Mechanics of Geomaterial Interfaces

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STUDIES IN APPLIED MECHANICS 42

Mechanics of Geomaterial Interfaces

STUDIES IN APPLIED MECHANICS Probabilistic Approach to Mechanisms (Sandier) 9. Methods of Functional Analysis for Application in Solid Mechanics (Mason) 10. Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates (Kitahara) 11. Mechanics of Material Interfaces (Selvadurai and Voyiadjis, Editors) 12. Local Effects in the Analysis of Structures (Ladeveze, Editor) 13. Ordinary Differential Equations (Kurzweil) 14. Random Vibration- Status and Recent Developments. The Stephen Harry Crandall Festschrift (Elishakoff and Lyon, Editors) 15. Computational Methods for Predicting Material Processing Defects (Predeleanu, Editor) 16. Developments in Engineering Mechanics (Selvadurai, Editor) 17. The Mechanics of Vibrations of Cylindrical Shells (MarkuP) 18. Theory of Plasticity and Limit Design of Plates (Sobotka) 19. Buckling of Structures-Theory and Experiment. The Josef Singer Anniversary Volume (Elishakoff, Babcock, Arbocz and Libai, Editors) 20. Micromechanics of Granular Materials (Satake and Jenkins, Editors) 21. Plasticity. Theory and Engineering Applications (Kaliszky) 22. Stability in the Dynamics of Metal Cutting (Chiriacescu) 23. Stress Analysis by Boundary Element Methods (Bala~, Sladek and Sl,Sdek) 24. Advances in the Theory of Plates and Shells (Voyiadjis and Karamanlidis, Editors) 25. Convex Models of Uncertainty in Applied Mechanics (Ben-Haim and Elishakoff) 26. Strength of Structural Elements (Zyczkowski, Editor) 27. Mechanics (Skalmierski) 28. Foundations of Mechanics (Zorski, Editor) 29. Mechanics of Composite Materials-A Unified Micromechanical Approach (Aboudi) 30. Vibrations and Waves (Kaliski) 31. Advances in Micromechanics of Granular Materials (Shen, Satake, Mehrabadi, Chang and Campbell, Editors) 32. New Advances in Computational Structural Mechanics (Ladeveze and Zienkiewicz, Editors) 33. Numerical Methods for Problems in Infinite Domains (Givoli) 34. Damage in Composite Materials (Voyiadjis, Editor) 35. Mechanics of Materials and Structures (Voyiadjis, Bank and Jacobs, Editors) 36. Advanced Theories of Hypoid Gears (Wang and Ghosh) 37A. Constitutive Equations for Engineering Materials Volume 1: Elasticity and Modeling (Chen and Saleeb) 37B. Constitutive Equations for Engineering Materials Volume 2: Plasticity and Modeling (Chen) 38. Problems of Technological Plasticity (Druyanov and Nepershin) 39. Probabilistic and Convex Modelling of Acoustically Excited Structures (Elishakoff, Lin and Zhu) 40. Stability of Structures by Finite Element Methods (Waszczyszyn, Cichoh and Radwaflska) 41. Inelasticity and Micromechanics of Metal Matrix Composites (Voyiadjis and Ju, Editors) 42. Mechanics of Geomaterial Interfaces (Selvadurai and Boulon, Editors) .

General Advisory Editor to this Series: Professor Isaac Elishakoff, Center for Applied Stochastics Research, Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, U.S.A.

STUDIES IN APPLIED MECHANICS 42

Mechanics of Geomaterial Interfaces

Edited by

A.P.S.

Selvadurai

Department of Civil Engineering and Applied Mechanics McGill University Montreal, Canada

M.J.

Boulon

Laboratoire 3S, IMG Universite Joseph Fourier Grenoble, France

1995

ELSEVIER Amsterdam- Lausanne- New York- Oxford- Shannon-Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN 0-444-81583-X 91995 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts ofthis publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed bythe publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper. Printed in The Netherlands

PREFACE The term

interface can be described as either a surface forming a common boundary

between identical material regions or as a surface which separates two distinct material regions. Interfaces are a common place occurence in many branches of engineering where either the material under consideration is endowed with distinct regions, or the material is designed with distinct regions to achieve an optimum performance. The term geomaterial interface refers to a distinct idealized surface between geomaterials such as rock, soil, concrete, ice and other metallic and non-metallic engineering materials. The subject of

geomaterial interfaces recognizes the

important influences of the interface behaviour on the performance of interfaces involving cementaceous materials such as concrete and steel, ice-structure interfaces, concrete-rock interfaces and interfaces encountered in soil reinforcement. During the past two decades, the subject of geomaterial interfaces has attracted the concerted attention of scientists and engineers both in geomechanics and applied mechanics. These efforts have been largely due to the observation that the conventional idealizations of the behaviour of interfaces between materials by frictionless contact, bonded contact, Coulomb friction or finite friction tend to omit many interesting and important influences of special relevance to geomaterials. The significant manner in which non-linear effects, dilatancy, contact degradation, hardening and softening, etc., can influence the behaviour of the interface is borne out by experimental evidence. As a result, in many instances, the response of the interface can be the governing criterion in the performance of a geomechanics problem. The primary objective of this volume is to provide a documentation of recent advances in the area of geomaterial interfaces. In the opinion of the Editors, the developments in the general subject area of geomechanics has matured to the point that reasonably comprehensive expositions could be provided to illustrate fundamental and experimental aspects of interface behaviour, constitutive modelling of interface response and, in particular, the adaptation of such responses in computational techniques, involving finite element, boundary element and distinct element methods for the solution of problems of technological interest. The volume consists of subject groupings which cover ice-structure interfaces, soil-structure interfaces, steel-concrete interfaces, mechanics of rock and concrete joints and interfaces in discrete systems. The first section, on ice-structure interfaces, examines the modelling of frozen soil-structure interfaces, and ice-structure interfaces and the contact zone behaviour between ice and structures, as characterized by failure in the form of crushing and flake development at the interface. Section 2, on soil-structure modelling, deals with the constitutive modelling of interfaces and the development of experimental procedures for the characterization of such constitutive models. This section also examines the development of finite element schemes and boundary element schemes for the examination of a vm'iety of non-linear soil-structure interface problems involving

vi embedded structures, contact mechanics and fractured surfaces with frictional constraints. Section 3 deals with the role of interface responses on the mechanical behaviour of steel-concrete interfaces. Novel applications in this area include the consideration of lattice network models for the modelling of fracture evolution in the concrete region, and structural models for the study of load transfer from embedded steel fibre-concrete interfaces. This section also includes a thorough examination of the important implications of experimental observations on the modelling of steel-concrete interfaces. The computational modelling of the steel-concrete interfaces presented in this section also considers the role of continuum damage and plasticity effects, cracking and other inelastic processes on the behaviour of structural components. The mechanical response of rock and concrete joints is discussed in Section 4 of the volume. The topics covered in this section include the development of experimental techniques for the characterization of rock joints, constitutive modelling and the implementation of such results in finite element and boundary element computations. The section also contains the application of finite element schemes for the study of rock slopes containing joints. The topic of concrete joints considered here is of particular importance to the computational modelling of joints encountered in concrete arch dams. In Section 5 a variety of special phenomena of interest to interfaces in discrete systems are examined. Topics, such as pore pressure effects in the interface response, are discussed, with special reference to landslide processes.

The mathematical and computational aspects of

frictional contact in collections of rigid and deformable media is discussed by appeal to a variety of examples. This section also contains a complete discussion of the modelling of interface localization in Cosserat continua which exhibit plasticity effects. The section culminates with a discussion of the contact conditions at particulate media where inter-particle effects include frictional and deformability effects. In its original concept, the volume was to have included a section devoted to fluid flow and porous media effects at interfaces; this section unfortunately could not be prepared due to unforeseen commitments on the part of the prospective contributors. The Editors hope that the volume will be a useful addition, as a benchmark reference, to the extensive literature dealing with mechanics of geomaterial interfaces. The consideration of a wider class of interfaces, their experimental analysis, constitutive modelling and computational implementation should be of particular interest to practising engineers and researchers interested in further developing this subject area. The importance of the general subject area need not be restricted to applications purely in the area of geomaterials; the subject matter has wider applications to problems encountered in bio-engineering, with particular application to the mechanics of prosthetic implants, thin film and substrate technology encountered in material

vii science and interfaces encountered in cementaceous ceramic components, multilayered structures, nanophase and nanocomposite materials and polymer-inorganic interfaces. It is hoped that researchers engaged in the geomechanics field can take advantage of these research opportunities and that those engaged in the wider applications can take advantage of the considerable experience and expertise developed in the field of geomechanics. A.P.S. Selvadurai McGill University Montreal, Canada

M.J. Boulon Laboratoire 3S IMG Grenoble, France

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ix ACKNOWLEDGEMENTS The authors would like to express their sincere thanks to the authors for their patience and understanding in the preparation of their contributions to this volume. The review process associated with each contribution involved the modification of a number of chapters. The assistance of the authors in this endeavour was invaluable.

The original concept for the

development of a volume devoted to Mechanics of Geomaterial Interfaces was first discussed by the Editors in the Spring of 1989. One of the Editors (A.P.S. Selvadurai) is grateful to the INPG for the award of a Visiting Fellowship to Laboratoire 3S IMG, CNRS, UniversitE Joseph Fourier, Grenoble, France, which enabled the development of the basic outline for the volume. The major part of the finalization of the content of the volume, and communications with prospective authors, was achieved during the visit of one of the Editors (M.J. Boulon) as an NSERC International Fellow to Carleton University in 1992/1993. The Editors are grateful to the Department of Civil and Environmental Engineering at Carleton University for the initial support provided in connection with the organization of this volume. They are also appreciative of the support provided by the Laboratoire 3S, IMG, Universit~ Joseph Fourier and the Department of Civil Engineering and Applied Mechanics at McGill University in making the final preparations towards the publication of the volume.. The cooperation of the Editorial Staff at Elsevier Scientific Publishers, Amsterdam, in the development of the volume is gratefully acknowledged. A number of the authors and other scientists and engineers kindly reviewed the Chapters, and their assistance has resulted in contributions of substantial merit. Finally, the preparation of this volume was greatly facilitated by the expert editorial assistance provided by Mrs. Sally J. Selvadurai, who completed much of the copy editing of the original and revised versions of each contribution and compiled the author and keyword (subject) indices and the general layout for the volume. She was also responsible for following up on the multitude of queries and communications that are usually associated with an undertaking of this nature. The Editors gratefully acknowledge her assistance.

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xi LIST OF CONTRIBUTORS E.E. Alonso, Technical University of Catalunya, Barcelona, Spain A. Alvappillai, American Geotechnical, Anaheim, CA, U.S.A. G.L. Bal~izs, Stuttgart University, Germany G. Beer, Technical University Graz, Austria M. Boulon, Universit6 Joseph Fourier, Grenoble, France I. Carol, Technical University of Catalunya, Barcelona, Spain J.L. C16ment, Ecole Nationale Sup6rieure de Cachan, Paris, France R.O. Davis, University of Canterbury, Christchurch, New Zealand C.S. Desai, University of Arizona, AZ, U.S.A. R.M.W Frederking, National Research Council of Canada, ONT, Canada P. Garnica, Universit6 Joseph Fourier, Grenoble, France A. Gens, Technical University of Catalunya, Barcelona, Spain J.-M. Hohberg, IUB Engineering Services Ltd., Berne, Switzerland M. Jean, Scientifique et Technique du Languedoc, Universit6 Montpellier, France L. Jing, Royal Institute of Technology, Stockholm, Sweden B. Ladanyi, Ecole Polytechnique, QC, Canada Z. Li, Northwestern University, IL, U.S.A. O. Merabet, INSA, Lyon, France A. Misra, University of Missouri-Kansas City, MO, U.S.A. M.E. Plesha, University of Wisconsin, WI, U.S.A. B.A. Poulsen, Center for Advanced Technologies, Kenmore, Australia H.W. Reinhardt, Stuttgart University, Gemlany J.M. Reynouard, INSA, Lyon, France D.B. Rigby, University of Arizona, AZ, U.S.A. K. Riska, Helsinki University of Technology, Espoo, Finland A.P.S. Selvadurai, McGill University, QC, Canada S.P. Shah, Northwestern University, IL, U.S.A. Y. Shao, Northwestern University, IL, U.S.A. D.S. Sodhi, U.S. Army Cold Regions Research and Engineering Laboratory, NH, U.S.A. O. Stephansson, Royal Institute of Technology, Stockholm, Sweden G.W. Timco, National Research Council of Canada, ONT, Canada

xii P. Unterreiner, CERMES, Ecole Nationale des Ponts et ChaussEes, Noisy-le-Grand, France J.G.M. van Mier, Delft University of Technology, The Netherlands I. Vardoulakis, National Technical University, Athens, Greece P.A. Vermeer, Stuttgart University, Germany A. Vervuurt, Delft University of Technology, The Netherlands M. Zaman, University of Oklahoma, OK, U.S.A.

xiii TABLE OF CONTENTS

Preface Acknowledgements

ix

List of Contributors

xi

ICE-STR UCTURE INTERFA CES Frozen Soil-Structure Interfaces B. Ladanyi Experimental Investigations of the Behaviour of Ice at the Contact Zone G.W. Timco and R.M.W. Frederking

35

An Ice-Structure Interaction Model D.S. Sodhi

57

Models of Ice-Structure Contact for Engineering Applications K. Riska

77

SOIL-STRUCTURE INTERFACES Modelling and Testing of Interfaces C.S. Desai and D.B. Rigby

107

Soil-Structure Interfaces: Experimental Aspects M. Zaman and A. Alvappillai

127

Soil-Structure Interaction" FEM Computations M. Boulon, P. Garnica and P.A. Vermeer

147

Boundary Element Modelling of Geomaterial Interfaces A.P.S. Selvadurai

173

STEEL-CONCRETE INTERFACES Lattice Model for Analysing Steel-Concrete Interface Behaviour J.G.M. van Mier and A. Vervuurt

201

Modelling of Constitutive Relationship of Steel Fiber-Concrete Interface S.P. Shah, Z. Li and Y. Shao

227

Steel-Concrete Interfaces: Experimental Aspects H.W. Reinhardt and G.L. Bal~izs

255

Steel-Concrete Interfaces: Damage and Plasticity Computations J.M. Reynouard, O. Merabet and J.L. ClEment

281

xiv

MECHANICS OF ROCK AND CONCRETE JOINTS Mechanics of Rock Joints: Experimental Aspects L. Jing and O. Stephansson

317

Rock Joints - BEM Computations G. Beer and B.A. Poulsen

343

Rock Joints: Theory, Constitutive Equations M.E. Plesha

375

Rock Joints" FEM Implementation and Applications A. Gens, I. Carol and E.E. Alonso

395

Concrete Joints J.-M. Hohberg

421

INTERFACES IN DISCRETE SYSTEMS Pore Pressure Effects on Interface Behaviour R.O. Davis

449

Frictional Contact in Collections of Rigid or Deformable Bodies: Numerical Simulation of Geomaterial Motions M. Jean

463

Interface Localisation in Simple Shear Tests on a Granular Medium Modelled as a Cosserat Continuum I. Vardoulakis and P. Unterreiner

487

Interfaces in Particulate Materials A. Misra

513

AUTHOR INDEX

537

SUBJECT INDEX

539

ICE-STRUCTURE INTERFACES

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

Frozen soil - structure interfaces

B. Ladanyi Ecole Polytechnique, C.P. 6079, Station A Montreal, Quebec, H3C 3A7, Canada

1. GENERAL The problem of frozen soil - structures interface behavior concerns many engineering problems, and in particular those in the fields of permafrost engineering and artificial ground freezing. For example, in the design of deep foundations and piles in permafrost, the interface behavior plays a major role, as does also that at the contact between frozen ground and lining in the design of artificially frozen shafts and tunnels. In addition, the interface problem is also of interest in connection with the performance of excavating tools and machines in permafrost. Adfreeze bond of frozen soil against a structure depends essentially on the physical properties of the soil, the characteristics of the interface, the temperature, and the type and the rate of loading. Under tensile loading, the interface behavior is governed mainly by the tensile adfreeze strength of pore ice. Under shear loading, in turn, the behavior depends on the shear strength of frozen soil, which is a granular composite material, composed of solid grains, ice and unfrozen water. The following gives a brief overview of some basic characteristics of frozen soils that may affect the behavior of interfaces.

2. CREEP AND STRENGTH BEHAVIOR OF FROZEN SOIL 2.1 Introduction From the point of view of the science of materials, frozen soil is a natural particulate composite, composed of four different constituents: solid grains (mineral or organic), ice, unfrozen water and gases. Its most important characteristic by which it differs from other similar materials, such as unfrozen soils and the majority of artificial composites, is the fact that under natural conditions its matrix, composed mostly of ice and water, changes continuously with varying temperature and applied stresses. In spite of the presence of unfrozen water, when ice fills most of the pore space, the mechanical behavior of a frozen soil will reflect closely that of the ice. The pore ice is usually of a polycrystalline type with a random crystal orientation. Under ordinary

conditions, its response to deviatoric stresses is governed by the motion of dislocations and can be represented by a power-law creep equation of the Norton-Bailey type. The yielding and failure of polycrystalline ice under a triaxial state of stress differs from most other materials, because under a high hydrostatic pressure, it first weakens and then eventually melts. On the other hand, when subjected to shear stresses at low hydrostatic pressures and at ordinary freezing temperatures, it shows a ductile yielding at low strain rates, but becomes more and more brittle as the strain rate increases (Mellor, 1979)[1].

2.2 Sources of strength On the basis of findings made by many previous investigators who studied systematically the shear behavior of frozen sands [1-6] and on the basis of their own investigations, Ting [7] and Ting et al., [8] concluded that the shear behavior of frozen sands is controlled essentially by the following four physical mechanisms:

(1) (2)

Pore ice strength, Soil strength, which consists of interparticle friction, particle interference and dilatancy effects, (3) Increase in the effective stress due to the adhesive ice bonds resisting dilation during shear of a dense soil, and (4) Synergistic strengthening effects between the soil and ice matrix preventing the collapse of soil skeleton. Based on these observations, Ting et al. [8] proposed also a schematic failure mechanism map, Fig.l, which expresses the fact that the simultaneous presence of various mechanisms depends on the volume fraction of sand in the ice/sand mixture. It is clear that, in addition to the soil density, the importance of any of these mechanisms in the observed strength of a frozen soil will depend also on such factors as temperature, confining pressure, and the deformation history. For example, in ice-rich soils, where the ice/soil ratio is high (over 1.38 in sands), most of the strength due to intergranular friction and dilatancy effects vanishes. 2.3 Dilataney hardening and softening effects When a two-phase granular mass, consolidated under hydrostatic pressure, is submitted to shear stresses, its initially stable structure will either collapse, if its density is low and/or if the confining pressure is high, or it will expand in the opposite case. If the pore filling matrix has a low compressibility, and if overall volume changes during shear are prevented, the shear will produce an increase in the matrix pressures in the first case, and a decrease in the second case. This will have as a result a decrease in intergranular stresses at low densities, and an increase of these pressures at higher densities of the granular mass, at least as long as the matrix bond remains unbroken. Because of these dilatancy-induced changes in intergranular stresses and the resulting softening and hardening effects on the material behavior, these phenomena have been termed in soil and rock mechanics "dilatancy softening" and "dilatancy hardening" effects. Available experimental evidence shows, however, that in a frozen sand at ordinary pressures and temperatures, the dilatancy hardening effect may exist only up to the strains of about 1%, after which the pore ice starts to break in a brittle manner under combined tensile and shear stresses.

2.4 Creep of frozen soil under constant deviatoric stress

When a frozen soil specimen is subjected to a constant deviatoric stress, it will respond with an instantaneous deformation and a time-dependent deformation. If the load is high enough, it will display a limiting strength. The basic creep curve consists of three periods of time during which the creep rate is: (I) - decreasing, (II) -remaining essentially constant, and (III) - increasing. These are often called periods or stages of primary, secondary and tertiary creep. For stresses lower than the long- term strength of the frozen soil, the second period, with the minimum creep rate, and the third period, with increasing creep rates may not develop. The shape of creep curves for frozen soils is influenced not only by the soil type, its density, ice saturation and temperature, but also by the applied stress-and strain-history. From a set of creep curves, each of them corresponding to a different deviatoric stress, but to the same temperature, confinement and strain history conditions, it is possible to obtain the basic theological curve for the soil and the tested conditions, by plotting the observed minimum creep rates against the applied deviatoric stress. For frozen soils in the usual temperature range, the curve has mostly a complex non-linear shape. Approximately the same theological curve can be deduced from constant strain rate tests by plotting the attained peak stress against the applied strain rate. There is ample experimental evidence in ice, frozen soils and high- temperature metals that a close correspondence exists between the peak stress observed at a given strain rate in a constant-strain-rate (CSR) test and the minimum strain rate in a constant-stress-creep (CSC) test. Or, as expressed by Mellor [1], the ratio (amax/d) in the former is approximately equal to the ratio (O'/~min) in the latter. In addition, in polycrystalline ice, in the two kinds of tests the above extreme ratios occur at about the same strain, which is also valid for the first peak in ice-cemented frozen sands. 2.5 Effect of ice content on strength

The mechanical behavior of frozen soils depends in a high measure on that of the pore ice which normally binds the grains together and fills most of the pore space. The strength of ice depends on many factors, the most important of which are temperature, pressure and strain rate, as well as the size, structure and orientation of grains. The strength of ice increases with decreasing temperature, and its mode of failure is strain-rate- dependent. With varying temperature and strain rate, its response to loading is found to vary from viscous to brittle. In permafrost soils ice exists at very high homologous temperatures, mostly above 90% of the fusion temperature, which limits its deformation mechanisms to a narrow area, characterized by a power- law creep, resulting mainly from the motion of dislocations [9]. Goughnour and Andersland [2] have studied the influence of sand concentration on strength of sand-ice mixtures at temperatures ranging from -4~ to -12~ When sand concentration was increased beyond 42% by volume, the influence of interparticle friction and dilatancy became apparent, while at lower concentrations strengths were only a little higher than those of pure ice. The strength increases only as long as the sand remains ice-saturated. When the ice fraction tends to zero, the strength of an unsaturated frozen sand decreases rapidly towards that for a dry sand.

2.6 Effect of normal pressure on strength

As mentioned in the foregoing, at sand concentrations higher than about 40%, the strength of frozen sand becomes a function of the strength of both the ice cement and the soil skeleton. It has been found, however, that these two sources of strength do not necessarily act simultaneously. This is due to the fact that the ice matrix, under normal pressure and temperature conditions, is much more rigid than the soil skeleton and attains its peak strength at much lower strains. As a result, when a relatively dense frozen sand is sheared in compression under a low confining pressure, it often shows two yield points: one at about 1% axial strain, and another at about 10% or more. The shape of the failure envelope of a frozen soil tends to be fairly complex, and it is expected to depend on the soil type, its density, and ice saturation, as well as on the temperature and strain rate. In addition, since there is as yet no method available for measuring intergranular stresses during shear of frozen soils, all the results can be plotted only in terms of total stresses. Based on experimental evidence of the last 20 years, there are certain common conclusions that can be drawn concerning the failure envelope of frozen soils 9 (1)

The shape of the failure envelope is approximately parabolic at relatively low temperatures and high strain rates. When the temperature increases and/or the strain rate decreases, the failure envelope shrinks and straightens up, with its slope slightly smaller than or equal to that of the same soil when unfrozen (Fig. 2). At very low strain rates (or very long times under stress), and/or at temperatures close to the melting point, the cohesion intercept tends to zero, and the remaining strength is then governed by intergranular stresses and mineral cohesion.

(2)

The ratio between the values of the uniaxial compressive and uniaxial tensile strengths depends strongly on the strain rate and temperature, and it varies from 1 at low strain rates up to about 5 at high rates of strain, similarly as in polycrystalline ice. This is so because the latter strength is much less rate- and temperature-sensitive that the former, at least in the brittle failure range.

2.7 Effect of strain rate on strength

In a wide area of strain rates, the behavior of a frozen soil will probably be similar to that found by Haynes et al. [10] for a frozen silt at -9.4~ shown in a log-log plot in Fig. 3. The observed rate sensitivity of peak strength of the frozen silt is found to be similar to that reported for polycrystalline ice by Hawkes and Mellor [11], the main difference being that for frozen silt the two strength lines separated at about 4 times higher strain rates than for ice. More generally, when the peak strengths obtained in such tests are plotted against the applied strain rates in a log-log plot, it is often found that the resulting line ("rheological curve")is not a continuous straight line, as assumed by the power-law creep equation of the type = Bo"

(I)

but that its slope, defined by: n = d(log ~)/d(log o), tends to be lower at low rates of strain, and higher at high strain rates.

For a dense frozen sand at low temperatures, n tends to be of the order of 10 or more, and it seems to be very little affected by temperatures below -5~ At higher temperatures, such as -2~ a break in the slope at a rate of about 10 .5 s 1 was observed by several investigators [12,13] (Fig. 4), reducing n to 5 or even 3 at low rates. Clearly, as mentioned earlier, in very ice-rich soils, the ice governs the behavior, and n = 3 closely approximates the results [14-16]. There are also indications that n decreases considerably with decreasing ice saturation, salinity [17] and when a cyclic loading is applied to a frozen sand [18]. As for the failure strain, it is generally found that lower temperatures and strain rates both reduce the failure strain. If the strain at the absolute maximum strength is considered, which may be either the first (ice-cement) or the second (friction) peak, the variation of the failure strain with strain rate will not necessarily be continuous, but may show a sharp drop at the brittle-plastic transition. Figure 5, taken from [13], shows a typical variation of the failure strain, which is seen to be remarkably constant in each of the two strain rate regions. 2.8 E f f e c t o f t e m p e r a t u r e

on strength

Because of its direct influence on the strength of intergranular ice, and on the amount of unfrozen water in a frozen soil, the temperature has a marked effect on all aspects of the mechanical behavior of frozen soils. In general, a decrease in temperature results in an increase in strength of a frozen soil, but at the same time it increases its brittleness, which manifests as a larger drop of strength after the peak, and an increase in the compressive over tensile strength ratio [19-21] (Fig. 6). Down to about -10~ the embrittlement effect of temperature is felt much more in a frozen sand or silt than in a frozen clay, which at that temperature still contains enough unfrozen water to keep it plastic.

3. ANALYTICAL REPRESENTATION OF CREEP AND STRENGTH DATA FOR FROZEN SOILS 3.1 C r e e p f o r m u l a t i o n s

In mechanics of frozen soil it is usually assumed that the total strain, e, resulting from a deviatoric stress increment, is composed of an instantaneous strain, Eo, and a delayed or creep strain, e(c), e = e o + e to)

(2)

In general, the instantaneous strain, eo, may contain an elastic and a plastic portion, but at usual service loads, excluding instantaneous failure, the plastic portion may be absent. The creep strain, in turn, is considered to be composed of a primary creep and a secondary or steady state portion, although the latter may sometimes be reduced to just an inflection point on the creep curve, preceding the tertiary creep. In practice, for relatively short-term processes like ground freezing, the strain e o in Eq.(2) is considered to be governed by the Hooke's law, while the creep strain, e (c) is usually defined by an empirical primary creep formulation. On the other hand, for long-term problems, such as the behavior of foundations in permafrost, the short-term

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rate,

, i01

10

i

2,o-,'

" ',o-,'

" ',o-,'

s -1

Figure 3. Relationship between strength and strain rate for a frozen silt (-9.4 deg.C) [10]. plastic

i

r~.l~

|~.

i

brittle I

o

-2~

~.~-:,,.

9.6oc 9- 1 0 ~

9- 1 5 ~ .1

1

10 -7

10"6

,

I

IO-S

'

,o-,

AXIAL STRAIN RATE,

,o~

Figure 4. Compressive strength vs. strain rate relationship for a frozen sand at different temperatures [13].

~

-'-'~'t'~

-6 ~ C

9 -10_* C 9 -15" C

I

I

10 -4 10 -3 AXIAL STRAIN RATE, S-I

Figure 5. Failure strain vs. strain rate relationship for a frozen sand [13, 25].

1 "

10-2

701

,

,

,

/ Machine speed: 6ol- A: 0.0423 cm/s / ~.. / B: 4.23 cm/s /

10~ o . ~ n _ ? T e n s i o n

,/

/

r

I

I .J ,o

]

0 " ""=---~=--'===----- = 0 -10 -20 -30 -40 -50 -60 Temperature, ~ Figure 6. Average strength vs. temperature relationship for frozen silt in uniaxial compression and tension tests [20]. response, including elastic, plastic and primary creep portions, is sometimes lumped together to form a "pseudo-instantaneous" plastic strain, EO) [22,23], which is defined as the intersection at the strain axis, when the slope at the minimum or steady-state creep rate is extrapolated back to t = O. In the latter case, for the portion of creep curves at and beyond the inflection point, but before tertiary creep, the total strain can be expressed by [22]: e = er + ~

t

(3)

where ~(~in = (dE(O/d0 is the minimum (or steady-state) creep rate, and t is the time. Based on the available experimental experience with frozen soils, and following [22] and [23], it is found that in Eq.(3) ~(~in can be conveniently expressed by a power-law approximation, ~)

--- ~o(o/%0

*

(4)

where, in Eq.(4), ace is the temperature-dependent creep modulus, corresponding to the reference strain rate, ~c, while n > 1 is an experimental creep exponent. The experimental parameters in Eq. (4) can be determined by plotting the creep test results in appropriate log-log plots, as explained in the [23]. Assuming the validity of the yon Mises flow rule and the volume constancy for all plastic deformations, including the creep strains, the power law of Eq. (4), adopted for the uniaxial case, can be generalized for the triaxial state of stress and strain, by expressing stresses, strains and strain rates in these equations by their "equivalent" values, defined by:

10 2 oe

= (3]2) SijSij = 3J 2

e ,2 - ( 2 / 3 ) %

(5)

% = (4/3)i 2

"~ = (2/3) eij " eij 9 %

- (4/3)t 2

(6)

(7)

where s O and eij are the deviatoric stress and strain tensors, respectively, while J2 and 12 are the second invariants of the stress and strain deviator tensors. The dot above a symbol denotes time rate. With this generalisation, Eq.(4) becomes: eemin

which is the well-known Norton-Bailey power-law creep equation, used extensively in the literature for steady-state creep formulation in high-temperature metals and ice. Written in tensor form, Eq.(8) becomes

:

1

(9) ij = 2 O e 0 ) ~ Oe0)

Sij

Because of the assumed validity of the von Mises law, leading to the above relations, the power-law of Eq.(9) becomes: - For cylindrical symmetry: "~) = ~ c [ ( O 1 - o 3 ) / o j " el

(10)

- For plane strain: el9e) = (v~/2)..~/~,[(o 1 - o 3 ) / o j *

(11)

- For simple shear: "~ = (e'l - ~'3) = 3Cn * l)12 ~'r (T' l O cO) n

(12)

These relationships make it possible to determine the creep parameters n, b, and ac~ from a series of simple laboratory tests. In the primary creep range, in turn, it is usually considered that the creep strain e (c) in Eq.(2) can be expressed as a product of independent stress-, time- and temperature-functions [22].

e(C) = fl (o) f2 (t) f3 (T)

(13)

11 A convenient form of such a primary (or transient) creep law is the Andrade's empirical law e (c) = A o n t b

(14)

which can be extended to three dimensions by assuming the validity of the von Mises flow rule as before. For example, Ladanyi and Johnston [24] write the law in the form: (r (15) e~ = (off Oc0)" (~:r b In Eq.(15), n, b, and or are three experimentally-determined frozen soil parameters, of which the last one, oa, denotes the reference stress corresponding to an arbitrary reference strain rate, ~r and to a soil temperature 0 = -T. In this respect, it is very important to note that, although the primary creep law can be transformed into rate sensitivity of strength law by simply putting b = 1 into the former, available experimental evidence shows that the creep parameters n and ar are usually quite different for the two cases, with n being much higher at failure than during primary creep. This behavior, which has been observed both in shear of frozen soils and in interface shear situations, is thought to be due to strain localisation effects which put doubt into the validity of von Mises generalization whenever an uniform straining is replaced by a shear surface or a shear zone formed within the specimen. The effect of temperature on creep of a frozen soil can be included in the value of the creep modulus ac0 by means of an empirical formula [23]: %0 = %0( 1 + 0/0c) w

(16)

where 0 c = 1~ w is an empirical temperature exponent, usually smaller than one, and a~o is the value of a~ obtained in unconfined creep tests, extrapolated back to 0~ as shown in [23]. Equation (15) represents the "time-hardening" formulation of the primary creep. The corresponding creep rate is in that case:

ic r

b.r

oco)n(blt) l-b

07)

If time t is eliminated from Eq.(15), one gets the "strain- hardening" form of the creep rate equation 0

(18)

Although the strain-hardening formulation offers a more accurate representation of the reality than the time-hardening one, the latter is nevertheless often prefered because it makes it possible to obtain closed-form solutions of some simple practical problems. In addition, as shown in [22], if a time-hardening assumption is adopted, any solution obtained for a steady-state creep law of Eq.(4) can be readily transformed into a transient creep form, by considering that the strain rate in Eq.(4) is the result of a differentiation

12 of strain, not with respect to time, t, but to an arbitrary time function, F(t). If F(t) = t b is selected for the time function, it is found that the transient creep form of a steady-state creep solution can be obtained by replacing everywhere ~c by (~r b, and t by t b [24].

3.2 Strength formulations Similarly as in unfrozen soils, the concept of failure in frozen soils includes both rupture and excessive deformation. Depending on type of soil, temperature, strain rate and confining pressure, the mode of failure may vary from brittle, similar to that in a weak rock, through brittle-plastic, with a formation of a single failure plane or several slip planes, to purely plastic failure without any visible strain discontinuities. The last type of failure by excessive creep deformation is typical for permafrost problems involving ground temperatures of only a few degrees below the melting point of ice. The creep strength is defined as the stress level at which, after a finite time interval, either rupture or instability leading to rupture (e.g., tertiary creep) occurs in the material. In compression testing of frozen soils, the creep strength is usually defined as the stress at which the first sign of instability occurs. In a constant-stress creep test, this condition coincides with the passage from steady-state to accelerated creep, or simply to the inflection point on the creep curve. On the other hand, in a constant-strain-rate compression test this condition corresponds to the first drop of strength after the peak of the stress-strain curve. Creep strength prediction consists in finding a relationship between the creep strength, aef, time to failure, tf, secondary or minimum creep rate, ~(c)_in, failure strain, ~ef, and temperature, 0 = -T. Compression creep testing of frozen soils often shows that the amount of permanent strain at the onset of tertiary creep is approximately constant for a given temperature and type of test. This behavior suggests that instability occurs when the total damage done by straining reaches a certain critical value. Although there is some experimental justification for using a constant permanent strain as a basis for the creep-failure criterion in frozen soils, this criterion is convenient for the design purposes, because it limits the total strain to values acceptable for the structure. In actual compression testing, when both constant-stress-creep tests and constant-strain-rate-compression tests results are available, it is most often found, as mentioned earlier, that this critical creep strain is approximately equal to the failure strain at the peak of stress-strain curves in the latter type of tests. For high-ice-content soils and for long time intervals, the plastic strain E(i) in Eq.(3) can be neglected relative to the creep strain portion, giving

tf -- e=d~c(O=/o~*

(19)

This makes it possible to write also the creep strength of a frozen soil as a function of time to failure, o a -- oce (ea/tf~c) TM or as a function of the minimum creep rate

(20)

13 ..(c)

Oef = Oco I,e:g~aia/~:c)t/n

(21)

if, for long time intervals, one defines .(c)

r

(22)

=r

If, on the other hand, a primary creep formulation of Eq.(15) is adopted, and considering that the true instantaneous strain can be neglected relative to the primary creep strain, the creep strength becomes _l/n

o d = o ~ %f (b/~: 0 ~

(23)

Clearly, for b = 1, Eq.(23) reduces to Eq.(21), but, as mentioned previously, the creep parameters are usually not the same in primary and steady state creep. The effect of normal, or confining, pressure on creep and strength of a frozen soil can be taken into account in several different ways [23,25,26]. For example, cold, ice-rich soils, containing too little unfrozen water to consolidate under confining pressure, tend to behave under triaxial test conditions like weak rocks, showing failure envelopes of a parabolic shape. Although these envelopes can reasonably well be described by second-degree parabolas, it is more customary in practice to approximate them, at least on the compression side, by a set of straight-line Coulomb envelopes, defined by Coulomb parameters c and r both of which may depend on time to failure (or strain rate) and temperature, as expressed by T = c(t,0) + o tan ~ (t,0)

(24)

As shown in [23], for an ice-rich, cold frozen soil, when ice bond still exists and where both c and r are affected by temperature and strain rate, a good approximation of the observed behavior, in terms of principal stresses, can be obtained by writing ( O 1 - O3) f =

(*l/~c) 1/n [Oc0 + o3(Nr - 1)]

(25)

where N ~ = the value of the flow factor Nr = tan 2 (45 + r f o r e = r which represents the slope angle of a Coulomb envelope at dl = ~c, i.e., at the same reference rate which also determines ecv In terms of r and or, Eq. (25) can be written as 1;f

=

(O + He)tan Oc

(26)

with H c = c cotOc = (Oc0/2N~)cot 1r2 (~c representing a set of straight-lines with varying slope angles, r through the same point 0'at H - H c (Fig.7a).

(27) all of them passing

14

TI

::~r-,'l >~r-'l>~I

,,

O'

,

0 (a) Hard

|---

oFrozen

0 (b) Plastic Frozen

.

~

Figure 7. Simplified failure envelopes for (a) Hard Frozen ("ice-rich"), and (b) Plastic Frozen ("ice-poor") frozen soils [26].

On the other hand, for frozen soils with large quantities of unfrozen water, or when consolidation is possible so that the confining pressure can be transfered to the soil skeleton, at least at failure, the angle r may remain approximately constant, while only the cohesion will be affected by temperature and strain rate. In that case, the shear strength can be well approximated by the equation ( o I - 03) f = of.(t,O) + o 3 ( N , - 1)

(28)

xf = c(t,O) + o tan ~

(29)

or

where r = const., which is represented by a set of parallel straight lines in the Mohr plot, Fig. 7b. In practical application, the main difference between Eqs. (25) and (28) is that, according to Eq.(25), when ~1 ~ 0, (a 1 - a3) f --. 0, i.e., there is no true long-term strength, which corresponds to the behavior of ice, while, according to Eq.(28), when ~ ~ 0, the strength tends to a finite value, the long-term strength, ( O 1 - O3)lt

=

03(N4~ -

1)

(30)

which is of a frictional character, as expected in dense and consolidated frozen soils.

15 3.3 Effect of salinity on frozen soil creep and strength

The strength of frozen soils depends strongly on their ice content. In saline soils, the volumetric ice content is a function of the salinity of pore water and the temperature, [27]. In general, it is found that increasing salinity of porewater, increases the creep rate and reduces the strength of frozen soil under otherwise comparable conditions [28-37]. 4. BEHAVIOR OF FROZEN S O I L - STRUCTURE INTERFACES 4.1 General As mentioned in the introduction to this chapter, the frozen soil - interface behavior, is affected not only by the soil type, its density, degree of saturation and unfrozen water content, but also by the type of structural material of the interface, as well as by the character and size of micro- and macro -asperities in the interface. In particular, the latter, combined with the rate of loading, will determine whether the shear failure will occur at the interface or within the soil. 4.2 Sources of information on interface behavior

Due to the importance of interface behavior for the design of piles and other buried structures in permafrost, the majority of available sources of information on this subject deal with a complex problem of adfreeze bond on pile-soil interface, with a special reference to its short and long term behavior. As a result, there is comparably more data available from field and laboratory testing of piles and anchors embedded in frozen soils than from systematic studies of frozen soilmaterial interface by direct shear tests. Some basic findings in the latter studies will be reviewed in the following. 4.3 Shear tests on interfaces Some early studies on the adfreeze strength of frozen soil-solid material interfaces have been mentioned in the Russian literature in the 1930's. More recently, Sadovskiy [38] published the results of a systematic study of frozen soil interface behavior. Using direct shear tests, he studied the adfreeze bond between 3 types of frozen soils (sand, silty sand, silty clay), ice, and 2 materials (concrete and metal). The study included also some tensile tests. Figure 8 shows a Mohr-Coulomb plot of typical results obtained in this investigation which was conducted at a temperature of-5.5~ General findings in this study can be summarized as follows :

(1)

Adfreeze strength of frozen soil to concrete increases with increasing moisture content to a certain maximum value, but then decreases when the soil is supersaturated with ice. The latter is nevertheless higher than adfreeze strength of pure ice to concrete in rapid shear.

(2)

Under natural conditions, where there is a heat flux between the pile and frozen ground, an ice film is often formed on the interface. The adfreeze strength is then affected by the ice film and its thickness. At short term, this ice film increases the adfreeze strength, but the opposite occurs at long term loads.

16

(3)

The tests performed at a temperature of-5.5~ under rapid shearing conditions (Fig. 10) show an increase of the peak adfreeze bond with the applied normal stress. However, after failure, the residual bond is only a small fraction of the original one, showing a typical brittle fracture behavior.

Similar results were also obtained by Roggensack and Morgenstern [39], who conducted a series of direct shear tests on both undisturbed and reconstituted samples of unfrozen and frozen silty clays. Figure 9 presents the strength envelope obtained from the tests on unfrozen clay, showing a peak effective friction angle of 26.5 ~ and a small effective cohesion intercept. Multiple shear reversals on both natural and pre-cut shear planes indicated a residual friction angle of 23~ Figure 10 shows the strength envelope for the same clay when frozen after consolidation ("CU"-tests). Similarly as in the Sadovskiy's study, it is found that freezing gives to the soil an increasing cohesion, but the angle of shearing resistance remains virtually unchanged. Inspection of the longitudinal sections after shear revealed that distinct horizontal and apparently continuous ice layers occupied what appeared to be the principal shear plane. The same phenomenon of moisture migration towards the shear plane was also observed in direct shear tests on an undisturbed frozen clay. Figure 11 summarizes the water content deviations from the average for each of the specimens tested, showing a large water content increase near the shear plane. This phenomenon of ice concentration in the shear plane under slow shear conditions, which is clearly also applicable to frozen soilrough interface shear, has important implications in connection with the long-term behavior of piles in permafrost. Weaver and Morgenstern [40] report the results of a series of direct simple shear creep tests on a variety of reconstituted frozen soils (sand, silt) and on ice. The test apparatus and conditions have made it possible to develop uniform shear strain within the soil when sheared between plates of different roughnesses, as long as the applied shear stress remained below the adfreeze strength of the interface. The tests were performed at about -I~ for duration of up to 45 days. An integral part of the study was the investigation of the effect of plate roughness on the creep characteristics of adfreeze bond. Three plates of different roughnesses were tested. Roughness was quantified using center line averages (CLA), defined as the average distance from peaks to valleys on the material surface. CLA values for the three upper plates were 0.0025, 0.125 and 5.0 mm. The roughness of the lower plate was 5.0 mm for all the tests. The study showed that for very rough aluminium plates (CLA = 5 mm) adfreeze bond failure is characterized by shearing along interfaces located within frozen soil, giving bond strengths comparable to the shear strength of the frozen soil. For explaining the load transfer mechanism for smooth (CLA < 0.0025 mm) plates, the authors hypothesized that a continuous liquid-like layer exists on the surface of internal boundaries of ice, as found by Jellinek [41]. Based on the observations on the interface behavior of ice with a smooth plate, the authors [40] proposed for frozen silt the failure mechanism shown in Fig. 12. In the model, the air voids occupy the interasperity space, but the soil particles create an additional frictional resistance along the shear plane. They consider this model to be applicable only if the freezing front progresses from the soil to

17

~" v/" X

;!.~

l_

i2 3 45-

f

. ~

/ !

1 i" I

i

, -1

4

.1 ,-'.2

I

I

i

I

__i~-T/-~-'--~'." - ~ , " .0 1 or, M~ 2

i,

Ice-Concrete Silty Clay - Concrete Silty Clay - Metal (Ice film) Silty C l a y - Metal (No ice film) lee-Metal

Figure 8. Adfreeze and tensile strengths of frozen soil and ice against different foundation materials (after [38]). ( I b / i n 2) ;.~

( I b / i n 2) ?oo

o,

io i

,

?o r

;

'

~ -~ Z .x

'

Z .=: =5o o'

- 265

r

- 7kNlm2!

'

b/O

40

8O r

;6o

DIRE SHEAR ON REMOULDED MOUNTAIN RIVER CLAY. SERIES 6

ii

DIRECT SHEAR ON REMOULDEO MOUNTAIN RIVER CLAY. SERIES &

E

4,J ~ I

3o I

J -

22

1

400 Pe~ Tell

s .m

IllmperilgrO

oe

,,i

c

9

1 0 tO 1.2 c

330 ~

.,

..-~o,, t~ < uJ o~ ;03

o o

50

DO on

NORMAL

~o STRESS

200

2~o

o

.

230

0

.

.

lO0

200

300

t,o%-

.... ~

v

12o

f

]

0

.

400

.500

10

600

( k N / m 2) ~n'

Figure 9. Direct shear envelopes for unfrozen Mountain River clay [39].

NORMAL

STRESS

(kN/m

2}

Figure 10. Direct shear envelopes for frozen Mountain River clay [39].

18

TEST FS-01

TEST F S - 0 2

w - 211.7% or~ ~ 252 kN/m 2

w

92~.~PI~ o n ', 1'!1 kN,m 2

TEST FS.-O~

TEST F S - 0 3

1

w - 24 9%

TEST F S - 0 6

j

TEST F S - 0 4

~. - 491 k N / m 2

w - 502%

., w-21~.ml,

on,,lmTkNIm2

~w-31.0%

- - - - - 3 i ~ - ~ - ~ . . . . . . . . .,, . . .

an-l~to2"21kN/m2

~1~~le~511~

,', w , . 31.2%

on - 3 7 0 k N / m 2 I-

TEST F S - 1 3

.

o

;

932.4%

J

,

o n - 217 kN/rn Z

TEST F S - 1 2

i

I

w - 300%

w

On'4-r~kNIm2

TEST FS-11

TEST PS-10

II. - 3167 k N I m 2

TEST FS-,Oe

- TEST FS-O7

w-~17.4% o n . , 6 O g k N / m " . . . . . . . . . . .

.

/ ....

w - 2S.0~

On - 311 k N / m 2

w-]~.ll% on-133kNIm2 . . . . . . . . . . .

,,

.

: ....

w - ~.8~

TEST F S - 1 5

TEST F S - 1 4

.

-

o n - 370 k N / m 2

on

,o 9129 kN/m 2

TEST F S - 1 6

On'192kNIm2

w-21~.2% O n ' ( l ~ T l o C J O 3 k N / m 2 ! w-21S.9% . . . . . . . . . -~ . . . . . . . . . . . . . . . . . .

Figure 11. Deviation from average water content (%) in the shear zone of sheared specimens of frozen soil [39].

400

--r ......

...

o~ 300

'Ce.t,,~'..,

o '~tlt

R i g i d I::qate

f

V

Directkon of .~lied Shear

J

r _

:7,::

.~ ....

--

Concrete

Sand-Wood,

_

(NRCC 1976)

" 3~rlc~"" "'.'>. ...~

c/~ . . . . ~ e /

-

,,., "'.. ~ e~ ~

,,or ~_~zs-.." .<%

~

"I3

,~."-..

,<

led

Figure 12. Shear failure mechanism between a smooth plate and frozen soil, after [40]

~--

"~,"<.~. -

~-

0 -5

,

,

-4

-3 -2 Temperature (C)

-1

0

Figure 13. Summary plot of published adfreeze strengths by 1981 [40].

19 the plate during preparation. If the freezing front advances in the opposite direction, then a thin layer of ice may form at the plate-soil interface, and the failure would be characterized by the ice, as observed earlier by Sadovskiy [38]. In summary, the adhesive shear failure mechanism was found to be a function of the ice content at the interface and the local geometry of the asperities on the interface. Subsequently, Weaver and Morgenstern [42] have used this information, combined with the available data on adfreeze strength from various literature sources, ( Fig. 13) to develop the following simple method for determining the long-term adfreeze strength r a for the design of piles in permafrost. The adfreeze strength is assumed to be related to the long-term shear strength of the same frozen soil, tit , by the relation 9o -- m ~ a

(31)

where rn characterises the type and the roughness of the interface. The long-term shear strength of a frozen soil is considered to be composed of frictional and cohesive components, following the Coulomb law 1:it = elt + o n tan t~k

(32)

where qt is the long-term cohesion of the frozen soil, o n is the normal stress and r is the friction angle at the interface. For piles embedded in permafrost they suggest to neglect the friction term because of a relatively low lateral normal stress, and to retain only xh " m %

(33)

Based on all available evidence on long-term shear and adfreeze strengths measured in various frozen soils and pile materials, Weaver and Morgenstern [42] suggest for the coefficient m the values of: m = 0.6 for steel and concrete piles, m = 0.7 for uncreosoted timber piles and 1.0 for corrugated steel piles. In order to check the validity of this method and the related assumptions, Ladanyi and Th6riault [43] carried out a series of direct double-shear tests, involving two types of frozen soils (sand and silt) and two kinds of interface materials (steel and aluminium). The tests were made at a temperature of-2 ~ C and at a shear velocity of 16.9 mm/day. In the tests, a special attention was paid to the sources of adfreeze bond and the conditions of its healing after shear failure. In both tested frozen soils when sheared alone in a double shear box with 63 mm diameter, the peak strength was attained at a displacement of about 2.5 mm, and 5 mm more were needed to attain the residual strength. The peak strengths of the two materials were represented by Coulomb straight lines with average values of shear parameters : c = 0.4 MPa, r = 17~ for frozen silt, and c = 1.28 MPa, r = 45 ~ for frozen sand. Figures 14 and 15 show a summary of these test results in a plot relating the ratios : rmg/rsp and rmJrsp , respectively, with the applied normal pressure, where rmp and rmr denote peak and residual adfreeze strengths of frozen sand against the metallic surface, and r sp denotes the peak strength of frozen sand at the same normal stress.

20 I1 will be seen in these figures that in such short-term tests 9 (1) (2)

the peak strength ratios obtained are rather low, being of the order of 10 % or less in the normal pressure range of 100 kPa, and they tend to increase with increasing normal pressure, following a slope of about 20~ in the case of steel, and about 28~ in the case of aluminium surfaces.

Bond healing tests In bond healing tests, each shearing test was stopped in the post- peak region, after a displacement of about 5 mm, to be resumed after different periods of time, under the same normal pressure. In frozen silt it was found that the percent fraction of shear strength recovered after waiting up to 95 hours was a function of normal stress. In the brittle failure range, i.e., for o n < 1 MPa, it was found that the amount of bond healing increased with increasing normal pressure, and that, under favourable conditions, it may even exceeded the initial peak strength of the material. This behavior of frozen silts is considered to be due to the presence of about 14 % of unfrozen water at the test temperature of 02~ Figure 16 shows the results of bond-healing tests carried out with frozen sand against a steel surface. In that case, a clear bond recovery with time was observed tending to a complete recovery for times over 300 hours and pressures over 200 kPa. Figure 17 drawn from Fig. 16 represents a tentative rate of bond recovery with time under normal pressures varying from 100 kPa to 500 kPa. This difference in post-failure healing, on one hand within frozen sand, and on the other, at the frozen sand-steel interface, is considered to be due to the pressure thawing of ice at the contact of sharp sand grains and the flat steel surface, resulting in water migration and refreezing in the pore space in contact with the metal. ImPlications for pile design In current pile design practice in frozen soils, it is usually considered that long-term pile capacity can be predicted by extrapolating either laboratory or field creep testing information to the service life of the pile. However, the stress relaxation phenomena in frozen soils point to the fact that, at long periods of time, the pile behavior may be governed by factors quite different from those obtained by simple extrapolation of shortterm data. It is postulated that at long term, the pile shaft capacity will depend not only on the long-term cohesion of the frozen soil, but also on the residual friction angle at the soil-pile interface, and the total original lateral ground stress. The probable long-term value of the shaft resistance will then be [43] : 9~ = m qt + o ~

tan ~lt

(34)

implying that at long term the frictional contribution to the shaft resistance may not be negligibly small with respect to the cohesive adfreeze bond. Effective lateral stresses may be considered only in those frozen soils which contain large quantities of unfrozen water, such as saline soils in offshore permafrost regions. This experimental investigation has confirmed many previous findings that, in the case of straight-shafted piles in permafrost, the adfreeze bond is essentially brittle, leading to

21

0.4' I Tmm_E

. PEAK

T

,

~sp

mp

0.3

sp

RES I DUAL

, PEAK 'v RES I DUAL

0.3 _~

mr

0.2

-

sp

~~

~.~

9 9

01 _

/

__/

9

9

I~ ~ r ~

. ~ ~

I

02

~

/ j/

7--

i

"

, ,

. ~,~../

v ~ .~-

0.1

I

0.5

~ 0~ V,~"~

/ / /

. ~.--

0

0

"~m~

S~V-~-_

5

I

/

/" ~'~,~

~

1.0

0.5

0

(Tn, MPa

Figure 14. Adffeeze bond of frozen sand against steel, expressed as a fraction of shear strength of frozen sand (-2 deg.C) [43].

~n, MPa

1.0

Figure 15. Adfreeze bond of frozen sand against aluminium, expressed as a fraction of shear strength of frozen sand (-2 degrees C) [43].

,[ 100

A

9

. . . .

% too

5o~a'/ r /

21 203

I*i ,/

3.1o7h

/

4.

61 h

5.

Oh

5y 0 [/'

0

,

5o

h

I

0.5

i. o n 2. 3. 4. 5.

q

1

1.0

.

~ , MPa

Figure 16. Shear tests with frozen sand against steel surface: Bond-healing as a function of pressure, with time as parameter (-2 deg.C) [43].

0 o

,1 100

=

500 400 300 200 i00

kPa kPa kPa kPa kPa

I

200

Time, h

300

Figure 17. Shear tests with frozen sand against steel surface" Bond healing as a function of time, with normal pressure as parameter (-2 deg.C) [43].

22 a large loss of pile capacity after only a small axial displacement. Although the test results indicate that a portion of the original cohesive bond may be recovered with time under favourable conditions, and especially in soils containing large amounts of unfrozen water, a determination of the long-term cohesive bond from short-term data by a simple extrapolation in time or strain rate, seems quite dubious for such piles. Frozen-unfrozen soil interface and effect of salinity Chamberlain [33] used direct shear tests to investigate the shear strength of frozen soil at the freezing front. He conducted the shear tests on sand and clay soil samples during frost front penetration. Samples were prepared with distilled water and seawater and tested in a direct shear box at shear plane temperatures ranging from 0~ to -5~ The unique character of the freezing zone is that it is a dynamic region of heat and moisture flow. In saline soils, the freezing zone is even more complicated, with soluble salts concentrating in unfrozen water films and brine pockets. The samples, sheared at a rate of about 0.03 mm/sec attained usually their peak strength at about 4 mm shear displacement. The results show that, for the clay the effect of salt water was essentially a shift along the temperature axis by an amount approximately equal to the freezing point depression of the seawater. For the sand, the effect was more complicated. In fact, for frozen clay, the maximum shear strength in both distilled water and seawater saturated samples fell on a narrow scatter band, when plotted against the unfrozen water content at the shear plane (Fig. 18). In general, the effect of salinity on frozen soil strength is complex, because the unfrozen water in frozen saline soils occurs in two different locations : one, adjacent to soil particles, reduces the strength of the ice-particle bond, and the other, in brine pockets, reduces the strength of the ice component. 5. INTERFACE BEHAVIOR FOR PILES IN PERMAFROST 5.1 Model pile tests In the design of piles in cold regions there are two problems where the interface behavior is of primary importance. The first one concerns the frost jacking of piles due to adfreeze stresses acting upwards on the portion of the pile traversing a frost-susceptible active layer, subject to frost heaving. The second problem, in turn, deals with the pile design, in which case the concern is how to transfer to the frozen soil the loads applied to the pile, by satisfying the requirements for the safety against failure and the allowable settlement or settlement rate conditions. Clearly, in the two problems, the tasks are quite opposite : In the frost-jacking problem, the main purpose is to minimize the adfreeze force or eliminate it completely by making the pile as smooth as possible. In the pile design, in turn, the interest is in maximizing the adfreeze strength by selecting an appropriate pile type, pile surface roughness and the method of installation. In addition, in the latter case, the long-term behavior of pile-soil interface under natural permafrost conditions is of great importance. Since the 1960's, many reports on pile-loading tests in permafrost [36,37, 44-50] as well as on frost-heaving forces on piles [51-55] have been published. In addition, several investigators have studied the pile-frozen soil interface problem by performing stress or displacement-rate-controlled tests on model piles in laboratory. Probably the most complete study of this kind was carried out at the National Research

23

600

I

1

I

M O R I N CLAY

N(kPa)

~ d - - ~ t o r 60 L 9

Seowoter

I .~

9

60

o-

400

--

it

i 2ool-~E

0

20 Wu,, Unfrozen

40 Woter Contlnt

Of Sheor P l o n e

(%)

Figure 18. Maximum shear stress vs. estimated unfrozen water content at shear plane, for clay [33].

lo|

I

I

'l

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"

4I

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~24

i

"--'-------

~ - 0.305 mm/min s-- 0 . 0 3 0 m m / m i n

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,

o

o~

C

l.o

l.~

"Oo,r<,;a,o<,

~

~.o

~.~

~,o

DI SPIAC E',IENI. mm

Figure 19. Load-displacement curves for model piles in frozen sand at -6 degrees C and at displacement rates from 0 . 0 2 to O.1 mm/min, for: A. Untreated fir, B. Painted steel, and C. Concrete [56].

