A semi-analytical solution for the transient response of one-dimensional unsaturated single-layer poroviscoelastic media

This paper develops a semi-analytical solution for the transient response of an unsaturated single-layer poroviscoelastic medium with two immiscible fluids by using the Laplace transformation and the state-space method. Using the elastic–viscoelastic correspondence principle, we first introduce the Kelvin–Voigt model into Zienkiewicz’s unsaturated poroelastic model. The vibrational response for unsaturated porous material can be obtained by combining these two models and assuming that the wetting and non-wetting fluids are compressible, the solid skeleton and solid particles are viscoelastic, and the inertial and mechanical couplings are taken into account. The Laplace transformation and state-space method are used to solve the basic equations with the associated initial and boundary conditions, and the analytical solution in the Laplace domain is developed. To evaluate the responses in the time domain, Durbin’s numerical inverse Laplace transform method is used to obtain the semi-analytical solution. There are three compressional waves in porous media with two immiscible fluids. Moreover, to observe the three compressional waves clearly, we assume the two immiscible fluids are water and oil. Finally, several examples are provided to show the validity of the semi-analytical solution and to assess the influences of the viscosity coefficients and dynamic permeability coefficients on the behavior of the three compressional waves.     

单层不可压缩饱和多孔介质一维瞬态响应精确解

基于Biot理论,考虑惯性、黏滞和机械耦合作用,假定固体颗粒和流体均不可压缩,得到了表面任意竖向荷载作用下单层饱和多孔介质一维瞬态响应的精确解。导出了以固体骨架位移表示的无量纲控制方程,并将边界条件齐次化。求解对应无黏滞耦合作用的特征值问题,得到一组满足齐次边界条件、关于空间坐标的正交函数基。利用变异系数法和基函数的正交性,得到一系列相互解耦的、关于时间的二阶常微分方程及相应的初始条件,并采用状态空间法求解常微分方程,得到位移分量。对整体平衡方程关于空间坐标积分,根据边界条件可确定总应力,并进而求得孔隙压力。通过算例验证所得解法的正确性     

Exact solutions for one‐dimensional transient response of fluid‐saturated porous media

Based on the Biot theory, the exact solutions for one‐dimensional transient response of single layer of fluid‐saturated porous media and semi‐infinite media are developed, in which the fluid and solid particles are assumed to be compressible and the inertial, viscous and mechanical couplings are taken into account. First, the control equations in terms of the solid displacement u and a relative displacement w are expressed in matrix form. For problems of single layer under homogeneous boundary conditions, the eigen‐values and the eigen‐functions are obtained by means of the variable separation method, and the displacement vector u is put forward using the searching method. In the case of nonhomogeneous boundary conditions, the boundary conditions are first homogenized, and the displacement field is constructed basing upon the eigen‐functions. Making use of the orthogonality of eigen‐functions, a series of ordinary differential equations with respect to dimensionless time and their corresponding initial conditions are obtained. Those differential equations are solved by the state‐space method, and the series solutions for three typical nonhomogeneous boundary conditions are developed. For semi‐infinite media, the exact solutions in integral form for two kinds of nonhomogeneous boundary conditions are presented by applying the cosine and sine transforms to the basic equations. Finally, three examples are studied to illustrate the validity of the solutions, and to assess the influence of the dynamic permeability coefficient and the fluid inertia to the transient response of porous media. Copyright © 2010 John Wiley & Sons, Ltd.     

Exact solutions for the one-dimensional transient response of unsaturated single-layer porous media

The one-dimensional transient response of unsaturated single-layer porous media is studied based on the theory of unsaturated porous media proposed by Zienkiewicz et al., and exact time-domain solutions are obtained for three types of nonhomogeneous boundary conditions. During the solution procedure, the nonhomogeneous boundary conditions are transformed into homogeneous boundary conditions. Then, the eigenfunction expansion method is utilised to obtain the exact solutions for these new boundary conditions. Several numerical examples are provided to investigate the propagation of compressional waves, and it is verified that three types of compressional waves exist in unsaturated porous media that contain two immiscible fluids.     

