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Understanding the Rheology of Concrete
Understanding the Rheology of Concrete
Dr. Nicolas Roussel
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Estimating, modelling, controlling and monitoring the flow of concrete is a vital part of the construction process, as the properties of concrete before it has set can have a significant impact on performance. This book provides a detailed overview of the rheological behaviour of concrete, including measurement techniques, the impact of mix design, and casting.Part one begins with two introductory chapters dealing with the rheology and rheometry of complex fluids, followed by chapters that examine specific measurement and testing techniques for concrete. The focus of part two is the impact of mix design on the rheological behaviour of concrete, looking at additives including superplasticizers and viscosity agents. Finally, chapters in part three cover topics related to casting, such as thixotropy and formwork pressure.
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2012
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Woodhead Publishing
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english
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384 / 373
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0857090283
ISBN 13:
9780857090287
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Woodhead Publishing in Materials
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PDF, 28.61 MB
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Understanding the rheology of concrete © Woodhead Publishing Limited, 2012 Related titles: Polymer modified bitumen (ISBN 9780857090485) This book, the first of its type, provides a comprehensive and indepth coverage of the science and technology of polymer modified bitumen. After an initial introduction to bitumen and polymer modified bitumen, the book is divided into two parts. Part I focuses on the preparation and properties of a range of polymer modified bitumen, including polymer bitumen emulsions, modification of bitumen with poly(urethanes), waste rubber and plastic and polypropylene fibres. Part II addresses the characterization and properties of polymer modified bitumen. Topics covered include rheology, simulated and actual long term ageing studies, the solubility of bituminous binders in fuels and the use of Fourier transform infrared spectroscopy to study ageing/oxidation of polymer modified bitumen. The book is an essential reference for scientists and engineers from both academia and the civil engineering and transport industries interested in the properties and characterisation of polymer modified bitumen. Fibrous and composite materials for civil engineering (ISBN 9781845695583) The application of fibrous materials in civil engineering is an important and interesting development. Steel is one of the most popular composite materials used for structural applications, however many types of steel structures are subjected to corrosion damage. The use of fibrous structures including knitted, woven and braided fabrics are becoming recognised as an attractive reinforcement due to their fibre orientation, conformability and potential of producing shaped reinforcements. This book reviews these fibrous materials with regards to their role in structural reinforcement. Key topics include short fibre reinforced concrete, natural fibre reinforced concrete and high performance fibre reinforced cementitious composites. Service life estimation and extension of civil engineering structures (ISBN 97; 81845693985) This book reviews ways of rehabilitating ageing reinforced concrete structures using fibrereinforced polymer (FRP) composites and how to assess their remaining service life. The book covers topics such as rehabilitation techniques for service life extension, probabilistic methods for service life estimation, rehabilitation and service life estimation of corroded structures and NDE/NDT of rehabilitated structures. Details of these and other Woodhead Publishing materials books can be obtained by: • • • visiting our web site at www.woodheadpublishing.com contacting Customer Services (email: sales@woodheadpublishing.com; fax: +44 (0) 1223 832819; tel.: +44 (0) 1223 499140 ext. 130; address: Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK) contacting our US office (email: usmarketing@woodheadpublishing.com; tel. (215) 928 9112; address: Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia, PA 191023406, USA) If you would like to receive information on forthcoming titles, please send your address details to: Francis Dodds (address, tel. and fax as above; email: francis.dodds@ woodheadpublishing.com). Please confirm which subject areas you are interested in. © Woodhead Publishing Limited, 2012 Understanding the rheology of concrete Edited by Nicolas Roussel © Woodhead Publishing Limited, 2012 Published by Woodhead Publishing Limited, 80 High Street, Sawston, Cambridge CB22 3HJ, UK www.woodheadpublishing.com Woodhead Publishing, 1518 Walnut Street, Suite 1100, Philadelphia, PA 191023406, USA Woodhead Publishing India Private Limited, G2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India www.woodheadpublishingindia.com First published 2012, Woodhead Publishing Limited © Woodhead Publishing Limited, 2012 The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the authors and the publishers cannot assume responsibility for the validity of all materials. Neither the authors nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Control Number: 2011939656 ISBN 9780857090287 (print) ISBN 9780857095282 (online) The publisher’s policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acidfree and elemental chlorinefree practices. Furthermore, the publisher ensures that the text paper and cover board used have met acceptable environmental accreditation standards. Typeset by RefineCatch Limited, Bungay, Suffolk Printed by TJI Digital, Padstow, Cornwall, UK © Woodhead Publishing Limited, 2012 Contents Contributor contact details Introduction ix xi Part I Measuring the rheological behaviour of concrete 1 1 3 Introduction to the rheology of complex fluids P. COUSSOT, Université ParisEst, France 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2 Solids Newtonian fluids Suspensions Fluids with slightly nonNewtonian character Yield stress fluids Thixotropy Viscoelasticity Conclusions 3 8 10 12 15 18 21 22 Introduction to the rheometry of complex suspensions 23 G. OVARLEZ, Université ParisEst, France 2.1 2.2 2.3 2.4 2.5 2.6 3 Rheometry Characterisation of simple yield stress fluids Characterisation of thixotropic yield stress fluids Advanced techniques for the study of local flow properties Notes References 23 32 39 47 59 59 Concrete rheometers 63 C. F. FERRARIS and N. S. MARTYS, National Institute of Standards and Technology, USA 3.1 3.2 3.3 Introduction Rotational rheometers for concrete Comparison of concrete rheometers 63 64 72 v © Woodhead Publishing Limited, 2012 vi Contents 3.4 3.5 3.6 3.7 Modeling of concrete rheometers Conclusions Acknowledgments References 74 79 79 80 4 From industrial testing to rheological parameters for concrete 83 N. ROUSSEL, Université ParisEst, IFSTTAR, France 4.1 4.2 4.3 4.4 4.5 5 Introduction The slump test family and its limits The LCPC BOX test Conclusions References 83 85 91 94 95 The rheology of cement during setting 96 S. GAUFFINETGARRAULT, University of Bourgogne, France 5.1 5.2 5.3 5.4 Hydration: chemical reactions and kinetics Rheology of cement pastes Parameters influencing mechanical efficiency of calcium hydrosilicate (CSH) References 96 101 108 111 Part II Mix design and the rheological behaviour of concrete 115 6 117 Particle packing and the rheology of concrete X. CHATEAU, Université ParisEst, France 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 7 Introduction Compacity and porosity Packing of monosize spheres Packing of identical particles Packing of nonidentical particles Particle packing models Fibre and grain mixture Effect of particle size distribution on rheology Conclusions References 117 118 118 121 124 127 138 138 141 141 Superplasticizers and the rheology of concrete 144 R. FLATT, Eidgenössische Technische Hochschule Zürich (ETHZ), Switzerland and I. SCHOBER, Sika Technology AG, Switzerland 7.1 7.2 Introduction Chemical characteristics of superplasticizers © Woodhead Publishing Limited, 2012 144 146 Contents vii 7.3 7.4 7.5 7.6 7.7 7.8 Physical characteristics of superplasticizers Superplasticizers and rheology: microscopic behaviour Superplasticizers and rheology: macroscopic behaviour Superplasticizers and cement chemistry Conclusions and outlook References 162 177 183 194 200 201 8 Viscosityenhancing admixtures and the rheology of concrete 209 K. H. KHAYAT, Université de Sherbrooke, Canada and Missouri University of Science and Technology, USA and N. MIKANOVIC, HeidelbergCement Technology Center GmbH, Germany 8.1 8.2 8.3 8.4 8.5 9 Introduction Chemical nature, classification and mode of action of viscosityenhancing admixtures Effect of viscosityenhancing admixtures on rheology of water–cement systems Effect of viscosityenhancing admixtures on stability of cementbased systems References 209 Fibre reinforcement and the rheology of concrete 229 210 214 223 226 S. GRÜNEWALD, Delft University of Technology, The Netherlands 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Introduction Fibres in cementitious materials Fibre rheology Rheology of fibre concrete Developments in fibre concrete and rheology Conclusions References Appendix: notation and symbols 229 231 235 243 249 251 252 255 Part III Casting and the rheological behaviour of concrete 257 10 259 Modelling the flow of selfcompacting concrete L. NYHOLM THRANE, Danish Technological Institute, Denmark 10.1 10.2 10.3 10.4 10.5 10.6 Introduction Homogeneous fluid approach (computational fluid dynamics) Distinct element method (DEM) Suspension flow Future trends Sources of further information and advice © Woodhead Publishing Limited, 2012 259 268 274 277 280 281 viii Contents 10.7 References 281 11 Thixotropy: from measurement to casting of concrete 286 N. ROUSSEL, Université Paris Est, IFSTTAR, France 11.1 11.2 11.3 11.4 11.5 11.6 Introduction Origin of thixotropic behaviour Thixotropy in practice Rheological models for cement paste Rheological models for concrete References 286 287 289 290 292 294 12 Understanding formwork pressure generated by fresh concrete 296 P. BILLBERG, Swedish Cement and Concrete Research Institute, Sweden 12.1 12.2 12.3 12.4 12.5 12.6 12.7 13 Introduction Factors affecting formwork pressure Relation of concrete structural behaviour at rest to lateral pressure Characterization of fresh concrete structure Modelling of lateral pressure Conclusions References Understanding the pumping of conventional vibrated and selfcompacting concrete 296 297 302 304 312 327 328 331 D. FEYS, Université de Sherbrooke, Canada 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 Introduction Theoretical background Recent experimental research Pressure loss in straight sections Pressure loss in bends and reducers Conclusions Acknowledgement References 331 332 339 343 350 350 351 351 Index 355 © Woodhead Publishing Limited, 2012 Contributor contact details (* = main contact) Editor and chapters 4 and 11 Chapter 3 N. Roussel IFSTTAR 58 Boulevard Lefebvre 75732 Paris CEDEX 15 France C. F. Ferraris* and N. S. Martys National Institute of Standards and Technology 100 Bureau Drive MS 8615 Gaithersburg MD 20899 USA Email: nicolas.roussel@ifsttar.fr Email: clarissa@nist.gov Chapter 1 P. Coussot Université ParisEst Laboratoire Navier (UMR CNRS) 2 Allée Kepler 77420 Champs sur Marne France Email: philippe.coussot@ifsttar.fr Chapter 2 G. Ovarlez Université ParisEst Laboratoire Navier (UMR CNRS) 2 Allée Kepler 77420 Champs sur Marne France Chapter 5 S. GauffinetGarrault Laboratoire Interdisciplinaire Carnot de Bourgogne Département I. R. M, Université de Bourgogne CNRS UMR 5209 Avenue Alain Savary – BP 47870 21078 DIJON CEDEX France Email: sandrine.gauffinet@ubourgogne.fr Email: guillaume.ovarlez@lcpc.fr ix © Woodhead Publishing Limited, 2012 x Contributor contact details Chapter 6 Chapter 9 X. Chateau Université ParisEst Laboratoire Navier (UMR CNRS) 2 Allée Kepler 77420 Champs sur Marne France S. Grünewald Department of Design and Construction Delft University of Technology Stevinweg 1 PO Box 5048 2600 GA Delft The Netherlands Email: xavier.chateau@gmail.com Chapter 7 Email: S.Grunewald@tudelft.nl R. J. Flatt* ETH Zürich Institute für Baustoffe (IfB) HIF Building, E 11 Schafmattstrasse 6 8093 Zürich Switzerland Email: flattr@ethz.ch