不可压饱和粘弹性多孔介质的变分原理及其无网格模拟

基于饱和多孔介质理论,在固相和液相微观不可压,固相骨架小变形且满足线性粘弹性积分型本构关系的假定下,建立了流体饱和粘弹性多孔介质动力响应的若干Gurtin型变分原理,包括Hu-Washizu变分原理.利用所建立的变分原理,导出了流体饱和粘弹性多孔介质动力响应无网格数值模拟的离散控制方程,此方程是一个关于时间的对称微分方程组,便于分析计算.作为数值例子,研究了流体饱和粘弹性多孔柱体的一维动力响应,数值结果揭示了流体饱和粘弹性多孔柱体中波的传播特性以及固相粘性的影响.     

流体饱和标准线性粘弹性多孔介质中的平面波

研究了流体饱和不可压标准线性粘弹性多孔介质中平面波的传播和反射问题．在固相骨架小变形的假定下,得到了粘弹性多孔介质中波动方程的一般解,讨论了弥散关系和波的衰减特性．结果表明：在流体饱和不可压粘弹性多孔介质中,仅存在一个耦合纵波和一个耦合横波,纵波和横波的波速、衰减率等取决于孔隙流体与固相骨架间的相互作用以及固相骨架本身的粘性．同时,研究了半空间自由边界上入射波(纵波、横波)的反射问题。得到了非均匀反射波的波速、反射系数、衰减率等的表达式及其相关的数值结果．     

不可压饱和粘弹性多孔介质固结问题的Gurtin型变分原理

基于多孔介质理论,在固相骨架和孔隙流体微观不可压,固相骨架小变形且满足线性粘弹性积分型本构关系的假定下,利用卷积积分的性质,本文首先建立了以固相骨架位移、孔隙流体相对速度和孔隙流体压力为宗量的流体饱和粘弹性多孔介质固结问题的一个Gurtin型变分原理.其次,利用Lagrange乘子法解除相关的变分约束条件,建立了流体饱和粘弹性多孔介质固结问题的若干广义Gurtin型变分原理,包括第三类的Hu-Washizu型变分原理.最后,简单讨论了等价初边值问题的相应变分原理.这些Gurtin型变分原理的建立不仅丰富了饱和粘弹性多孔介质的相关理论,而且为相关数值模拟方法,如有限元法、无网格法等的建立奠定了理论基础.     

饱和多孔弹性Timoshenko悬臂梁的拟静力弯曲

在经典单相Timoshenko梁变形和孔隙流体仅沿多孔梁轴向运动的假定下,基于不可压饱和多孔介质的三维理论,本文首先建立了横观各向同性饱和多孔弹性Timoshenko悬臂梁拟静力弯曲的一维数学模型,并给出了相应的边界条件。其次,利用Laplace变换及其数值逆变换,分析了端部不同渗透条件下,饱和多孔弹性Timoshenko悬臂梁在端部梯载荷作用下的拟静力响应,给出了饱和多孔Timoshenko悬臂梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶等随时间的响应曲线,并与饱和多孔Euler-Bernoulli悬臂梁的响应进行了比较,考察了梁长细比对弯曲的影响。数值结果表明：固相骨架与孔隙流体的相互作用具有粘性效应,梁弯曲的拟静态挠度具有蠕变行为,端部渗透条件对梁的弯曲响应有显著的影响,并且,饱和多孔弹性Timoshenko悬臂梁的拟静态响应亦存在Mandel-Cryer现象。     

不可压饱和多孔弹性简支梁的动力响应

在杆件弯曲小变形的假定下,考虑杆件的侧向变形因素,根据多孔介质理论,本文首先建立了不可压饱和多孔弹性梁弯曲变形时动力响应的控制方程。其次,基于所建立的控制微分方程,利用变量分离法,研究了两端可渗透的饱和多孔弹性简支梁在梁中间集中载荷作用下的动力响应,得到了不同物性参数下简支梁动态弯曲时挠度和孔隙流体压力等效力偶等随时间的响应曲线。研究发现由于孔隙流体和固相骨架的相互作用,不可压饱和多孔弹性梁挠度的动力响应具有粘性特征,同时,随着时间的增加,饱和多孔弹性梁的挠度、弯矩等最终趋于经典弹性梁的静挠度、弯矩,此时,孔隙流体压力为零,梁的固相骨架承担所有的外载荷。     