饱和土一维简谐响应解析解的求解和应用:I求解

对于饱和土一维简谐响应的u-w和u-p方程，通过消元分别将二阶微分方程组变为一个四阶的微分方程。求解此微分方程则得到一维简谐响应的通解。对于特定的边值问题，通过求解描述边界条件的线性方程组得到满足边界条件的解析解。此解析解可用于饱和土动力响应研究的诸多方面。     

An analytical solution for the transient response of saturated linear elastic porous media

The motions of fluid and solid phases in saturated porous media are coupled by inertial, viscous and mechanical interactions as described by Biot's equations. A one-dimensional exact analytical solution of the Biot's equations for the completely general solution of the transient problem in saturated, linear, elastic, porous media is presented. The problem is solved by using the Fourier series. The transient response of porous media is shown for typical material properties of a natural granular deposit and for different degrees of viscous coupling. The analytical results show the mechanics of dispersive wave propagation in saturated porous media and they should provide a useful comparison term for the existing numerical solution methods.     

Transient dynamic response of multilayered saturated media subjected to impulsive loadings

This paper presents a stable and efficient method for calculating the transient solution of layered saturated media subjected to impulsive loadings by means of the analytical layer element method. Starting with the field equations based on Biot's linear theory for porous, fluid‐saturated media, and the seepage continuity equation, an analytical layer element for a single layer is established by applying Laplace‐Hankel integral transform. The global stiffness matrix in the transform domain for a layered saturated half‐space subjected to a transient circular patch loading is obtained by assembling the layer elements of each layer. The displacements in the time domain are derived by Laplace‐Hankel inverse transform of the global stiffness matrix. Numerical examples are conducted to verify the accuracy of the method and to demonstrate the influences of type of transient loading, buried depth of loading, permeability, and stratification of materials on the transient response of the multilayered saturated poroelastic media.     

A semianalytical solution for one-dimensional transient wave propagation in a saturated single-layer porous medium with a fluid surface layer

One-dimensional transient wave propagation in a saturated single-layer porous medium with a fluid surface layer is studied in this paper. An analytical solution for a special case with a dynamic permeability coefficient k_f → ∞ and a semianalytical solution for a general case with an arbitrary dynamic permeability coefficient are presented. The eigenfunction expansion and precise time step integration methods are employed. The solution is presented in series form, and thus, the long-term dynamic responses of saturated porous media with small permeability coefficients can be easily computed. We first transform the nonhomogeneous boundary conditions into homogeneous boundary conditions, and then we obtain the eigenvalues and orthogonal eigenfunctions of the fluid–solid system. Finally, the solutions in the time domain are developed. As the model is one dimensional, geometric attenuation is absent, and only the attenuation in the saturated porous medium is considered. We can apply this model to analyse the influences of different seabed types on the propagation of acoustic waves in the fluid layer, which is very important in ocean acoustics and ocean seismic. This solution can also be employed to validate the accuracies of various numerical methods.     

横观各向同性含液饱和多孔介质黏弹性瞬态反应

在考虑横观各向同性含液饱和多孔介质固体骨架和流体可压缩性以及固体骨架的黏弹性特征下,基于横观各向同性含液饱和多孔介质u-w形式的三维动力控制方程,以固相位移u、液相相对位移w为基本未知量,综合运用Laplace变换、双重Fourier变换等方法,在直角坐标系下通过引入中间变量,将六元2阶动力控制方程组化为两组各含4个未知变量的常微分方程组,给出了直角坐标系下横观各向同性含液饱和多孔介质三维黏弹性动力反应的积分形式一般解;作为理论推导的验证,通过引入初始条件和边界条件,对横观各向同性含液饱和多孔介质半空间黏弹性瞬态反应问题进行了求解。解答的退化验证表明,所推导的理论解是正确的。     