Chapter 10 L. Nyholm Thrane Danish Technological Institute, Concrete Gregersensvej 2630 Taastrup Denmark Email: lnth@dti.dk I. Schober Sika Technology AG Corporate Research Tüffenwies 16 8048 Zürich Switzerland Chapter 12 Email: schober.irene@ch.sika.com Chapter 8 K. H. Khayat Université de Sherbrooke 2500 Boulevard de l’Université Sherbrooke Québec J1K 2R1 Canada Email: kamal.khayat@usherbrooke.ca P. Billberg Swedish Cement and Concrete Research Institute Drottning Kristinas väg 26 100 44 Stockholm Sweden Email: peter.billberg@cbi.se Chapter 13 D. Feys Université de Sherbrooke 2500 Boulevard de l’Université Sherbrooke Québec J1K 2R1 Canada Email: Dimitri.Feys@usherbrooke.ca © Woodhead Publishing Limited, 2012 Introduction Throughout its history, concrete has often been called ‘liquid stone’. Before turning into an artificial stone, concrete is indeed fluid enough, for a short period, to flow and fill a mould. This property has given the construction industry the ability to cast structural elements such as slabs, beams and columns. Whereas stones are dressed, wood is sawn and steel is shaped at high temperatures, only concrete gives architects the freedom to shape buildings at their will at the building site. ‘Liquid’ is, however, a very simple word to describe the full complexity of the behaviour of concrete in the fresh state. First, unlike simple liquids such as water or oil, concrete is made of elements of many different sizes (from several nanometres to a couple of centimetres) and of various types (organic or mineral) suspended in water. Each of these elements brings its own contribution to the behaviour of concrete in the fresh state. This contribution is strongly affected not only by mix proportions but also by temperature and the casting process itself. The vast family of industrial cementitious materials present such a variety of behaviours that their classification could seem unattainable. This is even more so for the potential to predict, if only qualitatively, their response in practical processing conditions. These industrial processing conditions are also very complex. They are transient flows with complex boundary conditions. Lubricating layers may appear, for instance, at the surface of the mould. Moreover, formworks often contain a dense network of reinforcing steel bars, the size of which is close to the size of the coarsest particles in the material. Only a successful casting process is able to guarantee the mechanical bond between the reinforcement bars and concrete and, from a more general point of view, the adequate mechanical and durability performance requirements of the concrete structural element. Moreover, unlike simple (Newtonian) liquids, the surface does not selflevel when concrete is poured in a mould. The material is able to support an amount of the stress generated by gravity without flowing similar to that of mayonnaise, paint or tomato ketchup. In the case of firm concretes, this property is so strong that vibration is often applied to the material to ease the casting process and allow for filling of the formwork. Only recently have concretes able to flow under their xi © Woodhead Publishing Limited, 2012 xii Introduction own weight without vibration while staying homogeneous (called ‘selfcompacting concretes’ or ‘selfconsolidating concretes’) been developed. This property may, however, prove useful in specific cases such as shotcrete where, after spraying, it is clearly desirable that the material stays on the wall in thick layers rather than flows down it. Time has also a strong effect on the behaviour of concrete in the fresh state. When left at rest, material consistency increases significantly with time. Some of these changes are reversible; their effects are erased by remixing in a truck for instance or by any type of strong shearing. Other changes, in particular the consequences of the hydration phenomenon, are irreversible and thus contribute to the longterm evolution of material properties (towards the solid state). No matter the origin of the evolution, it may strongly affect the way concrete is cast and the quality of the final structural element. For instance, two successively poured layers of concrete may not form a homogeneous material if the resting time between the pouring of the layers is too long. It is noticeable in the history of concrete technology that the development of processing methods has been faster than the development of the understanding of the material behaviour in the fresh state. Concrete has indeed been used extensively since the beginning of the twentieth century, whereas the modern science of rheology (i.e. the study of the flow of matter) and the developments in technology which have made it possible have only occurred in the last 50 years. What are the key objectives of research on the rheology of fresh concrete? Depending on scientific background, an academic answer could be ‘the ability to correctly measure and quantify the rheological properties of fresh concrete’ or ‘the understanding of the correlation between components’ proportions and rheological properties in the fresh state’ whereas a practitioner would probably answer: ‘the ability to predict whether or not a given concrete will correctly fill a given formwork’. The continuing challenge of understanding particle interactions in cementitious materials in the fresh state up to the engineering prediction of the casting processes lies at the interface of three disciplines: physics, chemistry and fluid mechanics. Physics and chemistry allow correlations to be made between mix design and macroscopic fresh properties. Fluid mechanics allows analysis of the tests used for the measurement of these properties and, in its most advanced form, for the prediction of industrial casting processes. Keeping the above in mind, we have chosen in this book to gather international experts from the three disciplines. Although their approaches are different, they all have in common the dedication of some or most of their research activities to understanding the fresh properties of cementitious materials. We hope that combining these contributions will form a useful basis for the understanding of the rheology of fresh concrete from mix design to casting processes. This book is divided into three parts. Following an introductive Chapter 1 on the rheology of complex fluids, chapters 2 to 5 focus on the way the rheological © Woodhead Publishing Limited, 2012 Introduction xiii behaviour of cementitious materials can be measured. Chapters 6 to 9 then review the main correlations that have been established between mix design and fresh properties. Finally, chapters 10 to 13 show how the knowledge of fresh properties can allow for the prediction of industrial casting processes. Although chapters 10 to 13 focus mainly on concrete industrial processes, chapters 1 to 9 can be useful for anybody interested in any other cementitious materials. Chapter 1 goes through the main rheological behaviour types. In particular, the most important behaviour types for cementitious materials (yield stress behaviour and thixotropy) are emphasised along with the importance of distinguishing the main interaction types within the material for evaluating the physical origin of its behaviour. Chapter 2 presents the most used experimental methods for the measurement of the rheological properties of complex suspensions such as fresh cementitious materials. It deals successively with the basis of rheometry, the measurements of the properties of simple yield stress fluids, the measurements of the properties of thixotropic yield, stress fluids, shear localisation and material heterogeneities. Chapter 3 introduces current rheometer technology for measuring the properties of fresh concrete or mortar while Chapter 4 presents some correlations between industrial tests such as the slump test or slump flow test and yield stress. Whereas the previous chapters focus on the properties of fresh cementitious materials within a casting process perspective, Chapter 5 shows how rheometry can also be a valuable tool for the nondestructive study of the evolution of the structure of cement pastes from mixing to setting. Chapter 6 recalls the main results concerning the effects of particles properties on packing characteristics. The dependence of the overall rheological properties of highly concentrated suspensions of polydisperse particles on the particle size distribution and particle blend composition is investigated. Chapter 7 focuses on the effects of deflocculating polymers on the fluidity of fresh cementitious materials. It first describes the chemical and physical characteristics of these molecules before detailing their modes of actions within a cement suspension. Chapter 8 deals with the high molecular weight watersoluble polymers used as viscosity agents in cementitious materials. After describing their chemical nature, it recalls the various ways they are able to improve the stability of cementitious materials. Chapter 9 focuses on how addition of fibres can affect the rheology of cementitious materials. It also shows how flowinduced fibre orientation can affect the mechanical properties of the hardened material. Chapter 10 gives an overview of the use of numerical tools to simulate the flow of fresh concrete. It presents the three main techniques: homogeneous fluid simulations, modelling of discrete particle flow (distinct element method), and numerical techniques to model particles suspended in a fluid along with examples of application and future trends. Chapter 11 focuses on the origin of the thixotropic behaviour of cementitious materials, before presenting models able the consequences of thixotropy on casting processes. In Chapter 12, existing models for the prediction of formwork pressure as a function of fresh properties of concrete are discussed. © Woodhead Publishing Limited, 2012 xiv Introduction The emphasis is on the effect of the structural buildup of fresh concrete and existing measurement techniques for this crucial parameter. Finally, Chapter 13 deals with the pumping of concrete. After a theoretical description of this specific flow, the influence of the nature of the pumped concrete, the material of the pipe, the total applied pressure, the temperature and bends and reducers are discussed. Nicolas Roussel © Woodhead Publishing Limited, 2012 1 Introduction to the rheology of complex fluids P. COUSSOT, Université ParisEst, France Abstract: This chapter considers the main types of rheological behaviour. The basic ways of modelling them (essentially in simple shear) and the physical origins of the behaviour of different material types are explained and typical real materials behaving in that way are mentioned. The most important behaviour types (yielding and thixotropy) for cementitious or concrete materials, the importance of flow regime (or ‘physical state’) with regards to the time scale of observation and the flow rate for predicting the behaviour type, the importance of distinguishing the main interaction types within the material for evaluating the physical origin of its behaviour are emphasised. Solids, simple fluids, suspensions, slightly nonNewtonian materials, yield stress fluids, thixotropy and viscoelasticity are reviewed. Key words: solid, liquid, suspension, yield stress fluid, thixotropy. 