0

4

8

12 16 20 Settlement. s, m m

Figure 20. Results of constant settlement rate tests with three types of model piles in frozen sand: A. Smooth, B. Tapered, and C. Corrugated (-5 deg.C) [62].

24

r

i 9 9 O 0

I

ilIll

I

I

I

I

u I IVy I

l

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10.0

r lul~i

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IC FIR CONCIIETE PAINI[D STEEL Cl(OSOTEO IC FIR

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i

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fill

--I--

B.C.

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ST(EL H-SECTION

,--, z z

,v

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9

oo.fo

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CROSS-HEAD SPttO. mmlmin

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I I|lII.

0

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Figure 21. Adfreeze strength vs. displacement rate relationship for different types of model piles in frozen sand (-6 deg.C) [56].

10.0

ADFRE(Z(

STRENGTH

IN ICE

(raid).

MPa

Figure 22. Comparison of adfreeze strength of model piles in frozen sand and in ice, at the same displacement rate (-6 deg.C) [60].

Council of Canada between 1978 and 1985 [56-60]. Reports on similar kinds of tests with model piles in frozen soils were also published by Andersland and Alwahhab [61], Ladanyi and Guichaoua [62], and Biggar and Sego [35]. Parameswaran [56,57] performed his model piles tests by pushing 76.2 mm diameter piles, made from various materials (treated and untreated wood, painted and unpainted steel and concrete) through a frozen-sand-filled rigid cubical box with 305 mm sides. In some of his tests [56] the piles were pushed through the soil at a constant displacement rate (0.0005 to 0.10 mm/min), while in others [57,58] a constant axial load was applied to the pile and its displacement response was observed. In addition, he studied also the effect of cyclic loading on piles in frozen sand [59], and compared the behavior of model piles in frozen sand with those embedded in ice [60]. A general problem of such piles tests, both reduced and full scale, is that the lateral normal stress acting on the pile is largely unknown. This limits the observations to the overall value of the adfreeze strength and its variation with displacement, displacement rate, and time, without a possibility to distinguish its adhesive and frictional components. Nevertheless, such information has a practical value for the design of straight-shafted piles for axial short-term loads. Some important conclusions that may be drawn from Parameswaran's investigations can be briefly summarized as follows.

(1)

The adfreeze bond of frozen sand to the pile (at -6~ behaves in a brittle manner, with large drop of strength after the peak (Fig. 19). Both the peak and the residual strength vary with the pile material and increase with surface roughness. (A similar strong effect of pile surface roughness is also shown in Fig. 20, from model pile tests in frozen sand reported in [62]).

25 (2)

1:f

When the peak strengths from such tests are plotted in a log (adfreeze strength) vs. log (displacement rate) plot (Fig. 21), the experimental points fall on a set of straight lines, each of them valid for a given pile material. The results show that the peak adffeeze strength r f can be expressed as a power-law function of the displacement rates =

A (~)"

(35)

The values of the peak adfreeze strength r f and the rate exponent n obtained by Parameswaran [56] are listed in Table 1. Table 1. Rate exponent n and peak adfreeze strengths r f from model pile tests, [56] Pile material

B.C. fir Concrete Painted steel Creosoted B.C. fir Spruce Unpainted steel

Exponent n

Adfreeze strength rf (MPa) at displacement rate of 0.0005 mm/min

4.48 4.63 6.02 5.37 10.23 9.66

1.140 0.525 0.497 0.403 0.960 0.527

(3)

In load-controlled tests under the same conditions, Parameswaran [57] found that the model piles had a typical creep response with primary, pseudo-steady-state and tertiary creep. (Which is quite comparable to the results of full scale tests or anchors in permafrost published by Johnston and Ladanyi [46]). When the minimum displacement rates from such tests were plotted against the average applied shaft stress in a log-log plot as in Fig. 21, the resulting straight lines gave the following rate exponents in Eq. (35) : n = 6.7 for painted steel, 8.1 for natural B.C. fir, and 9.1 for concrete. On the other hand, in long-term creep tests at low interface stresses (< 1.0 MPa) and at temperatures of-2~ and -2.5~ Parameswaran [58] found for different piles and for 3 different frozen soils (clay, silt and sand) that the relationship between the displacement and the time in the primary creep range could be represented by a power law of the form s = C tb

(36)

with b varying between 0.34 and 0.56 with an average value of 0.46.

(4)

As for the effect of alternating loads, already Li & Andersland [18] have found a strong decrease of shear strength of frozen sand under cyclic loading comparing with the strength measured under static loading. Similarly, Parameswaran [59] found that, when an alternating stress, with a frequency of 10 MHz was superimposed on a static stress the rate of settlement of model piles in frozen soils

26

(5)

at -2.2~ increased considerably. In particular, an alternating stress as small as 3 percent of the static stress caused a doubling of the displacement rate of uncoated wood and concrete piles in frozen sand. This effect must clearly be taken into account in the design of piles, based either on ultimate capacity or on maximum allowable settlement and settlement rate (see also in [64]). Finally, when the same type of model pile tests were carried out in ice [60], it was found that at -6~ both wood and steel piles has adfreeze strengths slightly higher in frozen sand than in pure ice (Fig.22), with their relative difference decreasing with the increase in adfreeze strength.

5.2 Creep and failure at the pile-soil interface In 1972, Johnston and Ladanyi [46] published the results of a field study of the behavior under pull-out loads of a series of short concrete piles. The piles were realized by inserting a corrugated steel rod in a pre-drilled hole in frozen soil and grouting the annulus around the rod. The resulting "rod anchors" with a diameter of about 15 cm, were embedded about 3 m in a varved frozen silt at an average temperature of-0.5~ After recovering the piles at the end of the testing period, it was found that their surface was very rough, which is thought to be due to the irregular surface of augered holes and probably also to unequal thawing of clay and silt layers of the varved soil during concrete curing. The piles were subjected to both sustained and step-wise increasing loads till failure. In general, the behavior of such a pile under a sustained load follows a typical creep curve, with well developed primary, secondary and tertiary phases (with secondary phase being reduced to an inflection point at higher stresses). By excavating trenches next to the piles after the end of the tests, and owing to the presence of horizontal dark-(clay) and light-(silt)-coloured layers in the varved frozen soils, it was possible to observe the soil deformations produced by soil creep at the interface (Fig. 23). For simulating the observed interface creep behavior, the authors used a simple theory of shear around an axially loaded cylindrical inclusion, originally used by Nadai [63] for evaluating the results of the needle viscosimeter in a power-law material. As shown in [46], when the material follows a power-law relationship between the shear strain rate y and the applied shear stress r, such as = ~,o

(~/T)*

(37)

in which n and r c (at an arbitrary reference strain rate Vc) are the experimentally determined creep parameters, the corresponding displacement rate s of the cylinder (as long as it is fixed to the soil) is given by (Fig. 24)

where a is the cylinder radius and r a is the uniform shear stress transmitted to the surrounding soil.

27

Figure 23. Creep deformation of a varved frozen silt at the interface with a corrugated grouted rod anchor, observed after a pull-out test (-0.5 deg.C) [46].

l ~FILLED

1,2~i~ ~o,~

P ~GROUTED

[ L

'o

_

dz

r

P

9

_._.~ 2o ~.5-'

Figure 24. Schema for the analysis of creep deformation of frozen soil in contact with a rough anchor pile [46].

28 The values of the exponent n found from grouted-rod pull-out tests in frozen varved silt at -0.5~ were 7.5 and 8.05, respectively. This equation can be transformed into a primary creep form valid for a step load a

S n-

(,~Jb)b (~.j,[.r

,ub

(39)

1

As mentioned earlier, this transformation does not imply that creep parameters are the same during primary and secondary creep, with especially n being generally much smaller in the former than in the latter. The validity of Eqs. (38) and (39) is limited by a critical displacement at which interface slip occurs. After the slip, the interface shear strength usually falls to a much smaller residual value. For very rough grouted piles, it was found [46] that slip started at a displacement of about 2 to 3 cm, but for smooth piles the critical displacement may be of the order of only a few millimetres (Figs. 19 and 20). 5.3 Effect of pile installation method Since most interface problem in permafrost concern the bearing capacity of piles, it will be of interest to discuss briefly the interface conditions as related to the pile installation method. Although, under certain favourable conditions, steel piles can be driven into relatively warm, fine-grained, frozen soils, pile driving in permafrost areas is usually considered to be difficult and expensive, because it requires special reinforced pile sections and heavy driving machinery. For that reason, most piles in frozen soils are installed by boring. Bored piles in permafrost are usually made of prefabricated concrete, steel or timber, and are mostly installed in pre-drilled oversized holes. The gap between the pile and the soil is filled with a compacted sand-water slurry, which is let to freeze, either naturally or artificially. These "Slurried Piles" have, therefore, their end-bearing on the natural frozen ground, but their lateral shaft resistance is related to the strength of the frozen slurry. Alternatively, a cement grout can be used for filling the annulus around the pile [36,37]. On the other hand, prefabricated piles can also be driven into predrilled holes of a slightly smaller diameter, or in previously heat-treated holes, in which case, their lateral resistance will depend on the properties of the natural ground. 5.4 Effect of pile roughness and shape When the pile surface is roughened there will be a corresponding increase in the pile capacity. Long [65] discusses the use of piles with tings or helix-type protuberances to mobilize the shear strength of the soil rather than relying on the adfreeze bond along the surface of a smooth pile. Thomas and Luscher [66] describe the use of a corrugator that is drawn up the inside of the pipe pile, after the backfill is installed but before it freezes, which produces a series of corrugations along the pile embedment length. The enhanced capacities of such piles were examined in field loads tests and reported in Luscher et al. [49] and Black and Thomas [48]. Work by Andersland and Alwahhab [61] showed that the introduction of lugs into a steel rod embedded in frozen sand greatly increased the pull out capacity of the rod. Sego and Smith [67] observed an increase of 100 % in adfreeze resistance for model piles in a sand slurry backfill when the surface of the pile was sandblasted to remove the black lacquer coating and to roughen the pile surface.

29 In order to get a better understanding of the behavior of piles embedded in permafrost, an experimental study was undertaken by Ladanyi and Guichaoua [62]. The study included a large number of both load- and settlement-rate controlled tests on three types of model piles : smooth and corrugated straight-shafted piles, and smooth tapered piles. Like in practice, the piles were installed in oversized holes in frozen sand, surrounded by compacted sand-water slurry, and left to freeze at -5~ There was no pile-soil contact at the pile end. In Fig. (19), the observed behavior of the three types of piles is compared at two different rates of settlement. Their response is seen to be drastically different. The smooth piles failed at low stress in a brittle manner, loosing all their strength after a displacement of about 2 mm. The shaft resistance of corrugated piles continued to climb up to about 2 MPa, reaching its peak at about 5 mm and decreasing slowly towards the residual, which after 12 mm displacement, remained still at 50 % of the peak strength. The tapered piles showed typically a small first peak at about 0.6 mm displacement, indicating a loss of adhesion as in smooth piles, but after this, their resistance continued to rise steadily without any sign of strength loss. Another pile test program was carried out by Biggar and Sego [36,37] in Iqaluit, NWT, Canada. The program consisted of uplift load testing of ten 114 mm diameter steel pipe piles with various surface modifications, (installed in 165 min predrilled holes and backfilled with clean sand), and four Dywidag bars (backfilled with Ciment Fondu grout). All tests were performed in saline permafrost, consisting of a dense clayey gravelly sand till. The ground temperature during load testing was -5.5~ + 1.0~ In the depth interval of interest for the tests, the moisture content of the soil was 6 to 7 % and its salinity varied between 15 and 22 ppt. The results of the tests show that considerable gains in the short-term pull out capacity of piles in saline permafrost can be realized by modifying the pile surfaces and (or) backfill materials. In a frozen sand backfill, welding four 12 mm rebar bracelets over the lower 1 m of the pile doubled the capacity of smooth piles. Sandblasting the pile surface gave a fourfold increase in capacity. Using a cement grout backfill increased the loadcarrying capacity to nearly 10 times that of smooth piles. The effect of changing the backfill from a sand slurry to grout was to shift the failure surface from the pile-backfill interface outward to the backfill-native soil interface, thus increasing the surface area over which the load was resisted, and fully mobilizing the shear strength of the native soil. Grout backfill was successfully used in permafrost soil in which the temperatures were -5.5 + I~ without heat damaging the prebored hole or preventing the grout curing. Average pile-backfill interface shear stresses recorded in these pile tests varied from 80 kPa for plain pipe to about 300 kPa for sandblasted pipe, which corresponds to the shear stresses between the backfill and the native soil of 54 kPa and 210 kPa. 6. NUMERICAL ANALYSIS OF FROZEN SOIL-STRUCTURE INTERFACES In the analysis of piles in permafrost, joint or interface elements play an important role in modelling the highly sheared soil layer at the interface. In an analysis of laterally loaded piles in permafrost, Foriero and Ladanyi [68] have used for that purpose two types of joint elements 9conventional thin-layer elements [69] incorporating creep parameters and thin-layer joint elements, incorporating a variable shear modulus.

30 Their FEM computation procedure starts from the static elastic solution at time t = 0, and is followed by a time marching scheme which advances the solution one timestep at a time. The solution takes into account the non-linear visco-plastic response of frozen soil, and yields rate-dependent soil reactions and pile displacements. CONCLUSIONS (1) The frozen soil-interface behavior is affected not only by the soil type, its density, temperature, degree of saturation and unfrozen water content, but also by the type of structural material of the interface, as well as by the character and size of micro- and macro- asperities in the interface. In particular, the latter, combined with the rate of loading, will determine whether the shear failure will occur at the interface or within the soil. (2) Due to the importance of interface behavior for the design of piles and other buried structures in permafrost, the majority of available sources of information on this subject deal with a complex problem of adfreeze bond on pile-soil interface, with a special reference to its short and long-term behavior. (3) Under natural conditions, when freezing front advances from the interface to the soil an ice film is often formed on the interface. The adfreeze strength is then affected by the ice film and its thickness. At short term, this ice film increases the adfreeze strength, but the opposite occurs at long term loads. (4) Under slow sheafing conditions, there is a tendency for moisture to migrate towards the shear plane, so that with time ice may accumulate in the shear zone. (5) Under stress and temperature conditions usually encountered in practice, interface shear occurs in a brittle manner, starting with ice bond failure at small displacements, followed by a large loss of strength and eventual mobilization of interface friction. Once the ice bond is broken, it can be recovered only under relatively high pressures acting for a sufficiently long time. (6) The shear displacement behavior of the interface shows a typical creep response, with primary creep at small displacements and steady-state creep after the primary ice bond has been broken. For steady state creep a power-law approximation, with rate exponents varying from about 5 to 10, as a function of interface material type and roughness, was found to be valid. REFERENCES 1. M. Mellor, Mechanical properties of polycrystalline ice. In: "Physics and Mechanics of Ice, IUTAM Symp., Copenhagen, (Per Tryde, Ed.), Springer, Berlin, (1979) 217-245. 2. R. R. Goughnour and O. B. Andersland, Mechanical properties of a sand-ice system. Proc. ASCE, 94(SM4), (1968) 923-950. 3. O. Andersland and I. A1Nouri, Time dependent strength behavior of frozen soils. Proc. ASCE, 96 (SM4), (1970) 1249-1265. 4. B. D. Alkire and O. B. Andersland, The effect of confining pressure on the mechanical properties of sand-ice materials. J. of Glaciology, 12(1973) 469-481.

31 5. E. J. Chamberlain, C. Groves and R. Perham, The mechanical behavior of frozen earth materials under high pressure triaxial test conditions. Geotechnique, 22(1972), 469-483. 6. F. H. Sayles, Triaxial and creep tests on frozen Ottawa sand. 2nd Int. Conf. on Permafrost, Yakutsk, North Amer. Contr. Vol. (1973)384-391. 7. J. M. Ting, The creep of frozen sands: Qualitative and quantitative models. Res. Rep. R81-5, Dept. of Civil Engrg., M.I.T., (1981). 8. J. M. Ting, R.T. Martin and C.C. Ladd, Mechanisms of strength for frozen sand. J. Geotech. Engrg., ASCE, 109, 10 (1983) 1286-1302. 9. M. F. Ashby and H.J. Frost, Deformation mechanism maps applied to the creep of elements and simple inorganic compounds. In: "Frontiers of Materials Science" (L.E. Murr and Ch. Stein, Eds.), Marcel Dekker Inc., New York (1975) 391-419. 10. F. D. Haynes, J. A. Karalius and J. Kalafut, Strain rate effect on the strength of frozen silt. USACRREL Res. Rep. 350 (1975), 27p. 11. I. Hawkes and M. Mellor, Deformation and fracture of ice under uniaxial stress. J. of Glaciology, 11(1972) 103-130. 12. V. R. Parameswaran, Deformation behavior and strength of frozen sand. Canad. Geotech. J., 17(1980) 74-88. 13.R.A. Bragg and O. B. Andersland, Strain rate, temperature and sample size effects on compression and tensile properties of frozen sand. 2nd. Symp. on Ground Freezing, Trondheim, 1 (1980)34-47. 14. E. C. McRoberts, T. C. Law and T. K. Murray, Creep tests on undisturbed ice-rich silt. 3rd Int. Permafrost Conf., Edmonton, 1 (1978) 539-545. 15. J. S. Weaver, Pile foundations in permafrost. Ph.D. Thesis, Univ. of Alberta, Edmonton (1979). 16. N. R. Morgenstern, W. D. Roggensack and J. S. Weaver, The behavior of friction piles in ice and ice-rich soils. Canad. Geotech. J., 17 (1980) 405-415. 17. K. W. Biggar, Adfreeze and grouted piles in saline permafrost. Ph.D. Thesis, Univ.of Alberta, Edmonton (1991). 18. J. C. Li and O. B. Andersland, Creep behavior of frozen sand under cyclic loading conditions. 2nd Symp. on Ground Freezing, Trondheim 1 (1980) 223-234. 19. F. H. Sayles and D. Haines, Creep of frozen silt and clay. USACRREL, Tech. Rep. 252 (1974) 50p. 20. F. D. Haynes and J. A. Karalius, Effect of temperature on the strength of frozen silt. USACRREL, Rep. 77-3 (1977) 27 p. 21. F. D. Haynes, Strength and deformation of frozen silt. Proc. 3rd Int. Permafrost Conf., Edmonton, 1 (1978)656-661. 22. J. A. H. Hult, Creep in Engineering Structures. Blaisdell Publ. Comp., Waltham, Mass, (1966) 115p. 23. B. Ladanyi, An engineering theory of creep of frozen soils. Canad. Geotech. J. 9 (1972) 63-80. 24. B. Ladanyi and G. H. Johnston, Behavior of circular footings and plate anchors embedded in permafrost. Canad. Geotech. J. 11(1974) 531-553. 25. B. Ladanyi, Mechanical behavior of frozen soils. Proc. Int. Symp. on Mechanics of Structured Media (A.P.S. Selvadurai, Ed.) B (1981) 205-245. 26. B. Ladanyi, Shallow foundations on frozen soil: Creep settlements. J. Geotech. Engrg., ASCE, 109, 11 (1983) 1434-1448.

32 27. D. E. Patterson, and M. W. Smith, Unfrozen water content in saline soils: Results using time-domain reflectometry. Canad. Geotech. J., 22 (1985) 95-101. 28. D. C. Sego, T. Schultz and R. Banasch, Strength and deformation behavior of frozen saline sand. 3rd Int. Symp. on Ground Freezing, Hanover, NH, 1 (1982) 11-19. 29. N. Ogata, M. Yasuda and T. Kataoka, Effect of salt concentration on strength and creep behavior of artificially frozen soils. Cold. Reg. Science & Technology, 8 (1983) 139-153. 30. J. F. Nixon and G. Lem, Creep and strength testing of frozen saline fine-grained soils. Canad. Geotech. J., 21 (1984) 518-529. 31. M. S. Nixon and G. M. Pharr, The effect of temperature, stress and salinity on the creep of frozen soil. J. of Energy Resources Technology, 106 (1984) 344-348. 32. L. J. Mahar and B. J. A. Stuckert, Strength and deformation behavior of frozen saline sand and gravel. Proc. Conf. ARCI'IC'85, San Francisco, ASCE, New York (1985) 557-565. 33. E. J. Chamberlain, Shear strength in the zone of freezing in saline soils. Proc. Conf. ARCTIC'85, San Francisco, ASCE, New York (1985) 566-574. 34. E. G. Hivon, Behaviour of saline frozen soils. Ph.D. Thesis, Univ. of Alberta, Edmonton (1991). 35. K. W. Biggar and D. C. Sego, The strength and deformation behaviour of model adfreeze and grouted piles in saline frozen soils. Canad. Geotech. J., 30 (1993) 319-337. 36. K. W. Biggar and D. C. Sego, Field pile load tests in saline permafrost, I, Test procedures and results. Canad.Geotech.J., 30(1993) 34-45. 37. K. W. Biggar, E. G. Hivon and D. C. Sego, Time dependent displacement of piles in saline permafrost. 6th Int. Permafrost Conf., Beijing, 2 (1993) 42-47. 38. A. V. Sadovskiy, Adfreeze between ground and foundation materials. 2nd Int.Conf. on Permafrost, Yakutsk, North. Amer. Vol. (1973) 650-653. 39. W. D. Roggensack and N. R. Morgenstern, Direct shear tests on natural fine-grained permafrost soils. 3rd Int. Permafrost Conf, Edmonton, 1 (1978) 728-735. 40. J. S. Weaver and N. R. Morgenstern, Simple shear creep tests on frozen soils. Canad. Geotech. J., 18 (1981) 217-229. 41. H. H. G. Jellinek, Ice adhesion. Can. J. of Physics, 40 (1962) 1294-1309. 42. J. S. Weaver and N. R. Morgenstern, Pile design in permafrost. Canad. Geotech. J., 18 (1981) 357-370. 43. B. Ladanyi and A. Th6riault, A study of some factors affecting the adfreeze bond of piles in permafrost. 5th Canad. Permafrost Conf., Quebec, Collect. Nordicana No.54, Laval Univ. (1990) 327-334. 44. F. E. Crory, Pile foundations in permafrost. First Int. Permafrost Conf., Lafayette, IN, 1963, NAS-NRC Publ. 1287 (1966) 467-472. 45. E. Penner and W. W. Irwin, Adfreezing of Leda clay to anchored footings and columns. Canad. Geotech. J., 6 (1969) 327-337. 46. G. H. Johnston and B. Ladanyi, Field tests of grouted rod anchors in permafrost. Canad. Geotech.J., 9 (1972) 176-194. 47. N. A. Tsytovich, The Mechanics of Frozen Ground, McGraw-Hill Book Co., New York (1975). 48. W. T. Black and H. P. Thomas, Prototype pile tests in permafrost soil. In: "Pipelines in Adverse Environments", ASCE (1978) 373-383.

33 49. U. Luscher, W. T. Black and J. F. McPhail, Results of load tests on temperature-controlled piles in permafrost. 4th Int. Permafrost Conf., Fairbanks, AK (1983) 756-761. 50. V. Manikian, Pile driving and load tests in permafrost. 4th Int. Permafrost Conf., Fairbanks, AK (1983) 804-810. 51. F. E. Crory and R. E. Reed, Measurement of frost heaving forces on piles. USACRREL, Tech. Rep. 145 (1965). 52. E. Penner, Uplift forces on foundations in frost-heaving soils. Canad. Geotech. J., 11 (1974) 323-338. 53. E. Penner and L. W. Gold, Transfer of heaving forces by adfreeze to columns and foundation walls in frost susceptible soils. Canad. Geotech. J., 8 (1971) 514-526. 54. L. Domaschuk, Frost heave forces on embedded structural units. 4th Canad. Permafrost Conf., Calgary (1982) 487-496. 55. J. B. Johnson and D.C. Esch, Frost jacking forces on H and pipe piles embedded in Fairbanks silt. 4th Int. Symp. on Ground Freezing, Sapporo (1985) 125-133. 56. V. R. Parameswaran, Adfreeze strength of frozen sand on model piles. Canad.Geotech.J., 15 (1978) 494-500. 57. V. R. Parameswaran, Creep of model piles in frozen sand. Canad. Geotech.J., 16 (1979) 69-77. 58. V. R. Parameswaran, Attenuating creep of piles in frozen soils. In: "Foundations in Permafrost", ASCE Spring.Conv., Denver (1985) 16-28. 59. V. R. Parameswaran, Displacement of piles under dynamic loads in frozen soils. 4th Canad. Permafrost Conf., Calgary (1982) 555-559. 60. V. R. Parameswaran, Adfreeze strength of model piles in ice. Canad. Geotech. J., 18 (1981) 8-16. 61. O. B. Andersland and M. R. M. Alwahhab, Bond and slip of steel bars in frozen sand. 3rd Int. Syrup. on Ground Freezing, Hanover, NH (1982) 27-34. 62. B. Ladanyi and A. Guichaoua, Bearing capacity and settlement of shaped piles in permafrost, llth Int. Conf on Soil Mech. & Found. Engrg., San Francisco (1985) 1421-1427. 63. A. Nadai, Theory of Flow and Fracture of Solids, Vol 2. McGraw-Hill Book Co., New York (1963). 64. J. S. Stelzer and O.B. Andersland, Model pile settlement behavior in frozen sand. J. of Cold Regions Engrg., ASCE 5 (1991) 1-13. 65. E. L Long, Designing friction piles for increased stability at lower installed cost in permafrost. 2nd Int. Permafrost Conf, Yakutsk, North. Amer. Contr. Vol., (1973) 693-698. 66. H. P. Thomas and U. Luscher, Improvement of bearing capacity of piles by corrugations. USACRREL, Spec. Rep. 80-40 (1980) 29-234. 67. D. C. Sego and L. B. Smith, Effect of backfill properties and surface treatment on the capacity of adfreeze pipe piles. Canad. Geotech. J., 26 (1989) 718-725. 68. A. Foriero and B. Ladanyi, FEM simulation of interface problem for laterally loaded piles in permafrost. Cold Reg. Sci. & Tech. (1994). 69. C. S. Desai, M.M. Zaman, J.G. Lightner and H. J. Siriwardane, Thin layer element for interfaces and joints. Int. J. Numer. Methods in Geomech., 8 (1984) 19-43.

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Mechanics of Georrmterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 1995 Elsevier Science B.V.

35

E X P E R I M E N T A L I N V E S T I G A T I O N S OF THE BEHAVIOR OF ICE AT THE CONTACT ZONE

G.W. Timco and R.M.W. Frederking National Research Council of Canada, Ottawa, Ont. K1A 0R6, Canada

The contact zone between ice and a material has important implications in several areas. Tests of ice friction have shown extremely low values. This has been attributed to the behavior of the ice in the contact zone. Tests with ice in which ice crushing has occurred have shown that the ice in the contact zone directly influences both the pressure and pressure distribution in the ice. Experiments have shown quite varied behavior in the contact zone. The characteristics of the contact zone have been investigated in several laboratory experiments as well as through medium-scale experiments in the field. These experiments and subsequent analysis have provided insight into the influence of the contact zone, and its important rheological properties. This paper reviews the experiments, analysis and results of these investigations into the behavior of ice at the contact zone. These experiments include friction tests, laboratory indentation tests, small-scale laboratory tests, the field indentation tests at medium scale, and the extrusion and flow of crushed ice at high pressures.

1. INTRODUCTION Ice plays an important part in several aspects of every day life in countries which have a boreal climate. The behavior of ice interacting with various materials has serious economic, scientific and even entertainment implications. This behavior is often directly controlled by the behavior of the c o n t a c t z o n e between the ice and the material. Over the past several years, there has been considerable effort placed in trying to understand this behavior. In this overview, some of the recent experimental evidence is described and interpreted in terms of understanding the contact zone in ice. The paper first looks at the contact zone with comparatively low pressures where the failure of the ice does not occur. This is the frictional behavior of ice. Next, the experimental evidence for higher loads where ice failure occurs is reviewed. In this case, the surface of the ice fails by crushing. Then, the surface layer is greatly disturbed from the original surface. Several examples of the significance of the work are described.

36 2. ICE FRICTION It is common knowledge that ice has a very low friction coefficient. This fact was used centuries ago in engineering projects, in, for example, the movement of large monolithic stones to sites surrounding the Ming tombs in northern China. The movement of the stones took place in the winter when the low friction of the snow and ice allowed easier movement of the stones. In everyday life, the low friction properties of ice are used in recreational activities such as skating, hockey and curling. On the other hand, driving an automobile on an ice-covered road can cause loss of control which may result in serious damage and loss of life. For some reason, the surface or contact zone of ice exhibits very low resistance to sliding. This, in spite of the fact that ice is a rather strong solid when loaded at reasonable loading rates. The reason for this phenomenon is not understood; however, the current hypothesis states that there exists a thin film of liquid water at the contact zone between the ice and the sliding material. Three explanations have been proposed to produce this liquid layer: pressure melting, melting due to frictional sliding, and the "liquid-like" surface properties of ice. Pressure Melting: When water freezes to become ice, it undergoes a first-order phase transformation. In this, there is a discontinuous change in energy (a latent heat), and discontinuities in the volume and molecular spacing; this occurs sharply at a particular temperature. For ice, an anomaly exists in the change in volume in that the solid phase is less dense than the liquid phase. The temperature at which this phase transformation takes place is a function of pressure. For a first-order phase transformation, the change in temperature (dT) with a change in pressure (dp) is governed by the Clausius-Clapeyron equation: dT dp

AV . . . . AS

0.074

C [ M N - m -2

(1)

where AV is the ice-water volume change and AS is the ice-water change in entropy (Barnes and Tabor 1966). Thus, pressure lowers the melting point of ice. Reynolds (1901) postulated that the pressure generated by a skate blade or sleigh runner on ice was sufficient to melt a thin film of water, thereby lubricating the interface and providing a low friction. A calculation of this for a typical skater, however, reveals that water can only form if the ice temperature is not lower than -1.1 ~ (De Koning et al. 1992). Melting due to Frictional Sliding: Bowden (1953) recognized that the pressure required to melt the ice would be too high for normal conditions, and he proposed that the water layer was produced by heat generated by the friction. The rate of heat supplied to the contact zone could be calculated from the energy flow caused by friction: P

= Ix/N v

(2)

where ~tf is the coefficient of friction, N is the normal force and v is the sliding velocity. Using reasonable values for a skater, De Koning et al. (1992) have shown that the temperature

37 rise at the surface of ice can be up to 7 ~ This local increase in temperature could melt the ice to produce the liquid layer at the contact zone. The "liquid-like" Surface Properties of Ice: Hobbs (1974) has discussed in detail the theory of the ice itself having an intrinsic "liquid-like" layer. This idea was originally proposed by Faraday (1859) and Tyndall (1859). It was suggested that the layer resulted from the sintering of the ice. Others (Weyl 1951; Nakaya and Matsumoto 1953) have suggested that the layer forms as a result of the molecular polarity whereby a transition layer forms. This layer has the properties of a liquid and lowers the free energy. At the present time there is no general agreement which theory is correct. However, there is agreement that the pressure melting approach seems to be incorrect, since the pressures which would be required to melt the ice are too high. There are a number of articles in the available literature which provide evidence for either of the other two theories. The difficulty in studying this problem relates to the extreme thinness of the liquid layer. There does not appear to be any hope of direct observation of the layer, and so its properties have to be inferred through indirect experimental approaches with accompanying theoretical explanations of the observations. Several investigators (for example Bowden and Hughes 1939; Bowden 1953; Niven 1959; Barnes et al. 1971; Oksanen 1981; De Koning et al. 1992) have measured the friction coefficient of ice against various materials. There is no intent to review these here except as the results relate to the interpretation of the contact zone in ice. The results of Bowden and Hughes (1939) which indicate that the thermal conductivity of a slider on ice influences the friction coefficient, lends support to the friction melting theory. This is also supported by later measurements by Bowden (1953) who showed that there is an inverse relationship between the friction and velocity. On the other hand, the experiments by Bowden (1953) also showed that solids with high contact angles with water gave lower friction. This supports the "liquid layer" hypothesis. Valeri and Mantovani (1978) report on measurements of the thermal expansion of ice with silicon strain gauges. They found that the response of the gauges closely followed the behavior of silicon at temperatures close to the melting point of ice. They interpret this as direct evidence that a liquid layer exists on ice. Further, De Koning et al (1992) report that ice which has been grown from a solution containing a small amount of a surface active agent exhibits lower friction. These surfactants would reduce the surface tension of ice causing a change in the surface equilibrium film thickness such that the liquid layer would be thicker. This would reduce the friction. This has been observed experimentally. There have been a number of approaches to analytically describe the liquid layer on the surface of ice in contact with another material. Most of these theories are based on thermodynamic arguments related to the surface free energy or chemical potential of the surface. The properties of the layer are usually assumed to vary throughout its thickness to reach the normal ice structure over a short distance. The layer arises due to the electrical anisotropy of the water molecule itself, which, at the surface, may have weak bonds to other water molecules. Niven (1959) has proposed, for example, that the molecules at the surface may act like "roller bearings". The molecular rotations at the surface would be possible because of the incomplete hydrogen bonds to lock the molecules in place. Fletcher (1973),

38 Rrrnkins and Miller (1973) and Gilpin (1979) have developed theories which predict the thickness of the layer as a function of temperature. Figure 1 shows the range of variation for these theories. In all cases, the theories predict small layer thicknesses (on the order of 1 to 100 rim), and a decreasing layer thickness with decreasing temperature. These and other theories have been reviewed by Gilpin (1979) and they will not be discussed here. Although there has been considerable interest and work in this area over several decades, there is still no clear understanding of the detailed mechanism for this liquid layer in the contact zone between ice and a sliding material. Ice is one of only a small handful of materials in which this liquid layer may exist (Fletcher 1973). 100

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Comparison of the calculated thickness of the "liquid-layer" of ice versus temperature based on models by Fletcher (1973), Rrmkins and Miller (1973) and Gilpin (1979).

3. ICE C R U S H I N G AT HIGH P R E S S U R E S

The frictional behavior of the contact zone between ice and a material changes if the applied loads and resulting pressures are high enough to cause the ice to fail by crushing. This results in the ice at the interface breaking into several small pieces. In this case, the properties of the contact zone play a key role in defining the pressures and pressure distribution which are developed between the ice and the material. The results have important implications, especially in cases where floating ice interacts with an offshore structure or an icebreaking vessel.

39 There has been a large number of experimental investigations in this area. Although the experimental approaches vary, the following objectives have been sought: What is the peak pressure which can be generated in the situation where an ice sheet crushes against a structure? 9

What is the pressure distribution in this case? What are the mechanisms at the contact zone which control the pressure and pressure distribution?

In this situation, the high stresses cause rapid disintegration of the ice in the contact zone. The ice exhibits a large range of microcracks which network and coalesce close to the ice surface. This results in the ice at the interface breaking into several small pieces. This region of broken ice is quite local and forms a layer of compact, pulverized ice pieces. Usually these ice pieces are extruded away from the interface as the indenting material advances further into the ice. This problem was addressed originally by Russian Scientists (Likhomanov and Kheisin 1971; Kheisin and Likhomanov 1973; Kurdyumov and Kheisin (1976)) in their analysis of impact tests on ice (see Figure 2). The pulverized ice pieces are under considerable pressure, and, in most cases they have to be squeezed out of the interface. This lead Kurdyumov and Kheisin to treat the pulverized layer as a pasty or powdery material which possess both viscous and plastic properties. Considerable effort has also been spent in trying to characterize the detailed properties of this crushed ice layer.

Figure 2.

Schematic illustration of the crushed ice layer which forms during the impact of an indentor impacting ice (after Kurdyumov and Kheisin 1976).

40 In the following sections, some of the experimental investigations into this problem are described. The list is not exhaustive; rather, it is meant to give the reader a flavor for the type of experiments which have been performed to date. The results of the experiments are discussed as they relate to the properties of the contact zone during ice crushing.

4. LABORATORY TESTS There has been several different types of laboratory experiments which have been performed to investigate the ice crushing process. These include indentation of floating ice sheets, crushing of small samples in a uni-axial press in a cold room, crushing/friction tests, and ice impact experiments. Each has lead to important information on the contact zone.

4.1 Friction/Crushing Tests Gagnon and Molgaard (1991a, 1991b) have developed an apparatus which measures friction forces on ice specimens under crushing normal loads. The apparatus consists of an ice specimen holder which is supported by an actuator (see Figure 3). The actuator A~ pulls the specimen against the outer surface of a hydraulically driven 8/~.x'~---" rotating wheel. The bottom of the specimen holder has a 50 mm circular view port. This window allows visual observation and video recording of the ice at the crushing interface. For the tests, rectangular prisms of freshwater ice were machined with a shallow 0 truncated peak on one side. This localized the initial contact in the central region of the sample.

G~ ~-~~~-

Figure 3.

, - - f~-'--

"-~

F H-

Crushing-friction apparatus of Gagnon and Malgaard (1991 a). A-hydraulic actuator; B-load cell; Cmild steel wheel (indentor); D-load cell; E- ice specimen; F-specimen holder; G-video camera; H-mirror.

At the start of a test, Gagnon and Molgaard (1991b) report that the ice specimen remained intact until sufficiently high load caused largea cracks to occur in the bulk sample. During this stress build-up, there was a transparent elliptical area where the indentor made contact with the ice. This area had a liquid-like appearance with occasionally a shiny ridge along the perimeter. Pressures based on the observed area and measured load were of the order of 90 MPa. When the crack occurred, the load on the

41 specimen dropped. With further advance of the actuator the load rose and the process repeated itself. Following each failure, the contact zone x;as seen to consist of an irregularly-shaped dark patch(es) that was in direct contact with the indentor (see Figure 4). This patch was surrounded by very tiny densely packed ice crystals with a powdery appearance, and small ice pieces which had resulted from spalling (see Figure 5a). Gagnon and Molgaard (1991b) have analyzed this situation in terms of the pressures on the ice. When the ice piece fractured and spalled, the resulting load dropped. The reduction in area contacting the indentor also greatly decreased, but in such a way that the pressure on the ice in contact with the indentor increased. This caused rapid pressure melting and fluid flow extrusion on top of the peakshaped undamaged ice (see Figure 5b). This thin layer of water flows rapidly, and in so Figure 4. Schematic of the contact zone as observed doing, it generates heat due to in the Gagnon and Molgaard (1991b) viscosity. As the flow leaves the experiments on crushing friction of ice. high pressure region and the freezing point is elevated (Equation 1), the flow refreezes. Thus, in this case there is indirect evidence that pressure melting occurs at the contact zone.

Figure 5.

Schematic illustration showing (a) the spalling process and (b) the pressure melting process in ice samples loaded using the crushing friction apparatus of Gagnon and Molgaard (1991 b).

42

4.2 Indentation of Floating Ice Sheets Tests of this type have been performed by several investigators (see e.g. Michel and Blanchet 1983; Timco 1986; K~'n/i and Muhonen 1990; Sodhi 1991; Muhonen et al. 1992). In a test of this type, a sheet of freshwater ice is grown in an ice tank. An indentor bar, which is connected to a load cell is pushed through the floating ice sheet. Figure 6 shows the test set-up used by Muhonen et al (1992) in their tests. In these tests, if the speed is high enough, and the aspect ratio (width of the indentor bar to the ice thickness) is not too large, local crushing of the ice occurs. Most tests have been performed with a single indentor bar, but more recent ones by Sodhi (1992; this volume) have used segmented indentors. The tests provide valuable information on the general process of ice crushing as well as the pressures generated during the crushing process.

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Schematic illustration of the test approach and instrumentation used by Muhonen et al (1992) for tests of ice indentation of floating ice sheets.

During the indentation process, microcracks occur in the ice in the region in front of the indentor. Figure 7 shows a typical cracking pattern in the ice, and the idealization of the ice sheet (Timco and Jordaan 1987; Jordaan and Timco 1988). The sheet consists of: Zone 1 in the far-field of virgin ice, Zone 2 ice which has microcracks, with some loss of stiffness, and Zone 3 ice which is immediately in front of the indentor. This ice, which is in the contact zone between the ice and the indentor is pulverized to the point where individual ice pieces exist. These ice pieces may move past each other and be ejected from the contact zone. In these indentor tests, the thickness (rl) of the pulverized layer (i.e. Zone 3) was estimated to be on the order of 1 to 5 mm.

43

Figure 7.

Schematic illustration and photograph of the damage in the ice in front of an indentor pushing through freshwater ice (after Jordaan and Timco 1988).

The movement of the crushed material is controlled by the local geometry of the interaction process. In the laboratory tests, the tests are designed such that the contact area remains relatively small. This is done since high loads can be generated during the crushing process and the laboratory equipment must be able to generate and withstand these loads. In analysis, it is usually assumed that the thickness of the pulverized layer (11) is constant both with width and through the thickness of the ice sheet at any point in time. Further, the pressure is also assumed to be constant across the width of the indentor. However, the pressure is assumed to vary throughout the thickness of the ice sheet. Jordaan and Timco (1988) assumed that the pressure was zero at the top and bottom of the ice sheet with a parabolic distribution across the thickness of the ice sheet. Then, the force on the indentor per unit width was found to be:

F

=

~

v

( --h)3

=

~

v

tt 3

(3)

where r - h / 11 is the ratio of the ice thickness to layer thickness, kt is the coefficient of viscosity of the crushed ice, and v is the indentor velocity. Thus, the force on the indentor was found to be inversely proportional to the third power of the crushed layer thickness. As the ice in this layer is ejected, the layer thickness decreases, and the force increases rapidly. Jordaan and Timco (1988) suggested that this force would increase until the ice in Zone 2 close to the indentor would be stressed to such a level that it would pulverize. This would

44 create a thicker crushed layer, and the load would decrease. With further advance of the indentor, the process would begin over again. In this analysis, the viscosity and thickness of the pulverized layer in the contact zone directly and significantly affects the load and pressure on the indentor.

4.3 Small-scale Indentation and Crushing A number of investigators have designed and used a small sub-press in a conventional uniaxial test apparatus to examine the ice crushing process (see e.g. Joensuu and Riska 1989; Fransson et al. 1991; Gagnon 1992). Figure 8 shows a typical test arrangement used for these tests (after Gagnon 1992). Usually a rectangular-shaped specimen is produced from ice of very high quality. The specimen is machined to have one "pointed" end; for example, in the shape of either a wedge or pyramid top. The ice piece is placed in a uniaxial compression test apparatus which can generate loads using a suitable actuator arrangement. In some tests, transparent platens are used. This, in combination with a mirror and video camera allows direct observations of the contact interface. In some tests (Fransson et al. 1991) a very high speed video camera has been used. This allows very detailed information of the contact zone region.

MATING PLATE FOR MTS FRAME \

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INSTRUMENTED CRUSHING PLATEN

When a load is applied to an ice specimen in the arrangement shown in Figure 8, the ice fails by crushing at the point of contact and by having larger ice pieces "flaking" or splitting from the interface area. The crushed ice is extruded away from the contact region. During Crushing apparatus with the first set of tests of this type by Joensuu and Figure 8. mirrors for viewing the Riska (1989), a new and interesting contact zone during smallphenomenon was observed. They observed an scale indentation tests (after apparent "line-of-contact" at the interface. This Gagnon 1992). contact line has subsequently been observed by others using this test arrangement. As well, Riska et al. (1990) found evidence of a similar line of contact on a ship, which had a transparent window, subjected to ice crushing in the Baltic Sea. This contact-line approach presents an alternate view to the "extrusion layer" approach discussed previously (see Figure 9).

45

INDENTOR

ICE SHEET ---~mmP,CONTACT LINE

PRESSURE DISTRIBUTION

Figure 9.

(b)

Schematic illustration showing the (a) extrusion approach and (b) the contact line approach to the indentation problem. A typical pressure distribution for the two cases is also shown in the figure.

Daley (1990) has developed an iterative failure model to describe the process. In the model, the ice in direct contact with the indentor is crushed, with a resultant constant pressure over this area. The shear stress generated in the intact ice is compared to the shear strength for different angles of shear planes running from the contact region to the edge of the ice sample. If the stress is equal to or exceeds the strength, an ice flake is formed and removed from the interface region. With the removal of the flake, the surface geometry is updated, and comparisons are again made in terms of the stress/strength relation. The process is iterative. This model gives a good explanation of the failure process, including the time-series behavior. Recently, Gagnon (1993) has used two electrical conductors to investigate the presence and thickness of a liquid in high pressure ice crushing events. Liquid was detected and furthermore, peaks occurred in the liquid sensor output when the load on the indentor dropped.

46 The upper limit for the liquid layer thickness was estimated by Gagnon (1993) to be approximately 3 ~tm when the load was increasing, and approximately 21 ~tm when the load on the indentor dropped. Similar to the tests using the friction/crushing apparatus described above, the contact between the platen and the ice consisted of low pressure zones of highly damaged crushed and/or refrozen ice, and transparent high pressure regions of undamaged ice.

5. MEDIUM-SCALE FIELD INDENTATION There have been a number of major field experiments performed to investigate the process of ice crushing with physically larger indentors than those possible in laboratory tests. In these tests, indentors with cross-sectional areas of 0.5 to 3 m 2 were used. This size is intermediate between the smaller laboratory tests, and the full-scale sizes of an offshore structure in ice. The first test of this type was performed in 1984 at Pond Inlet, N.W.T. in tunnels trenched into a grounded iceberg. The results of that program have been released (Masterson et al. 1992). A second, proprietary project was carried out at Byam Channel in 1985. For two consecutive years in 1989 and 1990, comprehensive test programs were carried out on multiyear ice at Hobson's Choice Ice Island (Frederking et al. 1990a; 1990b). These tests were directed towards understanding the pressures and pressure distribution which can occur during ice crushing events. As part of this research, extensive measurements and observations were made of the contact zone. A considerable amount of knowledge has resulted from these tests. In the 1989 project at Hobson's Choice Ice Island, a trench 65 m long, 3 m deep and 3 m wide was excavated in a multi-year ice floe which was attached to the Island. The ice indentation system consisted of a hydraulic actuator mounted on a skid to facilitate movement. The actuator had a capacity of 4.5 MN (450 tonnes), a stroke of 300 mm and a maximum velocity of 100 mm-s"(see Figure 10). A system of local pressure sensors were placed in the indentor from which information on the pressure and pressure distributions in the contact zone could be obtained. Figure 11 shows the ice face, dimensions of the indenter and locations of the local pressure gauges for one of the tests. A vertical strip of ice 120 mm wide initially contacted the flat indenter; the side surfaces of the ice had slopes of 1:3 as shown in Figure 1 l a. Figure 1 l b shows the initial and final contact widths of the ice face. The geometry and set-up of this test simulates plane strain extrusion of ice. The force-time curve (Figure 12a) is of a sawtooth shape, characteristic of crushing failure of ice. Measurement of local pressures, Figures 12b,c and d at positions located in Figure 1 lb show a striking in-phase relationship with each other and with the force of Figure 12a. On average, the pressure cells near the center of the indentation area (cells 4, 5 and 6) recorded substantially higher pressures than those near the edges. The average local pressure in the center (cells 4, 5 and 6) was about 15 to 20 MPa. The two side groups (cells 1, 2, 3, 7 and 8) had average local pressures in the range of 0 to 10 MPa. When plotted versus position, they showed generally triangular pressure distributions. Frederking and Sayed (1992) showed the distribution to be concave upwards for the extrusion phase and concave downwards for the load increasing phase (see Figure 13). An average pressure over the contact zone of 3 to 7 MPa, was calculated from the total force (Figure 12a) and a contact area which increased from 0.12 to 0.27 m 2 during the test. Thus it can be seen that the average pressure in the

47

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49

central part of the contact area is about 3 times the pressure for the whole area. Locally some very high individual pressures reached values of over 70 MPa. Such values approach pressure melting conditions for ice at about -10 ~ Examining pressures along the central group of sensors also showed a substantial variation over the 125 mm distance between them, differences of up to 30 MPa. There were even cases where the local pressure variations were greater. From Figure 12c it can be seen that cell 6 was recording zero pressure while cell 5 was experiencing a pressure of 50 MPa. This supports the concept of localized zones of high pressure, the so-called "hard spots". In the 1990 project a multi-element pressure panel of piezoelectric film sensors exhibited similar pressure values and variations of pressures (Muhonen and Koriseva 1990).

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Field observations have provided insight into the structure of the ice in the contact zone. Figure 14a shows this zone, as exposed by cutting out a section with a chain saw. The transition between the zone of crushed ice and the parent ice is generally quite distinct. The crushed ice is white in color (indicating the presence of many fracture surfaces which scatter light). Figure 14b shows an enlargement of a thin section of the area designated in Figure 14a. It can be seen that the crushed ice is very fine-grained; the parent ice has been fractured and ground into very small particles. Near the center, most of the large grains remain, although there are some signs of incipient damage on the grain boundaries. The area of relatively undamaged ice, as observed in cross-section, corresponds to a linear feature outlined on the contact face (Figure 14a). Inspection of the crushed ice layer after the test showed that near the edge of the contact area it was soft, and could be broken off easily by hand. Near the center of the area, however, the ice was quite solid even in areas where it had

50 been highly pulverized. More extensive mapping of the damage zone has been done by Meaney et al. (1992) and Sinha and Cai (1992). They again showed the variability in the crushed ice layer thickness. Gagnon and Sinha (1991) made observations of temperature which showed evidence for the occurrence of localized pressure melting at the interface between the indentor and the ice.

Figure 14.

Contact face for the ice at the end of one of the medium-scale indentation tests: (a) general view of the vertical face and horizontal cross section cut into the ice face, and (b) thin section of area A (after Frederking et al. 1990b).

51 6. B E H A V I O R OF CRUSHED ICE

All of the experiments described above have pointed towards the importance of the crushed ice layer in determining the pressure and pressure distribution. For this reason, a number of separate experiments have been performed and analyzed to investigate the behavior and properties of crushed ice. Sayed and Frederking (1992) performed experiments examining the two-dimensional flow between two converging parallel platens. The platens were 0.10 m by 0.20 m in size. Crushed ice particles were placed between the platens and they were pushed towards each other at a constant rate. One platen was instrunaented with pressure cells so pressures and pressure distributions could be determined. Further, flow patterns were observed through transparent side walls. These experiments showed that the pressure was strongly dependent on the thickness of the crushed ice layer. Further, the stress along the platens showed a strong distribution with high values in the center and low values near the edge. This spatial distribution had a pointed and concave shape, with a discontinuity in the center. This type of behavior is expected if the material behaves as a Mohr-Coulomb material. These tests were analyzed by Savage et al. (1992). Spencer et al. (1992) and Geotech and Memorial University (1989) report a similar type of experiment with, however, considerably larger platens and loading range. The load test frame consisted of a base frame and a hydraulic actuator. The piston of the actuator was coupled to an instrumented upper platen 0.76 m by 0.51 m. This platen contained nine pressure cells placed at strategic locations. The long edges of the platens were blocked to force flow in two directions only. The crushed ice material was squeezed out from between the two platens. The actuator was coupled to a 200 US G.P.M. servo valve and eight air-over-oil piston accumulators. These accumulators provided the large flow rates necessary for the tests. Average pressures and pressure distributions were determined for various conditions of grain size and loading rate. Typical pressure gradients were on the order of 5 to 10 MPa/mm. Jordaan and others (Memorial University 1989, Singh et al. 1993) have analyzed these results in detail. The extrusion of this ice was modelled using two different theoretical approaches: Mohr-Coulomb theory and theory of a viscous fluid. The Mohr-Coulomb theory, which was developed for granular materials provided a good model for extrusion at low pressures (less than 7 MPa). This result is in agreement with the tests by Sayed and Frederking (1992). In this low pressure region, the flow was along wedge-shaped planes, and the pressure distribution was "pointed" with a peak in the center of the platen. Typical values of cohesion ranged from 0.03 to 0.001 MPa; internal friction angle ranged from 31 ~ to 39~ and the wall friction angle ranged from 1.5 ~ to 3~ At higher pressures the Mohr-Coulomb approach was found to be inadequate. A better fit to the data was found when treating the crushed ice as a layer which flows as a viscous fluid. This would give a parabolic pressure distribution. Calculated values for the viscosity of the material gave 1.0 MPa-s at a rate of 2.5 mrn/s to 0.0009 MPa-s at 160 mm/s.

52 7. SUMMARY AND CONCLUSIONS

A considerable amount of information is currently available on the contact zone of ice. The experimental evidence indicates that ice appears to have a "liquid-like" layer which gives it its low friction behavior. Three theories have been advanced to explain this behavior: pressure melting, melting through sliding friction, and inherent liquid-like layer properties. Estimates of the thickness of this layer indicate that it varies with temperature, with a typical range of 1 to 50 nm. With increasing normal loads, ice can fail by crushing. In that case, the contact zone exhibits various types of behavior. Experiments have shown that a layer of pulverized small ice particles exist in the contact zone. The thickness of this pulverized ice is a function of several parameters. The properties of this layer affects the pressures and pressure distribution in the ice. Experimental evidence also shows that liquid water can be formed in this process with a layer thickness on the order of 3 to 21 ktm. REFERENCES

Barnes, P., Tabor, D. and Walker, J. 1971. The Friction and Creep of Polycrystalline Ice. Proc. Roy. Soc. Lond. A324, pp 127-155. Barnes, P. and Tabor, D. 1966. Plastic Flow and Pressure Melting in the Deformation of Ice I. Nature 210, pp 878-882. Bowden, F.P. 1953. Friction on Snow and Ice. Proc. Royal Soc. A217, pp 462-478. Bowden, F.P. and Hughes, T.P. 1939. The Mechanics of Sliding on Snow and Ice. Proc. Royal Soc. A, 172, pp 280-298. Daley, C. 1990. An Iterative Failure Process Model. Helsinki University of Technology Report 1990/M- 103, Espoo, Finland. De Koning, J.J., De Groot, G. and Ingen Schenau, J.V. 1992. Ice Friction during Speed Skating. Jour. Biomechanics 25, pp 565-571. Faraday, M. 1859. On Regelation, and on the Conservation of Force. Phil. Mag. 17, pp 162169. Fletcher, N.H. 1973. The Surface of Ice. in Physics and Chemistry of Ice. E. Whalley et al. Ed., pp 132-136, University of Toronto Press, Toronto, Canada. Fransson, L., Olofsson, T. and Sandkvist, J. 1991. Observations of the Failure Process in Ice Blocks Crushed by a Flat Indentor. Proceedings Port and Ocean Engineering under Arctic Conditions, POAC'91, Vol. I, pp 501-514, St. John's, Nfld. Canada. Frederking, R., Blanchet, D., Jordaan, I.J., Kennedy, K., Sinha, N.K., and Stander, E. 1990a. Field Tests of Ice Indentation at Medium Scale, Ice Island, April 1989. Client Report CR5866.1, IRC/NRC, Ottawa, Ont., Canada.

