The one-temperature field thermo-mechanical theory of fluid-filled porous materials

Summary The one-temperature field thermo-mechanical theory of a fluid-filled porous material with a linearly elastic solid and a Newtonian viscous fluid is established. The author's previous work, phenomenological constitutive theory for fluid-filled porous materials with solid/fluid outer boundaries, is introduced into a conventional mixture theory. Kinetical and kinematical quantities defined for a sample with a solid/fluid outer boundary, including interaction terms, correspond to those in the mixture theory directly from their definitions. The approximations involved in one to one correspondence between the constitutive equations in the mixture theory and those in the phenomenological theory are made clear.     

Mathematical framework for unsaturated flow in the finite deformation range

The presence of fluid in the pores of a solid imposes a volume constraint on the deformation of the solid. Finite changes in the pore volume alter the degree of saturation of a porous material, impacting its fluid flow and water retention properties. This intricate interdependence between the hydromechanical properties related to solid deformation and fluid flow is amplified when the deformation of the solid matrix is large. In this paper, we present a mathematical framework for coupled solid‐deformation/fluid‐diffusion in unsaturated porous material considering geometric nonlinearity in the solid matrix. The framework relies on the continuum principle of thermodynamics to identify an effective or constitutive stress for the solid matrix, and a water‐retention law that highlights the interdependence of the degree of saturation, suction, and porosity of the material. Porous materials are typically heterogeneous, making them susceptible to localized deformation. In this work, we consider random heterogeneities in density and degree of saturation as triggers of localized deformation in a porous material. Copyright © 2013 John Wiley & Sons, Ltd.     

A phase field model for the freezing saturated porous medium

In terms of the mixture theory and phase field theory, a phase field model is developed for the saturated porous medium undergoing phase transition. In the proposed model, it is postulated that during the phase transition of the porous medium, both the solid skeleton and pore fluid will undergo phase transition. The phase states of the solid skeleton and pore fluid are characterized by respective order parameters. To simplify the proposed phase field model, the temperatures and order parameters of the solid skeleton and pore fluid are assumed to be equal. The balance laws of the porous medium are given by the conventional mixture theory. The order parameter and the porosity of the porous medium are considered as internal variables and their evolution equations are determined by the entropy inequality of the porous medium. The constitutive representations for the stresses, entropies, heat fluxes, drag force and the evolution equations for the order parameter and porosity are derived by exploitation of the entropy inequality. To illustrate the proposed model, a concrete phase field model for the freezing porous medium is established in the paper. In the model, the memory effect associated with phase transition of the porous medium is taken into account by assuming Stieltjes integral for the strain energy of the porous medium. The constitutive representations for the above variables are then derived according to the proposed free energy expression for the porous medium. Finally, the boundary condition associated with the proposed model and the determination of some parameters involved in our model are discussed in the paper briefly.     

A plasticity theory for fluid-saturated porous solids

A theory of plastic flow is developed for an elastic-perfectly plastic porous solid with its intercommunicating void spaces filled with a viscous fluid. The constitutive equations are established where full use is made of the thermodynamical equations. The final result of the paper is the complete set of equations of the theory and the boundary conditions for the medium in question.     

Flow of compressible fluid in porous elastic media

A finite element formulation for flow of fluid in a porous elastic media has been derived from a Gurtin type variational principle. Biot's field equations for porous media have been used in which the constitutive relations include the compressibility of the fluid. It has been shown that by proper choice of the form of the compressibility of the fluid as a function of the state variables, this option of the formulation can be used to treat the partially saturated soil. The method is general with respect to geometry, boundary conditions and material properties. Finally, the results of a series of examples have been presented and compared with exact results to demonstrate accuracy and applicability.     

基于变分渐近法预测饱和多孔介质流固耦合性能的细观力学模型

饱和岩土类多孔材料内固、液相不同属性产生的各向异性和多孔微结构的不均匀性使得材料的细观力学特性计算变得十分复杂。为准确预测岩土类材料的有效弹性性能和细观应力-应变场,基于Biot多孔弹性介质理论,建立可描述岩土类多孔材料固液相运动的能量泛函和相应的多孔弹性本构关系;利用细、宏观尺度比作为小参数将能量变分泛函渐近扩展为系列近似泛函;以场变量波动函数为未知量,通过解决近似泛函的最小化问题(驻值问题)得到波动函数的解析解,从而建立逼近物理和工程真实性的细观力学模型,并通过有限元技术得以数值实现。多孔介质材料细观力学特性算例表明:与经典均匀化理论(将液体类比为具有较高泊松比的固体材料)相比,基于变分渐近均匀化细观模型预测的多孔介质材料细观力学特性更精确,尤其是能准确重构多孔微结构内局部应力-应变场分布,为损伤破坏、局部断裂分析奠定了坚实基础。      