饱和土地基在简谐荷载作用下的解析分析

基于多孔介质混合物理论,用解析的方法研究了不可压饱和土地基受到简谐荷载作用下的动力响应问题。利用Fourier积分变换求解耦合方程组,得到了二维饱和土介质在简谐荷载作用下的通解。针对表面透水的具有下卧基岩的饱和土层以及半无限饱和土地基的边界条件,获得了固体骨架位移、孔隙流体位移、固体骨架有效应力以及孔隙流体压力的积分形式解答,并通过数值算例分析了饱和土地基在简谐荷载作用下的响应。     

饱和黏弹性多孔介质中的平面波及能量耗散

研究了流体饱和不可压黏弹性多孔介质中的非均匀平面波及其能量流和能量耗散规律. 在流 相和固相物质微观不可压、固相骨架宏观服从积分型本构关系和小变形的假定下，利用 Helmholtz分解，得到了饱和黏弹性多孔介质中非均匀平面波的一般解以及纵波、横波相速 度和衰减率等的解析表达式，分析了平面波传播矢量和衰减矢量之间的关系. 数值结果表明 孔隙流体与固相骨架间的相互作用以及固相骨架的黏性对波的相速度、衰减率等有着显著的 影响. 同时，得到了饱和黏弹性多孔介质的能量方程，给出了能量流矢量和能量耗散率. 对 非均匀平面纵波和横波，推导了平均能量流矢量和平均能量耗散率的解析表达式.     

不可压饱和多孔弹性梁、杆动力响应的数学模型

基于多孔介质理论,首先建立了饱和多孔弹性杆件弯曲与轴向变形时动力响应的数学模型.其次,基于多孔弹性梁弯曲变形的数学模型,利用Laplace变换,分析了两端可渗透的饱和多孔弹性悬臂梁在自由端受阶梯载荷作用下的动静力响应,给出了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶等物理量随时间的响应曲线.发现不可压多孔弹性梁的拟静态响应亦存在Mandel-Cryer现象,多孔弹性梁的挠度具有与粘弹性梁挠度类似的蠕变特征,然而,其应力响应不同于粘弹性梁,随着时间的增加,梁拟静态响应的弯矩逐渐增加,并达到一个稳态值.这些结果有助于揭示植物根茎等力学行为的机理.     

多孔介质平板通道强迫对流中热局部非平衡时的热应力

根据微观不可压饱和多孔介质热-力-流相互作用的一般理论,在固相骨架小变形的假定下,考虑固相和流相相互作用的粘性耗散,研究了多孔介质平板通道强迫对流热局部非平衡的热应力问题.建立了问题的热-力数学模型,根据饱和多孔介质的平衡方程,在固相骨架只存在横向位移的假定下,求解了固相骨架的位移和相应的热应力,数值考察了各种物性参数对热应力分布的影响,讨论了热局部平衡模型的适用性.     

饱和粘弹性土层中悬浮桩的纵向振动

基于不可压饱和粘弹性多孔介质理论和一维杆件模型,利用虚拟土桩的概念,研究了不可缮压饱和粘弹性土层中悬浮桩纵向振动的动力刚度.通过势函数解耦和三角级数解法,得到了悬浮桩纵向谐调振动的半解析解和桩头动力刚度.在此基础上通过数值计算,研究了物性参数、几何参数等对刚度的影响,结果表明:悬浮桩纵向振动的动力刚度受桩长、土的软硬程度、激振频率等的影响,对比了细长桩和短粗桩的响应差异,这些结果可为悬浮桩-土相互作用的数值分析和工程设计提供参考.     

Dynamic and quasi-static bending of saturated poroelastic Timoshenko cantilever beam

Based on the three-dimensional Gurtin-type variational principle of the incompressible saturated porous media, a one-dimensional mathematical model for dynamics of the saturated poroelastic Timoshenko cantilever beam is established with two assumptions, i.e., the deformation satisfies the classical single phase Timoshenko beam and the movement of the pore fluid is only in the axial direction of the saturated poroelastic beam. Under some special cases, this mathematical model can be degenerated into the Euler-Bernoulli model, the Rayleigh model, and the shear model of the saturated poroelastic beam, respectively. The dynamic and quasi-static behaviors of a saturated poroelastic Timoshenko cantilever beam with an impermeable fixed end and a permeable free end subjected to a step load at its free end are analyzed by the Laplace transform. The variations of the deflections at the beam free end against time are shown in figures. The influences of the interaction coefficient between the pore fluid and the solid skeleton as well as the slenderness ratio of the beam on the dynamic/quasi-static performances of the beam are examined. It is shown that the quasi-static deflections of the saturated poroelastic beam possess a creep behavior similar to that of viscoelastic beams. In dynamic responses, with the increase of the slenderness ratio, the vibration periods and amplitudes of the deflections at the free end increase, and the time needed for deflections approaching to their stationary values also increases. Moreover, with the increase of the interaction coefficient, the vibrations of the beam deflections decay more strongly, and, eventually, the deflections of the saturated poroelastic beam converge to the static deflections of the classic single phase Timoshenko beam.     

