Granular and
Complex Materials
WORLD SCIENTIFIC LECTURE NOTES IN COMPLEX SYSTEMS Editor-in-Chief: A.S. Mikhailov, Fritz Haber Institute, Berlin, Germany
H. Cerdeira, ICTP, Triest, Italy B. Huberman, Hewlett-Packard, Palo Alto, USA K. Kaneko, University of Tokyo, Japan Ph. Maini, Oxford University, UK
AIMS AND SCOPE The aim of this new interdisciplinary series is to promote the exchange of information between scientists working in different fields, who are involved in the study of complex systems, and to foster education and training of young scientists entering this rapidly developing research area. The scope of the series is broad and will include: Statistical physics of large nonequilibrium systems; problems of nonlinear pattern formation in chemistry; complex organization of intracellular processes and biochemical networks of a living cell; various aspects of cell-to-cell communication; behaviour of bacterial colonies; neural networks; functioning and organization of animal populations and large ecological systems; modeling complex social phenomena; applications of statistical mechanics to studies of economics and financial markets; multi-agent robotics and collective intelligence; the emergence and evolution of large-scale communication networks; general mathematical studies of complex cooperative behaviour in large systems.
Published Vol. 1 Nonlinear Dynamics: From Lasers to Butterflies Vol. 2 Emergence of Dynamical Order: Synchronization Phenomena in Complex Systems Vol. 3 Networks of Interacting Machines Vol. 4 Lecture Notes on Turbulence and Coherent Structures in Fluids, Plasmas and Nonlinear Media Vol. 5 Analysis and Control of Complex Nonlinear Processes in Physics, Chemistry and Biology Vol. 6 Frontiers in Turbulence and Coherent Structures Vol. 7 Complex Population Dynamics: Nonlinear Modeling in Ecology, Epidemiology and Genetics
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World Scientific Lecture Notes in Complex Systems - Vol. 8
editors TAste
The Australian National University, Australia
T Di Matteo
The Australian National University, Australia
A Tordesillas
University of Melbourne, Australia
Granular and Complex Materials
'World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • BEIJING • S H A N G H A I • H O N G KONG • TAIPEI • C H E N N A I
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
GRANULAR AND COMPLEX MATERIALS Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-13 978-981-277-198-8 ISBN-10 981-277-198-0
Printed in Singapore.
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Dedicated to Francesca and Carmel
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Preface
The Science of Complex Materials is attracting an ever increasing interest and participation of researchers from a vast range of disciplines, including physics, mathematics, computational science, and virtually all domains of engineering. These lecture notes present invited lectures and selected contributions from the 20th Canberra International Physics Summer School and Workshop on Granular Materials, held at The Australian National University in Canberra, between the 4th and the 8th of December 2006. At the meeting, international experts from 21 different countries gathered together to debate and discuss the latest advances in the experimental, computational and mathematical analysis of granular materials and other related complex materials such as foams, porous media and cellular solids. The contributions in these Lecture Notes explore an array of problems reflecting recent developments in four main areas: • • • •
Characterisation and Modelling of Disordered Packings; Micromechanics and Continuum Theory; Discrete Element Method; Statistical Mechanics.
The common theme and, indeed, the driving force behind these investigations is the quest to unravel the connection between the microscopic and the macroscopic properties of complex materials. The microscopic structural properties of complex materials and disordered packings are introduced in the contribution by Weaire et al. which highlights intriguing similarities between granular materials and foams. An old unsolved question, “Why spheres poured in a container never pack denser than 64% of the total volume?” is addressed in an vii
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innovative way by Anikeenko et al. by introducing a set of geometrical tools to understand, classify and characterize the structure of disordered packings. Structural characterization of granular materials is also discussed in the contribution of Blumenfeld by using innovative entropic approaches. New techniques that have recently emerged from novel amalgams of established methodologies in mechanics are explored in several contributions. Vardoulakis et al. present a study that adapts “shallow water theory” to granular flows. Einav develops a new continuum theory on particle comminution in brittle granular materials by using thermomechanics and statistical mechanics. Tordesillas presents a novel continuum formulation that delivers constitutive laws expressed entirely in terms of particle scale properties by weaving together principles and techniques from thermodynamics, micromechanics, structural mechanics and micropolar theory. The contribution by Tejchman et al. explores size effects in finite element continuum simulations by using stochastic techniques. These continuum formalisms rely heavily on particle scale information. For this purpose, the effectiveness of discrete element method (DEM) remains unsurpassed. The contribution by Cleary shows what can be achieved in DEM simulations for industrial and geophysical applications, particularly in the areas of mixing, separation, excavation, comminution, storage, transport and sampling of granular materials. The contribution from Delaney et al. discusses cutting-edge developments in DEM combined with x-ray computer tomography to perform ‘virtual experiments’. The contribution by Behringer provides an overview of granular properties, with a particular focus on the dense granular states. An exploration of a range of phenomena, from force chains to non-affine motion, is illustrated through a series of experiments. These studies deliver new insights on the role of fluctuations at the microscopic and mescoscopic levels. Fluctuations are also the core of the contribution by Nicodemi et al. where a statistical mechanics approach that enables us to predict such fluctuations is constructed. An experimental validation of such an extension of thermodynamics to non-thermal systems is discussed in the contribution by Richard et al. where the dynamics of granular media submitted to gentle mechanical tapping is studied. These
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studies are crucial to understand the properties of stationary states and discuss the validity of such configurations as genuine ‘thermodynamic’ states. In conclusion, this volume presents a unique multidisciplinary panorama of the current research in complex materials. Our aim is to encourage open interdisciplinary discussions between physicists, engineers, mathematicians and computer scientists, in order to develop a common language and achieve complementary objectives. Tomaso Aste Antoinette Tordesillas Tiziana Di Matteo Editors
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Contents
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Chapter 1
Foam as granular matter D. Weaire, V. Langlois, M. Saadatfar and S. Hutzler
Chapter 2
Delaunay simplex analysis of the structure of equal sized spheres A.V. Anikeenko, N.N. Medvedev, T. Di Matteo, G.W. Delaney and T. Aste
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Chapter 3
On entropic characterization of granular materials R. Blumenfeld
43
Chapter 4
Mathematical modeling of granular flow-slides I. Vardoulakis and S. Alevizos
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Chapter 5
The mechanics of brittle granular materials I. Einav
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Chapter 6
Stranger than friction: force chain buckling and its implications for constitutive modelling A. Tordesillas
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Chapter 7
Investigations of size effects in granular bodies during plane strain compression J. Tejchman and J. Górski
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Contents
Chapter 8
Granular flows: fundamentals and applications P.W. Cleary
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Chapter 9
Fine tuning DEM simulations to perform virtual experiments with three-dimensional granular packings G.W. Delaney, S. Inagaki and T. Aste
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Chapter 10 Fluctuations in granular materials R.P. Behringer
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Chapter 11 Statistical mechanics of dense granular media M. Pica Ciamarra, A. de Candia, A. Fierro, M. Tarzia, A. Coniglio and M. Nicodemi
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Chapter 12 Compaction of granular systems P. Richard, F. Lominé, P. Ribière, D. Bideau and R. Delannay
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Denis Weaire∗ , Vincent Langlois, Mohammad Saadatfar and Stefan Hutzler School of Physics, Trinity College Dublin Dublin 2, Ireland ∗
[email protected] What has foam in common with granular matter? What can these two active research fields learn from each other, where do they overlap? We approach these questions by reviewing a wide range of foam theory (mostly simulation) and experiment, set in the context of granular matter research.
1. Introduction 1.1. History of foam research Foam and soap film research goes back to the dawn of modern science (Leonardo da Vinci,25 Robert Boyle12 ). In 1873 the blind Belgian scientist Joseph Plateau published his masterful account of his own researches and the subject’s previous history.73 Many of those older references are also to be found in the classic work of Mysels et al. on soap films.71 Later Lord Kelvin took an interest in the subject.94 The almostsimultaneous preoccupation of Kelvin with foams and Reynolds with granular media74 drew motivation from the same source: the structure of the ether. This was the all-pervading medium that 19th century physics required for the propagation of light waves. Remarkably, the same question re-emerges today in modern form (the structure of space-time on the Planck scale) and theorists in that subject use granular and foam language interchangeably! Foam structure, as first elucidated by Plateau, may be visualized/analysed from different perspectives, depending to some extent on the 1
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Fig. 1. From left to right: 1D foam consisting of regularly spaced parallel soap films (bamboo foam);22 2D foam confined between two glass plates; 3D foam (the vertical gradient in liquid fraction is due to gravity-driven drainage).
wetness (liquid volume fraction) : • • • •
a a a a
packing of bubbles tessellation of cells partitioning of space by films network of lines (Plateau borders)
Plateau border junction of 3 films
vertex or node intersection of 4 Plateau borders
Fig. 2. Elements of foam structure: from left to right, packing of bubbles; partitioning of space by films: the Weaire-Phelan structure; network of Plateau borders.
It is simpler in two dimensions, as Cyril Stanley Smith explained,84 but the two-dimensional (2D) system brings its own complications. For a start there are (at least) three varieties of ordinary 2D soap froths, as represented in Figure 3: • Hele-Shaw cell : one layer of bubbles confined between two plates;83 • Plate/liquid: the bubbles float in liquid under a plate;35,89 • Free-floating bubbles.14
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Fig. 3. Three types of 2D foams. From left to right: air bubbles confined between two glass plates; bubbles floating in liquid under a glass plate; monolayer of bubbles sitting at an air/liquid interface.
Note the contrast between the Bragg and Smith systems. Both were used to model grain growth in atomic crystals, but Bragg’s monodisperse bubbles, small enough to be roughly spherical, represent individual atoms, whereas Smith’s large polydisperse bubbles represent whole grains. Many foam
Fig. 4.
Left: Smith’s crystalline model83 . Right: Bragg’s bubble raft.14
experiments have been conducted with bubbles of diameters of a few millimetres (see 2.2). The effect of gravity is diminished for bubbles much smaller than the capillary length (about 1.6 mm for ordinary surfactants). Old ideas of how to make them monodisperse have been revived88 and are of great interest in microfluidics.30
1.2. Space and time scales Three different length scales are pertinent in foam physics: • the scale of the films where the physico-chemical properties of the stabilizing surfactants strongly influence the forces that determine dynamic properties of the foam; • the scale of the individual bubbles, ruled by mechanical equilibirum of the films;
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• the scale of the foam: smoothed continuum models can be built to model the foam as a non-newtonian fluid.23,49 surfactant molecules
air
Geometry dictated by − Plateau rules − Laplace law
hydrophobic part hydrophilic part
liquid
mean bubble radius surface tension liquid fraction liquid viscosity
air
Fig. 5. The three length scales: from left to right, a film of surfactant solution, an ensemble of bubbles, foam as a continuous medium.
Most phenomena take place on very different time scales as well. Typically they are as follows: • structural relaxations: a fraction of a second, • drainage to equilibrium under gravity: a few minutes, • coarsening, due to diffusion of gas: many minutes up to several hours. These different timescales are very convenient in separating effects in experiments. The properties of general interest include those that are essentially static or can be described quasi-statically, like structure, stability, elasticity, coarsening, quasistatic rheology, light scattering, electrical and thermal resistance. Increasingly, properties that are truly dynamic are addressed, including details of transformations and structural relaxation, rate-dependent rheology, drainage, convective instability, size segregation. The stability of a foam in relation to film rupture depends crucially on the chemistry of the foaming solutions. Common detergent foams are remarkably stable against film rupture, over periods of days, and are ideal for the study of many generic foam properties. Theory is sometimes complicated by the history-dependence of foam structure and hence its properties. Coarsening due to the diffusion of gas through the liquid cell walls eventually results in a foam whose average bubble diameter varies with the square root of time. However, the statistical properties of the bubble packing, such as the average number of faces per cell, remain the same.39,85 The unique polydisperse system reached after a long period of coarsening (the scaling state) is thus a useful choice for a
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standard system, to avoid the arbitrariness of samples otherwise created. Almost all of the above properties have closely equivalent counterparts for granular materials, so each field can be illuminated by the other. Is there a closer correspondence? Should we regard a gas bubble as a frictionless, compressible particle? Durian33 and others have tried to integrate foam theory with granular and atomic systems, by using a single idealised model for all three, as we shall explain below. Do foams offer any advantages to those interested in exploring generic properties of disordered systems? There may be some. In particular, foams have relatively well-defined and understood local structure and interactions. There is no solid friction, although there may be equally problematic Marangoni effects. Laboratory equipment can be rudimentary: glass plates, tubing, simple air and water pumps. High speed video with sophisticated image analysis is becoming common, but direct measurements may still be performed to great effect. Besides, many foams are partially transparent, so that some limited observation of their interior is possible. 1.3. Key physical parameters The key physical parameters describing an ordinary aqueous foam include those that characterize the liquid and its surface: • surface tension: in practice, its value is usually less of that for pure water (surface tension γ 13 γwater 24mN/m is often a good estimate). Many expressions for pressures, forces, energies, elastic moduli, yield stress are proportional to γ. In simple theories this is often the only parameter of interest and hence tends to disappear entirely in simulations. • bulk viscosity: it has been considered to control drainage and structural relaxation, but this is now questionable in some cases. • surface properties, such as surface viscosity and elasticity, are of great importance but are difficult to capture in simple formulae. Their role has been studied by physical chemists for a long time but are only now being properly integrated into foam physics. • a parameter that characterizes the permeability of films: it depends on the choice of gas and may be greatly reduced by an appropriate choice, in order to control coarsening by reducing the permeation of films. Further basic parameters characterise the foam structure:
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• the average bubble diameter d (or some other measure of size); • the liquid fraction φ (often a function of position), which tends to zero in the limit of a dry foam; • topological measures such as μ2 , the second moment of the distribution of the number of sides in a two-dimensional foam. This measures the spread of different types of cells (roughly corresponding to polydispersity), and crops up in various theories. How can we access quantitatively these structural parameters in practice? We might squash and two-dimensionalise a three-dimensional (3D) foam sample between two plates, in order to estimate d. It is also possible to infere to some extent the size distribution in the bulk from what is observable on the sidewalls. To measure the liquid fraction φ, we may weigh a sample, or use light scattering, electrical resistance, gamma ray absorption or Archimedes Principle. As for granular media, the complete structure can also be accessed by tomography techniques (see section 5). Remarkably, a lot of physics can be developed in terms of these few variables (leaving out surface viscosity and surface elasticity, if possible), after a few simplifying approximations. Foam physics is remarkably coherent and tractable at the level of ten percent accuracy, but becomes cluttered and obscured by a multitude of small effects if more precision is pursued. Most of the basic formulae that roughly capture physical properties can be found in the book of Weaire and Hutzler.95 A fuller appreciation of the present breadth of the subject may be gathered from the proceedings of the last EUFOAM Conference in Potsdam,66 or proceedings of previous European Foams Conferences.92,101 The next is to be held at Nordwijk in 2008.
1.4. Wet and dry foams The value of liquid fraction in a disordered foam can range between 0 (corresponding to an ideal polyhedral packing of bubbles) and approximately 0.36, where the latter value is the void fraction of a random packing of hard spheres (in two dimensions the equivalent value is approximately 0.16). If we call a foam wet if its (local) liquid fraction exceeds half of this value, the following estimate can be deduced for the thickness of the layer of wet foam lying on a pool of liquid,95 Wwet
l0 2 d
(1)
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where l0 is the capillary length, given by l0 =
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∼ 1.6mm for usual
surfactants (γ is the surface tension, ρ is the density of the liquid, g is the acceleration due to gravity). The number of bubble layers in a wet foam
Fig. 6. An illustration of the gradient of liquid fraction. Here P is less than 5. (Photograph by J. Cilliers, Imperial College London).
under gravity is thus given by dividing Wwet by d and we shall define this here as the dimensionless Princen number 2 l0 (2) P = d in honour of the late Henry Princen, pioneer of foam drainage and rheology. In the wet limit, a foam can be considered as a immersed granular material, with the differences that the grain weight is replaced by bubble buoyancy, and that the bubble surface is more deformable. On the other hand in the dry limit the foam can be seen as ai tessellation, rather like the Voronoi tessellation of a granular material. 1.5. Emulsions Emulsions that have comparable structural scales to those of typical foams behave quite similarly, and may be termed bi-liquid foams. For static properties, replacing gas by liquid and bubbles by droplets changes little apart from the terminology. Of course, there are bound to be some important dif-
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ferences in dynamic properties, wherever the viscosity of the enclosed liquid is significant. This is noticible in particular in so-called forced-drainage experiments, where the continuous phase is continuously replenished to avoid a drying out of the foam or emulsion.72 Whereas in foams the bubbles undergo convective rolls provided the rate of added liquid exceeds a certain threshold (cf. 2.3), in emulsions flow instabilities can take the form of density waves.43
Fig. 7. Drainage wave passing through an emulsion of oil in water (the dashed lines indicate the position of the wave front at successive times).43 In the case shown this results in a high water volume fraction(φ ∼ 0.5) with agitated motion of the oil droplets.
2. Static properties Different styles of modelling/simulation have been developed for static structures and quasistatic properties. They include • detailed representation of the structure (films, Plateau borders, junctions); • vertex models (not discussed here); • soft sphere/disk models. 2.1. Structure The first of the above models is a precise representation of foam structure, after it has been idealised, primarily by treating the films as infinitesimally thin. In two dimensions this was accomplished for a dry foam by Weaire and Kermode in 1984.97 Wet foam was similarly simulated by Bolton
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and Weaire9,10 (cf Fig. 15). Nowadays the specifically developed codes have been largely replaced by the 2D version of the Surface Evolver of Ken Brakke.15 Equilibration is usually straightforward, but some ques-
Fig. 8.
A 3D foam generated with Surface Evolver.
tions arise. What starting configuration is to be used? Most simulations start from a rather arbitrary Voronoi partition, as this is convenient. Is the resulting structure unique (for a given final topology)? The answer to the second question appears to be no, in general, but yes in practice for typical, finite samples. Brakke’s Surface Evolver15 also provides the standard procedure for equilibration in the case of 3D foams. Whereas the lines in 2D foams are always circular arcs, no such simplification is available in 3D, and a fine tessellation is needed to represent the surfaces. Helpful features of the Surface Evolver package include calculation of the Hessian matrix and its eigenvalues, which can help in characterising instabilities.100 The chief practitioner of the application of this methodology to foams has in recent years been Andy Kraynik. His extensive exploration of random equilibrium structures of monodisperse foams54 has revealed a wide range of possibilities, echoing the story of the various mondisperse hard-sphere packings in the theory of granular materials. This has provided a vindication of old experimental work of Matzke,65 who laboriously assembled foam samples by adding identical bubbles one at a time; he then analysed their contents with even greater labour. Some of us distrusted the detailed results of that unique effort, but Kraynik54 has now confirmed them.
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2.2. Crystallization Early observations have been made by Bragg and others13,14,47,59,60 of the crystalline ordering of a 2D raft made of small bubbles (see Fig. 4). This system has been used as an analogue to solid crystals and the interactions between bubbles have been modeled accordingly by a short-range repulsion (due to the bubble distortion) and a long-range attraction (due to the distortion of the water surface90 ). Some recent developments have been made by extending Bragg’s bubble raft concept into three dimensions. When the bubbles are typically smaller than 500μm, the Princen number becomes greater than 10, which allows equilibrium samples of wet 3D foams, with more than 10 layers. Remarkably, it has been observed that these small bubbles order spontaneously in three dimensions, with an apparent preference for FCC structure.88 The existence of stacking faults, grain boundaries, dislocations, all phenomena already identified in 2D bubble crystals, is yet to be investigated in detail. This spontaneous ordering is not observed in emulsions and granular media.
Fig. 9. Crystalline structure of bubbles in 3D: (a) A photograph of the surface of the foam. (b) A ray tracing simulation of a (111) fcc packing shows detailed agreement with the experiment.88
2.3. Drainage Drainage is closely analogous to sedimentation of suspensions, gas transport in fluidised beds, and water transport in wet soils. The liquid therefore passes through an essentially static structure, in the manner of liquid transport in a porous medium, but with an important difference. The local liquid
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fraction is not fixed but is to be determined to be consistent with liquid pressures: the foam breathes liquid in and out, with a change in Plateau border cross-sectional area. A simple dependence of rate of uniform flow (under gravity) on liquid fraction is often found experimentally: the flowrate Q goes as φ2 . This is rationalised in terms of Poiseuille flow, a consequence of high surface viscosity, so that the Plateau borders act as channels with a no-flow condition at their surface. Some surfactants produce a low surface viscosity instead, and a different theory is needed.52,53,91,94 The non-uniform flow through the foam can be described by the partial differential equation ∂φ ∂ 1 ∂φk+1/2 k+1 = −φ (3) ∂τ ∂ξ 2k + 1 ∂ξ where the parameter k characterizes the type of flow (φ = 1 for Poiseuille flow and φ = 1/2 for plug flow). Under forced drainage, the wetting front takes the form of a solitary wave.98 There is some connection with wetting fronts in soil mechanics, a branch of granular materials. In that field also, solitary waves describe wetting and are the solutions of kinematic equations, of a somewhat more complicated and uncertain form. Furthermore a convective instability appears at high enough flowrate.45 This seems roughly similar to instabilities in fluidized beds, but has a simpler form. It can also be compared to the convection rolls that appear in a shaken granular media above a certain threshold of acceleration.75 In the unstable regime there is size segregation in polydisperse foams, with smaller bubbles tending to collect at the bottom.46 This seems closely analogous to similar effects in granular materials.18 The fact that the convective instability occurs at a lower flow rate when the drainage column is tilted24 can be seen as analogue to the Boycott effect in sedimentation of suspensions.11,31 3. Dynamic properties 3.1. Rheology Continuum model As in the case of drainage, rheology has been treated with continuum models (see e.g64,93 ), most recently to analyse 2D shear experiments (see 3.2 and23,49 ). Instead of a detailed simulation, a coarse-grained description proceeds in terms of fields that represent the local average of liquid fraction, velocity, etc. Foam is a solid with well-defined elastic properties for
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low stress, but plastic at higher stresses, and it flows indefinitely above a certain yield stress. This central property is shared with granular material. Unlike the latter, foam has a very large elastic/plastic regime in terms of the strain range that may be explored before the yield stress is reached. There are simple formulae for elastic modulus and yield stress. In common with granular material, avalanches of topological changes can
Fig. 10. Early data44 showing avalanches of bubble rearrangements in wet foam due to applied shear. Left: Bubbles that change their nearest neighbour relationship are shaded. Right: The probability distribution of topological changes displays a long tail.
be observed (see Fig. 10), particularly in wet foams.44 There should also be force chains in the wet limit: they have been observed in simulations by Durian,34 but a method of detecting these experimentally has not been devised. Above the yield stress, i.e. once the foam flows, strain-rate dependent effects must be taken into account. They have proved difficult to capture convincingly in theory. The Bingham model is a useful heuristic device, in which the yield stress is added to a term proportional to the strain rate, as in a newtonian liquid. The effective viscosity can then be expressed as ηeff =
σ + ηp
˙
(4)
where σ is the shear stress, ˙ the strain rate and ηp the plastic yield. In reality, the introduction of a nonlinear viscous term (the HerschelBulkley model) seems necessary, but its validity and precise origins need to be explored. Note also that the Bingham model (eq. 4) applies only to shear strain that varies monotonically.
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HIIHFWLYH YLVFRVLW\ IHII
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HIIHFWRIFRDUVHQLQJ
%LQJKDPSODVWLF SODVWLFYLVFRVLW\
IS VWUDLQUDWH F Fig. 11.
Variation of effective viscosity with strain rate.
Soft-disk model In the case of a wet foam, another approach at the bubble scale can be useful. Douglas Durian investigated the rheological behaviour of a foam made of spherical bubblest by modelling their interaction with a heuristic model:33,34 the bubbles remain spherical but can overlap and repell each other elastically. This soft disk or sphere model is also the base of molecular dynamics simulations in granular materials and despite its simplicity has proven very efficient in reproducing the main feature of dry granular flows. However, the validity of this simple interaction model can be questioned.
ε
Fn
Fn
Fig. 12. Left: Elastic interaction between overlapping bubbles. Right: Topological rearrangement in a 2D foam under shear (reprinted figure with permission from.34 Copyright (1997) by the American Physical Society).
Theoretical calculations applied to special cases55,62,70 show that the true interactions are neither quadratic nor additive, even in the limit in which the bubbles are barely touching. In other words, the bubble-bubble potential
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depends on the confinement, and in particular the number of contacts of each involved bubble. More general cases (in which the environment of each bubble is not symmetrical) raise even greater problems, in any attempt to put soft-sphere models on a firm foundation for foams. Dissipation Dynamic properties require models that include dissipation, and the work of Durian brings these in straightforwardly by adding a viscous term to the spring force, which can only be accepted as a heuristic device. The true rate-dependent forces needed are very much a matter of debate, and include both hydrodynamic contributions that do not scale in a simple way (energy is dissipated due to shear flow of the viscous liquid within the soap films and Plateau borders), and surface contributions.17 In 2D there is another viscous force, of great importance. At least whenever there is at least one plate involved, wall drag is very important, and introduces effects not present in 3D foam, so the 2D model system must be an unreliable guide to its dynamic properties. What is the nature of this force ? This is the Bretherton problem,16 which has been subject to numerous investigations, experimental,20,87 theoretical28 and numerical80 (see also Fig. 13. Various power laws have been derived, rather than a force which is simply linear in the bubble velocity relative to the wall. This nonlinearity arises from the dependence of the local structure (in particular, the thickness of the film at the wall) on velocity.
Fig. 13. Left: Section of a bubble in a tube as modeled by Bretherton.16 Right: Deformation of the bubble and corresponding flow from numerical simulations.80
3.2. Shear banding Since the challenges of rheology and flow are so intractable, it is natural to have recourse to 2D model systems as a starting point. In the case of
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foams this has been the strategy of (among others) the groups of Glazier,1 Graner29 and Cantat.19 Amongst the main features, the phenomenon of shear-banding in sheared granular materials has been widely investigated.37,38,42,48,61 2D foam rheometers have been constructed by the groups of Debregeas27 and Dennin.57 In the first version, foam is sheared in a Couette (cylindrical) geometry (Fig. 14). It shows strong shear banding, localised at the inner boundary, with a velocity distribution decaying exponentially. The existence of a similar phenomenon in 3D is still subject to debate. Remaining in 2D, several explanations have been proposed. An early and initially convincing theory was based on a quasistatic description and detailed simulation;50 it is difficult to summarise, and remains problematical in several respects. A second and entirely different approach based on a continuum model
Fig. 14. From left to right: Experimental Couette setup for dry and wet foams; Exponential decrease of the tangential velocity across the gap (reprinted figures with permission from.27 Copyright (2001) by the American Physical Society); Prediction of the velocity profile by a continuum model.23
leads naturally to the shear banding in agreement with qualitative observation,23,49 and to several further results that seem to validate the model and have predictive power. Not only is this an essentially dynamic model, but it includes as its crucial ingredient the wall drag force which we have already pointed to as important in 2D. Apart from that the foam is modeled with the elementary Bingham model defined in section 3.1. The localisation length is determined solely by the viscous coefficient, together with the coefficient of wall drag, whereas neither coefficient appears in the previous theory. All is not clear yet, but we seem to be on the road to a full understanding. The comparison with granular experiments is intriguing (see the work of Behringer in this volume). In the latter case, much is similar. While a
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wealth of fine detail has been observed and analysed but the broader questions remain. Could the continuum model be relevant here? Experimental tests might help to resolve that question.
