Erosion and deposition of solid particles in porous media: Homogenization analysis of a formation damage

The aim of the paper is to model at a large scale, the formation damage in porous media by erosion and deposition of solid particles. We start from the equations governing the pore-scale processes of erosion, deposition, convection and diffusion. The macroscopic equivalent behaviour is investigated by using a homogenization method. Four characteristic models with different dominating phenomena at the pore scale are determined. The main results are twofold: first dispersion-deposition and dispersion-erosion phenomena are shown at the macroscopic scale for peculiar values of the dimensionless numbers; furthermore, and contrarily to phenomenological models, erosion and deposition generally occur in regions of intense and slow flow, respectively.     

Homogenization of delamination growth in an ACA flip-chip joint based on micropolar theory

In the present paper the delamination mechanism of a typical internal structure of the anisotropic conductive adhesive (ACA) interconnect for electronic packaging is modeled on the basis of micropolar theory and computational homogenization. The interface is treated as a finite Representative Volume Element (RVE), across which the macroscopic deformation is expressed in terms of regularized strong displacement and rotational discontinuities. For the microstructure of the RVE, the micro-macro kinematical coupling is considered as a Taylor series expansion in the regularized macroscopic discontinuities, and, connected to that, a discontinuous fluctuation field representing the microstructural variation is included to describe delamination on the microlevel. As to the microlevel delamination modeling, on the basis of the discontinuous fluctuation field, a damage coupled to slip and dilation formulation is used to model the interface degradation. The constitutive relations are established in a thermodynamic setting, where the interfacial free energy involves internal variables of damage and plastic deformation. The parameters of the interface are calibrated so that a predefined amount of fracture energy is dissipated in mode I. In the numerical example, the response of a planar interface is considered when it is subjected to the basic modes I-II and also the non-conventional rotational discontinuity mode. Case studies on fracture and geometry parameters have also been carried out. Finally, an uncoupled thermomechanical analysis of a microsystem involving a representative ACA microstructure has been made for the understanding of the microscopic delamination during a thermal cycling procedure.     

Micromechanical model of nonlinear deformation of shape memory alloys under phase and structure transitions

We propose a model of nonlinear deformation of shape memory alloys (SMA) under phase and structure transitions which reflects the basic characteristics of the phenomena that occur in these materials at the microlevel. The model takes account of the influence on the processes in SMA not only of the applied external macroscopic stresses but also of the microstresses randomly distributed in the material volume under study, with the characteristics of these distributions (mathematical expectation and root mean square deviation) treated as material constants. We also demonstrate that the results predicted by the model agree well with the experimental data for nickel titanium alloy samples.     

On the motion of a fluid-saturated porous solid

Two-parameter structure model of a porous solid is proposed as an approximation of a real porous structure and the macroscopic mass and momentum balance equations are derived for such a medium filled with liquid. The approach presented leads to the equations of motion for a fluid-saturated porous medium with coupling terms via cross-mass couplings. The linear form of these equations is equivalent to the well-known Biot equations.     

Generalized lattice‐BGK concept for thermal and chemically reacting flows at low Mach numbers

The lattice‐BGK method has been extended by introducing additional, free parameters in the original formulation of the lattice‐BGK methods. The relationship between these parameters and the macroscopic moment equations is analysed by Taylor series and Chapman–Enskog expansion. The parameters are determined from the macroscopic moment equations by comparisons with the governing equations to be modelled. Extensions are presented for the Navier–Stokes equations at low Mach numbers in Cartesian or axisymmetric coordinates with constant or variable density, for scalar convection–diffusion equations and for equations of Poisson type. The generalized lattice‐BGK concept is demonstrated by two applications of chemical engineering. These are the computation of chemically reacting flow through an axisymmetric reactor and of the transport and deposition of particles to filters under the action of different forces. Copyright © 2006 John Wiley & Sons, Ltd.     

钢-混凝土混合框架结构多尺度分析及其建模方法

多尺度计算是保证计算精度的同时最大限度降低计算代价的有效途径,在众多学科领域和工程问题中都得到了应用.在结构有限元多尺度分析领域,要解决的一个关键问题是如何实现局部微观模型与宏观结构模型之间的共同工作.为实现精细有限元模型在植入宏观结构模型时不同尺度模型界面的变形协调,提出有限元微观模型与宏观模型的界面连接方法,给出了轴向、横向和转角的约束方程.通过编制用户子程序,使该方法在有限元软件中得以实现,并通过简单的圆柱筒算例,对界面连接的合理性进行了验证.最后基于多尺度建模方法和复杂混合结构节点的精细模型,给出了钢-混凝土混合框架结构多尺度弹塑性时程分析的应用实例,结果表明多尺度计算可较好模拟节点的复杂边界条件.本文建议的界面连接方法可有效实现不同尺度模型界面的变形协调,为工程结构进行多尺度提供了条件.     