准饱和土中圆柱形衬砌的瞬态动力响应分析

内源瞬态荷载作用下圆柱形孔洞的动力响应解答是土动力学的经典问题之一。已有研究大都假设孔洞周围土体为理想弹性介质或完全饱和多孔介质。然而,实际工程中不存在完全弹性和完全饱和土体。分别视衬砌结构和周围土体为弹性材料和准饱和多孔介质（饱和度 95%）,根据牛顿第二定律、达西定律和Biot波动理论推导出准饱和土体的控制方程。根据边界条件导出衬砌和土体的位移、应力和孔隙压力的Laplace变换空间的解答。利用反Laplace变换数值计算方法,将解答转换为时域解。分析了饱和度对衬砌位移、应力和孔压的影响,结果表明,当95% 99%时,饱和度对径向位移和切向应力的影响较小;99% 100%时,饱和度对径向位移和切向应力的影响较大;但饱和度对孔隙压力的影响远大于对径向位移和切向应力的影响。得出位移、应力和孔压沿径向的衰减规律,当95% 99%时,饱和度对径向位移和切向应力沿径向衰减影响较小,99% 100%时,饱和度对径向位移和切向应力沿径向衰减影响较大,但饱和度对孔压沿径向的衰减影响远大于对径向位移和切向应力沿径向的衰减。     

An analytical solution for the transient response of saturated porous elastic solids

The general forms for the field equations governing the transient response of poroelastic media given by Biot and by Zienkiewicz are compared and relations between the material constants are obtained. A one-dimensional analytical solution is presented for the situation where the solid and fluid materials satisfy Biot'S dynamic compatibility relation. The transient response of porous media is illustrated for varying degrees of solid and fluid compressibility when subjected to step, cyclic and short duration spike surface tractions. The results obtained (for the special situation where the materials are dynamically compatible) exhibit the overall characteristics of wave propagation in porous media and will provide representative test problems which allow a quantitative evaluation of the accuracy of various numerical solution methods (e.g. finite element models).     

Two formulations for dynamic response of a cylindrical cavity in cross‐anisotropic porous media

Two formulations for calculating dynamic response of a cylindrical cavity in cross‐anisotropic porous media based on complex functions theory are presented. The basis of the method is the solution of Biot's consolidation equations in the complex plane. Employing two groups of potential functions for solid skeleton and pore fluid (each group includes three functions), the u–w formulation of Biot's equations are solved. Difference of these two solutions refers to use of two various potential functions. Equations for calculating stress, displacement and pore pressure fields of the medium are mentioned based on each two formulations. Copyright © 2009 John Wiley & Sons, Ltd.     

饱和半空间瞬态加载衬砌隧道的动力响应研究

工程中,地下衬砌隧道会遇到水压破裂压力、爆炸及突然开挖等瞬态荷载作用,将这些荷载理想化为作用在衬砌内边界上的均布瞬态荷载,研究圆柱形衬砌隧道在突加荷载、阶跃荷载和三角形脉冲荷载作用下的动力响应规律。根据Biot波动理论推导出半空间饱和介质的控制方程;视衬砌结构为弹性材料导出衬砌结构的控制方程。用极大半径凸圆弧近似半空间直边界,采用Graff加法公式进行坐标变换,将直角坐标表示的通解转化为极坐标表示的通解。根据边界条件导出衬砌和土体的位移、应力和孔隙压力的Laplace变换域的解答。利用反Laplace变换数值计算方法,将解答转换为时域解,得出3种瞬态荷载作用下衬砌隧道地面位移峰值、衬砌应力和孔隙压力的分布规律。     

二维饱和多孔介质因点汇诱发比奥固结的解析解

给出了有限二维饱和多孔介质因点汇诱发的Biot固结的一个解析解。其中假设多孔介质为均匀各向同性和线弹性,假设孔隙压力场符合第1类边界条件,数学模型采用可压缩多孔介质模型。利用傅里叶和拉普拉斯变换及相应反演获得了双重无穷项级数和形式的精确解。然后特别探讨了定流量点汇诱发的稳态解析解,并用文献现有解析解进行了验证。所提出的解析解适合于验证数值解,并可用于深入分析有限二维多孔介质的流-固耦合行为。     

Numerical analysis of 1‐D compression wave propagation in saturated poroelastic media

A numerical procedure is presented for the simulation of 1‐D compression wave propagation in saturated poroelastic media. The media are modelled as a two‐phase system consisting of compressible fluid and solids. Viscous coupling forces resulting from the relative motion between phases are characterized as Darcy type. The numerical procedure can account for effects of axial strain, nonlinear material behaviour, and various drained and undrained boundary conditions. Time integration is carried out explicitly and isothermal conditions are assumed. The method is capable of modelling shock wave fronts without introducing artificial viscosity. Numerical results are in close agreement with analytical solutions for several simplified cases and indicate that mass coupling may have important effects on fluid velocity and wave speed. Corresponding effects on solid velocity and wave speed are much smaller. Numerical results also indicate that damping occurs in a saturated poroelastic column and is dependent on the value of hydraulic conductivity. Copyright © 2007 John Wiley & Sons, Ltd.     