3.3. Dilatancy A recurrent theme in the description of granular materials is dilatancy, but it is often as vague as in Reynolds’ original and confusing accounts of it.74 It requires some clarification in seeking its analogue in foams. Very loosely, we may say that dilatancy is the tendency to expand when a material is sheared. But shearing may be static (elastic or partly plastic) or dynamic, above the yield stress. It would seem that we must at least distinguish elastic and dynamic dilatancy. In foams there is a quite well defined range of elastic distortion which can be well described by simulation and theory. It therefore offers the opportunity for an analysis of elastic dilatancy which presents little uncertainty. This was undertaken by Weaire and Hutzler,77,96 using 2D simulations. It should be noted that elastic dilatancy fits within the classical theory of elasticity, being merely a third-order effect; in general it is of arbitrary sign. Indeed it was found to be negative in the theory for dry 2D foams. The analogy with granular materials is to be found in wet foams: the effect rapidly becomes positive as liquid fraction increases. We may quantify the effect in a manner that is suggestive of possible experiments by asking: what is the difference of liquid fraction between two samples of which one is under static elastic shear, the other not, with contact between the two? The answer given was that the difference has a maximum of a few percent, 1 0.9 0.8
shear modulus G ( γ /A1/2 ) mu elastic / (gamma/root(A))
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Fig. 15. From left to right: Computer simulations of a two-dimensional foam with liquid fraction φ = 0.07 using the software PLAT:9 (a) unstrained; (b) under extensional strain = 0.23; Variation of shear modulus with liquid fraction.96
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halfway towards the wet limit (i.e. a liquid fraction of fifteen to twenty percent). Gravity complicates experimental design and the prediction is not as yet confirmed by measurement. Nothing has been predicted for dynamic dilatancy but it also seems a good candidate for measurement. In the case of granular materials, it is to be associated with the name of Bagnold,6 since his measurements were indeed for the dynamic case, and even produced a variation of dilatancy with shear rate. However, a recent critical review of this is skeptical.41 In parallel with other attempts to make sense of foams above the yield stress, this presents another opportunity for experiment and for comparison with granular materials. 4. Bubbles as soft grains ? Recently, Vanderwalle et al.7,21 revived the analogy between wet foams and granular materials by investigating bubble flows in the classical configuration of a 2D flow through a narrow aperture.8,40,67 To maintain a high liquid fraction in the foam it is necessary to avoid drainage and thus to keep the effective gravity low enough by confining the bubbles under a slightly inclined plate (Fig. 16). Within this setup, the small bubbles remain roughly spherical and move independently, thus behaving like deformable grains. Whereas in the case of granular media the grains undergo
Fig. 16. Experimental observation of the flow of bubbles through a narrow aperture (reprinted figures with permission from.7 Copyright (2006) by the American Physical Society).
solid friction and can therefore form contact arches, the bubbles experience only viscous forces. Thus Beverloo’s law giving the flow rate Q as a function
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of the particle size d and the aperture D reads differently:7 1/2 for solid grains g 1/2 (D − kd) Q∝ 1/2 3/2 D for bubbles (g sin θ) d −k
(5)
In Trinity College Dublin, another classical experiment for granular media has been reproduced with small bubbles: a monolayer of small bubbles is confined under a tilted rotating plate, thus forming a 2D rotating tumbler. This type of flow can be simulated by adapting Durian’s soft disk
Ω θ surfactant solution
Fig. 17. Bubbles in a rotating drum: experimental setup, snapshot of the bubbles during rotation, and numerical simulation.36
model and incorporating bubble inertia and wall friction. The viscous dissipation experienced by a single spherical bubble sliding along a plate has been investigated by Aussillous and Qu´er´e,5 who identified both Stokes and Bretherton components: Fd = aηV r + bκγ 1/3 (ηV )2/3 r2
(6)
where η is the dynamic viscosity of the bulk fluid, V the velocity of the bubble, κ−1 the capillary length and r the bubble radius. An important question is whether this formula, in particular the term for bulk friction, remains valid in the case of a system of bubbles. 5. Seeing inside foams (Computed Tomography) The dynamics and evolution of 2D foam structure are generally understood32,86,99 since the foam structure can be directly seen and dynamical processes (such as T1 and T2 transformations) are easier than the three dimensional ones to visualize and understand. In 3D, however, the challenge is to see and characterize the full inner structure of the foams which is not immediately visible from outside. Many techniques, primarily based on direct imaging, have been developed over the past few decades to probe the internal microstructure of complex materials.51,79 Direct measurement
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of the 3D structure of porous materials is now readily available from synchrotron and X-ray Computed Tomography (CT). These techniques provide the opportunity to measure experimentally the complex morphology of the microstructure of materials in three dimensions, in a non-invasive way, at resolutions down to a micron. One can then base calculations directly on the measured three-dimensional microstructure.2,3,26,63
Granular materials In recent years, X-ray CT has been employed in the field of granular materials to characterize the granular structure of single-sized hard spherical beads.2,3,76,81,82 This has allowed the researchers, for the first time, to investigate the static geometry of large packings of up to 150, 000 monosized hard spheres (See Fig. 18 and2 ). Attempts are under way to study the dynamics of the compaction process of elastic and deformable beads in threedimensions, a close analog to 3D wet foams, using X-ray tomography (see the paper of Saadatfar and others in this volume). This might shed light onto the understanding of topological changes of amorphous microstructured materials responding to applied stress, which bears close analogy with the deformation of 3D liquid foams (Fig. 18).
Fig. 18. Left: Reconstruction of a packing of 150,000 hard spheres obtained from tomography.2 Right: A 2D slice through the tomogram of a 3D packing of deformable rubber balls.
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Cellular Solids The physical and mechanical properties of cellular solids are a direct consequence of their complex microstructure. Linking properties to structure will lead to an understanding of how cellular solids can be optimised for given applications. This goal can be achieved by utilizing tomography to acquire the density map of the specimens (See Fig. 19 and78 ). Indeed, the manufacturing process of both open and closed cell foams are very similar to the packing of spheres in 3D as can be immediately deduced from figure 19.
Fig. 19. Images of open-cell aluminium foam (left) and close-cell polyurethane foam (right) obtained by tomography.78
Aqueous foams In the realm of aqueous foams, the first experimental investigation of a 3D analog of von Neumann’s law was carried out by means of optical tomography.68 The purpose of this experiment was to study foam structure and dynamics simultaneously by investigating morphology, topology, and dynamics of a 3D foam. Most recently X-ray tomography was used to study the evolution of initially 7000 bubbles in foams with liquid fraction φ between 0.1 and 0.2 percent.56 The main aim of such work is the determination of a growth law for bubbles, i.e. how/whether the number of faces of a bubble determines whether the bubble will shrink or grow during coarsening.
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6. Conclusions Drawing an analogy between foams and granular materials is certainly tempting and rewarding, and currently also en vogue, as exemplified by the publication of popular science books4,69 combining both themes (a collection of original articles on granular systems, foams, emulsions and supensions can be found in58 ). However, care needs to be taken when trying to pursue this analogy in great detail. The interactions between soft bubbles and hard grains are different, and in both cases not yet sufficiently understood. Idealised simulations and toy models have proved instructive, but the time has come to rebuild the foundations of the subject on more solid grounds. This is especially the case whenever dynamic effects are considered. 7. Acknowledgements Research is supported by ESA/ESTEC (14914/02/NL/SH and 14308/00/NL/SH) and Science Foundation Ireland (05/RFP/PHY0016). MS is supported by Irish Higher Education Authority, PRTLI, ITAC. DW thanks the ANU for hospitality during the period in which this review was prepared. References 1. Asipauskas, M., Aubouy, M., Glazier, J., Graner, F. and Jiang, Y. (2003). A texture tensor to quantify deformations: the example of two-dimensional flowing foams, Granular Matter 5, pp. 71–74. 2. Aste, T., Saadatfar, M. and Senden, T. (2005). Geometrical structure of disordered sphere packings, Phys. Rev. E 71, pp. 061302–061307. 3. Aste, T., Saadatfar, M. and Senden, T. (2006). Local and global relations between the number of contacts and density in monodisperse sphere packs, J. Stat. Mech. , p. P07010. 4. Aste, T. and Weaire, D. (2000). The pursuit of perfect packing (Institute of Physics Publishing, Bristol and Philadelphia). 5. Aussillous, P. and Qu´er´e, D. (2002). Bubbles creeping in a viscous liquid along a slightly inclined plane, Europhys. Lett. 59, pp. 370–376. 6. Bagnold, R. (1941). The physics of blown sand and desert dunes (Chapman and Hall, London). 7. Bertho, Y., Becco, C. and Vandewalle, N. (2006). Dense bubble flow in a silo: an unusual flow of a dispersed medium, Phys. Rev. E 73, p. 056309. 8. Beverloo, W., Leniger, H., and van de Velde, J. (1961). The flow of granular solids through orifices, Chem. Eng. Sci. 15, p. 260.
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9. Bolton, F. and Weaire, D. (1991). The effects of plateau borders in the twodimensional soap froth. i. decoration lemma and diffusion theorem. Phil. Mag. B 63, pp. 795–809. 10. Bolton, F. and Weaire, D. (1992). The effects of Plateau borders in the two-dimensional soap froth. I. General simulation and analysis of rigidity loss transition. Phil. Mag. B 65, pp. 473–487. 11. Boycott, A. (1920). Sedimentation of blood corpuscles, Nature 104, pp. 532–533. 12. Boyle, R. (1663). Experiments and Observations Touching Colours 1772, Works, Vol. 1 (Works, Rivington, Davis, etc., London). 13. Bragg, L. and Lomer, W. (1949). A dynamical model of a crystal structure. ii, Proc. Royal Soc. London Series A 196, pp. 171–181. 14. Bragg, L. and Nye, J. (1947). A dynamical model of a crystal structure, Proc. Royal Soc. London Series A 190, pp. 474–481. 15. Brakke, K. (1992). The surface evolver, Experimental Mathematics 1, http://www.susqu.edu/facstaff/b/brakke/evolver/. 16. Bretherton, F. (1961). The motion of long bubbles in tubes, J. Fluid Mech. 10, pp. 166–188. 17. Buzza, D., Lu, C.-Y. and Cates, M. (1995). Linear shear rheology of incompressible foams, J. Phys. II France 5, pp. 37–52. 18. Caglioti, E., Coniglio, A., Herrmann, H. J., Loreto, V. and Nicodemi, M. (1998). Segregation of granular mixtures in the presence of compaction, Europhys. Lett. 43, pp. 591–597. 19. Cantat, I. and Delannay, R. (2003). Dynamical transition induced by large bubbles in two-dimensional foam flows, Phys. Rev. E 67, p. 031501. 20. Cantat, I., Kern, N. and Delannay, R. (2004). Dissipation in foam flowing through narrow channels, Europhys. Lett. 65, pp. 726–732. 21. Caps, H., Traberlsi, S., Dorbolo, S. and Vandewalle, N. (2004). Bubble and granular flows: differences and similarities, Physica A 344, pp. 424–430. 22. Carrier, V., Hutzler, S. and Weaire, D. (2007). Drainage of bamboo foams, Coll. Surf. A , p. in press. 23. Clancy, R., Janiaud, E., Weaire, D. and Hutzler, S. (2006). The response of 2d foams to continuous applied shear in a couette rheometer, Eur. Phys. J. E 21, pp. 123–132. 24. Cox, S., Alonso, M., Weaire, D. and Hutzler, S. (2006). Drainage induced convection rolls in foams i: Convective bubble motion in a tilted tube, Eur. Phys. J. E 19, pp. 17–22. 25. da Vinci, L. (1506-1510). Codex Leicester. 26. Daerr, A. and Douady, S. (1999). Two types of avalanche behaviour in granular media, Nature 399, pp. 241–243. 27. Debr´egeas, G., Tabuteau, H. and di Meglio, J.-M. (2001). Deformation and flow of a two-dimensional foam under continuous shear, Phys. Rev. Lett. 87, p. 178305. 28. Denkov, N., Subramanian, V., Gurovich, D. and Lips, A. (2005). Wall slip and viscous dissipation in sheared foams: Effect of surface mobility, Colloids Surf. A 263, pp. 129–145.
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29. Dollet, B., Elias, F., Quilliet, C., Raufaste, C., Aubouy, M. and Graner, F. (2005). Two-dimensional flow of foam around an obstacle: Force measurements, Phys. Rev. E 71, p. 031403. 30. Drenckhan, W., Cox, S., Delaney, G., Holste, H., Weaire, D. and Kern, N. (2005). Rheology of ordered foams - on the way to discrete microfluidics, Colloids and Surfaces a-Physicochemical and Engineering Aspects 263, 1-3, pp. 52–64. 31. Duran, J. and Mazozi, T. (1999). Granular boycott effect: How to mix granulates, Phys. Rev. E 60, pp. 6199–6201. 32. Durand, M. and Stone, H. (2006). Relaxation time of the topological t1 process in a two-dimensional foam, Phys. Rev. Lett. 97, p. 226101. 33. Durian, D. (1995). Foam mechanics at the bubble scale, Phys. Rev. Lett. 75, pp. 4780–4783. 34. Durian, D. (1997). Bubble-scale model of foam mechanics: Melting, nonlinear behavior, and avalanches, Phys. Rev. E 55, pp. 1739–1751. 35. el Kader, A. A. and Earnshaw, J. C. (1999). Shear-induced changes in twodimensional foam, Phys. Rev. Lett. 82, pp. 2610 – 2613. 36. et al., V. L. (2007). in preparation . 37. Fenistein, D., van de Meent, J. and van Hecke, M. (2004). Universal and wide shear zones in granular bulk flow, Phys. Rev. Lett. 92, p. 094301. 38. Fenistein, D. and van Hecke, M. (2003). Wide shear zones in granular bulk flow, Nature 425, p. 256. 39. Glazier, J. and Weaire, D. (1992). The kinetics of cellular patterns, J. Phys.: Condens. Matter 4, pp. 1867–1894. 40. Hirshfeld, D., Radzyner, Y. and Rapaport, D. (1997). Molecular dynamics studies of granular flow through an aperture, Phys. Rev. E 56, p. 4404. 41. H¨ ohler, R. and Cohen-Addad, S. (2005). Rheology of liquid foam, J. Phys.: Condens. Matter 17, pp. 1041–1069. 42. Howell, D., Behringer, R. and Veje, C. (1999). Stress fluctuations in a 2d granular couette experiment: A critical transition, Phys. Rev. Lett. 82, p. 5241. 43. Hutzler, S., Pron, N., Weaire, D. and Drenckhan, W. (2004). The foam/emulsion analogy in structure and drainage, Eur. Phys. J. E 14, pp. 381–386. 44. Hutzler, S., Weaire, D. and Bolton, F. (1995). The effects of Plateau borders in the two-dimensional soap froth, III. Further results, Phil. Mag. B 71, p. 277. 45. Hutzler, S., Weaire, D. and Crawford, R. (1998). Convective instability in foam drainage, Europhys. Lett. 41. 46. Hutzler, S., Weaire, D. and Shah, S. (2000). Bubble sorting under forced drainage, Phil. Mag. Lett. 80, pp. 41–48. 47. Ishida, Y. (1976). in G. A. Chadwick and D. A. Smith (eds.), Grain boundary structure and properties (Academic press, London). 48. Jaeger, H., Nagel, S. and Behringer, R. (1996). Granular solids, liquids, and gases, Reviews of Modern Physics 68, pp. 1259–1273. 49. Janiaud, E., Weaire, D. and Hutzler, S. (2006). Two-dimensional foam rhe-
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ology with viscous drag, Phys. Rev. Lett. 93, p. 18303. 50. Kabla, A. and Debr´egeas, G. (2003). Local stress relaxation and shear banding in dry foam under shear, Phys. Rev. Lett. 90, p. 258303. 51. Kaveh, M., Mueller, R. and Iverson, R. (1979). Ultrasonic tomography based on perturbation solutions of the wave equation, Computer Graphics Image Processing 9, pp. 105–116. 52. Koehler, S., Hilgenfeldt, S. and Stone, H. (1999). Liquid flow through aqueous foams the node-dominated foam drainage equation, Phys. Rev. Lett. 82, pp. 4232 – 4235. 53. Koehler, S., Hilgenfeldt, S. and Stone, H. A. (2000). A generalized view of foam drainage: Experiment and theory, Langmuir 16, pp. 6327 –6341. 54. Kraynik, A., Reinelt, D. and van Swol, F. (2003). Structure of random monodisperse foam, Phys. Rev. E 67, 3, p. 031403. 55. Lacasse, M. D., Grest, G. S. and Levine, D. (1996). Deformation of small compressed droplets, Physical Review E 54, 5, pp. 5436–5446. 56. Lambert, J., Mokso, R., Cantat, I., P., R. Delannay, J. A. G. and Graner, F. (2007). Experimental growth law for bubbles in a wet 3d liquid foam, submitted http://arxiv.org/abs/cond-mat/0702685. 57. Lauridsen, J., Twardos, M. and Dennin, M. (2002). Shear-induced stress relaxation in a two-dimensional wet foam, Phys. Rev. Lett. 89, p. 098303. 58. Liu, A. and Nagel, S. (2001). Jamming and rheology (Taylor & Francis, London). 59. Lomer, W. (1949). A dynamical model of a crystal structure. iii, Proc. Royal Soc. London Series A 196, pp. 182–194. 60. Lomer, W. and Nye, J. (1952). A dynamical model of a crystal structure. iv. grain boundaries, Proc. Royal Soc. London Series A 212, pp. 576–584. 61. Losert, W., Bocquet, L., Lubensky, T. and Gollub, J. (2000). Particle dynamics in sheared granular matter, Phys. Rev. Lett. 85, pp. 1428–1432. 62. Mahadevan, L. and Pomeau, Y. (1999). Rolling droplets, Phys. Fluids 11, p. 2449. 63. Makse, H., Gland, N., Johnson, D. and Schwartz, L. (1999). Why effective medium theory fails in granular materials, Phys. Rev. Lett 83, pp. 5070– 5073. 64. Marmottant, P. and Graner, F. (2006). An elastic, plastic, viscous model for slow shear of a liquid foam, submitted http://arxiv.org/abs/condmat/0610261. 65. Matzke, E. (1946). Volume-shape relationships in variant foams. a further study of the role of surface forces in three-dimensional cell shape determination, Am. J. Botany 33, p. 58. 66. Miller, R. (ed.) (2007). EUFoam 2006 (Elsevier Science, Colloids and Surfaces A (in press)). 67. Mills, A., Day, S. and Parkes, S. (1996). Mechanics of the sandglass, Eur. J. Phys. 17, pp. 97–109. 68. Monnereau, C. and Vignes-Adler, M. (1997). Dynamics of 3d real foam coarsening, Phys. Rev. Lett. 80, pp. 5228–5231.
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69. Morsch, O. (2005). Sandburgen, Staus und Seifenblasen (Wiley-VCH, Weinheim). 70. Morse, D. and Witten, T. (1993). Droplet elasticity in weakly compressed emulsions, Europhysics Letters 22, p. 549. 71. Mysels, K., Shinoda, K. and Frankel, S. (1959). Soap Films (Pergamon Press, Paris). 72. P´eron, N., Cox, S., Hutzler, S. and Weaire, D. (2007). Steady drainage in emulsions: Corrections for surface plateau borders and a model for high aqueous volume fraction, Eur. Phys. J. E 22, pp. 341–351. 73. Plateau, A. (1873). Statique Exp´erimentale et Th´eorique des Liquides soumis aux seules Forces Mol´ eculaires (Gauthier-Villars, Paris). 74. Reynolds, O. (1886). Experiments showing dilatancy, a property of granular material, possibly connected with gravitation, Proc. Royal Institution of Great Britain Read February 12. 75. Ribi`ere, P., Philippe, P., Richard, P., Delannay, R. and Bideau, D. (2005). Slow compaction of granular systems, J. Phys.: Condens. Matter 17, pp. 2743–2754. 76. Richard, P., Philippe, P., Barbe, F., Bourl`es, S., Thibault, X. and Bideau, D. (2003). Analysis by x-ray microtomography of a granular packing undergoing compaction, Phys. Rev. E 68, 2, 020301. 77. Rioual, F., Hutzler, S. and Weaire, D. (2005). Elastic dilatancy in foams: a simple model, Coll. Surf. A 263, pp. 117–120. 78. Saadatfar, M., Arns, C., Knackstedt, M. and Senden, T. (2005). Mechanical and transport properties of polymeric foams derived from 3d images, Colloids and Surf. A 263, pp. 284–289. 79. Sakellariou, A., Sawkins, T., Senden, T. and Limaye, A. (2004). X-ray tomography for mesoscale physics applications, Physica A 339, pp. 152–158. 80. Saugey, A., Drenckhan, W. and Weaire, D. (2006). Wall slip of bubbles in foams, Phys. Fluids 18, p. 053101. 81. Sederman, A., Alexander, P. and Gladden, L. (2001). Structure of packed beds probed by magnetic resonance imaging, Powder Technology 117, p. 255. 82. Seidler, G., Martinez, G., Seeley, L., Kim, K., Behne, E., S.Zaranek, B. C. and Heald, S. (2000). Granule-by-granule reconstruction of a sandpile from x-ray microtomography data, Phys. Rev. E 62, p. 8175. 83. Smith, C. (1952). The shapes of metal grains, with some other metallurgical applications of topology, Metal Interfaces (ASM Cleveland) . 84. Smith, C. S. (1981). A search for structure (MIT Press, Cambridge). 85. Stavans, J. (1993). The evolution of cellular structures, Reports on Progress in Physics 56, pp. 733–789. 86. Stavans, J. and Glazier, J. A. (1989). Soap froth revisited: Dynamic scaling in the two-dimensional froth, Phys. Rev. Lett. 62, pp. 1318–1321. 87. Terriac, E., Etrillard, J. and Cantat, I. (2006). Viscous force exerted on a foam at a solid boundary: Influence of the liquid fraction and of the bubble size, Europhys. Lett. 74, pp. 909–915. 88. van der Net, A., Drenckhan, W., Weaire, D. and Hutzler, S. (2006). The
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crystal structure of bubbles in the wet foam limit, Soft Matter 2, 2, pp. 129–134. Vaz, M. and Fortes, M. A. (1997). Experiments on defect spreading in hexagonal foams, J. Phys. Condens. Matter 9, p. 8921. Vella, D. and Mahadevan, L. (2005). The Cheerios effect, Am. J. Phys. 73, pp. 817–825. Verbist, G., Weaire, D. and Kraynik, A. (1996). The foam drainage equation, J. Phys.: Condens. Matter 8, pp. 3715–3731. Vign`es-Adler, M. (ed.) (2005). Colloids and Surfaces A. Special issue: EUFoam 2004, Vol. 263 (Elsevier Science). Vincent-Bonnieu, S., H¨ ohler, R. and Cohen-Addad, S. (2006). A multiscale model for the slow viscoelastic response of liquid foams, submitted http://arxiv.org/abs/cond-mat/0609363. Weaire, D. (ed.) (1997). The Kelvin problem (Taylor and Francis, London). Weaire, D. and Hutzler, S. (1999). The Physics of Foams (Oxford University Press, Oxford). Weaire, D. and Hutzler, S. (2003). Dilatancy in liquid foams, Phil. Mag. 83, pp. 2747–2760. Weaire, D. and Kermode, J. (1984). Computer simulation of a twodimensional soap froth. ii. analysis of results, Phil. Mag. B 50, pp. 379–395. Weaire, D., Pittet, N., Hutzler, S. and Pardal, D. (1993). Steady-state drainage of an aqueous foam, Phys. Rev. Lett. 71, p. 2670. Weaire, D. and Rivier, N. (1984). Soap, cells and statistics: Random patterns in two dimensions, Contemp. Phys. 25, p. 59. Weaire, D., Vaz, M., Teixeira, P. and Fortes, M. (2007). Instabilities in liquid foams, Soft Matter 3, pp. 47–57. Zitha, P., Banhart, J. and Verbist, G. (eds.) (2000). EUFoam 2000: Foams, emulsions and their applications (Verlag MIT, Bremen, Germany).
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Chapter 2 Delaunay simplex analysis of the structure of equal sized sphere packings A.V. Anikeenko and N.N. Medvedev Institute of Chemical Kinetics and Combustion SB RAS, Russia T. Di Matteo, G.W. Delaney and T. Aste Department of Applied Mathematics, The Australian National University, 0200 Canberra, ACT, Australia We investigate the origin of the Bernal’s limiting density of 64% in volume fraction associated with the densest non-crystalline phase (random close packing limit) in equal sphere packings. To this end, we analyze equal sphere packings obtained both from experiments and numerical simulations by using a Delaunay simplexes decomposition. We show that the fraction of ‘quasi-perfect tetrahedra’ grows with the density up to a saturation fraction of ∼ 1/3 reached at the Bernal’s limit. Aggregate ‘polytetrahedral’ structures, made of quasi-perfect tetrahedra which share a common triangular face, reveal a clear sharp transition occurring at the density 0.646. These results are consistent with previous findings1 concerning numerical investigations.
1. Introduction Sphere packings have been used for centuries to model natural structures both at the atomic level and at macroscopic level.2 One of the main quests in these studies is to understand the nature of the transition between disordered and ordered-crystalline packings. It is known that for the densest packing of equal spheres, the fraction of volume occupied by the balls with respect to the total volume (packing fraction or density, ρ) is equal to ρ = √118 ≈ 0.74.2 Such maximal density can be realized in infinitely many ways, with two common examples being the face centered cubic lattice (fcc) and the hexagonal closed packed structure (hcp). All the maximally-dense packings are based on stacking close-packed 2-dimensional hexagonal layers 27
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of spheres. After the first and the second layers (a) and (b), there are two possibilities for the relative location of the third layer: it can be placed with the spheres in vertical correspondence with the ones of the first layer (a) or in an other position (c) that differs both from (a) and (b). A layered structure with maximal density can be built by stacking such layers in positions a, b or c with the only restriction that repetitions of two consecutive ‘letters’ (aa, bb, cc) must be avoided. The class of such maximally dense layered packings are known as Barlow packings; named after the mid 19th Century scientist who explored several possible stackings of spheres in an attempt to explain the atomic origin of crystal shapes. In terms of this ‘alphabet’ the hcp and fcc structures are the two simplest sequences being respectively: hcp = ababab... and fcc = abcabcabc... . They are the most common crystalline structures in atomic systems, like heavy metals, solid noble gases, and they are also commonly observed in colloids. However, these systems can also have non-crystalline phases. These are in general metastable states, and the atomic system will eventually relax into the crystalline phase, which is more stable from the thermodynamic point of view. On the other hand, particles of non-microscopic sizes (a-thermal systems with typical sizes above 50 μm, called under the general name of ‘granular materials’) reveal a strong tendency to avoid crystallization despite the fact that this is the most favorable state. To describe the nature of such non-crystalline packings and to understand the mechanisms that prevent crystallization is one of the major challenges in present day research on packings and granular materials. Empirical studies2–7 show that such packings can be produced at different densities in the rather broad range between the two limiting densities 0.55 (called random loose packing) and 0.64 (called random close packing). The fact that non-crystalline packings of equal spheres cannot be packed tighter than the limiting density of ∼ 0.64 was observed by J.D. Bernal in his experiments with steel balls.8 The microscopic origin of this bounding density is still unexplained. It is surprising that we still lack of a clear understanding of the structure of this disordered phase despite the relevance of non-crystalline packing problems to a broad range of applied and fundamental issues of scientific importance. Indeed, disorder is hard to classify as an equivalent to the ‘order parameter’ associated with structural properties is hard to identify. In disordered packings, each configuration is different from the others and the overall structure is an assembly made up of a very large number of different configurations that precisely match
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together as in a sort of jigsaw puzzle with a unique solution. On the other hand, these disordered packing are not completely random. Indeed, they present a very large number of repetitions: local configurations with very similar properties are found all over the packing, but such local ‘motifs’ are not identical and they are not positioned regularly. In these systems, traditional methods such as pair correlation function or the structure factor fail to give a clear characterization of the three-dimensional structural organization. Indeed, these are essentially one-dimensional measures that quantify the occurrence of characteristic lengths. Conversely, to study these threedimensional structures we must identify the three-dimensional ‘motifs’ in the atomic arrangements. To achieve this here we employ a description of the packing based on Delaunay simplexes. These simplexes are unambiguously and uniquely defined for any (regular or disordered) set of points in space. They define configurations of quadruples of ‘atoms’, and they are the simplest elements to which a three-dimensional packing can be reduced. Delaunay simplexes represent a mosaic covered space of a sample, so if we select simplexes with a given structural substance, the clusters of such simplexes give a design on the mosaic to reveal a structural motif.9,10 Recently Delaunay simplexes were applied in studying the structure of a large set of hard sphere packings at different densities.1 The paper considered the structure in terms of the content of the tetrahedral units (Delaunay simplexes of tetrahedral shape) and their local arrangements. Using a tetrahedrality criterion, derived from that used in Hales’ recent solution of the Kepler conjecture (which states that the densest possible packing of equal spheres occupies 0.74% of the volume as in the Barlow packings), it was observed that the volume fraction occupied by the tetrahedra increased with increasing density of the overall packing up to the Bernal limit. At this stage a distinct transition was observed, with the volume fraction occupied by clusters of tetrahedra (polytetrahedra) passing through a sharp maximum while the fraction of spheres involved in tetrahedra saturates. The position of this maximum was estimated at 0.646. In this work, we consider in further details the nature of this transition by comparing the numerical data with experimental observation. We confirm the polytetrahedral structure of disordered packings and retrieve the drastic behavior of the clusters of tetrahedra at the limiting density. Our results demonstrate that the structure of packings in physical experiments is very similar to the structure observed in ideal hard spheres packings.