Saturated Compressible and Incompressible Porous Solids: Macro- and Micromechanical Approaches

In this paper two complementary approaches are used to describe the mechanical behavior of saturated compressible and incompressible porous solids. The macroscopic investigation is based on the mixture theory, restricted by the volume fraction concept. In the micromechanical approach, a hierarchy of conditionally ensemble averaged fluid and solid phase momentum balance equations are derived for a simple model of quasi-static liquid saturated porous media. The ensemble averaged equations for both the phases agree remarkably well with the macroscopic results. A micromechanical basis for Terzhagi's effective stress concept is presented. In addition, an expression for additional partial solid stress modifying the effective stress principle, to account for deformability of solid materials, is also derived.     

Investigation of Donnan equilibrium in charged porous materials—a scale transition analysis

We propose a new theory describing how the macroscopic Donnan equilibrium potential can be derived from the microscale by a scale transition analysis. Knowledge of the location and magnitude of the charge density, together with the morphology of the pore space allows one to calculate the Donnan potential, characterizing ion exclusion in charged porous materials. Use of the electrochemical potential together with Gauss’ electrostatic theorem allows the computation of the ion and voltage distribution at the microscale. On the other hand, commonly used macroscopic counterparts of these equations allow the estimation of the Donnan potential and ion concentration on the macroscale. However, the classical macroscopic equations describing phase equilibrium do not account for the non-homogeneous distribution of ions and voltage at the microscale, leading to inconsistencies in determining the Donnan potential (at the macroscale). A new generalized macroscopic equilibrium equation is derived by means of volume averaging of the microscale electrochemical potential. These equations show that the macroscopic voltage is linked to so-called “effective ion concentrations”, which for ideal solutions are related to logarithmic volume averages of the ion concentration at the microscale. The effective ion concentrations must be linked to an effective fixed charge concentration by means of a generalized Poisson equation in order to deliver the correct Donnan potential. The theory is verified analytically and numerically for the case of two monovalent electrolytic solutions separated by a charged porous material. For the numerical analysis a hierarchical modeling approach is employed using a one-dimensional (1D)macroscale model and a two-dimensional (2D)microscale model. The influence of various parameters such as surface charge density and ion concentration on the Donnan potential are investigated.     

Boundary conditions for gases on a body surface

The influence of boundary conditions on the distribution of macroscopic parameters of various types is studied using the problem of heat transfer for monatomic and diatomic gases as an example. The calculations are made on the basis of model kinetic equations [1, 2]. A boundary condition that takes into account mutual transitions of rotational and translational energies of particles when they collide with a body surface is proposed for diatomic gas. Comparison with the experimental results made it possible to determine the coefficients appearing in the boundary conditions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 141–148, September–October, 1986.     

计及剪切影响的RC梁的非线性分析

为提高钢筋混凝土RC梁的计算效率和精度,提出了一种基于梁截面弯矩-曲率关系的宏观有限元方法,可用于各种跨高比RC梁的材料非线性分析。首先假定了混凝土和钢筋的非线性应力-应变关系,然后引入经过修正的Rodriguez截面模型,根据边界顶点把截面划分成若干梯形单元,利用quasi-Newton法求解由两个变量耦合而成的截面非线性平衡方程,由此建立RC截面的弯矩-曲率关系。在此基础上利用Timoshenko梁弯曲理论建立考虑横向剪切变形影响的RC梁的有限元分析模型。通过对试验梁的分析对比验证了所提出的分析方法的适用性。     

Multiscale modeling of a fluid saturated medium with double porosity: Relevance to the compact bone

In this paper, we develop a model of a homogenized fluid-saturated deformable porous medium. To account for the double porosity the Biot model is considered at the mesoscale with a scale-dependent permeability in compartments representing the second-level porosity. This model is treated by the homogenization procedure based on the asymptotic analysis of periodic “microstructure”. When passing to the limit, auxiliary microscopic problems are introduced, which provide the corrector basis functions that are needed to compute the effective material parameters. The macroscopic problem describes the deformation-induced Darcy flow in the primary porosities whereas the microflow in the double porosity is responsible for the fading memory effects via the macroscopic poro-visco-elastic constitutive law. For the homogenization procedure, we use the periodic unfolding method. We discuss also the stress and flow recovery at multiple scales characterizing the heterogeneous material. The model is proposed as a theoretical basis to describe compact bone behavior on multiple scales.     