53 Frederking, R.M.W., Jordaan, I.J. and McCallum, J.S. 1990b. Field Tests of Ice Indentation at Medium Scale, Hobson's Choice Ice Island, 1989. Proceedings IAHR 10th International Symposium on Ice, Vol. 2, pp. 931-944, Espoo, Finland, Frederking, R.M.W. and Sayed, M. 1992. Stress Distributions as a Reflection of Failure Processes during Medium-Scale Ice Indentation. Proceedings International Association for Hydraulic Research, IAHR Ice Symposium, Vol. 3, pp 1278-1288, Banff, Alberta, Canada. Gagnon, R.E. 1992. Heat Generation during Crushing Experiments on Freshwater Ice. in Physics and Chemistry of Ice, Hokkaido University Press, Japan, pp 447-455. Gagnon, R.E. 1993. Melt Layer Thickness Measurements during Crushing Experiments on Freshwater Ice. Journal of Glaciology (in press). Gagnon, R. E. and Molgaard, J. 1991a. Crushing Friction Experiments on Freshwater Ice. Proceedings IUTAM/IAHR Symposium on Ice/Structure Interaction, St. John's Nfld., Springer-Verlag, Berlin, pp 405-421. Gagnon, R. E. and Molgaard, J. 1991b. Evidence for Pressure Melting and Heat Generation by Viscous Flow of Liquid in Indentation and Impact Experiments on Ice. Annals of Glaciology, 15, pp 254-260. Gagnon, R.E. and Sinha, N.K. 1991. Energy Dissipation through Melting in Large-Scale Indentation Experiments on Multi-year Ice. Proceedings Offshore Mechanics and Arctic Engineering Symposium, OMAE'91, Vol. IV, pp 157-161, Stavanger, Norway. Geotech and Memorial University, 1989. Flow of Crushed Ice: Physical and Mechanical Properties and Behavior. Report 9538, Calgary, Alberta, Canada. Gilpin, R.R. 1979. A Model of the "Liquid-Like" Layer between Ice and a Substrate with Applications to Wire Regelation and Particle Migration. Jour. of Colloid and Interface Sci. 68, pp 235-251. Hobbs, P.V. 1974. Ice Physics. Clarendon Press, Oxford, U.K. Joensuu, A., and Riska, K. 1989. Contact Between Ice and Structure. Helsinki University of Technology, Laboratory of Naval Architecture and Marine Engineering Report M-88, Otaniemi, Finland (in Finnish). Jordaan, I.J. and Timco, G.W. 1988. Dynamics of the Ice-Crushing Process. Journal of Glaciology, Vol 34, No. 118, pp 318-326. K~rn~i, T. and Muhonen, A. 1990. Preliminary Results from Ice Indentation Test using Flexible and Rigid Indentors. Proceedings International Association for Hydraulic Research, IAHR Ice Symposium, Vol. 3, pp 261-275, Espoo, Finland.

54 Kheisin, D.E., and Likhomanov, V.A. 1973. An Experimental Determination of the Specific Energy of Mechanical Crushing of Ice by Impact. Problemy Arktiki i Antarktiki, Vol. 41, pp. 69-77. Kurdymnov V.A. and Kheisin, D.E. 1976. Hydrodynamic Model of the Impact of a Solid on Ice. Translated from Prikladnaya Mekhanika 12, pp 103-109. Likhomanov, V.A., and Kheisin, D.E. 1971. Experimental Investigation of Solid Body Impact on Ice. Problemy Arktiki i Antarktiki, Vol. 38, pp. 105-111. Masterson, D.M., Nevel, D.E., Johnson, R.C., Kenny, J.J. and Spencer, P.A. 1992. The Medium-Scale Iceberg Impact Test Program. Proceedings International Association for Hydraulic Research, IAHR'92, Vol. II, pp 930-966. Banff, Alberta, Canada. Meany, R., Kenny, S. and N.K. Sinha, 1991. Medium-Scale Ice-Structure Interaction: Failure Zone Characterization. Proceedings Port and Ocean Engineering under Arctic Conditions, POAC'91, Vol. I, pp 126-140, St. John's, Nfld. Canada. Memorial University, 1989. Flow of Crushed Ice: Physical and Mechanical Properties and Behavior. Memorial University of Newfoundland Report. St. John's, Nfld. Canada. Michel, B. and Blanchet, D. 1983. Indentation of $2 Floating Ice Sheet in the Brittle Range. Annals of Glaciology 4, pp 180-187. Muhonen, A. and Koriseva, J. 1990. Ice Pressure Measurements using PVDF-sensors. Proceedings POLARTECH, pp 526-538, Copenhagen, Denmark. Muhonen, A., K~a'n/i, T., Eranti, E., Riska, K. J/irvinen, E. and Lehmus, E. 1992. Laboratory Indentation Tests with Thick Freshwater Ice. Technical Research Center of Finland VTT Research Notes 1370, Espoo, Finland. Nakaya, U. and Matsumoto, A. 1953. Simple Experiment showing the Existence of a "Liquid Water" Film on the Ice Surface. Jour. Colloid Sci. 9, pp 41-49. Niven, C.D. 1959. A Proposed Mechanism for Ice Friction. Can. Jour. of Phys. 37, pp 247255. Oksanen, P. 1981. Friction and Adhesion of Ice. Proceedings International Association for Hydraulic Research, IAHR'81, Vol. II, pp 628-637, Quebec City, Quebec, Canada. Reynolds, O. 1901. Papers on Mechanical and Physical Subjects, 2, Cambridge University Press, Cambridge, UK. Riska, K., Rantala, H. and Joensuu, A. 1990. Full-Scale Observations of Ship-Ice Contact. Helsinki University of Technology, Laboratory of Naval Architecture and Marine Engineering Report M-97, Helsinki, Finland.

55 R6mkins, M.J. and Miller, R.D. 1973. Journal Colloid Interface Sci. 42, pp 103-111. Savage, S.B., Sayed, M. and Frederking, R.M.W. 1992. Two-dimensional Extrusion of Crushed Ice. Part 2:Analysis. Cold Regions Science and Technology 21, pp 37-47. Sayed, M. and Frederking, R.M.W. 1992. Two-dimensional Extrusion of Crushed Ice. Part 1:Experimental. Cold Regions Science and Technology 21, pp 25-36. Singh, S.K., Jordaan, I.J., Xiao, J. and Spencer, P.A. 1993. The Flow Properties of Crushed Ice. Proceedings 12th Offshore Mechanics and Arctic Engineering Symposium, OMAE'93, Vol. IV, pp 11-20, Glasgow, Scotland. Sinha. N.K. and Cai, B. 1992. Analysis of Ice from Medium-Scale Indentation Tests. National Research Council of Canada Report IME-CRE-LM-002, Ottawa, Ont., Canada. Sodhi, D.S. 1991. Ice-Structure Interaction during Indentation Tests. in Proceedings of IUTAM-IAHR Symposium, S. Jones et al. Ed., Springer-Verlag, Berlin, pp 619-640. Sodhi, D.S. 1992. Ice-Structure Interaction with Segmented Indentors. Proceedings International Association for Hydraulic Research, IAHR'92, Vol. 2, pp 909-929. Banff, Alberta, Canada. Spencer, P.A., Masterson, D.M., Lucas, J. and Jordaan, I.M. 1992. The Flow of Crushed Ice: Experimental Observations and Apparatus. Proceedings International Association for Hydraulic Research, IAHR'92, Vol. 1, pp 258-268. Banff, Alberta, Canada. Timco, G.W. and Jordaan, I.J. 1987. Time-series Variations in Ice Crushing. Proceedings 9th International Conference on Port and Ocean Engineering under Arctic Conditions, POAC'87, Vol. I, pp 13-20, Fairbanks, Alaska, U.S.A. Timco, G.W. 1986. Indentation and Penetration of Edge-Loaded Freshwater Ice Sheets in the Brittle Range. Proceedings Offshore Mechanics and Arctic Engineering, OMAE'86, Vol IV, pp 444-452, Tokyo, Japan. Tyndall, J. 1859. On the Physical Phenomena of Glaciers Part I. Observations on the Mer de Glace. Phil. Trans. Roy. Soc. 149, pp 261-278. Valeri, S. and Mantovani, S. 1978. The Liquidlike Layer at the Ice Surface: A Direct Experimental Evidence. Jour. of Chem. Phys. 69, pp 5207-5208. Weyl, W.A. 1951. Surface Structure of Water and Some of its Physical and Chemical Manifestations. Journal Colloid Science 6, pp 389-405.

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 1995 Elsevier Science B.V.

57

AN ICE-STRUCTUI~ INTERACTION MODEL

Devinder S. Sodhi U.S. Army Cold Regions Research and Engineering Laboratory 72 Lyme Road, Hanover, New Hampshire, USA 03755-1290

A theoretical model to simulate ice-structure interaction during intermittent crushing is developed on the basis of experimental results from indentation tests, which were conducted by pushing vertical, flat indentors into the edges of freshwater, floating ice sheets. An event during intermittent crushing comprises three phases: (1) a loading phase, (2) an extrusion phase and (3) a separation phase. The differential equations and solutions during each phase of interaction are presented along with conditions for termination of each phase. Besides simulating interaction during intermittent crushing, the model simulates the transition from intermittent to continuous crushing at high rates of indentation, as found during indentation tests. A few results from the model are presented to show the effect of various parameters on the velocity at which transition from intermittent to continuous crushing takes place and on the frequency of intermittent crushing.

1. INTRODUCTION Crushing of ice is a common mode of ice failure during an interaction between a moving sheet and a structure. Under certain conditions, the interaction results in severe structural vibrations. Peyton [1] and Blenkam [2] reported on ice-induced vibration of steel struc~zres in Cook Inlet, Alaska. M/i/itt~inen [3], Engelbrektson [4] and Nordlund et al. [5] have discussed the vibrations of lighthouses and channel markers in the Gulf of Bothnia while interacting with moving ice sheets. Jefferies and Wright (1988) reported on the vibration of Molikpaq, a 110-m-wide structure deployed in the southern Beaufort Sea, as a result of interaction with a multi-year sea ice floe. Many small-scale experimental studies have been conducted to increase understanding of the manner in which ice-struc~m interaction takes place. Theoretical models on ice-structure interaction can broadly be divided into two categories, according to the assumption made regarding the interaction force being proportional to either relative displacement (e.g. Matlock et al. [7]) or relative velocity (e.g., Blenkam [2]) between an ice sheet and a structure. For an extended discussion of this subject, the reader is referred to review papers by Sodhi [8] and M/i~itt/inen [9]. Recently, Eranti [10] has developed a model for ice-structure interaction based on smallscale tests by Muhonen et al. [11]. ice

58

The theoretical model presented in this paper is based on the results of indentation tests conducted by the author in freshwater ice at the U.S. Army Cold Regions Research and Engineering Laboratory (CRREL). A brief description of the experimental setup and the results will precede the presentation of the theoretical model and the solutions of equations that govern the motion of the structure during each phase of an interaction. Some results from simulations will also be presented.

2. EXPERIMENTAL STUDY For the experimental study; a single-degree-of-freedom system, which consisted of a mass and a spring, was chosen to simulate a vibrating structure (Fig. 1). The apparatus consisted of two horizontal plates attached to each other with four hinged links. The top plate was attached under the main carriage, spanning the test basin of the Ice Engineering Facility of CRREL. On the bottom plate, the supports for the indentor plate and lead weights of about 400-kg mass were mounted. As shown in Figure 1, a spring was also mounted between two brackets, one attached to the top plate and the other to the bottom plate. The spring consisted of two plates separated by rollers, and its stiffness could be changed by changing the span of the two plates by removing or installing the rollers. Indentor plates of different widths were mounted on three load cells so that the interaction force generated at the ice-structure interface could be measured independently of the structural motion. Besides measuring of the interaction forces, the displacements of the carriage and the indentor plate were measured separately with respect to a fixed datum. Further, the relative displacement of the two horizontal plates was measured. The acceleration of the bottom plate, on which the indentor assembly was installed, was also

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measured. These measurements provided a complete 15 - Max. Force = 11.761 kN set of data on the interaction, -~g ~ on which the model proposed z 10 .i 6 ~ in this paper is based. tO All experiments were done 4 ft. ~ 5 u_ with freshwater ice grown in the test basin, which mea, o ~-~ U.i sures 36 m long, 9 m wide 60 . Velocity = 0.6 mm/s and 2.6 m deep. The ice sheets E 50 had the columnar feattm~ of E 40 seeded ice. Ice sheets of differC a r r ~ 30 ent thicknesses (15-57 turn) E 20 ~ Ind~ntor I were grown during this proo 10 o_ gram. The apparatus carry._~ 0 E3 ing the indentor plate was -10 Structure pushed into the edges of 300 " floating ice sheets at different 200 " velocities (0.6-150 m m s-l). A 100 >,,, detailed description of the ex0 perimental setup and proced= -100 UJ ure can be found in Sodhi ~ y.AWc -200 [12]. -300020 40 60 80 Three modes of ice behav100 ior were observed during the Time (s) study: creep deformation at 15 Structure low indentation velocities & 10 with no vibrations, intermittent crushing at intermediate ~Q 5 indentation velocities with 01, ~, considerable vibrations, and 60 70 continuous crushing at high -10 0 10 20 30 40 50 Displacement (ram) indentation velocities with no vibrations. The stmcttwe ocFigure 2. Plots of experimental data of a test in which casionally vibrated at its natucreep deformation of ice took place. ral frequency (a situation referred to as resonance) at indentation velocities between those when intermittent and continuous crt~hing took place. During the resonant mode of interaction, the behavior of ice failure was intermittent, causing structural vibration. Typical records of the experimental data for the four modes of interactions are shown in Figures 2-5. Figure 6 shows part of the data in Figure 3 at an expanded time scale. Each figure shows time records of interaction force, displacements of the carriage and the indentor, the relative displacement of the structure with respect to the carriage, and the energy exchange between the ice and the stmcttm~. Force vs. displacement plots are also given at the bottom of each figure. v

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Figure 3. Plots of experimental data of a test in which icestructure interaction results in intermittent crushing. The effective pressure is defined as the measured force divided by the contact area, which is taken to be the product of indentor width and ice thickness. The maximum effective pressures at different indentation rates were found to differ by a factor of 3 to 5; high pressures (8-13 MPa) were measured at low indentation velocities (< 20 m m s-l), and low pressures (1.2-4.2 MPa) at high indentation velocities (> 100 mm s-a). Plots of measured effective pressure vs. the actual indentation velocity (Sodhi [13]) are shown in Figure 7, in which the data fall into two groups. During the experiments, a transition from intermittent crushing to continuous crushing took place at a certain velocity unique to the test conditions

61

of that day. Above that velocity, the interaction remained in the Max. Force = 11.3 kN continuous crushing mode. O9 O9 Energy exchanges during z 10 (1.)~ 83 a-a_ indentation tests (Sodhi [14]) o were computed using the exo 5 u_ O perimental data, as shown in uJ third plots in Figures 2-6. The energy supplied by the carriage 835 (XAWc) is partly stored in the E Vel. = 101.4 mm/s structural spring (PE), partly E 810 converted to kinetic energy C Carriage ~ S Indentor (KE), partly dissipated in deE 785 (1.) O forming and extruding the ice 83 I I I t I ~. 760 (X~Wi) and partly dissipated as CO Structure 121 heat (T_AQ) in the damping 735 mechanisms of the structure. 200 Except for the heat dissipation, PE F-"&W i all forms of energy were comv puted from the experimental r. 0 L data, and the heat dissipation C" UJ was computed from the energy balance using the first law of -200 thermodynamics. The main re5.8 5.9 6.0 6.1 6.2 6.3 sult of these computations reTime (s) Structure veals that intermittent crushing, 15 .--. T lndentor Carriage which resulted in ice-induced Z ~ 10 structural vibrations, took place o whenever there was an imbal~ 5 ance between the rates of work O! ! done by the carriage and the -10 0 10 20 30 40 50 indentor and that there were no Displacement (mm) vibrations when these rates of work were equal. Figure 4. Plots of experimental data of a test in which During intermittent crushthe structure vibrated at its natural frequency. ing (Figs. 3, 4 and 6), the interaction takes place in a cyclical manner. In each cycle or crushing event, the interaction force increases, causing a relative displacement between the carriage and the indentor. The relative displacement causes an imbalance between the energies supplied by the carriage and dissipated by the indentor, and the difference between the two is sto~-Kt in the structure as potential energy. This stored energy is later released after the ice fails at a certain effective pressure and causes the structure to execute transient or steady-state resonant vibrations (Figs. 4 and 6). In Figure 6, it can be seen that the interaction force was zero for a short period of time, indicating that a separation took place between the ice and the indentor. Test No. 110

15

62 Test No. 206 30 " Max. Force = 26.1 kN

12 L,-

::3

z 20 13_

o

o 10

,a

1.1_

800 E E

Vel. = 177.4 m

.i r

n

/

0 UJ

~

700

r'~ r

s

600 ~ , . ~ Indentor --~.....~-- - | Structure 500

I

1

~

I

I

~Wi

PE o

-1

4.62 4.82 5.02 5.22 Structure Time (s) 30 Indentor ~ v 20

5.42

5.62

~ 1 ~ ~ . . Carria2

O'

.

.

-30

0

30

.

.~

I

60 90 120 Displacement

.

150

.

180 210

Figure 5. Plots of experimental data of a test in which a transition from intermittent to continuous crushing took place as the ice speed was incrementally increased. The reason for different levels of effective pressures at low and high indentation rates is not exactly known, but microcracking observed during the experiments provides an indication of the deformation taking place in the ice in the vicinity of an indentor. The microcracks formed horizontally in the plane of an ice sheet as well as vertically along the columnar grains of the ice. At low indentation velocities, the microcracks formed within a radius of 0.5 m from the indentor. At high indentation velocities, no microcracks were observed around the indentor, which is an indication of localized deformation taking place near the indentor.

63 Separatior Extrusion

, ],

1 ,

Loading ~ I

Test No. 204

30

12

A

z,,e 20

r162

8

(1.} ~ - . S-a.

4

-,~

o

o 10

I.I..

9

-w

1130 E r Vel. = E 1100 '- 1070 E 1040 o

9

"

I

c

.~~--~... _ , .

~

-

-

w

.0

~:~

W

~

J

Indentor

1010 a

>,.,

"

LU

~

980 600 400 I 200 0 -200 -400 -600 10.90

I

30

T_.aWif

11.05

11.20

11.35

11.50

11.65

Time (s) . Indentor ..

Structure

._.z~"20

" e

10 I.!_

0" -30

i ~

()

30 60 Displacement (mm)

9O

Figure 6. Detailed plots of experimental data shown in Figure 3. In a later series of experiments (Sodhi [15]), microcracking during experiments at high indentation velocity was not particularly noticed. Similar observations were also made by Joensuu and Riska [16] who conducted experiments by impacting a moving glass plate against a block of ice. Absence of microcracking at high rates of indentation indicates that microcracks do not have time to form and that nonsimultaneous brittle failure of the ice sheet occurs across its thickness (spaUing) over many small contact areas. At low indenta-

64

16

III

I

l

"

I lllll

1

I

M

m m

RMFI

IlllllIIl

I IIIII

M M I~

IMI~IM~ MM M MM ~M

!

I

I

I IIII

R R

M

n v (I)

M s

E

cr~

EEE

CL E

.4-, o

'C_~ E EEE E i~C E E E E

iii

C IIII

I

I

I I IIII

0.1

1.0

I

I

I I IIII

10 Velocity (mm/s)

I

I

I I IIII

100

C I

I

I I IIII

1000

Figure 7. Plots of maximum (M,R,m) and average (S,r,C,E) effective pressures vs. indentor velocity in different modes of interaction (from Sodhi [13]). The symbols used in this figure denote the following interaction: M and E the failure and the extrusion pressure during intermittent crushing, R and r the maximum and the average pressure during resonant vibration, m and s the maximum and the steady-state pressure during creep deformation, and C the effective pressure during continuous crushing.

tion velocities, the contact between the indentor and the ice edge is over its entire area, and there is sufficient time for the ice to creep, which causes microcracking in the ice sheet ahead of the indentor.

3. THEORETICAL MODEL Though the experiments were conducted by pushing an indentor plate against the edge of a stationary floating ice sheet, the model developed here will be for a stationary structure with an ice sheet moving against and past it. The results can be transposed from one case to the other by superimposing a constant velocity without any change in the interaction forces. A schematic sketch for the ice-structure interaction model is shown in Figure 8. The model of a structure consists of a mass m, a spring of stiffness k and a viscous damping element of coefficient c. An ice sheet is assumed to be moving at a constant velocity v, and this is depicted in the model by specifying the velocity of a point in an ice sheet located very far from the indentor. The interaction force generated at the interface depends on the

6.5

~.~~=o X

y-v

Y

IF<,>! -I

i

I Grip does not slip during loading phase, but it slips during other phases.

Figure 8. Schematic sketch of the theoretical model. deformation and the crushing behaviors of ice. As shown in Figure 6, an event during intermittent crushing comprises three phases: (1) Loading phase in which the interaction force increases almost linearly with increase in the relative displacement between the ice and the structure. This phase continues until the ice fails, as indicated by a sudden decrease in interaction force. (2) Extrusion phase in which the structure moves forward at high velocity, extruding the pulverized ice in front of it. (3) Separation phase between the ice and the structtu~ in which the interaction force is zero while the structure executes a transient motion until the gap is closed at the beginning of a new cycle. Although the ice exists at a temperature very close to its melting temperature and the material properties are inelastic and rate dependent, the behavior of an ice sheet is assumed to be elastic during the loading phase. This assumption is supported by the experimental results of Cole [17], who conducted reversed direct stress (tensile-~:ompression) cyclic tests on freshwater ice. For a frequency of cyclic stress greater than I Hz, the stress-strain plots are almost linear with very little hysteresis. When a vertical, rigid indentor indents into an edge of a horizontal ice sheet, the pressure developed at the comers are very high according to elastic theory, as shown in Figure 9. However, such high pressures at the comers would deform the material and relax the stresses. Therefore, the pressure across the interface may be assumed to be uniform. Experimental measurements of effective pressure with a segmented indentor at low indentation velocity by Sodhi [15] support this assumption. From the elastic displacement (u) of the midpoint of an indentor (T'tmoshenko and Goodier [18]), an effective stiffness (ki) of an ice sheet of thickness h loaded along its edge over a width b by a uniform pressure p is given by ki = pb__hh =

u

~Eh

(2ln2_d_d+ l _ v ) b

= E__hh

(1)

[5

where E and v are the elastic modulus and Poisson's ratio (= 0.33) of ice, respectively. A point, located at a distance d from an indentor, is assumed to move at a constant velocity, as shown in Figure 9. For values of d / b = 50 and 500, we get [5 = 3.14 and 4.61, respectively. Bentley et al. [19] obtained values of elastic modulus for freshwater ice grown in the same

66

Figure 9. Plots of the interfacial pressure across the width of an indentor. basin in the range of 1-5 GPa. The stiffness ki can also be obtained from the plots of force vs. indentor displacement. If a value of E = 3 GPa is assumed, the slopes of force vs. indentor displacement plots give an approximate value of 13equal to 10. The fact that the theoretical and experimental values of 13are in the same order of magnitude is encouraging. Inelastic deformation also contributes to a higher value of [3from experimental data. The loading phase ends when the effective pressure reaches a critical value pf at which the ice fails and is incapable of supporting the imposed loads at that instant. The structure begins to move and to extrude the ice in front of it. As can be seen in Figure 6, the interaction force during the extnmion phase is more or less constant, and the relative velocity between the structure and the ice is high. In this model, the extrusion pressure (Pe) is assumed to be constant. The extrusion phase ends when the relative displacement between the structure and the ice sheet reaches a maximum; this happens when the relative velocity between them is equal to zero. If this condition is not satisfied, the interaction does not progress to the next phase, and a situation of continuous brittle crushing is assumed to have been reached. When the relative velocity of the structme with respect to the ice sheet decreases from a positive value to a negative value, the structure separates from the edge of the ice because it

6?

~y, y - x ,

z

Previous Event ~/ v..._L_ ~

Next Event

One Intermittent Crushing Event Loading

Phases:

-

- - I~

v

Extrusion Separation

-i-

i "o ._~ ~) r

o

J

to

tl

i

i2

i=

t5 o

"t3

r

Figure 10. Plots of the ice sheet displacement (y) and the penetration of a structure into an ice sheet (y-x) with respect to time. The prior penetration of structure into the ice sheet is given by a constant z which is incremented at the end of the extrusion phase, when the penetration ( y - x - z ) is a m a x i m u m .

moves faster than the point on the ice sheet which was in contact with the structure. In the separation phase, the structure executes a transient motion until the gap between the structure and the ice sheet is closed for the start of the next crushing event. In Figure 8, the displacements of the ice sheet (y) and the structure (x) are shown with respect to a fixed datum. In Figure 10, the displacement of the ice sheet y and the penetration of the structure into the ice sheet (y-x) are plotted for one event. Because the interaction force depends on relative displacement between the ice and the structure during an event, the penetration distance z during previous crushing events is subtracted from the displacement of the ice sheet in discrete steps after each event as shown in Figure 10. The discrete increment in the value of z is calculated at the end of an extrusion phase when the penetration (y-x-z) reaches a maximum value for that crushing event. As shown in Figure 10, the demarcation between the three phases is given by time instants to, tl, t2 and t3. The equations of motion of the structure during the three phases are (1) L o a d i n g p h a s e ( t o < t
(2)

(2) Extrusion phase (tl < t < t2), m ~ + c Yc + k x = Pe bh for y - :t > 0,

(3)

(3) Separation phase (t2 > t > t3), m Y + c ~ + k x = 0 for y - x - z < 0,

(4)

68 where the dots represent differentiation with respect to time. The conditions for the termination of each phase are indicated. The end of one phase is the start of the next phase. Sometimes, a separation between the structure and the ice may occur during the loading phase. This is taken into account by shifting the governing equation during the loading phase to that in the separation phase until the gap is closed again. Because the termination conditions are implicit, the instant when these conditions are satisfied is found by an iterative procedure, such as interval halving. The solution of these equations depends on the initial conditions of x and :~, which are equal to those at the termination of previous phase. The equations of motion can be rewritten in the following manner by defining elapsed time "~= t - tk, where k = 0,1 and 2, and the solution during each phase can also be written as given below: (1) Loading phase (0 < ~ < tl - to) ~f + 2 ~c COc:t + c0~ x = co2(yo + v~ - z), for ki ( y - x - z ) < pf b h .

(5)

x(l:) = kr(Yo- Zo + v ~ - 2~c v/%~)

+e

;c /o

Zo) c + v(1 (6)

+ e-~cCOcZcos COd'r,[xo-kr(Yo-Zo-2~cV/C0c)]

where I:= t - t o , C0~c= (k + k i ) / m , ~c = c/(2mCOc), COd= 1 / 1 - ~ 2 C0c,c o 2 = k i / m ,

Yo = Ylt=to'ZO= Zlt=to' Xo = XIt=to , YCo = YcI t=to, and

kr = ki/(k + k i).

(2) Extrusion phase (0 < z < t2- tl)

(7)

+ 2~ COn:~ + C02nx = 0C8C02, for y - x > 0

[( I

X(Z) = OC~+ e-r176 z Xl -0c5 cos C0d'~+

l{x,

+ ~r

(x I - 0c5)}sin (0d'r

where "~= t - tl, eL = Pe/Pf, 8 = pf b h / k , O)2n= k / m , ~ = c / ( 2 m COn),COd= 4 1-~ 2 COn, Xl = x lt= tl' and •

= :fit=t1"

(8)

69 (3) Separation phase (0 < 1:< t3- t2) +2~COn;t+0~2nX=0, for x - y - z < O X ('r

(9)

e-~C0n1:IX2 COSCOd~+ O~d (x2 + ~(OnX2)sin Od~]

(10)

where "r = t

t2 -

,

co2

=

k /m

,

~

=

c / ( 2 m COn), Okl

=

~/1 - ~2 (On, X2 = Xl t -t2" and:~2 =xl t =t 2"

AS mentioned earlier, the value of z remains constant during all phases, except at the end of the extrusion phase, when it is incremented by the penetration of the structure into the ice sheet (during the preceding loading and extrusion phase). The model presented here is similar to that of pushing a spring-block system sliding on a frictional surface. Because the sliding or kinetic coefficient of friction is less than that for static, a stick-slip motion results. At higher speeds, stick-slip motion does not take place; instead a steady-state, constant-velocity motion takes place. While the frictional force is always opposite to the d ~ o n of motion, the interaction force between an ice sheet and a structure can create only compressive stresses at the interface during an interaction. The interaction force is zero during the separation phase. Matlock et al. [7] proposed a model for ice-structure interaction in which they incorporated the first and third phase of the interaction presented above. They assumed the ice to be a set of cantilever beams at a certain spacing. Solutions of their model are given by Karr et al. [20]. At times, this model does not produce an interaction force record similar to those obtained from the indentation tests. Moreover, the energy dissipation in their model mostly takes place through the structural damping element, whereas most of the energy is dissipated during the extrusion phase in the present model. Solutions (6, 8 and 10) to equations (5, 7 and 9) are shown in Figures 11-13 to simulate the experimental results shown in Figure 3. In each figure, a phase plot between the normalized structure displacement (x/~)) and the normalized velocity (:~/C0n~) is also shown. The normalizing parameter 6 is the static displacement ( p f b h / k ) of the structure when the effective pressure in the contact area is equal to the failure pressure pf. The dashed horizontal line in each phase plot represents the normalized ice velocity (v/COn6). W h e n t h e phase plot (x - :~ curve) crosses this line, the interaction switches to the separation phase (Fig. 11 and 12); otherwise it remains in the extrusion phase (Fig. 13), simulating continuous crushing shown in Figure 5. Except for a few cycles in the beginning of the simulation, the phase plot remains constant, indicating that a steady-state, stable, cyclic situation is reached. Sometimes, the steady-state condition consists of two crushing events repeated endlessly, as shown in Figure 12. The velocity at which transition from intermittent to continuous crushing takes place depends on many factors, such as ratio (ki/k) of ice stiffness to that of structttre, ice failure pressure pf, ice extrusion pressure Pe and the structural damping and stiffness. Effects of parameters can be found by running simulation of the ice--structure interaction. The follow-

70 3O A

z

20

0

o

lO

10.4 1 10.8 I 11.21 1 1.6I 1 210

0

Time (s)

15 E

"-~(~

-

0:4

0~8

40

1~2

1.6

210

Time (s)

E :

x -20 t -40 | ,

0

~"

014

0.8

300

1.2

1.6

Time (s)

~

y

O"

"

0

0.4

0.8

i.6

i.2

Time (s)

0.3 _

2.0

2.o

_

0 C

8

t.,O v

-0.3

-0.5 -0.25

i

i

0

0.25

i

0.50 x/q5

i

0.75

!

1.00

Figure 11. Results of a simulation depicting one-cycle intermittent crushing: time-history plots (top), and phase plot of the velocity vs. the displacement of the structure (bottom). The following values of parameters and variables were assumed in the simulation: v = 0.15 m s"1, h = 0.03 m, b = 0.1 m, pf = 10 MPa, Pe = 3 MPa, E = 3.5 GPa, ki = Eh/lO = 10.5 M N rn -1, ~ = 0.1, m = 6 0 0 k g , k = 2 M N m -1.

71 3O z

~.

20

(I) rJ

t._

,,o

lO

[

0

9

20 15

0

-

]

]

] -

0.4

0.8 1.2 Time (s)

1.6

2.0

0.4

0.8

1.6

Z0

1".6

:;;.0

g X

0

40

1.2

Time (s)

o

~" - 2 o -40

-

0

014

0.8

400

1.2

Time (s)

E 300

Y

200

'

~.l O O ~ 9

0

-

-

0.4

0.3

~

f

"-

y-x

0

N

0.8

1~2

1.6

2.0

Time (s)

0

-0.3

-0.5 !

-0.25

0

!

0.25

!

0.50 x/8

i

0.75

!

1.00

Figure 12. Results of a simulation depicting two-cyde intermittent crushing: time-history plots (top), and phase plots (bottom). Same simulation as that in Figure 11, except for v = 0.2 m s-1.

?2 30 Z L

o

LL

20 10 0

014

0.8 1~2 Time (s)

1.6

210

0

0.4

0'.8 '1~2 Time (s)

1.6

210

15 E

10

E

5

X

0 --5

40

~

2o

~

0

E x9

-20 --40

0

. . . . . .

0.4

400 EN 300 200

Y / ~ ~ y _ x

~

100 0,

0

0.4-

9

0.8 1~2 Time (s)

1.6

2.0

i.6

;~.0

-~ Z

014

0.8 1".2 Time (s)

P(

0.2

r

3

0

oO v

-0.2

--0,4

i

0

,

!

0.25

0.50 x/6

0.75

Figure 13. Results of a simulation depicting continuous crushing: time-history plots (top), and phase plots (bottom). Same simulation as that in Figure 11, except for v = 0.21 rn s'1.

"73 0.5 B

ki/k 0.4

m

*

10.5

o

5.25

m

Vtr

0.3

9

-

2.625 m

[] 1.0

m

~[o n

9 0.5

0.2 n

0.1

m

J

0 0

I 0.2

i

I 0.4

I

I 0.6

I 0.8

Figure 14. Plots of the transition velocity vs. the damping ratio for different stiffness ratios (ki/k).

ing trends have been found by nmning a few simulations, in which the effective pressure for ice failure (pf) and extrusion (Pe) were constant. As shown in Figure 14, the transition velocity decreases with an increase of structural damping ratio ~ and with an increase of the ratio (ki/k) of effective ice stiffness to structural stiffness. The results shown in Figure 14 are lower than those given by K'Krn/i et al. [21], who postulated that the ice speed at which resonance occurs is almost equal to COn& An explicit expression for the frequency (~ of intermittent crushing cannot be found from the solutions (6, 8 and 10) because the initial conditions of each phase are determined by an iterative process by satisfying a condition at the end of the previous phase. The frequencyfwas determined from the time interval required to execute a steady-state, stable, cyclic event. Sodhi and Nakazawa [22] found that v/fvaries directly with ~, where ~=pfbh/ k) depends on the effective ice pressure, structure width, ice thickness and the stiffness of structure. The parameter v/(f~)) is the ratio of average penetration per cycle to the static deflection of structure at ice failure. Figure 15 shows plots of v/(f~)) vs. ki/k for different structural damping ratios. The structural damping has minor effect on the v/(f3), but the ice-structure stiffness ratio has a significant effect on the frequency.

4. CONCLUSION A brief review of experimental results on edge indentation experiments on floating ice sheets is given. Based on experimental results, a model is developed to describe the ice-

?4

i

I

i

I

i

i

I

i

I

0

9 0.10 o 0.15

[] mz~

9 0.20

0

[] 0.30

V

f~

o

g

9 0.40

~Q

a 0.50

1 m

0

I 0

i 2

!

I

i

I

4

i

6

I 8

i 10

t

ki/k

Figure 15. Plots of v/(f'6) vs. ki/k for various damping. structtwe interaction. Differential equations and their solutions are given for each phase of the interaction model. The model produces force and displacement time history plots that are similar to those obtained from indentation tests. The theoretical model simulates the transition from intermittent cn~hing to continuous crushing. Dependence of the transition velocity and the crashing frequency on different parameters is investigated from the results of model simulation.

REFERENCES

Peyton, H.R. (1968) Sea ice forces. Ice pressures against structures. National Research Council of Canada, Ottawa, Canada, Technical Memorandum 92, pp. 117-123. [2l Blenkarn, K.A. (1970) Measurement and analysis of ice forces on Cook Inlet structures. Proceedings, 2nd Offshore Technology Conference, Houston, Texas, U.S.A., OTC 1261, Vol. II, pp. 36,5-378. [3] M~i~itt'dnen, M. (1987) Ten years of ice-induced vibration isolations in lighthouses. In Proceedings, 6th International Conference on Offshore Mechanics and Arctic Engineering (OMAE), Houston, Texas, Vol. p. 261-266. [4] Engelbrektson, A. (1983) Observations of a resonance vibrating lighthouse structure in moving ice. Proceedings, 7th International Conference on Port and Ocean Engineering in Arctic Conditions (POAC), Helsinki, Finland, Vol. II, pp. 855-964. [51 Nordlund, O.P., ~ / i , T., and J~vinen, E. (1988) Measurements of ice-induced vibrations of channel markers. In Proceedings, 9th IAHR Symposium on Ice, Sapporo, Japan, Vol. 1, p. 537-549.

[1]

75 [61 Jefferies, M.G. and Wright, W.H. (1988) Dynamic response of "Molikpaq" to ice-structure interaction. Proceedings, 7th International Conference on Offshore Mechanics and Arctic Engineering (OMAE), Houston, Texas, USA, Vol. IV, pp. 201-220. [7] Matlock, H., Hawkins, W. and Panak, J. (1971) Analytical model for ice-structure interaction. ASCE Journal of Engineering Mechanics, EM4: 1083-1092. [81 Sodhi, D.S. (1988) Ice-induced vibrations of structures. Proceedings, 9th IAHR International Symposium on Ice, Sapporo, Japan, Vol. II, pp. 62,5-657. [9] M/i~itt'anen, M. (1988) Ice-induced vibrations of structures: Self-excitation. In Proceedings, 9th IAHR Symposium on Ice, Sappom, Japan,, Vol. 2, p. 658-665. [101 Eranti, E. (1992) Dynamic ice-structttre interaction: Theory and applications. Dissertation VTI' Publications 90, Technical Research Center of Finland, Espoo, Finland. [11] Muhonen, A., ~ / i , T. Eranti, E., Riska, K., J/irrinen, E. and Lehmus, E. (1992) Laboratory indentation tests with thick freshwater ice. VTI' Research Notes 1370, Volume I, Technical Research Center of Finland, Espoo, Finland. [12] Sodhi, D.S. (1991) Ice-structure interaction during indentation tests. Proceedings, IUTAM-IAHR Symposium on Ice-Structure Interaction (S. Jones, et al. Ed.). SpringerVerlag, Berlin, pp. 620--640. [13] Sodhi, D.S. (1991) Effective pressures measured during indentation on tests in freshwater ice. In Proceedings, 6th International Cold Regions Engineering Specialty Conference (published by American Society of Civil Engineers, New York N.Y.), Hanover, New Hampshire, February 26-28, p. 619-627. [14] Sodhi, D.S. (1991) Energy exchanges during indentation tests in freshwater ice. Annals of Glaciology 15:2 47-253. [15] Sodhi, D.S. (1992) Ice-structure interaction with segmented indentors. Proceedings, IAHR Ice Symposium 1992, Banff, Alberta, Canada, Vol. 2, pp. 909-929. [16] Joensuu, A. and Riska, K. (1989) J/i~n ja rakenttm v6il6nen Kosketoy (in Finnish), Helsinki University of Technolog~ Laboratory of Naval Architecttwe and Marine Engineering, Espoo, Finland, Report 17-88. [17] Cole, D.M. (1990) Reversed direct-stress testing of ice: initial experimental results and analysis. Cold Regions Science and Technology, 18(3): 303-321. [18] Tunoshenko, S. and CKnxtier,J.N. (1951) Theory of Elasticity. McGraw-HiU Book Company; New York, Second Edition. [19] Bentley, D.L., Dempsey; J.P., Sodhi, D.S. and Wei, Y. (1989) Fracture toughness of columnar freshwater ice from large-scale DCB. Cold Regions Science and Technology; 17: 1-20. [2o] Karr, D.G., Troesch, A.W. and Wingate, W.C. (1992) Nonlinear dynamic response of a simple ice-structure interaction model. Proceedings, 11th International Conference on Offshore Mechanics and Arctic Engineering (OMAE), Calgar~ Alberta, Canada, Vol. IV, pp. 231-237. [21] K'Krn/i, T. and Tunmen, R. (1990) A straightforward technique for analyzing structural response to dynamic ice action. Proceedings. 9th International Conference on Offshore Mechanics and Arctic Engineering, Houston, Texas, Vol. IV, pp. 135-142. [22] Sodhi, D.S. and Nakazawa, N. (1990) Frequency of intermittent crushing during indentation tests. Proceedings, IAHR Ice Symposium, Espoo, Finland, Vol. 3, pp. 277-289.

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 1995 Elsevier Science B.V.

77

Models of ice-structure contact for engineering applications Kaj Riska HELSINKI UNIVERSITY OF TECHNOLOGY Arctic Offshore Research Centre Tietotie 1, 02150 Espoo, Finland

1. INTRODUCTION

In most engineering applications where floating ice cover is encountered, it must be broken. This is the case when a ship proceeds in ice. When ice is forced against offshore structures by environmental driving forces, these are so large that the ice cover is broken. The failure mechanism of the ice cover may be global, i.e. by bending or buckling but the force itself is transmitted through the structure-ice interface. In many cases ice also fails at the interface and consequently the interface is not only transmitting the load but is also modulating it. Especially when inertial forces are involved, the events at the interface have an implication to the structural response. The theoretical modelling of the structure-ice contact interface has mainly concentrated on clarifying the contact load that the edge of ice cover can carry and modelling the ice failure at or beneath the interface. These models mostly have been pragmatic in the sense that the main parameters influencing the load have been included using regression techniques. The concept of interface has, however, a larger context than only the contact and here the restriction is to the phenomena at the contact. Recently it has been realized that the process of ice failure has a bearing on the dynamic response of the structure under the contact load. Some models of contact based on the ice failure process have been developed to answer particular problems such as the ice-induced vibrations. Research on the subject of structure-ice contact has not yet developed a commonly accepted base. Thus there is room for a review article, the aims of which are to describe the different contact models in the context of their backgrounds, and discuss their engineering implications. The structure-ice interaction forms the point of view on the contact models described. Frictional, adhesive and visco-elastic effect are ignored. The main source of material is published literature, but in some cases interpretation of some referenced work is made by the author. These passages are clearly identified for the reader as they may be considered controversial.

78 2. DESCRIPTION OF THE CONTACT PROBLEM

There are two basic cases of ice-structure contact: a mobile structure (ship) in a stationary ice field, and a stationary structure in a moving ice field. The side of a ship is usually inclined but structures in ice often have a vertical face at the water line. Sometimes it is important, due to different failure modes of ice, to distinguish between vertical and inclined structures when modelling the contact. A ship proceeding in level ice breaks the ice by first crushing the ice edge and then bending from the level ice a circular ice floe which turns under the ship. An illustration of ship-ice contact is shown in Fig. 1. where a photograph from a ship bow in level ice of about 30 cm thickness is shown. In the photograph the ship is proceeding down left and the crushed ice coming from the interface is very pronounced. Further, the circumferential and radial crack pattern due to vertical bending forces is clearly visible. When the contact force at the interface increases with increasing indentation, one of these cracks is activated to form an ice floe. The floes break off along the circumferential cracks and that creates a pattern of ice floes along the ship path which is termed the breaking pattern.

Figure 1. A photograph at the bow of a ship breaking level ice.

The above description of ship-ice contact can be used to idealize the failure modes of ice. The primary force on the ice cover comes through the contact, the other main external force being of hydrodynamic origin (bouancy and the added mass). The failure modes of ice edge

79 include the local crushing of ice and ice flaking close to ice edge. The ice cover fails mainly in bending. These are illustrated in Fig. 2. The sequence from the first contact to the formation of the bending crack is controlled by three forces" force to crush ice F c, force to form ice flakes F,, and force to bend ice F b. These depend on the ice thickness, contact width and height, ice tensile and shear strength, S T and S~ respectively and average contact pressure Pay" For a two dimensional case, the three forces are as follows

(i)

where h~ is ice thickness, h e is the contact height and B a typical load width (perpendicular to the plane). These proportionalities contain many assumptions which do not influence the sequence of ice failure. At the first contact, the ice edge is crushed until the contact height and thus the contact force has increased sufficiently that either a flake forms in thicker ice or a bending crack forms in thinner ice. The terms crushing and flaking require further definition; crushing refers to a process in which small ice particles are formed from intact ice and subsequently forced from the contact. Flaking refers to the formation of larger ice pieces due to cracks which run into the ice from the contact. The limiting thickness when flaking starts to dominate over bending is very roughly over 1 m.

Contact load Fc Bending cracks

"-....

Ice crushing" ~

g

Hydrodynamie reaction force

Ice bending

Figure 2. The main forces and icebreaking failure mechanisms: crushing, flaking and bending. The eqs. (1) must be modified if the situation is three dimensional; as in the case when the ship stem is breaking ice. This has a bearing when the total ice forces are investigated, but for the present purposes, the ice structure contact, the 2 and 3D situations are very similar. Briefly, the engineering contact problem is to determine the contact pressure P.~v and the

80 contact area A (in two dimensional case only the contact height he). Numerous experiments have indicated that the average pressure depends on many parameters such as contact area size and shape, indentation rate, ice temperature and salinity. The determination of these dependencies forms the broader contact problem. The other basic case for the ice structure contact is a vertical structure such as a pile against which an ice cover is pushed. This is illustrated in Fig. 3. Ice has been observed to fail in many different failure mechanisms. These include . * . . . . .

crushing micro cracking flaking circumferential cracking radial cracking cleavage cracking buckling.

Here the engineering contact problem is to determine the average contact pressure as the contact area is evidently the structure width by ice thickness. Actually this is not the case but within this geometrical contact area (Dxh~) there are areas of no pressure and areas of very high pressure. Thus the active contact area on which the ice pressure acts must also be determined. It has been observed that crushing and micro cracking occur simultaneously while flaking and micro cracking are exclusive to each other. The other four failure modes are more macroscopic in nature. When the structure is vertical there is no direct interaction between the local ice failure and a global failure.

Figure 3. Ice failure modes when a vertical structure indents level ice (modified from Sodhi [1] and Timco [2]).

81 The context in which any particular contact model is used is important. Engineering design is a synthesis of many particular models. The design may be judged to be as good as the weakest link in the reasoning chain. The modelling of the ice structure interface has implications for designing the local and global structures. Four different systems should be taken into account in establishing the loading on a vertical structure" . ice floe * structure * local structure * failure process.

Basically the modelling of the first three systems in the above list is accomplished by deriving the equations of motion, both for rigid body and elastic motions, for each of them separately. The modelling of the failure process involves basically the establishment of the failure load of the ice edge and its dependency on mainly the indentation rate. Fig. 4. shows schematically these subsystems. If the structure is inclined at the contact, then a fifth system, the global failure of ice sheet, should be taken into account. A wealth of literature exists on the description of structures under loads; the focus here is on modelling the failure process.

ICE

.

I

FLOE

FAILURE PROCESS

mi

F F

I

I

7

LOCAL STRUCTURE

f

STRUCTURE

F

.

ms

=-Us

////////I/////I/I/II//II/I////I/////////////////////

Figure 4. A sketch of the constituents of an ice structure interaction model.

3. MODELS BASED ON AVERAGE CONTACT PRESSURE

3.1. The Korzhavin Model The first decription of the contact pressure assumed it to be proportional to the compressive strength of ice. The model stems from observations of ice forces on bridge piers. The average pressure on the whole geometrical contact area is assumed to be

82 Pjv

= I ( D / h . t ) i n k S c (~;) ,

(2 )

where I(D/h i) is an indentation factor, k is a contact coefficient, and m is a shape factor (Korzhavin [3]). The contact area is termed geometrical as the total contact force is obtained by multiplying the pressure by Dh i. The contact coefficient is one for a perfect contact. The shape factor is taken as 1.0 for a flat structure and 0,9 for a circular cross section. The original Korzhavin formulation included a velocity term, (v/v0) ~3, where v0=l m/s, but usually the velocity is included in the strain rate dependency of the compressive strength of ice. A reference strain rate is defined for this purpose as E=v/4D (Michel & Toussaint [4]). The indentation factor accounts for the three dimensional nature of the stress in ice. Originally, Korzhavin related the indentation factor to the ratio of the width of the impacting ice floe and structural diameter D, this factor being 1 for narrow structures (wide ice sheets) and 2,5 for wide structures. Subsequent tests have suggested that the indentation factor depends rather on the aspect ratio D / h i. A widely accepted empirical formula is (Afanasev et al. [5])

x=

hi s--fi-+1,

D -~>1.

(3)

The Korzhavin formulation focused the attention of research on the indentation speed dependency and the aspect ratio effect. Less attention was paid to the indentation process which was assumed tacitly to proceed continuously at a constant force level provided that the indentation speed remained constant. A major assumption in treating the contact is that the structure is vertical. With this assumption the contact area is intuitively clear. Only recent investigations have questioned this. The strength of the Korzhavin equation is its wide empirical foundation. This makes it amenable to design purposes; it is also adopted in many standards (see e.g. API 1988 [6]). Theoretically, however, the use of Eq. (2) amounts to curve fitting in which the contact pressure has been made dimensionless using the compressive strength of ice. Some theoretical justification for this has been developed by investigating the indentation factor.

3.2. Theoretical Modifications of the Korzhavin Equation The indentation factor I has the main influence on the contact pressure once the contact for a flat indentor is perfect (k=l). It is clear that the indentation factor may then be one but values higher than one require some justification. These have been justified theoretically using the upper and lower bound theorems from plasticity analysis (Croasdale et al. [7]). To recall, the lower bound theorem makes use of a failure criterion and admissible stress state in ice while the upper bound theorem considers the velocity field of plastic deformation and equates the rate of external work to the rate of internal energy dissipation (Calladine [8]). The problem reduces now to finding an equilibrium stress state, which maximises the contact pressure and a velocity field which minimises it. Two main studies of the indentation factor using the plasticity theory have been performed. Croasdale et al. [7] used the Tresca failure criterion applicable, as they stated, for granular

83

and thus isotropic ice, and a very simple stress distribution (zero everywhere else than in a strip behind the indentor). The lower bound value of the indentation factor thus obtained is I=1 for all aspect ratios. The assumed velocity field resembles that due to flaking failure. The upper bound solution has a minimum for very small aspect ratios giving an indentation factor 1=2,6. From this cut off value the indentation factor decreases asymptotically towards one for a smooth indentor. Ralston [9] applied a modified von Mises failure criterion applicable for columnar grained ice and a more elaborate stress field. The modified failure criterion used was

f(oiJ)

ax [ (Oy-O'r) 2+(O'r-Oz) 21 +a3 (~176

-

(4,)

+~', ('~,.+~L) +a,~+a~ (o.+o,.) +~,,o. where a6=2(a~+2a3). The failure criterion above is given in a Cartesian coordinate system in which the xy-plane is the plane of the ice sheet and z-direction is vertical upwards. The failure occurs when f(cy~j)=l and stress states for which f(o~j)>l cannot be sustained. This macroscopic failure criterion does not warrant much attention as such mainly because it does not describe the failure process. It is given here as it served as the starting point for more elaborate failure criteria developed and discussed later. The velocity field used by Ralston was influenced by the early investigation of indentation by Hirayama et al. [10]. It is basically two dimensional in the plane of the ice cover. The indentation factor values obtained by upper and lower bound solutions are very close to each other: about 4 for small aspect ratios and decreasing towards 3 for larger aspect ratios. The use of plasticity analysis to determine the indentation factor provides a theoretical support for the empirical indentation factors like Eq: (3). This analysis also gives the maximum possible indentation factors reached with small aspect ratio. The drawback of the plasticity analysis is also evident; it is that ice is assumed to fail in a plastic fashion. Tests have shown that ice failure resembles plastic yielding at lower indentation rates, but at the same time, at lower rates the viscous behaviour of ice becomes more pronounced. The plasticity analysis gave a pragmatic justification for indentation factors but did not give more insight about the contact. It should be noted further that plasticity solutions are relatively easy to use in the two dimensional case where the structure is vertical. The next step in the development of contact models was to use the yield or failure criterion in conjunction with a correct stress field. But before discussing these developments the scale dependency of the average ice pressure should be introduced.

3.3. The Pressure-Area Curve The average ice pressure has been observed to decrease with increasing geometrical or rather apparent contact area. This observation holds true for very different contact geometries as shown in Fig. 5. Here it is important to distinguish between the actual contact area i.e. the area on which there is pressure, and the apparent contact area which is the area determined by the indentation depth and overall geometry of the ice and the structure at the contact. The apparent contact areas are indicated in Fig. 5. The average ice pressure given in Fig. 5. is obtained by dividing the measured total normal force by the apparent contact

84

area. When this is done in a wide variety of tests, a similar drop in pressure is obtained (Sanderson [11]). Another alternative to define the area is to use force gauges which have a different active area. This is done mainly in ice load measurements onboard ships. In this way the pressure drop with increasing gauge area is also noticed (Kuiala & Vuorio [121~

"

1

r--!

n

10 8

=

6

Q.

4

*J,

_

~,

-llr,g.

j

9

1

q9 O0

~0

o

.~

2

x~

~

<~9

!

lk"

TESTCONFIGURATION ~tr Ship bow penetrati~ a n~tiyear ice ric~ - Inclined I~ane crushing an ice edge 1

--

~

0.01

0.1

Contact

area

9

l

1

10

A-C.x

~-~>

[m 2]

Figure 5. The measured average contact pressures from two tests the geometries of which are also shown (Riska [13]). The plot in Fig. 5. has been done on a log-log scale. This is the common practice, as the way to describe the area dependency of ice pressure is to determine the exponent according to which the pressure depends on area in a power law. The exponents observed in different plots have varied between -0,2 to -0,5. The limits for the exponent are 0, corresponding to a constant pressure, a n d - 1 , corresponding to a constant point load. The range of contact area in different structure-ice interaction scenarios is from very small areas to about 10 m 2. Even in this range many diverse factors contribute. One factor arises from the definition of the area; the geometrical contact area could contain areas that are not loaded. This is suggested by pressure measurements with gauges having a diameter of 2 cm. These give pressures beyond 70 MPa (Frederking et al. [14]). A variant of this reasoning is the concept of contact zone according to which the contact is localized at any given time to a restricted area (Kry [15]). This idea is also incorporated tacitly in the Korzhavin formulation where the contact factor k could be varied with contact width. Another factor contributing to the pressure decrease with increasing areas is the Weibull-effect. Originally, the reasoning of Weibull was that larger volumes are more likely to contain a 'weak link' and thus larger test pieces are weaker in tension (Weibull [16]). The same logic may be extended to compressive failure and for area instead of volume, see e.g. Sanderson [11]. Basically the pressure-area curve is empirical. At best it gives some hints about the physical processes active at the contact during indentation. Its applicability lies in the design

85

of structural components. Each structural component (plating, frames, webs etc.) obtain their loading from a certain area determined by the spans and spacings of the elements. The pressure-area curve assists in determining the relevant loading for each of the structural elements. This practice has been adopted in many standards, e.g. Finnish-Swedish Ice Class Rules (Finnish Board of Navigation 1985 [17]) and Canadian Standards Association (1989) [18]. In the latter the pressure exponent adopted is -0,5.

3.4. The Concept of Nominal Ice Pressure When the body impacting an ice edge is inclined, such as a ship hull, the apparent or geometrical contact area is not constant during the indentation. The solution of the contact problem now includes both the contact pressure and the contact area. The total normal force Fn=PavA is the quantity which governs the development of the contact and the maximum penetration through the equations of motion of the ship and ice. The solution of the maximum penetration involves either the study of the dynamic bending of the ice cover, or solution of the motions of the structure and ice feature. For the solution of the motions the time trace of the load, including contact loads, is needed. The concept of nominal pressure was developed for this reason (Varsta [19]). The idea was developed by conducting two sets of indentation tests: one with water in the contact zone and the other without. The argument is now that the average pressure consists of two contributions: one is from solid ice-structure contact (dry contact, subscript c), and the other from contact in which there is a mixture of crushed ice and water between the intact solid ice and the structure (wet contact, subscript w). The average pressure is then assumed to be

where the empirical weighting factors kgw and kgc take into account the decrease of the average pressure when the indentation proceeds. The sum of these is always less than one. These include essentially the pressure-area effect and the redistribution of the contact types during the indentation. The theoretical coefficients kf account for the variation of pressure due to different relative stiffnesses in the structure. These are defined after the nominal pressures P,,w and P,,c are defined. The nominal pressure for the 'wet' case is defined to be the uniform pressure on the apparent contact area which produces a stress state in ice for which the macroscopic failure criterion gives value f((Yij)=l. A macroscopic failure criterion was developed for this purpose (Riska [20]). It is the modified Tsai-Wu criterion 3

3

i,J=l

i,J,k,l=l

which is a general power series up to the second power. The failure constants F~j and Gijkl may be determined from uniaxial tensile and compressive tests and some bi- or triaxial tests, e.g. (Frederking [21]). There are only seven nonzero constants for columnar grained ice and three for granular, isotropic ice.