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Delaunay simplexes
Fig. 1. Simplexes typical for dense packings of hard spheres: (a) perfect (it has T=0, dt = 0, δ = 0); (b) an example of ‘a boundary’ tetrahedral ∼ dq , δ ∼ 0.25); (c) perfect quartoctahedron (a quarter of T ∼ 0.018, dt √ with one edge 2 longer than the others, T = 0.05, dq = 0, dt = 0.179, δ text.
tetrahedron shape (with octahedron, = 0.41), see
The first step in the quest for a structural characterization is to identify local configurations and quantify their similarities and occurrence. There exists a general approach, used in geometry, which allows us to unambiguously select the closest quadruples for an arbitrary system of discrete points. Such an approach is the Voronoi - Delaunay tessellation (decomposition), well known both in physics11 and mathematics.12 This method exploits an evident geometrical fact that for each point in a set of points embedded in a given metric space it is always possible to distinguish the portion of space closest to such a point with respect to any other point in the set. This region is called the Voronoi polyhedron (cell, region) and the space-partition built from the assembly of all Voronoi cells is called the Voronoi tessellation or Voronoi diagram.12 For any Voronoi tessellation, there exists a dual tessellation called the Delaunay tessellation, which consists of Delaunay simplexes (irregular tetrahedra, in the general case)
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whose vertices are the quadruples of closest points in the set. The names of these constructions derive from the mathematicians that posed the mathematical foundations of the methods: G.F. Voronoi (1868-1908) explored in detail the properties of these tessellations by using analytical methods for lattice systems; whereas B.N. Delaunay (1890-1980) proved the correctness of Voronoi’s main theorems for points positioned at random in space.13–15 Shape characterization of simplexes In order to characterize the packing structure we first need to build a simple instrument to measure quantitatively the shape of each simplex. Several approaches have been suggested to characterize the proximity of a simplex to a perfect tetrahedron.9,16–19 In this paper we will discuss three different methods that embrace a significant range of possibilities. Edge differences, T -measure Let us start with a rather old and simple method in which the irregularity of the tetrahedron is quantified by summing over the average square of the simplex edge length differences9 1 2 T = (li − lj ) (1) 15l¯2 i<j
where li , lj are the lengths of the simplex edges, and ¯l is the mean edge length. In a perfect tetrahedron all edges have equal length and T is equal to zero. More generally, small values of T correspond to simplexes which are close to a perfect tetrahedron. Conversely, large values of T indicate significant deviations from regularity. We now want to identify a bound on the value of T which defines a class of tetrahedra that are regular enough and therefore can be considered ‘quasi-perfect tetrahedra’. In Refs.10,20 this measure was calibrated using the models of a fcc crystal at different temperatures (at different degrees of perturbation). It is known that this crystal structure (as any Barlow packing) can be reduced to a tiling with elementary tiles made of two perfect tetrahedra and one octahedron. At finite temperatures they are distorted, but as long as the crystal structure is retained the two main classes of Delaunay simplexes, tetrahedra and quartoctahedra (quarters of octahedra), are present (see Fig. 1). As a boundary value which identifies simplexes that are closer to perfect tetrahedra they choose Tb = 0.018. This value of Tb divides the two classes of simplexes in the calibrating models and is the
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value which we will use in this work. This can be compared to studies with a more physical viewpoint (see Refs.10,20 ), where the Delaunay simplexes, whose shape varies within the limits 0 ≤ T ≤ Tb , are associated with the tetrahedral configurations of atoms in heated fcc crystal. Procrustean distance, d-measure From the perspective of mathematical shape theory,21,22 the proximity of an arbitrary simplex to a given reference shape is estimated by the degree of coincidence upon their superposition. To this end, the total mean square deviation d2 between the corresponding vertices of the optimally superimposed simplexes can be calculated. The magnitude d is called the Procrustean distance between the two simplexes. Let {x1 , x2 , x3 , x4 } and {y1 , y2 , y3 , y4 } be the coordinates of the vertices of two simplexes. The square of Procrustes distance between such simplexes is: 4 1 2 2 d = minR,t,P ||yi − (Rxi + t)|| , (2) 4 i=1 where the minimum is calculated over all three-dimensional rotations R, the translations t, and all possible mappings between vertices of simplexes P . The measure d allows us to compare a simplex to any reference shape. For instance, it is possible to calculate the distance from a given simplex to both the perfect tetrahedron dt and to perfect quartoctahedron dq . According to this classification, a simplex can be considered as a ‘quasi-perfect tetrahedron’ if its Procrustean distance to a perfect tetrahedron is less than its distance to a perfect quartoctahedron, i.e. if the following condition is satisfied dt < dq .
(3)
For a given correspondence (mapping) between vertices, it is possible to calculate analytically the Procrustean distance and there are several algorithms to solve such least squares problems. For instance, one of the most efficient methods is based on computing the singular value decomposition of the derived matrix (see Ref.23 for details). Note that the Procrustean distances are a mathematically well defined distance measure, i.e. distance between equivalent simplexes is equal to zero, and the distance measured from simplex 1 to simplex 2 is the same as from 2 to 1. Thus the simplex with perfect tetrahedral shape has dt = 0, and the distance between perfect tetrahedron and quartoctahedron is equal to 0.17936.18
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Maximal edge length, δ-measure A very simple but effective way to determine how close an irregular simplex is to a perfect tetrahedron consists of calculating the length of the maximal edge emax . This method of selecting tetrahedral simplexes was used by Hales in his proof of the Kepler conjecture.19 This approach seems especially suitable for identical hard spheres of unit diameters, where the minimal possible length of the simplex edge is equal to 1. In this case, a value of emax close to unity unequivocally indicates that all edges are close to 1 and therefore the simplex is close to a regular tetrahedron. A convenient measure of the simplex shape is therefore the difference between the lengths of the maximal and the minimal edges: δ = emax −1.1 Small values of δ unambiguously indicate that the shape of the simplex is close to a perfect tetrahedron, while large values correspond to substantially distorted shapes. In the proof of the Kepler conjecture, Hales choose the maximal edge length 1.255 as the upper bound for ‘quasi-perfect tetrahedra’. In our notation this corresponds to δ = 0.255. It is important to remark that the δ-measure is strictly related to the two previous measures. One can verify that for dense packings of hard spheres all these measures pick practically the same tetrahedral simplexes. For disordered packing at density 0.64 we estimated that the conditions T < 0.018 and δ < 0.255 select the same simplexes with an overlap of 95%, and the coincidence rate increases with the onset of crystallization. Thus each of the measures reliably picks tetrahedra with shapes close to perfection. Some ambiguity is observed only for simplexes with boundary shapes, which are not critical for our analysis. 2. Models Computer simulations of sphere packings We study a large series of hard sphere packings with packing densities ranging from 0.53 to 0.71. Each packing contains 10000 hard spheres of equal radii in a box with periodic boundary conditions. The majority of the packings (more than 200) were obtained using a modified Jodrey-Torey algorithm that employs “repulsion” of overlapping spheres with gradual reduction of their radii.24–26 The initial configuration is a set of identical spheres uniformly distributed in the box. Overlapping of spheres is permitted at this stage. The algorithm attempts to reduce overlaps between spheres by shifting overlapping spheres and gradually shrinking of the radii. This is continued until all overlaps vanish. This algorithm can easily pro-
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duce not only disordered packings with densities up to the limiting value, but also more dense systems containing crystalline structures. It can lead easily to packings of a density of around 0.66. For higher densities, the result of an earlier run is used as a starting configuration, where the diameters are enlarged. This procedure can be repeated several times. In order to test the independence of structure on the algorithms used for packing generation we also computed a series of packings (about 70) in the range of densities from 0.54 to 0.67 by using the Lubachevsky-Stillenger algorithm.27 This algorithm employs a different procedure for densification of the packing. The initial configuration of spheres in this case is also random, however no sphere overlaps are permitted. The simulation then proceeds via event driven Newtonian dynamics in which the spheres are considered perfectly elastic. The radii of the spheres are gradually increased until a final “jammed configuration” is obtained. A principal control parameter of this algorithm is the growth rate for sphere radii. Small values of growth rates will result in crystallization as it is well-known for “physical” simulations of hard spheres.28,29 To avoid crystallization the growth rate should be rather large, forcing the packing into “jammed” non-crystalline structures.30,31 Physical experiments with sphere packings We tested the numerical results over a set of 6 experiments from a database of sphere packings obtained by X-ray Computed Tomography of large samples of disorderly packed mono-sized spheres. The experimental technique and some results were presented in detail in.7,32,33 These studies are the largest and the most accurate empirical analysis of disordered packings ever performed. At present, the entire database collects the coordinates (with precision better than 1% of the sphere diameters) of more than three million spheres from 18 samples of monosized acrylic and glass spheres prepared in air and in fluidized beds. The sample densities range from 0.56 to 0.64. In this work we will refer to samples A (ρ ∼ 0.586), B (ρ ∼ 0.596), C (ρ ∼ 0.619), D (ρ ∼ 0.626), E (ρ ∼ 0.630) and F (ρ ∼ 0.640). These samples are composed of acrylic beads in air contained within a cylindrical container.7,32,33 The geometrical investigation of the packing structure was performed over a central region at 4 sphere-diameters away from the sample boundaries.
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3. Results Fraction of quasi-perfect tetrahedra For each packing from our set of models and experiments we calculate all the Delaunay simplexes and selected the quasi-perfect tetrahedra shapes. Fig. 2 shows how the fraction of such tetrahedra depends on the packing density. One can verify that the general behaviors are comparable for all three measures of shape described above. Only the Procrustean distance (dt < dq ) tends to overestimate the fraction of tetrahedra at low densities. Indeed, this criterion can pick rather distorted simplexes that are far away from perfect tetrahedron, but are even farther from perfect quartoctahedron. Note each point on a curve represents an independent packing. The good coincidence of points at similar densities illustrates the representativeness of our computer models and highlights the agreement between experimental and numerical results. The fraction of tetrahedra rapidly grows with increasing density, reaching about 30% when approaching the critical value ρ ∼ 0.646. Interestingly, further increase of the density has little effect on the fraction of tetrahedra. Note that the fraction of quasi-perfect tetrahedra at ρ ∼ 0.646 is close to 1/3, which corresponds to the fraction of perfect tetrahedra in the Barlow packings. Such a coincidence of the fraction of quasi-perfect tetrahedra with the ones in the densest crystalline structure deserves special attention, as this can shed light on the physical meaning of the class of quasi-regular tetrahedra. However the problem is not simple: the question is what is the maximum fraction of tetrahedra which can be present in a dense disordered packing of equal spheres. It seems reasonable to conjecture that the fraction of 1/3 is an upper bound. However, a recent work34 seems to suggest that some classes of tetrahedral packings might reach larger fractions. Note also the body centered cubic (bcc) crystal consists of Delaunay simplexes which are all quasi-regular tetrahedra according to our criteria (Tbcc = 0.011, δbcc = 0.15). The fraction of tetrahedra depends also on the softness of the spheres. For instance, we have found that up to 40% of the tetrahedrons in a Lennard-Jones glass can be of the concerned quality. Fraction of polytetrahedral aggregates We have established that in disordered packings of equal sized spheres there is a rather large fraction of quasi-perfect tetrahedra which increases during densification and reaches a plateau around 30% when the limiting density is overcome. We now want to understand how these quasi-perfect tetrahe-
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Fig. 2. Fraction of Delaunay simplexes with tetrahedral shape as a function of packing density. Different curves correspond to different methods for selecting tetrahedra. From top to bottom: dt < dq , δ < 0.255, T < 0.018. Large symbols correspond to experiments. For simplification of the picture, only two methods are shown: dt < dq (large circles) and δ < 0.255 (large triangles). Vertical line marks the limiting density η = 0.646. Horizontal line marks the value of 1/3 that corresponds to the fraction of tetrahedra in the densest crystals.
dra can aggregate in more complex structures. We consider clusters built from three or more face-adjacent quasi-perfect tetrahedra and we call such structures polytetrahedra.1,35 Isolated tetrahedra and pairs of tetrahedra (bipyramids) are omitted as they are found in fcc and hcp structures. We can associate a graph to such polytetrahedra aggregates. In such a graph a vertex represents the centre of a quasi-perfect tetrahedron and a segment between two vertices is inserted when two quasi-perfect tetrahedra are sharing a face. Fig. 3 shows some of these graphs for local sphere packing configurations. In general, such graphs have the form of branching chains and five-edges cycles which combine in various “animals”.10,35 Mathematically speaking such a presentation of clusters of the selected Delaunay simplexes is called site-coloring on the Voronoi network . Indeed, because of the duality of the Delaunay and Voronoi tessellations, the center of any Delaunay simplex is a vertex of the Voronoi network, and a common
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Fig. 3. Examples of polytetrahedral aggregates (clusters of face-adjacent tetrahedra). a) three tetrahedra, b) a ring of five tetrahedra, c) a typical cluster for a dense disordered packing. The lower row shows the motives of the tetrahedra in clusters: the points mark the centers of tetrahedra and the lines indicate that they are adjacent through a common face. For cluster c) a skeleton of the graphs is also shown (dead ends are cut off).
segment is a Voronoi network edge connecting the neighboring vertexes.9,11 For disordered packings at low density a rapid growth is also observed (See Fig. 4). Upon approaching the Bernal’s critical density, the fraction of polytetrahedral aggregates also account for about 30% of all Delaunay simplexes. However, after the critical density the fraction of the tetrahedra belonging to polytetrahedral aggregates sharply decreases. This is a consequence of the formation of crystalline nuclei. Fig. 4 clearly demonstrates the polytetrahedral nature of disordered hard sphere packings. Thus we can say the transition from a lower density to higher density packing occurs via increasing the fraction of quasiperfect tetrahedral configuration and their coalescence into polytetrahedral aggregates. At the limiting density ρ ∼ 0.646 the fraction of quasi-perfect tetrahedra reaches its maximum. Above this point the polytetrahedral aggregates get gradually disassembled. This is a result of changing of the densification mechanism, i.e. crystalline nuclei where only single tetrahedra and bipyramids begin to appear. A reason why the mechanism of densification can be changed was explained in.1 At this density all spheres in the packing have been involved into the formation of tetrahedra. So a process of densification by means of formation of polytetrahedral nuclei becomes
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Fig. 4. Fraction of tetrahedra which are also part of polytetrahedra aggreagates vs. packing density. The behavior of these curves is quite different from the curves in Fig. 2.
exhausted. Fig. 5 demonstrates spatial distribution of polytetrahedral clusters inside our samples at density 0.64. For simplification of the pictures only skeletons of the clusters (see Fig. 3) are shown. Thus, after elimination of lineal clusters and cutting off all dead ends of the clusters, we see mainly aggregates of five-member rings. This picture reveals a “5-symmetry nature” of the disordered packings discussed by Bernal in his work.8 Visual analysis of these clusters shows that they are rather irregular. Note there are no clusters like dodecahedron (twelve 5-member faces) which could correspond to icosahedral local configurations of spheres. This fact is an additional argument that “icosahedral local order” is not typical for disordered packings of identical atoms.36–38 Note that we do not observe practically any 6member rings, although our class of tetrahedra allows distortions of shape to organize such rings (e.g. a part of the Delaunay simplexes in the bcc structure are arranged in such rings). In disordered packings, 6-member rings of tetrahedra seem not to be preferable.
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Fig. 5. Illustration of spatial distribution of the polytetrahedral clusters in packings of hard spheres at 0.64 in computer (left) and mechanical models (right). To simplify pictures only skeletons of the polytetrahedra are shown, see Fig. 3. Tetrahedral simplexes are selected according to the measure T < 0.018.
4. Conclusion We performed shape analysis of Delaunay simplexes for dense packings of identical hard spheres in a wide range of densities. Particular attention was focused on tetrahedral configurations of spheres, which are the characteristic feature of all dense disordered packings of spherical particles. We confirm the polytetrahedral structure of disordered packings and the drastic behavior of clusters of tetrahedra at Bernal’s limiting density (0.646). In disordered packings the tetrahedra prefer to coalesce via their faces to form locally dense aggregates (polytetrahedra) of various morphology. The important properties of such aggregates are, on one hand, their rather high local density, and on the other hand their incompatibility with crystalline structures. (In crystals they contact at edges or are organized in bipyramids). The fraction of polytetrahedral aggregates grows with the density of the disordered packing. Upon reaching the limiting density all spheres of the packing become involved in construction of the tetrahedra. Any further increase in density within this “polytetrahedral” principle of packing at this point becomes impossible. To reach higher density the “crystalline” principle has to be engaged.
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To study the sharp structural transitions considered in this work it is important that the basic tetrahedra is not perfect, i.e. not all spheres of the considered tetrahedral configurations contact each other, and the gaps between the neighboring spheres may be as large as δ ∼ 25% of the diameter of the sphere. These tetrahedra coincide with the class of quasi-perfect tetrahedra introduced by Hales. We also compared other measures to select tetrahedra. Specifically, we used the rather old measure of tetrahedrality (summing over the average square of the simplex edge length differences) and the Procrustean distance from mathematical shape theory. All measures demonstrate similar efficiency, i.e. all of them select practically the same simplexes. The class of selected tetrahedra, from the physical viewpoint, is those Delaunay simplexes whose shape varies within the limits as the tetrahedral configurations in heated fcc crystal. It was demonstrated that the structure of packings in experiments is very similar to the structure of ideal hard spheres with no friction. This seems to indicate that the structure of dense matter is determined first of all by impenetrability of atoms, and ultimately by geometric properties of the packings of non-overlapping spheres in three-dimensional space. Acknowledgements This work was supported by RFFI grants no.05-03-032647, no.06 03-43028 and partly from YS INTAS no. 04-83-3865 (to A.A.). TA and TDM acknowledge the partial support by the ARC discovery project DP0450292. References 1. A.V. Anikeenko, N.N. Medvedev, Polytetrahedral nature of the dense disordered packings of hard spheres Phys. Rev. Lett. 98 235504 (2007). 2. T. Aste and D. Weaire, The Pursuit of Perfect Packing, (Institute of Physics Publishing London 2000). 3. J. D. Bernal, A geometrical approach to the structure of liquids, Nature 183 141-7 (1959) . 4. J. D. Bernal and J. Mason, Co-ordination of Randomly Packed Spheres, Nature 188 910-911 (1960). 5. G. D. Scott, Radial Distribution of Random Close Packing of Equal Spheres, Nature 194 956-957 (1962). 6. G. Y. Onoda and E. G. Liniger, Random loose packings of uniform spheres and the dilatancy onset, Phys. Rev. Lett. 64 2727-2730 (1990).
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7. T. Aste, M. Saadatfar ,T.J. Senden, Geometrical structure of disordered sphere packings, Phys. Rev. E. 71 061302 (2005). 8. Bernal, J.D. The Bakerian lecture, 1962.The structure of liquids. Proc. R.Soc. Lond. A280 299-322 (1964). 9. Medvedev, N.N., Naberukhin Yu.I. Structure of simple liquids as a percolation problem on the Voronoi network, J.Phys.A:Math.Gen. 21L247-L252 (1988). 10. Y. I. Naberukhin and V. P. Voloshin and N. N. Medvedev. Geometrical analysis of the structure of simple liquids: percolation approach, Molecular Physics. 73(4) (1991) 917-936. 11. N. N. Medvedev. Voronoi-Delaunay method for non-crystalline structures. SB Russian Academy of Science, Novosibirsk. 2000 (in Russian). 12. Okabe A. Boots B., Sugihara K., Chiu, S. Spatial tessellations - concepts and applications of Voronoi diagrams Wiley 2000. 13. Voronoi, G. F. Nouvelles applications des parametres continus a la theorie des formes quadratiques. Deuxieme Memorie: Recherches sur les paralleloedres primitifs, J. Reine Angew. Math. 134 198-287 (1908); 136 67-181 (1909). 14. Delaunay, B. N. Sur la sphere vide. Proc. Math. Congr. Toronto Aug 11-16 1924. Univ. of Toronto press, Toronto, 1928, pp. 695-700. 15. Delaunay, B.N. Sur la sph`ere vide. A la memoire de Georges Voronoi. Izv. Akad. Nauk SSSR, Otd. Mat. i Estestv. nauk, No 7, pp. 793-800, 1934. (in Russian). 16. Kimura M., Yonezawa F., Nature of amorphous and liquid structyre computer simulation and statistical geometry, J.Non-Cryst.Solids 61-62 535 (1984). 17. R. M. Lynden-Bell and P. Debenedetti, Computational Investigation of Order, Structure, and Dynamics in Modified Water Models, J. Phys. Chem. B, 109(14) 6527 -6534 (2005). 18. A.V. Anikeenko, N.N. Medvedev and M.L. Gavrilova. Application of Procrustes Distance to Shape Analysis of Delaunay Simplexes. Proceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering 2006. Ed. B.Werner, IEEE Computer Society, 148-152 (2006). 19. Hales, T.C. Sphere packings, I. Discrete Comput. Geom. 17 1- 51 (1997). 20. Anikeenko A.V., Gavrilova M.L., Medvedev N.N. A Novel Delaunay Simplex Technique for Detection of Crystalline Nuclei in Dense Packings of Spheres, Lecture Notes in Computer Science (LNCS) 3480 816-826 (2005). 21. Dryden, I.L., Mardia, K., Statistical Shape Analysis (John Wiley & Sons, New York, 1998). 22. Kendall, D.G.; Barden, D.; Carne, T.K. Le, Huiling, Shape and shape theory, (Wiley, Chichester, 1999). 23. Umeyama S., Least-squares estimation of transformation parameters between two point patterns, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, NO. 4, April 1991, pp. 378-380. 24. Jodrey, W.S., Tory, E.M. Computer simulation of close random packing of equal spheres. Phys.Rev.A. 32, 2347-2351 (1985).
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25. Bezrukov, A., Bargiel, M. , Stoyan. D., Statistical Analysis of Simulated Random Packings of Spheres, Part. Past. Syst. Char. 19 111-118 (2002). 26. Lochmann, K., Anikeenko, A., Elsner, A., Medvedev, N., Stoyan, D. Statistical verification of crystallization in hard sphere packings under densification, Eur. Phys. J. B 53 67-76 (2006). 27. Skoge, M., Donev, A., Stillinger, F.H., Torquato, S. Packing hyperspheres in high-dimensional Euclidean spaces, Physical Review E 74 041127 (2006). 28. W. G. Hoover and F. H. Ree. Melting transition and communal entropy for hard spheres, J. Chem. Phys. 49 3609-3617 (1968). 29. Auer, S., Frenkel, D. Numerical prediction of absolute crystallization rates in hard-sphere colloids, J.Chem.Phys. 120(6) 3015-3029 (2004). 30. Rintoul, M.D., Torquato, S. Hard-sphere statistics along the metastable amorphous branch, Phys. Rev. E. 58 532-537 (1998). 31. Torquato S., Truskett T.M., and Debenedetti P.G. Is random close packing of spheres well defined?, Phys.Rev.Lett 84 2064-2067 (2000). 32. Aste, M. Saadatfar , A. Sakellariou, T.J. Senden, Investigating the Geometrical Structure of Disordered Sphere Packings Physica A 339 16-23 (2004). 33. T. Aste, Variations around disordered closed packings, Journal of Physics Condensed Matter 17 S2361–S2390 (2005). 34. J. H. Conway, and S. Torquato, Packing, tiling, and covering with tetrahedra PNAS 103 10612-10617 (2006). 35. Finney, J.L., Wallace, J. Interstice correlation functions: a new, sensitive characterization of non-crystalline packed structures, J. Non-Cryst. Solids. 43 165-187 (1981). 36. Frank, F.C., Supercooling of liquids. Proc. R. Soc. London. A 215 43-46 (1952). 37. Spaepen, F. Five-fold symmetry in liquids, Nature 408 781-782 (2000). 38. Ganesh, P. and Widom, M. Signature of nearly icosahedral structures in liquid and supercooled liquid copper, Phys. Rev. B. 74 134205 (2006).
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Chapter 3 On entropic characterization of granular materials
Raphael Blumenfeld Earth Science & Engineering, Imperial College, London SW7 2AZ, UK and Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK This chapter presents recent developments in entropic characterization of granular materials. The advantages of the formalism and its use are illustrated for calculation of structural characteristics, such as porosity fluctuations and the throat size distribution. I discuss the relations between the entropic formalism and stress transmission. It is argued that a new sub-ensemble of loading distributions is necessary, which introduces a tensor temperature-like quantity named angoricity.
1. Introduction: the entropic formalism The introduction of statistical mechanical methods to analyse assemblies of granular systems has led to the development of new concepts in the field. Most notable are the concepts of compactivity, the analogue of temperature, and of a volume function, the analogue of the Hamiltonian. The approach is based on a description of the entropy of the granular structure, namely, the statistics of configurations that a collection of grains can assume, given that they are confined to a container of a given volume V . This approach, developed originally by Edwards and collaborators,1 has been prompted by experimental observations2 of a reversible behaviour of post-vibrated granular beds (albeit with intriguing irreversible precursors). The reversibility, and more importantly the reproducibility of measurements of bulk properties, suggest the existence of an ensemble of equilibrium-like configuration states. Further support for the approach has come from numerical simulations.3 The formalism has many parallels with conventional statistical mechanics 43
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of thermodynamic systems. A central concept is a partition function Z = e−W({q})/X Θ ({q}) D{q} ≡ e−Y /X ,
(1)
that depends on: {q} - a complete set of degrees of freedom (DOF); W - a volume function that is the analogue of the Hamiltonian in thermodynamic systems and which depends on the DOF; Θ ({q}) - a probability density that describes the statistics of the DOF and imposes the constraint that the structure remain connected. Once these constraints, embodied in the form of δ-functions, are satisfied, the function Θ can be regarded as the conventional density of states. The scalar X ≡ 1/β, named compactivity,1 is the analogue of the temperature. The analogue of the free energy, named the free porosity, is Y = − ln Z/β 4 .5 The configurational entropy is S = β 2 ∂Y /∂β and the mean porosity is V = Y + XS = ∂(ln Z)/∂(β). Using this formalism, many other parallels can be, and have been, made between thermodynamic systems and granular systems. In spite of experimental and numerical testimonials, the general applicability of this approach has been controversial. In particular, it has not been clear how the idea of volumetric entropy can be used to understand macroscopic properties of granular systems. Nevertheless, the prospect of harnessing the power of statistical mechanical methods to the difficult problem of granular systems has been very appealing. In addition to lingering skepticism, a significant hurdle to a regular use of the formalism has been the lack of a suitable explicit volume function W that is both rigorously additive when summed over all grains and convenient for analytical and computational purposes. As a result, several approximations have been used in the literature, leading to model-sensitive results. The form of the volume function is significant both because it is the vehicle for the derivation of explicit estimates of structural properties and because it identifies the phase space that defines the structure of granular systems. This problem has been resolved recently both in 2D4 and in 3D,5 with the introduction of a new partition of he granular space. In this description, grains are replaced by representative polyhedra (polygons in 2D), constructed from the contact network of the original grains. The grain polyhedra surround cell polyhedra that represent the pores of the original structure. This description makes it possible to tessellate the space by topologically identical units, called quadrons.5 The details of the constructions and the tessellations are explained in figures 1 (2D) and 2 (3D) and their captions. In terms of these, a volume function can be identified
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as a sum over quadron volumes6 W =
Vq .
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(2)
q
The quadrons are the fundamental volume elements. A z-coordinated grain in 3D (2D) can be regarded as a composite of several quadrons - z in 2D and 6(z − 2) in 3D, a picture reminiscent of quarks in elementary particles.
grain g 00cg 11
r
00cg 11
R
g 111111111111111111 000000000000000000 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111 000000000000000000 111111111111111111
loop c
Fig. 1. The geometric construction around grain g in 2D. The vectors r cg connect contact points clockwise around each grain anticlockwise around each void. The vectors cg connect from grain centres to loop centres. The shaded quadrilateral is the quadron R associated with cg.
This construction resolves the problems of disentangling the geometrical correlations and identifying the DOF that describe the structure4 .5 The polyhedra are defined by their edge vectors r. Structural correlations arise from irreducible loops, each giving a dependent vector. The number of these loops is straightforward to calculate in 2D, using Euler’s relation,9 but not in 3D. These calculations are shown below. Another advantage of this description is that it allows a local characterization of the structure by a fabric tensor, as described in4 (2D) and in510 (3D). The fabric tensors are useful both for quantifying the quadron volumes and for modelling stress transmission in granular assemblies.11 As such, these tensors are natural descriptors of granular systems. 2. Calculations of volume-based structural properties Two dimensions Let V = W({q}) be the volume function of a system of N grains, described
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cgp 000000 111111 r111111 000000 000 111 000000 111111
111 000 000 111 000 111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 000 111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111
000000 111111 111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 11 00 000000 111111 000000 111111 000 111 000000 111111 00 11 000 111 000000 111111 00 11 000 111 000000 111111 00 11 000 111 000000 111111 00 11 000 111 000000 111111 00 11 000 111 00 11 000 111 00 11 0 1 000 111 0 1 00 11 0 1 000 111 0 1 00 11 000 111 00 11 000 111 00 11 000 111 00 11 Cell c 000 111 00 11 000 111 000 111
grain g
111 000 000 111 000 111 000 111 000 111 000 111 000 111 a
000 111 111 000 000 111 000 111 000 111 000 111 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 000 111 00 11 00 0011 11 00 0011 11 00 11 00 11 00 11 00 11 00 11 00 11
111 000 000 000111 111 000 111 000 111 000 111 000 111 000 111 000 111 000 111
1 0 1 0
1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111
1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111
1 0 0 1
1 0 0 1
1 0 0 1
c
111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111
1 0 0 1
d
00 11 00 11 111 000 00 11 00000 00 11 00011111 111 00 11 00000 11111 00 11 000 111 00 11 00000 11111 0 1 00 011 1 000 111 00 11 00 11 000 111 00 11 00 11 000 111 00 00 11 00011 111
b
Tetrahedron centroid
11 00 00 11
11 00 11 00
Cell centroid
111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111
11 00 00 11
11111 00000 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111
111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111
11 00 00 11
e
f
Fig. 2. The polyhedral representation in 3D. The edge vectors r cgp connect contact points around grains and the construction of a 3D quadron is shown in stages. The quadrons are non-convex octahedra (shown in f) that tessellate the space perfectly. Grain volumes are composites of quadrons.
by (2), which depends on a set of structural DOF {q}. The partition function is (3) Z = e−βW({q}) P({q})D{q} , where P({q}) is the density of states, i.e. the probability density of occurrence of particular configurations of {q}. Given a mean coordination number z¯, there are z¯N/2 contacts and z¯N vectors r. On using Euler relation, this gives z¯/2 irreducible loops, leaving z¯N DOF. Interestingly, this is also the number of quadrons and one can use the quadron volumes, Vq , to span the phase space. Absence of data for P({q}) led to several approximations4,5 but a recent study of it12 has revealed complex structure. In foam-like structure P({q}) is fitted well by a gamma distribution, P(Vq ) =
ba Vqa−1 e−bVq γ(a, bVmax ) − γ(a, bVmin )
;
Vmin ≤ Vq ≤ Vmax ,
(4)
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where Vmax 1/b, b is the inverse of a typical quadron volume and 3 < a < 4 and γ(a, A) is the incomplete gamma function.13 An ‘ideal quadron-gas approximation’ of uncorrelated quadrons gives z¯N Vmax
Z=
P(Vq )e−βVq dVq
.
