An expression for the permeability of anisotropic granular media

There are many expressions proposed for the permeability of isotropic media based on flow channel and pore size distribution concepts, but there are no such expressions for anisotropic media. In this paper an expression for the permeability of an anisotropic medium is proposed, which has been verified in the laboratory. The mechanism behind fluid flow through soil was investigated using microscopic computer simulations to propose an expression for macroscopic permeability. The soil was assumed to be a spatially periodic porous medium, and the Navier-Stokes equation was solved using the FEM with appropriate boundary conditions for several different arrangements of the porous medium. The basic variables influencing flow through soil at the microscopic level were identified as specific surface area, void ratio, particle shape, material heterogeneity and the arrangement of particles in a porous medium. A sensitivity analysis was carried out to obtain an expression for the permeability in terms of the above variables. The corresponding macroscopic variables for the above microscopic variables are average specific surface area, average void ratio, anisotropy, tortuosity due to material heterogeneity, and the arrangement of particles respectively. An expression for the directional permeability is proposed in terms of these variables for the most common occurrence of particles in a porous medium. For the verification of the proposed equation, the permeability values of a fine-grained sand were measured at different void ratios and were compared with those predicted by the proposed equation. The results show that the predicted permeability values from the proposed equation are very close to the measured values.     

A micromechanically derived anisotropic micropolar constitutive law for granular media: Elasticity

In the present work, it is attempted to derive a macroscopic constitutive law for the elastic deformation of granular materials, based on microscopic considerations. For the sake of simplicity, the solution is restricted to two dimensions, that is, a random assembly of infinitely extended cylinders. After examining pairwise contact interactions, the elastic energy rate of the assembly is retrieved in a discrete form. Introducing the probability density function of the contact orientations, the continuum form of the elastic energy density rate is evaluated as a function of generalized strains and curvatures, and their rates. The stresses and couple stresses result as dual variables to the generalized strain and curvature rates. Some properties of the resulting model are discussed, examples are presented and conclusions are drawn.Copyright © 2014 John Wiley & Sons, Ltd.     

A model of capillary cohesion for numerical simulations of 3D polydisperse granular media

We present a 3D discrete‐element approach for numerical investigation of wet granular media. This approach relies on the basic laws of contact and Coulomb friction enriched by a capillary force law between particles. We show that the latter can be expressed as a simple explicit function of the gap and volume of the liquid bridge connecting a pair of spherical particles. The length scales involved in this expression are analyzed by comparing with direct integration of the Laplace–Young equation. We illustrate and validate this approach by application to direct shear and simple compression loadings. The shear and compression strengths obtained from simulations reproduce well the experimental measurements under similar material and boundary conditions. Our findings clearly show that the number density of liquid bonds in the bulk is a decisive parameter for the overall cohesion of wet granular materials. A homogeneous distribution of the liquid within the bridge debonding distance, even at low volume contents, leads to the highest cohesion. The latter is independent of the liquid content as far as the liquid remains in the pendular state and the number density of liquid bonds remains constant. Copyright © 2007 John Wiley & Sons, Ltd.     

A hypoplastic model for granular soils incorporating anisotropic critical state theory

It is well known that soil is inherently anisotropic and its mechanical behavior is significantly influenced by its fabric anisotropy. Hypoplasticity is increasingly being accepted in the constitutive modeling for soils, in which many salient features, such as nonlinear stress-strain relations, dilatancy, and critical state failure, can be described by a single tensorial equation. However, within the framework of hypoplasticity, modeling fabric anisotropy remains challenging, as the fabric and its evolution are often vaguely assumed without a sound basis. This paper presents a hypoplastic constitutive model for granular soils based on the newly developed anisotropic critical state theory, in which the conditions of fabric anisotropy are concurrently satisfied along with the traditional conditions at the critical state. A deviatoric fabric tensor is introduced into the Gudehus-Bauer hypoplastic model, and a scalar-valued anisotropic state variable signifying the interplay between the fabric and the stress state is used to characterize its impact on the dilatancy and strength of the soils. In addition, fabric evolution during shearing can explicitly be addressed. Modifications have also been undertaken to improve the performance of the undrained response of the model. The anisotropic hypoplastic model can simulate experimental tests for sand under various combinations of principle stress direction, intermediate principal stress (or mode of shearing), soil densities, and confining pressures, and the associated drastic effect of different principal stress orientations in reference to the material axes of anisotropy can be well captured.     