86

The definition of the 'dry' pressure relies on the stress distribution due to a uniform normal forced displacement on the contact surface. As the calculated pressure rises to infinity at the comers of the contact area, some empirical reasoning must be used to determine the stress state at the point which is used for failure analysis (see Varsta [19], pp. 37-8). This is not elaborated upon here, because the contribution of the 'dry' contact is important only in the very beginning of the indentation and then goes to zero i.e. k~c--.0 with increasing indentation and k~w becomes the normalized pressure-area coefficient. The concepts of 'dry' and 'wet' pressure are illustrated in Fig. 6.

,,"

a) wet contact

!

///' 7'/

9

b) dry contact

Figure 6. Definition of 'wet' and 'dry' contact (Varsta [19]).

The calculation of nominal pressures requires the determination of the stress field and the application of the failure criterion (6). In most cases the contact geometry is so complicated that numerical methods, mainly the finite element method, must be used to determine the stress field. Fig. 7 presents the two dimensional finite element mesh used to determine the nominal pressures for a case where the structure (ship hull) is inclined 45 ~ towards vertical. These calculations were extended to three dimensions by defining a constraint coefficient C c. It gives the stress Oy so that Cc=0 corresponds to a plane stress state (i.e. O'y--.-~0) and Cc= 1 corresponds to a plane strain state. The results of calculations are shown with different inclination angles and Cc=0,8 in Fig. 8. The pressure through a layer of crushed ice, the 'wet' contact, seems to correlate well with the full scale measurements. This indicates that the force from the direct ice-structure contact is small compared to the uniform pressure distribution created by the crushed ice between the structure and intact ice. This conclusion will be examined later.

87 <~

Z

--

It

-

-

7

Figure 7. A finite element mesh to calculate the nominal pressures (Varsta [19]).

N~

15 " , , ~ ~

SHIRT TERM

+ 16RIM

T.--:I'C

O SlSU

T.=-3"C

,..max i.Jnw [MN/m ~]

T~['c] 10

-I 5 -1 |

-,<

~

~ -6 -3

-2 -1 |

10"

20"

30"

40"

50"

/3n Figure 8. The nominal 'wet' pressure presented together with some full scale values as a function of the inclination angle (Varsta [19]).

The long term results with IB Sisu in Fig. 8. are from (Vuorio et al. [22]), MT Igrim results are from (Korri et Varsta [23]), and IB Sisu short term results are from (Riska et al. [24]). The ice pressure was measured with a gauge of 20 cm diameter. The formulation in eq. (5) contains a factor which takes into account the relative flexibility of the indenting structure. There is a difference in local flexibility within the loaded area of

88

the structure if it is made of plate and frames because the plate deflects more under the load. The flexibility coefficients may be defined once it is assumed that the contact situation corresponds to the situation where a plate is forced onto a half space by line loads representing the frames. This situation results in a contact pressure distribution between the plate and the half space dependent on the characteristics of the half space. The situation is illustrated in Fig. 9. An elastic half space corresponds to the direct contact and a Winkler half space may be taken to correspond to the 'wet' contact.

F

F

F

J Ship hull . -

.

.

.

.

. .

.

.

.

.

-

.

H

, -

~

/ / S h i p hun ~__ x

iC3shed ICE

1~= b. p , ,

Figure 9. The contact of ice and a framed structure and the resulting contact pressure distribution.

The plate on an elastic foundation may be analyzed numerically, but the case of contact through crushed ice can be solved explicitly by assuming the contact to be width much wider than s, the frame spacing, and the ice (i.e. the Winkler half space) to be in a plane strain state. The derivation is shown here as it is an extension of the analysis in (Varsta [19], pp. 50-53). First consider an infinite plate with one frame i.e. the plate is loaded along the y-axis by a line force F. Then the pressure between the plate and ice is (Selvadurai [25], p. 51)

q(x)

F-[k coB_~_Xd~ -~ ~o -fi ~,

(7)

+_

D

where k is the foundation modulus, D=Et3/12(1-v2), t is the plate thickness and E and v are the elastic modulus and Poisson ratio of the plate, respectively. This expression may be integrated to give

89

1 -c~ = ~ F e (c o s ~ x + s i n ~ x )

q(x)

,

(8)

c~='~. The method of superposition gives now the final pressure distribution as mid

m

(9)

q~o~ (x) = q(x) + E q(x+ns) + ~ q(ns-x) n=l

/2=1

which expression can, by evaluating the series in closed form, be developed to qeoe (x)

=

P a v S ~ [BIc O s C a x c O s h ~ x + B a s i n ~ x s i 9.

+e -~x ( s i n ~ x + c o s ~ x ) sinh~s+sin~s BI = cosh~s-cos~s

Ba=

sin~s-sinh~s

r~x

], (10)

-i +i.

cosh~s-cos~8

If the maximum pressure is identified with the nominal 'wet' pressure, then the average pressure is obtained from (10), assuming that the contribution from direct contact is negligible, as kf w _

Paw

Pn, w

_

2

~s

cosh~s-cos~s

.

(ii)

sinh~s+sin~s

The parameter C 2 is proportional to t 3/4 with the proportionality constant dependent on the foundation modulus k. This may be obtained empirically or numerically, the value being of order 10 GN/m 3 (Varsta [19]). Thus, for normal frame spacings and plate thicknesses, the constant C2s is between 3 and 4 and the flexibility factor is about kfw=0,7. This decrease in pressure under plating compared to frames has been observed in some tests, see e.g. (Lindholm et al. [26], Watanabe et al. [27]). The Finnish-Swedish Ice Class Rules have adopted this formulation; the uniform design pressure used for frames is reduced for plating by 25 % disregarding at the same time the pressure shape. The empirical pressure-area curve and the nominal ice pressure concept may be combined to give a general expression for ice pressure (cf. Riska et Frederking [28])

p..- k ~ ( ~ A) opo..,

(12)

where A0=l m 2. The contact coefficient k and the area exponent c are still empirical constants. The value of these have been observed to be 0,1...0,3 and -0,3...-0,5 respectively. Further, it must be assumed that if the structure does not include any local stiffness variations i.e. it is uniformly stiff, then kfw-1.

90 The drawback of models which use a uniform nominal contact pressure is that they do not account for the ice failure mechanism. Consequently they contain many empirical constants the range of validity of which is restricted to the neighbourhood in parameter space of the testing situation. A further problem is that the array of major parameters influencing the contact pressure is not yet clear. The temperature and salinity of ice, contact area and aspect ratio together with the indentation speed are the main ones. The reliability of the average ice pressure formulations would be much improved if the physics of ice failure could be modelled even in a rudimentary way. More phenemenological models for the ice-structure contact are described in another chapter. Actually, only now is the contact given a proper attention. So far it has only been a geometrical concept.

4. MODELS OF THE CONTACT PROCESS

4.1 Introduction A multitude of experiments have shown that the ice in contact with a structure can fail in many ways, (see Fig. 3.). For vertical structures the major failure modes are, if the formation of occasional major cracks is ignored, crushing accompanied by microcracking, flake formation and an intermediate mode where both flaking and crushing with some micro cracking are present. The micro cracking is usually noticed when ice gets a whitish appearance due to small cracks of about the size of ice grains. The indentation speed has been proposed as the major parameter controlling these failure modes (Saeki & Ozaki [29], Sodhi [30]). More precisely, the situation seems to be such that the clear flaking situation is rare. With increasing indentation speed an intermediate failure type develops as shown in Fig. 10 (Saeki & Ozaki [29], Fransson [31]). The area where ice is crushed narrows and small flakes start to form; a flaking failure is reached eventually in high indentation speeds. Saeki & Ozaki [29] indicate further that the compressive strength of ice influences the amount of flaking. As the variation in compressive strength in their tests resulted from a variation in ice temperature, it may be conjectured that the brittleness of ice influences the amount of flaking.

\

[L_~. C~ing

Int~tiate

la.~ng

Figure 10. A sketch of the cross section through ice illustrating the crushing, intermediate and flaking failure (modified from Saeki & Ozaki [29]).

91 The mode of failure is naturally crucial in modelling the ice failure. It also has an influence on the average pressures as Fig. 11. shows. The low average pressures at higher indentation speeds (above 1 mm/s) where the flaking failure starts to dominate has been observed in laboratory tests (Riska et al. [32], Tuhkuri [33]). Ice pressure measurements with ships, where the average pressure has been deduced from a force on a certain gauge area, do not show speed dependency. One reason for this may be that the strength of ice cover in bending increases with increasing indentation speed due to hydrodynamic effects. These and many other observations have focused the interest in modelling on the crushing and the flaking failure modes. These models are described in this chapter.

9 Fresh water Ice o Sea Ice

o a. 10

E

8

w W

e 6

o

9

E

0 9

0

9 4 B

I11

.,, ~u

C)

9m~

0

..,

0

0,1

I

10

100

eml

- 9

9 ~=== =,1

9

talk 99 9 ,m-

.

== 9

9 -

1000

Indentation s p e e d [ m m l s l

Figure 11. The measured average ice pressures versus indentation speed using fresh water and sea ice (modified from Saeki & Ozaki [29] and Sodhi [30]).

4.2. Modelling the crushed ice

The visual outcome of a crushing failure is fine powdered ice on top of the ice sheet. The crushed ice must be produced somewhere; if it is assumed to be produced at the boundary between solid and crushed ice then the contact situation may be thought to be composed of a layer of crushed ice between the structure and solid ice as Fig. 12 shows. The concept of crushed ice layer together with a clear boundary between it and the solid ice can be traced to Lavrov [34]. The treatment of this problem here follows closely the presentation of Popov et al. [35]. The crushed ice is assumed to be a viscous incompressible fluid with viscosity ILl and thus its flow can be analyzed with the Navier-Stokes equation. The analysis is restricted to a planar case. If the inertial terms are neglected, the flow velocity in the z-direction is assumed to be negligible in the thin layer and Uxx is negligible compared to uzz, where u is the flow velocity in the x-direction and the subscripts refer to partial derivatives. Then the

92

Figure 12. The layer of crushed ice and the coordinate system used in analysis.

Reynolds equation for pressure in a squeeze film is obtained

aid ax

~U

(13)

- ~ ---~" az

The other equation needed is the continuity equation. If it is assumed that crushed ice is created at the solid ice boundary at a volume rate w then the equation of continuity or rather the conservation of mass equation is

w-H-

(14)

-~

If the thickness of the layer of crushed ice is constant then these equations can be solved for pressure as

W-/'I p ( x)

h~ _xa)

= 6 p --~s o ( - ~

( 15 ) .

The pressure equation may be considered as an equation for the thickness of the crushed ice by calculating the average pressure and equating it with the average pressure given by eq. (12): S

+ ~"c

c-2n

~'n,W~

3

= V,

(16)

where it has been assumed that w is equal to the indentation rate v. The constant C contains the obvious empirical parameters. This equation may be solved and used together with the equations of motion for the structure and ice in structure-ice interaction simulation. This has

93 not been done, mainly because the whole derivation contains many assumptions and still the description of the failure mechanism i.e. production of crushed ice is not based on physics. Despite this the application of this formulation where the layer thickness remains constant and the layer thickness and the viscosity are determined by experiments, e.g. Kheisin et Likhomanov [36], is used in many cases (Jordaan et al. [37]). A case where the layer thickness is not known a priori was analyzed by Kurdjumov & Kheisin [38]. One additional equation is needed and they assumed that p--H adding one more empirical constant. The form of the resulting pressure distribution is similar to the fourth root of eq. (15). The squeeze film has also been analyzed assuming the crushed ice to behave as a granular cohesive material obeying the Mohr-Coulomb yield criterion (Hallam & Pickering [39], Savage et al. [40]). The analysis is based on the depth-averaged equation of continuity (cf. Eq. (14)) and equations for conservation of linear momentum for an incompressible continuum (cf. Savage et al. [40]). The derivation of the contact pressure is quite lengthy and it suffices here to state that the resulting pressure on the structure is of form p(x)

- A~e - ~ ,

t 17 )

where the constants A 1 and A 2 contain material and geometrical constants. The solution (17) is very sensitive to the friction between the crushed ice and the structure or the solid ice boundary. If these planes are assumed smooth then the pressure distribution gets closer to that given in Eq. (15). The assumption of viscosity or Mohr-Coulomb behaviour of the crushed ice has been investigated experimentally. Experiments have shown that the pressure distribution in a squeeze film of crushed ice between two steel platens is concave (Sayed & Frederking [41 ]) and comparable to the distribution given by Eq. (17) but opposed to that given by eq. (15) which gives a convex distribution. The influence of the assumptions made in deriving the pressure distributions should, however, be clarified before any conclusions about the nature of crushed ice are made. The analysis of crushed ice assumes a clear boundary between the crushed ice and solid ice. Earlier it has been stated that the crushing type of failure is accompanied by extensive micro-cracking which gets less intense further away from the contact. The formation of stable cracks under increased loading has been long investigated (Gold [42]). When these results were used together with the theory of damage mechanics (see e.g. Krajcinovic [43]) a useful concept to analyse the microcracking medium was developed. In simple terms the theory defines a damage quantity, say fL the value of which is zero for an uncracked medium and one for a 'totally cracked' i.e. failed medium. For more details, see e.g. Santaoja [44] or Sjrlind [45]. The damage attains the value one at the crushed ice - micro cracked ice boundary and away from the boundary it decreases to zero. Combining the analysis of the crushed ice and the analysis of the micro cracked ice seems to provide a complete solution for the contact problem. There are, however, still problems. One is that for vertical structures this kind of modelling would produce a fairly constant force. This contradicts experiments where the force trace is usually composed of triangular shaped pulses. Another problem is that such analysis requires many empirical constants, the measurement of which is not a trivial matter.

94 4.3. The concept of finite crushing depth The observations made in the 1960's of ice failure against Cook Inlet offshore structures led to the conclusion that ice, when reaching failure, fails a finite depth into the ice (Matlock [46]). This assumption leads to the saw tooth type force trace and explains the resulting vibrations in the structures. A typical saw tooth type force trace from a laboratory test is presented in Fig. 13. Even though the original authors conceived the reason for the finite failure depth to be some kind of flaking, it has been applied in contact models where crushing and micro cracking dominates (Jordaan et al. [47], Comfort et al. [48], K~irn~i [49]). The sequence of indentation is, according to the finite failure depth concept, the following: The drop in the force trace corresponds to a jump of the crushed ice - micro cracked ice boundary some distance into the ice. After the jump when the indentation proceeds, the crushed ice is forced out from the contact and the force increases. When the average pressure corresponding to failure is reached, or when the damage reaches unity, ice fails again a finite distance into the ice and the cycle is repeated.

v=-50 nun/s, b=115 mm, M=2000 kg, k=20 kN/mm 240 200 160 ~

i

120

l

I

,Ji

! I..A ,Jl,/i i. }

Jjj/tAt ,.JJlll,,,JA, J IINIA,Illlildll t:1 I1t ,N IiIJ1114111141t f A/I/l/ill/l, AA/i,

~/v'Vi'1pv

y v ~lV~Vl I V~vl I V'i'vv'vvvv'~V

0

-80 0

1

2

3

4

t

5

6

7

(s)

Figure 13. A typical force trace from tests with vertical indentor (Muhonen & al. [50]).

The attraction of the finite failure depth concept is that it explains the observed heavy vibrations encountered in offshore structures. Once the magnitude of the jump of the boundary is tuned for a specific structure, vibrations result. The drawback of the finite failure depth concept is that the physical justification of the sudden failure of ice is not rigorous. Some kind of stick-slip mechanism has been suggested (Jordaan et al. [47]). There are other explanations for the vibrations. One is that the ice compressive strength in the Korzhavin ice pressure model is dependent on the indentation speed. Once the strength decreases with increasing speed this introduces a negative damping type of term in the equations of motion and leads to an instability (M~i~itt~inen [51]). Naturally the flaking model also gives rise to vibrations. These kinds of models are examined in the next section.

95 4.4. Flaking models The flaking type of ice failure (cf. Fig. 10) is commonly associated with shear cracks emanating from the contact zone. This conclusion was reached from observations of ice failure against Cook Inlet offshore structures (Matlock & al. [46]) and also from early laboratory tests (Croasdale et al. [7], Varsta et Riska [52]). However, the analysis of the macroscopic shear crack is indirect because the shear strength of ice is difficult to measure directly. Commonly it is determined using uniaxial strength results and from them calculating the shear strength using some failure criterion. Moreover, the analysis of one shear crack does not lead to a model of the contact process. The formation of one shear crack leads to a decrease in the contact height and thus in the contact force. If the contact pressure is taken as given by the Korzhavin equation, the average shear stress along any possible shear plane exceeds the shear strength only when the contact height is about the ice thickness. It required an interpretation of laboratory indentation tests and realization that local contact pressures may be high, up to 70 MPa (Frederking et al. [14]), to proceed in modelling the flaking process. The interpretation of laboratory indentation or rather crushing tests, is shown in Fig. 14 (Daley [53]). Ice specimens of a triangular shape were crushed in the tests. The flakes produced were shell shaped. Daley [53,54] idealized the process as a two dimensional one assuming the crack initiation be governed by the average shear stress on a straight plane from the middle of the contact. When this shear stress exceeds the shear strength given by the Coulomb failure criterion a flake forms. The contact sequence is now basically that after the formation of a shear crack the contact force is halved and then ice is again crushed as the penetration proceeds until the shear strength is exceeded again. A typical result from the contact process modelling is shown in Fig. 15.

nextf l ~

~ i o u s flakes

Figure 14. A sketch of shear flaking geometry as deduced from test of Joensuu et Riska [55], Daley [53]. The modelling of contact process was made practical by the conclusion that according to the two dimensional Coulomb failure criterion

96

1

Sc-S________~T

where z and p are the shear stress and normal stress in the shear plane and c and ~ are material parameters, there exists preferable failure planes. These are dependent on the edge at which the failure plane ends. If the ice piece edge geometry consists of line segments at angle 0 to horizontal (see Fig.15) then the direction of the failure plane is given by angle (cf. Fig. 15) which is -

n-i-A ---,12

(19)

During the process simulation it remains only to follow the force to initiate a failure plane to each edge of the ice specimen. When the shear stress is exceeded a flake develops.

force

flake crack

direct contact zone

Figure 15. Typical flaking geometry and force trace produced by the flaking model of Daley [53,54].

The simulations of the flaking process were shown to be sensitive to the ratio between the shear strength and the crushing pressure, k. The basic assumption was that the crushing pressure does not follow the macroscopic failure criterion but must be determined separately. The influence of k is illustrated in Fig. 16. When the shear strength and crushing pressure

97 are about the same then only a crushing type of failure results. Only when the crushing pressure is assumed to be high, flaking occurs. Daley [53] identified the crushing pressure with the pressure to create the phase change of ice from solid to liquid (about 100 MPa at 10~ This assumption has some evidence, Gagnon & Molgaard [56], Gagnon & Sinha [57], and is made plausible by the high local contact pressures measured (Frederking et al. [14], Joensuu et Riska [55]). The analysis proved further that the flaking process is chaotic. The controlling parameter for the chaotic behaviour is k, the shear to crushing strength ratio. This conclusion has repercussions to analysis of indentation and crushing tests and also for analysis of dynamic response under contact ice loads.

Figure 16. The influence of the parameter k on the flaking process Daley [54].

The flaking process, irrespectively if it is considered two or three dimensional, leads to a very narrow contact where the pressure is high. This kind of contact line has been observed visually in laboratory tests (Joensuu et Riska [55], Fransson et al. [58]) and in full scale icebreaker tests (Riska et al. [32]). The contact line provides an explanation to the pressurearea relationship. If the area on which the pressure is applied is exactly line-like (i.e. with

98

no width) along the edge of the level ice and the contact pressure is constant, say q, along the line, then the force along the edge is F=qD and the average pressure on the geometrical contact area would be Paw

F A

-

(20)

D q - --,q i f D - h ~ . Dh I

-

Thus for narrow structures the area exponent would be -0,5 i.e. the same as the exponent for pressure in eq. (12). Increasing the width of the contact line increases the exponent towards zero. This observation is illustrated in Fig. 17.

..............................................................................~........................................~.........................................) ................................................................................~.......................................i........................................ ................................................................................~........................................~.......................................

1 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-

.

.

.

.

.

.

.

-

...........................................................contact model for test 76 .............................................

pressure (MPa)

......................................i~.

~li'ii'"'/~ -X.al, Varsta

i ~[~' .......................................).......................................r ~

1

o.oool

..................)

pressure-areamodel fitted byRnska

o

o.ool

O.Ol area (m)

i [

i

o.1

1

Figure 17. Average contact pressure presented versus the apparent contact area from flaking simulation compared with data from Varsta [19] and Riska [131 Daley [53]. The above flaking model contains many assumptions, some of which influence the resulting process only quantitatively. One such is the assumption of the straight failure plane. This assumption was investigated by Kujala [59] and the force to produce failure along a curved failure plane was shown to be somewhat lower than that to produce a straight failure plane. The assumption that the plane originates from the center of the contact could be improved especially in view of the intermediate type of failure which combines the crushing and flaking modes. Further, the treatment of cracks as two dimensional should be improved and the use fracture mechanics instead of a macroscopic failure criterion studied.

99

5. CONCLUSION

The contact processes between a structure and a level ice sheet were divided into three types according to the ice failure type. These are the flaking, crushing accompanied with micro cracking and intermediate failure type combining the two first. That these types include all the failure modes of ice edge is not clear. Assuming so it would appear that once the model for crushing; extrusion of crushed ice accompanied with the damage modelling, and flaking; shear flakes and high contact pressure, is developed only refinements to these models are needed. This is, however, not the case. For one thing the analysis of cracks, crushed ice and the origin of crushed ice must be made physically rigorous. At present the analysis of flake formation, properties of crushed ice and the transition from damaged to crushed ice may be considered mainly empirical. This status of research is reflected in that the parameters controlling the failure type are obscure. There are three parameters commonly offered to explain the failure type. These are the speed of indentation v, aspect ratio of contact D/h~ and the apparent strain rate v/4D but the physical phenomena behind these parameters are not clear. One suggestion is that the brittleness of ice governs the failure type. Ice behaves in a more brittle manner at lower temperatures and higher actual strain rates. Thus the indentation speed and ice temperature should have a pronounced effect on the failure type. The influence of the contact width obscures this conclusion. The contact width may influence the average pressure through three dimensional effects, but it should not influence the failure type. Another open question is how to treat the intermediate type of failure in which both the flaking and crushing (extrusion & micro cracking) behaviour are present. In the intermediate type ice failure mode the flakes form at the edges i.e. closer to the top and bottom of the ice sheet. What is the parameter which controls where the flake forms is not known. At present we are, however, in a fortunate situation in that many indentation test series have been conducted in a careful manner. The reporting of these is also in the public domain. These results could serve as a spark to more physical modelling of structure-ice contact and verification of these models. The present review of ice-structure interfaces has concentrated on modelling the contact pressure between a flat but not necessarily vertical structure and level ice sheet. This sharp focus was necessitated by space limitation combined with the need to describe the models quite thoroughly. Consequently many interesting topics were not touched upon. The influence of a curved (circular) waterline cross sections was not discussed even though it may force three dimensional flaking. The ice type was limited to level ice thus leaving out the contact with rubbled ice or the consolidated layer in an ice ridge. Especially when ice is crushed, interesting frictional phenomena could occur. The aim of the author was to provide a review which touches the relevant developments in modelling the contact without drowning the reader in a wealth of details.

100 ACKNOWLEDGEMENTS This review reflects the research carried out about the ice-structure contact in the Helsinki University of Technology. The wealth of new results would not have been possible without an encouraging environment created by eagar young scientists. This environment made this review possible. Further, the author would like to thank the careful proof reading and comments of Mr. Brian Veitch and the fruitful comments of Mr. Jukka Tuhkuri and Prof. Petri Varsta.

REFERENCES 1. Sodhi, D. 1992. Ice-Structure Interaction with Segmented Indentors. Proc. of the IAHR Ice Symp., Banff, Canada, pp. 909-929. 2. Timco, G. 1986. Indentation and Penetration of Edge-Loaded Freshwater Ice Sheets in the Brittle Range. Proc. of the 5th Int. Symp. on OMAE, Tokyo, Japan, Vol. IV, pp. 444452. 3. Korzhavin, K.N. 1962. Action of Ice on Engineering Structures. USSR Acad. of Sci. Siberian Branch. CRREL Draft Translation No. 260, Hanover, USA, 1971. 4. Michel, B. et Toussaint, N. 1977. Mechanisms and Theory of Indentation of Ice Plates. J. of Glaciol., 19(1977)81, pp. 285-301. 5. Afanasev, V.P. & Dolgopolov, Y.V. & Shraishtein, Z.I. 1971. Ice Pressure on Individual Marine Structures. In 'Ice Physics and Ice Engineering', Israel Program for Scientific Translation, 1973, pp. 53-68. 6. API: American Petroleum Institute, 1988. Recommended Practice for Planning, Designing, and Constructing Fixed Offshore Structures in Ice Environments, RP 2N, Dallas, Texas, USA, 50 p. 7. Croasdale, K. et Morgenstern, N. et Nuttall, J. 1977. Indentation Tests to Investigate Ice Pressure on Vertical Piers. J. of Glaciol., 19(1977)81, pp. 301-312. 8. Calladine, C. 1969. Engineering Plasticity. Pergamon Press, Oxford, UK. 9. Ralston, R. 1978. An Analysis of Ice Sheet Indentation. Proc. of the 3rd IAHR Ice Symp. Lulefi, Sweden, Vol. 1, pp. 15-31. 10. Hirayama, K.-I. & Schwarz, J. & Wu, H.C. 1974. An Investigation of Ice Forces on Vertical Structures. Iowa Inst. of Hydraulic Research, Report No. 158, University of Iowa, Iowa, USA, 153 p. 11. Sanderson, T.J.O. 1986. A Pressure-Area Curve for Ice. Proc. of the 8th IAHR Ice Symposium, Iowa City, Iowa, Vol. 2, pp. 361-84. 12. Kujala, P. & Vuorio, J. 1989. Results and Statistical Analysis of Ice Load Measurements onboard Icebreaker Sisu in Winters 1979 to 1985. Winter Navigation Research Board, Report 43, Helsinki, Finland.

101 13. Riska, K. 1987. On the Mechanics of the Ramming Interaction between a Ship and a Massive Ice Floe. Technical Research Centre of Finland, Publications 43, Espoo, Finland, 86 p. 14. Frederking, R. & Jordaan, I. & McCallum, J. 1990. Field Tests of Ice Indentation at Medium Scale Hobson's Choice Ice Island, 1989. Proc. IAHR Ice Symp., Espoo, Finland, Vol. 2, pp. 931-44. 15. Kry, P. 1978. A Statistical Prediction of the Effective Ice Crushing Stresses on Wide Structures. Proc. IAHR Ice Symposium, LuleL Sweden, Vol. 1, pp. 33-47. 16. Weibull, W. 1951. A Statistical Function of Wide Applicability. J. of Appl. Mech., 18(1951 ), pp. 293-312. 17. Finnish Board of Navigation, 1985. Rules for Assigning Ships Separate Ice-Due Classes. Helsinki, Finland, 53 p. 18. Canadian Standards Association, 1989. General Requirements, Design Criteria, the Environmental Loads, Preliminary Standard $471-M1989, Toronto, Canada, 63 p. 19. Varsta, P. 1983. On the Mechanics of Ice Load on Ships in Level Ice in the Baltic Sea. Technical Research Centre of Finland, Publications 11, Espoo, Finland, 91 p. 20. Riska, K. 1980. On the Role of Failure Criterion of Ice in Determining Ice Loads. Technical Research Centre of Finland, Ship Laboratory, Report No. 7, Espoo, Finland, 31 p. 21. Frederking, R. 1977. Plane-Strain Compressive Strength of Columnar Grained and Granular Ice. J. of Glaciol., 18(1977)80, pp. 505-16. 22. Vuorio, J. et Riska, K. et Varsta, P. 1979. Long-Term Measurements of Ice Pressure and Ice-Induced Stresses on the IB SISU in Winter 1978. Winter Navigation Research Board, Rpt. No. 28, Helsinki, Finland. 23. Korri, P. & Varsta, P. 1979. On the Ice Trial of a 14 500 dwt Tanker on the Gulf of Bothnia. Proc. of the NSTM-79, The Society of Naval Architects in Finland (LARADI), Helsinki, Finland. 24. Riska, K. et Kujala, P. et Vuorio, J. 1983. Ice Load and Pressure Measurements onboard IB SISU. Proc. of the POAC '83, Helsinki, Finland, Vol. 2, pp. 1055-1069. 25. Selvadurai, A.P.S. 1979. Elastic Analysis of Soil-foundation Interaction. Elsevier Scientific Publ. Co., Amsterdam, 543 p. 26. Lindholm, J.E. & Riska, K. & Joensuu, A. 1990. A Contact between Structure and Ice, Results from Ice Crushing Tests with Flexible Indentor. Helsinki University of Technology, Ship Laboratory, Report M-101, Espoo, Finland. 27. Watanabe, T. & Yamamoto, K. & Yoshimura, N. 1983. Interaction between Ice and Stiffened Panel. Proc. of the POAC '83, Helsinki, Finland, Vol. 1, pp. 458-67. 28. Riska, K. & Frederking, R. 1987. Modelling Ice Load during Penetration into Ice. Report from Joint Research Project Arrangement I, Technical Research Centre of Finland & Transport Canada, Report TP 8237E, Espoo, Finland, 57 p.

102 29. Saeki, H. & Ozaki, A. 1979. Ice Force on Piles. Proc. of the IUTAM Physics and Mechanics of Ice Symposium, Copenhagen, Denmark, pp. 342-350. 30. Sodhi, D. 1991. Ice-Structure Interaction during Indentation Tests. In Proc. of IUTAMIAHR Ice Symp., eds. S. Jones et al., Spronger Verlag, Berlin, pp. 619-640. 31. Fransson, L. 1993. Video from indentation tests in the Baltic, winter 1993. 32. Riska, K. & Rantala, H. & Joensuu, A. 1990. Full Scale Observations of Ship-Ice Contact. Helsinki University of Technology, Ship Laboratory, Report M-97, Espoo, 54 p. 33. Tuhkuri, J. 1993. Laboratory Tests of Ship Structures under Ice Loading. Helsinki University of Technology, Ship Laboratory, Espoo, Vols. 1-3, 171 +318+478 p. 34. Lavrov, V. 1962. Questions of the Physics and the Mechanics of Ice. Transactions of the AARI, 247(1962). 35. Popov, Yu. & Faddeev, O. & Kheisin, D. & Yakovlev, A. 1967. Strength of Ships Sailing in Ice. Sudostronije Publishing House, Lenengrad, 228 p. 36. Kheisin, D. & Likhomanov, V. 1973. An Experimental Determination of the Specific Energy of Mechanical Crushing of Ice by Impact. Problemy Arktiki i Antarktiki, 41(1973), pp. 69-77. 37. Jordaan, I. & Maes, M. & Nadreau, J.P. 1988. The Crushing and Cleating of Ice in Fast Spherical Indentation Tests. Proc. of the 7th Int. Conf. on Offshore Mech. and Arctic Eng., Houston, Texas, USA, Vol. 4, pp. 111-116. 38. Kurdjumov, V. & Kheisin D. 1976. Hydrodynamic Model of Impact on Ice. Ptikladnaja Mekhanika, 12(1976) 10, pp. 103-109. 39. Hallam, S. & Picketing, J. 1988. Modelling of Dynamic Ice Loading of Offshore Arctic Structures. Proc. of the Polar Tech '88, Trondheim, Norway, Vol. I, pp. 235-248. 40. Savage, S. & Sayed, M. & Frederking, R. 1992. Two-dimensional Extrusion of Crushed Ice. Part 2: Analysis. Cold Regions Science and Technology, 21(1992), pp. 37-47. 41. Sayed, M. & Frederking, R. 1992. Two-dimensional Extrusion of Crushed Ice. Part 1: Experimental. Cold Regions Science and Technology, 21(1992), pp. 25-36. 42. Gold, L. 1972. The Failure Process in Columnar-grained Ice. National Research Council of Canada, Division of Building Research, Technical Paper No. 369, Ottawa, Canada, 108 p. 43. Krajcinovic, D. 1984. Continuum Damage Mechanics. Applied Mechanics Reviews, 37(1984), pp. 1-7. 44. Santaoja, K. 1990. Mathematical Modelling of Deformation Mechanisms of Ice. Technical Research Centre of Finland, Research Reports No. 676, Espoo, Finland, 215 p. 45. Sj61ind, S.-G. 1990. Continuum Damage Modelling of Polycrystalline Ice. Proc. of the IAHR Ice Symposium, Espoo, Finland, Vol. 1, pp. 449-463. 46. Matlock, H. & Dawkins, P. & Panak, J. 1971. Analytical Model for Ice-structure Interaction. J. of the Eng. Mech. Div. (ASCE), August 1971, No. EM 4, pp. 1083-1092.

103 47. Jordaan, I. & Kennedy, K. & McKenna, R. & Maes, M. 1991. Load and Vibration Induced by Compressive Failure of Ice. Proc. of the 6th Int. Specialty Conf., ASCE, Hanover, New Hampshire, USA, pp. 638-649. 48. Comfort, G. & Selvadurai, A.P.S. & Abdelnour, R. & Au, M.C. 1992. A Numerical Ice Load Model. Proc. of the IAHR Ice Symposium, Banff, Alberta, Canada, Vol. 1, pp. 243257. 49. K~irn~i,T. 1993. Finite Ice Failure Depth in Penetration of a Vertical Indentor into an Ice Edge. Annals of Glaciology, 19(1993). 50. Muhonen, A. & al. 1992. Laboratory Indentation Tests with Thick Freshwater Ice, Vol. II. Helsinki University of Technology, Ship Laboratory, Report M-122, Otaniemi, 397 p. 51. M~i~itt~inen, M. 1978. On Conditions for the Rise of Self-excited Ice-induced Autonomous Oscillations in Slender Marine Pile Structures. Winter Navigation Research Board, Res. Rpt. No. 25, Helsinki, 98 p. 52. Varsta, P. & Riska, K. 1977. Failure Process of Ice Edge Caused by Impact with Ship's Side. Proc. of the Ice, Ships and Winter Navigation Symposium in conjunction with the 100th anniversary of winter navigation in Finland, Oulu, pp. 235-262. 53. Daley, C. 1991. Ice Edge Contact, a Brittle Failure Process Model. Acta Polytechn. Scandinavica, Rpt Me 100, Helsinki,92 p. 54. Daley, C. 1990. Ice Edge Contact, an Iterative Failure Process Model. Helsinki University of Technology, Ship Laboratory, Report M-103, Espoo, Finland, 65 p. 55. Joensuu, A. & Riska, K. 1989. Ice and Structure Interaction [in Finnish]. Helsinki University of Technology, Ship Laboratory, Report M-88, Espoo, Finland, 57 p. 56. Gagnon, R. & Molgaard, J. 1990. Evidence for Pressure Melting in Indentation and Impact Experiments of Ice. Proc. of the Ice-ocean Dynamics and Mechanics Conf., Hanover, New Hampshire, USA. 57. Gagnon, R. & Sinha, N. 1991. Energy Dissipation through Melting in Large Scale Indentation Experiments on Multi-year Sea Ice. Proc. of the 10th Offshore Mech. and Arctic Eng. Conference, Vol. IV, pp. 157-161. 58. Fransson, L. & Olofsson, T. & Sandkvist, J. 1991. Observations of the failure Process in Ice Blocks Crushed by a Flat Indentor. Proc. of the POAC '91, St. John's, Canada, Vol. 1, pp. 501-514. 59. Kujala, P. 1993. Modelling of the Ice Edge Failure Process with Curved Failure Surfaces. Annals of Glaciology, Vol. 19.

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SOIL-STRUCTURE INTERFACES

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All fights reserved.

107

Modelling and testing of interfaces C. S. Desai and D. B. Rigby Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ 85721, USA

1. INTRODUCTION Many problems in geomechanics such as structures and foundations, and discontinuous rock masses involve contacts between two (dissimilar) materials, which are often called interfaces or joints. When such systems are subjected to static or dynamic loads, an interface can experience various modes of deformation such as slippage, debonding, rebonding and interpenetration that involve relative motions between points on the two neighboring materials. Development of accurate constitutive models for the characterization of the mechanical response of interfaces including such factors as adhesion, friction, roughness and asperities, irreversible deformations, existence of fluids and microstructural changes leading to hardening, and degradation or softening, is important for reliable solution of the associated boundary value problems. For calibration and determination of the parameters involved in a constitutive model, it is vital to develop and utilize appropriate laboratory or field test devices. Implementation of these advanced models in computational procedures such as the finite element and boundary dement methods toward reliable and robust solution require special considerations. 1.1. Scope The objectives of this chapter are to describe (1) a laboratory test device for static and cyclic testing of interfaces including measurement of pore water pressure, (2) the recently proposed disturbed state concept (DSC) for constitutive modelling of interfaces and joints including determination of relevant parameters, (3) validations of the model with respect to typical laboratory test data, and (4) briefly, computer implementation aspects of the DSC models. It is not intended here to provide a comprehensive review of the subject which is wide in scope. Other chapters in this book include reviews of many of the related topics; here, only those publications directly relevant to the scope of this chapter will be included.

2. LABORATORY TEST DEVICE The cyclic multi degree-of-freedom (CYMDOF) shear device was designed and developed by Desai [1,2] and used for testing and modelling interfaces and joints [3-6]. The CYMDOF

108 has been redesigned and modified including provision for the measurement of pore water pressures [6-9]; brief description is given below. The CYMDOF device is capable of simulating both direct and simple shear modes of deformation. Figures l(a),(b) show a photograph of the overall device and details of the circular test box, respectively. For direct shear testing, two material samples are placed one above the other in the two halves of the box. For simple shear testing, Fig. 1(b), a provision for simple shear deformation is made by confining the geologic specimen by annular smooth rings. The rings are coated with teflon to eliminate friction as much as possible. Under sheafing action a "thin" interface develops between the two materials (structural and geologic). For testing with measurements of pore water pressures in the interface, a rubber membrane is installed on the outside of the box and confined by a stack of specially designed rings. The test box is installed in a stiff 130 kN (30 kip) reaction frame, Fig. 1(a). The electrohydraulic control system is capable of applying cyclic loading with frequencies up to about 20 Hz. With the circular specimen of about 19 cm (7.50 inch) diameter, currently, a (constant) normal stress up to 2.8 MPa (400 psi) can be applied with shear stress of 3.8 MPa (550 psi) amplitude. A state of the art computer-controlled data acquisition and control system is used to run the tests. The applied loading in both horizontal and vertical directions can be programmed to be static, quasi-static, repetitive, or cyclic in nature, and testing can be performed under displacement- or load-control. A capability exists to enforce a constant variable stiffness load response in the vertical direction during interface testing. The quantities that can be measured during a test include the normal stress, shear stress, normal displacement, shear displacement, rotation of upper sample, and the pore fluid pressure. It is believed that the CYMDOF device provides significantly greater capabilities compared to devices available in the past.

3. DISTURBED STATE CONCEPT (DSC) In the context of the mechanical response of solids, the DSC is based on the idea that during deformation a material element, Fig. 2(a), can be considered to be a mixture of the material in two reference states: the "intact" state and the fully adjusted state (FAS). The "intact" is a relative state that represents the part of the material that deforms as a continuum. The microstructural changes result into transition of the intact material into the material as the FAS, which may be defined as the material that has "failed" and acts like a void, as in the classical damage concept [10], and can carry no stress at all, or it is in a liquid like state, in which it can carry hydrostatic or mean pressure but no shear stress, or it is in liquid-solid like state when it can deform in shear without change in volume but can continue to carry the stresses reached up to that state [11]. The last is often called the critical state [12]. Since the material at FAS is constrained by the surrounding intact material, it may not be appropriate to consider it to be like a void that cannot carry any stress at all. In the DSC, the observed behavior of the mixture is expressed in terms of the behavior of the material in the two reference states, intact and FAS. Details of the DSC, derivations of constitutive equations, calibration of parameters, validations for laboratory tests, and

109

(a)

Photograph showing overall setupN (~)

t___

___A

r - Con fining Rings /

F (.r-)

Merr~rvs~ Structural Mo teriol

m~ .:::+:i-'i~.'.lntei:~i:t~ Y-:.':+;,: Z +: [ ] L Geologic Material

m

=~

embrane Confining Rings

12.00 Dimensions in Inches

Co) Figure 1"

_1 7 P = Pore Pressure Trans du ce r L = LVOT

Details of Test Box Photograph and Details of CYMDOF

110 comparisons with such other concepts as damage and micromechanics, are presented elsewhere [11]. Here, a brief description for the solid material is presented, followed by details for application to interfaces and joints. The relative intact state excludes influence of factors that cause the disturbance. For instance, if microcracking, damage and softening occur, the response of the material without damage can be treated as that of relative intact state. Hence, for instance, with the theory of plasticity, if n number of parameters define the damage behavior, and rn number of parameters define associative hardening response without damage, n-m parameters cause the disturbance. Then, the associative plasticity model can describe the relative intact response [13]. The FAS response is assumed to be that of the liquid like or solid-liquid like state using the critical state concept. The disturbance function, D, acts as an interpolation function to describe the observed behavior in terms of behavior of relative intact and fully adjusted states. It can be expressed as

o -

(I)

po, pol, o,t)

where/j is an internal parameter such as plastic strain trajectory, plastic work or (ultrasonic) velocity, P0 and P0 are (initial) pressure and density, respectively, I denotes interparticle, intercluster or interface roughness, # is the temperature and t is time. A simple form of D is given by D = O , ( l - exp(-,ll)~ )

(2)

where Du represents disturbance at the residual state, Fig. 2(b), and X~ and ),2 are material parameters. Based on laboratory tests, D can be expressed as D

= A ~-A" AI-A c

(3)

where i, a and c denote intact, observed and fully adjusted or critical states, and A can be an observed quantity such as stress, (volumetric) strain and velocity. For the case of solid materials, the incremental observed stress is expressed as

.

,

(c

,)

doij : d o 0 + d D o i j - o 0

(c

+ D oq-oij

,)

(4)

where oii is the stress tensor and d denotes increment. If D = 0, Eq. (4) reduces to the continuum model, and the FAS does not carry any stress and Eq. (4) reduces to that in the classical damage model [10]. Applications of the DSC for the behavior of solid geologic materials (sands and clays) are given elsewhere [ 1l, 14-16].

111

(a)

Intact and fully adjusted materials

intact O

I D

9 R

observed R a

i

1-D

residual

O 1-D

i

fully d i s t u r b e d (critical state)

O

0

J, (b) Figure 2:

Schematic of disturbed state concept

Representation of Disturbed State Concept

v

O

Re

112 4. DSC FOR INTERFACES [20,21] The disturbed state concept is specialized for interfaces and joints by assuming that the interface zone can be simulated as a thin "smeared" layer between two materials, Fig. 3. Details of DSC for interfaces are given below. The relative total tangential or shear in a planar interface, Fig. 3, is given by It

= It r

+ It P

= It r

+ -UP + -~s

= u e

+ its

(5a)

where u c is the elastic part, uP is the plastic displacement of the contact asperities, u' is the slip displacement and u' is the total irreversible displacement. Similarly, the total normal displacement can be written as v = v e +~

+ v: - v e + v:

(5b)

4.1. Reference States For a given normal stress, tro, a part of joint reaches the critical state and critical shear displacement uo, where its shear resistance will be constant, and thereafter, it would experience shear displacements without change in shear and normal stresses, Fig. 4. Hence, as in the case of solid materials, the relations between the shear stress and normal stress at the critical, r ~ and tr,~ respectively, can be written as [17] 1; c

=

-"

Co On

(6)

where Co and m" are material parameters. The normal displacement (dilation) at the critical state, v ~, is expressed as [ 18]

where v ~ is the ultimate dilation when tr. = 0, Fig. 5, and ~, is a material parameter. Note that each tr, is associated with a critical normal displacement v~~ v2c, and so on. As in the case of solid materials, the intact state can be characterized by using the basic ~0 model in the hierarchical single surface approach [13]. Then the yield function, F, for solids is specialized to Fi as [19,20]"

(8) where % = hardening of growth function, % and q = ultimate parameters. A plot Fi in the a. vs r space is shown in Fig. 6. The hardening function is given by

= = a~/~'

(9)

where a~ and b~ are hardening parameters. The disturbance functions are defined for the shear stress and normal displacement as follows:

113

T (~:}

i

AsDe t,,,

1

I(a) ~otural

Ao

(b) Idealized

FullyAdjusted (Critical) f

I

(c) Intact and Critical Phases

Figure 3.

Natural and idealized joint with intact and critical states

l-.;

Intact ~."'---"'"~ Response ~

I

Y

I-*

/ .... Observed I //~e~oo~s~ ~

cO L

O tOO

~

Critical I Uc

Tangential Displacement

Figure 4.

Shear stress-relative shear or tangential displacement response for joint

114

U'3

c--

uo

(1)

On0 = 0

E E~ C

Q_ or)

Ul

on1 > 0

C

~2

U3

On2 > On1

G

E 0

Z

Tangential Displacement u Figure 5.

Normal slip vs tangential displacement curves at different normal stresses

a3

Figure 6.

Plot of Fj for joint/interface

115 %= = D , 1:" + ( I - D , ) ,

t

(lOa)

and v = = D,v"

(10b)

where v" and v = are irreversible normal displacement and normal displacement at the critical, respectively, and D. and D, are given by

D, = l -exp(-)q ~')

(lla)

and D,-1-exp(-

~v ~" )

(llb)

where x,, ~, r, and r,, are material parameters. Incremental forms of # and v' and use of Eq. (11) lead to incremental constitutive equations as

/ d''/ d

(12) = [I,'1-

where [ ~ q ' = [k'] ~ - 0e~' is the elastoplastic constitutive matrix for the observed behavior, which includes the constitutive tensor for the intact response. Here, e and p denote elastic and plastic, respectively. 4.2. Material Parameters [20,21] The material parameters for the intact and critical states and for the disturbance function are determined based on appropriate laboratory tests under normal and shear loading. Brief descriptions are given below. 4.2.1. Intact State Elastic: The shear and normal stiffnesses, K, and K,, are found from the unloading slopes of r vs u and r n vs v curves. These quantifies are included in [K'] =p, the elasto-plastic matrix, Eq. ( ) . Plasticity: The ultimate constants ~ and q are found from Fj = 0, Eq. (8), at the ultimate with a = 0: ~, = q~" o e

(13)

where ~= is ultimate asymptotic shear stress, Fig. (4). The parameter n is found based on the state of stress (~) when transition from compactive to dilative normal displacement occurs and aF/a~o - 0:

116

n

=

(14)

u

1-

"C

q

a t transition

Ton

The hardening or growth parameters al and bx are obtained by writing Eq. (8) as (15)

~ n a I - b l ~ n ~ o + n ~ n o m = ~n'A

and by solving.equations simultaneously for a number of points on the stress-displacement curves. Here A = ~aoq - r 2. 4.2.2. Critical State The values of Co and m', Eq. (6) are found from ~n'r

(16)

= ~nco + m * ~ n o n

_and by plotting tnr

L.

vs tncro. Similarly, plots of tnv ~ vs 0to, Eq. (7) yield values of Vo and

4.2.3. Disturbance Functions Taking logarithm twice in Eq. (lla), gives ~n[-~n(l-D)]

(17)

= ~nK, + r , ~ n ~ o

Plots of [-in(I-D,)] vs tn~D gives K, and 1",. Similarly, plots of [-in(I-D,)] vs tn~v, Eq. (11b) yields values of r~ and rv. D, and D, are given by D, -

Ti _ ~a

(18a)

Ti _ ~c

and Dv

=

VI - V a V i -V c

Parameters: below.

(18b)

The values of parameters for joint type A described subsequently are given

Intact Phase- 5;Modr

Elastic Constants: Plasticity:

K, = 5.6 Mpa/mm K, = 28.0 Mpa/mm al = 0.0656 bl = 1.42 n = 2.31 q = 1.91

117 Critical State co = 0.90 m" = 0.955 v ~ = 4.94 lq, = 1.21 Disturbance Function r, = 3.87 R, = 0.36 = 3.88 R~ = 0 . 9 0

5. VALIDATIONS Test results for various joints in concrete and rock, and interfaces between sand and concrete, sand and steel and mudstone and concrete were considered. These included tests (1) by Schneider [18] in three types of rock joints (Type C - smooth joint in limestone, Type A - rough in granite, and Type B - rougher in sandstone) under normal stress-controlled condition, (2) by Bandis et al. [22] for joints in simulated rock with different values of Joint Roughness Coefficients (JRCs), (3) by Desai and Fishman [19] for joints in concrete with different asperity angles under normal stress-controlled condition, (4) by Williams [23] for interfaces between concrete and mudstone with both normal stress and stiffness-controlled conditions, (5) by Uesagi and Kishida [24,25] for interfaces between concrete and sand and steel and concrete under normal stress-controlled condition. Details of static and cyclic testing, determination of parameters, and validation are given by Ma and Desai [21], Desai and Ma [20] and Navayogarajah et al. [26,27]. Here, only typical results are included. 5.1. Rock Joints

As indicated before, Schneider [18] performed tests on rock joints with three different roughnesses. Here, typical results for Type A joint in granite with intermediate roughness are presented; the material parameters for this joint are given above. Figure 7(a) shows back predicted shear stress, r vs. tangential displacement u at different initial normal stresses in comparison with the observed data. In Fig. 7(b) is shown typical comparison between back prediction and test data for normal (dilation) v vs. u results. This result and those for other tests by Schneider [18], Bandis et al. [22], Desai and Fishman [19] and Williams [23] showed highly satisfactory predictive capabilities of the DSC model (Desai and Ma [20]). 5.2. Interfaces

Static and cyclic tests on sand-concrete and sand-steel interfaces performed by Uesugi and Kishida [25,26] were considered. Here, the tests were performed by using a simple shear apparatus. The static behavior was simulated by using an approach similar to that described above; however, the influence of interface roughness was included by expressing roughness, R, as

118 Z.4,0 . . . .

Data:

==

9 ~rn -

1~o

~ ~

n - ~ - - - - ~ r ~

e

9

A

A

l.~4KPa

O-n- 1.25KPa

9 Crn-0.5 lYPa . .A A A

d

0 ~n-0.33YPa

Back'pr~flctions:

0.40

~o

-.

-

0.00

140

Tangential Displacement u (rnrn)

(a) 4.00

v Data: 9 Grn-0.81KPa

O ....,

Bar.~r~d~.ton: 2.OO

2:

1.00

0.00 ~ . t

o

~

140

Tangential Displacement u (rnrn)

(b)

Figure 7.

Comparisons of predictions and observations, data from Schneider: '2, (a) r vs. u, Type C joint; (b)u vs. u, Type C joint; ( c ) r vs. u, Type A joint; (d) u vs. u, Type A joint; (e) r vs. u, Type B joint; (f) ~, vs. u, Type B joint (21)

119

r

e

R-

1

if R,,~ < R.

(19a)

where the normalized roughness, R., is given by R. = R , ~ (L=0.2 ram)

(19b)

D,o (L = 0.2 mm) is the relative height of asperities between the highest peak and lowest trough over a length L = 0.2 ram, Dso is the mean diameter of sand particles and R~c~ is the critical normalized roughness. Then, the hardening function a in the plasticity model was defined as [26]: a = u exp(-a{,)

-~ -

~0

for

{o < {D

(20a)

G a = 0 for

(20b)

~D > ~a

where ~t," is the value of ~v at the peak shear stress, %, and "y and a are material parameters. Parameters .y and/iv" are expressed in terms of roughness R as 2 (21a)

Y=Pe

9, _

o 9

(.

~D = ~D1+

1

R]

(21b)

.)

~D2 R

The disturbance function, of which the classical damage function is a special case, is defined based on Eq. (3), and is expressed as D = D.(1 -exp(-A ~2) (22) where A is a material parameter which is expressed as a function of R [26,27]. 5.3. Cyclic Loading The cyclic behavior is simulated by defining the nonassociative hardening function otQ as

[ ( /1 where r is the associative hardening function and cd is its value at the start of shear loading, r is nonassociative parameter expressed as function of R as = ~1 + r'2R (23b) r~ and r2 are material parameters, and if/~ < /J,r~, tj~ =/j,; if/~ > /J,rt,/J~ = ~ ; and if/~v > 2/~n., /Jr = 2~n., ~n* is the value of ~v at the phase-change point during cyclic

120 loading and/Jr is the trajectory of volumetric plastic strains. The cyclic parameter fl controls compaction during reverse-loading, and is given by 12

=

1+121+u

(24)

where fl~ and t~ are parameters, and 3' is function of R, Eq. (19). Details of the above cyclic model, determination of parameters from laboratory test and validations are given elsewhere [26,27]. Typical results are given below.

5.4. Static Interface

Tests The material parameters for the steel-sand interface were obtained from the test results under cr = 78.4 kPa, Dr = 90%, and a = 98.00 Kpa and Dr = 90% with different roughnesses, and those for the concrete-sand interface were obtained from tests with cr = 98 Kpa and Dr = 90%. Figure 8 shows typical comparisons between the model predictions and test data for monotonic tests with cr, = 98 Kpa and Dr = 90% for different values of R. Figure 8 shows comparisons for cyclic tests with a, = 98 Kpa, Dr = 90% and R = 23 ftm. Both comparisons show very good predictive capability of the model.

6. INTERFACE MODELS A number of models have been proposed for interface behavior; they include spring, zero thickness constraint and thin-layer element models. Details of these and other models are given elsewhere in this volume. Here, a brief description of the thin-layer element [28] is included. The thin-layer interface element idea is based on the consideration that at a junction between two materials, there exists a distinct or smeared zone that exhibits behavioral modes different from the neighboring solid materials. In the case of soil-structure problems, it is often found experimentally (in the laboratory and field) that the shear transfer occurs in a thin layer of soil between the structural and geologic material, and f'mal "failure" may occur often in the thin layer. For rook joints with filler materials, a distinct joint exists. In the case of unfilled rook joints and metal contacts, the deformation process involves elastic and plastic strains, deformation in and damage done to breakage of asperities, leading to a thin smeared zone that represents the interface. Hence, it appears realistic to represent an interface as a thin zone, and consider relative motions such as slippage, debonding, rebonding and interpenetration as occurring in the thin zone. Indeed, the traditional zero thickness dement by Goodman et al. [29] in which weighted material properties are assigned to the hypothetical zero thickness material, the situation can be shown to be a special case of the thin-layer concept [30]. Indeed, the idea of thin-layer element takes a viewpoint that it is the constitutive behavior of the thin zone that is important, rather than the simulation through springs, zero thickness or constraint models, in which material characteristics are attached to hypothetical mechanical models. In this context, it is important to note that in most of such mechanical models, the major attention is given to the shear response, while the normal response is arbitrarily chosen through ad hoe values of normal stiffness during loading, debonding, etc. It is felt that as the behavior of most interfaces is nonlinear, the normal response in which the normal stiffness varies during loading, unloading and reloading, and is coupled with the shear

121

Predicted .... .......

-o-~-

_._. _._

--o--o-

Observed Rmax=3.8 /~m Rmax=9.6/~m Rmax=19 #m Rmax:=40

~rrt

o Rmax i

"~"'~~-~'~""'~ ~' "--- a

c5-

,...

~

~

~

.

i

--C--

.

= 40 .

__

.

.

/.tin

.

= t 9 ~ m

~.~

9.

~'

o.

~L

~

IO

h

9

o

--

9.6 /~m

o

~o-----<>~_o~_

,w~-,j,. = .3 . 8. ~ .. m.

t~

O - -

--

Q

. i

o

. -

,

.

.

.

i

. i

i

o .

.,-i

d O

'

Figure 8(a).

o.o

;.o

2.0

3.o u

4.0

~.o

6'.o

7.0

(mml

Comparison of model prediction with observation: cr = 98 kPa, D, = 90%, Steel-Toyoura sand interface. Monotonic loading (26,27)

122

Observed

',

Predicted

o-

t

i

;O

rid r

d-

-~o-

9

,(i

OI OI

all

N=2 ..... i5

N=2

I

I

I

_

. . . . .

15

i

t~

d Predicted

o

;

I1

1

_ ', >~

/

i/

I1

/

/

!

i

/2

to

'1

d I

,

-5

-2.5

0

U (mm)

Figure 8(19).

