Electromechanical response of (3–0, 3–1) particulate,fibrous, and porous piezoelectric composites with anisotropic constituents: A model based on the homogenization method

An analytical framework based on the homogenization method has been developed to predict the effective electromechanical properties of periodic, particulate and porous, piezoelectric composites with anisotropic constituents. Expressions are provided for the effective moduli tensors of n-phase composites based on the respective strain and electric field concentration tensors. By taking into account the shape and distribution of the inclusion and by invoking a simple numerical procedure, solutions for the electromechanical properties of a general anisotropic inclusion in an anisotropic matrix are obtained. While analytical forms are provided for predicting the electroelastic moduli of composites with spherical and cylindrical inclusions, numerical evaluation of integrals over the composite microstructure is required in order to obtain the corresponding expressions for a general ellipsoidal particle in a piezoelectric matrix. The electroelastic moduli of piezoelectric composites predicted by the analytical model developed in the present study demonstrate excellent agreement with results obtained from three-dimensional finite-element models for several piezoelectric systems that exhibit varying degrees of elastic anisotropy.     

The influence of void distribution on the yielding of an elastic-plastic porous solid

In a previous study, the authors proposed an anisotropic yield function for a porous metal. The present investigation extends the previous work by deriving a rate-type constitutive law based on the proposed yield function. A comparison is also made between yield loci derived from the yield function and those determined from an FE model. The FE calculations were performed for a flat plate, with a periodic array of cylindrical holes, under biaxial loading; and also for a porous solid with a periodic array of spherical voids under axial load and superimposed hydrostatic pressure. For the plate, the agreement between the yield function and numerical results was good, less so far the three-dimensional case. Good correspondence can be obtained by introducing a scalar multiplier in the pressure-dependent term in the yield function.     

Description of filtration dispersion on the basis of the finite-velocity diffusion model

The mechanism of dispersion of substances dissolved in a fluid as it moves through a porous medium (filtration dispersion) is examined and it is noted that the dispersion is mainly determined by the repeated division and coalescence of the threads in the pores. Considering the analogy between the space-time nonuniformity of the fluid velocity field for laminar motion in a porous medium and for turbulent motion in a channel, and also bearing in mind the effectiveness of the finite-velocity diffusion model for calculating turbulent diffusion, it is recommended that filtration dispersion be described on the basis of that model. A system of equations describing the dispersion of substances in a porous medium formed as a result of the hexagonal and cubic close packing of spheres of the same diameter is obtained. The results of a numerical solution for the two-dimensional and three-dimensional steady-state problems are presented. Simplified systems of equations that considerably reduce the computation time are proposed.     

Calculating the Anisotropic Permeability of Porous Media Using the Lattice Boltzmann Method and X-ray Computed Tomography

A lattice Boltzmann (LB) method is developed in this article in a combination with X-ray computed tomography to simulate fluid flow at pore scale in order to calculate the anisotropic permeability of porous media. The binary 3D structures of porous materials were acquired by X-ray computed tomography at a resolution of a few microns, and the reconstructed 3D porous structures were then combined with the LB model to calculate their permeability tensor based on the simulated velocity field at pore scale. The flow is driven by pressure gradients imposed in different directions. Two porous media, one gas diffusion porous layer used in fuel cells industry and glass beads, were simulated. For both media, we investigated the relationship between their anisotropic permeability and porosity. The results indicate that the LB model is efficient to simulate pore-scale flow in porous media, and capable of giving a good estimate of the anisotropic permeability for both media. The calculated permeability is in good agreement with the measured date; the relationship between the permeability and porosity for the two media is well described by the Kozeny–Carman equation. For the gas diffusion layer, the simulated results showed that its permeability in one direction could be one order of magnitude higher than those in other two directions. The simulation was based on the single-relaxation time LB model, and we showed that by properly choosing the relaxation time, it could give similar results to those obtained using the multiple-relaxation time (MRT) LB method, but with only one third of the computational costs of MRTLB model.     