1.1 Solids 1.1.1 Apparent behaviour We live in a solid environment made of ground, furniture, cars, etc, which fortunately keep their shape when we use them. If however we apply a force, larger than a critical value, to one of these objects, we reach a situation when they start to strongly deform or even break (actually for a force smaller than this critical value they deform slightly). For two different objects made of the same material this critical force is different, which means it is not an appropriate variable for describing the material’s behaviour as it is not intrinsic to it. In fact the appropriate variable is the stress, which is the ratio of the force to its surface of application. Finally the mechanical behaviour of solid materials is basically described in terms of the deformation induced by the stress applied and the critical deformation and critical stress at which the material flows or breaks. To better understand these concepts of stress and deformation it is useful to consider one of the most frequent and simple situations, namely the ‘simple shear’. In this case a piece of material is contained between two parallel solid planes which move relative to each other along one of their directions. The resulting deformation corresponds to the relative motion of parallel planes of materials (Fig. 1.1). The shear stress, τ, is the ratio of the force (F) applied to the section of the piece of material (S): τ = F/S. The deformation, ε, is defined as the ratio of the relative displacement of the solid planes Δx to the thickness H of the piece of material: ε = Δx/H. 3 © Woodhead Publishing Limited, 2012 4 Understanding the rheology of concrete 1.1 Principle of a simple shear: relative motion of material planes. In mechanics we are interested in the constitutive equation of the material, which is a relationship describing the intrinsic behaviour of the material, i.e. independently of the specific conditions to which it is submitted. For a solid undergoing a simple shear, the constitutive equation is mainly described in terms of the relationship between τ and ε, but in some cases the rate of displacement must also be taken into account (see Section 1.7). The simplest solid behaviour is the linear elastic behaviour, in which τ is simply proportional to ε: [1.1] In simple shear the coefficient of proportionality G in this relation is the elastic shear modulus. For a sufficiently large stress, the material no longer follows this type of behaviour but deforms widely. Here we can distinguish two main types of behaviour: for plastic materials the deformation tends to localise in some specific region within the material; for fragile materials there is a breakage into several parts (Fig. 1.2). 1.1.2 Microstructural origin Often we do not see what a material is made of and how it is organised at a local scale. Indeed most materials appear smooth or homogeneous at our scale of observation. The elements of materials that are of basic interest for our understanding of their mechanical properties are the largest elements which, by their distribution in space, play a critical role in the behaviour. These elements may be atoms, molecules, polymer chains, cells, clay particles or cement grains, which means that the size of the constitutive elements may range from few nanometres to several centimetres. The basic structure at the origin of a solid’s behaviour is the distribution of elements either linked to each other or jammed in a given space. An illustrative example is a vacuum pack of coffee beans: it is apparently solid because the beans are jammed in a limited volume, and its properties depend only on the properties of the beans and their distribution in space. The simplest material of this type is a solid crystal made of identical atoms or molecules distributed along periodic and symmetrical positions xi. The specificity © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 5 1.2 Two different possible (idealised) behaviour types of solid materials: elastic then fracture or plastic behaviour beyond a critical deformation. The drawings illustrate the different aspects of the material successively in the initial state, for a homogeneous deformation in the elastic regime, then either for a ductile or brittle behaviour. of such a system is that an atom (situated in x) interacts with its neighbours via strong van der Waals, covalent or ionic forces Fi, and as a function of the distance between the elements x − xi, so that this atom has a potential energy (E) defined . In the crystalline state each atom is in a position of force equilibrium by (Fig. 1.3), ΣFi = 0, which corresponds to a minimum of the potential energy. As a consequence each atom is in a well of energy from which it can escape only if a sufficiently large force (or a stress) is applied to it. In addition each element is submitted to a thermal agitation which in general induces small amplitude motions of this element around its equilibrium position. When a low stress is applied, each atom can be only slightly displaced from its initial position, thus reaching a new position of equilibrium: the atoms climb slightly along their potential well. This small displacement of each atom relatively to its neighbour finally induces a limited total displacement of the upper surface relative to the lower surface, and thus a limited deformation of the material. When the stress is released, the atoms fall back to the bottom of their potential well, i.e. they come back to their initial position. This explains the elastic behaviour of the material in the limit of small deformations. For a pure crystal of this type it is possible to relate the basic mechanical properties to the interactions at a local scale between neighbouring atoms. Indeed on one hand the relative displacement of neighbouring atoms induces a sample deformation, on the other hand it requires some energy supply to one atom for it to climb along its potential well. Finally this description explains how the stress increases as a © Woodhead Publishing Limited, 2012 6 Understanding the rheology of concrete function of the deformation at a local scale, which can be extrapolated at a macroscopic scale. For a sufficiently large stress some atoms in a crystal can get out of their potential well. This can induce mainly two types of phenomena: 1. A dislocation, i.e. the exit out of their potential well of a limited number of atoms along some planes. In such a case the displacement of this series of atoms is limited. The arrangement is locally broken but finally reformed with other atoms, probably taking advantage of some defects in the initial structure; at a macroscopic scale this corresponds to a plastic behaviour. 2. A breakage, i.e. the exit and separation of all the atoms along some surface through the material, leading to its breakage. After this process the material has in general lost its initial properties because the structure has been irreversibly modified. Other materials such as glass, some parts of rocks and some polymeric materials are in the glassy state (Fig. 1.3); the atoms or molecules are still densely packed as in the solid state but now the structure is disordered. Generally such materials appear as solids since, like crystals, they only deform slightly when subjected to a stress. However, for such materials, essentially because of the disordered structure, we do not have much information about the relation between the physics at a local scale and their mechanical properties. Indeed, this disorder implies that it is difficult to relate a local displacement to a macroscopic deformation because the 1.3 Different types of material structures associated with solid behaviour: (a) crystalline solid; (b) glass; (c) reticulated polymer; soft solid behaviour; (d) colloidal aggregate; (e) concentrated foam or emulsion. Different types of material structures associated with liquid behaviour: (f) simple molecular liquid; (g) polymer melt or solution; and (h) dilute or semidilute suspension in a liquid. For systems (a), (b) and (f) the arrows illustrate the thermal agitation of the elements while the material is macroscopically at rest. © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 7 way in which the stress distributes throughout the structure is a priori unknown. This is an active field of research. Due to their formation process most rocks at the surface of the Earth consist of a mixture of small homogeneous crystalline or glassy regions of various types dispersed in a matrix in a glassy state. This complex structure results from the way they have been formed, i.e. via a fast or progressive cooling of magmas containing various species possibly implying fractional crystallisation. There are also many solids, such as soils, woods, etc, with a heterogeneous structure at a local scale. Like most rocks they are made by the aggregation of solid elements. A rheophysical approach to their behaviour requires us to identify the mechanical characteristics of the main components and then to develop a micromechanical approach able to predict the rheological properties of heterogeneous solids. Finally we have various types of ‘soft’ solids such as gels and colloids (Fig. 1.3) made of mesoscopic elements dispersed in a liquid. If their concentration is sufficiently high these elements play a crucial role in the main rheological properties of the material; this is proved by the fact that it exhibits properties completely different from those of the interstitial fluid alone. Yet the energy of interaction between the colloidal elements is much lower than the interaction energy between two atoms. For example, while the typical van der Waals interaction energy varies inversely to the sixth power of the distance between atoms (which is very small within a solid), for two colloidal particles it approximately varies inversely to the second power of the distance between particles (which is much larger). This basically explains why the resistance to deformation of a system made of colloidal particles immersed in a simple liquid is much smaller than the resistance of a solid crystal. In some ideal cases (such as a pure colloid in crystalline order) it is possible to deduce a relationship between this resistance and the potential energy between the mesoscopic elements by following the same approach described above for pure solids. In a similar way, for gels made of a network of linked polymer chains in a simple configuration, it is possible to deduce an expression for the elastic modulus as a function of the chain characteristics. Indeed each chain behaves as a small rubber with a given stiffness which is proportional to the temperature and inversely proportional to the molecule length. As a consequence the deformation of the solid leads to a deformation of the network which induces an elongation or compression of each chain, which requires some force. Within the class of soft solids there are also all the pasty materials which behave as simple solids when they are too much deformed. These are for example muds, purées, sauces, toothpastes, mayonnaise and . . . fresh concrete. The fact that they behave as solids is readily observed when pouring them in a container: whereas for a liquid the free surface at rest is perfectly horizontal, it is generally at least slightly uneven for a pasty material and can keep the shape it has been given for the most ‘pasty’ among them. In general these systems contain a wide range of mesoscopic elements developing different types of interactions, as is typically the © Woodhead Publishing Limited, 2012 8 Understanding the rheology of concrete case for concrete including colloidal particles and coarse grains in a liquid. The structure is obviously also disordered and it is extremely difficult to establish a relationship between the local and the macroscopic behaviour. For some materials of simple structure such as concentrated foams or emulsions, a macroscopic deformation is intimately linked to the local deformations of each bubble or droplet, so that in the absence of other effects we find that the elastic modulus is proportional to the typical surface tension stress in such a material, i.e. the surface tension divided by the droplet or bubble radius. 