A continuum model for deformable,second gradient porous media partially saturated with compressible fluids

In this paper a general set of equations of motion and duality conditions to be imposed at macroscopic surfaces of discontinuity in partially saturated, solid-second gradient porous media are derived by means of the Least Action Principle. The need of using a second gradient (of solid displacement) theory is shown to be necessary to include in the model effects related to gradients of porosity. The proposed governing equations include, in addition to balance of linear momentum for a second gradient porous continuum and to balance of water and air chemical potentials, the equations describing the evolution of solid and fluid volume fractions as supplementary independent kinematical fields. The presented equations are general in the sense that they are all written in terms of a macroscopic potential Ψ$\Psi$ which depends on the introduced kinematical fields and on their space and time derivatives. These equations are suitable to describe the motion of a partially saturated, second gradient porous medium in the elastic and hyper-elastic regime. In the second part of the paper an additive decomposition for the potential Ψ$\Psi$ is proposed which allows for describing some particular constitutive behaviors of the considered medium. While the potential associated to the solid matrix deformation is chosen in the form proposed by Cowin and Nunziato (1981) and Nunziato and Cowin (1979) and the potentials associated to water and air compressibility are chosen to assume a simple quadratic form, the macroscopic potentials associated to capillarity phenomena between water and air have to be derived with some additional considerations. In particular, two simple examples of microscopic distributions of water and air are considered: that of spherical bubbles and that of coalesced tubes of bubbles. Both these cases are suitable to describe capillarity phenomena in porous media which are close to the saturation state. Finally, an example of a simple microscopic distribution of water and air giving rise to a macroscopic capillary potential depending on the second gradient of fluid displacement is presented, showing the need of a further generalization of the proposed theoretical framework accounting for fluid second gradient effects.     

Simulation of gas–solid particle flows over a wide range of concentration

A two-fluid model of gas–solid particle flows that is valid for a wide range of the solid-phase volume concentration (dilute to dense) is presented. The governing equations of the fluid phase are obtained by volume averaging the Navier–Stokes equations for an incompressible fluid. The solid-phase macroscopic equations are derived using an approach that is based on the kinetic theory of dense gases. This approach accounts for particle–particle collisions. The model is implemented in a control-volume finite element method for simulations of the flows of interest in two-dimensional, planar or axisymmetric, domains. The chosen mathematical model and the proposed numerical method are applied to three test problems and one demonstration problem. © 1998 John Wiley & Sons, Ltd.     

Kinetic models for dilute solutions of dumbbells in non-homogeneous flows revisited

We propose a two-fluid theory to model a dilute polymer solution assuming that it consists of two phases, polymer and solvent, with two distinct macroscopic velocities. The solvent phase velocity is governed by the macroscopic Navier–Stokes equations with the addition of a force term describing the interaction between the two phases. The polymer phase is described on the mesoscopic level using a dumbbell model and its macroscopic velocity is obtained through averaging. We start by writing down the full phase-space distribution function for the dumbbells and then obtain the inertialess limits for the Fokker–Planck equation and for the averaged friction force acting between the phases from a rigorous asymptotic analysis. The resulting equations are relevant to the modelling of strongly non-homogeneous flows, while the standard kinetic model is recovered in the locally homogeneous case.     

Formulation of a finite deformation model for the dynamic response of open cell biphasic media

The paper illustrates a biphasic formulation which addresses the dynamic response of fluid saturated porous biphasic media at finite deformations with no restriction on the compressibility of the fluid and of the solid skeleton. The proposed model exploits four state fields of purely kinematic nature: the displacements of the solid phase, the velocity of the fluid, the density of the fluid and an additional macroscopic scalar field, termed effective Jacobian, associated with the effective volumetric deformation of the solid phase.The governing equations are characterized by the property of being all expressed in the reference configuration of the solid phase and by the property of employing only work-conjugate variables, thus avoiding the use of a total Cauchy stress tensor.In particular, the set of governing equations includes a momentum balance equation associated with the effective Jacobian field. This equation, differently from the closure-equations proposed by other authors which express a saturation constraint or a porosity balance, is derived as a stationarity condition on account of a least-action variational principle.