A thermodynamical gradient theory for deformation and strain localization of porous media

In this work, a thermodynamically consistent gradient formulation for partially saturated cohesive-frictional porous media is proposed. The constitutive model includes a classical or local hardening law and a softening formulation with state parameters of non-local character based on gradient theory. Internal characteristic length in softening regime accounts for the strong shear band width sensitivity of partially saturated porous media regarding both governing stress state and hydraulic conditions. In this way the variation of the transition point (TP) of brittle-ductile failure mode can be realistically described depending on current confinement condition and saturation level. After describing the thermodynamically consistent gradient theory the paper focuses on its extension to the case of partially saturated porous media and, moreover, on the formulation of the gradient-based characteristic length in terms of stress and hydraulic conditions. Then the localization indicator for discontinuous bifurcation is formulated for both drained and undrained conditions.     

Analytical Determination of Flow Patterns in a Double Porosity Medium Containing Ellipsoidal Occluded Macro-Voids

This article presents the analytical study of fluid flow in a porous medium presenting pores of two different length scales: at the smallest or microscopic scale, the presence of connected voids confers a porous medium structure to the material investigated, while at the upper or mesoscopic scale, occluded macro-pores are present. This microstructure is employed to represent the progressive opening of inter-aggregate pore spaces observed in natural compacted montmorillonites polluted by heavy metal ions. Three-dimensional analytical expressions are rigorously derived for the pore fluid velocity and excess pore fluid pressure within the porous matrix, around an occluded ellipsoidal inter-aggregate void. The eccentricity ratio is employed to characterize the geometrical shape of the ellipsoidal void, while its size is characterized by the macro-porosity. Confrontations are made with numerical solutions in order to investigate the applicability of the analytical pressure and velocity solutions to microstructures of finite size.     

A semi-analytical model for the behavior of saturated viscoplastic materials containing two populations of voids of different sizes

This paper presents a micromechanical model for a porous viscoplastic material containing two populations of pressurized voids of different sizes. Three scales are distinguished: the microscopic scale (corresponding to the size of the small voids), the mesoscopic scale (corresponding to the size of the large voids) and the macroscopic scale. It is assumed that the first homogenization step is performed at the microscopic scale, and, at the mesoscopic scale, the matrix is taken to be homogeneous and compressible. At the mesoscopic scale, the second homogenization step, on which the present study focuses, is based on a simplified representative volume element: a hollow sphere containing a pressurized void surrounded by a nonlinear viscoplastic compressible matrix. The nonlinear behavior of the matrix, which is expressed using the results obtained in the first homogenization step, is approached using a modified secant linearization procedure involving the discretization of the hollow sphere into concentric layers. Each layer has uniform secant moduli. The predictions of the model are compared with the more accurate numerical results obtained using the finite element method. Good agreement is found to exist with all the macroscopic stress triaxialities and all the porosity and nonlinearity values studied.     

基于原位观测的泡沫金属细观与宏观压缩实验研究

采用原位观测的方法研究了脆性泡沫铝材料在压缩载荷下细观与宏观断裂破坏规律和吸能机理。针对多孔泡沫金属材料提出一种细观原位加载实验方法,采用特别设计与制备的试件,在S570扫描电镜下研究了特定胞孔在压缩过程中孔壁的失效顺序和破坏规律,并揭示了能量吸收的细观机理。对块体材料的宏观压缩实验表明,脆性泡沫铝是以多个断裂带的形式破坏。研究发现,孔壁缺陷和胞孔形态缺陷是诱发断裂带形成与发展的重要因素。依据尺寸效应对细观与宏观实验下泡沫铝的性能进行了比较。     

Flow Regimes in Packed Beds of Spheres from Pre-Darcy to Turbulent

Characteristics of flow regimes in porous media, along the processes of energy dissipation in each regime, are critical for applications of such media. The current work presents new experimental data for water flow in packed steel spheres of 1- and 3-mm diameters. The porosity of the porous media was about 35 % for both cases. The extensive dataset covered a broad range of flow Reynolds number such that several important flow regimes were encountered, including the elusive pre-Darcy regime, which is rarely or never seen in porous-media literature and turbulent regime. When compared to previous information, the results of this study are seen to add to the divergence of available data on pressure drop in packed beds of spheres. The divergence was also present in the coefficients of Ergun equation and in the Kozeny–Carman constant. The porous media of the current work were seen to exhibit different values of permeability and Forchheimer coefficient in each flow regime. The current data correlated well using the friction factor based on the permeability (measured in the Darcy regime) and the Reynolds number based on the same length scale. An attempt was made to apply recent theoretical results regarding the applicability of the quadratic and cubic Forchheimer corrections in the strong and weak inertia regime.     