1.2 Newtonian fluids We also live in a fluid environment: we are surrounded by air, sea, we need water at any time of our life, blood flows through our veins, etc. In contrast with solids these materials (fluids) may be deformed at will and flow as soon as some stress is applied to them. Here we focus on liquids, which correspond to a dense state of matter (Fig. 1.3). Let us for example consider a long conduit filled with water. When the conduit is horizontal the water does not move, but as soon as the conduit is inclined even slightly, the fluid starts flowing. The behaviour of many such materials is extremely simple: the velocity in the conduit is simply proportional to the applied pressure. If the conduit is long and if a large volume of fluid can be supplied from upstream, the flow will go on over a long time, implying that the deformation that the fluid can undergo without any modification of its properties is indefinite. This means that for fluids under usual conditions it is not the deformation which plays a critical role in the behaviour description but rather the flow rate. In a simple shear experiment (Fig. 1.1), we consider the ‘shear rate’, i.e. the ratio of the relative velocity (V) of the solid planes to the sample thickness (H): . γ = V/H. For this simple flow the shear rate is also the time variation of the . deformation: γ = (Δx/H∆t). Finally, for a liquid system, the constitutive equation is essentially sought in the form of the ‘flow curve’, i.e. the relationship between . the shear stress (τ) and the shear rate (γ ) in simple shear. Usual liquids made of small molecules such as water, oil, syrups, etc, are Newtonian materials. The shear stress is simply proportional to the shear rate: [1.2] where the coefficient of proportionality, µ0, is the material viscosity. The flow history has no impact on such materials. In particular they flow immediately at a rate proportional to the currently applied stress, and do not store any energy as a result of the deformation. Some other liquids (nonNewtonian) exhibit more complex behaviour which can be described by generalising the above approach, i.e. by considering the apparent viscosity (η), defined as: © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 9 [1.3] where η is not a constant but a function of the current shear rate. In fact the constitutive equation of a liquid as considered above remains valid as long as the flow is not too fast. Otherwise inertia effects, namely turbulence, may take place and the apparent behaviour of the material is more complex. Turbulence occurs when the ratio of inertia effects to viscous effects is sufficiently large. This may be appreciated from the value of the Reynolds number (Re): [1.4] in which ρ is the fluid density and H a characteristic length scale of the flow. When Re is much larger than 1, the flow is in general ‘turbulent’. When Re is much smaller than 1, the flow is ‘laminar’ and the constitutive equation as determined from usual rheological techniques is relevant for describing the material properties. This means that both the flow involved for the determination of the liquid properties and the flow which is to be described must be in the laminar regime. As for solids the atoms or molecules in a simple liquid are densely packed against each other. However, in contrast with solids, the elements in a simple liquid are not fixed in specific places but are submitted to a significant thermal agitation which induces fast displacements over a distance which may be of the order of the element size (Fig. 1.3). A useful way of describing the rheophysics of such systems consists of considering that, in a liquid at rest (due to the element packing and the interactions with their neighbours), the atoms or molecules are instantaneously in potential wells from which they can exit and move in any direction with the same probability as a result of thermal agitation. When a stress is applied to the material, this makes the elements on average climb along the potential well in some specific direction so that the probability of moving out in that direction is now larger than in the opposite one. The same phenomenon takes place for all the elements, finally inducing a relative motion of two neighbouring liquid planes along this direction at a rate which is a function of the probability of jump and the value of the stress applied. This gives rise to a continuous deformation of the material, i.e. a flow, at a rate function of the stress. In this scheme the basic result is that a flow is obtained whatever the value of the shear stress applied since even an extremely low stress destabilises the system. In addition it appears that for simple liquids made of atoms or small molecules such as water, oil, honey, glycerol, alcohol, etc, the shear rate induced is simply proportional to the shear stress, i.e. we have Newtonian materials. However, our understanding of the physical origin of this result is so far only partial. Such structure characteristics have an additional important consequence on the rheological behaviour of the material: due to the rapid agitation of the elements at our usual scale of observation the material does not retain any memory of its flow history. Its structure characteristics are thus constant and its rheological response to © Woodhead Publishing Limited, 2012 10 Understanding the rheology of concrete some stress is uniquely determined by the current stress level. This means that such fluids are not affected at all by the deformation they undergo, but this does not mean that no work is necessary to induce a flow. In fact some energy is required to push each molecule out of its potential. This energy is not stored in the material but is dissipated (in the form of a temperature increase) since, as soon as one stops the flow, the material is in the same mechanical state as before flow. This energy dissipation corresponds to the work supplied per unit time, which may be computed by integrating, over the sample volume, the product of the force applied to each plane and the relative velocity of two neighbouring planes. Finally, using the shear stress and the shear rate, the viscous dissipation per unit volume (P) can be expressed as: [1.5] 1.3 Suspensions The class of Newtonian materials is much wider than the class of pure materials. Indeed many materials are made of mesoscopic objects (of a size much larger than the size of the atoms or molecules composing the liquid) dispersed in a Newtonian liquid. These are ‘suspensions’. Whereas for a pure liquid the fluid layers glide over each other in simple shear, for a suspension the fluid layers follow more complex paths because of the presence of the elements. The result is a larger viscous dissipation for a larger concentration of elements but it can be shown that, as long as the distribution of elements in space remains isotropic and constant, the suspension is Newtonian. A critical parameter for appreciating the behaviour of suspensions is the volume fraction (ϕ) of elements in suspension, i.e. the ratio of the total volume of the elements to the total sample volume. When the volume fraction is sufficiently low, i.e. in the dilute regime, the elements are far from each other so that the liquid flow around each element is perturbed at a distance smaller than the separating distance between them (Fig. 1.4). We can say that the elements do not interact hydrodynamically. In that case the suspension generally has a rheological behaviour similar to that of the interstitial liquid, possibly with slightly larger rheological parameters. For a Newtonian suspending liquid, a dilute suspension of noncolloidal grains is Newtonian with a viscosity: [1.6] This expression constitutes an excellent approximation of reality for ϕ < 2%. At larger solid fractions the elements interact significantly so that on the behaviour of both the behaviour of the suspending liquid and the elements. For a suspension of noncolloidal grains in a Newtonian liquid, we still have a Newtonian material in the semidilute regime and various empirical formulae have been proposed to describe the suspension viscosity in that range. For example one may use the following expression (Krieger–Dougherty): © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 11 1.4 Relative viscosity of a suspension of uniform spheres in a Newtonian liquid as a function of the solid fraction. The schemes correspond to the different regimes of concentration (from left to right): dilute, semidilute, concentrated, compact. [1.7] which is relevant for representing the viscosity in a wide range of solid fractions (say up to 55%). Note that a critical parameter has been introduced in this equation, namely ϕm, the maximum packing fraction, which is the maximum volume fraction which can be reached with the grains. When the solid fraction approaches ϕm we are in the concentrated regime and the viscosity tends to infinity (Fig. 1.4). It may be considered that, for a higher concentration (compact regime), the structure is jammed and the material is solid. The maximum packing fraction is not a welldefined parameter as it depends on the spatial distribution of the elements, which may depend on the flow history. For uniform spheres it is around 55% if the particles are simply poured in a container and can be increased to 64% by a slight and long vibration. The crucial point is that the exact value to be used in Eq. 1.7 has a dramatic impact on the viscosity value in the concentrated regime, as may be seen in Fig. 1.4. Unfortunately there is not yet an appropriate procedure for determining the appropriate value of ϕm for use in Eq. 1.7. It seems that this value can only be determined empirically for each material from its viscosity vs. solid fraction curve: ϕm corresponds to the concentration at which the viscosity tends to infinity. Although it has several empirical aspects the approach described above appears to be very general in that it works for various grain types and size distributions. Indeed Eq. 1.7 can be used to estimate the viscosity of various types of materials made of elements in suspension in a simple liquid. For that purpose it is necessary to estimate the effective maximum packing fraction for the specific elements under consideration. For example, for a suspension of particles of large aspect ratios (the extreme case being fibres), the maximum packing fraction is much © Woodhead Publishing Limited, 2012 12 Understanding the rheology of concrete smaller than for uniform spheres. Conversely, when the grain size distribution is wide the maximum packing fraction is much larger than the typical value for uniform spheres (around 60%): for example it may be up to 90% or 95% for a grain size distribution ranging from a few microns to a few millimetres. Another problem is the fact that in the concentrated regime the particles may interact so strongly that the behaviour of the interstitial liquid becomes negligible. The rheological behaviour of the material is now highly dependent on the type of interactions between the suspended elements so that even for a suspension in a Newtonian liquid the Newtonian character may be lost. In this context, the description of the behaviour with the help of Eq. 1.7 is a rough approximation. The effective behaviour of this type of material is very complex, since depending on the flow history, the main source of viscous dissipation may vary from frictions between grains if a network of contact has formed throughout the sample (when the shear stress is more or less proportional to the normal stress) to hydrodynamic dissipations within the interstitial liquid if the grains are dispersed in such a way that they are separated by thin liquid films (when the shear stress is mainly proportional to the shear rate). The exact conditions under which the transition between these two regimes occurs are poorly identified. Moreover in such materials density heterogeneities develop easily as some particles can migrate slightly through the sample, which tends to increase the grain concentration in the more slowly flowing regions. This effect is critical because the transition between the two regimes can occur for a slight variation of concentration or a slight change of configuration (distribution of the grains in space). Now we are dealing with a material the behaviour of which must be described by taking into account the coupling between the local flow characteristics and the possible liquid transfer through the granular phase. For concentrated suspensions of colloidal particles, bubbles, droplets or polymers, the situation is different. The interactions between the elements play a major role in the behaviour of the system and there may be different flow regimes depending on the ratio of the intensity of these interactions to that due to simple hydrodynamic interactions (flow of the liquid). However, under usual conditions no migration of the elements leading to density heterogeneities can be expected. This is so either because we are dealing with systems in which the elements are squeezed against each other or because we are dealing with concentrated suspensions of small elements which by themselves form an equivalent porous medium of very small permeability. 1.4 Fluids with slightly nonNewtonian character 1.4.1 Shearthinning and thixotropy For Newtonian fluids the ratio of the shear stress to the shear rate is constant. Many other fluids have a nonNewtonian character: their apparent viscosity now varies with the shear rate and/or with the flow history. Such a character results © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 13 from the fact that, in contrast with Newtonian fluids, the origin of the viscous dissipation is now modified by the flow. This is particularly the case for suspensions of asymmetrical elements able to change their orientation or their shape during flow, or objects developing mutual interactions which may vary with the flow history. For example, a long object tends to align along the flow direction: on average it occupies this type of position more often than a direction perpendicular because, in the latter case, due to shear it rapidly rotates and reaches the direction of flow. In a slightly different way polymer chains tend to stretch along the flow direction. The apparent viscosity of the system is generally lower when the asymmetrical elements are aligned along the flow direction, because in this case, the perturbation of the flow due to the presence of the elements is smaller. As a consequence we can distinguish two types of effects on the mechanical behaviour. If this alignment develops more or less instantaneously for a given shear rate and depends significantly on shear rate, we will have a ‘shearthinning’ material for which the apparent viscosity decreases with shear rate (Fig. 1.5): [1.8] If the alignment takes some time to develop we will have ‘thixotropic’ effects, i.e. the apparent viscosity for a given shear rate varies in time: [1.9] From this example we see that shearthinning and thixotropy can be confused because they may find their origin in the same physical effect. Yet they are clearly associated with different mechanical effects: variation with flow rate for shearthinning and variation with time for thixotropy. Thixotropy is dealt with in more detail in Section 1.6. 