Axisymmetrical analytical solution for vertical vibration of end-bearing pile in saturated viscoelastic soil layer

Based on elasticity and the theory of saturated porous media, and regarding the pile and the soil as a single phase elastic and a saturated viscoelastic media, respectively, the dynamical behavior of vertical vibration of an end-bearing pile in a saturated viscoelastic soil layer is investigated in the frequency domain using the Helmholtz decomposition and variable separation method. The axisymmetrical analytical solutions for vertical vibrations of the pile in a saturated viscoelastic soil layer are obtained, and the analytical expression of the dynamical complex stiffness of the pile top is presented. Responses of dynamic stiffness factor and equivalent damping of pile top with respect to the frequency are shown in figures using a numerical method. Effects of the saturated soil parameters, modulus ratio of the pile to soil, slenderness ratio of pile and pile＇s Poisson ratio, etc. on the stiffness factor and damping are examined. It is shown that, due to the effect of the transversal deformation of the pile and the radial force of the saturated viscoelastic soil acting on the pile, the dynamic stiffness factor and the damping derived from the axisymmetrical solution are greatly different from those derived from the classical Euler-Bernoulli rod model, especially at some specific excitation frequencies. Therefore, there are limitations on applicability of the Euler-Bernoulli rod model in analyzing verticai vibration of the pile. More accurate analysis should be based on a three-dimensional model.     

Modeling motion-induced fluid release from partially saturated fibrous media onto surfaces with different hydrophilicity

Modeling the rate of fluid release from moving partially saturated nonwoven sheets in contact with a solid surface is a challenge, as the release rate depends on many parameters, some of which are difficult to quantify. In this paper, we report on a diffusion-controlled boundary treatment which we have developed to simulate fluid release from partially saturated porous materials onto surfaces with different hydrophilicy. The new boundary treatment considers the solid impermeable surface as a fictitious porous layer with a known fluid diffusive coefficient. Motion of the porous sheet on the surface is incorporated in the simulations by periodically resetting the saturation of the fictitious layer equal to zero, with a period obtained from the sheet’s speed of motion. Fluid transport inside the fibrous sheets is calculated by solving Richards’ equation of two-phase flows in porous media. Our numerical simulations are accompanied with experimental data obtained using a custom-made test rig for the release of liquid from partially saturated media at different speeds. It is demonstrated that the novel mathematical formulations presented here can correctly predict the rate of fluid release from moving fibrous sheets onto solid surfaces with different hydrophilicity as a function of time.     

Dynamics of Viscous Entrapped Saturated Zones in Partially Wetted Porous Media

As a typical multiphase fluid flow process, drainage in porous media is of fundamental interest both in nature and in industrial applications. During drainage processes in unsaturated soils and porous media in general, saturated regions, or clusters, in which a liquid phase fully occupies the pore space between solid grains, affect the relative permeability and effective stress of the system. Here, we experimentally study drainage processes in unsaturated granular media as a model porous system. The distribution of saturated clusters is analysed by optical imaging under different drainage conditions, with pore-scale information from Voronoi and Delaunay tessellation used to characterise the topology of saturated cluster distributions. By employing statistical analyses, we describe the observed spatial and temporal evolution of multiphase flow and fluid entrapment in granular media. Results indicate that the distributions of both the crystallised cell size and pore size are positively correlated to the spatial and temporal distribution of saturated cluster sizes. The saturated cluster size is found to follow a lognormal distribution, in which the generalised Bond number (\( Bo^{*} \)) correlates negatively to the scale parameter (μ) and positively to the shape parameter (σ). With further consideration of the total surface energy obtained based on liquid–air interfaces, we were able to include additional grain-scale information in the constitutive modelling of unsaturated soils using both the degree of saturation and generalised Bond number. These findings can be used to connect pore-scale behaviour with overall hydro-mechanical characteristics in granular systems.     

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media.