Analytical modeling and simulation of porous electrodes: Li-ion distribution and diffusion-induced stress

A new model of porous electrodes based on the Gibbs free energy is developed, in which lithium-ion(Liion) diffusion, diffusion-induced stress(DIS), Butler–Volmer(BV) reaction kinetics, and size polydispersity of electrode particles are considered. The influence of BV reaction kinetics and concentration-dependent exchange current density(ECD) on concentration profile and DIS evolution are numerically investigated. BV reaction kinetics leads to a decrease in Li-ion concentration and DIS. In addition, concentrationdependent ECD results in a decrease in Li-ion concentration and an increase in DIS. Size polydispersity of electrode particles significantly affects the concentration profile and DIS.Optimal macroscopic state of charge(SOC) should consider the influence of the microscopic SOC values and mass fractions of differently sized particles.     

Analysis of diffusion induced elastoplastic bending of bilayer lithium-ion battery electrodes

Bilayer electrode, composed of a current collector layer and an active material layer, has great potential in applications of in-situ electrochemical experiments due to the bending upon lithiation. This paper establishes an elastoplastic theory for the lithiation induced deformation of bilayer electrode with consideration of the plastic yield of current collector. It is found that the plastic yield of current collector reduces the restriction of current collector to an active layer, and therefore, enhances in-plane stretching while lowers down the rate of electrode bending. Key parameters that influence the elastoplastic deformation are identified. It is found that the smaller thickness ratio and lower elastic modulus ratio of current collector to an active layer would lead to an earlier plastic yield of the current collector, the larger in-plane strain, and the smaller bending curvature, due to balance between the current collector and the active layer. The smaller yield stress and plastic modulus of current collector have similar impacts on the electrode deformation.     

Numerical and analytical study to estimate the effect of two length scales upon the permeability of a fibrous porous medium

A fibrous porous medium with two length scales is modeled as a bed of porous cylinders aligned perpendicular to the flow of viscous fluid. The flow behavior is described using Stokes and Darcy flow equations in the regions around (higher length scale) and within the cylinders (lower length scale) respectively. The typical ratio of higher and lower length-scale regions enable us to invoke lubrication approximation and simplify the equations to develop a closed form solution for the overall permeability of this dual-scale porous medium. A parametric analysis is performed to explore the dependence of permeability on factors such as the volumetric ratio of higher and lower length-scale regions, permeability and size of inclusions in the smaller length-scale region. The analytical model is compared with the numerical results and the trend is compared with the experiments.     

The microstructural origin of strain hardening in two-dimensional open-cell metal foams

This paper aims at elucidating the microstructural origin of strain hardening in open-cell metal foams. We have developed a multiscale model that allows to study the development of plasticity at two length scales: (i) the development of plastic zones inside individual struts (microscopic scale) and (ii) the formation of plastic localization bands at the scale of the cellular architecture (mesoscopic scale). We address how plasticity at both scales contribute to the macroscopic yielding and strain hardening of cellular metals. One of the important results is that, in contrast to strain hardening in dense metals, strain hardening in cellular metals consist of a synergistic contribution of two sources: (i) strain hardening of the solid material (microscopic scale) and (ii) geometric hardening due to strut reorientation (mesoscopic scale). We show that the synergy of the two leads to an enhanced macroscopic hardening capacity. Our results are in qualitative agreement with experimental studies and elucidate the microstructural origin of plastic hardening in this class of materials.     

非平整、多孔介质海底上波浪传播的复合方程

为了反映近岸区域实际存在的多孔介质海底效应，并且考虑到波浪在刚性海底上传播模型的 最新研究进展，运用Green第二恒等式建立了波浪在非平整、多孔介质海底上传播的复合方 程. 假设水深和多孔介质海底层厚度均由两种分量组成：慢变分量，其水平变化的长度尺度大于 表面波的波长；快变分量，其水平变化的长度尺度与表面波的波长等阶，但其振幅小于表面 波的振幅. 另外，多孔介质层下部边界的快变分量比水深的快变分量小1个量级. 针对水体层和多孔介质层，选择Green第二恒等式方法给出了波浪传播和渗透的复合方程， 它在交接面上满足压力和垂直渗透速度的连续性条件,可充分考虑波数变化的一般连续性， 并包含了某些著名的扩展型缓坡方程.     