1.5 Main types . of flow curves represented in terms of the apparent viscosity (τ/γ ) as a function of the shear rate. © Woodhead Publishing Limited, 2012 14 Understanding the rheology of concrete Another possible origin of shearthinning is Brownian motion. Indeed, in a dilute suspension, the Brownian motion of colloidal particles leads to an average displacement of the particles from their initial position proportional to the square root of time. If the typical relative displacement of two particles induced by shear over a given time is much smaller, Brownian motion induces an additional viscous dissipation (as a result of the particle displacements through the liquid) which is much larger than that due to the mean shear flow. As a consequence the apparent viscosity at low shear rates in dilute colloidal suspensions is larger than at high shear rates. Finally the relative importance of Brownian motion and hydrodynamic dissipations may be appreciated from the Peclet number (Pe): [1.10] where b is the particle size, kB the Boltzmann constant and T the temperature. Brownian motion plays a significant role if Pe <<1. In addition, shearthinning effects may occur in moderate or concentrated suspensions as a result of variations in colloidal interactions with shear rate. From a general point of view this effect is poorly understood. It starts to find a relatively clear explanation (transition from a jammed to a liquid state) within the frame of concentrated suspensions exhibiting a yield stress (see Section 1.5), but in that case the shearthinning character is drastic since the apparent viscosity tends to infinity when the shear rate tends to zero. 1.4.2 Shearthickening A peculiar effect, namely shearthickening, sometimes occurs with suspensions of small particles of approximately homogeneous size. This is for example observed with a suspension of cornstarch in water: when gently mixing the material with a spoon it apparently reacts as a liquid of low viscosity, but if it is mixed more rapidly, it reacts as a liquid of very high viscosity and even can break like a solid. A similar effect is observed with concentrated suspensions of uniform small spheres; in this case it was initially suggested that this was an order–disorder transition which would explain the effect. Since then other explanations have been suggested such as the formation of ‘hydroclusters’, namely transient clusters of particles formed by hydrodynamic interactions. However, all the explanations of shearthickening rely on the general idea that there is some kind of jamming of the structure beyond some critical shear rate, a view which is close to the apparent macroscopic effect. This jamming is associated with the fact that the hydrodynamic force due to liquid flow between two approaching or separating particles diverges when the distance between the two solid surfaces tends to zero. There thus seems to exist a critical shear rate beyond which the structure formed by the particles in the liquid does not have enough time to relax during the macroscopic deformation: the imposed deformation tends to push the particles very close to each other, © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 15 leading to very high energy dissipation associated with a diverging apparent viscosity. Shearthickening may be seen as an effect opposite to shearthinning, which . would be described as dη/dγ > 0. In reality the trend is quite different: generally the apparent viscosity of shearthickening materials is more or less constant up to a critical shear rate value around which it starts to dramatically increase (Fig. 1.5) and it is not clear whether or not the measurements in the latter regime are relevant. There do not seem to be known examples of ‘smooth’ shearthickening, associated with a progressive increase of the apparent viscosity with shear rate. When such an effect has been observed, it could be explained by artefacts such as drying, particle migration leading to material heterogeneities or sedimentation. 1.5 Yield stress fluids Many industrial or natural materials have a twofold character, i.e. they behave as solids under some circumstances and as liquids otherwise. Actually these materials behave as solids when a stress smaller than a critical value is applied, so that they can keep the shape they have been given – a critical aspect for many applications. When a stress larger than the critical value is applied they apparently behave as liquids, i.e. they are able to flow more or less indefinitely as long as the stress is maintained. Examples of such materials include: • • • • • • • • creams, gels and clay suspensions used in cosmetics; various materials used in civil engineering such as ceramics slips, paints, mortars, plasters, cement pastes and fresh concrete; muds used for drilling or injections in the oil industry; foodstuffs such as marmalades, purées, soups, sauces, foams; food pastes used in the preparation of extruded cakes, crackers, biscuits, etc; sewage sludges; mineral waste suspensions from the mining industry; mud or debris flows in mountain streams. Some of these materials exhibit a timedependent behaviour, i.e. they are thixotropic, a behaviour illustrated by the fact that their resistance to flow increases after an increasing time at rest (see Section 1.6) – an aspect we will leave for now. Thus we are left with a material which essentially behaves as a solid, i.e. mainly elastic with possibly some viscoelastic effects, when the applied stress is smaller than the yield stress τc, and as a viscous liquid when the stress is larger than τc. In this liquid regime the apparent viscosity a priori has a specific aspect: in order to ensure the continuity with the solid regime it should tend towards infinity when the shear rate tends to zero, i.e. when the material stops flowing. This is indeed what is generally observed in practice: in a stress vs. shear rate diagram the data points form a horizontal plateau at low shear rates – at a level which can be considered as the yield stress level (Fig. 1.6). This implies that the apparent © Woodhead Publishing Limited, 2012 16 Understanding the rheology of concrete . . viscosity η is approximately τc /γ , which tends to infinity when γ tends to zero (Fig. 1.5). In general the Herschel–Bulkley model, which expresses as follows in simple shear: [1.11] in which k and n are material parameters, represents very well experimental data in a wide range of shear rate. In this model the three parameters depend on the material and n is generally between 0.3 and 0.5. The important point is that such a model can be very well fitted to experimental data for shear stress vs. shear rate over a range of several decades (typically four to five) of shear rates. As a consequence this model is able to represent the material behaviour from very slow flows (typically at shear rates of the order of 10−2s−1) to rapid flows (at shear rates of the order of 100s−1). This contrasts with the Bingham (n = 1) or even the Casson models still used in some cases but which generally fit the data over a much more limited range of shear rates. As a consequence, with the two latter models a specific set of parameters may provide appropriate predictions of the flow characteristics of a given material under some specific boundary conditions, but may be quite inappropriate for describing them under other circumstances. Although the structure of yield stress fluids relies on the arrangement of mesoscopic objects rather than on molecules, it is possible to use the micromechanical approaches developed for crystalline solids (see Section 1.1) to get some idea of the physical origin of the rheological parameters of such materials in the solid regime. In particular the elastic modulus may be related to the structure characteristics (interactions between elements) in some ideal cases and the yield stress value can be related to the critical stress needed to break the material structure. In contrast, for the liquid regime no relevant physical meaning has so far been provided for the additional parameters (k and n) and the general form of the model. Thus the Herschel–Bulkley model is nothing more than an appropriate fit to data. In particular it should not be considered that two separate sources of viscous dissipation exist within the material in the liquid regime: one related to the breakage of the network of interactions and giving rise to a constant stress term (the yield stress), and another being related to the hydrodynamic viscous dissipation due to the flow of the interstitial liquid and giving rise to a powerlaw function of the shear rate (the second term of the Herschel–Bulkley model). In fact the sum of these two terms describes a progressive increase in the total viscous dissipation as the shear rate increases, as a result of more complex relative motions of the elements including some liquid flows, and it happens that this increase can be well represented by such a function. This remark appears even clearer for some materials for which the yield stress associated with the solid– liquid transition differs from that to be used in the Herschel–Bulkley model for © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 17 1.6 Apparent flow curve of a yield stress fluid obtained from an increasing–decreasing stress or shear rate ramp. The increase corresponds to the dotted line and the decrease to the continuous line. the liquid regime. This is so for thixotropic materials (see Section 1.6) for which the apparent yield stress at rest varies with the flow history so that obviously no single yield value can be deduced and used to describe the liquid regime. The apparent flow curve is usually obtained from an increasing–decreasing stress or shear rate ramp (Fig. 1.6). In this case the sample is initially at rest in its solid regime. As a consequence the response at low shear rates in the increasing stage does not generally correspond to steady state in the liquid regime. The material remains in its solid regime as long as the total deformation it has undergone is below a critical value. The measurement system nevertheless provides some shear rate values computed over some finite time, which are then reported in the rheogram. This leads to the typical aspect of the curve represented in Fig. 1.6, with an initial steep slope in the increasing curve which in fact varies with the timing of increase of the shear rate or shear stress. In contrast the flow curve in the liquid regime obtained in the decreasing stage is relatively robust, i.e. independent of the flow history. Let us examine yield stress fluids more generally. They are usually made of a large number of elements suspended in a liquid and which develop significant mutual interactions. This is so for example for concentrated emulsions or foams in which the droplets or bubbles are squeezed against each other. When a small stress is applied the droplets or bubbles are only slightly deformed, and when the stress is released they get back to their initial state. When a large stress is applied the elements are sufficiently deformed to break the structure at least over a very short period of time. Then a similar (on average) structure reforms with the elements distributed differently. If the stress is maintained this process repeats indefinitely and the material deforms indefinitely: it flows like a liquid. Similar © Woodhead Publishing Limited, 2012 18 Understanding the rheology of concrete phenomena should occur with colloidal suspensions with particles developing mainly repulsive interactions. With attractive particles the situation is slightly different: they form a network of linked particles which may break if a sufficiently large stress is applied; then the broken material is made of flocs or elements which may be considered as individuals during some time, i.e. as long as they do not restore some link. The materials described above are model systems. Real materials are often concentrated mixtures of a lot of element types making it not so easy to distinguish which element type, and thus which interaction type, is at the micromechanical origin of the mechanical behaviour. The general characteristics of these systems nevertheless remain: • • • We are dealing with jammed, disordered structures in the solid regime. These structures continuously break and reform in the liquid regime. The disorder is similar in the liquid and the solid regimes. 