(5)
Vmin
Any volume-based structural property can be computed directly from (5). Three dimensions Euler relation is insufficient to determine the number of DOF in 3D. For illustration, consider foam-like structures, where every grain has z = 4 and is represented by a tetrahedron.5 Seen from within a pore, the ‘surface’ of a cell is made of the triangular faces of the ncg grains surrounding it, of the contacts between the triangles and of the ncf faces that the triangles enclose. The latter are the ‘throats’ that connect to neighbour cells. There are 3ncg /2 contacts and 3ncg edges on the surface, which, on using Euler relation, gives that a cell has ncf = 2 + ncg /2 throats. Summing over all cells, remembering that every throat is shared between two cells, gives that there are Nf = N + Nc throats, or 2(1 + N/Nc ) throats per cell. Every polyhedron edge is a 3D vector. The polyhedron of a z-coordinated grain has 2(z − 2) triangular faces and 3(z − 2) edge vectors. The interdependency of all the edge vectors is due to geometric correlation. As in 2D, every irreducible loop obviates one vector. In foam-like structures, three of the six edge vectors of every tetrhedron are dependent. Additionally, each of the N + Nc throats introduces a dependent vector. Therefore, there are in total 6N − 3N − Nf = 2N − Nc independent vectors, or 3(2N − Nc ) Nc Ndof = =3 2− (6) N N N DOF per grain. Unlike in 2D, this value is significantly lower than the number of quadrons per grain. The ratio Nc /N is a key quantity. For infinitely rough convex particles, it can be bounded by using the dual structure. In this structure, the duals of the grain polyhedra are the cell polyhedra and visa versa. The contact surfaces between the dual grains are the original throats. In 2D the dual of a statically determinate such structure is also statically determinate, but for frictionless grains, and it is plausible that the same applies in 3D. In general, the larger is z in the original system, the smaller the mean coordination number of the dual structure and visa versa. The lowest value that z¯ can assume is determined by the condition of mechanical equilibrium
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for infinitely rough grains, z¯ ≥ 4.14 The lowest mean coordination number for frictionless non-spherical grains is 12.14 This corresponds to the number of faces per cell in the original structure, zc = 2(N +Nc)/Nc , hence Nc /N ≤ 1/5. The opposite bound is obtained as follows. The highest packing density of identical smooth and rigid ellipsoids is achieved within a narrow range of aspect ratios where z¯ = 14.15 Polydispersity is expected to reduce z¯. This is supported numerically1617 and experimentally.18 Therefore, we get a low bound by setting zc = 14, Nc /N ≥ 1/6. Therefore, the mean number of grains surrounding a cell is between 20 and 24. Combining the bounds with relation (6), we get 27/5 ≤ Ndof /N ≤ 11/2 and the phase space is then Ndof = (5.45 ± 0.05)N -dimensional. Thus, the fraction of the quadrons needed to span the phase space is narrowly bounded between 9/20 and 11/24. Choosing these DOF to be uniformly distributed in space, reduces the correlations between them and makes the ideal quadron gas a better approximation to compute the partition function and volume-based structural properties.
3. Calculations of other structural properties Not all structural properties are convenient to compute with the quadrons as the DOF. An important example is the mean throat size, which is directly relevant to the permeability to fluid flow. The throat areas are difficult to express in terms of quadron volumes, but can be easily eritten in terms of the edge vectors r. From the considerations above, the number of DOF per throat is Ndof 33 3(2 − Nc /N ) 9 ≤ ≤ = 2 Nf 1 + Nc /N 7
(7)
and the mean number of edge vectors around a face is 5≤
6N 36 6 ≤ . = Nf 1 + Nc /N 7
(8)
The computation of effective throat sizes is as follows. A throat is a nonplanar polygon made of n triangles. The effective area to flow of a throat between cells c and c , whose centroids are A and A , can be estimated cc by the polygon’s projection onto a plane perpendicular to the vector R extending from A to A (see figure 3). Letting the polygon corners be ρi ,
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cc and placing the origin at i = 0, 1, ..., n − 1, ri = ρ i − ρ i−1 , ρ i = ri × R ρ 0 , the area of the projected throat is Athroat =
n−2 n−1 1 ρi × ρ j , 2 i=1 j=i+1
(9)
giving the effective throat size in terms of the edge vectors. The mean throat size is then 1 Athroat e−βW({r}) P({r})D{r} . (10) Athroat = ZNf throats
Computing similarly expectation values of higher powers of the sum in (9), we can construct the distribution of throat sizes to any required accuracy using techniques developed for the moment problem. centre − pore c centre − pore c’ A ρi
Rcc’ A’ ri
ρ
i−1
Fig. 3. A non-planar throat of the polyhedral structure is a non-planar throat. The cc extends between the centroids of the cells c and c which the throat connects. vector R The effective area to flow is estimated as the projection of the polygon onto a plane cc . perperndicular to R
The extension of this calculation for realistic throat size distribution, where the grain shape distribution is taken into consideration, is straightforward in the entropic formalism and will be described elsewhere. Other structural properties can be calculated similarly, e.g. the total surface area between the solid and the pore space, useful for estimating heat exchange.
4. The entropic formalism and mechanical stresses The discussion about the statistics of granular systems in mechanical equilibrium is not complete without taking into consideration the mechanical
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stresses that keep them in such states. The incorporation of stresses explicitly into the entropic formalism is a recent project in our group.19 Of particular interest is stress transmission in isostatic materials. Why these idealized systems are useful to understanding general granular materials has been discussed in20 .21 Stresses in planar systems are governed by ∂i σij = gj
(i, j = x, y) ;
pxx σyy + pyy σxx = 2pxy σxy
(11)
with σxy = σyx and g including external and body forces. The rightmost is a constitutive stress-structure relation, whose parameters pij characterize local structure. Their values were initially proposed empirically22 and statistically1 and eventually they have been derived from first principles. That derivation also highlighted their geometric interpretation on the grain level.11 Equations (11) can be solved under simplifying assumptions208 and they have been analysed rigorously recently.23 All analyses confirm that localized source loads give rise in such media to force chains, in agreement with experiments24 and simulations.25 Here, I discuss the relations between these solutions and the entropic formalism. The statistical and pure mechanical descriptions have developed largely independently, but they must be related. Measurements of contact force magnitude with exponential-tailed distributions led to statistical-mechanical based explanations that are independent of the solutions of (11)2627 .28 Disregarding these solutions necessitated introduction of assumptions that weakened the models and led to much controversy. This point can be illustrated in jamming of slowly sheared granular systems. The microstructure changes continuously in response to forces until the systems jams. We can now solve for the stress field in the jammed state, but it is the very stress solution which affected the structural characteristics. Thus, a full statistical mechanical description must include both the structural information and the boundary loading. It is important to recall here two important differences between granular and thermal systems. Measurements in the latter are normally done on time scales that allow the system sufficient time to explore a sufficiently large number of states to render the data typical. Granular systems are in principle quenched in a given state, especially for stress measurements. The only statistics involved here could be through ensembles of systems. Another significant difference is that eqs. (11) are hyperbolic, which is the reason for the chain solutions. This means that fluctuations in boundary force loads can be felt far from the load source. For example, consider a granular material pressed within a container by a flat plate of area S with a
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force F . Since the boundary of a granular pack is never flat then the plate presses on protruding grains differently than on their neighbours. These locally elevated forces act as localized load sources and give rise to chains. If the typical distance between such sources along the boundary is larger than the scale of resolution, or interest, then the particular distribution of chains is more significant than the mean pressure F/S. This suggests that: (i) care should be taken in the specification of the boundary loads and (ii) that one cannot ignore these fluctuations in the analysis of the statistics. To address this issue we expand the phase space of DOF to include an ensemble of all the possible loads19 .28 Consider an ensemble of all possible systems in mechanical equilibrium with given volume V and total boundary stress Π. Divide this ensembles into two sub-ensembles. One, the Πensemble, consists of all possible configurations of the particles that occupy a given volume V under a boundary loading of a given spatial distribution of force loads on the boundary. The other sub-ensemble, the V-ensemble, consists of all the possible realizations of boundary forces that add up to the boundary stress Π, such that the particular configuration of particles is kept fixed. Note that these are in fact all the stresses that are confined to within the so-called yield surface for this configuration. The partition function can now be written as Z=
e−W({q},{f })/X0 −F ({q},{f })/(V Xij ) Θ ({q}, {f }) D{q}D{f } ,
(12)
where {f } are the DOF describing the boundary loads, e.g. by the forces on every boundary grain, F is the force moment (from which stresses are defined) and the function Θ is a product of δ-functions requiring both that the particles are in contact and that Newton’s equations are satisfied. The variable X0 is the compactivity, previously denoted by X, and Xij are the components of a tensorial analogue of temperature defined via derivatives of the entropy S with respect to the boundary stresses Xij = ∂S/∂Πij .
(13)
The tensor Xij has been named angoricity.28 The use of this formalism is currently explored in our group. In particular, we are looking into developing a Boltzmann equation, taking into consideration both the tensorial structural description and the newly discovered stress solutions into consideration.
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References 1. Edwards S.F., IMA Bulletin 25, 94 (1989); Edwards S.F. and Oakeshott R.B., Physica D 38, 88 (1989); ibid. 157, 1080 (1989); Mehta A. and Edwards S.F., Physica A 157, 1091 (1989); Edwards S.F. and Grinev D.V., Phys. Rev. E 58, 4758 (1998). 2. Knight J.B., Fandrich C.G., Lau C.N., Jaeger H.M. and Nagel S.R., Phys. Rev. E 51, 3957 (1995); Nowak E.R., Knight J.B., Povinelli M.L., Jaeger H.M. and Nagel S.R., Powder Technol. 94, 79 (1997); Nowak E.R., Knight J.B., Ben-Naim E., Jaeger H.M. and Nagel S.R., Phys. Rev. E 57, 1971 (1998). 3. Metzger P.T. and Donahue C.M., Phys. Rev. Lett. 94, 148001 (2005); Makse H.A. and Kurchan J., Nature 415, 614 (2002); Barrat A., Kurchan, J., Loreto V., and Sellitto M., Phys. Rev. Lett. 85, 5034 (2000); Ono I.K., O’Hern C.S., Durian D.J., Langer S.A., Liu A.J. and Nagel S.R., Phys. Rev. Lett. 89, 095703 (2002); Fierro A., Nicodemi M. and Coniglio A., Europhys. Lett., 59, 642 (2002); Coniglio A., Fierro A., Nicodemi M., Ciamarra M.P. and Tarzia M., J. Phys.: Condens. Matter 17, S2557 (2005). 4. Blumenfeld R. and Edwards S.F., Phys. Rev. Lett. 90, 114303 (2003). 5. Blumenfeld R. and Edwards S.F., Euro. Phys. J., E 19, 23 (2006). 6. Note that the quadrons may not tessellate space perfectly when cells are extremely non-convex, as discussed in.7 Such cell may occur only in the presence of body forces, otherwise they cannot be in mechanical equilibrium. Even so, the probability of such cells is quite low and these cases are disregarded here. 7. Ciamarra M.P., Comment on “Granular entropy: Explicit calculations for planar assemblies”, Phys. Rev. Lett., in print (2007); Blumenfeld R. and Edwards S.F., reply to comment on “Granular entropy: Explicit calculations for planar assemblies”, Phys. Rev. Lett., in print (2007). 8. Blumenfeld R., New J. Phys. 9 (2007) 160. 9. See e.g. Coxeterc H.M.S., Regular Polytopes (Dover, New York, 1973). 10. Blumenfeld R. and King P.R., Entropy-mediated structure-permeability relations in skeletal porous materials Water Resources Res., submitted. 11. Ball R.C. and Blumenfeld R., Phys. Rev. Lett. 88, 115505 (2002). 12. Frenkel G., Blumenfeld R., Grof Z., King P.R., The structure, entropy and statistics of 2D granular systems, Phys. Rev. E., submitted. 13. Gradshteyn I.S. and Ryzhik I.M., Tables of Integrals, series, and Products, (Academic Press, San Diego 1979). 14. Ball R.C., in Structures and Dynamics of Materials in the Mesoscopic Domain, eds. M. Lal, R.A. Mashelkar, B.D. Kulkarni, and V.M. Naik (Imperial College Press, London 1999) 15. Donev A., Stillinger F.H., Chaikin P.M. and Torquato S., Phys. Rev. Lett., 92, 255506, (2004). 16. Aste T. and Weaire D., The Pursuit of Perfect Packing (Institute of Physics, Bristol, 2000) 17. Kraynik A.M., Reinelt D.A. and van Swol F., Phys. rev. E 67, 031403 (2003). 18. Matzke E.B., Am. J. Botany 33, 58 (1946).
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19. Blumenfeld R. and Edwards S.F., Compactivity and angoricity: granular stress statistics, in preparation. 20. Blumenfeld R., Phys. Rev. Lett. 93, 108301 (2004). 21. Blumenfeld R., in IMA Volume in Mathematics and its Applications 141 on Modeling of Soft Matter, eds. Maria-Carme T. Calderer and Eugene M. Terentjev, pp 235 (Springer-Verlag 2005). 22. Bouchaud J.P., Cates M.E. and Claudin P.J., J. Phys. II (France) 5, 639 (1995); Wittmer J.P., Claudin P., Cates M.E. and Bouchaud J.P., Nature 382, 336 (1996); Cates M.E., Wittmer J.P., Bouchaud J.P. and Claudin P., Phys. Rev. Lett. 81, 1841 (1998). 23. Gerritsen M., Kreiss G and Blumenfeld R., Stress chain solutions in twodimensional isostatic granular systems: fabric-dependent paths, leakage and branching, Phys. rev. Lett., submitted. 24. Liu C., Nagel S.R., Schecter D.A., Coppersmith S.N., Majumdar S., Narayan O. and Witten T.A., Science 269, 513 (1995); Vanel L., Howell D., Clark D., Behringer R.P. and Clement E., Phys. Rev. E 60 R5040 (1999). 25. Liffman K., Chan D.Y. and Hughes B.D., Powder Technol. 72, 225 (1992); Melrose J.R. and Ball R.C., Europhys. Lett. 32, 535 (1995). 26. Lovoll, Maloy K.J., and Flekkoy E.G., Phys. Rev. E , 5872 (1999); Corwin E.I., Jaeger H.M. and Nagel S.R., Nature 435, 1075 (2005); Coppersmith S.N., Liu C.H., Majumdar S., Narayan O. and Witten T.A., Phys. Rev. E 53, 4673 (1996); Rottler J. and Robbins M.O., Phys. Rev. Lett. 89, 195501 (2002); Majmudar T.S. and Behringer R.P., Nature 435, 1079 (2005). 27. Edwards S.F. and Grinev D.V., Granular Matter 4, 147 (2003). 28. Edwards S.F. and Blumenfeld R., in Physics of Granular Materials, ed. A. Mehta (Cambridge University Press, Cambridge 2007).
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Chapter 4 Mathematical modeling of granular flow-slides
Ioannis Vardoulakis and Sotirios Alevizos Department of Mechanics, Faculty of Applied Mathematics and Physics, National Technical University of Athens, Greece
In these lecture notes, we focus on the mathematical modeling of gravity-driven flows down a planar incline. The corresponding continuum theory is mostly inspired by the scientific tradition of openchannel hydraulics; hence a brief outline of this background is given as an introduction. In the second part, based on a recent publication of the authors, the pertinent balance laws are extended from Newtonian fluids to granular flows at moderate velocities. Finally, linear stability analysis techniques are presented as a tool for evaluating the regime of validity of the constitutive equations and the influence of the main open parameters of these models.
1. Introduction Granular materials have an obvious discrete nature. The transition from the discrete to the continuum constitutes an old-standing, open problem. Here we will try to follow an eclectic approach, harvesting useful ideas from both ends, without any a priori claim that the text is a rigorous mathematical treatise. Granular matter exists in various forms: solid, fluid and gaseous. Under gravity granular materials consolidate into a solid bed, characterized by quasi-permanent contacts among the grains. If loaded quasi-statically such granular packings behave mostly like solids. In a solid phase granular materials can be packed in various ways, and their porosity varies between a minimum- and a maximum value. Loosely packed, dry sands however may undergo a phase transition and liquefy
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quasi-statically; if their fabric has no “time” to adjust to the applied deformation rate (cf. di Prisco et al., 2000, Radjai & Roux 2002). Liquefaction of granular media can be triggered also by the action of the interstitial fluid (cf. Vardoulakis 1996 I & II, Vardoulakis 2004 I & II). Gravitationally driven granular flows lead also to fluidization of granular matter in places where grains slide and collide among each other. In such a fluidized state the porosity of the granular assembly is not substantially higher than the one controlling the loosest solid packing. However the grains in such states gain in mobility and exchange momentum through collisions. At higher kinetic energies, grains loose their permanent contacts and move in space very much like ultra dense gas molecules. Here we focus only on some aspects of gravity-driven flows down a planar incline. Since the corresponding continuum theory is mostly inspired by the scientific tradition of open-channel hydraulics (cf. Roberts, 1994), a brief outline of this background is used as an introduction to the main subjects touched upon in these lectures. 2. The continuum assumption Consider a granular material body, whose mass density ρ we want to determine. For example, we want to determine the density distribution of a natural soil deposit that may vary with depth z (Fig. 2). For the determination of the density at any given depth we choose first a 'sampling length' L around the sampling point at depth z . Secondly we measure the mass m ( z , L ) of the sample (with length L and crosssection A ) and finally, we compute the “average” mass density from the formula < ρ >=
m( z , L) LA
(2.1)
As can be seen from Fig. 2, the average density, as computed from Eq. (2.1), will be a function not only of position but also of the sampling length, < ρ > = ρˆ ( z , L) (2.2)
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Fig. 1. Connection between mean density and the volume from which it is derived.1
Fig. 2. In natural soil deposits the density is usually varying linearly with depth. 1 cf. L. Prandtl and O.G. Tietjens, Fundamentals of Hydro- and Aerodynamics, Dover, 1934.
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Fig. 3. A cubical REV centered at point Ρ ( x , y , z ) .
In general, we will find that for 'small' sampling windows one finds itself in region (I) of granular (molecular) fluctuations, whereas for very large L we are in region (III) of macroscopic variations (Fig. 1). There is however an intermediate region (II) where the average density is not affected by the sampling length. For example if the density variation is indeed linear with z , (Fig. 2), then from the trapezoidal rule it follows
< ρ > z ,L = ρ ( z)
(2.3)
According to Euler's original proposition in continuum mechanics we introduce the concept of the material point. Geometrically the material point is mapped in ℝ 3 on to the geometric center Ρ ( x , y , z ) of an elementary cube with dimension L (Fig. 3). This elementary material cube is called the “Representative Elementary Volume” (REV, Fig. 3). According to the remarks made above, we assume that this cubical REV is sufficiently large so that granular fluctuations are smoothened out and sufficiently small so that macroscopic changes are not affecting the result. For example, in a granular material the minimum dimensions of the REV are set by the grain size Dg . It is found that an assembly of about 103 sand grains possesses enough variability, so that averages are representative (cf. Vardoulakis, 1977).
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Following the above general procedure, the average value of a quantity like the density over the REV is mapped, according to Eq. (2.3) onto the material point that is located at the center of the corresponding REV. The density of a continuous medium is in general neither constant in space nor in time, i.e. the scalar density is a function of position and time, ρ = ρ ( x, y, z , t ) .Within the frame of continuum mechanics we will assume that such a density distribution function is at least piecewise continuous and that discontinuities or 'shocks' may occur along discrete surfaces.
3. The motion To each material point we assigned above a point P in space, e.g. the center of gravity of the corresponding REV. As illustrated in the previous section, the most primitive mechanical property, assigned to a material point, is its mass density. Thus, the mass of the material point is infinitesimal and it is computed from the value of the density field at that point times the volume that the REV is occupying, dm = ρ dV . The second important primitive property that is assigned to the material point of a continuum is that of its velocity. We may again imagine that the velocity of the material point is some kind of average velocity of the “grains” that occupy the corresponding REV, P ( x, y , z ) → v = v
REV
(3.1)
This theoretical basis for the study of motion of material bodies is attributed again to Euler, who studied the motion of the material points of a fluid along the so-called streamlines. We recall that the streamlines of the particles of a fluid constitute a double infinity of curves in 3Dspace with the following property: the (average) particle velocity vector at a given point in space and time is tangential to the streamline that is passing through that point at that time. Based on the work of Euler2, Lagrange3 observed that, since we are dealing with a triple infinity of particles (in 3D-space), it is meaningful 2 3
Leonhard Euler 1707-1783. Louis de Lagrange 1736-1813.
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in some occasions to consider instead of the streamlines the so-called pathlines, i.e. the triple infinity of particle paths in space. Accordingly in continuum mechanics we usually distinguish among the two viewpoints, the "Lagrangian" or particulate- and the "Eulerian" or spatial description of motion. Here we will adopt the Eulerian description.
4. The material time derivative In the frame of the Eulerian description of the motion all quantities are seen as functions of the spatial co-ordinates of the material points at current time and of the time variable. For example in an 1D setting, the velocity of a particle P which at time t occupies the position x is given as a function of x and t , v = v E ( x, t )
(4.1)
where the superscript E stands for Eulerian description. At a following time instant, t ′ = t + ∆t , the considered particle P has moved to a neighboring position, say, x′ = x + ∆u , where the incremental displacement of the particle is given by the particle’s velocity at the “event” ( x, t ) times the elapsed time interval, ∆u ≈ v E ( x, t )∆t . With the velocity field given, Eq. (4.1), the velocity of the considered particle P at this new position is, v′ = v E ( x′, t ′) = v E ( x + ∆x, t + ∆t )
(4.2)
E
If we want to compute the acceleration a = a ( x, t ) of the particle P we have to compare the velocity v = v E ( x, t ) of the particle at its original position to its velocity v′′ = v E ( x′, t ′) at its new position, v′′ − v a≈ (4.3) ∆t or, according to Fig. 4,
a≈
1 ( ( v′ − v ) + ( v′′ − v′) ) ∆t
(4.4)
The first term in the parenthesis on the r.h.s. of Eq. (4.4) is computed as follows:
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61
Fig. 4. The Eulerian description of the velocity field in time-space and the computation of the “material” time derivative of the velocity.
v′ − v = v E ( x + ∆x, t ) − v E ( x, t ) ≈
∂v E ∆x ∂x
(4.5)
and the second term as
v′′ − v′ = v E ( x + ∆x, t + ∆t ) − v E ( x + ∆x, t ) ≈ v E ( x, t + ∆t ) − v E ( x, t ) ≈
∂v E ∆t ∂t
(4.6)
Thus,
a=
∂v E ∂v E E + v ∂t ∂x
(4.7)
In fluid mechanics the above expression for the particle acceleration in an Eulerian description of the motion is called the “material” time derivative of the velocity field, because it accounts for the changes in the velocity as it is experienced by an observer who moves together with the particle (along the life-line of the particle). The material time derivative is denoted usually with a superimposed dot:
vɺ =
∂v E ∂v E + vE ∂t ∂x
(4.8)
The material time derivative consists of the local term, (∂v / ∂t ) , and the convective term, v(∂v / ∂x) . The latter is non-linear, since it consists of the product of the velocity and its spatial derivative. In solid
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mechanics’ applications, the convective term is usually negligible and therefore, the Lagrangian and Eulerian description of the continuum are identical. However, in fluid mechanics’ and granular flow mechanics’ applications, convective terms are essential and cannot be neglected.
5. Mass storage in open channel flow We consider mass balance in the case of flow of water in an open channel. Accordingly, we define the following quantities (Fig. 5): ρ = const. : the density of water4; B = const. : the width of the channel; h( x, t ) : the height of water along the channel; v( x, t ) : the “heightaveraged” fluid velocity; Q = hv : the fluid-discharge per unit width; mɺ = ρ BQ : the total water mass-flow across a vertical section. The massinflow and mass-outflow of water for a control volume between two given sections at x = a and x = b is mɺ in = ρ BQ E (a, t ), mɺ out = ρ BQ E (b, t )
(5.1)
Conservation of mass for the considered control volume yields the wellknown storage equation as follows: Let ∆mQ be the net mass-influx of water during the time interval ∆t between two neighboring sections at x = a and x = b = a + ∆x :
∆mQ = (mɺ in − mɺ out )∆t = ( ρ BQ E ( x, t ) − ρ BQ E ( x + ∆x, t ) ) ∆t ⇒ ∆mQ = − ρ B∆Q∆t , ∆Q = Q E ( x + ∆x, t ) − Q E ( x, t ) ≈
∂Q E ( x, t ) (5.2) ∆x ∂x
Let on the other hand ∆mh be the water mass stored inside this control volume. Mass storage during the considered time interval will cause an elevation of the free water surface by ∆h . Thus, the mass stored inside the control volume is computed as, ∆mh = ρ B (b − a)∆h = ρ B∆x ( h E ( x, t + ∆t ) − h E ( x, t ) ) ⇒ ∆mh = ρ B∆h∆x , ∆h = h E ( x, t + ∆t ) − h E ( x, t ) ≈
∂h E ( x, t ) ∆t ∂t
(5.3)
4 Notice that water, in problems with a free surface, is considered as an incompressible substance.
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63
Fig. 5. Water slice in open-channel flow: Any deficit in outflow results in an increase of the flow-height inside the considered control volume.
Since no mass is generated or produced inside the considered control volume, the mass balance equation
∆mQ = ∆mh
(5.4)
is expressing mass conservation. From the mass balance equation (5.4) and Eqs. (5.2), and (5.3) we obtain the well- known storage equation of open channel flow, −
∂ ∂h ( hv) = ∂x ∂t
(5.5)
where h = h E ( x, t ) is the elevation of the free water surface from the bottom of the considered water sheet and v = v E ( x, t ) is the mean fluid velocity, computed from the fluid discharge as, v=Q h (5.6)
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6. St. Venant’s “shallow water theory”
Fig. 6. Left: Forces acting on a control volume of a horizontally moving water sheet. The pressure is practically hydrostatic. Right: Fluxes and forces on water slice of a horizontally moving water-sheet. Base friction is neglected.
The shallow-water theory is traced to St.Venant (1850, 1871). Its successful application to open channel hydraulics is extensively presented in standard textbooks (Whitham 1974, Julien 1998). In order to reproduce St. Venant’s “shallow water” theory in its simplest form, we consider again an one-dimensional motion of an incompressible fluidsheet. In order to evaluate the balance of linear momentum equation, we assume that water is an inviscid fluid; i.e. a fluid that is capable to carry a pressure field, p = p E ( x, z , t ) and no shear stresses (Fig. 6). Let z be the vertical coordinate (Fig. 6). We assume that all accelerations in z-direction are negligible ( vɺz → 0 ), thus leading to a hydrostatic pressure field, p = patm + ρ w g ( h E ( x, t ) − z )
(6.1)
where patm is the atmospheric pressure, ρ w is the density of water and g = 9.81m / s 2 is the gravity acceleration.
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65
According to Newton’s law, the global dynamic equation for an infinitesimal "slice" of water between the cross-sections at two arbitrary positions a = x and b = x + dx (Fig. 6), is identical to the requirement that the acceleration in the flow-direction of the total mass of the considered slice of fluid is directly linked to the total force acting on that slice Pa − Pb + P0 sin ϕ = ( ρ hdx)vɺ (6.2) where h E ( x ,t ) E
P = P ( x, t ) =
∫
2 1 ρ w g ( h E ( x, t ) ) 2
p E ( x, z, t )dz = patm h E ( x, t ) +
0
(6.3)
is the total pressure force acting on the face of a slice at position x , 2 1 Pa = P E (a, t ) = patm h E (a, t ) + ρ w g ( h E (a, t ) ) 2 ∂P E Pb = P E (b, t ) = P E (a, t ) + dx = ∂x x = a
= P E (a, t ) + patm
∂h E ∂x
dx + ρ w gh E (a, t ) x=a
∂h E ∂x
(6.4) dx
x=a
and P0 is the total air-pressure force acting on the free surface, P0 sin ϕ = patm dx sin ϕ
(6.5)
In Eq. (6.5) ϕ denotes the inclination angle of the free surface at x = a ,
sin ϕ ≈ tan ϕ =
∂h E ( x, t ) ∂x
(6.6) x =a
Eq. (6.2) with Eqs. (6.4) to (6.6) lead to the local form of the dynamic equation in the horizontal direction, − ρ w gh ( ∂h ∂x ) = ρ w hvɺ
(6.7)
By introducing the definition of the acceleration as the material time derivative of the velocity, Eq. (4.8), we obtain the following dynamic equation
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−g
∂h ∂v ∂v = +v ∂x ∂t ∂x
(6.8)
Accordingly the set of governing equations consists of the storage equation (5.5) and the dynamical equation (6.8), which we summarize below:
∂h ∂h ∂v +v +h =0 ∂t ∂x ∂x
(6.9)
∂v ∂v ∂h +v + g =0 ∂t ∂x ∂x
(6.10)
These equations describe the propagation of waves in shallow watersheets. These waves are called also gravity waves, since gravity is the only force that drives the particles back to their original position.