On dispersive propagation of surface waves in patchy saturated porous media

Frequency-dependent velocity and attenuation for Rayleigh-wave propagation along a vacuum/patchy saturated porous medium interface are investigated in the low frequency band (0.1–1000 Hz). Conventional patchy saturation models for compressional waves are extended to account for Rayleigh wave propagation along a free surface. The mesoscopic interaction of fluid and solid phases, as a dominant loss mechanism in patchy saturated media, significantly affects Rayleigh-wave propagation and attenuation. Researches on the dispersion characteristics at low frequencies with different gas fractions in patchy saturated media also demonstrate a strong correlation between the Rayleigh-wave mode and the fast compressional wave. Especially, the strongest attenuation with the maximum value of 1/Q$1/Q$ for Rayleigh waves are obtained in the frequency range of 1–200 Hz. Numerical results show that the significant dependence of velocity and attenuation on frequencies and gas fractions presents a distinctive dynamical response of Rayleigh waves in the time domain.     

Permeability of Pore Networks Under Compaction

The characteristic pore length fixes the scale of permeability of a porous medium. For pore networks undergoing strong random compaction, this length becomes singular at transition porosities, revealing a change in the microstructure of the porespace controlling the transport. Nodal balances and lattice Boltzmann simulations of flow in pore networks under compaction show that the scaling between permeability and porosity changes near the transition porosities. Simulation results are compared with experimental permeability data from transparent two-dimensional micromodels of networks decorated with the same pore size distribution. Permeability?Cporosity data of media undergoing smooth compaction is well described by a single power law. Under strong compaction, however, the scaling between permeability and porosity is possible by traits only, the scaling exponent changes notably at given transition porosities. These behaviors are common to a wealth of permeability?Cporosity data thus far unexplained.     

Thermomechanical response of non-local porous material

A thermomechanical model of a porous material is presented. The constitutive model is based on the Gurson model, formulated within a thermodynamic framework and adapted to large deformations. The thermodynamic framework yields a heat equation that naturally includes the mechanical dissipation. To introduce a length scale, the Gurson model was enhanced through non-local effects of the porosity being taken into account. A numerical integration scheme of the constitutive model and the algorithmic stiffness tensor are derived. The integration of the plastic part of the deformation gradient is based on an exponential update operator, an eigenvalue decomposition is also being used to reduce the number of equations that need to be solved. The coupled problem that arises is dealt with by employing a staggered solution method. To examine the capabilities of the model, shear band formation in a thick disc and crack growth in a thick notched disc were investigated.     

Uptake of water from soils by plant roots

Uptake of water by plant roots can be considered at two different Darcian scales, referred to as the mesoscopic and macroscopic scales. At the mesoscopic scale, uptake of water is represented by a flux at the soil–root interface, while at the macroscopic scale it is represented by a sink term in the volumetric mass balance. At the mesoscopic scale, uptake of water by individual plant roots can be described by a diffusion equation, describing the flow of water from soil to plant root, and appropriate initial and boundary conditions. The model involves at least two characteristic lengths describing the root–soil geometry and two characteristic times, one describing the capillary flow of water from soil to plant roots and another the ratio of supply of water in the soil and uptake by plant roots. Generally, at a certain critical time, uptake will switch from demand-driven to supply-dependent. In this paper, the solutions of some of the resulting mesoscopic linear and nonlinear problems are reviewed. The resulting expressions for the evolution of the average water content can be used as a basis for upscaling from the mesoscopic to the macroscopic scale. It will be seen that demand-driven and supply-dependent uptake also emerge at the macroscopic scale. Information about root systems needed to operationalize macroscopic models will be reviewed briefly.     

Multiresolution continuum modeling of micro-void assisted dynamic adiabatic shear band propagation

A thermal-mechanical multiresolution continuum theory is applied within a finite element framework to model the initiation and propagation of dynamic shear bands in a steel alloy. The shear instability and subsequent stress collapse, which are responsible for dynamic adiabatic shear band propagation, are captured by including the effects of shear driven microvoid damage in a single constitutive model. The shear band width during propagation is controlled via a combination of thermal conductance and an embedded evolving length scale parameter present in the multiresolution continuum formulation. In particular, as the material reaches a shear instability and begins to soften, the dominant length scale parameter (and hence shear band width) transitions from the alloy grain size to the spacing between micro-voids. Emphasis is placed on modeling stress collapse due to micro-void damage while simultaneously capturing the appropriate scale of inhomogeneous deformation. The goal is to assist in the microscale optimization of alloys which are susceptible to shear band failure.     