Derived failure criteria for granular media

The failure criterion for a special rate-type material is derived using a method similar to that of Tokuoka.⁷ For particular choices of constitutive parameters it is shown that this failure criterion contains, as special cases, failure criteria suggested in four recent works dealing with elastic-plastic response of granular media.     

Deformation characteristics of inherently anisotropic granular media under repeated traffic loading: a DEM study

Acta Geotechnica - The cardioid-shaped loading path induced by moving traffic has been reproduced in the virtual element test of granular media using the discrete element method (DEM). The DEM...     

Physical model tests and discrete element simulation of shield tunnel face stability in anisotropic granular media

Acta Geotechnica - The stability of excavation face in shield tunneling plays a key role for construction safety. The ignorance of soil anisotropy in most previous studies would induce inaccurate...     

A frequency-domain analytical solution of Maxwell's equations for layered anisotropic media

An analytical solution of Maxwell's equations for layered anisotropic media is presented in a form which allows estimating the sought parameters by layer stripping without round-off accumulation. The solution in each layer is reduced to the standard procedures of solving a fourth-order algebraic equation, multiplication, addition, and inversion of second-order non-singular matrices. The algorithm has no limitations on layer thickness and is applicable to both very thick and very thin layers. The new numerical code is straightforward and can be easily parallelized.     

A stress and displacement discontinuity element method for elastic transversely anisotropic rock

The paper presents closed‐form solutions for stress and displacement influence functions for stress discontinuity (SD) and displacement discontinuity (DD) elements, for a two‐dimensional plane‐strain elastic, transversely anisotropic medium. The solutions for SD elements are based on Kelvin's problem and for DD elements on the concept of dipoles. Stress and displacement influence functions are derived for the following elements: constant SD, linear SD, constant DD, linear DD, square root DD, parabolic DD, constant DD surface, and linear DD surface elements. The formulations are incorporated into FROCK, a hybridized boundary element method code, and are validated by providing comparisons between the results from FROCK and the finite element code ABAQUS. A limited parametric analysis shows the effects of slight anisotropy on the stress field around the tip of a crack and of the orientation of the crack with respect to the axes of elastic symmetry. Copyright © 2014 John Wiley & Sons, Ltd.     

A micromechanics-based elasto-plastic model for granular media combined with Cosserat continuum theory

Acta Geotechnica - This paper is devoted to the micromechanical model of granular materials based on the Cosserat continuum theory. The generalised stress–strain relationships of a...     

A new digital method for three-dimensional stress analysis in elastic media

SummaryA New Digital Method for Three-Dimensional Stress Analysis in Elastic Media A new method for three-dimensional stress analysis in elastic media is outlined. According to this approach the problem is formulated purely in terms of conditions prevailing at the bounding surfaces of an elastic body. Stress and displacements on these surfaces, as well as inside the body, are expressed in terms of the superimposed effect of a set of basic solutions having suitable properties.The method is developed in matrix form. Some general properties of the governing matrices are established with a view to developing suitable numerical procedures.The convergence characteristics of two iterative schemes are considered.The paper concludes with observations concerning the generality of the approach.