7. “Shallow-water” model of granular flows In the previous sections we presented a brief outline of the shallow-water theory as applied to open-channel hydraulics. The adaptation of the shallow-water theory to granular flows started with the works of Savage (1979) and Savage & Hutter (1989). In recent years interest on granular flows has been revived through a series of publications, originating mainly from the fluid-mechanics, granular physics communities (Ancey et al. 1996, 2000, 2001, Pouliquen 1999 a & b, Duady et al., 1999, Forterre & Pouliquen 2003, da Cruz et al. 2005), where extensive reference to the pertinent literature can be found. The rheology of granular media in flow conditions is discussed extensively in the literature (see for example, Drake & Shreve, 1986, Ancey & Evesque, 2000). In summary, across a flow-slide one can distinguish among three regions, as shown in Fig. 7: (i) At the bottom of the slide a solid-like layer is forming, with thickness hs , that is sometimes called the frictional boundary layer. Due to the frictional boundary condition, the granular material in this layer behaves like solid sand. By that we mean a phase of a granular medium, where the grain-contacts are semi-permanent. Thus, its
Mathematical modeling of granular flow-slides
67
shear deformation-rate is small, if compared with the deformationrate of the overlying layers. (ii) The main bulk of the granular flow with thickness h f takes place in the middle layer and is called the frictional-collisional regime. (iii) On the top of the slide another, usually thin, layer of gaseous granular matter is forming, which is called the collisional boundary layer. In layer (ii) the granular material is fluidized, whereas in layer (iii) it behaves like a granular gas. If the flow is relatively slow the thickness of the upper gaseous layer is small, if compared to the total thickness of the flow-slide and in a first approximation it is neglected,
hg << hs + h f
, h = hs + h f + hg
⇒ h ≈ hs + h f
(7.1)
Assumptions about the structure of granular flows are usually corroborated by numerical experiments, using granular flow-dynamics simulators. In most cases the computational simulation of an infinite incline is addressed by the use of 2D particles and periodic boundary conditions in the flow direction (cf. Marroquin et al. 2006). Such simulations are used for example to compute the average velocity profile, which is at present hardly measurable. In that sense the aforementioned different phases of the flow of granular media in “slow” or “rapid” motion are also “observed” in the numerical simulations. Indeed, near the base of the slide, the velocity assumes comparably small values and then it increases monotonically with the height, up to the surface layer, where the aforementioned strong fluctuations of the velocity take place. In the literature different velocity profiles are reported. Using DEM simulations, Silbert et al. (2003) computed profiles for constant dip angles ranging from concave up to convex, depending only on the micromechanical material parameters used. Using the PIV technique in surface flows in rotating cylinders Jain et al. (2002) reported the same complex behavior of the velocity profile. Finally, Ancey (2002) reported a variation of the profile with the dip angle. Due to these uncertainties we keep as fact that we are dealing with shear-flows and as a first approximation we assume a linear velocity profile.
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Fig. 7. The three regimes of the granular flow-slide after Ancey & Evesque (2000): (1) basal layer (solid phase); (2) core layer (fluidized phase); (3) surface layer (“vaporized” phase).
for : 0 ≤ z ≤ hs 0 z − hs vx = for : hs ≤ z ≤ h = hs + h f vx ( x, t ) z = h h f
(7.2)
The profile (7.2) yields that the velocity at the top of the flow-slide is double its height-average,
vh ( x, t ) = vx ( x, t ) z = h = 2 v( x, t )
(7.3)
If the effects of the convective terms are neglected in the height averaging procedure, the normal to the main flow direction component of the velocity is set proportional to the time derivative of flow-height. Furthermore its is assumed to vary also linearly as, z − hs ∂h f vz = (7.4) h f ∂t For the case of a linear velocity profile, Eq. (7.2), the height-averaged acceleration in the flow direction is, ∂v 4 ∂v 4 v 2 ∂h f vɺ = + v − (7.5) ∂t 3 ∂x 3 h f ∂x
Mathematical modeling of granular flow-slides
69
From the above result we notice that within the frame of a “shallowwater” theory of granular flows, the height-averaged acceleration has a more complex structure than the usual “material time derivative”, which appears in open channel hydraulics, Eq. (4.8). We also notice that in the above consideration the density of the medium is assumed to be constant across a flow-slide,
ρ ≈ const.
(7.6)
This means that the volume changes due to dilatancy/contractancy in the solid bed or due to changes in the mean distance among the grains in the flow layer are considered as being of secondary importance5.
8. Mass conservation in granular flows We consider a flow-slide along an infinite incline with constant dip and with an erosion/deposition sheet forming at its base. By neglecting the base consolidation we arrive to a simple model, as depicted in Fig. 8, where the considered cross sections are taken to be normal to the original track surface. For a segment of the fluidized sheet, mass-flux deficit, mass-storage and erosion/deposition are expressed as ∂h f ∂ + (h f v) = e (8.1) ∂t ∂x The term e = e E ( x, t ) on the r.h.s of Eq. (8.1) is called the erosion speed. The erosion speed is set positive ( e > 0 ), if erosion of the basal layer takes place (mass intake). Thus, erosion results in a decrease of the solid bed height,
∂hs = −e ∂t
(8.2)
We remark that the consideration of a “mass generation” term represents an essential difference between the mass balance equations for granular flows, Eqs. (8.1) and (8.2), and the one holding in open channel 5 The assumption of constant density seems to be reasonable in boundary value problems with a free surface.
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Fig. 8. Mass balance in a small segment between the flowing and the resting material sheets.
hydraulics, Eq. (6.9). Notably that in the present setting the velocity of the particles in the solid bed is assumed to be negligible.
9. The dynamic equation of granular flow As already mentioned, within the shallow-water approximation it is usually assumed that the acceleration of the flowing mass normal to the flow is negligible. This assumption results in a static equilibrium equation in z -direction for the corresponding normal stress, yielding that the normal to the track stress-component, σ zz , is geostatic,
σ zz ( z ) = ρ g (h − z ) cos β , (hs ≤ z ≤ hs + h f )
(9.1)
Dynamic equilibrium is considered only the flow direction. Thus, in x -direction and for an infinitesimal flow-slice of height h f we obtain (Fig. 9):
ρ h f vɺ = −
∂ (h f < σ xx >) + ρ h f g sin β − τ ∂x
(9.2)
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71
Fig. 9. Dynamics of a slice of flowing granular material on a planar track.
where < σ xx > denotes the flow-height averaged normal stress in flowdirection. In order to remove the statical indeterminacy, the mean normal stress in flow direction, < σ xx > , is set proportional to the normal stress at the base of the flow-slide, < σ xx > = Kσ
(9.3)
σ = σ zz (hs ) = ρ gh f cos β
(9.4)
where
Rheological granular-flow tests are performed on a slider with constant dip angle, ( β = const. ). In this case with Eq. (7.5) the dynamic equation (9.2) with (9.3) becomes, 2 ∂v 4 ∂v 4 v 2 ∂h f g ′ ∂ h f + v − + K = b ∂t 3 ∂x 3 h f ∂x h f ∂x 2
(9.5)
where g ′ = g cos β and
(9.6)
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τ b = g ′ tan β − ; σ = ρ g ′ h f σ
(9.7)
The coefficient of proportionality K , as is introduced by Eq. (9.3) and is appearing in Eq. (9.5), is usually called the lateral earth-pressure coefficient. The terminology is borrowed from soil mechanics. In most numerical granular dynamics simulations it is found that K ≈ 1 . This result should be taken with some reservation, since in all DEM simulations known to us periodic boundary conditions are imposed. At any rate for K = 1 the dynamic Eq. (9.5) yields, ∂v 4 ∂v 4 v2 + v + g′ − ∂t 3 ∂x 3 hf
∂h f =b ∂x
(9.8)
10. Steady granular flows In order to evaluate the forcing term, Eq. (9.7), one needs additional information concerning the dependency of the shear stress to the various fields and dimensioned or dimensionless parameters which are introduced in the description of the granular flow problem. This is done usually by the selection of an appropriate basal friction law. Physical granular-flow experiments on a rough, inclined bed suggest that the frictional resistance to the flow, acting at the interface between solid and fluidized material consists of a static and a kinetic part,
τ = τ st + τ kin
(10.1)
The static part of the shear stress is expressed in terms of a constant static friction coefficient,
τ st = σµ st = ρ gh f cos β µ st , µ st = tan φst
(10.2)
where φst = const. is identified as a static friction angle, which is mobilized at the interface between the fluidized granular material and the solid deposit. In view of Eq. (9.7), the kinetic component of the shear stress, τ kin is also normalized by the normal interfacial reaction stress, σ . The (dimensionless) ratio τ kin / σ , is then identified as a kinetic friction coefficient,
Mathematical modeling of granular flow-slides
µkin =
τ kin τ = − µ stat σ σ
73
(10.3)
Accordingly the forcing term, Eq. (9.7), becomes, b = g ′ ( tan β − µ )
(10.4)
µ = µ stat + µkin
(10.5)
where
Notice that on the basis of Eq. (10.3) the kinetic friction coefficient can be determined experimentally from steady (non-accelerating) granular flows on an incline with constant dip angle β : Imagine an experimental set-up, where one can control the dip angle. From such a configuration the static friction coefficient is identified with the maximum inclination of the track for which no-flow is observed. With µ stat known, the kinetic friction as a function of β can be computed from the experimental data on the basis of Eq. (10.3), as soon as an equilibrium state is reached; i.e. as soon as the flow reaches constant velocity and height in the main bulk. This implies that the total acceleration vanishes and accordingly, b = 0 . As of today, the rheology of granular flows is not well understood. Thus, the structure of the constitutive equation for the kinetic friction is obtained by dimensional analysis and by the search for scaling laws that seem to fit satisfactorily the experimental (and numerical) data. In the pertinent literature we find that the kinetic friction coefficient is considered to be a function of a selected set of dimensionless numbers, which characterize granular flows. Such dimensionless numbers are for example the flow-height scaling factor: m f = h f Dg
(10.6)
and an appropriate Froude number of the flow6 Frf = v
gh f or FrD = v
gDg
(10.7)
with Dg denoting the (mean) grain diameter.
6 Notice that usually the Froude number of a flow is interpreted as the number that reflects the relative importance of inertial forces respective to gravity forces.
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For example Savage (1979) and later Ancey & Evesque (2000), motivated by open channel hydraulics practices, assumed that the kinetic component of the shear stress is proportional to the flow-height scaling factor and to a “dynamic fluid pressure”,
τ kin = A′m f ρ v 2 , A′ = const > 0
(10.8)
where A′ is a calibration constant. This assumption yields that the kinetic friction coefficient is a quadratic monomial of FrD ,
µkin = A′FrD2
(10.9)
We notice that some authors, using dimensional analysis considerations, argue that the kinetic friction in granular flows should depend on a particular combination of the above dimensionless numbers, called the inertial number (cf. Da Cruz et al. 2005, Pouliquen et al. 2005) Ι∝
Dg2
Γɺ
gh f
where Γɺ is the shearing rate for linear velocity profile, Γɺ = v h f
(10.10)
(10.11)
Notice that Ι f = Frf m f
(10.12)
The justification of the inertial number is usually done on the basis numerical simulations and on a time scaling consideration.
11. The Forterre-Pouliquen scaling In a recent paper Alevizos et al. (2007) modified the scaling laws, originally proposed by Forterre & Pouliquen (2003), in terms of the thickness of the flow layer. Starting from the originally proposed law for sand, Frh =
v h = −γ + α hstop gh
(11.1)
Mathematical modeling of granular flow-slides
75
40 35 30
hstop/Dg
25 20 15 10 5 0 27
29
31
33
35
37
39
41
43
45
β Fig. 10. Pouliquen’s “deposit” function hstop in relation to the dip angle of the incline.
where h is the total thickness of the flow and γ = 0.77 , α = 0.65 are empirical, fitting constants. The equilibrium deposit height hs ,eq is in turn identified in terms of Pouliquen’s hstop ; i.e. it is set that, hs ,eq = γ hstop α
(11.2)
where according to Forterre & Pouliquen (2003) hstop tan δ 2 − tan β = λ′ tan β − tan δ1 Dg
(11.3)
In Fig. 10 we plotted hstop , computed from Eq.(11.3), for the values of the various parameters, suggested by Forterre & Pouliquen (2003) for sand: λ ′ = 2.03 , δ1 = 27. , δ 2 = 43.4 . The singularity for β → δ1 signifies that δ1 should be identified with the angle of repose for the considered sand. By introducing the scaling factor η f = h f / hs , Eq. (11.1) yields, Frf =
v gh f
=α
hf hstop
1+
γ hstop α hf
(11.4)
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Based on this reinterpretation of the Forterre-Pouliquen scaling, the following law for the kinetic friction coefficient is proposed (Alevizos et al. 2007)
µkin
Frf γ hstop =C 1 + h f α h f
−1/ 2
(11.5)
where C=
λ′ ( tan δ 2 − tan β ) > 0 α
(11.6)
With the above result the expression for the total friction coefficient, Eq. (10.5), becomes
µ = µ stat + CI f
(11.7)
This model accounts for a decrease of the friction with increasing height of the granular sheet. This is a commonly argued subject in soil mechanics, where we find that at low stresses the Coulomb and the interfacial friction angles are increasing with decreasing normal stress (Vardoulakis & Sulem 1995). We remark also that in applications of the shallow-water theory to real-scale debris flows and avalanches, the height-scaling number m f (Eqs. (10.6) and (10.12)) is practically infinite, and with that we obtain 0 < Dg << h f ⇒ m → ∞ ⇒ µkin → 0, µ → µ stat = const.
(11.8)
Such a theory is the Savage-Hutter theory, where justifiably the concept of dynamic resistance is absent. Notice that in the Savage-Hutter theory there is a “dynamic” friction term, which accounts for the effect of centripetal forces acting on the base of the flow-slide in a track with variable topography. Obviously for flows on a planar track centripetal terms are absent. With the above assumption for µkin , Eqs. (11.5) and (11.6), the expression for the forcing term on the r.h.s. of Eq. (9.5) or (9.8) becomes,
1h γ h stop stop b = g ′ ( tan β − µ stat ) 1 − 1 + α hf α hf
−1/ 2
Frf
(11.9)
Mathematical modeling of granular flow-slides
77
12. An erosion-speed model In order to close the set of governing equations we need to establish a constitutive equation for the erosion-speed e , which appears in the mass balance Eqs. (8.1) and (8.2). Based on landslides observations, Hungr et al. (1984) proposed that the erosion speed is proportional to the flow velocity, i.e. e = − E v . This model is modified here as follows: First we recognize that, unlike in sediment transport hydraulics, the velocity profile in granular flows is non-constant over the flow-height and that no turbulent boundary layer can be assumed at the interface between the solid and fluidized granular phase. In that respect, we assume that erosion in shear granular flows is driven by the shearing rate7 and not by the velocity. Thus, for linear velocity profile we set that, (12.1) e ∝ Γɺ In addition to that we assume that erosion should dominate ( e > 0 ) in regions where the deposit is thicker than its equilibrium value, hstop . In the opposite case deposition should take place ( e < 0 ). Using dimensional analysis, above assumptions lead to the following constitutive equation for the erosion speed, e=E
v hf
γ hs − hstop , E = const. > 0 α
(12.2)
The constitutive equation (12.2) contains one free parameter, the erosion constant E , which should be calibrated experimentally.
13. The dynamic system We assume that the flow-slide has an evolving-in-time thickness h f with an erosion/deposition sheet forming at its base, whose thickness hs changes also with time until an equilibrium configuration is reached. At any instant the total thickness of the flow slide is h = hs + h f 7
(13.1)
One could argue here that, in reality, it is the particle spin that drives erosion and not the velocity gradient. In a non-Cosserat continuum, we usually assume that particle spin coincides with the continuum spin, which in the present case is half the velocity gradient.
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We apply now the mass-balance Eqs. (8.1) and (8.2), and the momentum-balance equation (9.5) in order to model the formation of a deposition sheet on an inclined planar track of infinite extend, meaning that all fields are independent of the x -coordinate. In this case, the governing balance equations yield the following autonomous dynamical system: h f e d hs = −e dt v b
(13.2)
with b and e given above by Eqs. (11.9) and (12.2), respectively. From the first two Eqs. (13.2) we obtain that during the considered transient process the total height of the flow-slide remains constant, h = h0 = const.
(13.3)
In order to discuss further the above set of evolution equations we introduce the following non-dimensional variables, hf t v t∗ = , v∗ = , ha∗ = (13.4) ( a = f , s, stop ) Dg Dg / g gDg With this change of variables, the set of governing equations transforms to the following autonomous set,
v∗ * γ * E h∗ − h* hs − α hstop ∗ ∗ ∗ s e(hs , v ) d hs 0 = = * dt ∗ v∗ b(hs∗ , v∗ ) 1 hstop v∗ B 1− α h0∗ h0∗ − hs*
(13.5)
B = cos β ( tan β − µ stat )
(13.6)
where
The system of Eqs. (13.5) can be readily integrated numerically for a given set of initial values, h*f (0) = h∗f 0 and v∗ (0) = v0∗ . As can be seen from Fig. 11 the equilibrium point is a fixed point of the dynamic system:
Mathematical modeling of granular flow-slides 0.50
79
Forterre-Pouliquen scaling law
0.45 0.40 0.35
v [m/s]
0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.000
fixed point 0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
h s [m]
Fig. 11. Numerically produced phase diagram of the erosion-deposition process. * e(hs∗ , v∗ ) = 0 ⇒ hs*,eq = γ hstop α * b(hs∗ , v∗ ) = 0 ⇒ veq =α
h0∗ h0∗ − hs*,eq ) ( * hstop
(13.7)
We notice also that the fixed point is independent of the value of erosion parameter E , which seems to influence primarily the speed at which the process reaches equilibrium. The mathematical analysis of the above dynamic system, Eqs. (13.5), yields that the stationary point is asymptotically stable. This property makes possible the interpretation of the experimental results as the relationship between the equilibrium-velocity and the equilibrium solid layer thickness-height, since the equilibrium solution is a fixed-point of the corresponding autonomous set of evolution equations. This means in turn that the equilibrium value for the thickness of the deposition sheet is consistently modeled by Pouliquen’s hstop .
14. The long wave-length linear stability limit For the determination of the regime of validity of the above model one can perform the linear stability analysis of the general system of partial differential equations, Eqs. (8.1), (8.2) and (9.8), for time growing
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I. Vardoulakis and S. Alevizos 3.00
2.50
unstable
Frf,cr
2.00
1.50
stable 1.00
0.50
0.00 0
0.5
1
1.5
2
2.5
3
3.5
4
K
Fig. 12. Stability regimes for granular flows down a planar track.
perturbations of the various fields, around the equilibrium points, Eqs. (11.2), (11.4) hɶ f , hɶs , vɶ = {C1 , C2 , C3 } exp [ikx + st ] , (14.1) k ∈ ℝ, s ∈ ℂ, Ci = const.
{
}
where the superimposed tilde denotes the perturbed field (cf. Roberts, 1994). The details of this procedure can be found in Alevizos et al. (2007). Assuming that the rest of the parameters are constant The corresponding stability threshold, i.e. the critical Froude number Frf ,cr of the flow, is slightly decreasing with the dip angle β ), and Frf ,cr ≃ 1.35
(14.2)
On the other hand as is shown in Fig. 12 Frf ,cr is strongly depending on the lateral earth-pressure coefficient K , which constitutes an open parameter of the problem at hand!
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81
15. Mathematical modeling of granular flow-slides: Some open questions
Fig. 13. Flow-slide in cylindrical topography: Deposition shock-wave recorded by Bassanou (2000).
The existence of kinematic waves in granular flows is speculated by some authors. For example, Rajchenbach (2002) reported the existence of kinematic waves in grain avalanches in a bi-dimensional model system. In their paper Forterre & Pouliquen (2003) stated that the flow is unstable, when “the velocity of the kinematic waves is larger than the velocity of the gravity waves”. In general, the existence or not of kinematic waves in granular flows remains an open question. The structure of the front of granular flow-slides (Pouliquen, 1999b) is another major open problem. Immaterial of the selection of the erosion/deposition law, the governing equations are hyperbolic in nature and lead to wave breaking solutions at the tip. Thus, in our opinion, the problem of the front is a completely different one, if compared to the considered problem of granular flows in the bulk. In the former, the effect of vorticity should be important. Most probably the best model at
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the tip is one that accounts for both velocity and vorticity, which in turn will lead inevitably to the scouring effect at the tip (negative base velocities) and to a motion of the bulk that resembles the unfolding of a carpet. Finally, considering the geometric characteristics of the Geolab Slider (Fig. 13), we recognize that for a granular flow slide in variable topography deposition takes place at the lower “elevations’ of the flowslide. The deposition manifests itself through the formation of a backwards moving deposition shock wave (Bassanou, 2000; Fig. 13). Deposition shock formation and its dynamics are not well understood today and are posing a number challenging questions.
References Alevisos, S. Vardoulakis, I., Stefanou, I. and Alonso-Marroquin F. (2007). Aspects of Mathematical Modeling of Granular Flow-slides. J. Fluid Mech., under review. Ancey, C., Coussot, P. and Evesque, P. (1996). Examination of the possibility of fluid mechanics treatment of dense granular flows. Mech. Coh. Frictional Mat., 4, pp. 305-319. Ancey, C. and Evesque, P. (2000). Frictional-collisional regime for granular suspen-sion flows down an inclined channel. Physical Review E, 62, (6), 8349-8360. Ancey, C. (2002). Dry granular flows down an inclined channel: Experimental investigations on the frictional-collisional regime. Phys. Review E, 65, 011304. Bassanou, M. Landslides Dynamics- Mathematical and Experimental Simulation of Debris Flow. Ph.D. Thesis, National Technical University of Athens, (2000). Da Cruz, F., Emma, S., Prochnow, M., Roux, J.-N. and Chevoir, F. (2005). Rheophysics of dense granular materials: Discrete simulation of plane shear flows. Physical review E, 72 (2): art. no. 021309, Part 1. De Saint-Venant, A.J.C.B. (1850). Mémoire sur des formules nouvelles pour la solution des problèmes ralitfs aux eaux courantes. C.R. Acad. Sc., Paris 31, pp. 283. De Saint-Venant, A.J.C.B. (1871). Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l’introduction des marées dans leurs lit. C.R. Acad. Sc., Paris 73, pp. 147-154. Di Prisco, C., Imposimato, S. and Vardoulakis, I. (2000). Mechanical modelling of drained creep triaxial tests on loose sand. Géotechnique, 50, 73-82. Drake, T.G. and Shreve, R.L. (1986). High-Speed Motion Pictures of Nearly Steady, Uniform, Two-Dimensional, Inertial Flows of Granular Material, J. Rheology, 30(5), pp. 981-993. Dressler, R.F. (1949). Mathematical solution of the problem of roll-waves in inclined open channels. Communications on Pure and Applied Mathematics, 2, pp. 149-194 Duady, S., Andreotti, B. and Daerr, A. (1999). On granular surface flow equations. Eur. Phys. J., 11, pp. 131-142.
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Forterre, Y. and Pouliquen, O. (2003). Long surface wave instability in dense granular flows. J. Fluid Mech., 486, pp. 21–50. Jain, N., Ottino, J.M. and Lueptow, R.M. (2002). An experimental study of the flowing granular layer in a rotating tumbler. Phys. Fluids, 14, pp. 572-582. Julien, P.Y., Erosion and Sedimentation. Cambridge Univ. Press, (1998). Marroquin, F.A., Vardoulakis, I., Herrmann, H.J. and Weatherley, D. (2006). Effect of rolling on dissipation of fault gouges. Physical Review E, 74, 031306. Pouliquen, O. (1999a). Scaling laws in granular flows down rough inclined planes. Phys. Fluids,11, pp. 542-548. Pouliquen, O. (1999b). On the shape of granular fronts down rough inclined planes. Phys. Fluids, 11, pp. 1956-1958. Pouliquen, O., Cassar, C., Forterre, Y., Jo P. and Nicolas, M. (2005). How do grains flow: towards a simple rheology for dense granular flows. In: Powders & Grains 2005, (ed. R. Garcia-Rojo, H. Herrmann & J. McNamara), pp. 859-865, Taylor & Francis Group. Radjai., F and Roux, S. (2002). Turbulentlike Fluctuations in Quasistatic Flow of Granular Media, Physical Review Letters, 89 (6), 064302-1. Rajchenbach, J. (2002). Dynamics of Grain Avalanches. Physical Rev. Letters, 88 (1), 014301-1-4. Roberts, A.J. A One-Dimensional Introduction to Continuum Mechanics, World Scientific, (1994). Savage, S.B. (1979). Gravity flow of cohesionless granular-materials in chutes and channels. J. Fluid Mech., 92, pp. 53-96. Savage, S.B. and Hutter, K. (1989). The motion of a finite mass of granular mate-rial down a rough incline. J. Fluid Mech., 199, pp. 177-215. Savage, S.B. and Hutter, K. (1991). The dynamics of granular materials from ini-tiation to runout. Part I: Analysis. Acta Mech., 86, pp. 201-223. Silbert, L., Landry, J., Grest, G. (2003). Granular flow down a rough inclined plane: Transition between thin and thick piles. Phys. Fluids, 15, pp. 1-10. Vardoulakis, I. Scherfugenbildung in Sandkörpern als Verzweigungsproblem. Ph.D. Thesis, Institute for Soil and Rock Mechanics, University of Karlsruhe, 70, (1977). Vardoulakis, I. (1996). Deformation of water saturated sand: I. Uniform undrained deformation and shear banding. Géotechnique, 46, pp. 441-456. Vardoulakis, I. (1996). Deformation of water saturated sand: II. The effect of pore-water flow and shear banding. Géotechnique, 46, pp. 457-472. Vardoulakis, I. (2004). Fluidization in artesian flow conditions: I. Hydro-mechanically stable granular media. Géotechnique, 54, No. 2, pp. 117–130. Vardoulakis, I. (2004). Fluidization in artesian flow conditions: II. Hydro-mechanically unstable granular media. Géotechnique, 54, No. 3, pp. 165–177. Vardoulakis, I. and Sulem, J. Bifurcation Analysis in Geomechanics, Blackie Academic and Professional, (1995). Whitham, G.B., Linear and non-Linear Waves, Wiley, (1974).
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Chapter 5 The mechanics of brittle granular materials
Itai Einav School of Civil Engineering, The University of Sydney, NSW, 2006, Australia
[email protected]
When considering the modelling of granular materials, use may be made of continuum mechanics principles if the granules are confined. The grain size distribution in these materials evolves with crushing. Therefore, in the continuum mechanics modelling of such matter, the grain size distribution should be taken into account as an internal variable function. In this chapter this idea is explored, first by adopting principles of variational calculus, but then a more accessible theory is motivated based on the concept of breakage.