1.6 Thixotropy We already mentioned this possible character of the behaviour, i.e. the dependence of the apparent viscosity on the flow history. It is for yield stress fluids that thixotropy is most often observed and can play a significant role. Since yield stress fluids exhibit two behaviour regimes it is necessary to examine how thixotropy takes place in each. 1.6.1 Solid regime Thixotropic effects in the solid regime can be well characterised from the increase of the elastic modulus in time as measured for small deformations. However, there is no general model for describing these evolutions and that can be considered as applicable to any material. Very roughly speaking the trends could be described by a logarithmic function of the time, i.e. G ∝ ln t/t0, which at least illustrates that the rate of increase of the elastic modulus decrease with the time of rest. Another possibility for appreciating the thixotropy in the solid regime is to observe the increase of the apparent yield stress in time. A similar rough logarithmic variation is observed. Qualitatively it is clear that the increase in strength results from the fact that the jamming increases or the interactions strengthen. For attractive colloidal systems this is related to the increase of the links in time: clear pictures of this effect have been obtained for simple colloidal systems in two dimensions, showing how aggregates progressively increase in size in time after the particles have been initially dispersed in the liquid. In this case the particles wander through the liquid as a result of Brownian motion and aggregate when they encounter other particles. However, even for such an ideal system, there is no model describing this kinetic © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 19 process and there is no obvious relationship between the structure characteristics and mechanical strength. 1.6.2 Liquid regime In the liquid regime thixotropy is observable by the fact that the apparent viscosity decreases in time while the boundary conditions are fixed: starting from the material at rest the shear stress decreases when the shear rate is imposed or the shear rate decreases when the shear stress is fixed. The importance of this decrease depends on the initial state: it is more marked for a longer time of rest. More generally a progressive decrease of the apparent viscosity in time is generally observed when one changes the level of shear rate imposed. These effects are associated with the fact that a specific steady state structure corresponds to each shear rate level, and it takes some time to reach. This phenomenon has a clear impact on the measurements with the help of the usual procedure for flow curve determination, i.e. increase then decrease of the shear rate or shear stress (Fig. 1.7). The level of the plateau (after the initial progressive increase) increases with the preliminary time of rest, then the fluid is liquefied and finally reaches the flow curve associated with a smaller time of rest. Once again there is no general picture of the evolution of the structure state with shear rate either from a mechanical or a physical point of view. For simple suspensions of attractive particles it seems reasonable to assume that, as the imposed shear rate is increased, the network of linked particles breaks in smaller flocs, such that it might be possible to establish some correspondence between the average floc size and the shear rate. Some attempts have been made in that way 1.7 Apparent flow curve for an increasing–decreasing stress or shear rate ramp after different times of rest: the initial plateau level increases with the time of rest so that the apparent yield stress increases, but the decreasing flow curves are identical. © Woodhead Publishing Limited, 2012 20 Understanding the rheology of concrete but there is a lack of effective knowledge of the processes that occur, so that in general fitting parameters are finally used in such models. It is worth emphasising that, although a unity of description would be useful, the physical processes at work in thixotropy processes in the solid and the liquid regimes are likely significantly different. Usually, the evolution of the state of the structure is described with the help of a single parameter, λ, and the rheological parameters are assumed to vary with this parameter. Besides, the time evolution of the state of structure of the material λ is described using a kinetic equation in which the rate of variation of λ is expressed as the difference between a rate of natural restructuring f and a rate of destructuring g: [1.12] However, the simple yielding behaviour described so far has some problems. First of all it is surprising to have materials behaving as solids below a critical stress and flowing homogeneously like simple liquids just beyond this critical stress. Indeed, for usual solids the flow start is associated with a breakage or strong localisation of shear in the material. Yield stress fluids would thus be special solid materials able to suddenly liquefy beyond a critical stress. Besides, the horizontal plateau observed for thixotropic yield stress fluids (Fig. 1.7) for an increasing ramp of stress or shear rate suggests that some flow instability may occurs: a material exhibiting such a plateau in its flow curve can hardly flow steadily at a shear in the range of the plateau. Actually it seems that mainly two classes of materials exist. Physical gels, concentrated emulsions or foams seem to behave mainly as simple yield stress fluids. This is likely to be because the network of interaction associated with the yielding behaviour results from the jamming of the elements squeezed against each other and is broken during shear over successive short periods of time so that the stress values needed to induce and maintain flow are very similar. Moreover there is no significant restructuring effect at rest. Thus there is a unique yield stress and the stress needed to maintain flow in the liquid regime increases from this value as the shear rate increases. In contrast for materials containing a significant fraction of attractive colloidal particles, the strength of the material in the solid regime can be significantly larger than that in the liquid regime because the network of interactions is broken in the latter state. This explains the plateau in the increasing part of the flow curve observed after a sufficient time of rest (Fig. 1.7). For such materials, shearbanding (i.e. a strong heterogeneity in the shear rate distribution) may develop for apparent shear rates along the plateau in the flow curve. However, in practice it is not clear that this effect has a strong impact on the macroscopic characteristics of the flow. Indeed a model fitted to the apparent flow curve (including a plateau) and assuming homogeneous flow appears to provide a sufficient approximation for describing most practical cases. © Woodhead Publishing Limited, 2012 Introduction to the rheology of complex fluids 1.7 21 Viscoelasticity 1.7.1 Solids Until now we have assumed that when the material is submitted to a given stress the steady deformation in the solid regime or the steady state shear rate in the liquid regime are reached instantaneously if the structure is not modified. In reality there are necessarily some time effects associated with viscous dissipation in the solid regime and storage of elastic energy in the liquid regime which may need to be take into account. Let us consider the solid case. A characteristic time is needed before the final deformation is almost reached. This finds its origin in viscous effects which are coupled or added to the elasticity of the structure. An illustrative picture of this situation is a gel made of polymers chains linked to each other and immersed in a liquid. For a given stress applied, the final deformation reached after some time depends strictly on the elastic modulus of the polymer network (which we assume here to be purely elastic). As the system is deformed the liquid is sheared so that there are also viscous effects, which may be described with the help of the apparent viscosity of the suspension of polymers in the liquid. Finally the total stress, . needed to impose a deformation γ leading to an instantaneous shear rate γ , can be written as: [1.13] This equation may easily be solved for example when the stress applied is fixed to some value. It is found that the deformation increases exponentially up to its asymptotic value τ0/G with a characteristic time µ/G. Thus, when the typical time of observation or experiment is larger than µ/G, the material essentially appears as an elastic solid. When, on the contrary starting from rest, one looks at the behaviour over timescales much smaller than µ/G, the material appears as a Newtonian material. For other material types the origin of viscous effects may be more complex, involving not only viscous dissipations in the interstitial liquid but also in the interactions between the elements of the structure. More complex behaviour than that described above may then be found but the general principle is the same. 1.7.2 Liquids In liquids also the steady state shear rate may not be reached immediately because initially part of the supplied energy is stored in the form of elastic energy. Let us for example consider the case of bubbles dispersed in a simple liquid. When a shear stress is applied to the system it tends to shear the liquid at a shear rate depending on its apparent viscosity, but also tends to deform the bubbles up to some deformation associated with the balance with surface tension effects. We assume that the liquid phase is a Newtonian fluid and the bubbles form a © Woodhead Publishing Limited, 2012 22 Understanding the rheology of concrete homogeneous material of elastic modulus G. Under these conditions the total deformation induced after some time can be written as: [1.14] . Under fixed shear rate (γ 0) this equation is easily solved and implies that the shear . stress decreases exponentially down to its asymptotic value µγ 0 with a characteristic time µ/G. Thus, when the typical time of observation or experiment is larger than µ/G, the material essentially appears as a simple liquid. When, on the contrary starting from rest, one looks at the behaviour over timescales much smaller than µ/G, the material appears as an elastic solid. Note that µ/G is also the characteristic time of relaxation of the system: if one stops the steady state flow at some time the bubbles will get back to their equilibrium shape with the characteristic time µ/G. The origin of elasticity storage in materials is the elongation of polymer chains, the deformation of flocs of colloidal particles, the deformation of droplets, etc. These effects can be modelled by more or less sophisticated models (in particular involving a large number of relaxation times), but the general principle as described above remains the same. 1.8 Conclusions When considering the various types of possible interactions between the components of a real complex fluid and the various corresponding effects on the macroscopic behaviour, it is often very difficult to be sure of any interpretation of rheological data (macroscopic) in terms of physicochemical interactions. In that aim it is preferable to use model systems in which the number of possible interactions and their effects on the macroscopic behaviour is limited. For real systems it is important to keep in mind the above background for simple or model systems, which can suggest reasonable interpretation. For all systems it is critical to carry out experiments that provide relevant data concerning their rheological behaviour. This behaviour needs to identify clearly the rheological regime in which the system is (solid, liquid, transient) and to control any artefact of the measurements which would affect the material homogeneity, the material volume or the assumed velocity field in the rheometer. This is the subject of the next chapter. © Woodhead Publishing Limited, 2012 2 Introduction to the rheometry of complex suspensions G. OVARLEZ, Université ParisEst, France Abstract: This chapter deals with experimental methods for the measurement of the rheological properties of materials, with a focus on complex suspensions such as fresh concrete. The various behaviours studied in this chapter are described in Chapter 1. The first section provides the basis of rheometry. The second section deals with the measurements of the properties of simple yield stress fluids. The third section is devoted to the measurements of the properties of thixotropic yield stress fluids. The phenomena of shear localisation and shear banding are discussed in the fourth section, together with material heterogeneities. Key words: rheology, rheometry, yield stress fluids, elastic modulus, yield stress, constitutive law, thixotropy, shear localisation, shear banding, migration. 