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ZusammenfassungNeues digitales Verfahren zur dreidimensionalen Spannungsanalyse in elastischen Medien In der vorliegenden Arbeit werden die Grundlagen eines neuen Verfahrens der dreidimensionalen Spannungsanalyse beschrieben.Die Behandlung des allgemeinen Problems der Gebirgsmechanik, d. h. die Bestimmung der Spannungs- und Verschiebungsverteilungen in der Umgebung von untertägigen Hohlräumen, mit Hilfe konventioneller Verfahren — wie z. B. der Methode der Finiten Elemente — hat den Nachteil, daß der gesamte Gebirgskörper berücksichtigt werden muß. Dies hat zur Folge, daß sich realistische dreidimensionale Probleme auch mit modernsten Digitalrechnern nicht mehr innerhalb annehmbarer Rechenzeiten lösen lassen. Das hier vorgestellte Verfahren vermeidet diese Schwierigkeiten durch eine Formulierung, die ausschließlich die Randbedingungen an den Begrenzungsflächen des Mediums verwendet. Spannungen und Verschiebungen an diesen Flächen sowie im Inneren des Gebirgskörpers werden als Überlagerungseffekt einer Reihe von Grund-Lösungen mit geeigneten Eigenschaften ausgedrückt.Aus Gründen der Einfachheit wird in dieser Arbeit nureine solche Grund-Lösung verwendet. Diese beschreibt die Wirkungen einer Einzelkraft, die an einem Punkt eines unendlichen Mediums angreift. Mit Hilfe dieser Lösung kann die Wirkung eines Aggregates derartiger Kräfte in Matrizenform ausgedrückt werden.Läßt man eine Einzelkraft an jedem infinitesimalen Element einer kontinuierlichen Fläche angreifen, dann können die Spannungs- und Verschiebungsverteilungen im Körper mit Hilfe der an dieser Fläche herrschenden Randbedingungen beschrieben werden. Begrenzt die Fläche einen geschlossenen Hohlraum, so trennt sie das Medium in einen inneren und einen äußeren Teil. Auf diese Art entstehen zwei Problemklassen: Während der innere Bereich die Spannungsanalyse von Ingenieurbauten beinhaltet, entspricht der äußere Bereich dem Hohlraumproblem der Gebirgsmechanik.Die beschriebene Formulierung mittels Matrizen ist für beide Bereiche gültig. Es wird jedoch gezeigt, daß die entsprechenden Matrizen charakteristisch unterschiedliche Eigenschaften haben. Diese Beobachtung ist von grundlegender Bedeutung für die Entwicklung numerischer Lösungsverfahren. Die Konvergenz-Charakteristiken zweier Iterativ-Schemata werden in Betracht gezogen.Zusammenfassend wird noch einmal die generelle Gültigkeit des Verfahrens betont. Weitere Grund-Lösungen und ihre Anwendung auf spezielle Probleme werden erwähnt.

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RésuméUne nouvelle méthode pour la solution de problèmes à trois dimensions dans la théorie de l'élasticité Les auteurs élaborent une nouvelle méthode pour la solution de problèmes à trois dimensions dans la théorie de l'élasticité.En Mécanique des Roches, lorsque les excavations sont situées dans une masse infinie de roches, les méthodes courantes, telle que les éléments finis, ont le désavantage d'avoir à opérer sur une quasi-infinité d'éléments. Il en résulte que l'analyse de configurations réelles à trois dimensions devient impossible même avec les plus grands ordinateurs modernes. La nouvelle méthode évite ces difficultés en reformulant le problème entièrement en fonction des conditions régnant aux limites de la configuration. Les tensions et les déplacements aux limites, ainsi qu'à l'intérieur de la masse, sont produits par les actions superposées d'un ensemble convenablement choisi de fonctions — solutions élémentaires.Pour alléger l'exposé on n'a employé qu'une seule fonction — solution dans cette publication. Elle décrit l'action d'une force concentrée en un point d'un milieu infini. Les actions d'une distribution de pareilles forces concentrées peuvent s'exprimer sous forme matricielle.En appliquant une force concentrée à chaque élément infinitésimal d'une surface continue située dans le milieu on arrive à exprimer les tensions et les déplacements en un point quelconque en fonction des conditions régnant sur cette surface.Si cette surface est fermée elle divise le milieu en une région intérieure et une région extérieure, ce qui donne lieu a deux classes de problèmes. Le problème intérieur se présente surtout en Résistance des Matériaux, tandis que le problème extérieur se recontre en Mécanique des Roches.Le formalisme matriciel mentionné ci-dessus est applicable dans les deux régions. Cependant il apparaît que les matrices associées ont des propriétés foncièrement différentes. Cette constatation est de première importance lors de l'élaboration de méthodes de solutions numériques ayant recours à l'ordinateur.Deux procédés de calculs appropriés sont analysés dans cette publication.Dans leurs conclusions les auteurs mettent de nouveau en relief la généralité de la méthode. Quelques exemples de fonctions — solutions élémentaires, différentes de la fonction — solution (force concentrée) employée dans cette publication, sont cités.