1. Introduction Continuum mechanics prove useful for modelling confined granular materials. For capturing the critical behaviour in shear, the Coulomb friction law is an excellent starting point. To account for the further critical behaviour in compression, critical state soil mechanics (CSSM) is often employed. CSSM was originally developed for clays1. When applied to sand, or to any other brittle granular matter, the theory falls short. In these materials the grain size distribution (gsd) is an evolving property. Therefore, in the continuum modelling of brittle granular matter, the gsd has to be an internal variable function. This seemingly simple concept allows the clarification of many open questions, and addressing problems that CSSM can not treat, beyond traditional geotechnical modelling; for example, the tracking of the grain sizes makes our theory suitable to any discipline that deals with confined
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comminution, including geophysics, geology, powder technology, mineral processing, agriculture, the food industry, and pharmaceutics. In the next section I explore a variational mathematical theory that consistently accounts for evolving grading. This approach is specialised in Sec. 3 to derive the breakage mechanics theory2. This ensures we obtain: (i) a cleaner route for discovering new physical results, and (ii) a rather more practical engineering approach. In Sec. 4, a possible link is observed between the granules’ surface energy, the amount of energy they store, and the way they self-organise in the system. In Sec. 5, a key result is extracted by deriving an expression for the granular agglomerate’s ‘critical comminution pressure’ – i.e., the pressure for the initiation of substantial crushing, often identified in geotechnical engineering as the pre-consolidation pressure. 2. Modelling evolving grading: the mathematical approach Let p(x) be the grain size distribution (gsd) by mass, then:
Λ (∫ p( x )dx − 1) = 0
(1)
where Λ is a Lagrange multiplier that will be of use later. The above equation restricts the gsd to be a proper density function. The effect the evolving grading has on the behaviour of granular materials can be explored by introducing the gsd as an internal variable function in a thermomechanical analysis. For that purpose, we first look at the effect of this function on the Helmholtz free energy potential Ψ*. The motivation of using ‘*’ next to Ψ will be clearer soon. The free energy function is a state potential, and in our system being a functional of p(x), as well as any other kinematic internal variables:
Ψ * = Ψ * (ε , p( x ) ) = 0
(2)
We assume that Ψ* depends merely only on a notional macroscopic strain ‘ε’, in addition to the gsd. This is made deliberately simple to focus on studying the effect of using p(x) as an internal variable. Denoting the ‘fractional energy’ as the energy stored in a given size fraction x by ψ = ψ (ε , x ) , and appreciating the additivity of energy:
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87
Ψ * = ∫ψ (ε , x ) p( x )dx
(3)
However, since the expression in Eq. (1) equals zero, and depends on the system variable p(x), the free energy may also be written by
Ψ = Λ + ∫ (ψ (ε , x ) − Λ ) p( x )dx
(4)
This alteration is necessary for constraining p(x) to be a density function, i.e., requiring that ∫ p ( x )dx ≡ 1 at any time. To be able to prove that this is always correct, we still need to show that Λ is truly a non-zero Lagrange multiplier. If we can not show this, then there would be no guarantee that ∫ p ( x )dx ≡ 1 . The idea of constraining the distribution density functions is often used as part of the statistical mechanical analysis of rigid granular materials3. Since Ψ is a functional its difference is given by:
δΨ =
∂Ψ δε + ∫ (ψ (ε , x ) − Λ )δp( x )dx = 0 ∂ε
(5)
where δp(x) denotes the variation of the gsd, caused by the fragmentation. If at any time grain crushing does not occur, p(x) remains constant and the second term in the right-hand side of Eq. (5) vanishes, leaving us purely with the first elastic deformation term. However, if grain crushing occurs, the energy stored in the systems is affected. Furthermore, grain crushing means size reduction and an increase to the surface area, suggesting energy consumption in the sense of fracture mechanics4. For that purpose we need to introduce an additional ~ potential: the ‘increment of dissipation’ potential Φ . Only the increment of dissipation can be considered as a potential, since it is non-integrable (i.e., the notion of ‘dissipation’ Φ can not be defined). The tilde sign ‘~’ ~ was used instead of ‘δ’ to denote this fact. By accepting that Φ must be non-negative and should also depend on the changing grading δp(x), we write (in accord with Eq. (3)):
~
~
~
~
Φ = Φ (ε , p( x ), δp( x ) ) = ∫ φ (ε , p( x ), δp( x ), x )dx ≥ 0 ~
(6)
where φ = φ ( x ) defines the amount of energy which is being dissipated from the fraction x. Excluding rate effects and considering quasi-static events only, one can consider the Euler’s theorem for first-order homogeneous functions. In our problem
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~ ∂φ (x ) Φ =∫ δp( x )dx = ∫ γ p ( x )δp( x )dx ≥ 0 (7) ∂δp( x ) ~ We denote γ p ( x ) = ∂φ ( x ) / ∂δp ( x ) as the ‘fractional dissipative crushing force’ arising from the change in gsd , δp (x ) , locally at x. ~
Modern Thermomechanics suggests5 that
~
σ : δε = δΨ + Φ
(8)
where the first term refers to the mechanical work supplied through the boundaries of the granular agglomerate. Combining this equation with Eqs. (5) and (7), and considering Ziegler’s orthogonality condition5, gives the definition of stress σ and γ p (x ) :
σ=
∂Ψ ∂ε
γ p ( x ) = Λ − ψ (ε , x )
(9) (10)
It is still not exactly clear what Λ means. However, we can define the statistical ‘average dissipative crushing force’ Γ, as the average of the ‘fractional dissipative crushing force’, simply by using the gsd as a weighing function (as is often used in statistical mechanics):
Γ = ∫ γ p ( x ) p( x )dx = Λ − ∫ψ (ε , x ) p( x )dx
(11)
such that using Eqs. (1) and (4) (12) Now we see that Λ is the sum of the energy stored in the system and the average dissipative crushing force, so analogous with Eq. (8), Λ reflects crushing mechanical work. Since the result is basically non-zero, the only way to satisfy Eq. (1) is having ∫ p ( x )dx ≡ 1 , which is what we needed in order to prove the suitability of using the constraint. The advantage of the above formulation is in introducing the gsd as an evolving property without employing any radical assumption, while adhering to formal mathematical derivation. However, the disadvantage is in the difficulty of keeping attached to developing physical models. For example, for completing model derivation, an explicit expression for the increment of dissipation has to be derived (i.e., an expression that is a
Λ =Ψ + Γ
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function only of ε, p(x) and δp(x)). Of course this expression should be physical at the same time, and this leaves an open challenge. 3. Modelling evolving grading: the physical approach In this section I would like to refer to the formulation of Breakage Mechanics2,6. In presenting this Theory, I will make one sacrifice in deviating from the rigor mathematical derivation above. This would help us in developing accessible physical models. The sacrifice I make is in replacing the need to impose Eq. (1) simply by assuming a unique linearscaling relation for the gsd:
p( x ) = (1 − B ) p0 ( x ) + Bpu ( x )
(13)
where p0 ( x ) and pu (x ) stand for the initial and ultimate gsd, and B is a scaling scalar internal variable called ‘breakage’, that grows from zero to one. The concept of an ultimate gsd is now well established, and this is often assumed as a fractal distribution (i.e., a density distribution by mass which indicates that the numbers of particles is inversely proportional to their size). The literature has many theoretical and experimental justifications to the concept of an ultimate gsd7. Since we basically have both p0 ( x ) and pu (x ) , the use of p(x ) as an internal variable function is replaced by the internal variable B. Clearly since both ∫ p0 ( x )dx = 1 and ∫ pu ( x )dx = 1 , we guarantee that ∫ p( x )dx ≡ 1 at any time, releasing us from needing to impose Eq. (1). An important feature is that B is a measurable internal variable, as described by Einav2, and as portrayed in Figure 1(a). Let us rewrite Eq. (3), by adopting (13):
Ψ = (1 − B )Ψ 0 + BΨ u
(14)
Ψ 0 = ∫ψ (ε , x ) p0 ( x )dx ,
(15)
Ψ u = ∫ψ (ε , x ) pu ( x )dx
(16)
where
are the macroscopic free energy averages of the fractional energy based on the initial and ultimate gsd. In conjunction, the general form of the increment of dissipation is now replaced by
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~ ∂Φ δB = EBδB ≥ 0 Φ = Φ (ε , B, δB ) = ∂δB ~
~
(17)
where we consider Euler’s theorem for first-order homogeneous ~ functions for quasi-static deformations, and use E B = ∂Φ / ∂δB . Combining Eqs. (14) and (17) with (8), and adopting Ziegler’s orthogonality condition, this time we have
σ=
∂Ψ ∂ε
(18)
E B = Ψ 0 −Ψ u
(19)
EB is termed the ‘breakage energy’ – i.e., the energy that is contained in the system for breaking particles, or for shifting the gsd, from the initial stage of loading till ultimate conditions. However, what we really like to know is how much energy is reserved in the system for breaking the particles, at any given moment, i.e., from the current state to the ultimate state. Therefore the ‘residual breakage energy’ is defined:
EB* = EB (1 − B ) = Ψ −Ψ u
(20)
While the breakage energy EB relates to the stripy area in Figure 1(a), EB is linked to the stripy area in Figure 1(b).
Percent finer: %
*
(a)
(b)
Fig. 1. Breakage measurement and evolution law2,6,8. The left figure (a) portrays the measurable definition of breakage. The right diagram (b) presents the breakage propagation criterion for granular materials.
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4. Surface energy, fractional energy & self organisation Which form should the fractional energy ψ (ε , x ) take? Numerically, this could be experimented using DEM simulations, first by squeezing particles in a box, then calculating the total stored energy in a given fraction size, then averaging to get ψ (ε , x ) . However, we can examine this question theoretically. The larger the granule’s surface area the more forces will be transmitted through it on average. Therefore, considering a force chain mesh containing only average forces, the fractional energy should scale with the granule’s surface area of the given fraction2. For example, in a system of spheres we can expect to have:
ψ (ε , x ) = ψ r (ε )( x/xr )2
(21)
where ψ r (ε ) is the fractional energy in a reference particle size xr. Two-dimensional DEM simulations validate the above theoretical explanation2, with data collapsing to a unique line, independent of the gsd, although with an exponent that shows slightly higher scaling with the surface area (in three-dimensional simulations, the exponent may therefore be expected to get slightly higher than two). Which kind of conclusion can we make from this appreciation of the universality scaling? Similar to fluids, the surface area of solids relates linearly to their surface energy γ (not to be confused with γp from before). Denote γr and γ(x) as the total surface energy of the reference grain size and of a grain with a size x. The surface energy of particles scales with their surface area; for example, the surface energy of spheres scales proportionally to their diameter squared, as in Eq. (21):
γ (x ) = γ r ( x/xr )2
(22)
Therefore we might expect that when particles are confined as an agglomerate, they will tend to self-organisation such that on average:
ψ (ε , x ) /ψ r (ε ) = γ (x ) / γ r
(23)
This may suggest that while interacting with the neighbors under confined conditions, an individual granule acts as an attractor with its ‘attraction potential’ being his total surface energy. But the neighboring particles also act as attractors, so the overall tendency of the system would be to self-organised to transmit energy in the most dispersed way
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(in the line of maximising entropy), and then resisting more to crushing. This suggestion, however, is somewhat speculative and requires further questioning, particularly because the surface energy denotes energy that has been released in the past while the link to the way they organise is based on the stored energy potential that relates to their current state. 5. Fracture propagation criterion The advantage of the formulation in Sec. 3, i.e., the analysis with breakage, is in its ability to allow more accessibly to adopt physical concepts when developing models. Developing a complete model—a model that can respond to any loading scenario—is beyond the scope of the current article. For that purpose, a reference is given to a future publication9. In this section, however, I will extract a key result from Einav8 by deriving a critical comminution pressure for confined brittle granular assemblies – i.e., the pressure for the initiation of substantial crushing, often identified as compressive yielding or the preconsolidation pressure. The concept is analogous to the critical tensile strength of a plate with an embedded crack by Griffith4. Griffith suggested that the weakening of material by a crack could be treated as an equilibrium problem in which the reduction of stored energy, when the crack propagates, could be equated to the increase in surface energy due to the increase in surface area. Here, the derivation is based on the energy principles of breakage mechanics rather than the energy principles of fracture mechanics. The increment of the residual breakage energy is given by differentiating Eq. (20):
δE B* = δEB (1 − B ) − E BδB
(24)
this increment is represented schematically by the crescent gray area in Figure 1(b), and relates to the loss of the amount of breakage energy that was available to crush particles. We postulate that the energy that is dissipated from the system during an increment should equate to the residual breakage energy loss:
~
Φ = E BδB = δE B*
(25)
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Integrating Eqs. (24) and (25) gives the breaking yielding criterion: yB = EB(1−B)2− Ec ≤ 0
(26)
where the critical breakage energy, Ec, is introduced as a constant of integration8. The equality sign denotes breakage process and the inequality refers to elastic deformations prior to crushing. Combining Eqs. (15), (16), (19), and (21) gives:
EB = ϑΨ 0
(27)
where ϑ = 1 − Ju/J0 is the criticality proximity parameter, being a function of the second order moment of the initial and ultimate gsd (i.e., of J0 = ∫ x 2 p0 ( x )dx and Ju = ∫ x 2 p0 ( x )dx ). Using (14):
EB =
ϑΨ 1 − ϑB
(28)
At the beginning Ψ = Ψ0 and B = 0. Assuming linear compression elasticity, we have Ψ = p2/2K (with p being the pressure and K the effective bulk modulus of the agglomerate). The critical comminution pressure is derived using the equality in (26):
pcr =
2 KEc
ϑ
(29)
This relation bears striking similarity to Griffith’s critical tensile strength. It describes, in a succinct way, the interrelation between three important aspects of the fracture process in confined brittle granular matter: (1) the material, by the critical breakage energy Ec (an analogue to Gc in Griffith’s expression) and Bulk modulus K (an analogue to the Young’s modulus E); (2) the pressure level pcr (an analogue to Griffith’s tensile strength σcr); and (3) the geometry of the particles given by the grains surface area through ϑ ( an analogue to the initial crack length ‘a’ in Griffith’s analysis). The formula explains why brittle granular materials undergo isotropic hardening as they crush in confined conditions. This is simply because as particles break, the initial gsd is gradually pushed towards the ultimate distribution. Effectively, the denominator in Eq. (29) becomes smaller and pcr gets larger.
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Further details as to how this theory can be adjusted for deriving the critical comminution pressure in more realistic agglomerates is given by Einav8, considering both the non-linearity nature of the effective elasticity and the initial agglomerate porosity. 6. Conclusions The grain size distribution of confined brittle granular materials is an evolving property. Two continuum mechanics theories have been defined for taking this fact into account. Initially a rigorous variational mathematical theory is defined, but this is followed by a more accessible physical approach. An observation is made showing that both the granules’ surface energy and the amount of energy they store, scales with their surface area, and may explain how they self-organised in the system. Finally, a simple critical comminution pressure formula is derived, showing striking resemblance to Griffith’s fracture tensile strength, and explaining the phenomenon of isotropic hardening in brittle granular materials. References 1. 2. 3. 4. 5. 6. 7. 8. 9.
Schofield A.N., Wroth C.P. 1968. Critical state soil mechanics. London: McGrawHill. Einav I. 2007a. Breakage mechanics. Part I- theory. J. Mech. Phys. Solids. 55(6), 1274-1297. Shahinpoor M. 1980. Statistical mechanical considerations on the random packing of granular materials. Powder technology, 25, 163-176. Griffith A.A. 1921. The phenomena of rupture and flow in solids. Phil. Trans. R. Soc. A. 221, 163-198. Ziegler H. 1983. An Introduction to Thermomechanics. North Holland, Amsterdam (2nd edition). Einav I. 2007b. Breakage mechanics. Part II- modelling granular materials. J. Mech. Phys. Solids. 55(6), 1298-1320. McDowell G.R., Bolton M.D. and Robertson D. 1996. The fractal crushing of granular materials. J. Mech. Phys. Solids. 44(12), 2079-2102. Einav I. 2007c. Fracture propagation in brittle granular matter. Proc. Roy. Soc. A. (Submitted). Einav I. 2007d. Soil mechanics: breaking grounds. Phil. Trans. R. Soc. A. (Invited to the next Triennial Christmas Issue).
Chapter 6 Stranger than friction: force chain buckling and its implications for constitutive modelling
Antoinette Tordesillas Department of Mathematics and Statistics University of Melbourne 3010 Victoria, Australia
[email protected]
A recently developed thermomicromechanical continuum formulation has paved the way for the construction of a new breed of constitutive laws without need for phenomonelogical parameters above the particle scale. Particle group behavior is key to this formulation and new insights into the role of force chain buckling in constitutive response are presented.
1. Introduction This investigation is part of an overall effort focused on the development of theoretical techniques for micromechanical constitutive models of granular media. The key objective is physical transparency in the formulation – specifically, one that captures governing physics across multiple length scales and results in a constitutive law that is free of any phenomenological quantities above the fine particle scale. Thus, model parameters are expressed solely in terms of particle-scale properties. Here we present a new approach that is based on a systematic study of material behaviour at multiple length scales within the framework of: Thermomechanics, Discrete Element Method, and Structural Mechanics (see Figure 1). Based on a homogenisation scheme that is on the scale of a particle and its first ring of neighbours [1,2], a generalised set of constitutive relations for dense granular systems is constructed using the 95
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principles of Thermomechanics [3,4]. These relations are expressed in terms of observable macroscopic state variables (e.g. strain) as well as a set of internal variables and their corresponding evolution laws. A long standing challenge in Thermomechanics is the physical interpretation of these internal variables [5,6]. For inelastic continua, internal variables are used to represent dissipative mechanisms which, in Classical Plasticity, are lumped together in a single property – the plastic strain [7]. However, a purely micromechanical formulation demands that the fundamental origins of dissipation be explicitly identified in order for the internal variables to be given clear physical meaning. To achieve this, we turn to the Discrete Element Method [8]. DEM simulations are undertaken to identify and characterise dissipative mechanisms emerging inside the deforming granular medium. During deformation, selforganized structures emerge at multiple length scales. These structures are subject to instabilities, and are then responsible for the loss of stored free energy in the system. In this study, their respective contributions to energy dissipation are quantified, and the predominant contributor is subsequently modelled within the framework of Structural Mechanics. The internal variables and their evolution laws are derived from a stability and failure analysis of this structure’s evolution. Continuum Mechanics
internal variables; evolution laws
nonaffine deformation
increasing strain
DEM
mesostructure
Structural Mechanics
Fig. 1. Elements of the theoretical framework for development of constitutive laws. Centre inset shows confined buckling of a force chain – a key mesostructural mechanism. Stored elastic potential energy accumulates at all of the contacts along the chain. Buckling of the chain leads to a collective release of this energy and is thus a predominant source of dissipation.
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The specific objective of this paper is to identify the role that buckling force chains plays in energy dissipation and its implications for constitutive modelling. The paper is arranged as follows. In Section 2, we briefly summarise our method for constitutive development. Herein, we identify two formalisms for internal variables and their evolution laws, and briefly discuss lessons learned from validation studies of associated constitutive models. In Section 3, we present results from DEM studies that highlight the link between local measures of nonaffine deformation and dissipation. We then describe a new study that probes mechanisms for dissipation in Section 4 and present new insights on the role of force chain buckling. Finally, concluding remarks and future research directions are given in Section 5. 2. The thermomicromechanical approach In common with other approaches in micromechanics, the thermomicromechanical technique developed in [1,2,9,10] proceeds in three steps. Step 1 involves the projection scheme, which relates the continuum deformation to the underlying particle motions. Step 2 relates the motions of the particle centres to the forces and moments at the contacts via a contact law [11]. Step 3 connects the contact forces and moments to the Cauchy stress or force per unit area, σ ij , and couple stress or torque per unit area, µ ij , via the stored free energy,
Ψ = Ψ(εε, κ ,α α ) , as follows:
σ ij =
∂Ψ ∂Ψ , µij = ∂ε ij ∂κ ij
(1)
where the stored free energy Ψ is expressed in terms of the micropolar deformation quantities of strain, εij , and curvature, κ ij , as well as a set of internal variables α = ( α 1 , α 2 ,....., α n−1 , α n ) . Each internal variable may be in the form of a scalar, vector, or tensor. For inelastic materials, these internal variables are tied to dissipative mechanisms responsible for the loss of stored free energy, and are thus central to robust formulations of energy flow and the projection scheme. For each internal variable
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introduced into the analysis, an evolution law is required to close the governing system of equations that satisfy
∂Ψ ⊗ δα αi ≤ 0 ∂α αi
(2)
in accordance with the 2nd Law of Thermodynamics (the symbol ⊗ denotes an appropriate inner product operator according to the order of the internal variable). A detailed exposition of this approach can be found in [1,2,9,10], and references cited therein. A key challenge when applying micromechanical techniques to constitutive modelling is the physical meaning ascribed to the internal variables [5,6,9], which carry information on dissipation. Two dissipative mechanisms observed in quasi-statically loaded densely packed granular systems are frictional slip and buckling of force chains. The former occurs on the contact scale, the latter on the mesoscale. Below we summarise key findings from past studies which deal with two different formalisms for the internal variables and evolution laws in constitutive development of densely-packed cohesionless granular assemblies. Readers interested in the details of these formulations are referred to [9,10]. (a) Dissipation on the micro or contact scale At the micro or contact scale, the dissipative mechanism for cohesionless granular systems is friction. Thus, a constitutive model can be developed in which plastic slip is represented as an internal variable [10]. The resulting constitutive model, however, over predicts stability and fails to capture the defining behavior of the material: i.e. softening under dilatation. The model’s inability to reproduce macroscopic softening is because the model has no means of dissipating energy associated with the normal contact forces. Specifically, although the model predicts loss of contacts in the direction of extension, the normal contact force continues to grow in the direction of most compressive principal stress. The amount of energy dissipated through slip at the contacts is much less
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than that accumulated at the contacts from the steady growth of the normal contact forces as loading proceeds. This results in the stress ratio increasing monotonically, even in the presence of plastic slip and loss of contacts in the direction of extension. Viewed from the standpoint of the force chain network, this result signifies that the force chains, which initially align themselves with the major principal stress axis, continue to sustain a steady increase in load even under continuing loss of lateral supporting contacts. The reasons for this are: (a) there is nothing in this formalism that limits the mechanism on the normal forces at the contacts in the direction of applied compression, and (b) slip only limits the tangential force at the contacts but in the absence of softening. (b) Dissipation on the mesoscale Numerous DEM simulations suggest that dissipation in granular systems transcends the contact scale and that contact dissipation through friction, does not constitute the sole source of dissipation in quasi-statically loaded cohesionless assemblies, e.g. [12-14]. Instead, irreversible microstructural rearrangements, inherent in unjamming transitions on the mesoscale (i.e. unjamming of particle clusters), hold the key to understanding energy flow in deforming granular systems. In a recent study [9], internal variables were introduced in the form of nonaffine deformation associated with group behavior. While previous methods have tied the normal and tangential motions at the contacts to a single strain, this new approach decoupled these motions and supplemented the degrees of freedom of the system by introducing three distinct internal variables as follows: pt et p e εijpn = εij − εen ij , ε ij = ε ij − ε ij , κ i = κ i − κ i
(3)
where εijpn , εijpt , κ ip represent the internal variables, εij denotes the total et e strain, and εen ij , εij , κ i are the elastic strains defined as
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fn ft et en ∆u = n = 2Rεij n i n j , ∆u = t = 2R(εetij t i n j + Rκ ei n i ), k k 1 ∆ω e = r 2 m = 2Rκ ei n i kR en
(4)
such that ∆u en , ∆u et denote respectively the elastic normal and tangential relative displacements at a contact, ∆ω e is the elastic relative rotation at a contact; f n , f t are the normal and tangential component of the contact force, respectively; m is the contact moment; k n ,k t ,k r are the elastic spring constants, R is the radius of the particle and n i is the contact normal vector. The resulting constitutive laws for a monodisperse granular system are given by: σ ij =
NR 2 n {k (δijδkl + δikδ jl + δilδ jk )(εkl − εklpn )+ k t (3δikδ jl − δijδkl − δilδ jk )(εkl − εklpt )} 4V
µi =
NR 4 t k + k r )(κ i − κ ip ). ( V
(5)
where N denotes coordination number, V is the area of the Delaunay polygon for the particle and δij is the Kronecker delta. Thermodynamics of internal variables is used to derive a set of evolution laws for εijpn , εijpt , κ ip based on curve-fitted DEM data and observations of unjamming transitions of mesoscale particle clusters in DEM simulations. The resulting evolution laws are, strictly speaking, phenomenological: they contain parameters that have no explicit link to the underlying particle-scale properties. However, an important clue to the underpinning micromechanics did emerge from this analysis. In essence, the evolution laws suggest a mesoscale mechanism (i.e. group behaviour) whose shear and volumetric strain are tied together in a way that describes softening under dilatancy and satisfied the 2nd Law of Thermodynamics. The constitutive model based on these evolution laws successfully reproduced not only the shear band thickness and angle, but also the emergent evolution of contact and contact force anisotropies inside the shear band that is consistent with force chain buckling. Thus,
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this recent study highlighted the significance of group behavior in granular systems, particularly that of unjamming of particle clusters caused by instabilities such as the buckling of force chains. 3. Evolution of dissipative structures To identify the origins of dissipation, we examined unjamming transitions on the mesoscale, and the role that buckling force chains plays in these events, using DEM [8]. The DEM model is described in detail in [14]. Simple elastic-plastic contact laws based on Hooke’s law for normal interactions, and Hooke’s law in conjunction with Coulomb’s law for both tangential interactions and contact moment are used. A system of 5098 densely packed, polydisperse, cohesionless circular particles was subject to quasistatic boundary-driven biaxial compression. Material behavior was examined for a range of sliding and rolling friction coefficients, µs and µr, respectively. In all of these simulations, shear banding occurred, a process that commenced just before the peak stress ratio. Here, we present results for the test with µs= 0.7, µr = 0.05: this system exhibits trends representative of those observed for all other friction values. In this test, the softening regime evolved in the presence of a single shear band: essentially steady-state evolution of this band commenced at the end of the third drop in the stress ratio, at which point the band was fully-developed and the two outer regions were essentially in rigid-body motion. In Figure 2, we show the strain evolution of dissipation rate together with the potential energy and stress ratio. The greatest peaks in dissipation coincide with the concurrent drops in the stress ratio and potential energy in the softening regime. Throughout the strain-softening regime, consecutive cycles of unjamming-jamming events were observed, reminiscent of slip-stick motion observed for other granular systems, e.g. [15]. During each drop in stress ratio (or potential energy), unjamming occurred: initially jammed, solid-like structures comprising of percolating force chain particles amidst weak network particles became unstable and underwent large rearrangements. To quantify the nonaffine motion that occurred during unjamming, the deviation of the actual particle motion from that dictated by local measures of strain and curvature was computed in accordance with the
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technique developed in [16]. These measures of nonaffine deformation are on the scale of a particle and its first ring of neighbours, in accordance with the homogenization scheme underpinning the constitutive approach presented in Section 2 (see, also [1,2,9,10]. Using the Delaunay triangulation, an equivalent continuum for a discrete assembly is constructed. The local strain experienced by a single particle with respect to its first ring of neighbours is defined by:
εij =
1 ∑ (pic + pic +1 )e jk 3 (lkc +1 − lkc ), 2V c
(6)
where the sum is taken anticlockwise over the set of branch vectors associated with the particle, ejk3 is the Levi-Civita symbol, lic is a vector joining the centres of the reference particle and its cth neighbour, V is the area of the Delaunay polygon for the particle and pic is a vector defined as:
p ci = uic − ui + eij 3 l cj ω ,
(7)
where ui and u ic denote the displacement of the reference and contacting particle, respectively, and ω is the rotation of the reference particle. The difference between the actual particle displacement and that dictated by the microstrain is given by:
∆pic = pic − εij l cj .
(8)
Thus, the following scalar quantity provides a measure of the nonaffine strain experienced by each particle:
∆εNonAff =
1 2V
∑ ∆pɺ ( l c
c +1
− l c + lc − l c−1 .
c
)
(9)
To isolate the effects of particle rotations on nonaffine deformation, a local curvature can be defined for a particle
κi =
1 ∑ (φ c + φ c +1)eij 3 (l cj +1 − l cj ) 2V c ∈ B
(10)
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where the relative rotation at a branch vector is such that φ c = ω c − ω . The difference between the actual relative rotation and that dictated by the local curvature is thus
∆φ c = φ c − κ i lic .
(11)
Accordingly, a measure of the nonaffine curvature experienced by each particle with respect to its first ring of neighbours may be written as
∆κNonAff =
1 2V
∑ ∆φɺ ( l c
c∈B
c +1
− l c + l c − l c−1 .
)
(12)
The strain evolution of the global average of the local measures of nonaffine deformation in equations (9) and (12) is shown in Figure 3. A marked increase in the peaks in the global average values of local nonaffine strain and curvature can be observed during unjamming events in the softening regime. A comparison of Figures 2 and 3 indicates that, on the macroscopic scale, dissipation strongly correlates with local nonaffine deformation. To determine whether this correlation exist also on the local scale, we examined the spatial distributions of nonaffine deformation for each unjamming event. Results are presented in Figures 4 and 5 for the third unjamming event: these correspond to the third peak in the dissipation rate (or third drop in stress ratio). Local nonaffine deformation over six distinct stages of unjamming, i.e. stages A-F, with axial strains varying from 0.055 to 0.057, is presented. Apart from rattlers, i.e. a particle trapped inside its first ring of neighbours, which carry no force and do not contribute to dissipation [17], nonaffine deformation is mainly confined to the shear band. As the regions on either side of the band are in essentially rigid body motion, dissipation is thus confined to the shear band and correlates with the local nonaffine deformation. 4. Force chain buckling and nonaffine motion DEM simulations and experiments suggest that global unjamming nucleates from the buckling of a few force chains in the band e.g. [14,15,18,19]. This trigger event causes constituent particles and
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surrounding weak network particles to become mobilised. The mobilised particles, and concurrent weakening confinement, destabilise and induce buckling of adjacent force chains. This process continues, quickly spreading along the band, before causing the two outer regions on either side to unload elastically. From the standpoint of energy flow, buckling of force chains leads to a collective release of stored potential energy that is accumulated at all of the contacts, concomitant with the steady growth of contact forces, along the force chain prior to its collapse (see Figure 6). The ensuing dynamic rearrangements of particles induced by buckling and collapse of force chains are responsible for the nonaffine deformation, and provide the mechanism for energy dissipation at two length scales: friction at the contact scale and irreversible structural rearrangements at the mesoscopic scale. Consistent with our homogenisation scheme on the scale of a particle and its first ring of neighbours, we examined the contributions of 3-particle segments of force chains to the total drop in potential energy and particle-scale nonaffine deformation during unjamming events. This analysis made use of a recently developed technique for identifying, for a given strain state, which particles are in force chains as opposed to those in the complementary weak network [20,21]. The idea is to derive a single vector for a particle, i.e. particle load vector, out of its contact forces. The following tensor is computed for each particle
σ*ij = ∑ ˆric f jc
(13)
c
where rˆi c represents the components of the unit normal vector pointing from the centre of the particle to the point of contact. The largest eigenvalue of this tensor and its associated eigenvector correspond to the magnitude and direction of the particle load vector. The direction of force transmission is represented by the direction of the eigenvector, such that groups of particles whose particle load vectors line up, and whose particle load vector magnitude is above average, constitute a force chain.
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To establish the predominant mechanism governing dissipation during the unjamming stages in the strain-softening regime, we proceed as follows for each unjamming event – i.e. drop in macroscopic stress ratio. Bearing in mind that dissipation is mainly confined to the shear band, we focus on the group of particles responsible for the drop in potential energy: i.e. force chain and weak network particles in the band that have sustained a drop in the magnitude of their particle load vector: call this Group A. Out of group A, we identify all possible 3-particle segments belonging to those force chains that have buckled during the unjamming event.