<Emphasis Type="Italic">A Priori</Emphasis> Parameter Estimation for the Thermodynamically Constrained Averaging Theory: Species Transport in a Saturated Porous Medium

The thermodynamically constrained averaging theory (TCAT) has been used to develop a simplified entropy inequality (SEI) for several major classes of macroscale porous medium models in previous works. These expressions can be used to formulate hierarchies of models of varying sophistication and fidelity. A limitation of the TCAT approach is that the determination of model parameters has not been addressed other than the guidance that an inverse problem must be solved. In this work we show how a previously derived SEI for single-fluid-phase flow and transport in a porous medium system can be reduced for the specific instance of diffusion in a dilute system to guide model closure. We further show how the parameter in this closure relation can be reliably predicted, adapting a Green’s function approach used in the method of volume averaging. Parameters are estimated for a variety of both isotropic and anisotropic media based upon a specified microscale structure. The direct parameter evaluation method is verified by comparing to direct numerical simulation over a unit cell at the microscale. This extension of TCAT constitutes a useful advancement for certain classes of problems amenable to this estimation approach.     

Effects of Medium Permeability Anisotropy on Chemical-Dissolution Front Instability in Fluid-Saturated Porous Media

This paper deals with the theoretical aspects of chemical-dissolution front instability problems in two-dimensional fluid-saturated porous media including medium anisotropic effects. Since a general anisotropic medium can be described as an orthotropic medium in the corresponding principal directions, a two-dimensional orthotropic porous medium is considered to derive the analytical solution for the critical condition, which is used to judge whether or not the chemical dissolution front can become unstable during its propagation. In the case of the mineral dissolution ratio (that is defined as the ratio of the dissolved-mineral equilibrium concentration in the pore-fluid to the molar concentration of the dissolvable mineral in the solid matrix of the fluid-saturated porous medium) approaching zero, the corresponding critical condition has been mathematically derived when medium permeability anisotropic effects are considered. As a complementary tool, the computational simulation method is used to simulate the morphological evolution of chemical dissolution fronts in two-dimensional fluid-saturated porous media including medium anisotropic effects. The related theoretical and numerical results demonstrated that: (1) a decrease in the medium anisotropic permeability factor (or ratio), which is defined as the ratio of the principal permeability in the transversal direction to that in the longitudinal direction parallel to the pore-fluid inflow direction, can stabilize the chemical dissolution front so that it becomes more difficult for a planar chemical-dissolution front to evolve into different morphologies in the chemical dissolution system; (2) the medium anisotropic permeability ratio can have significant effects on the morphological evolution of the chemical dissolution front. When the Zhao number of the chemical dissolution system is greater than its critical value, the greater the medium anisotropic permeability ratio, the faster the irregular chemical-dissolution front grows.     

Drag reduction of turbulent channel flows over an anisotropic porous wall with reduced spanwise permeability

The direct numerical simulation(DNS) is carried out for the incompressible viscous turbulent flows over an anisotropic porous wall. Effects of the anisotropic porous wall on turbulence modifications as well as on the turbulent drag reduction are investigated. The simulation is carried out at a friction Reynolds number of 180, which is based on the averaged friction velocity at the interface between the porous medium and the clear fluid domain. The depth of the porous layer ranges from 0.9 to 54 viscous units. The permeability in the spanwise direction is set to be lower than the other directions in the present simulation. The maximum drag reduction obtained is about 15.3% which occurs for a depth of 9 viscous units. The increasing of drag is addressed when the depth of the porous layer is more than 25 wall units. The thinner porous layer restricts the spanwise extension of the streamwise vortices which suppresses the bursting events near the wall. However, for the thicker porous layer, the wall-normal fluctuations are enhanced due to the weakening of the wall-blocking effect which can trigger strong turbulent structures near the wall.     