饱和土表面在水平集中荷载作用下的瞬态反应

分析了水平集中荷载作用在半空间饱和土表面时的瞬态问题。利用Laplace-Hankel变换对非轴对称Biot固结方程进行解耦，利用Laplace-Hankel数值逆变换得到半空间饱和土在时域内的数值解。退化到线弹性中的解与文献中的结果进行比较，验证了文中结果的正确性和数值逆变换的可靠性.可以用于研究地震工程中地震波的传播以及饱和土与结构之间相互作用等问题。     

Seismic response analysis of the deep saturated soil deposits in Shanghai

The quaternary deposits in Shanghai are horizontal soil layers of thickness up to about 280 m in the urban area with an annual groundwater table between 0.5 and 0.7 m from the surface. The characteristics of deep saturated deposits may have important influences upon seismic response of the ground in Shanghai. Based on the Biot theory for porous media, the water-saturated soil deposits are modeled as a two-phase porous system consisting of solid and fluid phases, in this paper. A nonlinear constitutive model for predicting the seismic response of the ground is developed to describe the dynamic characters of the deep-saturated soil deposits in Shanghai. Subsequently, the seismic response of a typical site with 280 m deep soil layers, which is subjected to four base excitations (El Centro, Taft, Sunan, and Tangshan earthquakes), is analyzed in terms of an effective stress-based finite element method with the proposed constitutive model. Special emphasis is given to the computed results of accelerations, excess pore-water pressures, and settlements during the seismic excitations. It has been found that the analysis can capture fundamental aspects of the ground response and produce preliminary results for seismic assessment.     

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.     

Analytical solution for longitudinal seismic response of long tunnels subjected to Rayleigh waves

The longitudinal seismic response of a long tunnel subjected to Rayleigh waves is investigated in this paper. The tunnel is assumed to be infinitely long, has a uniform cross section, and rests on a viscoelastic foundation. The free-field deformation under Rayleigh waves traveling parallel to the tunnel axis is decomposed into two directions, namely, the axial motion and the vertical motion, and transformed into dynamic loads imposed on the tunnel. Based on the Fourier and Laplace integral transform techniques, the governing equations of tunnels are simplified into algebraic equations, and the analytical solutions are obtained with the convolution theorem. The final solutions of the tunnel responses in terms of deflection, velocity, acceleration, axial force, bending moment, and shear force are investigated. The proposed solution is verified by comparison of its results and those from the finite element program ABAQUS. Further parametric analysis is carried out to investigate the influence of soil-structure relative stiffness ratio and wave frequency on dynamic longitudinal responses of the tunnel.     

Mode of a spherical cavity’s thermo-elastodynamic response in a saturated porous medium for non-torsional loads

The thermo-hydro-elastodynamic model (THED), in which the thermo-osmosis and thermal-filtration phenomenon in a two-phase porous thermoelastic medium can be considered, was previously presented by the authors [Ganbin Liu, Kanghe Xie, Rongyue Zheng. Model of nonlinear coupled thermo-hydro-elastodynamics response for a saturated poroelastic medium. Sci China (Ser E) 2009;52(8):2373–83] and is used in this paper to investigate the thermo-elastodynamic response of a spherical cavity in a saturated poroelastic medium when subjected to a time-dependent non-torsional thermal/mechanical source. The Separated Variable Method is introduced, and the non-zero displacement potentials are expanded in terms of the Legendre polynomial. Solutions for the displacement, temperature increment, pore pressure and stress are obtained in the domain of the Laplace transform. Numerical results are also performed for different modes and are compared with the results of the thermoelastic model (TED) to ascertain the validity and the difference between these two models.