Fig. 2. Evolution with axial strain of the potential energy Epot, rate of dissipation D. Included is the stress ratio expressed as the sine of the mobilized friction angle φ: peak of sinφ = 0.55 at εyy= 0.037.
∆εNonAff
∆κNonAff
sinφ
εyy
sinφ
εyy
Fig. 3. Evolution with axial strain of the stress ratio, sinφ, together with: (left) the global average of the particle-scale nonaffine strain 〈 ∆ εNonAff 〉 and (right) global average of the particle-scale nonaffine curvature 〈 ∆κNonAff 〉.
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A
B
C
D
E
F
Fig. 4. Distribution of local particle-scale nonaffine strain in stress ratio: white (zero) to black (high).
∆εNonAff
A
B
C
D
E
F
Fig. 5. Distribution of local particle-scale nonaffine curvature drop in stress ratio: white (zero) to black (high).
across the third drop
∆κNonAff
across the third
Buckling force chain and its implications for constitutive modelling
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Increasing strain
Fig. 6. Birth and death of a force chain: shown is a three-particle segment of the force chain and its confining weak network particles. Confined buckling of the force chain is a nonaffine mode of deformation.
For each of these buckled segments, we compute: a) the buckling angle θ =(θi-θf)/2, where θi, θf denote the angle between the branch vectors at the start and end of unjamming, respectively (solid and dashed line in Figure 7), and, for the central particle of the segment, b) the magnitude of the drop in potential energy and c) the nonaffine strain/curvature. We then consider groups of central particles belonging to segments whose buckling angle is such that θ>β. For each group B(β), we compute the average drop in potential energy, and nonaffine strain and curvature. We then normalize the average drop in potential energy and nonaffine strain/curvature for each group B(β) by the corresponding average of these quantities from Group A. The plots of these normalised quantities with the threshold angle β are shown in Figure 7. As the variational trends for these normalized quantities against β are qualitatively the same across each drop in the strain-softening regime, we averaged these across the first three drops in stress ratio (or first three unjamming events). The number of segments in B(β) diminishes with increasing β (data not shown). Figure 7 shows that central particles in buckling segments sustain above average levels of nonaffine strain/curvature and potential energy drop, and hence dissipation. Furthermore, the greater the degree of buckling θ, the greater are the drop in potential energy and nonaffine strain/curvature.
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∆εNonAff R ∆κNonAff ∆E pot
θi θf
β [°] Fig. 7. (Left) Illustration of the buckling angle θ = (θi-θf)/2 > 0. Variation with β of mean behaviour of central particles in buckled segments in group B(β) normalized by mean behaviour of all particles in shear band that have sustained a drop in potential energy. Mean behavior is shown for nonaffine strain and curvature, and magnitude of drop in potential energy. R denotes the average radius of the particles inside shear band. Note these quantities have been averaged across the first three unjamming events.
5. Conclusions Dissipation in granular systems transcend the contact scale. This study has highlighted the key role that force chain buckling plays in constitutive response. The mesostructural mechanism of force chain buckling subject to lateral confinement of weak network particles was found to be the predominant source of energy dissipation. As force chains generally align themselves with the major principal stress axis, they accumulate stored potential energy prior to collapse, concurrent with the steady growth of contact forces in this direction. However, force chains are prone to collapse via buckling due to loss of lateral confinement. Buckling leads to a coordinated release of the stored energy, manifesting itself on the macroscale via a drop in stress ratio and a concurrent peak in dissipation. This study suggests that this dissipative mechanism can be injected into constitutive formulations via local measures of the nonaffine deformation associated with the confined buckling and failure of force chains.
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Acknowledgments The author is grateful to her students Mr Sudhir Raskutti and Mr David Rafferty for assistance in the preparation of the figures and especially to Dr Stuart Walsh, Ms Maya Muthuswamy, Ms Nana Liu and Mr Stephen McAteer for numerous insightful discussions on granular systems that have served as inspiration for countless explorations of the rich complexities of this material. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
[11] [12] [13] [14] [15] [16]
[17] [18] [19] [20] [21]
A. Tordesillas, S.D.C. Walsh and B. Gardiner, BIT Num. Math. 44 539 (2004). S.D.C Walsh and A. Tordesillas, Acta Mechanica 167 145 (2004). K.C. Valanis, Acta Mechanica 116 1 (1996). I.F. Collins and G.T. Houlsby, Proc. R. Soc. London, Ser. A 453 1975 (1997). S. Nemat-Nasser and M. Hori, Micromechanics: Overall properties of heterogeneous materials (Elsevier, Amsterdam, 1993). I.F. Collins, Géotechnique 55 373 (2005). J. Lubliner, Plasticity theory (New York: Macmillan 1990). P.A. Cundall and O.D.L. Strack, Géotechnique 29 47 (1979). S.D.C. Walsh, A. Tordesillas and J.F. Peters, Granular Matter (in press) (2007). A. Tordesillas and S.D.C. Walsh, in Powders and Grains 2005: Proceedings of the 5th International Conference on the Micromechanics of Granular Media 1, edited by R. García-Rojo, H.J. Herrmann and S. McNamara (A.A. Balkema, Rotteredam, 2005) pp 419-424. K.L. Johnson, Contact Mechanics (Cambridge Univ. Press, 1985). J.-N. Roux, Phys. Rev. E 61 6802 (2000). N. P. Kruyt and L. Rothenburg, J. Stat. Mech. P07021 (2006). A. Tordesillas, Philosophical Magazine (accepted). N. Hu and J.F. Molinari, J. Mech. Phys. Sol. 52 499 (2004). S.D.C. Walsh, M. Muthuswamy and A. Tordesillas, “A definition of micropolar strain and curvature in discrete granular media”, in preparation. G. Marty and O. Dauchot, Phys. Rev. Lett. 94 015701 (2005). M. Oda and H. Kazama, Géotechnique 48 465 (1998). A.L. Rechenmacher, J. Mech. Phys. Solids 54 22 (2006). J.F. Peters, M. Muthuswamy, J. Wibowo and A. Tordesillas, Phys. Rev. E 72 041307 (2005). M. Muthuswamy and A. Tordesillas, J. Stat. Mech. P09003 (2006).
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Chapter 7 Investigations of size effects in granular bodies during plane strain compression
J. Tejchman* and J. Górski† Faculty of Civil and Environmental Engineering Gdańsk University of Technology, Poland 80-952 Gdansk-Wrzeszcz, Narutowicza 11/12 *
[email protected] †
[email protected]
The paper deals with numerical investigations of size effects in granular bodies during quasi-static plane strain compression under plane strain conditions and constant lateral pressure. For a simulation of the mechanical behaviour of a cohesionless granular material during a monotonous deformation path, a micro-polar hypoplastic constitutive relation was used which takes into account particle rotations, curvatures, non-symmetric stresses, couple stresses and the mean grain diameter as a characteristic length. Deterministic calculations were carried out with an uniform distribution of the initial void ratio for geometrically similar granular specimens of six different sizes. To investigate a statistical size effect, a Latin hypercube method as one of Monte Carlo reduction methods was applied. The random distribution of the initial void ratio was assumed to be spatially correlated. Truncated Gaussian random fields were generated using an original conditional rejection method. Some general conclusions were formulated.
1. Introduction One of the salient characteristics of the behaviour of granular and brittle materials is a size effect phenomenon, i.e. experimental findings vary with the size of the specimen. In general, the shear resistance in granular material1,2,3,4,5 and tensile strength in brittle ones6,7,8,9,10 increase with 111
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decreasing specimen size during many experiments including strain localization. The specimen ductility (ratio between the energy consumed during a shearing or fracture process after and before the peak) also grows. Thus, the results from laboratory tests which are scaled versions of the actual structures cannot be directly transferred to them. Two main size effects can be defined: deterministic and statistical. The first one is caused by strain localization which cannot be appropriately scaled in laboratory tests. Thus, the specimen strength increases with increasing ratio lc/L (lc – characteristic length of microstructure influencing both the thickness and spacing of localized zones, L – specimen size). This feature is strongly influenced by the pressure level in granular materials. The statistical effect is due to the presence of the randomness of the local material strength caused by number of weak spots whose amount usually grows with increasing specimen size. Thus, the specimen strength diminishes with increasing specimen size. In dynamic problems, this effect can be called a stochastic one. Up to now, the size effects are still not taken into account in the specifications of most of design codes for engineering structures. For quasi-brittle and brittle materials, there exist only few reliable approaches to the size effect phenomenon. For example, two deterministic size effect laws by Bazant and Planas7 allow to take into account a size difference by determining the tensile strength for prenotched structures and structures without an initial crack. The material strength is bound for small sizes by the plasticity limit whereas for large sizes the material follows linear elastic fracture mechanics. In the case of a statistical size effect, the most known is the Weibull or the weakest link theory11 based on a distribution of flaws in materials. It postulates that a structure is as strong as its weakest component. When its strength is exceeded, the structure fails since the stress redistribution is not considered. This model is not able to account for a spatial correlation between local material properties. Another approach to size effect was proposed by Carpintieri et al.12 which was based on the multifractality of a fracture surface which increased with spreading disorder of the material in large structures. In this approach, the material strength is bound for small and large sizes by the plasticity limit. According to Bazant and Yavari,13 the cause of size effect is energetic-statistical not fractal. The
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numerical calculations of a size effect in concrete specimens under tension with a microplane material model show14 that a statistical strength (obtained from random sampling) can be larger than a deterministic one in small specimens in contrast to large specimens which rather obey the weakest link model. The difference between a deterministic material strength and a mean statistical strength grows with increasing size. The structural strength exhibits a gradual transition from Gaussian distribution to Weibull distribution at increasing size.15 In the case of granular materials, no reliable size effect laws were proposed due to the fact that a performance of laboratory tests is more complex than in brittle materials. In addition, the effect of pressure is very pronounced. Deterministic size effects have been studied in granular materials intensively by numerous researchers using a FE-method based on enhanced continua including a characteristic length of microstructure16 and a strong discontinuity approach.17 In turn, in the case of statistical size effects, non-linear calculations are only few.18,19,20 The intention of the numerical simulations was to investigate a deterministic and statistical effect in cohesionless granular materials like sand with consideration of shear localization under quasi-static conditions by using a finite element method.21 These effects were investigated during a plane strain compression test under constant lateral pressure which is one of the most important laboratory geotechnical tests to measure the properties of granular materials.22,23,24 A finite element method with a micro-polar hypoplastic constitutive model25,26 was used which is able to describe the essential properties of granular bodies during shear localization in a wide range of pressures and densities during monotonous deformation paths. The deterministic calculations were performed with an uniform distribution of the initial void ratio in dense sand for 6 geometrically similar specimens of different sizes changing from 10×35 mm2 up to 320×1120 mm2 under constant lateral confining pressure of 200 kPa. The statistical size effect analyses were carried out with spatially correlated homogeneous distributions of an initial void ratio for two specimen sizes of dense sand: 40×140 mm2 (called medium size specimen) and 320×1120 mm2 (called large size specimen). The truncated Gaussian random fields were generated using a conditional rejection method.27,28 The approximated results were
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obtained using a Latin hypercube sampling method29,30 belonging to a group of reduction Monte Carlo methods.31 This approach allowed us for a significant reduction of the sample number without loosing the accuracy of calculations. Due to the lack of experimental data, the FEresults could not been compared with laboratory tests except of results for the specimen size of 40×140 mm2 which were confronted with corresponding laboratory experiments performed by Vardoulakis22 and Vardoulakis et al..32 This paper continues the research presented by Tejchman and Górski33 where a deterministic and statistical size effect were investigated during quasi-static shearing of an infinite granular layer between two very rough boundaries under constant vertical pressure using the direct Monte Carlo method and two reduction approaches: stratified sampling and Latin hypercube sampling. The calculations have shown that the solution of random nonlinear problems on the basis of several samples is possible. The most efficient reduction method is the stratified sampling with intervals described by equal probabilities. The deterministic size effect is rather small. The shear resistance at peak and at residual state decrease slightly with increasing ratio of the layer height ho and mean grain diameter d50. However, the material brittleness strongly increases with increasing ho/d50. The mean random shear resistance at peak is always smaller as compared to this with the deterministic uniform distribution. It diminishes with increasing ho/d50. Thus, the statistical size effect is stronger than the deterministic one. The weakest link principle always applies due to that shear localization forms in the horizontal weakest layer. The insight into physical mechanisms of the size effects is of a major importance for civil engineers to extrapolate experimental findings at laboratory scale to results which can be used in real field situations. Since large structures are far beyond the range of failure testing, their design must rely on a realistic extrapolation of testing results with smaller specimens. The present paper is organized as follows. In Section 2, a micro-polar hypoplastic model is briefly described. Section 3 deals with the simulation of discrete random fields. The information about the finite element discretisation and boundary conditions is given in Section 4. The
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numerical results of the deterministic and statistical size effects are presented and discussed in Section 5. Finally conclusions are given in Section 6. 2. Micro-polar hypoplastic model Despite a discrete nature of granular materials, their mechanical behaviour can be reasonably described by principles of continuum mechanics using an elastoplastic34,35,36 and hypoplastic37,38,39,40 approach. Non-polar hypoplastic constitutive models41,42,43 describe the evolution of the effective stress tensor depending on the current void ratio, stress state and rate of deformation by isotropic non-linear tensorial functions. The tensorial functions can be obtained according to the representation theorem by Wang.44 The constitutive models were formulated by a heuristic process considering the essential mechanical properties of granular materials undergoing homogeneous deformation. A striking feature pertinent to hypoplasticity is that the constitutive equation is incrementally linear in deformation rate. The hypoplastic models are capable of describing a number of significant properties of granular materials: non-linear stressstrain relationship, dilatant and contractant volumetric change, stress level dependence, density dependence and material softening. A further feature of hypoplastic models is the inclusion of critical states, i.e. states in which a grain aggregate can deform continuously be deformed at constant stress and a constant volume. In contrast to elasto-plastic models, a decomposition of deformation components into elastic and plastic parts, the formulation of a yield surface, plastic potential, flow rule and hardening rule are not needed. The hallmark of these models are their a simple formulation and procedure for determining material parameters with standard laboratory experiments. The material parameters are related to the granulometric properties of granular materials, such as grain size distribution curve, shape, angularity and hardness of grains.45 A further advantage lies in the fact that one single set of material parameters is valid for a wide range of pressures and densities. An exhaustive review of the development of hypoplasticity can be found in Wu40, Wu and Kolymbas46 and Tamagnini et al..47 To increase the application range, a hypoplastic constitutive law has been extended by an elastic strain
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range,48 anisotropy,49,26 viscosity,50 soils with low friction angles51 and clays.52 It has been shown that hypoplastic constitutive models without a characteristic length can describe realistically the onset of shear localization, but not its formation.53 A characteristic length can be introduced into hypoplasticity by means of a micro-polar, non-local and second-gradient theory.53,16 In this paper, we adopted the micro-polar theory. A micro-polar model makes use of rotations and couple stresses, which have clear physical meaning for granular materials. The rotations can be observed during shearing and but remain negligible during homogeneous deformation.54 Pasternak and Mühlhaus55 have demonstrated that the additional rotational degree of freedom of a micropolar continuum arises naturally by mathematical homogenization of an originally discrete system of spherical grains with contact forces and contact moments. A micro-polar continuum considers two linked levels of deformations at two different levels, i.e.: micro-rotation at the particle level and macrodeformation at the structural level.56 For the case of plane strain, each material point has three degrees of freedom: two translations and one independent rotation (Fig. 1). The gradients of the rotation are related to the curvatures, which are associated with the couple stresses. The presence of the couple stresses gives rise to a non-symmetry of the stress tensor and a presence of a characteristic length. The constitutive relationship between the rate of stress rate, the rate of couple stress, the strain rate and the curvature rate can be generally expressed by the following two equations:25,57,26 o
σ ij = Fij ( e,σ kl ,mi ,d klc ,ki ,d 50 ) , o
mi = Gi e,σ kl ,mi ,d klc ,ki ,d 50 .
(
)
(1) (2)
The Jaumann stress rate and Jaumann couple stress rate in the above equations are defined by o
•
σ ij = σ ij − wikσ kj + σ ik wkj
(3)
and o
•
mi = mi − 0.5wik mk + 0.5mk wki .
(4)
Investigations of size effects in granular bodies
117
a)
b) Fig. 1. Plane Cosserat continuum: a) degrees of freedom (u1 - horizontal displacement, u2 vertical displacement, ωc - Cosserat rotation), b) stresses σij and couple stresses mi at an element.
The functions Fij and Gi in Eqs. (1) and (2) represent isotropic tensorvalued functions of their arguments; σij is the Cauchy stress tensor, mi is the couple stress vector, e denotes the current void ratio, dklc is the polar strain rate and ki denotes the rate of curvature vector: d ijc = dij + wij − wijc ,
and
ki = w,ic .
(5)
The rate of deformation tensor dij and the spin tensor wij are related to the deformation velocity vi as follows: d ij = (vi , j + v j ,i ) / 2,
wij = (vi , j − v j ,i ) / 2,
( ),i = ∂ ( ) / ∂ xi .
(6)
The rate of Cosserat rotation wc is defined by c w21 = − w12c = wc
and
c wkk =0.
(7)
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J. Tejchman and J. Górski
For moderate stress level, the grains of granular materials can be reasonably assumed to be incompressible. In this case, the change of void ratio depends only on the strain rate: ∗
e = (1 + e ) d kk .
(8)
For the numerical calculations, the following micro-polar hypoplastic constitutive equation are considered:25 o
σ ij = f s [ Lij (σ^ kl ,m^ k ,d klc ,kk d 50 ) + ^
+ f d N ij (σ ij ) d klc d klc + kk kk d502 ]
(9)
and o
^
^
mi / d50 = f s [ Lci (σ kl ,m k ,d klc ,kk d 50 ) + ^
2 ], + f d Nic (mi ) d klc d klc + kk kk d 50
(10)
wherein the normalized stress tensor σˆ ij is defined by
σ^ ij =
σ ij σ kk
(11)
ˆ i is defined by and the normalized couple stress vector m ∧
mi =
mi , σ kk d50
(12)
wherein d50 is the mean grain diameter of soil. The scalar factors fs = fs (e, σkk) and fd = fd (e, σkk) in Eqs. 9 and 10 describe the influence of density and stress level on the incremental stiffness. The factor fs depends on the granulate hardness hs, the mean stress σkk, the maximum void ratio ei and the current void ratio e: h 1 + ei fs = s nhi ei
1− n
ei σ kk − e hs β
(13)
with α
1 1 e −e 1 hi = 2 + − i0 d 0 . c1 3 ec0 − ed 0 c1 3
(14)
Investigations of size effects in granular bodies
119
In the above equations, the granulate hardness hs represents a reference pressure similar to the atmospheric pressure, the coefficients α and β express the dependence on density and stress level respectively, and n denotes the compression coefficient. The multiplier fd represents the dependence on the relative density: α
e − ed fd = (15) . ec − ed The relative density in the above expression involves the void ratio in critical state ec, the minimum void ratio ed (the densest packing) and the maximum void ratio ei (the loosest packing). In a critical state, granular material experiences continuous deformation while the void ratio remains unchanged. The current void ratio e is bounded by the two extreme void ratios ei and ed. Based on experimental observations, the void ratios ei, ed and ec are assumed to depend on the stress level σkk (Fig. 2): n ei = ei0 exp − ( −σ kk / hs ) ,
(16)
n ed = ed 0 exp − ( −σ kk / hs ) ,
(17)
n ec = ec0 exp − ( −σ kk / hs ) ,
(18)
wherein ei0, ed0 and ec0 are the values of ei, ed and ec at σkk = 0, respectively.
a)
b)
Fig. 2. Relationship between void ratios ei, ec and ed and mean pressure ps in a (a) logarithmic and (b) linear scale (grey zones denote inadmissible states).
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J. Tejchman and J. Górski
For the functions Lij, Nij, Lic and Nic, the following specific expressions are used:25 ^
^
^ Lij = a12 d ijc + σ ij (σ kl d klc + m k kk d 50 ) ,
(19)
^ Lci = a12 ki d 50 + a12 m^ i (σ^ kl d klc + m k k k d 50 ) ,
(20)
^ ^ N ij = a1 σ ij + σ ij* ,
(21)
^ N ic = a12 ac m i,
(22)
where ^
^
a1−1 = c1 + c2 σ kl* σ lk* [1 + cos(3θ )], cos(3θ ) = −
6
^
^ * ^ * 1.5 pq pq
^
^
* * (σ kl* σ lm σ mk )
(23) (24)
[σ σ ] with c1 =
3 (3 − sinφc ) , 8 sinφc
c2 =
3 (3 + sinφc ) . 8 sinφc
(25)
The parameter φc is the friction angle in critical state and the parameter θ * denotes the Lode angle in the deviatoric plane at σˆ ii = 1 , and σˆ ij denotes the deviatoric part of σˆ ij . The micro-polar parameter ac in Eq.(22) can be correlated with the grain roughness. This correlation can be established by studying the shearing of a narrow granular strip between two rough boundaries.25 It can be represented by a constant, e.g. ac=1-5, or connected to the parameter a1−1 , e.g. ac=(0.5−1.5)× a1−1 . The parameter a1−1 lies in the range of 3.0-4.3 for the usually critical friction angle. The constitutive relationship requires the following ten material parameters: ei0, ed0, ec0, φc, hs, β , n ,α, ac and d50. The parameters hs and n are estimated from a single oedometric compression test with an initially loose specimen (hs reflects the slope of the curve in a semi-logarithmic representation, and n its curvature). The parameters α and β can be determined from a triaxial or plane strain test with a dense specimen and trigger the magnitude and position of the peak friction angle. The critical
Investigations of size effects in granular bodies
121
friction angle φc can be determined from the angle of repose or measured in a triaxial test with a loose specimen. The parameters of ei0, ed0, ec0 and d50 are obtained from conventional index tests (ec0 ≈ emax, ed0 ≈ emin, ei0 ≈ (1.1-1.5)emax). The calibration procedure was given in detail by Herle and Gudehus.45
3. 2D random field analysis using Latin hypercube sampling In the case of nonlinear calculations the only reliable solution is the direct Monte Carlo method. Contrary to stochastic finite element codes, the Monte Carlo method does not impose any restriction to the solved random problems. Using this method, a FE deterministic program can be implemented in the probabilistic analysis without any improvements. It should be pointed out that formulation of a stochastic finite element code concerning the micro-polar hypoplastic law is very difficult (if not impossible). The only limitation of the Monte Carlo method is the time of calculations. It is estimated that to reproduce exactly the input random data, as much as 2000 random samples should be used.58,28 Any nonlinear calculations for such number of initial data are, however, impossible due too long computation time. To determine the minimal but sufficient number of samples (which allows to estimate the results with a prescribed accuracy), a convergence analysis of the outcomes has been proposed.58,33 It has been estimated that in case of various engineering problems only tens, for example 50 realizations should be considered. A further decrease of sample numbers can be obtained using the Monte Carlo reduction methods. In Tejchman and Górski,33 a stratified and Latin sampling method were considered. The calculations has proved that using these reduction methods the results can be properly estimated by several realizations. Following the conclusion, in this work the Latin sampling method was applied. According to the Latin hypercube sampling method, the random field realizations were chosen in a strictly defined manner. First, the initial set of random samples was generated in the same way as in the case of the direct Monte Carlo method. Next, the generated samples were classified according to chosen parameters, for example: norms of the random vectors, mean values, the changes of the vector signs, and others. The
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J. Tejchman and J. Górski
samples were arranged according to this classification. The whole space of the samples was divided into subsets of equal probability and numbered. The Latin hypercube sampling method combined at random each subset number with other subset numbers of the remaining variables only once. From each defined in this way subset, only one sample was chosen for the analysis. The input data of the considered problem is a set of random fields describing the initial void ratio eo. A truncated Gaussian random field is applied
eo = eo (1 + νβ ( x1 , x2 ) ) ,
(26)
where eo is the mean value of the initial void ratio, ν = seo / eo is the coefficient of variation, seo describes the standard deviation of the mean value, and β ( x1 , x2 ) stands for the normalized homogeneous random field. Randomness of the initial void ratio was described by the following homogeneous correlation function59
K ( x1 , x2 ) = se2o × e
− λx1 ∆x1
(1 + λx1 ∆x1 )e − λx 2∆x2 (1 + λx2 ∆x2 ),
(27)
where ∆x1 and ∆x2 is are the distances between two field points along the horizontal axis x1 and vertical axis x2, λx1 and λx2 are the decay coefficients characterizing a spatial variability of the specimen properties (i.e. describe the correlation between the random field points) while the standard deviation seo represents the field scattering. The random fields were generated using a conditional rejection method proposed by Walukiewicz et al..27 According to his method a discrete random field was described by multidimensional random variables defined at mesh nodes. The random variable vector of initial void ratio eo (m × 1) was divided into blocks consisting of the unknown eou (n × 1) and the known eok ( p × 1) elements (n + p = m) . The covariance matrix K (m × m) and the expected values vector eo (m × 1) were also appropriately split:
e n eo = ou , eok p
K12 n K K = 11 , K 21 K 22 p
e n eo = ou . eok p
(28)
The unknown vector eou was estimated from the following conditional truncated distribution
Investigations of size effects in granular bodies
f t ( e ou eok ) = (1 − t )
−m / 2
( det K c )
−1/ 2
( 2π )
−m/ 2
123
×
1 T −1 e − e K e − e ( ) ( ) , ou oc c ou oc 2 (1 − t )
× exp −
(29)
where K c and eoc are described as the conditional covariance matrix and conditional expected value vector:
K c = K11 − K12K −221K 21 ,
(30)
eoc = xu + K12 K −221 ( eok − eok ) .
(31)
The constant t is the truncation parameter t=
seo ⋅ exp(− se2o 2) 2π erf ( seo )
(32)
with se
1 0 x2 exp( − )dx . (33) ∫ 2 2π 0 According to the conditional rejection method, any mesh point value is generated using the values calculated earlier. When large problems are solved, such approach is inefficient. Therefore, a “base scheme” was defined.27 The scheme covered a limited mesh area (hundred points) and only these points were used in the calculations of the next random values (Fig. 3). The simulation process was divided into three stages. First, the four-corner random values were generated (Fig. 3a) using an unconditional method. Next, all random variables in the defined base scheme (dotted rectangle in Fig. 3) were generated, one by one, using the conditional method (Fig. 3b). In the third stage, the base scheme was appropriately shifted, and the next group of unknown random values was simulated (Fig. 3c). The base scheme was translated so as to cover all the field nodes (Fig. 3d). It should be pointed out that this approach allows for generation of practically unlimited random fields (thousand of discrete points). To describe the discrepancies between the theoretical (Eq. 27) and generated fields the, following global Ger and local Ver errors are calculated:27 erf ( se0 ) =
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J. Tejchman and J. Górski
N
simulated value unknown values simulated in one base scheme step
d
a)
3
4
1
2
known values
b)
M
c
known values not included in the base scheme calculations
the next step of the base scheme shifting
d)
c)
28
Fig. 3. Successive coverage of the field points with the moving propagation scheme
Ger
(
ˆ K − K
ˆ = K, K
)
m
Ver
(
)
× 100% ,
(34)
) ×100% ,
(35)
K
kii , kˆii = ∑ i =1
(k
ii
− kˆii
(k )
.
ii
ˆ is the estimator of the covariance matrix K where K ˆ = K
1 NR (eˆ oi − eˆo )(eˆ oi − eˆo )T , ∑ NR − 1 i =1
eˆo =
1 NR ∑ eˆ oi . NR i =1
(36)
eˆ o is the estimator of the random vector eo , eˆo describes the mean value of eˆ o , NR denotes the number of realizations, K = (tr (K ) 2 )1/ 2 is the matrix norm, kii and kˆii denote the diagonal element of the covariance ˆ , respectively. matrix K and its estimators K
Investigations of size effects in granular bodies
125
4. FE-input data 4.1. Deterministic calculations The FE-calculations of a plane strain compression test (assuming an uniform distribution of the initial void ratio eo) were performed with 6 different sand specimen sizes bo×ho (which were geometrically similar): 10×350 mm2, 20×700 mm2, 40×140 mm2, 80×280 mm2, 160×560 mm2 and 320×1120 mm2. The specimen depth was l = 1.0 m due to plane strain conditions. The specimen dimensions of 40×140 mm2 were similar as in the experiments by Vardoulakis22 and Vardoulakis et al..32 In all cases, 896 quadrilateral elements divided into 3584 triangular elements were used. The quadrilateral elements composed of four diagonally crossed triangles were used to avoid volumetric locking due to dilatancy effects.60 Linear shape functions for displacements and the Cosserat rotation were used. The integration was performed with one sampling point placed in the middle of each element. To properly capture shear localization inside of the granular specimen, the size of the finite elements se should be not larger than five times mean grain diameter d50.61 For the specimen sizes changing from 10×35 mm2 up to 40×140 mm2 this condition was fulfilled (se ≤ 5×d50). However, for the specimen sizes larger than the size 40×140 mm2, this condition was violated (e.g se = 40×d50 for the size of 320×1120 mm2). Thus, these FE-results were mesh-dependent; the mesh dependence increased with increasing specimen size. A further increase of the amount of finite elements would too drastically increase the computation time. In turn, a remedy in the form of a remeshing technique (which is effective in such cases)62 was not the aim of these FE-analyses. A quasi-static deformation in sand was imposed through a constant vertical displacement increment ∆u prescribed at nodes along the upper edge of the specimen. The boundary conditions implied no shear stress imposed at the smooth top and bottom of the specimen. To preserve the stability of the specimen against horizontal sliding, the node in the middle of the top edge was kept fixed. To simulate a movable roller bearing in the experiment,32 the horizontal displacements along the specimen bottom were constrained to move by the same amount. Thus, no imperfections were used to induce shear localization with an uniform
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J. Tejchman and J. Górski
distribution of eo. The vertical displacement increments were chosen as ∆u/ho = 0.0000025. Some 3000 steps were performed. As the initial stress state, a K0-state with σ22 = γdx2 and σ11 = K0γdx2 was assumed in the specimen; x2 is the vertical coordinate measured from the top of the specimen, γd = 16.5 kN/m3 denotes the initial volume weight and K0 = 0.50 is the earth pressure coefficient at rest (σ11 horizontal normal stress, σ22 - vertical normal stress). Next, constant confining pressure of σc = 200 kPa was prescribed. For the solution of a non-linear equation system, a modified NewtonRaphson scheme with line search was used. The global stiffness matrix was calculated with only line terms of the constitutive equations (Eqs. 1 and 2). The stiffness matrix was updated every 100 steps. In order to accelerate the convergence in the softening regime, the initial increments of displacements and Cosserat rotations in each calculation step were assumed to be equal to the final increments in the previous step. The procedure was found to yield a sufficiently accurate solutions with a fast convergence. The magnitude of the maximum out-of-balance force at the end of each calculation step was found to be smaller than 2% of the calculated total vertical force of the granular specimen. Due to the presence of non-linear terms in deformation rate and material softening, this procedure turned out to be more efficient than the full NewtonRaphson method. The iteration steps were performed using translational and rotational convergence criteria. For the time integration of stresses in finite elements, a one-step Euler forward scheme was applied. The calculations were carried out with large deformations and curvatures using the so-called “Updated Lagrangian” formulation due to their effect on the results during plane strain compression.16 4.2. Statistical calculations (Latin hypercube sampling) When a stochastic distribution of the initial void ratio was assumed, 2 different specimen sizes were used: 40×140 mm2 (medium size) and 320×1120 mm2 (large size). In the probabilistic calculations, the same assumption as in the case of deterministic analysis were applied. The mean value of the initial void ratio was eo = 0.60 (initially dense sand). One chose a small standard deviation seo = 0.05 to generate the random
Investigations of size effects in granular bodies
127
Random vector max-min gaps
fields (Eq. 26) with λx1 = 1 and λx2 = 3. Thus, a strong correlation of eo in a horizontal direction and a weak correlation of eo in a vertical direction was assumed. The initial void ratio scattering in the specimen was also limited by the pressure dependent void ratios eio (upper bound) and edo (lower bound) (Eqs. 16 and 17). The truncation parameter t of the Gaussain field (Eq. 32) allows to fulfill these conditions. The midpoint method was applied which approximated the random field in each finite element by a single random variable defined as the value of the field at its centre. Thus, the dimension of the random field was the same as the finite element mesh, i.e. m = 16 × 56 = 896 points. The following dimension of the base scheme was applied c = d = 16 (Fig. 3). Using the algorithm described in Section 3, 2000 field realizations of the initial void ratio were generated. The global Ger and local (variance) Ver errors of the generation were calculated (Eqs. 34 and 35): Ger = 2.96% , Ver = 1.26% . Next, the generated field were classified according to two parameters: the mean value of the initial void ratio and the gap between the lowest and the highest value of the initial void ratio. The joint probability distribution (so called “ant hill”) is presented in Fig. 4.