Pore-Network Modeling of Isothermal Drying in Porous Media

In this paper we present numerical results obtained with a pore-network model for the drying of porous media that accounts for various processes at the pore scale. These include mass transfer by advection and diffusion in the gas phase, viscous flow in the liquid and gas phases and capillary effects at the liquid--gas interface. We extend our work by studying the effect of capillarity-induced flow in macroscopic liquid films that form at the pore walls as the liquid--gas interface recedes. A mathematical model that accounts for the effect of films on the drying rates and phase distribution patterns is presented. It is shown that film flow is a major transport mechanism in the drying of porous materials, its effect being dominant when capillarity controls the process, which is the case in typical applications.     

Theoretical Analysis of Anion Exclusion and Diffusive Transport Through Platy-Clay Soils

Diffusive transport through geosynthetic clay liners and engineered compacted clay landfill liners is the primary mechanism for mass transport from well-engineered modern landfills. For this reason, accurate estimates of diffusion coefficients for clay soils are essential for the design of engineered liner systems. A long-standing theoretical problem is the role of anion exclusion on the estimation of diffusion coefficients for ionic solutes migrating through charged porous media. This paper describes the steady-state solution of a fully coupled set of transport equations modeling ion movement through a permanently charged platy-clay soil. The microscale analysis takes into account the actual diffusion coefficient for each ion species, ion-pairing (as required by electroneutrality of the solution), as well as anion exclusion and cation inclusion ,arising from the permanent charge on clay particles. To render the problem tractable, the theoretical analysis focuses on an extremely small two-dimensional unit cell in an ideal, saturated, two-phase porous medium. The analysis presented here is limited to a particular geometrical example, but this example is sufficiently general for characteristic behaviours of systems of this kind to be identified. Most importantly, new insight is gained into the mechanism of ion migration through a charged platy-clay soil. The numerical results obtained from this study show that the identification of macroscopic transport quantities such as effective diffusion coefficients and membrane potentials from diffusion cell tests using standard diffusion theory only hold for a specific system. While ion exclusion behaviours are often referred to in the literature, as far as the authors are aware there has been no previous detailed microscale analysis of their role in steady-state diffusion through a charged platy-clay soil.     

Generalized Taylor-Aris moment analysis of the transport of sorbing solutes through porous media with spatially-periodic retardation factor

Taylor-Aris dispersion theory, as generalized by Brenner, is employed to investigate the macroscopic behavior of sorbing solute transport in a three-dimensional, hydraulically homogeneous porous medium under steady, unidirectional flow. The porous medium is considered to possess spatially periodic geochemical characteristics in all three directions, where the spatial periods define a rectangular parallelepiped or a unit-element. The spatially-variable geochemical parameters of the solid matrix are incorporated into the transport equation by a spatially-periodic distribution coefficient and consequently a spatially-periodic retardation factor. Expressions for the effective or large-time coefficients governing the macroscopic solute transport are derived for solute sorbing according to a linear equilibrium isotherm as well as for the case of a first-order kinetic sorption relationship. The results indicate that for the case of a chemical equilibrium sorption isotherm the longitudinal macrodispersion incorporates a second term that accounts for the eflect of averaging the distribution coefficient over the volume of a unit element. Furthermore, for the case of a kinetic sorption relation, the longitudinal macrodispersion expression includes a third term that accounts for the effect of the first-order sorption rate. Therefore, increased solute spreading is expected if the local chemical equilibrium assumption is not valid. The derived expressions of the apparent parameters governing the macroscopic solute transport under local equilibrium conditions agreed reasonably with the results of numerical computations using particle tracking techniques.     

Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids

The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile porous solids is investigated. The Gologanu–Leblond–Devaux’s (GLD) analysis of an rigid-ideal plastic (von Mises) spheroidal volume containing a confocal spheroidal cavity loaded axisymmetrically is extended to the case when the matrix is anisotropic (obeying Hill’s [Hill, R., 1948. A theory of yielding and plastic flow of anisotropic solids. Proc. Roy. Soc. London A 193, 281–297] anisotropic yield criterion) and the representative volume element is subjected to arbitrary deformation. To derive the overall anisotropic yield criterion, a limit analysis approach is used. Conditions of homogeneous boundary strain rate are imposed on every ellipsoidal confocal with the cavity. A two-field trial velocity satisfying these boundary conditions are considered. It is shown that for cylindrical and spherical void geometries, the proposed criterion reduces to existing anisotropic Gurson-like yield criteria. Furthermore, it is shown that for the case when the matrix is considered isotropic, the new results provide a rigorous generalization to the GLD model. Finally, the accuracy of the proposed approximate yield criterion for plastic anisotropic media containing non-spherical voids is assessed through comparison with numerical results.     

Effective Diffusivity of Porous Materials with Microcracks: Self-Similar Mean-Field Homogenization and Pixel Finite Element Simulations

We investigate the influence of distributed microcracks on the overall diffusion properties of a porous material using the self-similar cascade continuum micromechanics model within the framework of mean-field homogenization and computational homogenization of diffusion simulations using a high-resolution pixel finite element method. In addition to isotropic, also anisotropic crack distributions are considered. The comparison of the results from the cascade continuum micromechanics model and the numerical simulations provides a deeper insight into the qualitative transport characteristics such as the influence of the crack density on the complexity and connectivity of crack networks. The analysis shows that the effective diffusivity for a disordered microcrack distribution is independent of the absolute length scale of the cracks. It is observed that the overall effective diffusivity of a microcracked material with the microcracks oriented in the direction of transport is not necessarily higher than that of a material with a random orientation of microcracks, independent of the microcrack density.     

A Two-Equation Analysis of Convection Heat Transfer in Porous Media

This paper presents a two-equation analysis on the convection heat transfer in porous media based on the modeling developed by Carbonell and Whitaker (1984). The porous system under consideration is bounded by two parallel walls and heated uniformly from one side surface. The Darcy flow is imposed and the fully developed heat transfer is assumed. General solutions, which take into account the additional convective and conductive terms, are obtained for the temperature fields and the Nusselt number. The detailed studies are presented for the porous systems characterized by consolidated and unconsolidated circular unit cells. The results show that, for the consolidated unit cell case, a prediction without the additional convective term overestimates the heat transfer, while for the unconsolidated unit cell case, this effect is negligible. The additional conductive terms are also examined and found to act conventionally as part of the conductive terms.     

A Potential Formulation of Non-Linear Models of Flow through Anisotropic Porous Media

A theoretical analysis, based on the search for a normal dissipation potential, is performed in order to generalize the empirical non-Darcy one-dimensional flow models to 3-D flows through anisotropic porous media. In an abstract framework, it is proven that a large number of heuristic non-linear equations governing the multidimensional flow through isotropic porous media can be derived starting from a potential strictly related to the mechanical power dissipated by the fluid. Such a formulation allows to define, for the tensor permeability case, a wide class of filtration models according to the Onsager's generalized theory of dissipative mechanical systems. A consistent generalization to anisotropic permeability case of polynomial flow models is proposed. Both primal and dual mixed variational formulations associated to the proposed quadratic and incomplete cubic flow models are introduced and discussed.     

Rapoport-leas two-phase flow model for anisotropic porous media

The Rapoport-Leas mathematical model of two-phase flow is generalized to include the case of anisotropic porous media. The formula for the capillary pressure, which specifies the relationship between the phase pressures, contains a scalar function of a vector argument. In order to determine the scalar function, the capillary pressure tensor and the tensor inverse to the tensor of characteristic linear dimensions are introduced. The capillary pressure is determined by the contraction of the second-rank tensors with a unit vector collinear to the phase pressure gradients, also assumed to be collinear. It is shown that the saturation function introduced for isotropic porous media (Leverett function) can be generalized to include anisotropic media and is now determined by a fourth-rank tensor. Generalized expressions for the Leverett and relative phase permeability functions are given for orthotropic and transversely isotropic media with account for the hysteresis of the phase permeabilities and capillary pressure.     