2.5

5

-5

5

u (turn')

Comparison of model prediction with observation: a = 98 kPa, D, = 90%, Steel-Toyoura sand interface. R , ~ = 23/~m, cyclic loading (26,27)

123 behavior, plays a vital role in realistic characterization of the interface response. As the thinlayer element allows definition of both normal and shear response and their coupled effects, e.g., through the use of the DSC described above, it is possible to represent the interface behavior more realistically. Another important attribute of the thin-layer modelling with the DSC is that the constitutive responses of the surrounding materials and the interface are characterized by using the same mathematical framework. Here, in the computer (finite element) procedures, the thin-layer interface element is formulated by treating it as a quadrilateral or brick element in the same manner as the finite element equations for the solid neighboring elements are formulated. Thus, the thin-layer approach provides consistent formulation for solid and interface element. As a result, the need in some previous studies of using different models, say, elastoplastic for soils and bilinear elastic for interfaces, is eliminated.

6.1. Implementation Implementation of the thin-layer element has been achieved in static and dynamic finite element procedures [3,28,31-35]. Here, the same improved drift correction procedures are used for both the solids and interfaces [31-35].

7. ACKNOWLEDGMENTS Parts of this research were supported from Grant No. MSM 8618901/914 and CE 9320256 from the National Science Foundation, and No. AFOSR 830256 from the Air Force Office of Scientific Research, Bolling AFB. The review of research presented herein represents a continuing effort in which a number of persons have participated and contributed. Some of the recent results included herein are based on the contributions by Drs. K.L. Fishman, Y. Ma and N. Navayogarajah.

REFERENCES

Q

Q

4qt

Q

g

C.S. Desai. A Dynamic Multi-Degree-of-Freedom Shear Device, Report No. 80-36, Dept. of Civil Eng., Virginia Tech, Blacksburg, VA, USA (1980). C.S. Desai. Behavior of Interfaces Between Structural and Geologic Media, State-ofthe Art Paper, Proc. Int. Conf. on Recent Advances in Geotech. Earthquake Eng. and Soil Dynamics, St. Louis, MO (1981). M.M. Zaman, C.S. Desai and E.C. Drumm. An Interface Model for Dynamic SoilStructure Interaction, J. Geotech. Eng., ASCE, 110(9) (1984) 1257-1273. E.C. Drumm and C.S. Desai. Determination of Parameters for a Model for Cyclic Behavior of Interfazes, J. of Earthquake Eng. & Struct. Dyn., 114(1) (1986). C.S. Desai and B.K. Nagaraj. Modelling of Normal and Shear Behavior at Contacts and Interfaces, J. of Eng. Mech., ASCE, 114(7) (1988). D.B. Rigby and C.S. Desai. Cyclic Shear Device for Interfaces and Joints with Pore Water Pressure, Report to NSF, Dept. of Civil Eng. & Eng. Mechs., Univ. of Arizona, Tucson, AZ 85721, USA (1988).

124

@

@

0

10. 11.

12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22.

23. 24.

N.C. Samtani and C.S. Desai. Constitutive Modelling and Finite Element Analysis of Slowly Moving Landslides Using the Hierarchical Viscoplastir Material Model, Report to NSF, Dept. of Civil Eng. & Eng. Mechs., Univ. of Arizona, Tucson, AZ 85721, USA (1991). D.B. Rigby. Testing and Modelling of Saturated Clay-Steel Interfaces and Application in Finite Element Dynamic Soil-Structure Interaction, Ph.D. Dissertation, Dept. of Civil Eng. & Eng. Mechs., Univ. of Arizona, Tucson, AZ 85721, USA, under preparation. A. Alanazy. Testing and Modelling of Sand-Concrete Interfaces Under ThermoMechanical Loading, Ph.D. Dissertation, Dept. of Civil Eng. & Eng. Mechs., Univ. of Arizona, Tucson, AZ 85721, USA, under preparation. L.M. Kaclmov. Introduction to Continuum Damage Mechanics. Martinus Nijhoft Publishers, Dordecht, The Netherlands (1986). C.S. Desai. The Disturbed State Concept as Transition through Self-Adjustment Concept for Modeling Mechanical Response of Materials and Interfaces, Report, Dept. of Civil Eng. & Eng. Mechs., Univ. of Arizona, Tucson, AZ, USA (1992). K.H. Roscoe, A. Schofield and C.P. Wroth. On Yielding of Soils, Geotechnique, 8 (1958) 22-53. C.S. Desai, S. Somasundaram and G. Frantziskonis. A Hierarchical Approach for Constitutive Modelling of Geologic Materials, Int. J. Num. Analyt. Meth. Geomech., 10 (1986) 225-257. G.W. Wathugala and C.S. Desai. "Damage" Based Constitutive Model for Soils, Proc., 12th Canadian Conf. of Appl. Mech., Ottawa (1989). S.H. Armaleh and C.S. Desai. Modelling and Testing of a Cohesionless Material Using the Disturbed State Concept, J. of Mech. Behavior of Materials, 5(3) (1994). D.R. Katti and C.S. Desai. Modelling and Testing of a Cohesive Soil Using the Disturbed State Concept, J. of Eng. Mech., ASCE, Tentative Approval (1993). J.A. Archard. Elastic Deformation and the Laws of Friction, Proc., Roy. Soc. London, A243 (1958) 190-205. H.J. Schneider. The Friction and Deformation Behavior of Rock Joint, Rock Mech., 8 (1976) 169-184. C.S. Desai and K.L. Fishman. Plasticity Based Constitutive Model with Associated Testing for Joints, Int. J. Rock Mech. Min. Sc., 28(1) (1991) 15-26. C.S. Desai and Y. Ma. Modelling of Joints and Interfaces Using the Disturbed State Concept, Int. J. Num. Analyt. Meth. Geomech., 16 (1992) 623-653. Y. Ma and C.S. Desai. Constitutive Modelling of Joints and Interfaces by Using Disturbed State Concept, NSF Report, Dept. of Civil Eng. and Eng. Mechs., Univ. of Arizona, Tucson, AZ 85721, USA (1990). S. Bandis, A.C. Lumsden and N.R. Barton. Experimental Studies of Scale Effects on the Shear Behavior of Rock Joints, Int. J. Rock Mech. & Min. Sci., 18 (1981) 121. A.F. Williams. The Design and Performance of Piles Socketed into Weak Rock, Ph.D. Disser., Dept. of Civil Eng., Monash Univ., Melbourne, Australia (1980). M. Uesugi and H. Kishida. Influential Factors of Friction Between Steel and Dry Sands, Soils & Foundations, 26(2) (1986) 33-46.

125 25. 26. 27.

28. 29. 30. 31. 32. 33. 34. 35.

M. Ucsugi and H. Kishida. Frictional Resistance at Yielding Between Dry Sand and Mild Steel, Soils & Foundations, 26(4) (1986) 139-149. N. Navayogarajah, C.S. Dcsai and P.D. Kiousis. Hierarchical Single Surface Model for Static and Cyclic Behavior of Interfaces, J. of Eng. Mech., ASCE, 118(5) (1992) 990-1011. N. Navayogarajah. Constitutive Modeling of Static and Cyclic Behavior of Interfaces and Implementation in Boundary Value Problems, Ph.D. Disser., Dept. of Civil Eng. & Eng. Mechs., Univ. of Arizona, Tucson, AZ 85721, USA (1990). C.S. Desai, M.M. Zaman, J.G. Lightner and H.J. Siriwardane. Thin-Layer Element for Interfaces and Joints, Int. J. Num. Analyt. Meth. Geomech., 8 (1984) 15-43. R.E. Goodman, R.L. Taylor and T.L. Brekke. A Model for the Mechanics of Jointed Rock, J. Soil Mechs & Found. Eng. Div., ASCE, 99(10) (1974) 833-848. K.G. Sharma and C.S. Desai. An Analysis and Implementation of Thin-Layer Element for Interfaces and Joints, J. of Eng. Mech., ASCE, 118(12) (1992) 545-569. C.S. Desai, G.W. Wathugala, K.G. Sharma and L. Woo. Factors Affecting Reliability of Computer Solutions with Hierarchical Single Surface Constitutive Models, Int. J. Computer Meth. in Appl. Mech. and Eng., 82 (1990) 115-137. C.S. Desai, K.G. Sharma, G. Wathugala and D. Rigby. Implementation of Hierarchical Single Surface 6o and ~ Models in Finite Element Procedure, Int. J. Num. Analyt. Meth. Geomech. , 15 (1991) 649-680. C.S. Desai and L. Woo. Damage Model and Implementation in Nonlinear Dynamic Problems, Int. J. Comp. Mech., 11(2/3) (1993) 189-206. G.W. Wathugala and C.S. Desai. Constitutive Model for Cyclic Behavior of Cohesive Softs I: Theory. J. of Geotech. Eng., ASCE, 119(4) (1993) 714-729. C.S. Desai, G.W. Wathugala and H. Matlock. Constitutive Model for Cyclic Behavior of Cohesive Softs II: Applications, J. of Geotech. Eng., ASCE, 119(4) (1993) 730-748.

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

127

Soil-Structure Interfaces: E x p e r i m e n t a l Aspects

Musharaf Zaman I and Arumugam Alvappillai z

1Professor, School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, OK 73019, USA 2Staff Engineer, American Geoteehnical, 1250 N. Lakeviwe Avenue, Suite T, Anaheim, CA 92807, USA

The subject of interfaces has attracted the attention of many researchers from various areas including soil-structure interfaces, rock joints, reinforced concrete and masonry. Development of this subject can be divided into several broad categories, namely, fundamental and experimental aspects, constitutive modeling, implementation of constitutive models into computational techniques, and application to boundary value problems. This chapter presents the state-of-the-art on the experimental research works on the soil-structure interfaces. The development in testing of rock and concrete joints are discussed separately in other chapters. 1. INTRODUCTION An accurate modeling of soil-structure interfaces is very important to obtain realistic solutions to many soil-structure interaction problems. Both analytical and experimental studies conducted in recent years have shown that the interface behavior has great influence in overall structural response, particularly under dynamic loads. In the past, many interface models have been developed in conjunction with the numerical procedures designed to solve soil-structure interaction problems. The material parameters involved in these models have to be determined from appropriate laboratory and/or field tests to represent the interface behavior in a realistic manner. Experimental research work has also played a vital role in the validation of constitutive models and numerical techniques. The following sections review the development of various testing devices and techniques used in the experimental investigation of interfaces under static and dynamic (cyclic) loading conditions.

128 2. REVIEW OF INTERFACE TESTING UNDER STATIC LOADING CONDITIONS Most of the earlier work on interface testing was performed by using the conventional direct shear device. With an increasing need for better understanding of the interface behavior, a number of refinements and modifications have been made to the direct shear device to improve its shortcomings. In addition, several other sophisticated devices have also been developed. In this section, a review of available testing devices is presented along with test results under static loading conditions. 2.1 Direct Shear and Triaxial Devices

Direct shear device is often used to obtain numerical values for the interface or joint parameters under static loadings. In direct shear tests, two specimens usually having different properties are placed in the shear box in contact with each other. The load-displacement histories are recorded by gradually applying the shear loads for a constant, desired normal load. Direct shear tests were performed by Potyondy [1] to determine the interface behavior between several soils and construction material such as steel, concrete and wood. Sand, clay and a mixture of sand and clay were utilized with different moisture content. Surface roughness of the construction materials were also varied. Test results showed that the skin friction was lower than the shear strength of the soil for all the interfaces tested. It was also found that the skin friction was a function of soil type and moisture content, surface roughness and intensity of the normal load. The strength ratio (f,) defined by 6/~ where 6 is the interface friction angle and ~ is the angle of internal friction of the soil, was found to be 0.89 for smooth concrete dry sand interface and 0.99 for rough concrete - dry sand interface. Kulhawy and Peterson [2] conducted a series of direct shear tests on sandconcrete interfaces. A uniform sand and a well-graded sand were utilized in the testing at three different densities. The tests were performed at various levels of interface roughness; (i) smooth, (ii) intermediate rough, and (iii) rough. A glass plate was used to obtain smooth interface while intermediate rough and rough surfaces were prepared by using different fine aggregates. In addition, to represent actual field conditions in the laboratory, tests were also performed on samples made by pouting concrete directly onto a prepared sand sample. Both sand and concrete were allowed to cure without disturbance until tested. A roughness parameter depends on the gradation of the soil and the aggregate in the concrete was used to define the interface roughness. Test results indicated that the shear stiffness was nonlinear and dependent on normal stress. Residual strength of the interface ranged from 95% of the peak strength in the loose state to 85% in the dense state. The strength ratio, f,, was found to be ranging from 0.78 to 1.0 for smooth interfaces and 0.93 to 1.0 for rough interfaces. Tests performed with sand-concrete specimens where the concrete was poured directly in the sand specimen, showed that the shear failure surface occurred within the sand at a distance of 1 to 2 times of D100 from the interface, whereD100

129 is the maximum particle size. A number of testings, both in the laboratory and in the field, has been conducted to date to study the soil-structure interaction effects and its influence on the behavior of embedded piles. Direct shear and modified triaxial devices were used in the laboratory testings. A study conducted by Mohan and Chandra [3] on pries embedded in clay soils showed that the skin friction ratio, defined as the ratio of interface shear resistance to the undrained shear strength of the soil, varied from 0.7 to 0.8 in direct shear tests. However, the actual pile tests yielded the skin resistance from 0.45 to 0.54. Coyle and Sulaiman [4] investigated the effects of void ratio, saturation and lateral pressure on skin friction acting on piles. The laboratory tests were conducted using a small steel pile embedded in the sand. A large triaxial shear device capable of testing samples up to 6 in. in diameter and 12 in. in height was utilized in the testing (Figure 1). The laboratory test results showed that the skin friction increased with density, with confining pressure and with saturation provided free drainage. However, a field study conducted on instrumented piles driven into a saturated sand revealed that the skin friction decreased as depth below ground surface increased. This behavior in the field was thought to be due the load carried by the pile tip.

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The behavior of bored piles was studied by O'Neill and Reese [5] in a field and laboratory testing program. The distribution of sheafing resistance was measured

130 during the field tests in which full-sized instrumented bored piles were load tested in the stiff clay formation. A series of laboratory direct shear tests were conducted on interfaces consisting of mortar poured directly on sand. These test results indicated that the failure occurred from 0 to 0.5 in. into the soil. The skin friction ratio obtained from these tests compared favorably with those obtained from field tests. Desai and Holloway [6] and Desai [7] reported a series of direct shear test results on concrete-sand interfaces for various sand densities. The parameter D10 for the sand tested was 0.20. The strength ration, f#, was found to be ranging from 0.87 to 0.89. These test results were used to predict the behavior of piles via finite element procedure. Clemence and Brummund [8] conducted a series of direct shear tests to define the interface characteristics of pier concrete surface and the sand. The laboratory tests included on both smooth glass and rough concrete surfaces in contact with sand at various densities and were performed in a 2.5 in (64 mm) diameter circular direct shear device under varying confining pressures. The results from interface tests were represented in a hyperbolic form and used to estimate the pile behavior. To study the procedure outlined to include the skin friction in the design of drilled piers, a large-scale model test was conducted on a concrete pier. The instrumented model pier with 16 in. (410 mm) in diameter and 15 ft (4.6 m) long was constructed in a large diameter test pit (see Fig.2). A series of tests were conducted on this pier by applying both eccentric and concentric axial loadings to determine the contribution of skin friction on the pier capacity. A number of other laboratory and field investigations has been conducted in the past to study the influence of interface behavior on pile responese ( Tomlinson [9], Flatte [10]).

Figure 2.

A Large-scale model test on an instrumented pier (After Clemence and Brummund [8])

131

Interface testing was also reported in other areas of soil-structure interaction problems. Clough and Duncan [11] conducted direct shear tests on composite specimens consisting partly of sand and partly of concrete. Results from the interface testing were utilized to analyze the retaining wall behavior. The sand used in the tests had a D10 of 0.15 ram, a uniformity coefficient of 1.7 and consisted of subrounded, subangular particles. The concrete specimens were east against steel coated with form-release compound to obtain a surface representative of a concrete wall east against steel forms. Two series of tests were performed after the concrete was cured for 7 days and 28 days. Results obtained for those two concrete curing periods were found to be essentially the same. The peak angle of wall friction was found to be 33 degrees, resulting in a friction ratio (ft) of 0.83. Lo et. al. [12] developed laboratory tests for measurement of strength parameters for both well-bonded and unbonded concrete-rock contacts at the damfoundation interface. Complete strength envelope was obtained for well-bonded contacts between concrete and rock by performing triaxial compression and extension tests and direct tension tests on samples recovered at the dam-foundation interface of Saunders Dam located in Ontario. Triaxial compression test conducted on a eonerete/dolostone contact specimen showed shear failure along contact surface. Failure occurred with clear separation along the contact surface during direct extension test. Triaxial extension test on concrete/limestone specimen resulted in failure by tensile fracture along a 0.3mm thick shale seam at about 10 mm below contact surface. For unbonded contacts, the simple shear tests were employed to determine the interface characteristics. Test results were presented for concrete/gneiss contact for three normal stress of 345 kPa (50 psi), 690 kPa (100 psi) and 1380 kpa (200 psi). The most of the above referenced studies utilized direct shear device in the laboratory testing of interfaces. Although a direct shear test is relatively easy to perform, there are several inherent limitations of this test: (i) The direction of critical stress can be inclined to the direction of shearing, (ii) maximum shear stress can be greater than the measured shear stress parallel to the axis of the shear box, and (iii) distribution of shear stress and shear strain are nonuniform. Due to these limitations and others, new laboratory devices and techniques have been developed.

2.2 Ring Torsion Device

Yoshimi and Kishida [13] conducted interface testing on dry sand-steel using a ring torsion apparatus (Figure 3). In the ring torsion apparatus, the specimens are in the shape of annular ring. The inside diameter of this device is 240 mm (9.45 in.). The width of the ring is 24 mm (0.95 in.) which is small compared to its diameter. With the application of normal load at the interface, torque is applied to one of the rings and the angular deformations are measured. The advantage of this device is that a relatively uniform state of stress and strains can be obtained at the interface. One disadvantage of this device is that it is difficult to prepare uniform soil mass in a ring shape. Set up of this test is also complicated and time consuming.

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The interface testing by Yoshimi and Kishida [13] using a ring torsion apparatus included for a wide range of surface roughness at initial densities of sand. Test results showed that the coefficient of friction of smooth interface was independent of normal stress. The frictional resistance and volumetric behavior of the interface were primarily governed by roughness of the metal surface.

2.3 Simple Shear Device A simple shear type apparatus was used by Uesugi and Kishida [14] and Uesugi [15] for interface testing between steel and dry sand. This apparatus provides a contact surface of 400 mm xl00 mm between steel and sand. By providing the steel

133 specimen larger than the contact surface, the interface area was kept constant during the experiment even when sliding occurs. The sand specimen was prepared in a container consisting of stacked aluminum plates. The horizontal surfaces of these plates were coated with Teflon to allow sand to deform with minimum frictional resistance. With this apparatus, the sliding displacement at the interface can be obtained with distinction from the displacement due to shear deformation of sand. The schematic diagram of the simple shear apparatus is shown in Figure 4. Four different types of sands, namely, Fujigawa sand, Fukushime sand, Glass Beads and Toyoura sand were used in the testing. Low-Carbon structural steel was machined to make a rectangular specimen with the dimensions of 500 mm in length, 150 mm in width and 40 mm in thickness. Test results showed that the type of sand and the steel surface roughness are influential factors on interface behavior. The normal stress and the mean grain size were not found to be significant. For smooth steel surface, sliding occurred along the steel-sand interface while for rough surface, shear failure of sand mass took place. The amount of sand particles crushed during the test was found to be proportional to the sliding distance of the frictional surface. 1070mm

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134

Kishida and Uesugi [16] reviewed various types of interface testing devices and looked at their strengths and limitations. Table I gives a comparison of various devices used in the interface testing as reported by Kishida and Uesugi [16]. Of the available testing devices, a ring torsion device was thought to be most ideal because of its endless interface. However, this device requires extreme care in sample preparation and testing. A simple shear, on the other hand, can be operated with much less difficulty although a non-uniformity in the interface stresses cannot be avoided. Comparison of test results obtained from both the ring torsion device and the simple shear device indicated good agreement in the interface behavior. The experimental studies discussed so far are limited to static loading condition. In the following section, the development of interface testing in the dynamic (cyclic) loading conditions is discussed. Table 1.

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Advantages and Disadvantages of Interface Testing Apparatuses (After Kishida and Uesugi [16])

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3.0

Interface Testing Under Dynamic loading Conditions

Most of the testing devices considered in this section can be used for both dynamic and static loading conditions. For this reasons, the following discussion concerns with static as well as dynamic interface testing. 3.1 Annular Shear Device Brummund and Leonards [17] proposed a test device called annular shear device, to conduct interface testing under static and dynamic loading conditions. The schematic diagram of this device is shown in Figure 5. nn

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In the annular shear device, interface is introduce as the circumference of a circular rod which is inserted coaxially into a cylinder of soft. The soil is enclosed in a light

136 rubber membrane. The shearing stress is applied through an axial load to the rod while the application of a vacuum to the membrane results in normal stress to the sample. Brummund and Leonards [17] performed a series of static and dynamic tests using this testing device to determine static and dynamic coefficients of frictions. Dynamic tests were carried out by applying an impulse load to the rod. In these tests, two types of sands with rounded particles and the angular particles were utilized. Three different rod materials, steel, smooth mortar and rough mortar were used to represent the construction materials. Some tests were performed with lubricants such as graphite and teflon applied to the rod. For smooth concrete - sand interface, the static and dynamic coefficients of friction were found to be 0.59 and 0.67, respectively. The static and dynamic coefficients of friction were 0.74 and 0.82 for rough interface. One advantage of the annular shear device is that it is geometrically similar to embedded piles and steel reinforcements in concrete. However, it is uncertain that a uniform state of stress is achieved at the interface. Also, any eccentricity in placing the rod in the sample will cause irregular stress distribution across the interface.

3.2 Ring Shear Device Huck et. al. [18] developed a ring simple shear device for interface testing under static and dynamic loadings. This device ( Figure 6) can provide relatively uniform shear stress distribution. The soil specimen is annulus of 17.7 cm in outside diameter and 12.7 cm in inside diameter. The specimen is permitted up to 2.5 cm in thickness and is confined by membranes. Provision is made for pore pressure measurements at the interface. The soil sample is loaded at top and bottom by two concrete samples, resulting in two interfaces. Huck et. al. [18] conducted interface testing for two interface roughnesseswith two different soil types, Ottawa sand and kaolin clay. It was found that the interface behavior followed Mohr-Coulomb criteria with the increasing shear intercept with increasing surface roughness. Huck eL al. [18] also proposed an analytical model based on both deterministic and statistical approaches for characterizing interface behavior.

3.3 Simple Shear Device Simple shear device similar to the one discussed in section 2.3 was used in the interface testing under cyclic and monotonic loads (Uesugi [15], Eguchi [19]). A schematic diagram of the test device is depicted in Figure 7. The device can be used for steel-sand and concrete-sand interfaces. Sand is contained in a stack of rectangular frames. The height of the sand specimen can be changed by stacking the required number of frames. This device can also be converted into shear box type apparatus by replacing the rectangular frames by 18 mm thick steel box (Figure 7c).

137

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Cyclic and monotonic tests were conducted for concrete-sand and steel-sand interfaces. Three different sands, Toyoura sand, Fujigawa sand and Seto sand, were used at 90% initial relative density. Tests were conducted for normal stresses of 98 kN/m z and 490 kN/m 2. Cyclic test results on rough steel-sand interfaces indicated strain softening behavior at the beginning of the cycle. However, no more softening effect was noticed after the loading cycle was reversed. The ultimate shear stress ratio (~/o) was essentially same with number of cycles. For smooth interfaces, the shear stress ratio increased and reached a steady value with the number of cycles. Navayogarajah et al. [20] used these results to propose a constitutive model to describe the interface behavior subjected to static and cyclic loadings.

138

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139 3.3 Direct Shear Device

AI-Douri and Poulos [21] conducted a series of direct shear tests on sandmetal interfaces under both static and cyclic loadings. The effects of interface surface roughness, overburden pressure, void ratio, size and shape of grains and saturation were studied. Six different calcareous sands with carbonate contents between 89 and 94% and specific gravity ranging from 2.72 to 2.77 were used in the testing program. Aluminum and steel blocks were used in the direct shear box to study the interface behavior between the sand sediments and construction materials. Tests were conducted in modified shear boxes. The modified shear device (Figure 8) prevents leakage of soil particles from the gap between the top and bottom halves of the shear box. Static and Cyclic tests were performed on both soft-soft and soil-metal interfaces. Static shear tests indicated that both the internal friction angle and soilmetal interface friction angle were decreased as the initial void ratio increased. It was observed in the cyclic testing that the shear stress decreased as the number of cycles increased. The increasing normal stress, void ratio, displacement amplitude and angularity of sand particles increased the compressibility of sand. The cyclic test results gave the indications of degradation of pile skin friction under cyclic loading. The larger the volume change under cyclic loading the greater the amount of degradation will result.

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Schematical Diagram of Modified Shear Box (After AI-Douri and Poulos [21])

140 3.4 Cyclic Multi-Degree-of-Freedom (CYMDOF) Device A direct shear type device called cyclic multi-degree-of-freedom (CYMDOF) was developed by Desai [22] for smile and cyclic testing of interfaces and joints. This device has undergone continuous refinement and modifications by Desai and his coworkers to improve its capability to predict realistic interface behavior. The CYMDOF device consists of a loading frame designed to withstand a vertical or horizontal load up to 30 tons (270 kN). Normal and horizontal loads are applied through an electro-hydraulic control system of two actuators with a maximum capacity of 7 tons (62 kN). Cyclic loads can be applied in a sinusoidal form with a maximum frequency of 5 Hz. The interface tests under the translational mode are performed by using translational shear box. Figure 9 shows the details of this shear box. The bottom and top parts of the shear box have square cross-sections with the dimensions of 16 X 16 in. (41X41 cm) and 12 X 12 in. (31 X 31 cm), respectively. Concrete or other materials such as ballast and rock is place in the bottom half and the sand is placed in the top half of the shear box. A rubber membrane is used to avoid leakage of materials through the gap between the edges of top and bottom samples.

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141 A torsional shear box is used to test the interface to the rotational degreesof-freedom about the axis normal to the interface. It consists of an annular specimen with the sand placed in the lower part. Concrete forms the top part of the specimen.The static or cyclic loads are applied to the top part of the specimen. Nagaraj [23] reported a preliminary development of a rocking device to simulate partial bonding and debonding at the interface under rotational deformation about the horizontal axis. In the present form, this rocking device cannot be used to test sand-concrete interface due to instability in the system. However, it is reported that the testing could be performed with the interfaces formed by stiff materials. Using CYMDOF device, Drumm [24], Zaman [25], Zaman eL al. [26], Desai et. al. [27] and Drum and Desai [28] reported a series of test results on sandconcrete interface under both static and cyclic loading conditions. Tests were conducted on the Ottawa sand-concrete interface for varying amplitude of displacement, amplitude of shear stress and normal stress, number of loading cycles, initial relative density of sand. It was found that the interface response was not significantly affected by the frequency. The shear stiffness was shown to increase with the number of loading cycles, corresponding to an increase in sand density. Nagaraj [23] and Desai and Nagaraj [29] studied the normal behavior and the combined normal and shear behavior of the sand-concrete interfaces by conducting a series of laboratory tests using CYMDOF device. Tests were performed under static and cyclic loading conditions. In cyclic normal load tests, the effects of the magnitude of the initial normal load and the amplitude of the sinusoidal normal load were studied while the frequency and the initial density of the sand were kept constant at 1.0 Hz. and 15%, respectively. Cyclic tests under combined normal and shear stresses were performed by providing a varying shear displacement in a sinusoidal form and a cyclic normal stress. The behavior of interface under static normal stress indicated a nonlinear response during initial loading and unloading while a linear relation during reloading. Interface behavior under cyclic normal stress was found to be dependent on applied initial normal stress, the amplitude of stress and the number of loading cycles. The behavior under combined cyclic normal and shear stresses showed increasing shear stress for given shear displacement amplitude with increasing normal stress and the number of loading cycles. Modifications have been made to the CYMDOF device in recent years to improve its capability. A new system capable of supporting cyclic testing at 30 Hz. and measuring pore water pressure has been designed and constructed ( Rigby [30]). The apparatus consists of an 3 in. thick and 7.5 in. diameter upper sample and 3 in. thick and 9 in. diameter lower sample. A limitation of this device is that the upper sample must be a solid. With this device, pore water pressure can be introduced and measured at the interface. Samtani [31] used this modified apparatus to test the interface between creeping landslide mass and the base rock. The lower sample was made of clay soil obtained from an actual slowly moving landslide at Villarbeney in Switzerland. A soft rock representative of the base of landslide was used in the upper sample. This soft rock was fabricated in the laboratory by using a mixture of Villarbeney clay, sand and cement with the proportions 40% clay, 20% sand and 40% cement by weight and cured for 15 days. To simulate the field conditions, slow drained tests were

142 conducted in the laboratory at normal pressures of 15, 30 and 50 psi. The shear displacement was applied at a rate of 0.0005 in/min. (0.0127 mm/min.). Test results indicated a monotonic increase in shear strength and monotonic decrease in vertical displacement Both shear strength and displacement approached to a finite value. In all the tests, failue occurred in the soil, below the actual rock-soft contacL The CYMDOF device has also been used to study the behavior of simulated rock joints under quasi-static and cyclic loadings (Fishman [32], Fishman and Desai [33]). The simulated specimens were cast in concrete with varying surface geometries and tested under different levels of normal stress and amplitudes of cyclic displacements. A number of other studies have been reported in the past by using a variety of testing equipments and methodologies to study the behavior of rock and concrete joints under the static and dynamic loading conditions. These studies are not included herein since they are discussed in detail in other chapters.

3.5 Shaking Tables The dynamic interface behavior between concrete and other construction materials was also studied by shake table tests (Aslam et.al. [34]). Large concrete blocks having dimensions of 3 ftx 2 ftx 1 ft (0.91m x 0.61m x 0.30m) were assembled with various materials forming interfaces in a 20 ft x 20 ft (6.1m x 6.1m) shake table that can generate sinusoidal displacement or reproduce earthquake ground motions. Test results indicated a wide range of variation of 0.18 to 0.60 in dynamic coefficient of friction between concrete-concrete interface. This variation was due to variability in concrete finish, cement contact and strength reduction caused by wear from sliding. The frequency of the sinusoidal loading was found to be of little influence in the dynamic friction coefficient. Of the other type of interfaces tested the dynamic coefficients were found to be (i) 0.26 to 0.30 for concrete-plywood interface (ii) 0.10 to 0.15 for concrete-teflon interface and (iii) 0.09 to 0.12 for concrete-graphite interface. An advantage of shaking tables is that the actual earthquake motions can be simulated during testing. However, the testing device is very expensive. It is also difficult to control normal loading at the interface.

REFERENCES J.G. Potyondy, Skin Friction Between Various Soils and Construction Materials, Geotechnique, 11, No.4 (1961) 339. F.H. Kulhawy and M.S. Peterson, Behavior of Sand-Concrete Interfaces, Proc. 6th Pan American Conf. on Soil Mech. and Found. Engrg., 2 (1979) 225. .

D. Mohan and S. Chandra, Frictional Resistance of Bored Piles in Expansive Clays, Geotechnique, 11, No.4, (1961) 194.

143

0

H.M.Coyle and I.Sulaiman, Skin Friction for Steel Piles in Sand, J.Soil Mech. and Found., ASCE, 93, 6 (1967) 261. M.W. O'Neill and L.C. Reese, Behavior of Bored Piles in Beaumont Clay, J. of Soil Mech. and Found. Div., ASCE, 98, 2 (1972). C.S. Desai and D.M. Holloway, Load-Deformation Analysis of Deep Pile Foundations, Proc. Symp. Appl. Finite Elem. Meth. Geotec. Engrg., Vicksburg, Mississippi (1972).

0

C.S. Desai, Finite Element Method for Analysis and Design of Piles, Misc. Paper S-76-21, U.S. Army Engr. Waterways Exp. Station, Vicksburg, Mississippi (1976). S.P. Clemence and W.F. Brummund, Large-Scale Model Test of Drilled Pier in Sand, J.Geo. Div., ASCE, 101, 6, (1975) 537. M.J. Tomlinson, The Adhesion of Piles in Clay Soils, Proc.4th Int. Conf.Soil Mech. and Found. Engrg., London, (1957).

10.

K. FlaRe, Effects of Pile Driving in Clay, Can. Geo. J., 9, 1 (1972) 81.

11.

G.W. Clough and J.M. Duncan, Finite Element Analyses of Retaining Wall Behavior, J. Soil Mech. Found. Div., ASCE, 97, 12 (1971).

12.

K.Y. Lo, T. Ogawa, B. Lukajic and D.D. Dupak, Measurement of Strength Parameters of Concrete-Rock Contact at the Dam-Foundation Interface, Geotechnical Testing Journal, 14, 4 (1991), 383.

13.

Y. Yoshimi and T. Kishida, A Ring Torsion Apparatus for Evaluating Friction Between Soil and Metal Surfaces, Geotechnical Testing Journal, GTJODJ, 4, 4 (1981) 145.

14.

M. Uesugi and H. Kishida, Influential Factors of Friction Between Steel and Dry Sands, Softs and Foundations, 26, 2 (1986) 33.

15.

M. Uesugi, Friction Between Dry Sand and Construction Materials, Dissertation, Tokyo Institute of Technology, 1987.

16.

H. Kishida and M. Uesugi, Tests of the Interface Between Sand and Steel in the Simple Shear Apparatus, 37 12 (1987), 45.

17.

W.F. Brummund and G.A. Leonards Experimental Study of Static and Dynamic Friction Between Sand and Typical Construction Materials, ASTM,J. Testing and Evaluation, 1, 2 (1973) 162.

144 18.

P.J. Huck et al., Dynamic Response of Soil/Concrete Interfaces at High Pressure, Report No. AIWL-TR-73-264 by IITRI for Defense Nuclear Agency, Washington, D.C. (1974).

19.

M. Eguchi, Frictional Behavior Between Dense Sand and Steel Under Repeated Loading (in Japanese), Thesis, Tokyo Institute of technology, 1985.

20.

N. Navayogarajah, C.S. Desai and P.D. Kiousis, Hierarchical Single-Surface Model for Static and Cyclic Behavior of Interfaces, J. of Eng. Mech., ASCE, 118, 5 (1992) 990.

21.

R.H. Al-Douri and H.G. Poulos, Static and Cyclic Direct shear Tests on Carbonate sands, Geotechnical Testing Journal, 15, 2 (1992) 138.

22.

C.S. Desai, A Dynamic Multi Degree-of-Freedom Shear Device, Report No.836, Dept. of Civil Engrg., Virginia Tech., Blacksburg, VA (1980).

23.

B.K. Nagaraj, Modeling of Normal and Shear Behavior of Interface in Dynamic Soil-Structure Interaction, Dissertation, University of Arizona (1986).

24.

E.C. Drumm, Testing, Modeling and Application of Interface Behavior in Dynamic Soil-Structure Interaction, Dissertation, University of Arizona (1983).

25.

M.M. Zaman, Influence of Interface Behavior in Dynamic Soil-Structure Interaction Problems, Dissertation, University of Arizona (1982).

26.

M.M. Zaman, C.S. Desai and E.C. Drumm, Interface Model for Dynamic Soil-Structure Interaction, J. of Geotech. Eng. ASCE, 110,9 (1984) 1257.

27.

C.S. Desai, E.C. Drumm and M.M. Zaman, Cyclic Testing and Modeling of Interfaces, J. of Geotech. Eng., ASCE, 111, 6 (1985) 793.

28.

E.C. Drumm and C.S. Desai, Determination of Parameters for a Model for the Cyclic Behavior of Interfaces, J. Earthq. Eng. and Str. Dyn., 14 (1986) 1.

29.

C.S. Desai and B.K. Nagaraj, Modeling for Cyclic Normal and Shear Behavior of Interfaces, J. of Eng. Mech., ASCE, 114, 7 (1988) 1198.

30.

D.B. Rigby, Cyclic Shear Device for Interfaces and Joints with Pore Water Pressure, Thesis, University of Arizona (1988).

31.

N.C. Samtani, Constitutive Modeling and Finite Element Analysis of Slowly Moving Landslides Using Hierarchical Model, Dissertation, University of Arizona, (1991).

145 32.

ICL. Fishman, Constitutive Modeling of Idealized Rock Joints Under QuasiStatic and Cyclic Loading, Dissertation, University of Arizona (1988).

33.

ICL. Fishman and C.S. Desai, A Constitutive Model for Hardening Behavior of Rock Joints, Second Int. Conf. on Const. Laws for Eng. Mat., Tucson, Arizona, (1987).

34.

M. Aslam, W.G. Godden and D.T. Scalise, Sliding Response of Rigid Bodies to Earthquake Motions, NTIS (1975).

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon(Editors) 9 1995 Elsevier Science B.V. All rights reserved.

147

Soil-structure interaction: F E M computations M. Boulon=, P. Garnica ~ and P.A. Vermeer ~

aLaboratoire 3S, Universit6 Joseph Fourier, B.P. 53, 38041 Grenoble Cedex 9, France b Institfit f'tir Geotechnik, University of Stuttgart, Pfaffenwaldring 35, D-70569 Stuttgart, Germany The present paper reviews some aspects of the soil-structure interface behaviour at the element level and the numerical integration of the corresponding interface constitutive models. The principles of the finite element analyses related to boundary value problems involving contact between solids and consequently interface elements are presented. Some applications to piles under tension loading are presented to illustrate the results of these procedures. 1. I N T R O D U C T I O N The numerical modelling of soil-structure interaction in the static range of loading is highly dependent on the type of constitutive law used to simulate the contact between the surrounding deformable bodies, irrespective of the numerical method (e.g., the finite element method, the boundary integral method, the distinct dement method). In the early numerical analyses related to metal forming, behaviour of rock masses, behaviour of piles and other structures embedded in soils, the constitutive equation for unilateral (no tension) frictional contact was mainly a rigid perfectly plastic Coulomb law, and the penalty function technique was considered as an excellent tool for applying these contact (constraint) conditions. But the comparison between experience and modelling has often proved to be disappointing, especially for problems involving small stress levels. Since that time, Geomechanics has pointed out that the contact zone (interface) frequently undergoes a complete change in structure during a shearing load. This change is due more or less to the granular nature of the soil, which allows for localized dilatancy or contraction (according to the density and the local stress level) after small shearing movements, and for degradation of the friction at large tangential relative displacements. The framework of elasto-plasticity has been usually used for representing the mechanical behaviour of interfaces. Relative movements between bodies in contact are described either by a local high velocity gradient (thin layer) or by a kinematic discontinuity. In both cases, the development of a non-linear behaviour within the interface is drastically rapid compared with that which can develop in the bodies in contact, inducing a very slow rate of convergence of the global solution. Except for models of thin layers, the well known joint elements, incorporating the interface constitutive equations, are different in nature from volume elements since the best sampling points for stresses are the nodes (instead of intermediate Gauss points for volume elements). This difference induces some unexpected oscillations of the stresses during simulation, and requires a special way of

148 numerical integration within the joint elements. Since full plasticity frequently occurs in interfaces, and since the degradation mentioned above (partially due to grain crushing) acts as a local softening, sophisticated methods of resolution of the resulting non linear system of algebraic equations are required at the structural level (arc length control). After a brief survey of the situations where interfaces are needed for numerical simulation of soil-structure interaction, the authors describe some aspects of the soil-structure interface behaviour at the element level and the numerical integration of the constitutive models of interfaces, according both to the elastoplastic and to the rate type framework. Then the principles of the finite element analysis related to interface elements and some applications to piles are presented. 2. S O I L - S T R U C T U R E I N T E R F A C E B E H A V I O U R , W H E R E , W H E N A N D HOW ? In geotechnical engineering distinction can be made between soil-soil interface problems and soil-structure interface problems. Road embankments, fiver emba.nkments and free excavations are examples of soil-soil interface problems. Shallow foundations, deep (pile) foundations, tunnels and earth retaining structures are examples of soilstructure interface problems. Finite element analyses are carried out for both types of problems. As yet it is not a very common method of analysis in geotechnical engineering, but its use is continuously in the increase. By far, most applications are in the field of soil-structure problems, where predictions of displacements are often more important than that for soil-soil problems. In the following some typical soil-structure problems will briefly be reviewed. Shallow f o u n d a t i o n s : In recent years the finite element analyses were typically performed for special projects such as nuclear plants, offshore structures and high rise buildings. However, due to the development of user-friendly finite element codes the field of application is now widening towards shallow foundations of smaller structures. The contact surface between the soil and the structure can be subjected to various conditions, ranging between fully rough and frictionless. For working loads slippage tends to be small as these loads are most usually just a fraction of the failure loads. Therefore the use of interface elements is of limited importance for assessing displacements. On the other hand, these elements are mandatory when loading such a structure up to failure (Figure

1). T u n n e l s : In recent years many finite element analyses of tunneling problems have been performed and one may expect an on going activity in this field. Indeed finite element analyses are needed to assess the effect of new tunnels on existing foundations and vice versa. As yet few published analyses involve interface elements between the tunnel lining and the soil (e.g. Duddeck [1]); this might be justified by the fact that there is little slipping on the lining. Nevertheless we expect a growing application of interface elements in this field to anticipate possible slippage. E a r t h r e t a i n i n g s t r u c t u r e s : The title earth retaining structure covers a wide range of different structures. Classical gravity structures such as massive walls, L-walls and cofferdams (Figure 2a) are regularly used. For analysing displacements of such structures one needs the finite element method, as there is hardly an alternative. In comparison to shallow foundations deformations may be large and slipping may also be expected under working loads. Hence, the use of interface elements accounting for slippage and

149

o .

,~

.

~

(b)

(a)

(c)

(d)

Figure 1" Application of interface elements; (a) geotextile or geogrid; (b) slender sheetpile wall; (c) diaphragm wall; (d) shallow foundation.

m

w m

I (a)

(b)

(c)

(d)

Figure 2: Earth retaining structures. possible gap development is highly important. In contrast to gravity structures, sheet pile walls, diaphragm walls and soldier pile walls are mostly propped of anchored. In combination with modem grout anchors (Figure 2b) a complex soil-structure problem is obtained, as anchors may yield by slipping between the grout body and the surrounding soil. Such yielding can be conveniently modelled by means of interface elements. Reinforced earth walls (Figures 2c and 2d) with a fine spacing of geotextiles or geogrids can be considered as a gravity structure. F o u n d a t i o n piles: As yet the finite element method is rarely used to predict loadsettlement curves for foundation piles (e.g. Randolph [2]). Instead, in-situ load tests are commonly used and the combined approach (i.e. both test loading of in-situ piles and finite element analyses) is non-routine. To date combined projects have only been carried out for small model piles in research programs. No doubt very precise predictions of load-settlement curves can hardly been expected, since the unknown horizontal soil stresses tend to dominate this problem, but the method might well be calibrated and improved on the basis of test data. In this paper considerable attention will be devoted to the tension pile problem. For this particular problem the deformation is concentrated in a thin zone of intense shearing soil around the pile. It is a pure form of soil-structure interaction that cannot be analysed without the use of interface elements at the soilpile contact. Obviously, the analysis of a tension pile is quite similar to the analysis of

150 a vertical grout anchor. Remark: It is well known that in the context of elasticity re-entrant corner points of structures create singularities. From our point of view, at these corner points, it is advisable to extend the line of interface elements slightly into the mesh by adding one extra interface element. This is indicated by the extended dashed fines in Figure 1. The extra interface elements must obviously get full soil strength, and not the reduced wall friction. For details on this topic the reader is referred to Van Langen and Vermeer [3].

DFtEC~SHEAR P o t y o n d i (1 961 ) D e s a l et al.(1 9 8 5 )

.NdNLJI.A~~ Brumund & L e o n a r d s (1973)

I : t N G ~ Yoshlml & K l s h i d a (1 9 8 1 )

Figure 3" Direct shear configurations for experimental study of interface behaviour.

3. M O D E L L I N G OF B A S I C P H E N O M E N A TERFACE BEHAVIOUR

IN SOIL-STRUCTURE

IN-

3.1. I n t e r f a c e s v a r i a b l e s The following sign conventions will be used in this paper: normal stress is considered positive in compression, and dilatancy (increase in volume) is considered positive. Let be t and [u], respectively, be the stress vector acting on an interface and the relative displacement vector across the boundaries of the interface. Their time derivatives are denoted t" and [u_"]. The axes of expression of these vectors are two axes s and t belonging to the tangent plane of the interface, and n being orthogonal to this plane. The number of components considered in the ensuing is C (C = 3 in the 3D case and C = 2 in the 2D case). 3D: 2D:

3.2.

t-- {7" fin} T , [U] = {[W] [lt]} T, t_''- {'/" O'n}T,

ILL]}T

E x p e r i m e n t a l results

In laboratory, the experimental study of soil-structure interface behaviour is carried out following one of the three configurations showed in Figure 3. The direct shear box (Figure 4, after Boulon [4]) is up to now the best suitable tool for studying such a type of behaviour. In-situ devices as the penetrometer or the vane-test apparatus partly bring out some information on direct shear mechanisms. The penetrometer works as an annular direct shear test and the vane-test mesures the localized soil-soil direct shear.

151

Figure 4: The classical direct shear test. For example, results of direct shear tests between Hostun sand (dh0 - 0.74ram, ID = 0.9) and a rough material are presented in Figure 5. The rough material is constituted by some sand glued on a steel plate. The tests at constant normal stress (S) and at constant volume (V) have to be considered as extreme direct shear paths. These tests outline the kinematic and static aspects of dilatancy, but also the contractive effects of shearing load that points out and intense grain crushing due to the localisation of plastic energy dissipation within the interface. 3.3.

Constitutive models

Two major kinds of constitutive equations are used for modelling the soil-structure interface behaviour, often associated with the finite element method. The first one considers the soil-structure interface as a thin continuum (Desai [5], Ghaboussi [6]); the thickness of the interface elements should then be specified. These models are discussed in section 4.3 related to interface elements. In the second approach, the interface zone is replaced by a two-dimensional continuum (Boulon [7], Gens [8]), subjected to kinematic discontinuities and exhibiting tangential as well as normal displacement jumps (relative displacements). Following this approach, the interface thickness is not a constitutive parameter, since it is directly embodied in the constitutive equations chosen for modelling the interface behaviour. This implies that the interface is considered as a zone of zero thickness where kinematic discontinuities take place. In the ensuing some constitutive laws belonging to this class are discussed. The interface equation is the matrix relation _d between the two vectors t" and [~] for any possible state of the interface material, and for any tangent loading of this material, the relative displacements being assumed to be small: t_"= d(state, tangent loading).[u']

(1)

The matrix _d consists in diagonal and off diagonal terms: d=

[k,,

kns

k,. ] in t h e 2 D

knn

J

case.

(2)

152 a (kPa)

[u](mm)

1100._

.80

_

.62

_

880.

_

660.

_

.44

__

440.

_

.26

_

220.

_

.08

_

r

\

I

__~__, (v) ! (s)

t

§ / /

.00

_

-.10

.00

4.0

8.0

12.

16.

1100._

880.

880.

_

660.

_

4~).

_

220.

_

440.

I;

220. "+-t-~,, " : : : : :': : :

: ,~: : : : :

.00

.00

m

.00

4.0

8.0

12.

16.

20.

8.0

12.

16.

_

.00

20.

[w](mm)

(kPo)

T

1 IO0._

660.

4.0

.00

[~](~)

I" (kPa)

_

20.

/ /

Y 220.

n

440.

660.

880.

1100.

[~](r.m) . (kPo) Figure 5" Classical dense sand-rough material interface behaviour. S: test at constant normal stress; V: test at constant volume.

The diagonal terms are the internal shear ( k , ) and the internal normal (k=,,) stiffness. The off diagonal terms kon and k,o express the coupling between shear and normal phenomena. This coupling is activated by the well known dilatancy taking place during shearing of a dense material. An incremental or tangent path is described by the 2C (C = 3 in 3D and C = 2 in 2D) components of the vectors t_"and [u_']. Generally, we can prescribe C components or relations of the incremental path (see Examples 1, 2 and 3 presented later), the C remaining components being given by the constitutive equations. For some special incremental path a component number smaller than C is only allowed to be prescribed (Example 4).

Examples of i n c r e m e n t a l p a t h s (2D case, C = 2) Example 1: direct shear at constant a,. prescribed path: response deduced from the constitutive equation:

[tb] = [~-~o], &, = 0 [~] = - ~-~.[~-o],/" = [k,, _ k,..k.,k..].[~---]

E x a m p l e 2: direct shear at constant volume.

153

I~~l~

initialstate On

Figure 6: Special direct shear path from a plastic state of stresses.

prescribed path: response deduced from the constitutive equation:

[tb] = [~"o], [u] = 0 § = ko,.[~-o], an = kno.[W"o]

E x a m p l e 3: direct shear at prescribed external normal stiffness ke. prescribed path: response deduced from the constitutive equation:

[tb] = [~-"o], ~ = ke [u] = k.-k.. [6-ol,

= [k.. +

k.,~.kn.

E x a m p l e 4: a special direct shear path (pseudo-oedometric) from a plastic state of stress (Figure 6). prescribed data: &n = -c%0, normal unloading, #n0 > 0) additional data (Figure 6) § = b~.tan6 response deduced from the constitutive equation: /- = -tan&ano, [~] = k..t'nS'k"~176176 .a,~o-

0]

The elastic interface models ([22]) exhibit only diagonal terms:

d = 0

in the 2D case.

(3)

k,n

For that reason, they axe not able to simulate the dilatancy phenomenon. As a consequence, with elastic models there is no difference between incremental paths at constant normal stress and at constant volume. In the non-linear elastic models proposed by HoUoway [9] and Desai [10], [11] the coupling between normal and tangential phenomena are omitted, thus dilatancy description is absent. Most interface models proposed in the literature are elastoplastic (Ladanyi and Archambault [12], Ghaboussi [6], Carol [13], Gens [8], Boulon & Nova [14]). The last improvements, incorporate the hardening law (Gens [8], Aubry [151, Boulon & Jarzebowski [16]) in order to model the cyclic behaviour. Some three dimensional aspects have been studied by some authors (Carol [13]). All these models are based on classical concepts of soil mechanics, like the critical state theory, the dependency upon the

154

T f

~'~~

on

":';;i

J

oo~ f on

(b)

(a)

On

(c)

Figure 7: Coulomb type yield surface and extensions. effective isotropic pressure, the stress-dilatancy relationship and the analogy between volume and interface variables. Another family of interface constitutive equations is the incrementally non-linear relationships of the interpolation type (Darve [17]). For this class of models it is assumed that the behaviour of the interface material for a particular class of loading path is known. These paths have been called "elementary direct shear path". For these particular loading paths the behaviour is described by analytical expressions and an interpolation rule is required in order to compute the incremental response for any unknown incremental loading path. 3.4. An elasto-plastic interface law In this section an elastic perfectly plastic interface law is briefly described. Usual concepts are used like the summation of elastic and plastic relative velocity, an elastic constitutive equation (see hereabove), a yield criterion f and a plastic potential g. The yield criterion is defined by the wall friction angle ~ of the interface and its adhesion a. The plastic potential which allows for the flow rule uses the interface dilatancy angle ~.This leads to the Mohr-Coulomb type functions: (4) (5)

f = r - a, tan6 9 = r - a, tanO

A switch coefficient is introduced; thus a takes the values 0 (elasticity) or 1 (etablished plasticity); the resulting tangent behaviour is described by:

&~

ko, + k , ~ . T

akook~tan~b

k,~.k~ + k~,T.(1 - a )

[6]

(6) where T = t a n S t a n ~ . The constitutive matrix (fight hand side) is singular in the full plasticity range (a = 1), and non-symmetric for ~ ~ if, as usual for soils. The above simple model suffers obviously from some shortcomings. The first shortcoming is the allowance for tension in the case of cohesive materials. Tensile stresses up to

155

1;

j [u]

(a)

(b)

-reality s a+Gn!tan f

[wl

[w]

Figure 8: Performance of Mohr-Coulomb model in test with constant normal stress. a magnitude of a.cotan~ can occur, as illustrated in Figure 7a. This can be improved by adding the extra tension cut-off yield surface of Figure 7b or by introducing a new yield surface as illustrated in Figure 7c. Mathematical formulas for the latter type of yield surface have been proposed by Gens et al [8] and by Bonnier [18]. The use of a curved yield surface ( f = 0) requires a non-linear plastic potential function, g. The tension cut-off is generally combined with an associated flow rule in the tension range and with a yield vertex at the intersection of the two surfaces. Both the tension cut-off and the curved yield surface model debonding, i.e. gap development between the structure and the soil, but within the concept of perfect plasticity immediate rebonding is predicted upon reloading; this is obviously incorrect and concepts of hardening and softening are needed to model proper rebonding in cyclic loading (gooijman and Vermeer [19]). The second shortcoming of the present model is the continued dilatation during shearing, at least for r > 0. In reality dilatation will disappear as soon as the soil particles have reached a critical void ratio, as illustrated by the dashed line in Figure 8b. For improvement, a dilatancy cut-off criterion as described by Vermeer [20] can be added for the so-called advanced Mohr-Coulomb model. In addition to a tension cut-off and to a dilatancy cut-off, this model also involves non-linear elastic stiffnesses:

ko. = k.o(<,,,/o-,,o)"

k,,,, = k,,,,o(O-,,/<,,,o):

(7)

where a,o is an arbitrary reference pressure and k,oo, k , , o a n d n are constants. For very accurate analyses of soil-structure interaction, one obviously need accurate data for these constants. For engineering purposes, one tends to use the exponent n = 0 and the elastic stiffnesses are neither related to measurements. Instead, ko~ and kn, are treated as penalty coefficients; they are taken as large as possible without degrading the numerical convergence of the equilibrium iterations in non-linear finite element analyses. For r = O, the elastoplastic behaviour of the interface becomes independent from [tb] and is reduced to:

As in linear elasticity, there is no difference between the incremental paths at constant normal stress and at constant volume. In this paper restriction is made to perfect plasticity, but hardening-softening plasticity is needed to model interface behaviour in

156 more details. For such studies, the reader is for instance referred to Aubry [15] and Boulon [16], [14].

3.5.

A n incrementally non-linear interface law

For the ensuing the general principles are presented. The basis of a three dimensional soil-structure interface model in specific material axes and the experimental device used to obtain the model parameters have been developed by Boulon [7]. The law is considered as a one to one non-linear relation between an incremental loading [6]dt and the corresponding incremental response t" dr, where dt is an infinitesimal time step. It is useful to normalize the set of incremental loadings and responses from an initial state by:

II[~]lldt

([~,1~ + [,~]~)'/~a,~

=

(9)

Then the direction of the incremental loadings appears as: {,x ~,}z _

1

_ ~_ll[hll----T{['i'] [~1}~

(10)

All incremental loadings belong to an unit circle in the (A, p) space. The normalized response corresponding to (A, p) is therefore:

{~ .}r = l--k-{+ ~.}r II[~]II

(11)

It is assumed that the interface material exhibits a continuous incremental behaviour for the loading direction (A,/~). Only the case of a non-viscous material is examined here: as a consequence, the relation between the set of possible [~_] and/_ from an initial state is of degree one versus their components; the directional incremental parameters A and p therefore allow the various loadings and responses to be mapped in the corresponding incremental spaces. It is then possible to build the approximate incremental response by interpolation, only knowing a small number N of incremental responses to basic incremental loading paths:

r/

= ~w, i=1

r/i

-.

b.

=

b. ([tb], [~],initial state)

(12)

where Wi are interpolation functions. Several kinds of interpolation functions (polynomial, trigonometric) were investigated (Boulon & Garnica [21]). The basic direct shear paths which can be performed in the laboratory correspond to tests at constant normal stress (S), at constant volume (V) or pseudo-oedometric -oedometer after shearing- (PO); loading and unloading can be taken into account. (V) and (PO) paths have a fixed position on the unit circle in the (A,p) space (Figure 9) -except the fully plastic (PO) path- while the point corresponding to the (S) paths changes during a test (because of an evolution from contraction to dilatancy). The incremental paths are obtained by derivation of analytical expressions representing r and [u] as functions of [w] for the (S) test, r and a, as functions of [w] for the (V) test, and a, as function of [u] for the (PO) test.