0.24 0.22 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.58
0.59 0.6 0.61 0.62 Random vector mean values
0.63
Fig. 4. Selection of 12 pairs of random samples using Latin hypercube sampling: 1 – 10, 2 – 1, 3 –11, 4 – 5, 5 – 5, 6 – 6, 7 – 2, 8 – 4, 9 – 3, 10 – 12, 11 – 9, 12 – 7.
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J. Tejchman and J. Górski
One dot in Fig. 4 represents one random vector described by its mean value and the difference between its extreme values. The two variable domains were divided in 12 intervals of equal probabilities (see vertical and horizontal lines in Fig. 4). Next, according to the Latin hypercube sampling assumptions, 12 random numbers in the range 1-12 were generated (one number appeared only once) using the uniform distribution. The generated numbers formed the following 12 pairs: 1 – 10, 2 – 1, 3 –11, 4 – 5, 5 – 5, 6 – 6, 7 – 2, 8 – 4, 9 – 3, 10 – 12, 11 – 9 and 12 – 7. According to these pairs, the appropriate areas (subfields) were selected (they are presented as rectangles in Fig. 4). From each subfield only one realization was chosen and used as the input data to the FEM calculations. In this way the, results of 12 realizations were analyzed. 5. FE-results 5.1. Deterministic size effect Figures 5-6 show results for an initially dense sand during plane strain compression for 6 different specimen sizes (geometrically similar): 10×35 mm2, 20×70 mm2, 40×140 mm2, 80×280 mm2, 160×560 mm2 and 320×1112 mm2 with the uniform distribution of the initial void ratio eo = 0.60 and lateral confining pressure σc = 200 kPa. Fig. 5 presents the evolution of the normalized vertical force P/(σcbol) against the normalized vertical displacement of the top boundary u2t/ho (the force is related to the initial specimen width). The deformed FE-meshes for different sizes at u2t/ho = 0.075 with the distribution of the distribution of equivalent total strain ε = ε ij ε ij are shown in Fig. 6. Tab. 1 includes the information about the maximum normalized vertical force P/(σcbol), mobilized overall internal friction angle at peak φp, vertical strain corresponding to the maximum force u2t/ho and normalized shear zone thickness ts/d50. The mobilized overall internal friction angle was calculated from the formula including the principal stresses on the basis of the Mohr’s circle σ −σ2 φ = arcsin 1 (37) σ1 + σ 2 with σ1 = P/(bl) and σ2 = σc (b – actual width).
Investigations of size effects in granular bodies
8
8
P/(σcbol)
6
a
f d
4
b
6 c 4
e
2 0
129
2
0
0.02
0.04
0.06
0.08
0 0.10
t
u2/h0 Fig. 5. Evolution of vertical normalized forces versus vertical normalized displacements of the top edge during compression with a dense specimen (eo=0.60) for different specimen sizes: a) 10×35 mm2, b) 20×70 mm2, c) 40×140 mm2, d) 80×280 mm2, e) 160×560 mm2, f) 320×1120 mm2 (with uniform distribution of eo).
a)
b)
c)
d)
e)
f)
Fig. 6. Deformed FE-meshes with the distribution equivalent total strain measure during compression with a dense specimen for different specimen sizes (with uniform distribution of the initial void ratio eo=0.60): a) 10×35 mm2, b) 20×70 mm2, c) 40×140 mm2, d) 80×280 mm2, e) 160×560 mm2, f) 320×1120 mm2) (specimens are not proportionally scaled).
130
J. Tejchman and J. Górski
Table 1. The values of maximum normalized vertical force P/(σcbl), internal friction angle at peak φ, vertical strain corresponding to the maximum vertical force u2t/ho, normalized shear zone thickness and ts/d50 (uniform distribution of initial void ratio with eo=0.60). ts/d50
at peak
u2t/ho at peak
6.10
45.92
0.0250
7
Specimen size bo×ho
P/(σcbol) at peak
10×35 mm2 20×70 mm2
φ p [o ]
6.04
45.71
0.0240
11
2
6.01
45.62
0.0236
14
80×280 mm2
5.99
45.55
0.0233
28
160×560 mm
5.97
45.48
0.0231
46
320×1120 mm2
5.94
45.38
0.0229
46
40×140 mm
2
The resultant vertical force on the specimen top increases first, shows a pronounced peak, drops later and reaches slowly a residual state. The strength and ductility increase with decreasing specimen size. The mobilized friction angle at peak φp varies from: φp = 45.38o (32×1120 mm2) up to φp = 45.92o (10×35 mm2). The vertical strain corresponding to the peak grows with decreasing specimen size (from 2.29% up to 2.50%). The mobilized internal friction at residual state is practically the same for all specimen sizes; it is approximately φres = 34o at u2t/ho = 10%. The obtained results of internal friction angles at the peak and at the residual state, and the corresponding normalized vertical displacements of the sand specimen for the specimen of 40×140 mm2 (φp = 45.6o, φres = 34o, u2t/ho = 0.0236) are in a satisfactory agreement with corresponding laboratory results with Karlsruhe sand carried out by Vardoulakis22 and Vardoulakis et al.,32 where the dimensions of the sand specimen were: ho = 140 mm, bo = 40 mm and l = 80 mm, respectively. The experiments with dense sand (eo = 0.55-0.60) resulted in the following values of φp = 45o-48o and φcr = 32o-33o at σc = 200 kPa. However, the calculated stiffness is too high before the peak (in the hardening region). The thickness of the internal shear zone appearing inside of the specimen (at mid-point of the specimen) increases with increasing specimen size; it varies between 7×d50 up to 46×d50. The thickness of the
Investigations of size effects in granular bodies
131
shear zone for the specimen sizes larger than 40×140 mm2 is certainly influenced by the mesh discretisation (as it was noted in Section 4). The thickness of the shear zone was determined on the basis of shear deformation and Cosserat rotation. To define the edges of the shear zone, one assumed that the Cosserat rotation larger than 0.1 occurred in the shear zone. The calculated thickness for the specimen of 40×140 mm2 , ts = 14×d50, is similar as the observed thickness, ts = 15×d50, during plane strain compression tests with dense sand (eo = 0.60) at σc = 200 kPa.22 The calculated inclination of the shear zone (about θ = 53o-54o) is also similar as in the experiment with dense sand (55o-60o). The thickness of the shear zone on the basis of void ratio is slightly larger since each dense granulate undergoes dilatancy before shear localization occurs. 5.2. Statistical size effect
5.2.1. Medium sand specimen (40×140 mm2)
P/(σcbol)
Figs. 7-9 present the results obtained for the same set of N=12 random fields in a medium size specimen (40×140 mm2) using Latin hypercube sampling of Fig. 4. 8
8
6
6
4
4
2
2
0
0
0.02
0.04
0.06
0.08
0 0.10
t
u2/ho Fig. 7. Evolution of 12 load-displacement curves during compression with a dense specimen for different random fields of eo ( eo =0.60, specimen size 40×140 mm2).
J. Tejchman and J. Górski
132
a)
g)
b)
h)
c)
i)
d)
j)
e)
k)
f)
l)
Fig. 8. Deformed FE-mesh with the distribution equivalent total strain measure during compression with a dense specimen for 12 different random fields of eo ( eo =0.60, specimen size 40×140 mm2).
The evolution of the normalized vertical forces P/(σcbol) against the normalized vertical displacement of the top boundary u2t/ho and the deformed FE-meshes at u2t/ho=0.075 with the distribution of equivalent total strain ε = ε ij ε ij are shown in Figs. 7 and 8, respectively. Tab. 2 includes the values of the maximum normalized vertical force P/(σcbol), mobilized overall internal friction angle at peak φ p, vertical strain corresponding to the maximum force u2t/ho and normalized shear zone thickness ts/d50 for 12 random fields. In addition, the evolution of void ratio in the specimen (Fig. 9) is shown for the field of Fig. 8b. The evolution of the normalized vertical force is quantitatively the same as for an uniform distribution of eo. For 3 cases, the residual vertical normalized force (related to the initial specimen width) is higher
Investigations of size effects in granular bodies
133
since the interior shear zone hits a top boundary (Figs. 8a, 8c and 8i) influencing its actual width (in the remaining cases, a shear zone intersects the vertical sides). The shear zone develops inside of the specimen somewhere at the weakest spot depending on the initial distribution of eo (Fig. 8). The shear zone width changes between 15×d50 and 19×d50 (on the basis of the Cosserat rotation).
a)
b)
c)
d)
e)
f)
Fig. 9. Evolution of void ratio field during plane strain compression test (random field of Fig. 8b) at: a) u2t/ho=0.007, b) u2t/ho=0.014, c) u2t/ho=0.021, d) u2t/ho=0.028, e) u2t/ho=0.036, f) u2t/ho=0.071 (specimen size 40×140 mm2).
During deformation (Fig. 9), first, a pattern of shear zones can observed in the sand specimen. Next, strain localization continues to localize within a single zone. The shear zone becomes well visible at u2t/ho=2% (Fig. 9c). In the specimen region beyond the shear zone, small changes of void ratio are visible what is in agreement with experiments by Yoshida et al..23 The estimated expected value and standard deviations were respectively: for the friction angle at peak φˆp = 44.17o and sˆφ p = 0.77, the normalized horizontal displacement of the top boundary u1t/ho t corresponding to the peak uˆ1 / ho =0.0204 and sˆut / ho =0.0014 and the 1
134
J. Tejchman and J. Górski
normalized shear zone thickness tˆs / d 50 =15.9 and sˆts / d 50 =1.7. The mean random internal friction angle at peak ( φˆp = 44.17o) is by 1.2o smaller than this with the uniform initial void ratio φ p = 45.38o. In turn, the mean random shear normalized zone thickness (ts/d50 = 15.9) is by 15% larger than this with the uniform initial void ratio ts/d50 = 14.0 (due to a smaller softening rate). The mobilized internal friction at residual state (related to the actual specimen width b) is approximately φres = 33.0o at u2t/ho=10%. Table 2. The values of maximum normalized vertical force P/(σcbol), internal friction angle at peak φ, vertical strain corresponding to the maximum vertical force u2t/ho, normalized shear zone thickness and ts/d50 for different random fields ( eo =0.60, bo=40 mm, ho=140 mm). ts/d50
at peak
u2t/ho at peak
6.05
45.75
0.0206
14
V-02-891
5.70
44.54
0.0190
14
V-03-1441*
5.73
44.66
0.0194
17
Nr
P/(σcbol) at peak
V-01-1673*
φ [ o]
V-04-1396
5.43
43.55
0.0188
17
V-05-1121*
5.95
45.40
0.0215
14
V-06-806
5.34
43.23
0.0183
14
V-07-932
5.55
43.99
0.0197
14
V-08-780
5.64
44.32
0.0217
17
V-09-1892*
5.58
44.10.
0.0215
19
V-10-1412
5.47
43.69
0.0199
17
V-11-1120
5.40
43.45
0.0211
17
V-12-138
5.37
43.32
0.0231
17
Mean value
5.60
44.17
0.0204
15.9
0.77
0.0014
1.7
Standard 0.22 deviation *- shear zone intersecting the top edge
5.2.2. Large sand specimen (320×1120 mm2) Next, a large sand specimen of 320×1120 mm2 was subject to a similar FE-analysis (Fig. 10). The results are qualitatively the same as for a
135
Investigations of size effects in granular bodies
P/(σcbol)
medium sand specimen (however, their scattering is smaller). For only 2 cases, shear zone hits a top boundary and the residual vertical force is then higher than in the remaining specimens. 8
8
6
6
4
4
2
2
0
0
0.02
0.04
0.06
0.08
0 0.10
t
u2/ho Fig. 10. Evolution of 12 load-displacement curves during compression with a dense large specimen of 320×1120 mm2) for different random fields of eo ( eo =0.60) a) with respect to a deterministic size effect.
The estimated expected value and standard deviations were respectively: for the friction angle at peak φˆp = 44.61o and sˆφ p = 0.217o, the normalized horizontal displacement of the top boundary u1t/ho t corresponding to the peak uˆ1 / ho =0.00204 and sˆu t / ho =0.0007 and the 1 normalized shear zone thickness tˆs / d 50 =76 and sˆts / d 50 =10.04. The mean random internal friction angle at peak ( φˆp = 44.61o) is by 0.8o smaller than this with the uniform initial void ratio φ p = 45.38o. The mean random shear normalized zone thickness (ts/d50=76) is significantly larger than this with the uniform initial void ratio ts/d50 = 46. The mobilized internal friction at residual state is approximately φres = 33.3o at u2t/ho=10%.
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J. Tejchman and J. Górski
6. Conclusions The following conclusions can be drawn from non-linear FEinvestigations of a deterministic and statistical size effect in granular bodies during plane strain compression under constant lateral pressure: •
• •
•
•
•
•
a) with respect to a deterministic size effect: The deterministic size effect is small. The difference in the mobilized internal friction angle at peak and at residual state is only 0.5o-1.0o. The shear resistance at peak decreases with increasing specimen size. The material brittleness strongly increases with increasing specimen size. The thickness of the shear zone increases with increasing specimen size (due to a decreasing rate of softening). b) with respect to a statistical size effect: The mean random shear resistance at peak is always smaller as compared to this with the deterministic uniform distribution. The difference in the mobilized internal friction angle at peak is 0.8o-1.2o depending on the specimen size. Thus, the statistical size effect is slightly stronger than the deterministic one. For the assumed stochastic parameters, the weakest link principle is valid. The thickness of the shear zone in the random layer can be larger than this at the deterministic uniform distribution of the initial void ratio. During deformation, first, a pattern of shear zones can observed in the sand specimen. Next, strain localization continues to localize within a single zone. The shear zone can hit the top boundary. The mean thickness of the shear zone and its inclination in the random specimen are insignificantly influenced by a stochastic distribution of the initial void ratio.
The numerical calculations of a statistical size effect in granular bodies during plane strain compression will be continued. First, they will be carried out with different decay coefficients and standard deviations in both directions of the specimen (using a truncated Gaussian random field to describe the distribution of the initial void ratio). Next, more realistic random fields will be assumed on the basis of X-ray Computed
Investigations of size effects in granular bodies
137
Tomography.63,64 Finally, the FE-calculations will be performed with rigid footings on soil where a noticeable deterministic size effect takes places.2,3 References 1.
2.
3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
15.
16. 17. 18. 19. 20. 21.
E. Wernick, Tragfähigkeit zylindrischer Anker in Sand unter besonderer Berücksichtigung des Dilatanzverhaltens, Publication Series of the Institute for Rock and Soil Mechanics, University Karlsruhe 75 (1978). F. Tatsuoka, S. Goto, T. Tanaka, K. Tani and Y. Kimura, Deformation and Progressive Failure in Geomechanics (eds.: A. Asaoka, T. Adachi and F. Oka), Pergamon (1997). J. Tejchman and I. Herle, Soils and Foundations, 39, 5 (1999). J. Tejchman, Soils and Foundations, 44, 4 (2004a). J. Tejchman and J. Górski, Granular Matter (in press) (2007a). Z. P. Bazant and E. P. Chen, Applied Mechanics Reviews, 50 (10) (1997). Z. P. Bazant and J. Planas, Fracture and size effect in concrete and other quasibrittle materials. CRC Press LLC (1998). M. R. A. van Vliet, Size effect in tensile fracture of concrete and rock, PhD thesis, University of Delft (2000). J. Chen, H. Yuan and D. Kalkhof, in Report Nr.01-13, Paul Scherrer Institute (2001). C. Le Bellego, J. F. Dube, G. Pijaudier-Cabot and B. Gerard, European Journal of Mechanics A/Solids. 22 (2003). W. Weibull, Journal of Applied Mechanics, 18, 9 (1951). Carpinteri, B. Chiaia and G. Ferro, in: Size effect of concrete structures, eds. M. Mihashi, H. Okamura and Z. P. Bazant, E&FN Spon (1994). Z. P. Bazant and A. Yavari, Engineering Fracture Mechanics 72, 1-31 (2005). Z. Bazant and S. D. Pang, in Computational Modelling of Concrete Structures, EURO-C 2006 (eds.: G. Meschke, R. de Borst, H. Mang and N. Bicanic), Taylor and Francis (2006). M. Vorechovsky and D. Matesova, Computational Modelling of Concrete Structures, EURO-C 2006 (eds.: G. Meschke, R. de Borst, H. Mang and N. Bicanic), Taylor and Francis (2006). J. Tejchman, Computers and Geotechnics 31, 8 (2004b). R. A. Regueiro and R. I. Borja, Int. J. Solids and Structures 38, 21 (2001). M. A. Gutierrez and de Borst, R. Computational Mechanics (S. Idelsohn, E. Onate, E. Dvorkin, eds,), CIMNE, Barcelona (1998). G. A. Fenton and D. V. Griffiths, J. Geotech. Geoenvironment. Eng., 128, 5 (2002). Niemunis, T. Wichtmann, Y. Petryna and T. Triantafyllidis, in Proc. Int. Conf. Structural Damage and Lifetime Assessment, Rome (2005). K. J. Bathe, Finite element procedures in engineering analysis. New Jersey: Prentice-Hall. Inc. Eaglewood Cliffs (1982).
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Vardoulakis, Scherfugenbildung in Sandkörpern als Verzweigungsproblem, PhD thesis, Institute for Soil and Rock Mechanics, University of Karlsruhe, 70 (1977). T. Yoshida, F. Tatsuoka and M. Siddiquee, Localisation and Bifurcation Theory for Soils and Rocks, eds.: R. Chambon, J. Desrues and I. Vardoulakis, Balkema, Rotterdam (1994). Desrues and C. Viggiani, J. Num. and Anal. Methods in Geomechanics, 28, 4 (2004). Tejchman and G. Gudehus, J. Num. and Anal. Methods in Geomechanics, 25 (2001). J. Tejchman and A. Niemunis, Granular Matter, 8, 3-4 (2006). H. Walukiewicz, E. Bielewicz and J. Górski, Computers and Structures, 64, 1-4 (1997). J. Górski, Non-linear models of structures with random geometric and material imperfections simulation-based approach. Monography, Gdansk University of Technology (2006). Z. P. Bazant and K. L. Lin, J. Structural Engineering ASCE, 111, 5 (1985). Florian, Probabilistic Engineering Mechanics, 2 (1992). J. E. Hurtado and A. H. Barbat, Archives of Computational Method in Engineering 5, 1, 3–30 (1998). Vardoulakis, M. Goldscheider and G. Gudehus, Int. J. Num. Anal. Meth. Geom. 2 (1978). Tejchman and J.Górski, Intern. Journal for Numerical and Analytical Methods in Geomechanics (in press) (2007b). P. V. Lade, Int. J. Solid Structures. 13 (1977). P. Vermeer, In Proc. Int. Workshop on Constitutive Relations for Soils (eds. G. Gudehus, F. Darve, I. Vardoulakis), Balkema (1982). M. Pestana and A. J. Whittle, In. J. Num. Anal. Meth. Geomech., 23 (1999). J. Desrues and R. Chambon, Ingenieur Archiv, 59 (1989). F. Darve, E. Flavigny and M. Megachou, Int. J. Plasticity, 11, 8 (1995). D. Kolymbas, A rate-dependent constitutive equation for soils, Mech. Res. Comm., 6 (1977). W. Wu, Publication Series of the Institute of Soil and Rock Mechanics, University Karlsruhe, 129 (1992). G. Gudehus, Soils and Foundations, 36, 1 (1996). E. Bauer, Soils and Foundations, 36, 1 (1996). P. A. von Wolffersdorff, Mechanics Cohesive-Frictional Materials, 1 (1996). C. C. Wang, J. Rat. Mech. Anal., 36. Herle and G. Gudehus, Mechanics of Cohesive-Frictional Materials, 4, 5 (1999). W. Wu and D. Kolymbas, In. D. Kolymbas (ed.), Constitutive Modeling of Granular Materials, Heidelberg, Springer (2000). C. Tamagnini, C. Viggiani and R. Chambon, In. D. Kolymbas (ed.), Constitutive Modeling of Granular Materials, Heidelberg, Springer (2000). Niemunis and I. Herle, Mechanics of Cohesive-Frictional Materials, 2 (1997). Tejchman, E. Bauer and W. Wu, Acta Mechanica, 1-4 (2007). G. Gudehus, Granular Matter, 8 (2006).
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57. 58. 59. 60. 61. 62. 63. 64.
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Herle and D. Kolymbas, Computers and Geotechnics, 31 (2004). D. Masin, Int. J. Numer. and Anal. Meths. in Geomech., 29 (2005). T. Maier, Numerische Modellierung der Entfestigung im Rahmen der Hypoplastizität, PhD Thesis, University of Dortmund (2002). Oda, Powders and Grains, Rotterdam, Balkema (1993). E. Pasternak and H.-B. Mühlhaus, Computational Mechanics – New Frontiers for New Millennium (eds.: S. Valliappan S. and N. Khalili), Elsevier Science (2001). H. Schäfer, Versuch einer Elastizitätstheorie des zweidimensionalen ebenen Cosserat-Kontinuums. Miszellaneen der Angewandten Mechanik, Festschrift Tolmien, W., Berlin, Akademie-Verlag (1962). Tejchman, Computers and Geotechnics 33, 1 (2006). E. Bielewicz and J. Górski, International Journal of Non-linear Mechanics, 37, 4–5 (2002). Przewłócki and J. Górski, Probabilistic Engineering Mechanics, 16 (2001). E. Groen, Three-dimensional elasto-plastic analysis of soils, PhD Thesis, Delft University (1997). Tejchman and E. Bauer, Computers and Geotechnics, 19(3) (1996). W. Ehlers and T. Graf, Bifurcations and Instabilities in Geomechanics (eds.: J. Labuz and A. Drescher), Swets and Zeitlinger (2003). Sheppard, M. Knackstedtr, T. Senden and M. Saadatfar, in Proc. 20th Canberra International Summer School and Workshop on Granular Material (2006). T. Aste, T. Di Matteo, M. Saaadatfar, M. Schroeter, T. J. Senden, and H. Swinney,. in Proc. 20th Canberra International Summer School and Workshop on Granular Material (2006).
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Chapter 8 Granular flows: fundamentals and applications
Paul W. Cleary CSIRO Mathematical and Information Sciences Private Bag 33, Clayton, Vic, 3169. Australia
DEM allows the prediction of complex industrial and geophysical particle flows. The importance of particle shape is demonstrated through a series of simple examples. Shape controls resistance to shear, the magnitude of collision stress, dilation and the angle of repose. We use a periodic flow of a bed of particles to demonstrate the different states of granular matter, the generation of dilute granular flow when granular temperature is high and the flow dependent nature of the granular thermodynamic boundary conditions. A series of industrial case studies examines how DEM can be used to understand and improve processes such as separation, mixing, grinding, excavation, hopper discharge, metering and conveyor interchange. Finally, an example of landslide motion over real topography is presented.
1. Introduction Particle scale simulation of industrial and geophysical flows using DEM (Discrete Element Method) offers the opportunity to better understand fundamental flow dynamics and the importance of particle properties. It can generate insights that enable better designs for process equipment and for risk management of extreme geophysical flows, such as landslides. DEM simulation involves following the motion of every particle (coarser than minimum resolvable size) and modelling each collision between the particles and between the particles and their environment. It
141
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P.W. Cleary
was first applied to geotechnical problems [1]. Over the next two decades it has grown in popularity and has been used to study many particle flow problems. See review articles by Campbell [2] and Walton [3] for more information on rapid granular flows and the use of DEM to help understand these. For typical examples of DEM modelling used to understand engineering and geophysical type flows, see [4, 5, 6, 7, 8]. Here we discuss the importance of particle shape and use a series of simple flows to demonstrate its key effects. We then briefly present a series of industrial examples of DEM modelling and finally show predictions of a landslide on real topography. 2. Discrete Element Method The general methodology is now well established and is described in review articles [2, 9, 3]. We use a classical linear-spring and dashpot collision model, which is described in more detail in Cleary [10, 8]. Briefly, the particles are allowed to overlap and the amount of overlap ∆x, and normal vn and tangential vt relative velocities determine the collisional forces via a contact force law. There are a number of possible contact force models available in the literature, approximating the collision dynamics to various extents, see [3, 11] for details. There is little definitive evidence to support the use of any one of these models over any of the others for the materials found in most industrial and geophysical flows (predominantly rocks). In particular, there is little evidence to suggest that the choice of the contact model actually affects the flow dynamics in large scale systems. The normal force:
Fn = −kn ∆x + Cnvn ,
(1)
consists of a linear spring to provide the repulsive force and a dashpot to dissipate a proportion of the kinetic energy. It is restricted to being positive to prevent unphysical attractive forces at the end of collisions when the first term is small and the later term is larger and negative. The maximum overlap between particles is determined by the stiffness kn of the spring in the normal direction. Typically, average overlaps of 0.10.5% are desirable, requiring spring constants of the order of 104-106
Granular flows
143
N/m in three dimensions. The normal damping coefficient Cn is chosen to give the required coefficient of restitution ε (defined as the ratio of the post-collisional to pre-collisional normal component of the relative velocity), and is given in [10]. The tangential force is given by: Ft = min µFn , kt ∫ vt dt + Ct vt ,
{
}
(2)
where the vector force Ft and velocity vt are defined in the plane tangent to the surface at the contact point. The integral term represents an incremental spring that stores energy from the relative tangential motion and models the elastic tangential deformation of the contacting surfaces, while the dashpot dissipates energy from the tangential motion and models the tangential plastic deformation of the contact. The total tangential force Ft is limited by the Coulomb frictional limit µ Fn,, at which point the surface contact shears and the particles begin to slide over each other. Particles are typically represented in DEM as spheres which have strong quantitative drawbacks, so we choose to represent them as superquadrics. These shapes were first used in two dimensions by [12, 13] and in three dimensions by [8]. In their principle reference frame, they are specified by: m
m
m
x y z + + =1 a b c