2.1 Rheometry 2.1.1 Simple shear In order to predict the behaviour of materials in any complex flow, which a priori involves strains in all directions of space, it is necessary to know the relationship between the deviatoric stress tensor σij and the strain rate tensor dij (which is, for example, σij = 2ηdij for a Newtonian fluid of viscosity η).1 Such a tensional relationship is very hard to derive from experimental measurements. The objective of rheometry is to simplify the problem as much as possible by measuring the flow properties of materials in the case where there is only one nonzero component of the strain rate tensor, namely simple shear. Another simple case is extensional flow, which provides a complementary means of characterisation of materials but is not studied here (see Macosko, 1994). Note also that new methods have recently been developed to investigate the behaviour in controlled threedimensional flows (Jop et al., 2006; Ovarlez et al., 2010). Thanks to its simplicity, simple shear allows us to characterise and understand the flow behaviour of complex fluids. Moreover, such characterisation can be used directly to predict the flow behaviour in many practical situations (pipe flows, flows down a slope, etc). Nevertheless, in more complex cases, this characterisation still has to be generalised in a tensorial form. In simple shear (Fig. 2.1), shear occurs only through the relative motion of parallel layers of the material. Noting x the direction of flow and y the direction of the spatial variations of the velocity Vx(y), the only nonzero component of the 23 © Woodhead Publishing Limited, 2012 24 Understanding the rheology of concrete 2.1 Schematic representation of a simple shear experiment. strain rate tensor in simple shear is dxy = dyx. It is then convenient to define the shear rate as which is thus the velocity gradient. If γ͘ is constant along the y direction, simple shear is said to be homogeneous. Three components of the stress tensor can be nonzero in simple shear: the shear stress τ = σxy, and the first and second normal stress differences N1 = σxx − σyy, N2 = σyy − σzz. The purpose of rheometry is to characterise the dependence of the shear stress τ (and, if necessary, of the normal stress differences) on the shear strain γ and/or the shear rate γ͘ in simple shear. Homogeneous simple shear is theoretically achieved by putting a sample of the material between two infinite parallel plates separated by a distance H (named ‘gap’) and imposing a difference of velocity V between the two plates. The material behaviour is then characterised by measuring the relationship between two macroscopic quantities: the force F(V) that has to be applied to enforce a flow characterised by the velocity V. In the ideal conditions of the homogeneous simple shear of Fig. 2.1, the shear rate and the shear stress are obtained through γ͘ = V / H and τ = F / S (S being the plates’ surface, supposed to tend to infinity). The macroscopic measurements F(V) then provide directly the constitutive law τ (γ͘ ). Note also that, in some materials, a nonzero normal force FN has to be applied to maintain a constant gap during shear; in the ideal conditions of Fig. 2.1, FN provides a measurement of the total normal stress Σzz = FN / S as a function of γ͘ . As two normal stress differences may be nonzero, it is seen here that a single experiment in simple shear is not sufficient to obtain these two quantities. It is shown below that the use of two different geometries allows the two quantities to be measured. In order to impose simple shear with a finite geometry, most experiments use rotational flows. The main problems of rheometry are then to: • • find conditions in which homogeneous simple shear can be achieved; relate in a general case (homogeneous and heterogeneous shear), the measured macroscopic quantities (rotational velocity, torque, normal force) to local quantities (shear rate and stresses) and thus to the material constitutive behaviour τ (γ͘ ). It can be seen below that, in many cases, only approximate equations exist. It is thus vital to know their domain of validity and to be able to choose the equations appropriate for a given material in a given situation. © Woodhead Publishing Limited, 2012 Introduction to the rheometry of complex suspensions 25 2.2 Standard geometries: parallel plates, cone and plate, Couette (Coussot, 2005). The three main geometries that achieve simple shear flow are cone and plate, parallel plate and Couette (Fig. 2.2). Simple shear is also achieved in capillary flow and inclined plane geometry. These are not studied here but details can be found in, for example, Macosko (1994) and Coussot (2005). Below we present the basic equations and their origin, and discuss the application to complex fluids; detailed calculations can be found in Macosko (1994). 2.1.2 Cone and plate In a cone and plate geometry, the material is put between a cone and a disk of the same symmetry axis and the same radius R. The cone tip is truncated to avoid friction between the cone and the bottom plate. The cone virtual tip is supposed to touch the plate. The bottom plate is fixed, and shear is imposed by rotating the cone around the symmetry axis at a velocity Ω, resulting in a torque T. In these conditions, in spherical coordinates, the only nonzero component of shear is dθφ, which a priori depends on the angle only θ. From the stress equilibrium equation, it can then be shown that the shear stress τ = τθφ (in spherical coordinates) is roughly homogeneous for small cone angles: the shear stress variation from the plate to the cone is of order of 0.5% for a cone angle θ0=4°, which explains why cone of angles lower than 4° are always chosen. Because the shear stress τ is homogeneous and uniform in the whole gap, for materials which are characterised by a univalent relationship between stresses and strain rates (which excludes shearbanding materials), the shear rate γ͘ is constant in the whole gap. With a noslip boundary condition at the cone and plate surfaces, the shear rate can then simply be computed. The velocity at a radius r on the cone surface is Ωr; the velocity is zero on the plate surface, while the gap size at this radius is r tanθ0 © Woodhead Publishing Limited, 2012 26 Understanding the rheology of concrete leading to . This equation simply relates the shear rate to the applied macroscopic rotational velocity Ω. The torque T resulting from a constant shear stress τ exerted on the cone surface is . Note that the measurement of the normal force Fz exerted on the cone also provides: [2.1] To conclude, the main advantage of cone and plate geometry is that it achieves almost perfect homogeneous simple shear, provided the angle is small and that slip is avoided on the surfaces. Then, from the torque/rotational velocity relationship T(Ω), one can simply extract the constitutive behaviour τ (γ͘ ) thanks to reliable equations that are true whatever the material constitutive law is: [2.2] [2.3] This means that, together with small gap Couette geometry (see Section 2.1.4), this geometry is in general the best choice for material characterisation. However, it should be noted that the truncature of the cone tip is usually a few tens of microns and that the gap at the edge for a 2°/40 mm diameter cone is 0.6 mm. This implies that this geometry cannot be used for many suspensions; it can be used for colloidal suspensions whose elements’ size is of a few microns at most. Moreover, special care should be taken with shearbanding materials, for which two material states may coexist for a given shear stress. In this case, the approximate homogeneity of stress for small angles is still valid (it comes from stress equilibrium equation, independently of the material behaviour), but the shear rate need not be homogeneous anymore (a basic analysis of this case is found in Sections 2.3.1 and 2.4.3; details are found in Coussot, 2005; Ovarlez et al., 2009). Other problems may occur with thixotropic systems: at yield, the material may experience strong rejuvenation near the cone (where the shear stress is slightly higher) and stay solid near the plate; subsequent analysis of the properties from macroscopic measurements in such conditions are then likely to be incorrect. 2.1.3 Parallel plates In a parallel plate geometry, the material is put between two disks of same symmetry axis and of same radius R, separated by a distance H (named the ‘gap’). © Woodhead Publishing Limited, 2012 Introduction to the rheometry of complex suspensions 27 The bottom plate is fixed and shear is imposed by rotating the upper plate around the symmetry axis at velocity Ω, resulting in a torque T. In these conditions, in cylindrical coordinates, the only nonzero component of shear is dzθ which a priori depends on both r and z. From the stress equilibrium equation it can then be shown that the shear stress τ (r) = τzθ (r) does not depend on z. This is the only information we have about the shear stress: in contrast with a cone and plate geometry, τ (r) cannot be computed directly from the torque only. For materials characterised by a univalent relationship between stresses and strain rates, this also means that the shear rate γ͘ (r) depends only on the radius r. With a noslip boundary condition at both plate surfaces, the velocity at a radius r is Ω r on the upper plate and zero on the lower plate, leading to: [2.4] In other words, simple shear is achieved at any radius r in a parallel plate geometry, but shear is heterogeneous along the radial direction. Equation 2.4 also means that, from the knowledge of the applied macroscopic rotational velocity Ω, the local shear rate in the gap is known whatever the material constitutive law is. To compute the shear stress from the torque measurements, there are two possibilities. First, it can be shown that the shear stress τ (R) at the edge of the geometry (r=R) can be computed exactly as: [2.5] When associated with the shear rate computed at the same radial position R, Eq. 2.5 provides τ (γ͘ ) data from T(Ω) data. However, as this expression involves the derivative of the torque with respect to the rotational velocity, many accurate T(Ω) data have to be recorded to allow for the computation of τ (γ͘ ) data. It is thus not very convenient to use. Another possibility consists of using expressions that are strictly valid only for a given material and to use them for any material. For example, for a viscous fluid τ (r) = ηγ͘ (r), one gets Combining this equation with . makes it possible to extract a τ (γ͘ ) data from a single T(Ω) measurement. These couple of data can a priori be computed at any radial position r. This position can be chosen so as to minimise the error made in the material characterisation when using these equations for nonNewtonian materials. It has been shown that the error is minimised at r≈3R/4 for materials of constitutive law τ Ɱ γ͘ n with n<1.4; combining τ (3R/4) and γ͘ (3R/4) then leads to a correct evaluation of their behaviour within 2% error (Macosko, 1994). This explains why the choice of computing the shear stress and shear rate at 3R/4 is made by default in most rheometer software. © Woodhead Publishing Limited, 2012 28 Understanding the rheology of concrete Note that the measurement of the normal force exerted on the upper plate also provides τ θθ + τ rr − 2τzz = N1 − N2 (Macosko, 1994). If normal stress differences are important, full determination of the stress tensor in simple shear then implies a need to use both parallel plate and cone and plate geometries. To conclude, parallel plate geometries are characterised by a heterogeneous but controlled shear rate in most situations provided slip is avoided on the surfaces. Standard approximate equations can be used in many situations to directly relate a torque/rotational velocity measurement T(Ω) to a data point of the constitutive behaviour through: [2.6] [2.7] It should be noted that the shear rate computed in that way is the exact value at 3R/4 whatever the material constitutive law, whereas the shear stress is exact only for Newtonian fluids. Finally, note that in contrast to a cone and plate geometry, the gap is not fixed. It can be as high as a few millimetres (the main limitation being that gravity effects have to remain negligible to allow for the above equations to be derived; Coussot, 2005). Parallel plates can then be used for suspensions whose elements size is of 200 µm at most (e.g. mortars with fine particles). Shear being heterogeneous, special care should be taken with yield stress fluids at low shear rate: shear localisation is then likely to appear (Coussot, 2005) and is more difficult to take into account than in a Couette geometry (see below). 