157

11

/

.v,u<

i

s,,

I

V,I

~>

S,

V

PO,I V: constant volume S: constant normal stress

, ~ PO,u

PO" pseudo-oedometric I-loading u "unloading

Figure 9: Incremental normalized spaces, loading and response Finally the constitutive matrix can be obtained via Euler's theorem for homogeneous functions (of order one here):

//I

[~]

/

4. F.E. A N A L Y S I S : P R I N C I P L E S

In this section we shall describe the basic principles for modelling the behaviour of two drained bodies in contact, with friction along a surface called soil-structure interface. 4.1.

N u m e r i c a l t r e a t m e n t of the contact p r o b l e m with friction

Let be f~ the space domain corresponding to the two bodies in contact, F, the part of their external boundary undergoing prescribed tractions t_-,F, their potential contact surface during the time interval (ri, r l ) . For the time step (m-l, r.), being in the range ri < r~-I < r < r~ < r l , the principle of virtual work yields:

]~. r~.(L. - L._,)dr

(14)

V ~_2 virtual field of velocity kinematically admissible, and where: a_~ is the pseudo-vector corresponding to the stress tensor, is the pseudo-vector corresponding to the time derivative of virtual strain at time r, t, is the stress vector acting on the interface, fir is the prescribed traction vector, f_.~ is the body force vector.

158 After discretization the real velocity field at time r inside the element f/e is defined by: =

H='fZ,

(15)

where U., is the nodal velocity vector at time 7 and H e is the matrix of spatial interpolation functions. The real strain rate field yields (Be: derivation matrix): (16)

s = B~

In the same way, the relative velocity field is defined as:

[,_]:. = ~.u_".

(i7)

where b~" is the matrix allowing for the calculation of relative velocity from absolute velocity. Using the pseudo-elasticity (or auxiliary) matrices D==~(volume) and =0~" (interface), equation (14) becomes (after linearization with respect to time):

K==o (U__. - U__._,) = ( F . - F . _ , )

+ (L_~ - ~ _ o, )

-

(L_. -

(18)

L_._,)

with

(zg)

- ~ -0, 1 =

~ t ~0 . - ~ . _0, ) d r + ~ ~.er,

Tb-,.(to = ,~r.

-

o

~._,)ar

]

(21)

.

A n iterative form of equation (18) is preferred. The subscript i denotes the number of the equilibrium iteration of the nth load step:

K=o {(~ - U._,)' - (~ - ~_,)'-~} = (F_. - ~_,) -(L_. - ~_,)'-'

(23)

The matrix K==o allows control of the equilibrium iterations. W h e n convergence is reached, the internal real nodal forces (L, - L_,_I)~-I equilibrate the external nodal forces (F, - F_,_I). (L__,- L_,_I)~-I is calculated by local integration of the real changes in stresses between T, and T,_~.

159 4.2.

Global loading

The designers are interested in obtaining the equilibrated solutions taking into account the behaviour of the volume and of the interfaces at the time steps (1,2, .... , n 1, n,..., f). They are also interested in minimizing the number of time steps, and at the same time obtaining a well converged solution at each time station. There is a compromise to be achieved between the size of the loading steps and the rate of convergence. Generally the first loading step is choosen very small in order to make the convergence easier. The same technique is applied if there is a change in the global loading direction. Most of the calculations are performed with the modified Newton-Raphson technique (i.e., without recalculating K==0). But in case of highly non-associative constitutive responses (volume and interface) the true Newton Raphson technique seems to be better. The presence of interfaces in a boundary value problem reduces the speed of convergence specially when debonding appears, because there is a large difference between the tangent stiffness of an open joint and of a closed joint. Riks's method, which allows control of the convergence in real time and to adapt the size of the loading step to its level of non linearity, is a very useful tool for optimizing the number of loading steps. 4.3.

Interface e l e m e n t s

Interface elements are numerical entities used in the finite element technique for modelling kinematic discontinuities that are present in some boundary value problems. In numerical practice, two types of interface elements are used: a reduction from a classical volume finite element (thin layer interface elements) and a surface element which is a zero-thickness joint element. For the first type of element (Ghaboussi et al [6], Desai et al [5]), the soil-structure interface is considered as a thin continuum and the thickness of the interface should be specified. This may cause some problem since the real thickness is often unknown and is also very small compared to the other dimensions of the region. It is even unclear which type of test should be performed to determine the constitutive law of the material within those elements: the soil should in fact be subjected to very large deformations for modelling this interface behaviour. No doubt that the simplest approach to model soil-structure interfaces is to use thin continuum elements. This would indeed circumvent the use of special elements and one would not have to extend computer codes to include elements that create unnecessary problems for pre-, and post-processing routines. Unfortunately this leads to some complications. In non-associated plasticity the limit load may be influenced by initial stresses and by the sequence of loading, at least for frictional soil. The classical zero-thickness interface elements are the joint elements of Goodman [22] and Goodman & Dubois [23]. An important improvement is due to Zienkiewicz et al [24] with the so called "no-tension" elements. A generalization for axisymmetrical problems was presented by Heuze & Barbour [25]. An important concept brought out from these works is the notion of shear path at prescribed normal stiffness, implying that an external normal stiffness can be different from the internal one. The soil-structure interface behaviour considered as a surface property is a pragmatic approach used in order to model very complex phenomena (large strains, large rotations, large relative displacements and rotations of the grains themselves) occuring between the two boundaries of a shear band inside the soil along a structure. Many interface

160 IL

9

9

a

9 . y

(b)

(c) 2 9

10 v

~

v

v

f

(a)

1

-1

/ 0

Figure 10: Elements used in the P L A X I S

+1

code (volume and interface).

elements are presented in the literature. For example, the 2D finite element code P L A X I S (Vermeer [20]) uses 15 or 6 noded triangular elements (Figure 10a) for the discretization of the volume. The corresponding interface elements are 10 or 6 noded joint elements (figure 10b). The geometrical transformation is linear (Figure 10c), and contains a rotation matrix R. This rotation matrix R is built with the normal and tangential unit vectors of I',. The auxiliary stiffness matrix is written as-

= [{=+x Tb__~.TR ~=o"~ bW'det(J)cl~ L

d~=-I

(24)

--

where J is the Jacobian of the geometrical transformation and ~=o~"is the local auxiliary constitutive matrix of the interface expressed in the local axes. The choice of the spatial integration method has recently caused considerable controversy. If we correctly use a priori any numerical integration method (Gauss, NewtonCotes, Lobatto, ...), it must induce correct results provided that the discretisation is adapted to the gradient of relative displacement along the interface. Nevertheless Hohberg [26] and Gens & al [8] have showed that the best results for stresses are obtained using a Newton-Cotes integration. Van Langen [27] observes that the stress oscillations observed when using Gauss integrations are produced by complex equivalent nodal tansfers between volume and interface elements. This problem disappears if the element density is large enough (Schellekens et al [28]). From our point of view, the stress sampling error is also a function of the integration method. In the ensuing the Newton-Cotes method is used, since in case of interface constitutive equations involving the relative velocity, the better sampling points are the nodal points 5. N U M E R I C A L

INTEGRATION

OF CONSTITUTIVE

MODELS

The techniques for local integration of interface constitutive models are discussed in this section. Most time integration methods for constitutive equations use a constant increment. The increment size is initially selected considering that the difference between integration with this size and the one with a lower size should be negligible. For the i *h equilibrium iteration of the global loading step n, equation (22) contains the term (t_y,- _t,._l ) that needs a local time integration between times rn-] and rn (dr: infinitesimal step of time):

161

(t,.

t_,._,)'-' = f"

-

s

(25)

n--1

The relative velocity [u'] is considered as constant during this loading step n, then equation (25) can be transformed: (tn - t,~-i

J~:L~---'r~+d[tt]~')i

(26)

Formula (26) shows that the time integration is replaced by an integration on a path defined by the history of relative displacements, which is a classical exercise for constitutive models, at constant or adaptive increments. Several integration algorithms have been proposed in such a way that consistency, stability and objectivity conditions are satisfied (Belytschko [29], Charlier [30]). However it is not rare that the elastic prediction gives a stress state beyond the yield surface. Because such stress states are physically inadmissible it is classical to use return schemes in order to map the elastic prediction on the yield surface. Other methods such as the sub-increment technique allow for improvement of the result of this integration. In case of incrementally non linear interface constitutive equations we prefer to use an adaptive local integration method (Rouainia [31]) where the size of the increment depends on the curvature of the local stress path and allows for the reduction of the number of increments and at the same time to control the precision. The stress vector t is expanded in a Taylor series at time r, assuming [6] to be constant for this global loading step:

t_,.+d,. = t_,. + dr_,. + apt_,.

(27)

with 1

dr, = g .d[~] ;d~t, = ~ d (d,).dh]

(2S)

We use an explicit integration at first order versus time providing that the second order term is negligible compared to the first order term:

IId2t, lloo << IIdLIIoo

(29)

where I1.11oois the infinite norm. The second order term can be approximated numerically by:

d2t, ~- dt,+d, -dt,

(30)

and equation (27) becomes: -1=

p

Ilat_,+~, at, IIoo < < II~t, lIoo -

1

(31)

The precision of integration is then characterized by the index 1/p. If the norm IId[~]lloo gets lower, the term p increases and the criterion (31) is satisfied. By fixing an interval of allowable precision, it is possible to regulate the increment size of local integration step. The adaptive step size criterion is finally written as:

162

1 Pmaz

_< _1 _< 1 P

(32)

Pmin

It should be highligted that the concept of integration of the interface constitutive equation can be enlarged to situations after debonding. In this case, the continuation of the integration is simply written: dt_~ -- 0

6.

F.E. A P P L I C A T I O N S

6.1.

(33)

AND RESULTS

N u m e r i c a l s i m u l a t i o n s of pull-out t e s t s in calibration c h a m b e r

Extensive research on the behaviour of interfaces between granular soil and structures has been developed since the early 8O's by Boulon [32] with a special emphasis on the application to the prediction of lateral friction along piles (Boulon & Foray [33]). It has been considered that pile tests in a calibration chamber could been used for two purposes: 1. as a boundary problem to check numerical simulations, 2. as a physical model to give practical design values. Experimental visualisations have shown that during the penetration or during a loading stage of a pile in sand, large shear displacements are localized in a very thin zone of about ten times the mean grain diameter of the sand, close to the pile shaft. This suggest that the mechanism governing the pile-soil contact can be modelled by a direct shear test, between the soil and a rough material. In this section we present numerical models (Rouainia [34]) of puU-out tests in a small prototype calibration chamber (26 cm in internal diameter, 50 cm in height) designed and tested by Eissautier [35] to study the effects of lateral boundary conditions on friction elements of 3.2 cm in diameter. Modelling has been made with the incrementally non linear model of interface behaviour and with the elastoplastic model.

F.E. mesh s y s t e m The mesh system consists of 16 volume elements (12 Gauss points by element) for the soil and four elements for the pile. We use four interface elements. Figure 11 shows the geometry, the boundary conditions and the finite element mesh of this problem. The computations are carried out into two stages: 1. oedometer initialisation of the stress state due to the surface load t- for a weightless material, within the soil mass (a,r = aeo = K o a z z ) , and within the interface ( a n = K o a z , ) , r = 0 -moulded pile-. K0 is a function of the in-situ density and of the stress history, and az, = t-. 2. history of vertical displacement applied to the head of the pile. The loading step is 0 . 1 r a m .

163

Figure 11" Geometry, boundary conditions and mesh for the calibration chamber problem (axial symetry).

case 1 case 2 case 3

ID

eo

0.9 0.3 0.9

0.55 0.73 0.55

~ (kPa) Ko Eloo(MPa) 589 589 284

0.4 0.4 0.4

65 38 65

v 0.27 0.27 0.27

r 42 42 42

e0 " initial void ratio, E10o : Young modulus measured from a triaxial test under a lateral stress of 100 kpa, r : friction angle. ID : relative density index Table 1" Mechanical parameters utilized of the Hostun sand (ds0= 0.74 ram). Typical results obtained for the initial conditions and for the mechanical parameters shown at Tables 1 and 2 are presented. For the first computation, the unbalanced stresses at the integration (Newton-Cotes) point number 129 within one of the interface elements are presented in Figure 12. A high convergence rate is always observed excepted for the first global loading step corresponding to the initialisation of the stress state . This convergence is generally obtained after six equilibrium iterations per loading step.

6.2. Local paths The calculations resulting from the elastoplastic interface model show a continous increase in the local mobilized friction, resulting from the constant dilatancy angle

164

[u]+ ~+,1~,o (drld[w])so (drld[w])vo

~,0 124 354 707 1061

0.70 0.62 0.45 0.35

15 7 4 2

8 lO 12 14

,5.p

~

50 I00 145 160

6.,.

28 ~ 47 ~ 30 ~ 27 ~ 41 ~ 28 ~ 26 ~ 38 ~ 27 ~ 26 ~ 35 ~ 26 ~

a,0 (kPa): initial normal stress, [u] + (mm): m a x i m u n of the normal relative displacement during a direct shear test at constant normal stress (S), a + (kPa)" m a x i m u n of the normal stress during a shear test at constant volume (V) (dr/d[w])so ( k P a / m m ) : initial slope during a (S) test, (dr/d[w])vo ( k P a / m m ) : initial slope during a (V) test, 6: wall friction angle at constant volume, 6,p: m a x i m u m friction angle during a (S) test, 6or: residual friction angle during a (S) test. Table 2: Main mechanical p a r a m e t e r s of soil-pile interface

650

-L,.-

400

9~

150 ,

-I00

!

.

0

v

'.

_.,..,llmlml~l)-inll'

.

.

,

9 i

*

I.

j

9 ,

-

.

9 v

[

I

I,

J

I

J

v

f

f

!

!

I

l""I.

I00 200 300 Cumulated equilibrium iterations

il

9 i,

I

I

w

,

i

v

400

Figure 12: Local convergency at the integration point n u m b e r 129 at interface elements for the case 1. 1[ vii = lit i - ~[[2, t i and ~ , respectively real and pseudo-elastic stress vectors at the equilibrium iteration n u m b e r i.

165 assumed in the plastic range. An advanced Mohr-Coulomb model based on the critical state theory allows to overcome this shortcoming (Vermeer [20]). Figures 13 to 15 show the local paths at some integration points within interface elements for each of the three calculations performed.

(~)

(b) (kPa)

a

(kPo)

1000._

IOO0.

_

/

800.

600.

_

~~/;

6OO.

400.

_

200.

_ ~

800.

-~".,

13g

;

,..___~ 148 3--0 El) l a w

EXPERIMENT

400.

1

0.0

~

200.

_, 0.0

.00 2.O

4.O

6.0

8.0

10.

(c)

2.0

.C

[w](mrn)

[ul(mm)

4.0

.07

t

(d)

BOO. _

7

.o,

~"41

~

-.05

_ .00

10.

[wl(mm)

1000._

-.o2

8.0

-r (kPo)

.10

.o,

6.0

2.0

4.0

6.0

8.0

10.

[wl(mm)

600.

_

400.

_

200.

_ _

.00 _ .00

I

~r162

~

~___.el4s

/

S

X" ""-1 ~

200.

400.

iO0.

800.

1000.

a

(kPo)

Figure 13: Local paths at integration points 139, 144 and 148 (case 1). ID = 0.9, a,, = 5 8 9 k P a : evolutions (a) of the normal stress, (b) of the shear stress, (c) of the normal relative displacement; (d) stress path; [w]" local relative displacement. The evolution of the normal relative displacement (curves c) during shearing shows the classical interface behaviour between dense sand and a rough material: an initial contraction, followed by a dilatancy after a tangential relative displacement of around 1 mm. This dilatancy induces an increase in normal stress (curves a) because the local paths nearly correspond to paths at prescribed normal stiffness. At high stress levels a contraction phenomenon has been generally observed after a tangential relative displacement of around 5mm. This phenomenon is partially due to progressive grain crushing during shearing. The evolution of shear stress is compared with the experimental mesurements (Figures 13b to 15b).

166 At the integration point 148 (at the head of the pile), small values for normal and shear stresses seem to indicate a local tendency to debonding. Generally, the local unit friction computed by the incrementally non linear model is higher than the unit friction modelled by the elastoplastic model, because the wall friction angle of the elastoplastic model was choosen as the residual value (large tangential relative displacements). In contrast, the experiment and the first model exhibited a peak unit friction of approximately the same magnitude.

(~) a

(b) (kPo)

750.

r 9 (kPo)

I--,-]

_

,I

6 ~ o

_

450.

_

300.

_

14---+t-

k-.._~t4. ~ f ~

2.0

4.0

6.O

8.O

[~](mm)

300.

_

200.

_

.07

~. .... .it48

.04

~

/, _ _

I

/

f ~

_.o2 -.05 2.0

4.0

6.0

8.0

10.

[~](mm)

" t - - - I- 1 4 8

$._..~El)law ~---0~fl=KRIMEICr

2.0

4.0

6.0

8.0

10.

[wl(mm)

(d) (kPo) _

i

i

139,144

600.

_

450.

_

300.

_

150.

_

.00 .00

~.

_

750.

~.____ 139

~...~I~ /

.00

r

_ I

.01

_

10.

[.](ram)

(c) .10

400.

.00

_ .00

_

100. _

__

.00

500.

14---+1~

I

_ .00

150.

300.

450.

600.

750.

o"

(kPo)

Figure 14: Local paths at integration points 139, 144 and 148 (case 2). ID = 0.3, 589kPa: evolutions (a) of the normal stress, (b) of the shear stress, (c) of the normal relative displacement; (d) stress path; [w]: local relative displacement. a,, =

6.3.

Global

results

Comparison between the computed and mesured load transfer along the pile during loading are showed on Figures 16 and 17. The curves corresponding to the small loads are quite different because the initial negative friction due to the installation of the sand around the pile has not been simulated in the calculation. The curves related to

167

(a) a

(b)

~- (kPo)

(kPo) 1000._

500.

_

800.

_

400.

_

600.

_

300.

_

400.

_

200.

_

200.

_

100.

_

139,144

i+--.+J~

/

!

/

~._

y ,

/,

.00

_

.00

~

B

2.0

4.0

6.0

8.0

,/

, /

_

.00

10.

159,1, ._+148

/ m

2.0

4.0

8.0

6.0

10.

[w](r~m)

[,,](mr.)

(c)

i]

(d)

[.](ram)

-r

.10

(kPo) 1000.

_ ,..

i

I

139,144

;/ .02

-.02

--

coo.

F"k'-....

_

h___4.s48

600. _

r

4-1~1 o,,..

-- -

400.

_ f

-.06

-.10

_ .00

2.0

4.0

6.0

8.0

10.

[w](mm)

200.

_

.00

_

l

f

j

/

_

( /

200.

400.

600.

800.

1000.

a (kPo)

Figure 15: Local paths at integration points 139, 144 and 148 (case 3). ID = 0.9, a,, = 2 8 4 k P a : evolutions (a) of the normal stress, (b) of the shear stress, (c) of the normal relative displacement; (d) stress path; [w]" local relative displacement. the maximum load (nearly linear in all cases) show a difference in head load of about 25 to 30%, since the influence of the debonding is not properly modelled. The elastic properties of the sand itself were elastoplastic with a mean modulus not influenced by changes with respect to the stress level. Non linear elastic properties function of the mean stress levels improve the results (Vermeer [20]). Figure 18 shows the load-displacement curve of the model pile, resulting from the computation (two models of interface) and from the experiment. It should be noticed that the range of peak value corresponds to a mean value of the coefficient of earth pressure (K) acting on the pile (for a loose sand the value of K is about (3 in our case). This value is about ten for a dense sand. That means that the initial normal stress acting on the shaft has been magnified by a factor 12 (loose sand) to 20 (dense sand). This magnification corresponds to the well known phenomenon of apparent friction.

168

a) computation

,/ i~

1.0

/

V !

' .60

C~ S

,

:

.80

/

00 9

i"

.60

[w]=O.Smm ~___+[w]=|.Omm

o

.... '0

~

[w]=2.0mm [w]=2.Smm

I ~

LF.......~ [ w ] 20.

30.

t

!

~

/

i

Ii

9

i/:)

,,

.40

[w]=O.Smm

2.,"""

+___+[w]= t.omm ,. ......~ [ w ] = l . S m m [w]=2.0mm

.20

I

c ~ [,,1==~.5==

/

,.._...,.[w]=3.smm + .......+ [w]=4.5mma

.oo

,~5mm [L 40.

i

I

h~"---~ [w]=3.Smm J

10.

,

.80

s.."

i~t.t....~;'-

:

"i

/,'/, / :::/:Y i "

.

.

,'

li /Z~: r//~,.u ~:

.00

._

/ "/ /

I: : :~ I://,:.

.20

1.0

/ ,,:/

"

/

9

/Ir"'

.40

a) experiment

50.

10.

.00

20.

'

I

30.

40.

1-

50.

Load transfer Qz (kPa)

Load transfer Qz (kPa)

Figure 16: Comparison between the computed and the measured load transfer curves. Case 1" ID = 0.9, cr~ = 5 8 9 k P a .

a) experiment

a) computation 1.0

t ,

.80

'/,

1.0

/

/

I : / / : / ' :: / " ;i/ ,'." /.:/. /'

.60

a S .40

/,/

If:/j .2O

.00

.00

5.0

;

.80

10.

/."

/

:i

I//

//

o:

I /~'

.60 a N

[w]=O.5mm .i.----i.[w]= l . O m m ~........ -~ [w]= l . S m m 3)....(D [ w ] = 2 0 m m [w]=3 5 m m -~"-- -",t- [ w ] = 4 . S m m

i

i

15.

20.

Load transfer Oz (kPa)

.20

_ 25.

},//,'/

.40

.00

[w]=O.Smm

+___+ [w]= 1.onun

Z7 g .00

5.0

~. ........~ [w]= 1.5mm .... ~E) [ w ] = 2 . 0 m m

[w]=3.smm ~___~[w]=4.smm i 10.

15.

20.

25.

Load transfer Qz (kPa)

Figure 17: Comparison between the computed mid the measured load transfer curves. Case 2: Io = 0.3, a~ = 589 kPa.

169 25. Z .2s 0

20.

O' "(3

l::l 0

15.

C 0

..."

......... ;,4~ .~'': ................. -)(~J

10. C

.4 /

.....

O ~-~'"

5.0

,"

I

, ,"

INL law -t----+ EP law ~_........_~EXPERIMENT

.00

:

I

.00

2.0

4.0

6.0

Head displacemenf

8.0

10.

(mm)

Figure 18: Load versus prescribed displacement" a comparison. Case 2: ID = 0.3, a~ = 589 kPa. 7. C O N C L U S I O N The design of structures subjected to soil-structure interaction and specially to contact with friction should be tackled using soil-structure interface constitutive equations. These laws differ from the laws for soils due to three main features: the size of the relative displacements and of relative rotations between grains, the high level of dilatancy and contraction under shearing, and the presence of an intense degradation effect resulting from localisation in the pattern of a shear band. The elastoplastic interface constitutive equations are easy to use but do not modelize all these effects. The incrementally non linear interface constitutive equations seems to be more versatile for modelizing all these interdependent phenomena. Nevertheless, their use remains difficult at the local integration level as well as the global (structural) level. Special techniques of adaptive step were developped for this purpose and still need improvement. The recent progress in modelling have shown that some numerical techniques (virtual thickness, penalty methods,...) have to be interpreted from a constitutive equations point of view (for more simplicity and clarity). At the present time, the finite element method should be complemented by accurate constitutive equations, specially for interfaces, and we can hope that in a near future, this research area will be able to progress from a qualitative description toward a quantitative one. 8. A C K N O W L E D G E M E N T S The work described in this paper was supported in part by the " G R E C O G6omat6riaux" and by the " E U R O - G R E C O Geomaterials" (soils, concretes, rocks)-contract n o SC1"0073 C(EDB)-. This support is kindly acknowledged.

170 REFERENCES

1. Duddeck H. (1991), "Application of numerical analyses for tunnelling", Int. J. for Num. and Anal. Meth. in Geomech., vol.15, pp. 223-239. 2. Leong E.C., and Randolph M.F. (1991), "Finite element analyses of soil plug response in open-ended pipe piles ", Int. J. Num. Anal. Meth. in Geomech., 15(2), pp. 121-141. 3. Van Langen H., and Vermeer P.A. (1991), "Interface elements for singular plasticity points", Int. J. Numer. Anal. Meth. Geomech., Vol. 15, 301-315. 4. Boulon M.(1989), "Basic features of soil-structure interface behaviour", Computers and Geotechnics, 7, pp 115-131. 5. Desai C.S., Drumm E.C., and Zaman M.M. (1985) "Cyclic testing and modelling of interfaces". Journal of Geotechnical Eng. Division ASCE, 111, n~ 793-815. 6. Ghaboussi J., Wilson E.L., and Isenberg J. (1973) "Finite elements for rock joints and interfaces", Journal Soil Mechanics and Found. Div. ASCE, 99, SM10, 833-848. 7. Boulon M.(1988), "Contribution g la m6canique des interfaces sols-structures. Application au frottement lateral des pieux". M6moire d'habilitation, Universit6 Joseph Fourier, Grenoble, France. 8. Gens A., Carol I., and Alonso E.E. (1989) "An interface element formulation for the analysis of soil reinforcement interaction", Computers & Geotechnics, 7, 115-131. 9. Holloway D.M., Clough G.W., and Vesic A.S. (1975) "The mechanism of pile-soil interaction in cohesionless soils", Research Report, Duke University, Durham, USA. 10. Desai C.S. (1974) "Numerical design analysis for piles in sand", Journal Geotechnical Eng. Div. ASCE, 100, GT6, pp. 613-635. 11. Desai C.S. (1981) "Behaviour of interfaces between structural and geologic media", Int. Conf. on recent advances en geotechnical earthquake engineering and soil dynamics, Saint Louis, USA, pp. 619-638. 12. Ladanyi B., and Archambault G. (1970) "Simulation of shear behaviour of a jointed rock mass", Rock mechanics, Theory and practice: proc. l l t h symposium on rock mechanics, WH Somerton eds, Am. Inst. of Mining, Metal and Petroleum Eng, Berkeley USA, pp. 105-125. 13. Carol I., Gens A., and Alonso E.E., (1986) " Three dimensional model for rock joint", IInd Int. Conference on Numerical Methods in Geomechanics. Ghent, Belgium, pp. 179-189. 14. Boulon M., and Nova R., (1990) "Modelling of soil-structure interface behaviour, a comparison between elastoplastic and rate type law", Computers and Geotechnics, 9, 21-46. 15. Aubry D., and Modaressi A. (1990) "A constitutive model for cyclic behaviour of interfaces with variable dilatancy", Computers & Geotechnics, 9, 47-58. 16. Boulon M., and Jarzebowski A., (1991) "Rate type and elastoplastic approaches for soil-structure interface behaviour: A comparison", Proc. of the 7th Int. Conf. on Computer Methods and Advances in Geomechanics, (G.Beer, J.R. Booker and J.P. Carter eds.), Balkema pub., pp. 305-310, Cairns, 6-10 May. 17. Darve F. (1987) "Les lois incr6mentales non-lin6aires", Manuel de rh~ologie des g6omat6riaux, Presse Ecole des ponts et chauss~es, Paris, France. 18. Bonnier P.G. (1993), "Testing, Modelling and Numerical Analysis of the Mechanical Behaviour of Bituminous Concret", Doctoral thesis, Delft University of technology.

171 19. Kooijman A.P., and Vermeer P.A. (1988), "Elastoplastic analysis of laterally loaded piles", Proc. 6th Int. Conf. Num. Meth. Geomech., Vol 2, 1033-1042, !nnsbruck. 20. Vermeer P.A. (1993) "PLAXIS version 5: a finite element code for soil and rock plasticity", A.A. Balkema / Rotterdam / Brooldield. 21. Boulon M., and Garnica P.(1990), "Constitutive interpolation and soil-structure directionaUy dependent interface law". Proc. of the second ENUMGE, Santander, Spain, september, pp 45-56. 22. Goodman R.E., Taylor R.L., and Brekke T.L. (1968) "A model for the mechanical behaviour of jointed rocks", J. Soil Mech. and Found. Div., ASCE 94(3), 637-659. 23. Goodman R.E., and Dubois J. (1972) "Duplication of dilatancy in analysis of joint rocks", J. Soil Mech. Div., ASCE 98(4), 399-422. 24. Zienkiewicz O.C., Best B., Dullage C., and Stagg K.G. (1970) "Analysis of non-linear problems in rock mechanics with particular reference to jointed rock systems", Proc of 2nd Int. Conf. on Rock Mechanics, Belgrade, Yougoslavia, pp.8-14. 25. Heuze F.E., and Barbour T.G. (1982) "New models for rock joints and interfaces", Journal Geotech. Eng. Div. ASCE, 108, GT5, pp 757-776. 26. Hohberg J.M. (1990) "A note on spurious oscillations in FEM joint elements", Earthquake Eng. Str. Dyn., 19(5), 773-779. 27. Van Langen H. (1990) "Numerical analysis of soil-structure interaction", PhD Thesis, Delft University of Technology, The Netherlands. 28. ScheUekens (1991) "Interface elements in finite element analysis", Internal Report, T.U. Delft nr25-2-90-17. 29. Belytschko T. (1983) "An overview of semidiscretization and time integration procedure", Computational Methods for Transient Analysis, (Belytschko T. &~Hughes T.J.R. Eds.), North Holland Pub., Amsterdam, The Netherlands, pp. 1-63. 30. Charlier R. (1987) "Approche unitize de quelques problSmes non-lin6aires de m6canique de milieux continus par la m6thode des ~16ments finis", Thesis, Universit6 de Liege, Belgium. 31. Rouainia M., Garnica P., and Boulon M. (1992) "Constitutive model of soil structure interface behaviour and integration by an adaptive step", Third Int. Conf. on Computational Plasticity (COMPLAS III), Barcelona, Spain, pp. 738-750. 32. Boulon M.(1988), "Numerical and physical modelling of piles behaviour under monotonous and cyclic loading", Modelling soil-water structure interactions, (P.A. Kolkman, J. Lindenberg and K.W. Pilarczyk eds.), Balkema pub., Delft, 1988. 33. Boulon M., and Foray P.(1986) "Physical and numerical simulation of lateral shaft friction along offshore piles in sand", IInd International Conference on Numerical and Offshore Piling, Nantes, 21-22 may, pp. 127-147. 34. Rouainia M. (1992) "Mod61isation de l'interaction sol-structure par 616ments finis de haute pr6cision et loi d'interface incr6mentale non-lin6aire. Application aux pieux', Thesis, USMG-INPG, Grenoble, France. 35. Essautier M. (1986) "Frottement lat6ral des pieux en milieu pulv6rulent", Thesis, I.N.P.G., Grenoble, France.

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Mechanics of Geomaterial Interface. A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All fights reserved.

173

Boundary element modelling of geomaterial interfaces A.P.S. Selvadurai Department of Civil Engineering and Applied Mechanics, McGill University 817 Sherbrooke Street West, Montreal, Qc, Canada, H3A 2K6

The present paper reviews the application of boundary element techniques to the study of non-linear processes which occur at both soil-structure and geological interfaces. The influences of the interface behavior are demonstrated through specific examples dealing with soil-structure interfaces and surfaces of fractures which exhibit non-linear phenomena. 1. I N T R O D U C T I O N There is a large group of problems in geomechanics where the non-linear processes are localized and primarily restricted to interacting distinct material regions. Geomaterial interfaces are examples of such problems where processes such as interface friction, cohesion, slip, separation, dilatancy and degradation can have a dominant influence on the transfer of loads between interacting material regions. The influences of such interfaces are important to the accurate modelling of the performance of a variety of problems associated with, for example, anchorages, load-transferring devices such as tension piles and caissons embedded in geological media, earth retaining structures, such as shafts and tunnels and fracture surfaces in rock masses [Figure 1]. In a majority of such interactions, the non-linear processes are restricted to identifiable surfaces which separate a region into zones with predominantly elastic characteristics. The laws governing the interaction of interfaces can include a variety of phenomena which can be classified according to Coulomb friction, finite friction, interface plasticity, damage and asperity degradation, visco-plasticity and creep. Alternatively, interface responses can be derived from field studies and laboratory investigations. The analysis of non-linear interface phenomena can be approached via a variety of analytical and numerical methods. The use of analytical techniques for the study of interfaces and contact problems have primarily focussed on the investigation of the response of either frictionless or bonded interfaces. When the interfaces are frictionless, interface separation or the unilateral nature of the contact regions needs to be considered. Such unilateral contact surfaces can include regions which are either advancing or receding or stationary. Several important aspects of the unilateral contact problem related to elastic media have been examined in the literature in mechanics and accounts of such contact problems are given by Dundurs and Stippes [1], de Pater and Kalker

174

,6,

ear interface 9

X-cylindrical anchor

r

lexible strip anchors

S

foundation

I.:;z";'::':.'.'i .I

.~:. ~

~ non-linear interface

/ fault with non-linear~ interface

cylindrical pipeline

cutting wedge undation

non-linear interface

non-linear terface

" -"~dilatant ~ interfaces crack

Figure 1. Geomechanics problems involving non-linear geomaterial interfaces

175 [21, Duvaut and Lions [31, Selvadurai [4], Gladwell [5], and Kikuchi and Oden [6]. When the interface exhibits frictional phenomena, the analytical treatment of an interface problem presents a complex exercise in mathematical analysis. Some limited progress can be made in the analytical solution of frictional contact problem involving circular contact regions (see e.g. Spence [7] and Turner [8]). In general, the analytical treatment of frictional contact problems can be attempted only in a limited number of situations iflvolving highly simplified interface and loading configurations. The limitations and complexities of the analytical methods have therefore encouraged the application of numerical methods of stress analysis such as finite element and boundary element schemes for the solution of interface problems with materially non-linear interface phenomena. Special interface elements have been employed quite extensively in the finite element modelling of contact and interface problems in the structural mechanics and geomechanics [see e.g. Goodman et al. [9], Ghabbousi et al. [10], Zienkiewicz et al. [11], Fredriksson [12], Hermann [13] and Desai and Nagaraj [14]. The boundary element methods offers a further numerical approach to the study of non-linear interface phenomena. The method is particularly effective in situations where the boundary regions in which the non-linear phenomena occur can be identified a priori. The effectiveness of the numerical procedure is also emphasized in situations where the non-linear interfaces occur between predominantly elastic regions. Boundary element technique has been applied for the study of a number of technologically important problems involving nonlinear interface contact. For example, Andersson [15] and Andersson and Allan-Persson [16] have applied boundary element techniques to the study of frictional contact and separation problems. Similar problem involving frictional contact, separation and slip at frictional interfaces have been considered by Selvadurai and Au [17-20] and Selvadurai [21,22]. Crotty and Wardle [23] have also applied boundary element schemes to examine geomechanics problems in which the planes of weakness exhibit non-linear characteristics. Selvadurai and Au [24-26} have applied the boundary element technique to study the influence of non-linear effects at the surfaces of cracks in geological media. The scope of this paper is to examine the application of boundary element techniques for the study of both axisymmetric and two-dimensional problems involving non-linear interfaces. The essential aspects of the theoretical formulation are briefly reviewed. The applications of the method to several interface problems of geomechanics interest are presented. 2. T H E B O U N D A R Y

ELEMENT

METHOD

In the ensuing we shall present, for completeness, a brief account of the essential features of the boundary element method as applicable to multiple geomaterial regions D (~ which interact through interfaces which exhibit non-linear phenomena. The media are assumed to exhibit isotropic linear elastic properties such that the incremental form of the boundary integral equation governing an individual region D (~ can be written as

Cij'aj

+

IF (p)

"uj

/r (p) *(P) P)

where i, j = 1, 2, 3 (or z, y, z); the Greek superscript or subscript refers to the material -"(P) and/bJp) are the incremental values of the boundary displacements and region p; uj

176

tractions respectively; u~} p) and P~.(P) are the corresponding fundamental solutions (see ~.g.,[27,2S]) given by

u i,(jp ) _ -

1 16~Gp(1 -

vp)r { ( 3 - 4.p)~,j + r,ir,j}

(2)

and

pi}(p) _ - 87r(1 -1 up)r 2 {[(1 - 2yp)6ij + 3r,ir,jlr,n - (1 - 2Up)[r,,nj - r,jni]}

(3)

respectively where r is the distance between the source and field points; ni are the components of the outward unit normal vector to F(P); 6ij is Kronecker's delta function; G a and yp are respectively the shear moduli and Poisson's ratios. Also in (1) cij = 6ij/2 if the boundary is smooth. The results (1) to (3) can be developed for each subregion D(P)(p = 1,2, ...m) of a multiphase material region separated by non-linear material interfaces or boundaries S(Z)(/~ = 1,2, ...n). The bodies in contact can be subjected to the following types of boundary conditions. (a) Prescribed displacements on boundary $1 where 0

(4)

(b) Prescribed tractions on boundary $2 where

(5) (c) Interface conditions on boundary $3 where

P, -

R, +

(6)

where/~i are incremental residual tractions and K~j are stiffness coefficients (frictional, elastic-plastic, dilatant, etc.) of the interface. After a boundary element discretization, the equation (1) can be reduced to a matrix equation of the form

[H]{fl}=[G]{15}

(7)

Where [H] and [G] are the boundary element influence coefficient matrices. If the configuration of the boundary and the interface conditions are defined at any level of deformation, one can obtain the final system equations in the form

[A]{ILI} =

{B}

from which the boundary or interface unknowns can be determined.

(s)

177 3. I N T E R F A C E

CONSTITUTIVE

RELATIONSHIPS

The constitutive behaviour of a geomaterial interface is influenced by a variety of phenomena which include the mechanical properties of the regions in contact, the response characteristics of the material regions at the macro- and micro-scale and the rate at which tractions are transmitted between the interfaces (see e.g. Bowden and Tabor [29] and Johnson [30]). Although the macro- and micro-structural phenomena at geomaterial interfaces are both varied and complex, it is convenient to model the interface constitutive behaviour in a phenomenological sense. In its simplest form, the response of interface can be characterized by either completely smooth or completely bonded behaviour with elastic-plastic, Coulomb friction, finite friction, dilatant and damaging phenomena occupying intermediate positions. Further type of interface responses can include strain hardening and strain softening phenomena and time dependent effects of a viscoelastic, viscoplastic or creep nature. In this section, we shall present a treatment of the interface constitutive modelling procedure which can be applied to interfaces which exhibit Coulomb friction or dilatant friction with interface degradation which results from the breakup of asperities at a contact zone. The interface constitutive relationships are defined in terms of the incremental relative displacements /k i and the corresponding incremental tractions ii. We now assume that the corresponding relative interface displacements are composed of an elastic or recoverable component A!e) and an irrecoverable or plastic (or irreversible) component/k} p) i.e.

"~i- A!e)+ A!p)

(9)

where for an interface i, (or j) can be assigned notations applicable to the local interface coordinates. For convenience, we shall denote i , j = x, y, z and the normal to the interface is assigned the z-coordinate. The elastic component of the incremental displacement is related to the interface tractions by the linear constitutive law

i, =

(lo)

where kij are the linear stiffness coefficient of the interface. In order to establish the irreversible incremental deformations, it is necessary to define the stress level at which yielding phenomena occurs at the interface. 3.1. T h e C o u l o m b I n t e r f a c e For interfaces which exhibit Coulomb friction, the yield function F is given by F -(t~

+ t~y) 89+ #tz - 0

(11)

where ti are the total value of the tractions and # is the coefficient of friction at the interface. At the limit given by (11) the interface will slip and the irreversible slip displacement can be obtained from a flow/slip rule identical to that used in the classical theory of plasticity (Hill, [31]), i.e.,

178

Ots

(12)

where J~ is a proportionality factor known as the plastic/slip multiplier and ~ is the plastic/slip potential given by

r = (t~ + t,)2 89

(13)

By substituting (12) into (9) and then into (10) we obtain

ar i,- k,j(/xj- ~-~)

(~4)

When the interface displacements result in slip, the incremental yield function 2~ can be expressed by

OF. t, = o Ot~

(15)

From (14) and (15)we obtain 1

OF

(16)

where the scalar function ~b can be written as

OF

r

0r

ktm~0tm

(171

Assuming that th~ irreversible part of the incremental deformation/k! p) is governed by a relationship of the type

i,

= k! 7)/~j

(is)

it can be shown that

k!;')-kiJ

r1

Oe~ kitk,~j OF

&m

(19)

Therefore, as with the numerical implementation of classical associative or nonassociative plasticity phenomena, once the yield function (F) and the slip potential (r are known it is possible to define k!~p). For example, for the values of F and r given by (11) and (13) respectively, and for the special case when kx. = kvv = kt;

and all other kq = O, (19) can be expressed in the form

(20)

179

1 [k] (~p) --(t2x + t2 )

/

ktt 2 -kttxt v

-ktt~ty ktt2~

0

0

-#kntx(t 2 + ty) -#k~ t y ( t x2 + ty) 2 89

(21)

kn 2 + ty) 2

3.2. T h e d i l a t a n t interface The Coulomb frictional interface model can be generalized to accommodate the influence of the frictional interfaces which exhibit dilatancy associated with asperity effects. Particularly in the modelling of fractured geomaterial interfaces encountered with concrete, rock and other brittle materials, the dilatant friction model accounts for the opening of a fracture surface during shear (see e.g. Goodman et al. [9], Barton and Stephansson [32], Rossmanith [33], Shah and Swartz [34]). An idealized physical model of a dilatant interface is shown in Figure 2. The asperities are represented by periodic ridges with angle c~ and the surfaces of the ridges exhibit Coulomb friction characterized by a coefficient of friction #. Y

Coulomb plane ~L x

Figure 2. The dilatant interface The basic procedures outlined in the previous section can be extented to include interfaces which exhibit dilatant phenomena. The dilatant interface model proposed by Plesha and Belytschko [35] is based on an asperity model in which the asperity angle is defined by c~ (Figure 2). The corresponding forms for the yield function F and the plastic potential r take the form

F=[{tzsina+(t

2 + t y2) 89cosa} 2] 89+ # [t cos _ (t2z + t 2 ) -}sinai

(22)

- [{tz sinc~ + (t 2 + t2y) 89cos c~}2] 89

(23)

and

respectively. As can be observed, when the asperity angle a --~ 0, equation (22) and (23) reduce to (11) and (13) respectively. The asperity degradation effects can be accounted for by a relationship of the form

c~- c~oexp[-CW (p)]

(24)

180 where c~0 is the initial asperity angle, C is a constant which signifies degradation of the interface (i.e. a material parameter) and W(p) is the total plastic work of the tangential forces at the interface, which, in the incremental form is given by

(25) 3.3. A d h e s i v e p h e n o m e n a The basic methodologies outlined in the preceding section can be extended to include interface phenomena which exhibit both frictional and adhesive effects. For example, adhesive effects can be characterized by a Mohr-Coulomb model which possesses both cohesion (c) and internal friction (r where # = tanr 3.4. C o n t a c t a n d s e p a r a t i o n p r o c e s s e s Within an increment of loading, processes such as separation, re-contact, slip and adhesion can exist. These interface conditions can be interpreted in terms of the changes in the stiffness properties of the interface. (a) Separation During the loading process, it is possible that the normal contact traction at the interface becomes a tensile stress. In such a case, the boundaries at the interface will separate and the boundary condition at location will be given by the S2-type (equation

(5)). (b) Re-contact Also, re-contact may be possible when the relative displacement across the separated interface is greater than the initial gap. Then, the boundary condition in the direction of re-contact will be changed from an S2-type to an Sl-type. (c) Slip The condition for slip is given by the yield function (11) and when this condition is reached, the expression in (21) will be applied as a S3-type boundary condition. (d) Adhesion When the stress condition at the interface does not violate the yield criterion (11), the boundary condition at the interface will be of the S3-type and the expression in (10) will be used. With each increment and iteration, all the above four conditions must be checked in order to obtain a stable condition at the interface. 4. L O C A L I Z E D I T E R A T I V E

SOLUTION PROCEDURES

For the boundary condition given by (4) to (6), We can rewrite the matrix equation (7) in the form

(26)

181

where the superscripts (i) i = 1,2,3 indicate the types of boundary condition. For a non-linear interface problem, one should apply an efficient solution technique to analyze the incremental and iterative matrix equation (26). A procedure can be summarized as follows. As the total boundary consists of linear and non-linear types, We can apply the elimination to the linear portion of the boundary constraint i(1) )

-- { B } + [(~(3)]{i~(3)}

[A, I2I(3) - G(3)K(eP)] (

/~(3)

(27)

where [A] is the reduced version of [ - G ( 1 ) , H (2)] and it is an upper triangle type of matrix; {13} is the reduced form of the right-hand side vector from known boundary values and [I2I(3)] and [G(3)] are their corresponding reduced forms. Now, (27) can be separated into two relations. The first relationship, corresponding to (S1) and ($2), is

0

A2 -

/)2

(~

which is a back-substitution form of the solution of {i (1)} and {A (2)} if {A (3)} is known. Notice that equation (28) can be used at any level of increment when the boundary condition on ($3) is determined. The second relation is an uncoupled equation for {~(3)}, i.e. (29) which has the unknowns on boundary $3 only. Therefore, at any increment level, (29) can be applied in an iterative manner in order to determine a configuration of boundary $3 . By such a procedure, the non-linear boundary problem is solved by a localized iterative procedure and the overall BEM system matrix is factorized only once for any number of increments. 5. A P P L I C A T I O N S The basic procedures outlined in the previous sections are used to examine certain problems of interest to geomechanics applications. 5.1. P l a n e crack w i t h a frictional interface r e g i o n We examine the problem of plane crack of length 2b which contains a central interface region of length 2a. The plane crack is located in an elastic medium of infinite extent and subjected successively to an initial axial stress a0 and an increasing shear strain 7 (Figure 3). The central interface region exhibits Coulomb friction with # = 0.1, 0.3 and 0.5. The objective of the analysis is to examine the manner in which the interface condition at the crack surface influences the stress intensity factor at the crack tip. In these studies the singular behaviour of the stress field at the crack tip is modelled by

182 employing the singular traction quarter - point elements proposed by Cruse and Wilson [36], where the displacement and tractions take the following forms 2

[A,; t,] - ~ [bm~ ; cm~ (m;~)]

(30)

rn--0

where bm and cm the displacement side of the crack of crack problem

are constants. The stress intensity factors can be obtained by applying correlation method which utilizes the displacement of nodes on either (see e.g. Figure 4). The accuracy of the boundary element modelling has been extensively investigated in the literature [37,38].

~Y I

~Y I ! 0

I

region of frictional constralnt---~

I I I I

I region of I frictional I constraint--~:

~/---crack . _ _ .

2b

. . . .

.

.

.

.,,/.--crack

.

Lt

I~

2b

.~1

T

Figure 3. Crack with a frictional interface region Y /

Figure 4. Crack tip node configuration The incremental values of the stress intensity factors are given by 2G

/~'I - (k + 1) and

'

I

L

I~

"r

{4[hy(B)-/ky(D)] + by(E)-/ky(A)}

183

2c

gII

(k + 1)

--

{ 4 [ h x ( B ) - hx(D)] + z ~ x ( E ) - hx(A)}

(32)

where the locations A, B, D and E are shown in Figure 4. I is the length of the crack tip element; k = ( 3 - 4v) for plane strain and k = (3 - v)/(1 + v) for plane stress problem. Owing to the asymmetry of the problem (due to the shear loading) and the assumed Coulomb behaviour of the interface region, the stress intensity factor KI associated with 7 is always zero. The numerical results for the crack shearing mode stress intensity factor KII derived the nonlinear scheme is shown in Figure 5. The corresponding analytical results for the extreme limiting cases involving the single crack (/~ ---, 0 ) and double crack (# ~ co) are given by Rooke and Cartwright [39]. The results presented in Figure 6 and 7 correspond to the case where the central interface region exhibits dilatant frictional characteristics corresponding to the Plesha-Belytschko model described in section 3.2. The initial asperity angle s0 = 4 ~ and the degradation parameter C is a variable.

0-05

~o

tilllli 7"

C

b o.

2a

o.o, t

o.o

-0 .o.s "--

-

-

~

y

Itn

= - - - -

/ = 0-1

/

9i 0 3

0x

I

o

oo,

.

0-01

0-00 0

, I0

, 20

j 30

y

40

I 50

60

9103

Figure 5. Crack shearing mode stress intensity factor - Coulomb interface region 5.2. A w e d g e i n d e n t a t i o n of a cracked layer

We consider the problem of the plane strain frictional-adhesive indentation of a notch in an elastic layer of finite extent. The base of the notch contains a crack of finite length (Figure 8). The interface between the indenting wedge and the elastic medium is characterized by an interface friction property (#) and cohesion (c). The rigid wedge is subjected to a central load 2P. The angle of the wedge and the length of the crack at the root of the notch are variables in the problem. The Figure 9 shows the boundary

184 0-05

+,+;'+++ 20

0-04

r

~---

"--~- = O- I

7"

Zb ~ 9

Y

/

.o. ~__, = O.5 9 3

~

kn

I 3

0-03

Kxz

.,,y ..o.+,/ ~

G,/;'g 0-02

Xe c;~~

4'0

0-01

0-00 I0

0

20

40

30

50

60

7" = 103

Figure 6. Crack shearing mode stress intensity factor - Dilatant interface region

0-005

%

+~+++;1, 2a

0-004

" .o.8 ---~--= O- I

~ o~,,o~ "

- k~n-

,o 3

0-003

-

Kz

ftff.tff

G,/'~"

~-,o'----~

0-002

~

/

~-,o~X / \ ,+ o,o,o~

0-001

::

0-000 0

I0

20

30

40

50

60

7' 9103

Figure 7. Crack open mode stress intensity factor - Dilatant interface region

185

2P (G,u) H t

Figure 8. Wedge indentation of a cracked layer

/~-TOTAL

LOAD e @

~

FREE

SURFACE

26.5

INTERFACE

:50

5)

,/ J

1 q

1

30 NODES 31 ELEMENTS

71

Figure 9. Boundary element discretization

186 element discretization adopted in the solution of the problem. The Figure 10 illustrates the manner in which the flaw opening mode stress intensity factor at the crack tip is influenced by the magnitude of the load (2P), the coefficient of interface friction (#) between the indenting wedge and the elastic region and the cohesion (c/G) at the interface. The results clearly indicate the importance of the non-linear characteristics of the interfaces in the analysis problems where such interfaces play an active role in fracture mechanics considerations. 5.3. A c o n t a c t p r o b l e m for a c r a c k e d layer We examine the problem of an elastic layer of finite dimensions containing a crack, which indents an elastic layer region with differing elastic properties (Figure 12). The indenting layer contains a crack of finite length extending across its width. The contact between the two layers exhibits frictional characteristics. The interface between the layers is subjected to an initial precompression ~0 along its entire length. The indentation is caused by the centrally placed line load P. The objective of the incremental analysis is to establish the manner in which the stress intensity factor at the crack located in the indenting layer is influenced by the line load P and the frictional characteristics of the interface. The Figure 13 illustrates the boundary element discretization employed in the analysis. The Figure 14 shows the variation in the stress intensity factor KI as a function of the applied load and the coefficient of friction at the interface. It may be observed that linear variations in K1 are obtained for the two extreme situations approaching either bonded (# ~ oc ) or frictionless (# ~ 0 ) conditions. For arbitrary values of #, the stress intensity factor exhibits a non-linear variation. The results for the stress intensity factor shown in Figure 14 are applicable to the case where the indenting layer and underlying medium exhibit identical elastic characteristics. The Figure 14 also contains results for situations where the separation can be induced at the layerelastic medium frictional interface. The separation at the frictional interface contributes to only a marginal decrease in the stress intensity factor. The Figure 15 presents results for the situation where the layer indents an elastic medium with higher (relative) elastic stiffness. Other examples of the application of boundary element techniques to contact/fracture problems are given by Selvadurai [40] and Aliabadi et al. [41]. 5.4. C i r c u l a r a n c h o r at a p r e c o m p r e s s e d geological i n t e r f a c e We consider the problem of a rigid plate anchor of diameter 2a and thickness 2d which is located at a pre-fractured plane which is subjected to a precompression stress ~0 [42]. The boundary element scheme is used to study the load-displacement relationship of the anchor which is subjected to an in-plane translation. The problem is examined in two stages. (i) The initial effects due to wedging action are determined iteratively to establish the extent of the zone of separation at the pre-fracture geological interface. In this instance, the frictional affects at the interface between the anchor and the elastic medium are suppressed. (ii) The anchor is subjected to an incremental in-plane load. The application of the incremental in-plane load will in general result in a perturbation of the zone of separation. In the analysis, however, the radius of the zone of separation is kept constant. It is assumed that the interface between the plate anchor and the elastic medium exhibits Coulomb frictional phenomena. The results for the in-plane load (/9o) vs. displacement (~i) are shown in Figure 16. It is evident that the limit

187

,.,I -/I // /

/

~.6 L

2P

"~ - - - - - ~ I

O.

~V

26.

,--

/

oA ~ . o .

?

/.

O-o~ ,-o~ / 0.4

,

BOND!

,

.0 O.

0 0.10

0

l 0 30

0 20

~ 0 40

1 0 50

l 0 60

1 0 70

1 0.80

I 0.90

4P 103 GH

2

Figure 10. Crack open mode stress intensity factor due to wedge indentation

2P

1.6

~

///--- ( G, v )

,"

/

~ . .,~"

3

,.o- o.~6~"

o.~ -~.o--

~

v

90.3

,?,

"'~

/

~"-'/ ~'~176 "~-IU

/

/f

/

/

j j

j

J Jj

y j

f

J

0.2

0 0

O. I0

0 20

0 30

0 40

0.50

0 60

1 ,,. 0 70

1 0 80

I 0.90

J-%.,o' GH

Figure 11. Crack open mode stress intensity factor due to wedge indentation

188

HT"~~ I

/ ,/'~/

_

/~-INTERFACE WITH NON-LINEAR pHENOMENA

9

L/

-ELASTIC LAYER(Gi ui)

Figure 12. A cracked elastic layer on an elastic medium

F--'-- INTERFACE WiTH NON-LINEAR .~___/~_1 PHENOMENA

;

-

~

L I .

.

(o,,~,) ~ x ~ - ,o,,~,)

.

.

~

_! I

29 ELEMENTS 64 NODES Figure 13. The boundary element mesh for cracked elastic layer

189

, .s 0.0:3

0.02

- !E ' ~ "I

~/

KI ~.0"3 -E;,' O.OI

0

0.05

0.10

0.15

0.20

0.25

16P G2H2 Figure 14. The variation of

KI for the elastic layer on an elastic m e d i u m P

J_ 9

0.O3

_

~

~

v

INTERFACE FRICTION (.=)

a

)

0.02 l[!

H

GtCE

~,

4

o-=

W2 ~ v2 a 0.3

0.01

0

0

0.05

0.10

0.15

0.20

16P

6,. 2 Figure 15. The variation of K1 for the elastic layer on a rigid s t r a t u m

0.25

190 load for the disc anchor and the transition from elastic behavior to the attainment of the limit load is influenced by the far field stress which acts normal to the fracture plane. This result is consistent with results expected for interface anchor capacity in the presence of Coulomb frictional phenomena at the interface.

z t ~ t

cro ~tt

0.8 2d 0.7

~;;'.:~

= po - " - ! . .~-----~--frictional TI.'-..-~.OF...'~: 1 ,nterfece ~ - - anchor

0.6 (T~ G

p 9tan 3 0 ~ 0.5

,._....__ (To = O. 2 G

'V= 0 - 3

Po

kf -~-= I0 0"4

ao ~--= 0.1

kn

.~-9 = I0 7 d ~"

0.3

----

O'o ~ - 90 . 0 5

90 . 0 5 % ="

90"01

0"2

\

0"1

0

0"05

0-10

0"15

(To 90-001 G

0"20

0"?.5

0"30

a

Figure 16. Load-displacement relationship for disc anchor with frictional interface 5.5. A c y l i n d r i c a l a n c h o r p r o b l e m

In section 3.1 and 3.2 attention was focussed on the definition of interface constitutive responses which were defined in terms of conventional plasticity formulations. An alternative to the formalized plasticity approach is to employ constitutive responses derived from experimental observations. Results of such investigation are given by Boulon et al. [43], Plytas [44] and Hoteit [45]. For example, Hoteit [45] conducted an extensive

191 program of experimental research for the purposes of defining the interface behaviour between dense and loose Hostun sand (Labanieh [46]) and rigid plates which are classified as either rough or smooth. The roughness of the plates is derived from Hostun sand which is bonded to it. The smooth plates are highly polished machined surfaces. The results of the experimental studies [45] include the influence of the normal stiffness on the shear response of the interface. This is in contrast to the conventional study of the interface shear responses of soils in which such responses are evaluated at either constant normal stress or at constant volume conditions. With increase in normal stiffness, the interface also experiences particle crushing or asperity degradation. The results of the experimental data can be used to develop analytical expressions, albeit empirical, for the non-linear, shear responses at the interface. For example, referring to a system of cylindrical polar coordinates (r,O, z), the shear traction (Pz) vs the relative shear displacement (A~) at the interface between a cylindrical anchor and the surrounding elastic medium can be described in term of non-linear relationship of the form shown in Figure 17.