Hydrodynamic permeability of membranes built up by spherical particles covered by porous shells: effect of stress jump condition

This paper concerns the flow of an incompressible, viscous fluid past a porous spherical particle enclosing a solid core, using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. At the fluid–porous interface, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. No-slip and impenetrability boundary conditions on the solid spherical core have been used. The hydrodynamic drag force experienced by a porous spherical particle enclosing a solid core and permeability of membrane built up by solid particles with a porous shell are evaluated. It is found that the hydrodynamic drag force and dimensionless hydrodynamic permeability depends not only on the porous shell thickness, particle volume fraction γ and viscosities of porous and fluid medium, but also on the stress jump coefficient. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta–Morse’s condition). Some previous results for the hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.     

Mechanics of porous elastic materials containing multiphase fluid

A continuum theory of mixtures for a porous elastic solid saturated by immiscible viscous fluids is presented. The theory includes micro-inertial effects for the local fluctuation in volume fractions of the solid and fluid constituents. Gradients of volume fraction of both the elastic solid and fluid constituents are included in the constitutive variables. Equations governing the macroscopic motion are developed and show that the present theory contains both Biot's equations and multiphase Darcy flow through porous media as special cases.     

Stokes flow past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition

The solution of the problem of symmetrical creeping flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition is investigated. The Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream function formulations are used. As boundary conditions, continuity of velocity and surface stresses across the porous surface and Kuwabara boundary condition on the cell surface are employed. Explicit expressions are investigated for both inside and outside flow fields to the first order in a small parameter characterizing the deformation. As a particular case, the flow past a swarm of porous oblate spheroidal particles is considered and the drag force experienced by each porous oblate spheroid in a cell is evaluated. The dependence of the drag coefficient on permeability for a porous oblate spheroid in an unbounded medium and for a solid oblate spheroid in a cell on the solid volume fraction is discussed numerically an and graphically for various values of the deformation parameter. The earlier known results are then also deduced from the present analysis.     

饱和多孔介质一维瞬态波动问题的解析分析

采用基于混合物理论的多孔介质模型,提出了饱和多孔介质一维动力响应的初边值问题。利用拉氏变换和卷积定理,分别得到了边界自由排水时在任意应力边界条件和任意位移边界条件下瞬态波动过程的解析表达。几种典型的数值算例同时给出了两类边界条件下瞬态波动过程中多孔固体的位移场、应力场和孔隙流体的速度场、压力场。结果表明,饱和多孔介质的波动过程是多孔固体和孔隙流体中以同一速度传播的两种波动的耦合过程,时效特性分析也揭示了饱和多孔介质固有的表观粘弹性性质。     

Topology optimization of creeping fluid flows using a Darcy–Stokes finite element

A new methodology is proposed for the topology optimization of fluid in Stokes flow. The binary design variable and no‐slip condition along the solid–fluid interface are regularized to allow for the use of continuous mathematical programming techniques. The regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by Darcy's law. Fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the Stokes and Darcy equations. The mixed formulation of the new Darcy–Stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems. Convergence to the no‐slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid–fluid interface are not needed. The optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature. The advantages of the Darcy–Stokes approach include that it uses existing stabilization techniques to solve the finite element problem, it produces 0–1 (void–solid) topologies (i.e. there are no regions of artificial material), and that it can potentially be used to optimize the layout of a microscopically porous material. Copyright © 2005 John Wiley & Sons, Ltd.     

Plasticity of compressible/dilatant rocklike materials

An elastic/plastic constitutive equation is proposed for porous solid materials such as rocks, porous metals, etc. The aim is to describe, with a general constitutive equation, such typical phenomena for these materials as volume compressibility and dilatancy. The starting points are the experimental data obtained in triaxial tests and in uniaxial confined compressibility tests. It is shown how this constitutive equation can actually be determined. The concept of a dilatancy threshold boundary is introduced by a mathematical definition, as are the concepts of compressibility and dilatancy. A numerical example is given for granite.     

On the linear problem of swelling porous elastic soils with incompressible fluid

In this paper we consider the linear theory of swelling porous elastic soils in the case that the fluid is incompressible. The formulation belongs to the theory of mixtures for porous elastic solids filled with fluid and gas. It proposes some new mathematical difficulties. Continuous dependence results on initial conditions and supply terms are obtained in the general case. Logarithmic convexity method is used in case of fluid saturation. Structural stability with respect two constitutive coefficients is also obtained in the case of fluid saturation.     

