):1917-1920.
[7]DOMENICO S N.Effect of water saturation on seismic reflectivity of sand reservoirs encased in shale[J].Geophysics,1974,39(6):759.
[8]VARDOULAKIS I,BESKOS D E.Dynamic behavior of nearly saturated porous media[J].Mechanics of Materials,1986,5(1):87-108.
[9]黄义,张引科.非饱和土本构关系的混合物理论(Ⅱ)——线性本构方程和场方程[J].应用数学和力学,2003,24(2):124-137.HUANG Yi,ZHANG Yin-ke.Constitutive relation of unsaturated soil by use of the mixture theory(Ⅱ)—linear constitutive equations and field equations[J].Applied Mathematics and Mechanics,2003,24(2):124-137.
[10]蔡袁强,李保忠,徐长节.两种不混溶流体饱和岩石中弹性波的传播[J].岩石力学与工程学报,2006,25(10):2009-2016.CAI Yuan-qiang,LI Bao-zhong,XU Chang-jie.Analysis of elastic wave propagation in sandstone saturated by two immiscible fluids[J].Chinese Journal of Rock Mechanics and Engineering,2006,25(10):2009-2016.
[11]徐长节,史焱永.非饱和土中波的传播特性[J].岩土力学,2004,25(3):354-358.XU Chang-jie,SHI Yan-yong.Characteristics of wave propagation in unsaturated soils[J].Rock and Soil Mechanics,2004,25(3):354-358.
[12]弗雷德隆德D G,拉哈尔佐H.非饱和土土力学[M].陈仲颐,张在明,陈愈炯,等译.北京:中国建筑工业出版社,1997.
[13]BROOKS R H,COREY A T.Hydraulic propertied of porous media[C]//Hydranlic Paper 3.Fort Collins:Colorado State University,1964:3-27.
[14]陈龙珠,陈胜立,梁发云.饱和地基竖向振动的衰减特性[J].上海交通大学学报,2002,36(3):376-381.CHEN Long-zhu,CHEN Sheng-li,LIANG Fa-yun.Vibration attenuation of saturated strata by vertical surface loads[J].Journal of Shanghai Jiaotong University,2002,36(3):376-381.
[15]BOLZON G,SCHREFLER B A,ZIENKIEWICZ O C.Elastoplastic soil constitutive laws generalized to partially saturated states[J].Geotechnique,1996,46(2):279-289.
[16]陈国兴,谢君斐,张克绪.土的动模量和阻尼比的经验估计[J].地震工程与工程振动,1995,15(1):73-84.CHEN Guo-xing,XIE Jun-fei,ZHANG Ke-xu.The empirical evaluation of soil moduli and damping ratio for dynamic analysis[J].Earthquake Engineering and Engineering Vibration,1995,15(1):73-84.
[17]尚守平,熊伟,杜运兴,等.饱和场地土动力特性试验研究[J].岩土力学,2008,29(1):23-27.SHANG Shou-ping,XIONG Wei,DU Yun-xing,et al.Experimental study of dynamic characteristics of saturated soils[J].Rock and Soil Mechanics,2008,29(1):23-27.
[18]蔡袁强,孟楷,徐长节.成层地基在轴对称荷载下的稳态振动分析[J].岩土力学,2004,25(12):1994-1998.CAI Yuan-qiang,MENG Kai,XU Chang-jie.Steady-state vibration analysis of layered saturated soils under axisymmetric cyclic loading[J].Rock and Soil Mechanics,2004,25(12):1994-1998.3.02.52.00-5m)地表排水、排气地表不排水、不排气1(/)0,r(zu1.51.00.500 2 4 6 8 10r/m
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