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Liquefaction and cyclic mobility model for saturated granular media

A new constitutive law for the behaviour of undrained sand subjected to dynamic loading is presented. The proposed model works for small and large strain ranges and incorporates contractive and dilative properties of the sand into the unified numerical scheme. These features allow to correctly predict liquefaction and cyclic mobility phenomena for different initial relative densities of the soil. The model has been calibrated as an element test, by using cyclic simple shear data reported in the literature. For the contractive sand behaviour a well‐known endochronic densification model has been used, whereas a plastic model with a new non‐associative flow rule is applied when the sand tends to dilate. Both dilatancy and flow rule are based on a new state parameter, associated to the stiffness degradation of the material as the shaking goes on. Also, the function that represents the rearrangement memory of the soil takes a zero value when the material dilates, in order to easily model the change in the internal structure. Proceeding along this kind of approach, liquefaction and cyclic mobility are modelled with the same constitutive law, within the framework of a bi‐dimensional FEM coupled algorithm developed in the paper. For calibration purposes, the behaviour of the soil in a cyclic simple shear test has been simulated, in order to estimate the influence of permeability, frequency of loading, and homogeneity of the shear stress field on the laboratory data. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerically obtained vortices in granular media

With our meshfree numerical code SPARC (Soft PARticle Code), which is based on strong solutions of the equations of equilibrium, we were able to derive vortex patterns ("turbulence") in deformations hitherto believed to be homogeneous. The formation of such vortices demonstrates the nonuniqueness of the corresponding boundary value problem. We present some evidence that such vortices can be related with ptygmatic folds, which are observed in rock.     

Poroelastic coefficients for anisotropic single and double porosity media

Acta Geotechnica - Closed-form expressions for poroelastic coefficients are derived for anisotropic materials exhibiting single and double porosity. A novel feature of the formulation is the use of...     

Modelling the elastic behaviour of granular materials

A review of the literature indicates that the elastic behaviour of granular materials is isotropic and that Poissony's ratio is constant, whereas Young's Modulus, the bulk modulus and the shear modulus vary with the mean normal stress and the deviatoric stress. A nonlinear, isotropic model for the elastic behaviour is developed on the basis of theoretical considerations involving the principle of conservation of energy. Energy is therefore neither generated not dissipated in closed-loop stress paths or in closed-loop strain paths. The framework for the model consists of Hooke's law, in which Poission's ratio is constant and Young's modulus is expressed as a power function invlving the first invariat of the stress tensor and the second invariant of the deviatoric stress tensor. The characteristics of the model are described, and the accuracy is evaluated by comparison with experimental results from triaxial tests and three-dimensional cubical triaxial tests with a variety of stress paths. Parameter determination from unloading–reloading cycles in conventional triaxial compression tests is demonstrated, typical parameter values are given for granular materials and extension of the model to soils with effective cohesion is described.     

Description of deformation process in inherently anisotropic granular materials

The main focus in this work is on modeling of mechanical response of granular materials that display inherent anisotropy. Both the experimental and numerical investigations are described. First, the results of direct shear as well as drained/undrained triaxial tests that involve crushed limestone with elongated angular‐shaped particles are reviewed. Afterward, a mathematical framework is presented for modeling of elastic/ inelastic deformation that incorporates the multi‐laminate approach. The deformation is monitored on a set of randomly oriented planes, and the formulation incorporates the thickness of the shear band that is associated with sliding/separation process. A systematic procedure for identification of material functions/ parameters is outlined that is based on the results of direct shear tests, and the framework is later applied to simulate the behavior under triaxial conditions. The results of numerical simulations are compared with the experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical derivation of water retention for random monodisperse granular media

The mechanics of water retention in unsaturated granular media is of critical importance to a broad range of disciplines including soil science, geotechnical engineering, hydrology and agriculture. Fundamental to water retention is the relationship between degree of saturation and suction, referred to as the water retention curve (WRC). The majority of WRC models are empirically based and seldom incorporate physically meaningful parameters. This study presents an analytical model for the WRC that considers separate contributions from fully filled pores and partially filled pores containing liquid bridges. A recently established unique k-gamma pore volume distribution function for randomly assembled monodisperse granular materials is adopted to determine the contributions of fully filled pores. Calculation of the contribution of residual pore water retained in partially filled pores is undertaken by representing pores as individual cells shaped as platonic shapes of various sizes and determining the volume of all liquid bridges suspended between particles within the pore cells. Weighting factors for the various cell types are obtained from the pore volume distribution to determine the relative contribution of different pore cell geometries to the total residual pore water. The combined model accurately describes experimental data for monodisperse spherical glass beads for both wetting and drying, even though the underlying assumptions do not reflect exactly the complex, interconnected and highly irregular geometry of the pore space. A single parameter provides the lateral shift between the wetting and drying curves. The results of this study provide a geometric understanding of the mechanisms of water retention in granular media.