P, CP I o,

.

.

.

.

.

.

I I I I

(Az)mo *

Az

Figure 17. Interface shear response at the cylindrical anchor The shear response is characterized by the experimentally derived material parameters r [Az]maz , [Pz]max, and w. These parameters are influenced by the interface normal stiffness kT~ and the interface normal stress ano. Similar procedures can be established to describe the relationship between the normal stress (Pr) and relative normal displacement (At). For example for a rough plate - dense Hostun sand interface, the normal stiffness and initial normal stress dependent parameters ~, (Az)max, etc, characterizing the shear response can be expressed in the empirical forms 1

8 = 0.0267an0 - 2.875(krT)5 + 290

(Pz)ma~ = {0.76 - 0.016(krr) 89 }an0 + 6.5k)r

(33)

192

O'nO

1

(Az)ma x -- {2.276 -- 0.017(krr) 1}1--~ + O'19(krr)5 + 1.12

1

w -- {0.165 + O.O0049ano}(krr) ~ + 4.2 where anoiS expressed in kPa and kr, is expressed in k P a / m m . Although the relationship between the normal stress/9, and the differential displacement Ar is not evaluated in the experimental investigation, it could be assumed that P ~ - kr, Ar

(34)

/

conditions

int:rface

free

surface

v

9

.

J ..

I

80 elements'.

~.9

" o

i

L 9 B

r.;i//////,

L f~

L,, - 3 0 2a R a -", 4 0 0

'

:

~

-I/////////////

:

2 . A ........

J v--,

Figure 18. Boundary element discretization of the embedded anchor problem Selvadurai and Boulon [47] incorporated the empirical interface responses in the incremental boundary element formulations, to examine the behavior of a rigid cylindrical anchor which is embedded in frictional contact with a geological medium, such as a cemented granular soil of finite extent. The boundary element discretization of the anchor-geological medium system is shown in Figure 18. For purposes of the presentation of the numerical results the geometric aspect ratio of the anchor is a the value I/2a = 10. The geological medium exhibits a predominately linear elastic response (shear modulus Gs; Poisson's ratio us) beyond the immediate vicinity of the interface region. The geological medium has unit weight ~, = 17 k N / m 3 and in the boundary element modelling the influence of the self weight is represented by a linearly varying normal stress at the anchor-geological medium interface. The analysis does not account

193 for the stress changes that are associated with the installation of the anchor. The radial stiffness can be expressed in terms of the pressuremeter modulus ( E p ) which itself can be related to the shear modulus of the geological medium and the diameter of the anchor region. k,.~ - k -

Ep ;

(35)

a

Ep = 2(1 +

~)Vo(~p/AV)

where V0 is the cavity volume, Ap is the pressure change and A V is the change in volume occurring during the application of Ap. The linear variation of in-situ radial stress at the anchor-geological medium interface is computed by employing a coefficient of at rest earth pressure defined by the elastic estimate K0 = V s / ( 1 - Vs). The Figure 19 illustrates the quasi-static uplift load cycling behaviour observed in anchor. The results indicate characteristic trends in load cycling behaviour observed in anchor elements in which the interface non-linear responses exhibit degradation with increased relative slip.

Gt = IO s k Po % = 0-30

~

kPo k = io a mm L --=lO 20

id onchor

pile

2n

(MN)

t

o

!

|

I

2

!

l

3

Figure 19. Load-displacement response for the embedded cylindrical anchor

194 6. C O N C L U S I O N S The examination of the mechanics of geomaterial interfaces is important to the accurate modelling of the performance of geotechnical structures. The non-linear phenomena at such localized interfaces can have a significant influence on the load-displacement behaviour of tension structures such as anchor plates, anchor piles, soil nails and earth reinforcement. Non-linear interface phenomena can also influence the behaviour of cracks with partial closure and soil-foundation interaction problems with interface friction. Numerical schemes offer the most convenient techniques, for examining a variety of non-linear _phenomena associated with separation, adhesion, friction, dilatant friction and asperity degradation at an interface. This paper demonstrates that the boundary element method can be successfully applied to examine non-linear phenomena at material interfaces. The particular advantage of the boundary element method is that interfaces on which the non-linear constraints are prescribed can be conveniently modelled via this approach. In instances where the interacting media are elastic, the interfaces are modelled as boundary constraints between regions. The versatility of the boundary element method is established by appeal to a variety of examples which deal with cracks, embedded anchor regions and contact problems with non-linear interfaces. 7. A C K N O W L E D G E M E N T S

The work described in this paper was supported in part by a Natural Sciences and Engineering Research Council of Canada Grant A3866. The work was performed at the Department of Civil and Environmental Engineering at Carleton University. The author is grateful to Dr. M. C. Au, formerly, Research Associate, Department of Civil and Environmental Engineering, Carleton University for assistance with the numerical computation. The author is grateful to Mr. Jun Hu for assistance with the preparation of the typescript. REFERENCES

[1] Dundurs, J. and Stippes, M. (1970) Role of elastic constants in certain contact problems, J. Appl. Mech. 37, 965-970. [2] de Pater, a. D. and Kalker, J. J. (Eds.) (1975) The Mechanics of Contact Between Deformable Media, Proc. IUTAM Symposium, Enschede, Delft Univ. Press, Delft. [3] Duvaut, G. and Lions, J. L. (1975) Inequalities in Mechanics and Physics, Springer Verlag, Berlin. [4] Selvadurai, A. P. S. (1979) Elastic Analysis of Soil-Foundation Interaction: Developments in Geotechnical Engineering, Vol. 17, Elsevier Scientific Publ. Co., Amsterdam. [5] Gladwell, G. M. L. (1980) Contact Problems in the Classical Theory of Elasticity, Sijthoff and Noordhoff, The Netherland. [6] Kikuchi, N. and Oden, J. T. (1988) Contact Problems in Elasticity, Soc. Industrial and Appl. Math. Phil., Pa. [7] Spence, D. (1968) Self similar solutions to adhesive contact problems, Proc. Roy. Soc. Ser. A. 305, 58-80.

195

Is] [9] [10] [11] [12] [13] [14]

[15] [16] [17] [ls] [19] [20] [21] [22] [23] [24]

Turner, J. R. (1979) The frictional unloading problem on a linear elastic halfspace, J. Inst. Math. Applics., 24, 439-469. Goodman, R. E., Taylor, R. L. And Brekke, J. L. (1968) A model for the mechanics of jointed rock, J. Soil Mech. Foundn. Div., Proc. ASCE, 94, 637-659. Ghabbousi, J., Wilson, E. L. and Isenberg, J. (1973) Finite elements for rock joints and interfaces, J. Soil Mech. Fdn. Div, Proc. ASCE, 99, 833-848. Zienkiewicz, O. C., Best, B., Dullage, C. and Stagg, K. (1970) Analysis of non-linear problems in rock Mechanics with particular reference to jointed rock systems, Proc. 2nd Congress, Int. Soc. Rock Mech., Belgrade, Yugoslavia, 3, 501-509. Fredriksson, G. (1976) Finite element simulation of surface non-linearites in structurn mechanics, Comp. and Struct., 6, 281-290. Hermann, L. R. (1978) Finite element analysis of contact problems, J. Eng. Mech. Div., Proc. ASCE, 104, 1042-1057. Desai, C. S. and Nagaraj, B. K. (1986) Constitutive modelling for interfaces under cyclic loading, in Mechanics of Material Interfaces, (Selvadurai, A. P. S. and Voyiadjis, G. Z., Eds.) Studies in Applied Mechanics, Vol. 11, Elsevier Scientific Publ. Co., 97-108. Andersson, T. (1981) The boundary element method applied to two-dimensional contact problems with friction, in Boundary Element Methods, Proc. 3rd Int. Seminar, Irvine, Califonia, (Brebbia, C. A. Ed.) Springer Verlag, Berlin. Andersson, T. and Allan-Persson, B. G. (1983) The boundary element method applied to two-dimensional contact problems, Ch. 5 in Progress in Boundary Element Methods (Brebbia, C. A. Ed.), Comp. Mech. Publ, 2, 136-157. Selvadurai, A. P. S. and Au, M. C. (1985) Response of inclusions with interface separation, friction and slip, Proc. 7th. Int. Conf. on Boundary Elements (Brebbia, C. A. and Maier, G., Eds.), Como. Italy, Springer Verlag, 14.109-14.127. Selvadurai, A. P. S. and Au, M. C. (1987) Boundary element modellin_g of interface phenomena. Topics in Boundary Element Research (Brebbia, C. A. Ed.) Ch. 5, 112-126, Springer Verlag, Berlin. Selvadurai, A. P. S. and Au, M. C. (1986) Boundary element modelling of dilatant interfaces, Proc. 8th. Int. Conf. Boundary Element Meth. Eng., Tokyo. Selvadurai, A. P. S. and Au, M. C. (1988) Mechanics of non-linear interfaces, Proc. Can. Soc. Civil Eng. Annum Conf., Calgary, Alta., III, 320-334. Selvadurai, A. P. S. (1988) Non-linear material interface: A boundary element approach, Proc. IUTAM Symposium on Advanced Boundary Elements (T. A. Cruse) San Antonio, Texas, Springer Verlag, Berlin, 389-396. Selvadurai, A. P. S. (1993) Mechanics of a rock anchor with a penny-shaped basal crack, Int. J. Rock Mech. Min. Sci. and Geomech. Abstr. 30, 1285-1291. Crotty, J. M. and Wardle, L. J. (1985) Boundary element analysis of piecewise homogeneous media with structural discontinuities, Int. J. Rock Mech. Min. Sci. and Geomech. Abs., 33, 419-427. Selvadurai, A. P. S. and Au, M. C. (1988) Cracks with frictional surfaces, a boundary element approach, Boundary Elements (Brebbia, C. A. Ed.), Southampton, 3,211-

196

[251

Selvadurai, A. P. S. and Au, M. C. (1989) Crack behaviour associated with contact problems with non-linear interface constraints, BETECH' 89, Proc. 4th Int. Conf. Boundary Element Methods, Windsor, Ont., (Brebbia, C. A. and Zamani, Eds.), 3-17. [261 Selvadurai, A. P. S. and Au, M. C. (1989) Some aspects of non-linear interfaces in geomechanics: boundary element modelling, Numerical Models in Geomechanics, NUMOG III, Proc. Int. Symp., Niagara Falls, Ont. (Pietruszczak, S. and Pande, G. N., Eds.), Elsevier Sci. Publ. London, 463-471. [27] Brebbia, C. A. (1978) The boundary Element Method for Engineers, Pentech Press, London. [2s] Banerjee, P. K. and Butterfield, R. (1981) Boundary Element Methods in Engineering Science, McGraw-Hill, New York. [2o] Bowden, F. P. and Tabor, D. (1964) Friction and Lubrication of Solids, Vol. I and II, Oxford Univ. Press, Oxford. [3o] Johnson, K. L. (1985) Contact Mechanics, Combridge Univ. Press, Combridge. Hill, R. (1960) The Mathematical Theory of Plasticity, Oxford Univ. Press, Oxford. [32] Barton, N. and Stephansson, O. (Eds.) (1990) Rock Joints, Proceedings of the Regional Conference, Oslo, A. A. Balkema, The Netherlands. [33] Rossmanith, H. P. (Ed.) (1990) Mechanics of Jointed and Faulted Rock, Proc. Int. Conf. Vienna, A. A. Balkema, The Netherlands. [34] Shah, S. P. and Schwartz, S. E. (Eds.) (1989) Fracture of Concrete and Rock. Proc. SEM-RILEM Int. Conf., Houston, Texas, Springer Verlag, Berlin. [35] Plesha, M. and Belytschko, T. (1986) On the modelling of contact problems with dilation, in Mechanics of Material Interfaces, Studies in Applied Mechanics, Vol. 11, (Selvadurai, A. P. S. and Voyiadijs, G. Z. Eds.) Elsevier Scientific Publ. Co., The Netherlands. [36] Cruse, T. and Wilson, R. B. (1977) Boundary integral equation method for ealstic fracture mechanics, AFOSR, TR-78-0355, 10-11. [aT] Blandford, G. E., Ingraffea, A. R. and Liggett, J. A. (1981) Two dimensional stress intensity factor computation using the boundary element method, Int. J. Num. Meth. Eng. 17, 387-404. [as] Smith, R. N. L. and Mason, J. C. (1982) A boundary element method for curved crack problems in two dimensions, in Boundary Element Method in Engineering (Brebbia, C. A. Ed.) Springer Verlag, 472-484. [3o] Rooke, D. P. and Cartwright, D. J. (1976) Compendium of Stress Intensity Factors, HMSO, London. [4o] Selvadurai, A. P. S. (1991) Non-linear interfaces and fracture mechanics, Wiss, Z. Hoch. Arch. Bauwes, B, Weimar, 37, 21-23. [41] Aliabadi, M. and Brebbia, C. A. (Eds.) (1993) Computational Methods in Contact Mechanics, Computational Mechanics, Publ., Southampton. [42] Selvadurai, A. P. S. and Au, M. C. (1987) Non-linear interactions in flat anchors embedded in geological media, Proc. 2nd Int. Conf. on Constitutive laws for Engineering material, Theory and Appl. Tucson, Arizona, (Desai, C. S. and Krempl, E. Eds.) Elsevier Sci. Publ. II, 1093-1107.

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[43] [44]

[45] [46]

[47]

Boulon, M., Hoteit, N. and Marchina, P. (1988) A complete constitutive law for soil structure interfaces, Proc. VI, Int. Conf. Num. Meth. Geomech., Innsbruck, Austria. (Swoboda, G. Ed.) I, 311-316. Plytas, C. (1985) Contribution (i l~tude exp~rimentale et numerique des interfaces sols granulaires-structurees, Th~se, Universitd de Grenoble, France. Hoteit, N (1990) Contribution fi l~tude du comportement d'interface sable-inclusion et application au frottement apparent, Thdse, Universit~ de Grenoble, France. Labanieh, S. (1991) Critical state and constitutive parameter identification. Proc. 3rd Int. Conf. Constitutive laws for Engineer material: Theory and Applications (Desai, C. S. Krempl, E., Frantziskonis, G. and Saadatmanesh, A. Eds.) ASME Press, 739-742. Selvadurai, A. P. S. and Boulon, M. (1992) Boundary element modelling of the mechanics of a near surface cylindrical anchor, Numerical Models in Geomechanics, Proc. 4th Int. Symp. Num. Models in Geomech. NUMOG IV, Swansea, U. K. (Pande, G. N. and Pietruszczak, S. Eds.) 2, 629-643.

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STEEL-CONCRETE INTERFACES

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

LATTICE

MODEL

FOR ANALYSING

201

STEEL-CONCRETE

INTERFACE

BEHAVIOUR J.G.M. van Mier and A. Vervuurt Delft University of Technology, Department of Civil Engineering, Stevin Laboratory, P.O. Box 5048, 2600 GA Delft, The Netherlands

In this paper the application of a newly developed lattice model to the bond between steel and concrete is shown. In the model, the concrete is modelled as a triangular framework. The heterogeneity of the concrete is implemented by projecting the lattice on top of a generated particle structure, and by .assigning different properties to the lattice beams appearing in the various phases of the particle structure. Fracture is simulated by removing in each load-step the beam element with the highest stress over strength ratio. The model is applied to bond of steel to concrete. Three different examples are shown: the effect of ribs on cracking in the bond zone, a detailed analysis of a miniature bond-slip test [ 1], [2], and the pull-out of a steel anchor from concrete. All problems are treated as plane stress problems, which implies that longitudinal cracking is omitted. In all the analyses, adhesion between the steel and the concrete is modelled through a very low tensile strength of the beam elements in the lattice. In the example of the fibbed bar and the anchor pull-out, experimental crack data are available, and the results of the simulations are in accordance with experiments. The analysis of the miniature bond-zone test can be used for the formulation of a macroscopic bond-slip relation. The assumption is again that cracking in the concrete layer which is in contact with the steel bar is responsible for much, if not all, of the observed slip of the steel bar. It seems that friction is activated only when the bond layer is sufficiently cracked.

1. I N T R O D U C T I O N Interfaces play an important role in concrete technology and reinforced concrete structural design. In the material itselL interfaces are encountered at the meso-level between the matrix and the aggregate particles. In reinforced concrete the interface between steel and concrete is essential for a good performance of the structure. In both cases, i.e. for the behaviour of the interface between aggregate particles at the meso-level and the steel-concrete interface, three different types of bonding must be considered. These are, physical-chemical interaction between the two interfaces (adhesion), mechanical interlock and frictional stress-transfer. Let us first consider the aggregate-matrix interface. In general chemical bonding seems of importance only when reactive aggregates are used, see for example [3]. Porosity of the aggregates seems to play a major role in the development of the interface between aggregate and cement paste, [4]. The interface zone between the aggregate and the cement paste is

202 rather porous when normal dense natural aggregates (such as fiver gravel) are used, see for example in Mindess [5]. However, when more porous aggregates are used, for example artificial products such as lytag, a more tight bond zone can develop (e.g. [6]), due to the suction of water from the cement paste at the interface. Mechanical interlock of aggregate particles seems more effective for irregular and flaky types of aggregates. This is mainly of importance when aggregate interlock is modelled [7]. In that case the resistance of the interface (which is now a crack) to in-plane shear or tension normal to the interface plane is enhanced. Note that in aggregate interlock experiments at small crack openings often secondary cracking may develop inclined to the main crack, [8]. The development of these secondary cracks is in many cases triggered by irregularities in the crack plane. As shown in [9], when such secondary cracking appears, friction on the interface seems of less importance, and a simple tensile fracture model suffices to describe the observed phenomena. For the bond between steel and concrete, adhesion (chemical-physical interactions) seems of less importance. The steel can be compared to the non-porous natural aggregates such as fiver gravel. Mechanical interlock and friction seem to be the major load-transfer mechanisms for steel to concrete interfaces. These latter mechanisms are often enhanced by using special ribbed reinforcing bars, or in anchoring technology by using anchors with a special geometry. For example the bond between ribbed reinforcing bars and concrete leads to a completely different stress-transfer mechanism and distinct crack paterns develop in the concrete near the ribs on the reinforcing bar as shown clearly in the classical experiments of Goto [ 10] and the more recent tests of Otsuka [ 11 ]. The details of the interaction between aggregates and cement matrix at the meso-level, or between a steel bar or anchor and concrete can be better understood using numerical modelling techniques. The sole use of experiments is often too limited, or even some phenomena like internal cracking are very difficult to visualise. Early work on the analysis of bond-slip of reinforcing bars was done by Ingraffea et al. [12]. At that time the fracture models were still too limited to allow for a full fledged analysis of the problem. In their analyses, Ingraffea and co-workers, ignored the appearance of longitudinal cracking, and the simulations were stopped as soon as the first primary crack appeared. Recently, more detailed analyses were carried out by Rots [13]. He used the smeared crack model of DIANA to simulate cracking in a tensile specimen with a central reinforcement bar, which is a very popular test to measure bond-slip relations. The analyses of Rots were a great improvement: a radial symmetric analysis was carried out, allowing for longitudinal cracking [14]. The bond zone between steel and concrete can be looked upon in different manners. The most common point of view is to consider the concrete and steel as two independent operating materials, and to lump the bond-slip behaviour in a zone with zero thickness between the steel and the concrete. In most models, spring type elements are used with a specific bond-slip behaviour. The interface springs are in the ideal case three-dimensional springs [1 ], [2]. Another point of view is to assume that most of the slip between steel and concrete originates from the cracking that takes place in a narrow zone of concrete just outside the steel bar. Earlier it was hypothesized that the extent of such a zone could be approximately equal to the diameter of the reinforcing bar [ 1]. The bond problem could then also be tackeled in numerical codes by assuming a zone of a single diameter size around the steel bar. Different constitutive properties for the concrete in this zone would have to be defined then. This idea was worked out in [1] and [2], and a special detailed bond-slip test was designed for determining the properties of this rather extensive bond-zone. Independent

203 of which point of view is taken, both methods may profit from a detailed analysis of the cracking in the bond-zone. In such an approach the concrete and the steel bar must be modelled to a high degree of accuracy in order to mimic the load-transfer from steel bar to concrete as closely as possible. This means that also the ribs on the steel bar must be modelled. With the development of the lattice model such an approach has become possible. The lattice model was designed for simulating cracking in concrete at the grain level [15], [16], [ 17]. In view of the results obtained with the model for cracking in plain concrete, it was expected that good results could also be obtained for modelling steel-concrete interactions. The only limitation for the present model is that no frictional slip is modelled, but this seems no objection for analysing the first stage of steel-concrete interactions where cracking is the main cause for the observed behaviour. In this paper, first the lattice model is outlined. Next the capabilities of the model in analysing fracture of plain concrete are shown. Subsequently detailed analyses of steel-concrete interactions are treated.

2. OUTLINE OF THE LATTICE MODEL In the lattice model, the continuum is discretized in a network of brittle breaking beam elements. A similar procedure was already proposed in 1941 by Hrennikov [ 18], who used large trusses to solve problems of elasticity. At the time that Hrennikov presented his model, computational capabilities were insufficient, and the approach was not followed. Recently, mainly through activities in theoretical physics (Herrmann & Roux [19]), the approach has received new attention, and modelling fracture has become possible. The major step made by Herrmann [20] was to use a frame model rather than a truss model. Fracturing was simulated by removing beam elements from the frame as soon as the specified failure strength was reached. After removal, the complete frame was relaxed until the next beam could be removed. The model proposed by Herrmann was recently linked to a finite element code [ 16], [17], [ 18]. An important change was made, namely, the square lattice originally proposed by Herrmann (figure la) was replaced by either a regular triangular lattice or by a lattice with random beam lengths [21], see figure l b and 1c. This has the important implication that the macroscopic Poisson's ratio (i.e. the Poisson' s ratio of the total lattice) can be adjusted quite accurately to the Poisson's ratio of concrete (or rock). Note that Hrennikov's original truss-

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Figure 1. (a) Regular square lattice after Herrmann [20], (b) regular triangular lattice, and (c) triangular lattice with randomly distributed beam lengths.

204 model would always lead to a Poisson's ratio of 0.33, which is clearly much to high for geomaterials.

2.1. Implementing Heterogeneity It is essential that heterogeneity is included in the model. Up till now, several different approaches have been followed for introducing disorder in the model. In our first model [22], heterogeneity was implemented by assigning different limit strengths to each of the beam elements in a regular triangular lattice. In the first simulations a normal distribution of beam strengths was assumed, but clearly other distributions could be implemented quite easily. In the second approach [ 15], first a particle structure of concrete is generated, whereafter it is projected on top of a regular triangular lattice as shown in figure 2. The beam elements falling inside the aggregates, or in the matrix, or at the interface between the aggregate and matrix are given the properties of these distinct phases (figure 2b). Thus, essentially, concrete is modelled as a three-phase material. It should be noted however that this approach is only valid for simulations where tensile fracture prevails. When a composite like concrete (under the assumption that the stiffness of the aggregates is higher than the stiffness of the matrix) is loaded in compression, most likely a load-transfer mechanism as scetched in figure 3 will prevail. The aggregate particles will form a three dimensional stack, and compressive forces are transfered between the different particles, leading to splitting forces in the horizontal direction. Thus tensile cracking will still occur, but frictional effects will become more important as well. Clearly the aggregate interactions are neglected in the present two-dimensional approach. However because of computational limitations we have to live with this simplification. b

o

n

d

~

matrix(M)

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Figure 2. (a) Particle overlay, and (b) assigning different properties to beam elements falling in either of the three phases. The method of particle overlay as shown in figure 2 is of course not limited to the regular triangular lattice. The same method can be applied using other types of lattices such as the random lattice of figure l c. Especially this combination is appealing, because in the regular lattice sometimes the cracks seem to follow the mesh lines, [21 ]. In the random lattice this problem is eliminated. The random lattice is constructed starting from a regular square grid as shown in Figure 4. The method was developed by Mourkazel and Herrmann [24]. In each box of the grid (box size s x s mm), a point is selected at random, with uniform distribution. Subsequently the random lattice is defined by connecting always the three points which are closest to each other.

205 direction of external loading

"

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Figure 3. Load-transfer mechanism in a concrete with stiff aggregate particles, after [23]. $

LL

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/ Figure 4. Construction of a lattice with random beam length. The random lattice can also be used directly, i.e. an overlay with a generated particle structure is not required. When the random lattice is used without particle overlay, disorder is caused by the differences in beam length. If this disorder is strong enough to mimic the structure of concrete is debatable. Of the three methods descibed above, in particular the particle overlay method, in combination with either the regular triangular lattice or with the random lattice yields very good results for concrete specimens subjected to either uniaxial tensile or combined tensile and shear loadings.

2.2. Fracture Law and Model Parameters As mentioned before, fracture is simulated in the lattice model by removing in each loadstep the beam element with the highest stress over strength ratio. The effective stress in a beam is the maximum stress in the outer fibres of the beam following = F/A + o~ 9 {IMil, IMjl}m~ / W

(1),

where F is the normal force in the beam, M~ and Mj are the bending moments in the nodes i and j of the beam, A = b . h the cross-section of the beam, and W = b-hE/6 the section modulus. The factor (x is introduced for regulating the amount of bending that is taken into account. In general 0 < o~ < 1. The coefficient cx mainly influences the tail of the softening diagram as shown in [25]. After a beam has been fractured, simply a new linear elastic

206 analysis is carried out using the reduced lattice. The analyses are carried out with a standard finite element package. The parameters that have to be specified in the model are quite limited. Contrary to macroscopic fracture models no softening function has to be introduced ([13], [26]), but only single valued limit strength values have to be specified for the various constituents, viz. aggregate, matrix and bond zone. In principle the following parameters must be given: tensile strength f~, f~, ftb, Young's Modulus E a, Era, F_,b, beam length 1~, beam size b 9 h, coefficient tx, where the subscript 'a' stands for aggregate, 'm' for matrix and 'b' for bond zone between aggregate and matrix. The determination of some of these parameters is quite straightforward. The influence of beam length ~ on the global reponse was systematically investigated, see [17], [25]. It was found that the length ~ of the beams in the lattice should be smaller than one third of the diameter of the smallest aggregate particle in the concrete structure. If a larger length is chosen, the effect of the small particles will not be included. As recently demonstrated, the small particle effect is quite substantial in the tail of the softening diagram (see [27] and the next section). The cross-section of the beams, and the values for the Young's moduli are related to one another. In general we first select realistic values for the Young's moduli, i.e. values obtained from macroscopic experiments are used. It remains of course questionable if this is allowed. In most analyses the Young's moduli are E a = 70 GPa and E m = E b = 25 GPa. Now the beam dimensions have to be adjusted to obtain a reasonable value for the stiffness of the complete lattice. The procedure is as follows. First the thickness of the beams is set to the thickness of the element that is to be analysed. Next the height of the beam is chosen such that the stiffness of the complete lattice matches the stiffness of a real concrete element with the same composition. For the regular triangular lattice with particle overlay (aggregates between 3 and 8 mm included) this results in a beam height h = 0.68.~. This means that relatively high beams are used. The main problem so far has been the choice for the strength of the beam elements f~, ftb and ftm" In fact it would be better to specify the ratio's fJftb and fJftb, because these determine the amount of disorder in the microstructure. Note that always the strength is tuned to a uniaxial tensile test on a standard laboratory specimen of 50 x 60 x 150 mm. This tuning procedure is essential, and has to be repeated always when a new type of lattice is used, for example when a random lattice based on a larger grid-size s (figure 4) has been selected. Up till now the procedure has been to use strength values for mortar and rock for the matrix and aggregates respectively. For the bond strength a low value was selected based on the macroscopic bond tests of Rehm et al. [3]. These values had to be multiplied by a factor 13 in order to arrive at the same peak strength in the above mentioned standard tensile test. In addition, it should be mentioned here that the tensile strength of, at least mortar and rock, is size dependent (see for example [28]). An unrestricted use of macroscopic strength values is therefore probably not allowed, but because of a lack of relevant data this simplification has

207 been used throughout this paper. In all the physical basis for the limit strength parameters is still under debate. The large advantage however of the new approach is that only single valued parameters have to be specified, which makes the derivation of the parameters from a tuning analysis on the standard tensile test inherently more simple. For more details concerning the determination of the model parameters, see refs. [17], [21]. Obviously, many lattice elements have to be included in a calculation if the fracture response of concrete is to be mimiced in great detail. Therefore, in general, only the area of the specimen where cracks are expected to grow is modelled as a lattice, the remainder of the geometry is modelled with simple plane stress continuum elements that are available in the finite element package that is used to solve the model. Up till now only plane stress problems have been treated because of computational limitations. However the same procedure can be transfered directly to three dimensions.

3. INTERFACE B E T W E E N A G G R E G A T E AND MATRIX In the past few years the model has been tested extensively for concrete and sandstone laboratory scale specimens subjected to tensile or combined tensile and shear loading, see references 9, 15, 16, 17, 21, 25, 27 and 29. The great strength of all these simulations was that correct crack patterns could be found for many different loading cases, without changing the initially determined model parameters (from the standard tenmsile test as described in the previous section). In this paragraph some results of simulations of uniaxial tensile tests are shown. Morover, some recently obtained experimental crack data will be shown, which confirm the capability of the model to mimic real crack behaviour of concrete. As mentioned before, the amount of detail included in the generated particle structure is of utmost importance. In Figures 5 and 6, the load-displacement diagrams and the crack patterns obtained in simulations of the standard tensile test are shown. The dimensions of the prism are 60 x 150 mm, with a thickness of 50 mm. The specimen is double notched at half height; the notches are 5 mm wide and 5 mm deep sawcuts. The nodes at the bottom are supported in vertical direction, and free to move in horizontal direction, except for the middle node. A uniformly increasing vertical displacement is applied at the upper edge of the

4t

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Figure 5. Stress-crack opening diagrams in uniaxial tension. Comparison between numerical simulations with and without particles between 2 and 3 mm, and an experiment on a normal concrete with 8 mm maximum aggregate.

208

Figure 6. Crack growth for the two simulations of Figure 5: (a), (b) simulation with small particles between 2 and 3 mm included, and (c), (d) without particles smaller than 3 mm. The two initial stages (a), (c) are the crack patterns at 200 beams removed, (c) and (d) are the crack patterns when 300 beams are removed. specimen, thereby keeping the bottom and upper edges parallel to each other during the complete loading to fracture. Only the part where cracks are expected to grow is modelled with the random lattice with a grid size s = 1 mm. Two different analyses were carried out, i.e. the first with aggregate particles bewteen 3 and 8 mm included in the mesh, the second with all particles between 2 and 8 mm included. The particle distribution was according to a Fuller curve. The load-displacement curves are shown in Figure 5, where they are compared to an experimental result. The boundary conditions in the test were exactly the same as those in the simulations. For each of the simulations two crack patterns are shown, viz. the stages where 200 beams and 300 beams are removed. The respective locations are indicated in the load-displacement diagrams. In the analyses, a similar softening behaviour was found as in the experiment, except that the behaviour just beyond the peak is somewhat more brittle for the simulations. The effect of including smaller particles in the mesh is clearly visible: when more small particles are included a more 'ductile' response is obtained, especially in the tail of the softening diagram. Another reason for the observed discrepancy is the fact that the three-dimenional fracture process in the laboratory test is simulated in two dimensions. This means that a crack in the simulations always must grow through the depth of the entire specimen. Obviously this does not occur in experiments, as clearly shown for example in [30]. The simulations indicate that near the peak microcracking takes place around the larger aggregate particles: both Figures 6a and 6c indicate that debonding occurs first. Again this seems to be in agreement with experimental observations [30], [31]. In the tail of the softening diagram, larger crack branches have developed in the specimen, but the prisms are still not separated in two parts. Intact material ligaments connect the two parts. These so-called crack face bridges were

209 recently also observed in tensile tests, both by using a vacuum impregnation technique, [30], [32], and by using an optical microscopy technique [33]. Some examples of crack face bridging in 2 mm mortar, 12 mm lytag lightweight concrete (containing up to 4 mm sand) and 16 mm normal concrete are shown in Figure 7. The tests revealed a correlation between maximum size of the stiff aggregate particles included in the concrete and the amount of stress that can be transferred in the tail of the softening diagram [32].

P (kN)

!\.~

I

0

o

20

40

60

80 100 ~- w (~tm)

(a)

Figure 7. (a) Load-crack opening diagrams for three different concretes" 2 mm mortar, 12 mm lytag and 16 mm concrete, (b) crack face bridging at 100 ~tm in 2 mm mortar (curve no.I), (c) in 12 mm lytag (curve no.2), and (d) in 16 mm concrete (curve no.3), [33]. The effect of small particles can be recognised especially in figure 7c. In this photograph quite some branching and bridging, especially around small sand particles is visible. It will be evident that the simulations are not correct when the small particles are omitted from the model. However, computational limitations generally will enforce an upper bound for the amount of particles that can be included in the model. The bond between aggregate and matrix is one of the most important parameters in the model. The bond strength is most likely very low for normal concrete, [3]. In the present analyses, ftb = 1.25 MPa. Surprisingly however, the effect of a variation of the bond strength by a factor of 10, only results in an increase of the macroscopic stress by 30 %, see [29]. This indicates that global strength is not only controlled by bond strength, but more likely by matrix strength.

4. I N T E R F A C E B E T W E E N STEEL AND C O N C R E T E

4.1. Analysis of mechanical interlock of ribbed rebars Good bond between concrete and steel is essential when reinforced concrete is considered. Slip in this zone is often seen as a combination of adhesion, frictional stress

210 transfer and mechanical interlock of the ribs, e.g. [34]. In terms of the lattice model presented earlier, it can be stated that bond between steel and matrix it basically the same as bond between aggregate and matrix. Mutual differences in behaviour are expressed by different strength and stiffness of the beam elements in the model. According to Goto [10] failure of ribbed rebars occurs through the growth of internal cracks among cracks that are visible at the surface of the concrete. Experiments (figure 8) show that internal cracks are formed near the compressive side of a lug at angles of about 60 o to the bar axis.

Figure 8. Photograph of internal cracking around reinforcing steel, after Goto [10]. Vos [34] was among the first to study this local crack behaviour near steel reinforcement bars, using axisymmetric simulations of a single rib. The simulations of Vos have recently been repeated with the lattice model. The main purpose is to conf'trrn whether the internal

Figure 9. (a) Representation of a slip layer section according to Vos [34] and (b) lattice model for the slip layer section showing the area where the random lattice is overlaid with a generated grain structure. The part of the element where no cracks are expected to grow is modelled using normal 4 noded plane stress elements.

211 cracking can be simulated with a tensile fracture criterium for the lattice beams only. The dimensions of the so called "slip layer section" that he analysed are shown in figure 9. Figure 9b shows the finite element mesh of one slip layer section, partly modelled with a twodimensional lattice. Because of the small size of the section, a regular square grid size of 0.1 mm was used for the random lattice (figure 4). The grain structure projected on top of it includes particles with a minimum diameter of 0.2 mm and a maximum of 2 mm. The adhesive strength of the steel-matrix interfaces was relatively low, viz. 0.5 MPa. It should be mentioned that this value is somewhat arbitrary and that the effect of this adhesive strength is not examined. The strength and stiffness of the remaining beam elements was kept the same as in the simulations presented before. In the finite element calculation the upper (horizontal) boundary was allowed to move in a horizontal direction only, the vertical displacements of this edge were coupled to the vertical displacement of point A such that the upper boundary translated parallel to the lower boundary. A linear elastic spring representing the radial stiffness of the outer surrounding concrete was connected at the upper left comer (figure 9). In total two plane stress simulations were carried out: the first with k--O, and the second with an infinite stiffness of the elastic spring (k_-oo). The left and right (vertical) boundaries were free to move. Loading was achieved by a prescribed horizontal displacement of the upper left comer A. Results The results indicate that the failure mechanism depends strongly on the boundary conditions. In figure 11 the crack history for each of the two simulations is given. The amount of broken beam elements is also given for each stage in the fracture process. It can be seen that there are no significant differences at the beginning of the fracture process (up to step 250). The simulations show that the concrete part of the slip layer section is separated from the steel bar almost immediately. It is remarkable that, at the right side a continuous crack is formed around a grain particle, in spite of the much lower strength of the steel-matrix interfaces. Because of failure of the elements in the interfacial zone, slip between the rebar and the concrete occurs and a compressive strut can form near the lug of the reinforcement bar. For the situation where k=0 an internal crack is formed parallel to this strut. However the angle of this localized crack to the bar axis is less than 60 o, the value measured in Goto's tests [10]. For k=oo no localized crack is formed until 1250 beam elements are broken. It seems that to simulate a fully localized crack pattern in this situation, the lattice has to be chosen larger. Nevertheless the angle to the bar axis seems larger than for k=0. Obviously the stiffness of the elastic spring has an important effect on the slope of the internal cracks. The crack growth observed in the simulations presented in this paper are similar to the patterns found by Vos. The slope of the cracks predicted in his simulations is also less than 60 o. The actual k-value will be somewhere between the two limit cases examined here, i.e. between k=0 and k=oo. A condition that was not examined but may also influence the failure mechanism is the rotational stiffness of the right and left edge of the slip layer section. When a continuous bar is considered, there will definitely be some lateral resistance against rotation of the vertical edges. It is expected that increasing the stiffness of these edges will lead to a more steep slope of the internal cracks. After all the internal cracks in figure 8 show a decreasing slope near primary cracks. Another point that has not been investigated is the influence of the rotational stiffness of the upper (horizontal) boundary. In both simulations this boundary was only allowed for a restricted parallel displacement which implies inf'mite

212 rotational stiffness. In reality however also some rotation will be possible. Considering the effect of stiffness of the horizontal and vertical edges of the slip layer section it might be interesting to investigate the same situation in which all boundaries are supported by distributed linear elastic springs, instead of one single (vertical) spring at the upper left comer.

Figure 10. (a) Crack history for a simulation with k=0 and (b) k=~. The number of broken beam elements of each loading step is given by the numbers between brackets. 4.2. Local bond tests A disadvantage of the results of the bond-slip simulations as outlined in the previous paragraph is that it is very difficult to extrapolate them to a real situation. Comparison is impossible, not with any experiment nor with any practical situation. As mentioned in the introduction, a detailed bond test was developed at the Institute for Building Research (TNO) in order to define the effective properties of the (cracked) bond-layer [1 ], [2]. In these local bond tests the assumption was made that slip is mainly caused by micro cracking in a small zone around the rebar. An axisymmetric concrete specimen was developed contained in a hollow aluminum tube with a centred rebar. The rebar was either fibbed or smooth with a diameter of 16 mm. The thickness of the concrete layer was taken equal to the radius of the rebar and resembles the slip layer around the rebar. The aluminum tube with a thickness of

213 6.3 mm represents the confinement of the surrounding concrete which is assumed to be uncracked. With the lattice model this situation has now been analysed. However we had to confine ourselves to a plane stress simulation, because a full fledged three dimensional analysis would require a too large computational effort. The two dimensional mesh used for the numerical simulation for rebar and tube is given in figure 11 a. To obtain perfect bonding between concrete and tube, a grove is created at the inner side of the aluminum tube. A cross sectional view of the tube used in the experiment is given in figure 1 lb.

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Figure 11. (a) Finite element mesh, (b) cross section, and (c) lattice model with generated grain structure used to simulated the detailed experiments [ 1]. Considerable attention is given in modelling the axisymmetric experiment. To obtain the same radial stiffness in the plane stress simulations as in the experiments springs were attached at the outer side of the tube. The stiffness of the springs was determined using the tube geometry and linear elastic properties of the tube that was used in the tests. For the simulations of the concrete part of the specimen a regular lattice was used (figure 1 l c). Particles with a minimum diameter of 2 mm were projected on top of the lattice. The maximum diameter was taken a quarter of the diameter of the rebar, i.e. 4 mm. To model the tests with high accuracy a small beam length was taken (0.75 mm). Especially the grove in the tube and the ribs of the rebar made it necessary to use a lattice with such a small beam size. The boundary conditions are also given in figure 1 l a. Loading was achieved for F2---0.5*F1 and RA+RB---0.5 *F~. The stress and displacement measurements in the experiments

214 were not taken axisymmetric in contradiction with the geometry of the specimen. To mimic the situation as good as possible deformations were measured at 30 different nodes in the finite element mesh. Five nodes at regular distance were selected at the outside of the tube and the transition between tube and concrete and between concrete and rebar. Four different situations were examined to qualify the bond slip behaviour. First simulations of a ribbed and smooth rebar were performed where no special strength was assigned to the beam elements in the interfacial zone, i.e. the transition between concrete and rebar. Thereafter also the case was examined where the interfacial (or adhesive) strength was taken equal to the interface strength as chosen in the example of paragraph 4.1 where only a rib was modelled (i.e. ft--0.5 MPa).

Results The results of the detailed tests presented in [1], [2] are very limited. The failure mechanism is hard to study because of the presence of the tube which encloses the whole concrete specimen. Qualification of the cracks therefore was not possible. Conclusions were extracted only from the various stress-deformation relationships. The horizontal and vertical stress was plotted as a function of the horizontal and the vertical displacement of the bond zone. However because the extensometers were attached to the tube surface no direct measurements were taken at the aluminum tube interface. The results given in [ 1] show very unstable behaviour of all load-displacement curves. The maximum radial bond stress measured in the experiments was 1 and 6 MPa for the smooth bar and ribbed bar respectively when a tube of 6.3 mm was used. For a diameter of the rebar of 16 mm and a length of the bond zone of 50 mm this results in a force (F2-F1) of 5 and 30 kN respectively. When the results of the simulations are compared with the experiments, only a very small resemblance is found. Two explanations can be given. First the experiments were tested under cyclic loading which is not possible with the lattice model. The computed load-displacement curves can therefore be compared only with the envelope curve of the cyclic experiment. The second point is mentioned earlier and concerns the deficiency of the predicted relation between the applied load and displacements. Both points make that precaution has to be taken in drawing any definite conclusions. However mutual comparison between the various simulations remains very well possible. Because the grain structure is kept equal for all simulations, differences in strength of the interfacial zone as well as the presence of the ribs at the rebars can be studied. Figure 12 shows the crack pattern of each simulation in an advanced stage of the fracture process. To emphasize the crack pattern in the lattice of beam elements no grain structure was plotted in figure 12 and 13. The rib geometry is shown schematically in these figures. The grain structure which was used is shown in figure 4.4c and was not changed during the simulations. When no interfacial strength is assigned (figure 12a and 12b) not very much differences can be observed in the crack patterns of the smooth and the ribbed rebar. The adhesive strength is quite large (between 1.25 and 10 MPa) and seems to override the effect of the lugs. These inclined cracks seem to nucleate everywhere along the bar irrespective of the location of the lugs. The angle of the cracks with the rebar is approximately 60 o. Because the strength of the interface elements seems to play a major role, also simulations were performed where a low adhesive strength (ft--0.5 MPa, see figure 12c and 12d) was assigned to the beam elements connected to the steel rebar. The behaviour during fracturing is comparable with the simulation of one single rib. First the concrete is torn loose

215

Figure 12. (a, b) Crack patterns for smooth and ribbed rebar where no interfacial strength is specified (i.e. it can vary between 1.25 and 10 MPa) and (c, d) for a interfacial strength of 0.5 MPa. from the rebar, compressive struts are formed near the lugs of the ribbed reinforcing bar and continuous cracks are formed parallel to these struts (figure 13). When a smooth bar is used, the bar is separated almost immediately but now hardly any diagonal cracks are formed. The fact that the angle of the cracks corresponds with the findings of Goto [10] and Otsuka [11] confirms the statement made in the previous chapter where only one single rib was modelled. Not only the outer surrounding concrete at the top of a slip layer section influences the failure mechanism but also the neighbouring sections and fibs. A more thorough parameter study of the interfacial properties should be carried out to arrive at more reliable results. The simulations demontrate however perfectly clear the essence of the bond-slip behaviour. The global behaviour of a reinforcing bar depends strongly on the microcracking near the interfacial zone between concrete and rebar.

Figure 13. Crack history for a ribbed rebar with an adhesive strength of 0.5 MPa for the beam elements is the interfacial zone. The numbers between brackets exhibits the amount of broken beams.

216 The influence of the adhesive strength is confirmed by the load-displacement responses shown in figure 14. In this figure the load F1 is plotted versus the average lateral deformation. The lateral deformation is calculated as the displacement of the inner side of the aluminum tube minus the displacement of the transition between concrete and rebar. Hardly any differences are observed between a smooth and ribbed rebar when no interfacial strength is specified. However when an adhesive strength of 0.5 MPa is used for the interfaces the behaviour of the smooth rebar becomes considerably more ductile (figure 14b). The fact that the curves in figure 14b are interrupted at a low lateral deformation is owing to the interfacial properties of the beams in these simulations. At this stage almost all elements at the interface of concrete and rebar are torn loose. 3.0

r, [kN] ,,

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2.5 2.0

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1.5 1.0 0.5 0.0 0.0

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ft,int~rfoce-- 0.5 N / m m ribbed

ooth 5:0

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(b)

Figure 14. Load-slip diagrams of local bond simulations. (a) With interface strength equal to concrete strength (1.25 MPa < ft < 10 MPa), and (b) interface strength ft---0.5 MPa.

5. ANALYSIS OF ANCHOR PULL-OUT IN CONCRETE 5.1. Description of the problem A last example of steel concrete interactions comprises the analysis of anchor pull-out in concrete. Both simulations and experiments were cawed out. The selected geometry was according to the round robin proposal of RILEM TC 90-FMA [35]. First the problem will be outlined whereafter the results of experiments and simulations are compared. The tests and simulations focus on the failure mechanism and the deflection of the upper edge of the anchor (point A in figure 15). Part of the proposal dealt with the effect of specimen size, which is directly related to the embedded depth of the anchor. Furthermore the support span was varied, and the effect of lateral confinement was studied. In this paper only attention is given to the effects of confinement on the pull-out behaviour and size effects in the unconfined situation are shown. The effects of varying the support span is discussed in more detail in [21], [36]. For the experiments a dedicated test set-up was developed in the Stevin Laboratory as illustrated in figure 16. To minimize the influence of friction in the supports double hinged tensile bars were used to support the specimens. This method of supporting the specimens allows for horizontal translations and is already used successfully [25]. The average displacement of the anchor head was measured using four LVDTs. This signal was used as

217

5t

-o.~j___~p I o in t A 3t

k

a

o 0 0 0

~.o

0 v -U

/ 6d (300,600,900) t=d/10 " - ~ ~ ft=3 MPa GF=100 Nrn/m 2 Ec=30 GPa (fc=30 MPa) Figure 15. Round robin geometry and material properties proposed by Elfgren, Eligehausen and Rots [35].

Figure 16. Schematic view of the testing apparatus.

feed back in the closed-loop servo-controlled system. Lateral confinement was applied in the same manner in both the simulations and the experiments. Both edges of the specimen were connected with steel bars. These bars were fixed to stiff steel beams to assure a uniform distribution of confining forces at the specimen edges. A prestressing force of approximately 1 kN was applied to prevent the beams from slipping. The simulations presented in this paper were carried out with a random lattice with constant strength of the beam elements. For the specimens with an embedded depth of the anchor of 50 mm a cell size of the regular square lattice of s=5 mm seemed sufficient. Larger specimens (d=100 mm and d=150 mm) have been simulated with s=10 mm because of computational limitations. The calculation time would take more than a month on a SUN SPARC 2 workstation when a random lattice with a square lattice size of 5 mm (s=5) was used for the largest specimen.

5.2. Experimental results In the experiments it was observed that the global failure mechanism remains the same for specimens of different size as well as for specimens with different support spans. Figure

218 17 shows three cracked specimens of different size (d=50, 100 and 150 mm), with a support span of twice the embedded depth of the anchor (a=2d). At one side of the bolt this crack grows towards the support, along the compressive diagonal. At the other side however the crack propagates in an extended line in the opposite direction, i.e. downwards from the bolt and towards the edge of the specimen. Similar patterns were also found for unconfined specimens with different support spans (i.e. a---d and a=d/2, [36]. Asymmetric crack patterns for small support spans (a=d/2) were also observed by Helbing et al. [37]. n

m

oo

I

i

m

!I i

! J

(a)

(b)

(c)

Figure 17. Failure mechanisms for unconfined specimens. When the edges of the specimen are obstructed to move away from each other (due to confinement), only the post-peak regime is influenced. Because confinement is activated only when the edges of the specimen are driven apart, no significant differences are observed until the peak in the load-displacement response is reached. Figure 18 shows that the peak-load for an unconfined specimen is approximately the same as a specimen loaded with confining bars. 60 ] 50

F, [kN]

60

| / / ",.-..~ - ~"'n." - , ] / /

30

F, [kN]

Q=d 50

i

30

20

20

1o

1o

o

o 0

200

400

600

800

deformetion [~m]

1000

0

200

400

660

860

-I000

deformation [#m]

(a)

(b) ........

confined

unconfined

Figure 18. Load-displacements curves for specimens with a support span of (a) a--d and (b) a=2d.

219 Due to crack localisation beyond the peak a higher load is required to maintain the prescribed deformation rate. This behaviour results in a second peak in the load-displacement curve. When the support span increases this second peak will be observed in a more advanced stage of the fracture process, i.e. at a larger displacement of the anchor head. The effect of confinement on the crack pattern is illustrated in figure 19. Because of the horizontal confining forces, the cracks are not longer allowed to grow asymmetric as in the unconfined situation (figure 19a). This results in more symmetric crack patterns. As soon as the cracks have reached the supports, they start to bifurcate (point A in figure 15). This behaviour was also found in tests performed by Karihaloo [38]. The compressive state of stresses caused by the confinement to crack growth parallel to the compressive strut which is formed, i.e. horizontal. The increasing stiffness of the reinforcement bars leads globally to a more ductile behaviour, as shown in figure 18. This phenomena will also be discussed in the next paragraph. i

m

f

(a)

(b)

Figure 19. (a) Final crack pattern for an unconfined and (b) a confined specimen with a support span of a=2d. Bridging and bifurcating of the cracks can be observed at different scales. To demonstrate this phenomena, surface cracks were optically measured with a QUESTAR remote measurement system. The system consists of a high resolution (1.1 ~tm), long distance microscope fixed on a tilt-head. Three stepper motor take care of motions in three orthogonal directions. The measurement system operates at a distance of approximately 150 mm of the specimen surface. Illuminating the specimen is achieved by two fibre optic arm connected to a light source. For more specific information the reader is referred to [33]. After the fracture process was taped, a video printer was used to print the relevant images. At the moment the system is upgraded for automatic scanning a specific area at the specimens surface. After digital storage of the pictures, image processing with programmable software should give more detailed information of the fracture processes in two phase materials. Figure 20 shows some results of this real time measuring technique for the anchor bolt experiments. Cracks growing in area B (figure 20) are shown at three different stages of the fracture process. At the macro level branching of the cracks can be observed near the supports of an unconfined specimen (point A in figure 15). When the cracks are observed at the meso level also

220 bifurcating cracks and crack-face bridges are found. Vacuum impregnation tests on single edge notched concrete plates [30], [32] loaded in tension show also these crack-face bridges.

,

(b) 2mm Figure 20. Crack face bridging and bifurcating cracks observed in area B with optical microscope. The black spots in the crack patterns 1, 2 and 3 are surface pores.

5.3. Simulations The crack patterns observed in the experiments can be simulated very well with the lattice model. The failure mechanism shows an asymmetric crack pattern (figure 21a), resembling the experiments. This failure mechanism can be explained by the rigid body I

(a)

[ I

Co)

Figure 21. (a) Final crack pattern of an unconfined specimen simulated with the lattice model. (b) Rigid body kinematics of a cracked specimen.

221 kinematics of a cracked specimen (figure 21b). To obtain similar types of failure in the simulations and experiments it is very important that the simulations are in accordance with the experiments. Both in the experiments and the simulations, the supports are allowed to translate in horizontal direction. In a set up where the specimen is supported in compression (for example between roller bars), it is more likely that friction is introduced in the specimen. This leads to a loading case which is more or less similar to the loading situation of a confined specimen. 30

F, [NN]

20 10

0

0

260

460

660

860

1000

deformation [Nm] Figure 22. Load-displacement curve of a specimen with an embedded depth of the anchor of 100 mm. The load-displacement response predicted by the lattice model shows a too low peak load and a less ductile post-peak response as expected from the tests (figure 22). This brittle behaviour can be explained by neglecting three dimensional effects in the plane stress simulations. Also neglecting too much detail in the lattice influences the brittleness of the numerical model, [27]. A too low peak-load may be caused by the fact that a too coarse lattice is used for properly introducing the concentrated load near the anchor bolt. Locally this leads to high stresses in this bonding zone which results in premature breaking of the beam elements. Note that the model has proven capable in predicting the correct peak load for specimens loaded in tension and combined tension and shear [9], [ 17]. However in those cases a more refined lattice could be used. The bond-slip behaviour is best outlined by varying the stiffness of the confining bars, as mentioned before. Therefore three simulations for both a=d and a=2d were carried out. The stiffness of the confining bars was taken k=0, k=500 MPa and k=oo respectively. The other variables were kept constant. The lateral confinement was applied in the same way in the numerical analyses as in the experiments: both edges were restricted to move apart by connecting them with steel bars. These bars were connected to steel beams modelled at the left and right edge of the specimen. In figure 23 the load-displacement responses are plotted for three simulations with varying stiffness of the reinforcement bars. The case k=500 MPa corresponds to the situation in the experiments. It can be seen that there is a considerable influence in the post peak behaviour using zero (k--0), partial (k=500 MPa) or infinite confinement (k=oo). The global load-displacement response tends to be more brittle for increasing stiffness, as would be expected. For infinite confinement the load even starts to increase after a slight drop. For smaller support spans the load starts to increase earlier than for a large span, as can be concluded by comparing figure 23a and 23b.

222

F [kN]

F [kN] 40 I

d-150, a-d

I,, I

ltl i I o

'*"/t~. I, ,/~,r

30

/

,'~ /

,,!

/t ~

"1

/

-

40

d-150, o-2d

30 i I/,

20

9

'

.

s"

s"

I

20

i, ~

It

"1

I

t s

10

10

0

5'0

160

(a)

1,50

260

250

0

0

5'0

160

260

250

deformation [/xm]

deformation [#m]

k-O

1,50

k-500 N/mm

2

(b)

. . . . . . . . k=infinite

Figure 23. Influence of confinement on the load-displacement response for different support spans, i.e (a) a=d and (b) a=2d.

6. CONCLUSIONS In this paper several examples of the application of a newly developed lattice model to steel-concrete interface problems are shown. With the model basic crack phenomena in plain concrete can be simulated quite accurately. Initial debonding and crack face bridging in the softening branch of the stress-crack opening diagram of a uniaxial tensile bar, appear as a natural consequence of the modelization. A brittle fracture law for the matrix and the bond layer between aggregate and matrix suffices. The simulations indicate that concrete is a multiscale material, and even crack growth phenomena around the small particles have an enormous effect on the global response and can therefore not be neglected. The model was succesfully applied for the simulation of bond-slip behaviour of ribbed rebars. The cracking of the concrete layer near the lugs on the rebars is in agreement with observations by Goto [ 10]. The inclination of the cracks that nucleate near the lugs is strongly dependent on the amount of confinement on the interfacial zone. This confinement is either caused by the concrete surrounding the interfacial zone, or alternatively by some external load. The results of the bond-slip simulations indicate that the lattice model can be used for analysing macroscopic bond-slip behaviour between steel and concrete. However, it should be kept in mind that boundary condition effects play a major role on bond-slip behaviour. A reliable bond-slip model should therefore include these effects. An alternative approach may (in the near (?) future) be to model the bond-slip behaviour at the meso-scale. The advantage of the 'brittle' lattice model is that only single valued parameters are needed for simulating realistic crack phenomena. It is the author's opinion that a good model for interface behaviour should describe the physics of the problem as accurately as possible. The lattice model is a first step towards such an approach.