2.1.4 Couette geometry In a Couette geometry, the material is put between two coaxial cylinders. In most cases, the outer cylinder is fixed and shear is imposed by rotating the inner cylinder of height H around the symmetry axis at a velocity Ω, resulting in a torque T. Note that the distance of the inner cylinder to the bottom of the cup is usually chosen to be much larger than the gap size to ensure that the contribution to the torque from shear at the bottom is as negligible as possible; these end effects can be corrected with appropriate procedures if necessary (Macosko, 1994). In these conditions, in cylindrical coordinates, the only nonzero component of shear is drθ which a priori depends on r only. From the stress equilibrium equation, it can be shown that the shear stress τ (r) = τrθ (r) is known everywhere in the gap and is given by: [2.8] © Woodhead Publishing Limited, 2012 Introduction to the rheometry of complex suspensions 29 i.e. the shear stress is heterogeneous and controlled. On the other hand the shear rate γ͘ (r) is a priori unknown. This is different from the case of a parallel plate geometry, where the shear rate is heterogeneous and controlled while the shear stress is unknown. Two cases have to be distinguished: thin gap Couette geometry and wide gap Couette geometry. A Couette geometry is said to have a thin gap when the shear stress can be considered as homogeneous within 1%. From , this i.e. that the gap size Ro − Ri is smaller than 0.5% of the means that inner radius Ri. It then implies that, for materials characterised by a univalent relationship between stresses and strain rates (which excludes shearbanding materials), the shear rate is almost constant in the whole gap. With a noslip boundary condition at the surface of both cylinders, the shear rate can then simply be computed as: [2.9] while the shear stress is: [2.10] Other cases are referred to as wide gap Couette geometries. In this case, as for a parallel plate geometry, an exact equation exists (Coussot, 2005): [2.11] However, Eq. 2.11 is not very convenient to use as it does not provide directly the value of the shear rate at a given position, but the difference between the shear rates at the inner and outer cylinders. Nevertheless it is used in the simple case of yield stress fluids close to yielding, where shear localisation occurs (see Section 2.4.3). It then reduces to , which automatically takes shear localisation into account (Jarny et al. 2008; Ovarlez et al., 2008; Coussot et al., 2009). It is more usual to use approximate equations. As for a parallel plate geometry, one computes the shear rate distribution for Newtonian materials and uses the γ͘ (r) equation obtained in that way for any material. To extract τ (γ͘ ) data from a single T(Ω) measurement, a conventional choice is to compute the shear stress τ (r) and the shear rate γ͘ (r) in the middle of the gap. Standard equations used to relate a torque/rotational velocity data T(Ω) to a constitutive behaviour data point τ (γ͘ ) are – finally, when computed in the middle of the gap R = (Ro + Ri)/2: [2.12] © Woodhead Publishing Limited, 2012 30 Understanding the rheology of concrete [2.13] Other approximate equations can be used, depending on the radius where shear stress and shear rate are computed; at the inner radius, one gets and . Note that flows in a Couette geometry do not provide normal stress measurements unless specific normal force probes are put on the outer cylinder surface (Deboeuf et al., 2009). The main advantage of Couette geometries for complex fluids is that the gap can be large. It is thus the preferred geometry for suspensions made of large particles such as fresh concrete. However, unless the inner cylinder radius is very large, a wide gap implies a very heterogeneous shear stress distribution which poses the problem of shear localisation and shearinduced migration – see below. 2.1.5 Vane tool Concentrated suspensions are often studied with a vaneincup geometry; see Barnes and Nguyen (2001) for a review. In this geometry, the inner cylinder of the Couette geometry is replaced by a vane tool, which consists of N thin blades centred on a thin shaft (usually, N = 4 or 6). Compared with the coaxial cylinders Couette geometry, it a priori offers two main advantages. First, it allows the study of the properties of structured materials with minimal disturbance of the material structure during the insertion of the tool. Second, it is supposed to avoid wall slip (Saak et al., 2001) because the material sheared in the gap of the geometry is sheared by the (same) material that is trapped between the blades. Consequently, it is widely used to study the behaviour of pasty materials containing large particles as fresh concrete (Koehler et al., 2006; Estellé et al., 2008; Wallevik, 2008). The streamlines in a vaneincup geometry are generally not circles. This implies that the above equations for a Couette geometry cannot a priori be applied in this case. In practice, for a Newtonian material, the torque exerted on a vane tool is lower than that exerted on a cylinder of same inner radius driven at a same rotational velocity; it is typically 10–30% lower for standard geometries (Sherwood and Meeten, 1991; Ovarlez et al., 2011). Nevertheless, a practical method, known as the Couette analogy (Bousmina et al., 1999; Estellé et al., 2008), can still be used to infer constitutive laws from macroscopic measurements with a vane tool. It basically consists of calibrating the geometry with linear (Hookean or Newtonian) materials. The equivalent inner radius Ri,eq of a vaneincup geometry is defined as the radius of the inner cylinder of a Couette geometry that would have the same torque/shear stress and velocity/shear rate conversion factors for a linear material. For any material, all macroscopic data are then analysed as if the material was sheared in a Couette geometry of inner cylinder radius Ri,eq. The nonlinearity (which affects the flow field) is sometimes accounted © Woodhead Publishing Limited, 2012 Introduction to the rheometry of complex suspensions 31 for as it is in a standard Couette geometry (Estellé et al., 2008). This approach may finally provide constitutive law measurements within a good approximation (Baravian et al., 2002). Note that for suspensions of coarse particles, particle depletion may be induced very rapidly by shear near the blades. It results in a kind of wall slip near the blades (Ovarlez et al., 2011), whereas the vane tool is mainly used to avoid this phenomenon. This suggests that a coaxial cylinders geometry with properly roughened surfaces is preferable, when possible, when studying flows of pasty materials with large particles. A detailed analysis is provided in Section 2.4.5. 2.1.6 Geometry factors for yield stress measurements Some of the equations derived in parallel plate and Couette geometries are only approximate: they are chosen to provide the best possible characterisation of a wide range of materials with very different nonlinear rheological properties. When studying a given kind of material, one question that arises is the accuracy of these equations for specific quantities. Below we discuss the case of yield stress measurements in yield stress fluids. In a parallel plate geometry, at yield τ (r) = τY everywhere on the upper plate. Together with this yields: [2.14] It is remarkable that Eq. 2.14, which here is exact for yield stress measurements, is the same as Eq. 2.8, which is optimal for flow properties. Equation 2.8 can then be used reliably together with Eq. 2.7 to evaluate all material properties. In a Couette geometry, yield first occurs on the inner cylinder, where stress is maximal. This implies that at yield, τ (Ri) = τY , and thus: [2.15] This means that, in a wide gap Couette geometry, the best choice of geometry factors for yield stress measurements may not be the same as those for flow properties (see for example Eq. 2.11 and Eq. 2.12). In a vaneincup geometry, cylindrical symmetry seems to be recovered in yield stress fluids near yielding (Keentok et al., 1985; Yan and James, 1997; Savarmand et al., 2007). This would mean that Ri,eq = Ri can be used for yield stress measurements and one may thus compute exactly the yield stress with: [2.16] in this specific situation. This point is discussed in more detail in Section 2.4.5. © Woodhead Publishing Limited, 2012 32 Understanding the rheology of concrete 2.1.7 Experimental problems Most of the experimental problems found in rheometric experiments come from the main hypotheses made when computing shear stress/shear rate relationships from torque/rotational velocity measurements. First, one has to ensure that the flow and the material are homogeneous; these specific issues are dealt with in Section 2.4. Then, a noslip boundary condition is assumed at the interfaces of both tools. Rough surfaces are usually used to ensure there is no wall slip in the experiments. Moreover, the absence of wall slip in parallel plate and Couette geometries can be checked by performing experiments with two different gaps (Coussot, 2005). If different results with two different gaps are obtained, this means that the noslip hypothesis made in the analysis is not correct, i.e. there is wall slip. Another important problem in rheometry arises from the fact that the applied torque is the sum of the torque applied to the material and the inertial torque applied to accelerate/decelerate the rotating tool. One thus has to be particularly careful when applying ramps and oscillations (see Section 2.2.1). Although the inertial torque can theoretically be accounted for, this may pose accuracy problems when it is dominant. This makes high frequency oscillatory measurements particularly difficult. For more details on experimental problems, see Coussot (2005). 2.2 Characterisation of simple yield stress fluids Yield stress fluids (see Chapter 1) are materials that flow steadily only when the applied shear stress τ in simple shear is higher than a yield stress τY . In addition, the behaviour of ‘simple’ yield stress fluids is basically independent of the flow history. Different methods are presented below devoted to characterising the flow properties of yield stress fluids when τ ≥ τY and their solid properties when τ ≤ τY, and to determining the value of τY accurately. 2.2.1 Steady flow properties Steady flow properties in simple shear are characterised by the stationary relationship τ = f(γ͘ ) between the shear stress τ and the shear rate γ͘ . These data can be obtained in both stress controlled and shear rate controlled experiments. Creep tests Creep tests consist of applying a series of constant shear stresses τi over a long period of time, usually after a preshear (Fig. 2.3). For, τi ≥ τY a yield stress fluid reaches steady flow at a shear rate γ͘ i after a time teq which depends on τi and the material. The data τi = f(γ͘ i) then define the flow curve of the material. For τi < τY, © Woodhead Publishing Limited, 2012 Introduction to the rheometry of complex suspensions 33 2.3 Schematic representation of a creep experiment. one should observe unsteady flow with a vanishing shear rate , which defines the yield stress τY . This is illustrated in Section 2.3.1 for thixotropic yield stress fluids. Of course, a series of constant shear rates can also be applied until a constant shear stress is reached. These last experiments are commented on in more detail in the case of thixotropic materials. But although performing creep tests is a very accurate method for characterising materials, it is quite time consuming and does not provide a lot of data. Ramps and sweeps The independence of the behaviour of simple yield stress fluids on flow history usually implies that, when the ap