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 theory of macrodispersion for the scale-up problem

Dispersion is the result, observable on large length scales, of events which are random on small length scales. When the length scale on which the randomness operates is not small, relative to the observations, then classical dispersion theory fails. The scale up problem refers to situations in which randomness occurs on all length scales, and for which classical dispersion theory necessarily fails. The purpose of this article is to present non-Fickian, theories of dispersion, which do not assume a scale separation between the randomness and the observed consequences, and which do not assume a single length scale.Porous media flow properties are heterogeneous on all length scales. The geological variation on length scales below the observational length scale can be regarded as unknown and unknowable, and thus as a random variable.We develop a systematic theory relating scaling behavior of the geological heterogeneity to the scaling behavior of the fluid dispersivity. Three qualitatively distinct regimes (Fickian, non-Fickian and nonrenormalizable) are found. The theory gives consistent answers within several distinct analytic approximations, and with numerical simulation of the equations of porous media flow.Comparison to field data is made. The use of Kriging to generate constrained ensembles for conditional simulation is discussed.     

Multiresolution Wavelet Scale Up of Unstable Miscible Displacements in Flow Through Heterogeneous Porous Media

We describe scale up of geological models of field-scale porous media using a new method based on the wavelet transformations. The porous media of interest contain broadly-distributed and correlated permeabilities. Wavelet transformation of the permeability field of such porous media coarsens the geological model from smallest to largest length scales, drastically reduces the number of equations to be solved, preserves the important information on the permeability field at all the relevant length scales, and yields numerical results for any fluid flow property that are as accurate as those that are obtained with the highly detailed geological model of the same porous media. To test this method, we carry out extensive computer simulations of unstable miscible displacement processes and the associated viscous fingering phenomenon in highly heterogeneous porous media, both with the fine-scale geological model and the coarsened model. Excellent agreement is found between the results of the two sets of simulations.     

Effect of electrical conductivity of flow on current distribution in a magnetogasdynamic channel with allowance for the Hall effect

The effect of a magnetic field on the current distribution on a plane continuous anode situated opposite the cathode in a rectangular magnetogasdynamic channel with an external magnetic field was experimentally investigated. The distributions of the charged-particle density and the electron temperature near the outlet end of the electrodes were measured. The distribution of electrical conductivity in the flow was calculated. The electron density distribution along the channel is attributed to ambipolar diffusion of plasma to the walls. For an interpretation of the current distribution results, the method of integral relations in a linear approximation was used to solve the problem of a constant-velocity flow of a gas with variable electrical conductivity across a magnetic field in a plane magnetogasdynamic channel of constant cross section formed by electrodes of finite length and insulators. The Hall effect was taken into account. Experiments in which the effect of an external magnetic field on the current distribution on plane sectioned short electrodes in a magnetogasdynamic accelerator was investigated were described in [1]. In the present investigation, continuous long electrodes were used. These electrodes prevented the side effects due to coupling of the current to the ends of the electrode sections and helped to reveal some features of the current density distribution on the anode.     

Size effects and idealized dislocation microstructure at small scales: Predictions of a Phenomenological model of Mesoscopic Field Dislocation Mechanics: Part II

In Part I of this set of two papers, a model of mesoscopic plasticity is developed for studying initial-boundary value problems of small scale plasticity. Here we make qualitative, finite element method-based computational predictions of the theory. We demonstrate size effects and the development of strong inhomogeneity in simple shearing of plastically constrained grains. Non-locality in elastic straining leading to a strong Bauschinger effect is analyzed. Low shear strain boundary layers in constrained simple shearing of infinite layers of polycrystalline materials are not predicted by the model, and we justify the result based on an examination of the no-dislocation-flow boundary condition. The time-dependent, spatially homogeneous, simple shearing solution of PMFDM is studied numerically. The computational results and an analysis of continuous dependence with respect to initial data of solutions for a model linear problem point to the need for a nonlinear study of a stability transition of the homogeneous solution with decreasing grain size and increasing applied deformation. The continuous-dependence analysis also points to a possible mechanism for the development of spatial inhomogeneity in the initial stages of deformation in lower-order gradient plasticity theory. Results from thermal cycling of small scale beams/films with different degrees of constraint to plastic flow are presented showing size effects and reciprocal-film-thickness scaling of dislocation density boundary layer width. Qualitative similarities with results from discrete dislocation analyses are noted where possible.We discuss the convergence of approximate solutions with mesh refinement and its implications for the prediction of dislocation microstructure development, motivated by the notion of measure-valued solutions to conservation laws.