A dynamic coupling numerical model for pollutant transport under wave condition

In order to study the diffusion, migration, and distribution of pollutants among overlying water-body and porous seabed under wave conditions, a dynamic coupling numerical model is proposed. In this model, the coupling between wave field of overlying water-body and seepage of porous bed, the capture and release of pollutants in porous media, and the transport process between the two different regions are taken into account. We use the unified equations for pressure correction and pollutant concentration to solve the numerical model, which avoids repeated iteration on the interface boundary. The model is verified by several case studies. Afterwards, the processes involving release of pollutant from porous seabed and transportation to overlying water-body under different wave conditions are investigated. The results show that the water depth, wave height,and wave period have great influences on the release, capture, and transport processes for phosphorus pollutant.     

On Determining the Percolation Reynolds Number and the Characteristic Linear Dimension for Ideal and Fictitious Porous Media

Various versions of representations of the percolation Reynolds number for porous media with isotropic and anisotropic flow properties are considered. The formulas are derived and the variants are analyzed with reference to model porous media with a periodic microstructure formed by systems of capillaries and packings consisting of spheres of constant diameter (ideal and fictitious porous media, respectively). A generalization of the Kozeny formula is given for determining the capillary diameter in an ideal porous medium equivalent to a fictitious medium with respect to permeability and porosity and it is shown that the capillary diameter is nonuniquely determined. Relations for recalculating values of the Reynolds number determined by means of formulas proposed earlier are given and it is shown that taking the microstructure of porous media into account, as proposed in [1, 2], makes it possible to explain the large scatter of the numerical values of the Reynolds number in processing the experimental data.     

A linearity‐preserving cell‐centered scheme for the heterogeneous and anisotropic diffusion equations on general meshes

In this paper a finite volume scheme for the heterogeneous and anisotropic diffusion equations is proposed on general, possibly nonconforming meshes. This scheme has both cell‐centered unknowns and vertex unknowns. The vertex unknowns are treated as intermediate ones and are expressed as a linear weighted combination of the surrounding cell‐centered unknowns, which reduces the scheme to a completely cell‐centered one. We propose two types of new explicit weights which allow arbitrary diffusion tensors, and are neither discontinuity dependent nor mesh topology dependent. Both the derivation of the scheme and that of new weights satisfy the linearity‐preserving criterion which requires that a discretization scheme should be exact on linear solutions. The resulting new scheme is called as the linearity‐preserving cell‐centered scheme and the numerical results show that it maintain optimal convergence rates for the solution and flux on general polygonal distorted meshes in case that the diffusion tensor is taken to be anisotropic, at times heterogeneous, and/or discontinuous. Copyright © 2010 John Wiley & Sons, Ltd.     

Influence of frictional anisotropy on contacting surfaces during loading/unloading cycles

This paper presents numerical investigations on the loading and unloading of a three-dimensional body in frictional contact with a rigid foundation. The evolution of the sliding process during loading/unloading cycles is analyzed. The important case of anisotropy is examined along with the effect of the sliding rule. The solution algorithm is based on a variational inequality which combine the contact problem and the frictional problem. The numerical results of the punch problem show the hysteretic and irreversible behavior occurring when friction is anisotropic.     

双相各向异性介质弹性波场有限差分正演模拟

从双相各向异性介质模型出发,以Boit理论为基础,推导了斜方晶系各向异性介质-阶弹性波动方程,引入固、流体密度比和孔隙几何参数,将Biot方程系数简化为测量简单、物理意义明确的物理量,采用交错网格技术建立了各向异性孔隙介质波动方程的高精度差分格式,并首次对这类差分格式的频散特性和稳定性作了详细分析讨论,解决了计算稳定性和边界反射问题,与解析解的对比以及理论模型的数值模拟都表明,该方法不仅大大降低了计算量,提高了正演速度,并且具有良好的稳定性和精确性。