Direct numerical simulation of saturated deformable porous media using parallel hybrid Lattice–Boltzmann and finite element method

Numerical techniques for modeling saturated deformable porous media have mainly been based on mixture theory or homogenization techniques. However, these techniques rely on phenomenological relationships for the constitutive equations along with assumptions of homogeneous and isotopic material properties to obtain closure. Direct numerical simulations of the multiphasic problem for flow in deformable porous media avoid such assumptions and thus can provide significantly accurate understanding of the physics involved. They serve as a tool to investigate the constitutive relationships in complex geometries. They also allow the validation of the existing mixture theory models and determine their limitations. In this work, a parallel hybrid method using Lattice Boltzmann Method (LBM) for fluid phase and Finite Element Method (FEM) for solid phase is used for direct numerical simulation of saturated deformable porous media. The method provides a number of unique features including scalability on distributed computing necessary for such a problem. The method has been validated for modeling fluid–structure interactions in complex geometries against a number of experimental and analytical solutions. Further some challenging problems has been chosen to show the capability of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

Viscous vortex core generation

Summary The unsteady flow of a viscous incompressible fluid between two porous co-axial circular cylinders is analysed when the outer cylinder is impulsively set into rotation. When there is radial inflow, vorticity is transferred from an unsteady boundary layer initially over the outer cylinder to a steady boundary layer over the inner cylinder.With 1 Figure     

Boundary layers in fibrous composite materials

Many studies in the theory of composite materials are based on the homogenization approach, which consists of the substitution of the original heterogeneous medium by a homogeneous one with certain effective properties. Though this procedure works well for the entire composite solid, it cannot be applied in the vicinity of the outer boundary. The transmission of an external load applied at the boundary to the inner domain of the material occurs by a redistribution of stresses between the constitutive components (inclusions and matrix) and involves strong singularities in the local stress field, which may result in microscopic failure of the composite structure. In the present paper, we propose an approximate analytical procedure, allowing determination of the stress?Cstrain field in the vicinity of the outer boundaries of fibre-reinforced composite materials. It is also shown that controlled decrease in bonding between the components leads to a more uniform redistribution of local stresses, which can essentially reduce the risk of failure.     

一维分数导数粘弹性饱和多孔介质层的稳态响应

固相骨架的应力-应变关系利用分数导数粘弹性Kelvin 模型来描述,在流相和固相微观不可压以及小变形的假定下建立了分数导数粘弹性饱和多孔介质层一维稳态响应的数学模型和运动控制方程,求得了分数导数粘弹性饱和多孔介质层一维稳态响应的固相位移和液相位移。通过数值算例分析了分数导数的阶数对稳态响应的影响。研究结果表明:固相位移和液相位移随频率的增大逐渐趋于零,在低频时,分数导数的阶数越大固相位移和液相位移越大。     

Elastoplasticity theory for porous materials

A continuum theory is derived for the modeling of elastoplastic work-hardening porous materials. The theory provides a set of constitutive relations which, using the properties of the inelastic matrix, determines by an incremental procedure the overall response of the porous solid to various types of loading. In the elastic region, effective elastic moduli of the porous material are obtained. Comparison with theoretical and experimental results are given.     

A macroscopic description of the quasi-static behavior of granular materials based on the theory of porous media

Granular materials fall into the class of porous media. But in contrast to materials like foams and sponges their structure is discontinous on a microscopic level. For this reason the particles may undergo independent displacements and rotations. This is the classical kinematics which may be captured by a micropolar or Cosserat theory on the macroscopic level. The goal of this paper is to combine the theory of porous media as a macroscopic theory dealing with multi-phase systems and the micropolar theory describing extended kinematics and taking care of the discountinous structure of granular media on the micro scale. The resulting micropolar theory of porous media may be used to describe the quasistatic behavior of granular materials. In the present contribution thermodynamically consistent constitutive relations for the elastic response of a dense granular matrix material saturated by a viscous pore fluid are given and applied to some boundary value problems which demonstrate the physical relevance of the proposed model. Received: 9 June 1999     

BEM modeling of saturated porous media susceptible to damage

This paper deals with the numerical analysis of saturated porous media, taking into account the damage phenomena on the solid skeleton. The porous media is taken into poro-elastic framework, in full-saturated condition, based on Biot’s Theory. A scalar damage model is assumed for this analysis. An implicit boundary element method (BEM) formulation, based on time-independent fundamental solutions, is developed and implemented to couple the fluid flow and two-dimensional elastostatic problems. The integration over boundary elements is evaluated using a numerical Gauss procedure. A semi-analytical scheme for the case of triangular domain cells is followed to carry out the relevant domain integrals. The non-linear problem is solved by a Newton-Raphson procedure. Numerical examples are presented, in order to validate the implemented formulation and to illustrate its efficacy.