223 Finally, the paper contains an example of the pull-out problem. In this case not only numerical simulations were carried out, but parallel to the analyses several experiments were performed in a specially designed loading frame. Again the model predicts realistic cracking response. The load-displacement response is less accurate, which can be explained from a too coarse modelling. Future developments in computer technology, viz. the development of more powerful computers and new fast solution procedures, would probably allow for more detailed, and therfore more accurate simulations of the pull-out problem.

ACKNOWLEDGEMENT

The pull-out experiments were carried out by Mr. A.S. Elgersma. The authors are indebted for his expert help. The discussions with Dr. E. Schlangen are greatly acknowledged.

REFERENCES

1.

Dragosavi6, M. and Groeneveld, H., Concrete Mechanics - Local Bond, part I Physical Behaviour and Constitutive Consequences, and Part II Experimental Research, Research Report BI-87-18/19, TNO Building and Construction Research, Rijswijk, The Netherlands, (1988). Dragosavi6, M. and Groeneveld, H., Bond Model for Concrete Structures, in Proceedings of the Int'l. Conf. on Computer Aided Analysis and Design of Concrete Structures, Split, Yugoslavia, Pineridge Press, Swansea, UK, (1984). Rehm, G., Diem, P. and Zimbelmann, R., Technische Mi3glichkeiten zur Erh/3hung der Zugfestigkeit von Beton, Deutscher Ausschuss fiir Stahlbeton, Heft 283, Berlin, 1977. Maso, J.C. (editor), Interfaces in Cementitious Composites, Proceedings of the RILEM Conference held in Toulouse, October 1992, Chapman & HalI/E~FN Spon, London/New York, (1992). Mindess, S., Interfaces in Concrete, in Materials Science of Concrete I (ed. J.P. Skalny), The American Ceramic Society, Inc. Westervill (OH), (1989) 163. Zhang, M.H. and Gjorv, O.E., Cement & Concrete Research, 20 (1990) 610. Paulay, T. and Loeber, P.J., Shear Transfer by Aggregate Interlock, in Shear in Reinforced Concrete, Volume I, ACI SP-42, American Concrete Institute, Detroit, (1974) 1. Van Mier, J.G.M. and Nooru-Mohamed, M.B., Fracture of Concrete under Tensile and Shearlike Loadings, in Fracture Toughness and Fracture Energy - Test Methods for Concrete and Rock, H. Mihashi, H. Takahashi and F.H. Wittmann eds., Balkema, Rotterdam, (1989) 549. Nooru-Mohamed, M.B., Schlangen, E. and Van Mier, J.G.M., Advanced Cement Based Materials, 1 (1993) 22. 10. Goto, Y., A CI Journal, 68 (1971) 244. 11. Otsuka, K., X-ray Technique with Contrast Medium to Detect Fine Cracks in Reinforced Concrete, in Fracture Toughness and Fracture Energy - Test Methods for Concrete and Rock, H. Mihashi, H. Takahashi and F.H. Wittmann eds., Balkema, Rotterdam, (1989) 521.

224 12. Ingraffea, A.R. Gerstle, W.H., Gergely, P. and Saouma, V., J. Structural Engineering (ASCE), 110 (1984) 871. 13. Rots, J.G., Simulation of Bond and Anchorage: Usefulness of Softening Fracture Mechanics, in Applications of Fracture Mechanics to Reinforced Concrete, A. Carpinteri ed., Elsevier Applied Science Publishers, London/New York, (1992) 285. 14. Tepfers, R., Magazine of Concrete Research, 31 (1979) 3. 15. Schlangen, E. and Van Mier, J.G.M., Cement & Concrete Composites, 14 (1992) 105 16. Van Mier, J.G.M. and Schlangen, E., Journal of the Mechanical Behaviour of Materials, 4 (1993) 179. 17. Schlangen, E., Experimental and Numerical Analysis of Fracture Processes in Concrete, PhD thesis, Delft University of Technology, (1993). 18. Hrennikov, A., Journal of Applied Mechanics, (1941) A169. 19. Hen'mann, H.J. and Roux, S. (eds.), Statistical Models for the Fracture of Disordered Media, Elsevier, Amsterdam (1990). 20. Herrmann, H.J., Patterns and Scaling in Fracture, in Fracture Processes in Concrete, Rock and Ceramics, J.G.M. van Mier, J.G. Rots and A. Bakker eds., Chapman & HalI/E&FN Spon, London/New York, (1991) 195. 21. Vervuurt, A., Van Mier, J.G.M. and Schlangen, E. (1993), accepted for Materials & Structures (RILEM). 22. Schlangen, E. and Van Mier, J.G.M., Boundary Effects in Mixed Mode I and II Fracture of Concrete, in Fracture Processes in Concrete, Rock and Ceramics, J.G.M. van Mier, J.G. Rots and A. Bakker eds., Chapman & HaI1/E&FN Spon, London/New York, (1991) 195. 23. Reinhardt, H.W., Anspriiche des Konstrukteurs an den Beton, hinsiuchtlich Festigkeit unde Verformung, Beton, 5 (1977) 195. 24. Mourkazel, C. and Herrmann, H.J., A Vectorizable Random Lattice, Preprint HLRZ 1/92 (1992). 25. Schlangen, E. and Van Mier, J.G.M., International Journal of Damage Mechanics, 1(4) (1992), 435. 26. Hillerborg, A., Modeer, M. and Petersson, P.-E., Cement & Concrete Research, 6(6) (1976), 773. 27. Van Mier, J.G.M., Schlangen, E. and Vervuurt, A., Analysis of Fracture Mechanisms in Particle Composites, in Micromechanics of Concrete and Cementitious Composites, C. Huet ed., Presses Polytechniques et Universitaires Romandes, Lausanne (1993), 159. 28. Carpinteri, A. and Ferro, G., Apparent Tensile Strength and Fictitious Fracture energy of Concrete: A Fractal Geometry Approach to Related Size Effects, in Fracture and Damage of Concrete and Rock (FDCR-2), H.P. Rossmanith ed., Chapman & Hall/E&FN Spon, London/New York, (1993) 86. 29. Schlangen, E. and Van Mier, J.G.M., Numerical Study of the Influence of Interfacial Properties on the Mechanical Behaviour of Cement-Based Composites, in Interfaces in Cementitious Composites, J.C. Maso ed., Chapman & Hall/E&FN Spon, London/New York, (1992) 237. 30. Van Mier, J.G.M., Cement & Concrete Research, 21(1) (1991) 1. 31. Hsu, T.T.C., Slate, F.O., Sturman, G.M. and Winter, G., ACI Journal, 60 (1963), 209. 32. Van Mier, J.G.M., Fracture Process Zone in Concrete: A Three Dimensional Growth Process, in ECF8 Fracture Behaviour and Design of Materials and Structures, D. Firrao

225 ed., EMAS Publishers, Warley, UK, (1990) 567. 33. Van Mier, J.G.M., Crack face bridging in Normal, High Strength and Lytag Concrete, in Fracture Processes in Concrete, Rock and Ceramics, J.G.M. van Mier, J.G. Rots and A. Bakker eds., Chapman & Hall/E&FN Spon, London/New York, (1991) 27. 34. Vos, E., Influence of Loading Rate and Radial Pressure on Bond in Reinforced Concrete. A Numerical and Experimental Approach, PhD thesis, Delft University of Technology, (1983). 35. Elfgren, L., Eligehausen, R. and Rots, J., Round-Robin Analysis and Tests of Anchor Bolt - Invitation, in RILEM TC-90 FMA Fracture Mechanics of Concrete - Applications, L. Elfgren ed., Lule~ (1991). 36. Vervuurt, A., Schlangen, E. and Van Mier, J.G.M., A Numerical and Experimental Analysis of the Pull-Out Behaviour of Steel Anchors Embedded in Concrete, Report 25.5-93-1/VFI, Delft University of Technology, (1993). 37. Helbing, A., Alvaredo, A.M. and Wittman F.H., Round-Robin Tests Of Anchor Bolts, in RILEM TC-90 FMA Fracture Mechanics of Concrete - Applications, L. Elfgren ed., Luleh (1991) 8" 1. 38. Karihaloo, B.L., Testing of Anchor Bolts in Plane Stress and Axisymmetric Geometries, in Fracture and Damage of Concrete and Rock (FDCR-2), H.P. Rossmanith ed., Chapmann & Hall/E&FN Spon, London, (1993) 191.

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

227

Modelling of Constitutive Relationship of Steel Fiber-Concrete Interface Surendra P. Shah, Zongiin Li and Yixin Shao NSF Science and Technology Center for Advanced Cement Based Materials, Northwestern University, Evanston, IL 60208, U. S. A.

ABSTRACT In this chapter, the modelling of constitutive relationship of steel-concrete interface is discussed. Studies of the interfacial zone between steel and bulk cementitious matrix demonstrated that the properties of this special region could be much different from the behavior of the bulk matrix and thus should be used as a physical background for the interface modelling. Various characterizations of steel-concrete interface are reviewed. In general, the constitutive relationship of steel-concrete interface can be divided into two categories according to the consideration of adhesion. If one assumes that adhesion exists on a part of the interface, then the constitutive relationship can be established by treating the interface as a bonded-debonded case. Otherwise, so called slip-based interface model, in which the constitutive equation is based on the bond stress-slip relationship, can be applied. For the bonded-debonded interface model, the constitutive equations are established by dividing the interface, corresponding to the bonded and the debonded regions. For the debonded case, three different methods of describing frictional force, i.e. constant frictional force distribution, Coulomb frictional force distribution and cohesive frictional force distribution, are examined. In this chapter, the constitutive relationship for bonded-debonded interface is emphasized and two criteria for interface debonding, stress based criterion and fracture parameter based criterion, are inspected. The various constitutive relationships are further evaluated by comparing them with the recently conducted Moire Interferometry test results. The comparison supports the bonded-debonded characterization for steel-concrete interface and shows the limitation of the modelling of frictional force in a debonded region. I. MICROSTRUCTURAL CHARACTERIZATION OF STEEL-CEMENT INTERFACE An interfacial zone existing between the bulk matrix and the steel was first observed in 1970's[I][2]. Since then various studies have been conducted to investigate the microstructure characteristics of the inteffacial zone to meet the demand of appropriate modelling of the constitutive relationship for steel-cement interface. The investigations reported in the literature include microscopic observation, such as when Scanning Electronic Microscopy (SEM) is used to distinguish the chemical components and the physical features for the interface, and micromechanical observation when the indentation is employed to measure the mechanical properties of the interface.

228 1.1 Optical observations of interracial zone A direct observation of the existence of an interfacial zone is shown in Fig. 1. A backscattered electron image of a polished surface is illustrated. Denser regions emit a larger number of electrons and appear brighter. It can be seen from the figure that the color of the interfacial zone is much darker than the color of the bulk matrix. This implies that the interfacial zone has higher porosity than the bulk matrix does. Hence, the interface is a weak transition zone.

Figure 1

A backscattered electron image of steel cement interface

Pinchin and Tabor[ 1] carried out a split test by using cylindrical paste specimens with a single steel wire embedded in the center to study the interfacial zone in detail. The specimens were first split to expose the steel-cement interface. The exposed surfaces of the steel and the cement paste were observed with SEM. They reported that the interfaces were rich in CH and had a porous material made up of C-S-H and some ettringitc. The quantity of the enrichment of Ca(OH)2 near the steel-cement interface was also examined by X-ray diffraction. The interface material powder was obtained by drilling the hole remaining in the matrix after pulling out of the wire. The X-ray diffraction pattern of the interfacial material was compared with that obtained from the bulk matrix of the same specimens. It was demonstrated that the peak height ratio of [Ca(OI~ + CaCO3]/Si was 28% - 42 % higher for the powder from the interface than for the powder from the bulk matrix. However, Pinchin and Tabor did not discuss the orientation of the CH at the steel-cement interface and it was difficult to resolve from their micrographs whether it tended to form a f'dm or not.

229 The structure of the steel-cement paste interface was also investigated by A1-Khalaf and Page[2] using cylindrical specimens, the bottom half of which was made of steel and the upper half of cement paste. These specimens were broken in uniaxial tension, and it was found that the fracture occurred in the paste close to the interface for young specimens and fight at interface for old specimens. The exposed surfaces were studied by SEM. They reported that the paste side of the interface consisted of a discontinuous CH layer (< 1/zm thick) that replicated the topography of the steel surface, with C-axis oriented normal to the steel surface. The CH exhibited a dendritic growth and in many cases crystal boundaries could be observed. In addition to this layer, there was frequent evidence of a C-S-H gel of various morphologies, and well developed crystals formed at different orientations. The C-SH gel was not densely packed near the interface. It occasionally could be observed adhering to the steel surface, but usually it was confined to the paste side of the interface. Bentur et al.[3] also carded out the experiments using a compact tension specimen reinforced with a steel fiber. The specimen was ground and polished to expose the fiber to a depth corresponding to about half the steel-fiber diameter. This surface was kept upwards for observations in the SEM. They found that the interfacial zone surrounding the fiber was substantially different from the "bulk" paste further away from the fiber surface. As schematically shown in Fig. 2, the interfacial zone consisted of (a) a thin (1 or 2/~m thick) duplex film in actual contact with the reinforcement, (b) outside of this zone, a layer about 10 to 30/~m in thickness, which, in reasonably well hydrated systems, was largely occupied by relatively massive calcium hydroxide crystals, with occasional interruptions of more porous regions, and (c) again, outside of this zone a highly porous layer parallel to the interface. This layer gradually became denser as it approached the bulk matrix. The

~ . t -,'k.x "/'\'-,,: \.., .~,,Z >',.~,>.'~ 9 s ' . ; . I ". x~,.. 'L._~'.x. ,.~,'.~. - Z:,'x~, ~ " ,~ ~" ~ "/.. ~', "~.I_, "-,,_.

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Schematic of fiber-cement interface, [3].

230 interfacial zone also included regions where no CH was deposited, but contained rather a porous mass of C-S-H, ettringite, and individual CH crystals in contact with the duplex film. The mechanism of the formation of the interfacial zone was studied by Wei et al.[4]. They verified that the formation of this special region began during the mixing operation. At this time, a film of water formed on the surface of the steel fiber. During mixing, cement particles moved into the water film, but the concentration of particles remained smaller than the rest of the bulk materials. The lowest concentration of cement particles was closest to the steel fiber surface. The concentration gradually increased as the distance from the fiber surface increased. Cement solutes diffused simultaneously with cement particles into the boundary film of water. When the matrix had hardened, further migration of cement particles was prevented and the interface transition zone was formed. From the above mentioned observations, it can be concluded that the interface between steel and bulk cement matrix contains a rather porous mass of C-S-H, ettringite and discontinuous CH crystals in contact with the steel surface. Due to this characteristic, it can be inferred that the interface is a thin weak layer with low modulus. When we talk about the properties of the interface, usually it implies the properties of this thin layer. Hence, the interface can de defined as a thin layer of matrix separating the steel surface and bulk matrix in the present study. 1.2 Micromechanical behavior of interfacial zone To study the micromechanical properties for the interfacial zone, Wei et al.[4] conducted microhardness tests. The microhardness was measured with a microhardness tester. A Vickers indentation shape was used. The microhardness was measured five to seven times at each points. The measurements were made at locations where the particles were hydrated. The microhardness of the weakest point, for the 0.5 mm diameter smooth steel fiber used, was between 25 and 35 microns from the surface of the fiber. The width of the interface transition ring was approximately 80 microns. For distances greater than 80 microns from the surface of the steel fiber, the microhardness was constant for a given water-cement ratio (see Fig. 3). w/C_ A similar test was carded Ir ( I ) 0130 out by Pinchin and Tabor[I]. They obtained the hardness 11] profiles from the surface of steel 1 wires in a bulk of the cement paste. Profiles from their sealed and water cured single fiber 141 specimens were measured at the - 40 (5) age up to one month. Their hardness profiles clearly showed a decreased hardness between 0.15 mm to approximately 0.75 mm from the wire surface for 0 E.O 40 60 80 I00 120 O.m~crons both curing conditions. Each point was the result of a The microhardness measurement minimum of 8 measurements on Figure 3 results of interfacial zone, [4]. 3 different samples. The

231 decrease in hardness from 70 kg/mm2 in the bulk to about 30 kg/mm 2 within 0.2 mm of the steel wire surface reflected an increase in porosity from 16% to 25 %. The differences of mechanical properties between the interfacial zone and the bulk matrix are also demonstrated by recently conducted uniaxial tension tests of steel wire reinforced cementitious composites using Moire interferometry technique[5]. Fig. 4 shows the locally enlarged moire fringe patterns for a steel fiber and the surrounding matrix. Since Moire interferometry is a deformation measurement technique, the observation of the tiny interface zone can be made only after the relative displacement between the fiber and the matrix along the interface has taken place. Shown in Fig. 4 is picture taken from the debonded region, in which the discontinuity of deformation at the interface is clearly demonstrated. The difference of fringe density can be used to distinguish the fiber, the interface and the bulk matrix. Fringes on the fiber surface have sharp edges while fringes in bulk matrix material are undefined. The sparse bulk matrix fringes tilt at a distance somewhere away from the fiber. The fringes between the tilted horizontal fringes and the fiber edge are paralleled along the longitudinal fiber direction. These denser parallel fringes indicate the high shear deformation of the interface zone. The thickness of the interface turns out to be 40-70/~m. 1.3 Summary In the steel reinforced cementitious composites, there exists an interfacial zone between the steel and the bulk matrix. The chemical, physical and mechanical properties of

Figure 4

Interfacial zone in composite reinforced with 1.3 % steel fiber, [5].

232 this region are different from that of the bulk matrix, forming the third phase in this kind of composite. It is obvious that an appropriate modelling of constitutive relationship of steelcement interface should take this fact into account. 2. SLIP-BASED CONSTITUTIVE MODELLING 2.1 Definition The slip-based models assume that the distribution of the shear stress along the interface only depend on the relative movement between the steel and the matrix. If there are no displacement discontinuities on an interface, there will be no shear stress. Hence, it can be inferred that the slip-based type of modelling ignores the possibility of bonding where the displacement continuity exists across the interface. Consequently, the slip-based interface model can be interpreted physically as an interface without adhesion but with an interface traction caused by surface roughness and friction. For one dimensional case, the mathematical description of the slip-based model can be expressed as:

U=UI-U~

(1)

and

9f ~ , , = ~ v )

zpz,,,-o

u~o

(2)

U--O

where Uf and U m are fiber and matrix displacements along X direction, rf and r , are the shear stress on the fiber and on the matrix, as shown in Fig. 5.

Figure 5

Displacement and stress parameters on an interface

233 2.2 Examples The general description for constitutive relationship of steel-matrix interface is given in eqn. (2). Usually, for a slip-based model, one assumes that the shear stress is a function of displacement discontinuity. Thus, the key point in developing such a model is how to derive the expression for displacement discontinuity. Jiang et al. [6] studied the transfer of forces across the interface by bond. They developed a one-dimensional model to describe the bond-slip relationship. A parabolic bond stress distribution was assumed as, ~9 -'~o[1

-(1-4/) 2]

(3)

where r was the bond stress at a distance x from the end of steel bar, ro was the peak bond stress, and I was the length of the steel bar between the two primary cracks. The value of ro was considered as a material property and could be obtained by using the following formula, (4)

o =0.034o,u,t(1-0.01 o~)

where tr,t~a is the steel stress. They used this model to calculate the stresses in a steel bar, the slip in concrete, the tensile stiffening, and total elongation of reinforcing bars. Nammur and Naaman [7] developed a continuous slip-based interface model. They used a piece-wise linear stress-slip relationship as shown in Fig. 6. For the linear range, they assumed that,

9(x)=~U(x)

(5)

T ,db,

,~ U Slip dlaplsmmmt Figure 6 Shear stress-slip relationship

where, r was the shear stress and U was the slip. K represented bond modulus. They assumed that the variation of slip U be equal to the difference between the strain in the fiber and the strain in the matrix. Hence,

dV__~x)_~.,Cx) t'Ix)e,.Cx)

(6)

After several subsequential manipulations, they finally derived the expression a s ,

9(x)-

xP,X

si.hCXx)

(7)

where Pt = total force; X = the ratio of the force acting on the fiber end to the total force;

234 d = fiber diameter ; L = fiber embedment length; and

(8)

~.

=

I +E,,,(I_Vt)jd

with Vf as the fiber volume ratio. The primary interest of Nammur and Naaman was to describe the initial part of the steel wire pull-out process and was not to distinguish the slip hardening or the strain softening effects. The slip hardening and slip softening phenomena were discussed by Wang et al.[8]. They pointed out that for steel wire the pullout frictional stress, rf, often decreases with slip (softening phenomenon) while in the synthetic fiber pullout, the frictional stress, rf, was found to increase with slip (hardening phenomenon). These different responses may be attributed to the abrasion mechanisms of the different fibers during the process of pull out. If a fiber surface becomes smoother due to friction, slip softening will be observed. On the other hand, if a fiber surface becomes rougher due to slip, slip hardening will be observed. Wang et al. [9] excluded the perfectly bonded interface from the analysis since they argued that the introduction of a perfectly bonded interface typically had little significance on the pullout load-displacement relationship in cementitious composites. They assumed that the frictional bond strength be a quadratic function of the slippage,

~(U)=ao+al U+a2U2

(9)

where ao, al and a2 were constants which were determined empirically by curve fitting. The additional examples of slip-based type of model can be found from Tepfer's work[10] and Yankelevsky' s work[ 11]. The major drawback for this type of constitute relationship is the fact that adhesion effects are ignored; these play a significant role at the interface as we will discuss later. Also, these models do not include the third phase which has different mechanical properties from the bulk matrix as we mentioned earlier.

3. BONDED-DEBONDED TYPE OF CONSTITUTIVE RELATIONSHIP

3.1 Boundary conditions The bonded interface requires not only the interfacial traction continuity but also the displacement continuity. This requirement can be expressed as, for the one dimensional case,

xf-- x=

v:= u.

(10)

The above description implies that there exists an adhesive bond between the steel

235 and the matrix. The adhesive bond is often denoted as elastic bond. It is necessary to assume a criterion beyond which the adhesive bond will be broken and a bonded interface will be changed to a debonded interface. On a debonded interface, the displacement continuity no longer applies, although the interfacial traction continuity is still satisfied. The mathematical descriptions can be written as,

,! = I:.,

(II)

Uf >Urn 3.2 Constitutive relationship for a bonded interface As mentioned earlier, a bonded interface requires both displacement and traction continuity. In other words, steel and interface will deform together. Thus, to describe the constitutive relation for shear stress, a reference point away from the interface should be predetermined for measuring the relative displacement. Based on experimental observations of interface mentioned earlier in this chapter, Stang et al.[12] modelled the interfacial zone as a thicknessless elasto-plastic layer (see Fig. 7) and the bulk matrix as a rigid body. Hence, the relative movement between interface and bulk matrix can be determined with a clear physical meaning. The relationship between the shear force per length acting at the interface and the displacement in the shear lag at the interface is given by, q = KU

(12)

where q is shear force per unit length. U is the displacement and K is the shear stiffness for shear lag. By substituting this equation into the differential equation, EIAya-q=O

(13)

The mathematical model for interface

Figure 7

the explicit expression for displacement can be obtained as, V(x) =

where

P

r

(14)

236

(15)

Lawrence[13] models the constitutive relationship for the bonded interface by using a concept of virtual displacement. He assumes that,

17=K(UI-U~

(16)

where K is some constant, Ue is the virtual displacement in the longitudinal direction of the steel fiber at a point in the steel fiber a distance x from the embedded end if the matrix has the same elastic properties as the steel, and U= is the virtual displacement of the matrix at the same point, if the steel is replaced by the matrix. When a bonded interface is subjected to an external disturbance, the shear force and energy increase in the interface. It is natural that some debonding criteria must apply to the steel-cement interface beyond which the bond interface will become a debonded interface. 3.3 Criteria for debonding Two approaches can be used to establish the debonding criteria for the steel-cement interface. One is the stress based criterion in which the initiation of debonding requires the shear stress to exceed a given value of the shear strength. The other is based on fracture mechanics principles using the energy release rate criterion. Such formulation is based on the assumption that the propagation of the debonding zone requires a certain energy and that debonding will occur only when the energy flowing into the interface exceeds the value of the specific resistance energy. The first approach is typified by Lawrence[13], Gopalaratnam and Shah[14], Naaman and Shah.[15], Greszczuk[16], Takaku and Arridge[17], Mashall and Oliver[18], Hseuh[19][20][21][22], Stang et al. [12] and Li et al. [23]. The representative works of the fracture mechanics approach include those of Gumey and Hunt[24], Bowling and Groves[25], Outwater and Murphy[26], Alkinson et a1.[27], Stang and Shah[28][29], Hutchinson and Jensen[30], Stang et al. [12] and Li et al.[23].

3.3.1 Strength based criterion for debonding In general terms a strength based criterion for debonding is based on the assumption that debonding takes place only when the maximum stress at the interface reaches a critical value, i.e. the bond strength of the interface. For a traditional shear lag type of analysis, usually only one strength parameter is used in the failure criterion and can be expressed as follows,

17=17y

(17)

where ry represents the interface shear strength. One of the earliest investigations of the strength based debonding criterion was reported by Greszczuk[16] in 1969. His analysis was based on a shear lag theory which assumed that the extension stresses in the matrix and the shear stress in the fiber were

237 negligible. The shear stress distribution along the fiber was characterized by two parameters: the average shear stress and the maximum interfacial stress, which occurs at the matrix face. Greszczuk found the ratio of these two values was a function only of fiber embedded length and expressed as: l:y= % # c ~ ( 0 t L )

(18)

where ry =the shear strength of a interface; r,, =the average shear stress of a interface; L = fiber embedded length; and c~=(2G/bREf) ~ in which G = Shear modulus, b=effective width of interface, R = fiber radius, Er = fiber Young's modulus. It can be seen from the equation that as L approaches zero, r~, approaches ry. The bond in Greszczuk's analysis was assumed to be entirely adhesional. When the maximum shear stress exceeded the interfacial shear strength, complete instantaneous debonding occurred. The possible contribution of frictional resistance on the debonded interface was not considered in Greszczuk's work. In contrast to Greszczuk's model, Lawrence [13] assumed that partial debonding might occur during the process of the fiber pull-out and once the interface debonded the effect of friction between a sufficiently long fiber and the matrix could support an increasing load. In this model, the nature of the debonding process depends not only on embedded length, but also on the ratio of the elastic interfacial shear strength to the frictional shear force as illustrated in Fig. 8.

P

P

P

Ak

v

d

End displacement

End displacement

End displacement

a) T y = Tf b) T y > T f > 0 r lrf=0 Figure 8 Effectof frictional stresson steel-cementinterfacedebondingandpull-out

238 The maximum fiber load required to achieve complete debonding and imtiate pull-out was given by:

pf,, =, ~V~_~Ktanhx a_L 2

L Xmax>'~ "

(19)

where, r r = the shear strength of the interface rf = the frictional shear strength of the interface

N = ~A~_~A./(F~A=-F~,))

= CK(EmAm-EfAf)/F~AfEmA,. C is the circumference of the fiber in contact with matrix, and K is some constant. Takaku and Arridge[17] conducted a pull-out test with a stainless steel wire and compared the test results with their analytical model. The model developed by them was similar to Greszczuk's. An equation for the distribution of shear stress along the fiber was derived based on shear lag analysis. Accordingly, the maximum interracial shear stress was achieved where the fiber exited the matrix. The expression obtained by them, in the ratio of ~-.~against ry, gave,

r.y=r..,,~.cah~.

(20)

where c~ was a parameter depending on the geometry of the system and x, was the point at which debonding was just developed. Takaku and Arridge assumed that when this stress exceeded the interfacial shear strength, debonding initiated and continued catastrophically. When the fiber was completely debonded, the pull-out process began and the frictional resistance to fiber movement was considered. The frictional stress was found to decrease with increasing embedment length, attributed to Poisson's contraction of the fiber. The pull-out model of Lawrence has been extended by Laws[31]. She pointed out that the frictional bond played an important role in composite property. The theoretical loaddisplacement would generally fall into one of two classes after the debonding initiation, depending on the magnitude of the frictional resistance. In the first case, where the frictional resistance was negligible, the debonding was catastrophic. In the second case, the load might increase with progressive debonding if frictional resistance was large enough. Laws also considered that friction might contribute to the interfacial strength of a debonded region, increasing the adhesional bond strength. The expression obtained by Laws for interfacial strength took the form of,

239 (21)

q, = 13[,~

where q~ represented the shear force per unit length; P was fiber force; Xowas the boundary of bond-de2mnded zone; L was embedded length; and/3 was a geometry parameter. Phan-Thien et a1.[32] presented an asymptotic and numerical solution for the elastic pull-out of a fixed plane surface. In their study, an expression was derived for the pull-out load vs. displacement relationship for perfectly bonded fiber. From this, they developed the shear stress distribution formula as, GU 1

/~ln--~ +LlnZ(12 L~, - L ) - 4 ( 1 - v )

+(1-v)ln(1 +L)]

(22)

where i, was the Poisson's ratio and z represented the fiber longitudinal direction. It should be pointed out that these expressions required numerical solutions and the process was somewhat complicated. Gopalaratnam and Shah[14] examined the fiber puN-out problem in considerable detail. Assuming a purely elastic stress transfer between the fiber and matrix, they obtained an equation for the interracial shear stress. The interface might be partially debonded, and a constant frictional shear stress was assumed to act over this region. An expression was derived for the load capacity as a function of inteffacial crack length and of the ratio of elastic to frictional shear stress,

2n;zYsinh~mL+ nRLI:/(I -m)cosh~mL p

(23)

_

(1 -g)cosh~mL+a where m was the ratio of the bonded length m the total embedded length, a and 13 took the form of a =

AIEf A.E.

(24)

and (25)

240 Using equation (23), a critical crack length could be determined by setting the derivative equal to zero. Similar to the work of Lawrence, Gopalaratnam and Shah suggested that the nature of the debonding process depend on the value of the frictional shear stresses. If the frictional shear stress was greater than zero, the interface might debond in a stable manner until the critical crack length was reached. If the frictional stress was equal to zero, then the debonding would become catastrophic. The behavior predicted by this model agreed satisfactorily with experimental results for steel fibers. Stang et al. [12] expressed this criterion in terms of force per length and assumed that debonding started once the force per length, q, reached a critical value qy. The solution obtained by them was as follows,

q=(P-qf a)~ cosh(tox)

O.~x<(L-a)

sinh[~ (L-a)]

q =q,,

(203

x>(L-a)

where ~o is defined in equation (15). It is clear from these expressions that the maximum value for q is reached at x =L-a. The debonding would propagate as soon as external force P reached the value of debonding force, Pd, at which q reached oo .

qy=(Pa-q!a)~cx~[ t~(L-a)]

(27)

3.3.2

Fracture based criterion for debonding The fracture based criterion for debonding is characterized by the assumption that the propagation of the debonding requires a certain energy and that this energy is a characteristic for the bond between fiber and matrix. A fracture mechanics approach to the problem of interface debonding treats the debonding region as a Mode H type of crack. It is further assumed that the crack will propagate as soon as the energy supplied by an external force exceeds the surface resistance energy of steel/matrix interface. Outwater and Murphy[26] used an energy balance argument similar to the Griffith theory of brittle fracture to predict the debonding stress, trd :

(28)

where Ef is the Young's modulus of the fiber or wire, R is its radius and I' is the fibermatrix interfacial energy. Stang and Shah[28] also applied the Griffith-type criterion for the crack-growth load to investigate the single fiber pull-out problem for both the elastic fiber case and the rigid fiber case. The equation they derived to estimate the critical tensile stress was,

241

od=2v,,l e,z 2r co-v/r/R

(29)

for the case of elastic fiber, and

1 +v,, [In(

)-1] R

(30)

for the case of rigid fiber, where ~ represents composite Young's modulus and Vf is the steel fiber volume ratio. However, it should be noticed that the two fracture mechanics based criteria mentioned above did not take the frictional stress into account. Considering that the shear stress (frictional) was still transmitted across the debonded region of the interface behind the debonding front, an increment of stress on the wire was required to propagate the debonding front. Bowling and Groves[25] therefore modified the formula proposed by Outwater and Murphy [26],

ssTR§:j,R 2 a

(31)

where re was the interfacial frictional stress, assumed to be constant in the debonded region and a was the debonded length. Atkinsonet al. [27] conducted a combined theoretical and experimental study of a rod pull-out problem. In their analysis, the principles of fracture mechanics were applied to the pull-out of a stiff rod from an incompressible matrix. A finite dement analysis of the problem demonstrated several significant features of the stress distribution. Singularities occurred at two regions where the normal stress dominated; at the rod tip, and at the matrix surface where the rod exited. It was therefore reasonable to assume that an interfacial crack initiate at one of these two locations. The experimental investigation showed that the pull-out behavior was always controlled by crack initiation and propagation from the matrix surface. The finite element results also indicated that the stress field along the fiber was dominated by shear stresses. Therefore, a shear failure along the interface could be expected. Following the fracture mechanics approach, the strain energy release rate was calculated as a function of interfacial crack length. Wells and Beaumont [33] studied the processes of fiber pull-out and debonding from a viewpoint of energy absorption. They proposed a model for the fracture of brittle fibers under non-uniform stress, predicting the probability of fracture sites. They also concluded that the energy absorption in composite was dependent on the length of debonding and pullout. Following the arguments of Wells [33] and Beaumont and Stang and Shah [28] that the fracture mechanics approach to the fiber debonding problem was preferred to the shear

242 stress-based approach, Gao, Mai and Cotterell[34] presented a new fracture mechanics-based debonding criterion by including the friction in the debonded region. For the frictional controlled debonding, they developed the expressions for critical fiber debonded length and end displacement at the point of critical pullout force Pc as

(32)

where a~ is critical debonded length, P~ is the critical force, Po is "frictionless" debonding load and X and K are constants determined by material properties; and Uc is defined as

Uc= RZEjA[[Pr K ~14 P-Pc 1-2KVfl

/~-Pc]~ [

t

(33)

From the analytical results obtained they defined several basic concepts such as the friction zone, strong fibers and weak fibers. Simple relations between debonding load, debonding depth and fiber-end displacements were also obtained. Stang et al.[12] also introduced a fracture mechanics based criterion for debonding which included the effect of frictional stress on the debonded interface. The solution was given as:

where Gc~t was the critical energy release rate. This expression can be reduced to the solution of Outwater and Murphy [26] in the case of vanishing friction and large embedment length compared to both fiber radius and length of the debonded zone. 3.3.3 Summary This section briefly reviewed several of the existing debonding criteria. The criteria can be divided into two categories. The first approach is based on shear lag theory and assumes that failure is initiated when the shear strength of the interface is exceeded. The second approach is based on fracture mechanics and assumes that the interfacial crack will propagate as soon as inteffacial energy reaches a critical release rate. In addition to the different basic approaches, the theories differ in the consideration of a debonded interfacial region and the assumed shear stress distribution in this region.

3.4 Frictional shear stress distribution in a debonded interface Generally speaking, there are three kinds of models to describe the constitutive relationship for the debonded interface: constant frictional stress model, Coulomb type of frictional stress model and slip-based frictional stress model. For a constant frictional stress model, the frictional stress distribution is assumed to

243 be a constant along the whole debonded area. This kind of assumption has been used by Bowling and Groves[25], Lawrence[13], Hutchinson and Jensen[30], Marshall and Oliver[18], Gray[35], and Stang et al. [12]. The expression for constant frictional stress model is as follows,

~=~

(35)

where rf is the frictional shear stress. Perhaps for a one dimensional model the friction is primarily due to asperity. Although the assumption of constant friction is not close to reality, as discussed later, it is still being used by researchers to simplify the formulations. In some models a Coulomb type of failure criterion has been used. In this kind of criterion, two parameters, C and #, are introduced,

"~=C-I~O

(C-I.=o)zO

(36)

where C is a measure of the adhesion and IL is the frictionalcoefficient. In the simplified shear lag analyses of Pinchin and Tabor[36], Beaumont and Aleszka[37], Aboudi[38], Hseuh[19][20], and Gao et al. [34], the displacement perpendicular to the interface does not enter the analysis. Thus, in these cases the following relations on the interface are applied, If

I~fl f

--

Om

I~.I

= =

= f - p o t.

(37)

where the contact stresses af and or.,,are related to volume changes (shrinkage, thermal mismatch) external load and Poisson's ratio for the matrix and the steel material. The effectsof Poisson's ratioon the contact stressesand thus on the frictionalstresses were investigated in detailby Hsueh[39][40] who studied both fiber pullout and push down (indentation) theoretically and experimentally in a fiber reinforced ceramic composite. The direction of the frictional surface traction is evident as long as the loading and the direction of sliding is monotonic in time and in fact all the above relations are valid under the implicit assumption that

dU-O>o

(38)

dt with

u_-vFv "

r

However, in more general cases the direction of the frictional stress might change. It can be assumed that the direction of the shear stress is determined from the direction of the displacement discontinuity rate, as suggested by Abudi[38] for a more general composite material system. The suggested relation between displacement discontinuity rate and direction

244 of the frictional stress is given by:

asn(1)

= aSh(O)

(4o)

where sign(f) refers to the sign of f. A relation for f which introduces the Coulomb frictional law and at the same time ensures the correct direction for r and distinguishes between contact and non-contact has been introduced by Abudi[38]:

f-sign( (.l)(C- ~ ol)H(C - ~ oI)

(41)

where H is Heaviside's function, given by

H(x) __(O 1

for for

(42) x>0

Note that the type of description used by Abudi implies that tensile stress can be transferred across the debonded interface. The slip-based model introduced in the previous section can be directly used to describe the frictional shear stress distribution in the debonded region. No further discussion is necessary here. 0

INTERPRETATION OF INTERFACIAL PROPERTIES BASED ON P U I J . ~ U T EXPERIMENTS

Many approaches have been proposed to interpret the experimental results of the pullout tests. Most methods presented in the literature are only for interpretation of the strength based criterion parameters. A review has been given by Gray[35]. Note that the conventional methods described by Gray are not convenient because the parameters in the formulae have to be determined by fitting the experimental data. Also, the material parameters obtained from the experiments reported by Gray are significantly affected by the method of interpretation employed. The shear strength, ry, obtained can range from 3.1 MPa to 98 MPa. In this section, two recently proposed methods are reviewed. These methods are generally simpler and more objective. One of these methods is developed by Li et al. [23]. In this method, they used three material parameters for both stress based criterion and fracture based criterion. They are ~, ry (or qy) and rf (or qf) or w, P and rf (or qf), respectively. In this approach, the fiber is treated as an elastic material with a modulus El, the bulk matrix is considered rigid, and the stiffness of the negligibly thick interface layer is related to r (see Fig. 7 and Eqn. 15). The interface stiffness parameter, w can be determined from the initial slope of the experimental load-end displacement curve by solving the equation,

U(L) =cothta/.,(E/4/a) P

(43)

245 The interfacial yield parameter, qy, the interfacial frictional forces, qp and the specific energy, s can not be directly obtained from a load-slip experimental curve. A method which utilizes the maximum load, P*,,~, and the slip displacement corresponding to P'M is proposed. These test results are significantly easier to determine than other responses such as onset of debonding. Note that the maximum load generally occurs under partial debonding conditions. Three equations obtained are: (44)

qy = qyCosh 2(~(L-a))

(45) q!

p~

aco + Sinh(co(L-a))Cosh(co(L-a))

=

0.5(coa)2 + Cosh2(~(L-a)) + ~a Sinh(~o(L-a))Cosh(~(L-a))

(40

e a + Sinh(e(L-a)) Cosh(coCL-a)) - O'*(Z) JEf,t~j6} =0

where U'(L) represents the end displacement corresponding to the peak load.

6.00E-5 ..,..

Critical point; Ucrit E ._~

m 5.75E-5--

TEEL WIRE S P E C I M E N 14-DAY C U R I N G L = 12.0 mm

G)

-

5.50E-5 0.000

I

i

421/m

i

I

0.004

1

I

I

0.008

I

!

0.012

Debonding length (m) Figure 9

Critical value of end displacement and correslxmding debonding length

246 Since the only unknown in eqn. (46) is debonding length, a, its solution will result in the length of the debonded zone at the peak load, P',~. Furthermore, the parameter study of displacement, U(L), and debonding length, a, in eqn. (46) demonstrates that there exists a critical value of U(L)c at which the multiple roots of debonding lengths occurs as shown in Fig. 9. Hence, for the values of U(L) greater than U(L)o, the debonding length is determined by using the value of U(L)c. After obtaining the value of debonding length, a, from eqn. (46), at the peak load, P'.~, a back substitution of this value into eqn. (45) will yield the frictional shear bond strength, oz. Moreover, the back substitution of oz into eqn. (44) gives the value of %. By defining n as the number of steel fibers, and R as the radius of a fiber, shear strength measures which are independent of fiber diameter can be defined as ry and rf according to the following equations:

q, 2xR

n

(47)

2nR n Using the fracture energy approach, parameters I' and q/can be obtained from the experimental data based on the same approach. Two formula used to calculated I', and q/ are obtained as:

2E/ A/ p P = ( q__f)2 [Cosh,t(t~(L_a)) _ Cosh2(t~(L_a))]

(4a)

cO

*

q:- P~

~

a . + SinTi(.(L-a))

Cosh(.(Z,-a))

(49)

It is interesting to note that the relationship between q/and P'~ obtained using a fracture energy approach is exactly the same relationship obtained using maximum shear stress criterion Eq. (45). Hence, the procedures to determine the debonding length, a, and back substitution to obtain shear frictional bond strength q/would be identical to the maximum shear strength approach with the exception that the maximum shear strength approach uses eqn. (44) to calculate qy while the fracture energy approach applies eqn. (48) to calculate 2Ef Af p I'. Naaman et al. [41][42] have proposed another method to obtain the inteffacial parameters, the bond modulus r, the bond strength ry, the constant frictional bond stress rf, the value of the end slip U0 at which the bond is assumed to deteriorate; and the decaying frictional parameters lj and ~/, describing the deteriorating frictional zone. The bond modulus r is determined from the slope of the linear ascending portion of the pullout curve, which can be determined graphically. To get the values of rf and ry, they

247

proposed to solve a system of three nonlinear equations in three unknowns. The three unknowns are rf, ~y and Xp, which is defined by:

Xp=e-ua-~P

(SO)

where )~ is a constant, I is the embedded length and a~ is the debonding length at the peak load. The system of three equations in three unknowns is l

~~.r

t~(x

Pp(O-1)[

1 2 ~)x,-~

',

~-x~

---0

(51)

:5

(sz)

V~(xp ~(xp x +/]z(O-2)-t][ x +/] aj.

X ~ +/] -~t

1-Xp §

A.e..

(53)

-u,

Once above equations have been solved simultaneously, the values of rf and ry can be found from,

t,

~Y=2xR

(54)

zf 2xR Comparing these two methods, the first one looks simpler since one does not need to solve three nonlinear equations simultaneously. Also it provides the interpretation of the fracture parameters. The advantage of the second method is that it gives a more general description for the post peak response due to the decaying frictional parameters ~ and ~/.

248 5.

EVALUATION OF CONSTITUTIVE RELATIONSHIP BY MOIRE INTERFEROMETRY TECHNIQUE 5.1 Verification of existence of bond Fig. 10 shows the moire fringe patterns obtained from uniaxial tension tests of steel wire reinforced cement matrix composites[5]. Continuous fringes mean a continuous

Figure 10

Bonded zone along interface after the end of multiple cracking (Vf = 1.3 %)

249

displacement field. The interface debonding is characterized by the fringe discontinuity at the interface. At the center part of the pictures, however, fringes are continuously crossing the interface of the steel wire and matrix, which means that the boundary between the steel wire and the matrix undergoes the same displacement. It indicates that a perfect bond does exist at the interface. This is true even at relatively high composite stress which corresponds to a strain value four times as large as the one when the multiple cracking stops (Fig. 10 b). The distribution of the shear stress along the interface between two major cracks is plotted in Fig. 11. It is seen that the shear stress reaches the peak at the boundary of bondeddebonded regions. The peak moves into the bonded region as the load increases. The peak values are close for different loading levels. According to these experimental observations, it can be concluded that the notion of shear strength make sense. In addition, the bondeddebonded type of constitutive relationship for steel cement interface should be more preferable than the slip-based models.

4"~176 1 (Applied 2.00_ t A

Stress ~

5"3MPa

~

I o

g

R

J,

w

w

~ o.~

l-

Fiber volume

-4.00 0.00

Figure 11

ratio: 1.3%

i I

'

2.00

I

4.00

'

I

6.00

'

I

8.00

Distancebetweencracks(ram)

'

10.00

Distribution of interface shear stress

5.2 Verification of frictional stress modelling One of the important relationships in steel cement composite modelling is the constitutive relationship between interface shear stress and interface slip. Since the distributions of interface slip and the interface shear stress can be obtained from the moire intefferometry tests, the relations between the two quantifies are readily available. Fig. 13 shows the shear stress and slip relationship along the whole debonded interface at the different loading levels for the composites with different wire volume ratios. The shear stress reaches its maximum at zero slip and drops gradually with the increasing value of the slip. Fig. 12a shows a larger interface slip than that shown in Fig. 12b. It is obvious that the points close to the boundary (between bonded and debonded zones) carry a large shear stress

250 4.00 (I): Fiber volume ratio: 1.3% CO

n

3OO

Z

(n (n cn

L a) u)

ca 2.00

/

8 < ~ 1.00 rm.

o.oo , , , , I , , , , , 1 , , , , , I , , , , , I , , , , , I , , , , , I , , , , , i , , , , , 0.00

2.00

4.00

6.00

8.00 10.00 12.00 14.00 16.00

Interface slip (micron) 4.00 (b): Fiber volume ratio: 5.9% r

n

3.

L_

I ~ _AppUed m

"i

A

tU

]

10.3 M I : ~ 16.6 MPaJ 23.4 MPaJ

IU

o.oo

;,,,,1,,,,,I,,,,,I,,,,,1,,,,,I,,,,,I,,,,,I,,,,,

0.00

2.00

4.00

6.00

8.00 10.00 12.00 14.00 16.00

Interface slip (micron)

Figure 12

Fdctional shear stress - slip relationship

251 and undergo a small slip, while the points close to the matrix cracks carry a small shear stress and undergo a large slip. Frictional shear stresses are not constant either along the whole debonded interface or at different loading stages. Only after the wire sustains large slip do the frictional shear stresses tend to reach uniformity (Fig. 12a). The approximate constant frictional shear stress can be estimated by averaging the shear stresses in debonded region at different applied loading levels. The results show that the average frictional shear stress is 1.18 MPa for the stx~imen group with a V, of 1.3%, and 1.42 MPa for the specimen group with a Vf of 5.9 %. Although the frictional shear stresses in both composites differ somewhat, they are still of the same order of magnitude. 6. CONCLUSIONS The constitutive relationship for steel-cement interface has been reviewed in this chapter. The microstructural study shows that a porous and weak interfacial zone does exist between the steel and the bulk matrix and should be considered as the third phase in the interface constitutive modelling. Two categories of the constitutive models for the interface can be found in the literature, the slip type and the bonded-debonded type of model. It is shown from the recently conducted experiments that the bonded-debonded model more accurately represents the physical observations of the interface. In the bonded-debonded model, two kinds of debonding criterion, strength based and fracture mechanics based, are discussed. The strength-based criterion for debonding is based on the assumption that the interfacial bond has a critical value and that is a material parameter. On the other hand, the fracture mechanics based model takes the toughness parameter or energy release rate as the material parameter. It is generally agreed that frictional stresses act along the debonded steelcement interface. Three types of models, constant frictional stress, Coulomb type of frictional stress and slip-based frictional stress models, are described. According to the results of recently conducted moire experiments on steel-cement interface, it seems that the slip-based frictional stress model is the most realistic. Two methods of interpreting the results of pullout tests are introduced. The interfacial material properties can be obtained by using these methods. References: I. Pinchin, D. J. and Tabor D., "Interfacial phenomena in steel fiber reinforced cement I: Structure and strength of interracial region", Cem. Concr. Res., 8, (1978), 15-24. 11

0

0

11

A1 Khalaf, M. N. and Page, C. L., "Steel/mortar interface: mierostructural features and mode of failure", Cem. Concr. Res., 9, (1979), 197-208. Bentur, A., Diamond, S. and Mindess, S., "The microstructure of the steel fibercement interface", J. Mater. Sci., 20, (1985), 3620-26. Wei, S., Mandel, J. A. and Said, S., "Study of the interface strength in steel fiberreinforced cement-based composites", ACI J., 83, (1986), 597-605. Shao, Yixin, Li, Zongjin and Shah, Surendra P., "Matrix cracking and interface debonding in fiber-reinforced cement-matrix composites', accepted for publication, J. Advanced Cement Based Materials, (1993).

252

0

"

"

.

Jiang, D. H., Shah, S. P. and Andonian, A. T., " Study of the transfer of tensile forces by bond", ACI J., 81, (1984), 251-59. Nammur, G. and Naaman, A. E., "Bond stress model for fiber reinforced concrete based on bond stress-slip relationship", ACI Mater. J., 86, (1989), 45-57. Wang, Y., Li, V. C. and Backer, S., "Analysis of synthetic fiber pull-out from cement matrix", (Mindess S. and Shah S. P., Eds) Bonding in Cementitious Composites, MRS Symposium Proceedings, 114, MRS, Pittsburgh, (1988), 159-65. Wang, Y., Li, V. C. and Backer, S., "Modelling of fiber pull-out from a cement matrix", Int. J. Cem. Comp. Ltwt. Contr. 1013], (1988), 143-49.

10.

Tepfer, R., "Cracking of concrete cover along anchored deformed reinforcing bars', Mag. Concr. Res., 31, No. 106, (1979), 3-12.

11.

Yankelevsky, D. Z., "Bond action between concrete and a deformed bar- a new model", ACI J., 82, (1985), 154-61.

12.

Stang, H., Li, Z. and Shah, S. P., "The pullout problem -- the stress versus fracture mechanical approach', ASCE, J. Engng Mech., 116 [10], (1990), 2136-50.

13.

Lawrence, P. J., "Some theoretical considerations of fiber pull-out from an elastic matrix', J. Mater. Sei., 7, (1972), 1-6.

14.

Gopalaratnam, V. S. and Shah, S. P., "Tensile failure of steel fiber reinforced mortar", ASCE, J. Engng Mech., 113 [5], (1987), 635-52.

15.

Naaman, A. E. and Shah, S. P., "Pull-out mechanism in steel fiber reinforced concrete', ASCE, J. Struct. Div., 10218], (1976), 1537-48.

16.

Greszczuk, L. B., "Theoretical studies of the mechanics of fiber matrix interface in composites', ASTM STP 452, (1968), 42-58.

17.

Takaku, A., and Arridge, R. G. C., "The Effect of Interfacial Radial and Shear Stress on Fiber Pull-Out in Composite Materials', J. Phys. D: Appl. Phys., 6, (1973), 2038-47.

18.

Marshall, D. B. and Oliver, W. C., "Measurement of interfacial mechanical properties in fiber reinforced ceramic composites', J. Am. Ceram. Soe., 70[8], (1986), 542-48.

19.

Hsueh, C.-H., "Interfacial friction analysis for fiber-reinforced composites during fiber push-down (indentation)', J. Mater. Sci., 25, (1990), 818-28.

253 20.

Hsueh, C.-H., "Fiber pullout against push-down for fiber-reinforced composites with frictional interfaces", J. Mater. Sci., 25, (1990), 811-17.

21.

Hsueh, C.-H., "Some Considerations of evaluation of interracial stress from the indentation technique for fiber-reinforced ceramic composites", J. Mat. Sci. Lea,. 8, (1989), 739-42.

22.

Hsueh, C.-H., "Elastic load transfer from partially embedded axially loaded fiber to matrix", J. Mat. Sci. Lea., 7, (1988), 497-500.

23.

Li, Z., Mobasher, B. and Shah, S. P., "Characterization of interfacial properties for fiber-cementitious composites", J. Ame. Cer. Sot., 7419], 1991, 2156-64.

24.

Gumey, C. and Hunt, J., "Quasi-static crack propagation", Phil. Trans., Roy. Sot., 299, Ser. A, (1967), 508-24.

25.

Bowling, J. and Groves, G. W., "The debonding and pull-out of ductile wires from a brittle matrix", J. Mater. Sci., 14, (1979), 431-42.

26.

Outwater, J. P. and Murphy, M. C., "On the fracture energy of unidirectional laminate", in proceedings of 24th Annual Technical Conference of Reinforced Plastic/Composites Division, The society of the Plastics Industry Inc., Composite Div., New York, (1969), (Paper No. llC).

27.

Atldnson, C., Avila, J., Betz, E. and Smelser, R. E., "The rod pull-out problem, theory and experiment", J. Mech. Phys. Solids, 3013], (1982), 97-120.

28.

Stang, H. and Shah, S. P., "Failure of Fiber-Reinforced Composite by Pull-Out Fracture", J. Mater. Sci., 21, (1986), 953-57.

29.

Stang, H. and Shah, S. P., "Fracture Mechanical Interpretation of the Fiber/Matrix Debonding Process in Cementitious Composites", (Wittmann, F. H. Eds.), Fracture Toughness and Fracture Energy of Composites, Elsevier, (1986), 513-23.

30.

Hutchinson, J. W. and Jensen, H. M., "Models of fiber debonding and pullout in brittle composites with friction", Mech. Mater., 9, (1990), 139-63.

31.

Laws, V., "Micromechanical aspects of the fiber-cement bond', Composites, 13, (1982), 145-51.

32.

Phan-Thien, N., Pantelis, G. and Bush, M. B., "On the elastic fiber pull-out problem: asymptotic and numerical results", J. Appl. Math. Phys. (ZAMP), 33, (1982), 251-65.

33.

Wells, J. K. and Beaumont, P. W. R., "Crack-tip energy absorption processes in fiber composites", J. Mater. Sci., 20, (1985), 2735-49.

254 34.

Gao, Y. C., Mai, Y. W., and Cotterell, B.,'Fracture of Fiber-Reinforced Materials', J. Appl. Math. Phys., 39, (1988), 550-72.

35.

Gray, R. J., "Analysis of the effect of embedded fiber length on fiber debonding and pullout from an elastic matrix, Part 1, Review of theories', J. Mater. Sci., 19, (1984), 1680-91.

36.

Pinchin, D. J. and Tabor, D., "Inelastic behavior in steel wire pull-out prom portland cement mortar', J. Mater. Sci., 13, (1978), 1261-66.

37.

Beaumont, P. W. R. and Aleszka, J. C. "Polymer concrete dispersed with short steel fibers', J. Mater. Sci., 13, (1978), 1749-60.

38.

Aboudi, J., "Micromechanical analysis of fibrous composites with coulomb frictional slippage between phases', Mech. Mater., 8(2&3), (1989), 103-15.

39.

Hsueh, C.-H., "Interfacial debonding and fiber pullout stresses of fiber reinforced composites', Mater. Sci. Eng. A 123 [1], (1990), 1-11.

40.

Hsueh, C.-H., "Evaluation of interfar shear strength, residual clamping stress and coefficient of friction for fiber reinforced ceramic composites', Acta Metal, 3813], (1990), 403-9.

41.

Naaman, A. E., Nammur, G. G., Alwan, J. M. and Najm, H. S., "Fiber pullout and bond slip. I: analytical study', ASCE, J. Struct. Engng., 11719], (1991), 2769-90.

42.

Naaman, A. E., Nammur, G. G., Alwan, J. M. and Najm, H. S., "Fiber pullout and bond slip. II: experimental validation', ASCE, J. Struct. Engng, 11719[, (1991), 2791-2800.

Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All fights reserved.

S t e e l - C o n c r e t e Interfaces:

255

Experimental Aspects

H.W. Reinhardt and G.L. Balhzs Stuttgart University, Institute 70550 Stuttgart, Germany

of

Civil

Engineering

Materials,

Pfaffenwaldring

4,

1. INTRODUCTION Cementitious materials such as c o n c r e t e - similarly to natural s t o n e s - are rather brittle and have an inherent weakness in resisting tension. Reinforcement is applied to provide equilibrium and control of deformation after cracking. The interaction, which is often refered to as bond, between concrete matrix and reinforcement is an essential property influencing the behaviour of the material. Without interaction it would not be able to perform as a composite material. In early studies, a perfect bond between steel and concrete without any relative displacement was assumed. Test results from the fifties [ 1] indicated, however, that the bond forces on the interfaces are basically governed by the relative displacement (slip) of the initially coincident steel and concrete cross-sections. Furthermore, several parameters, such as geometry of the interface (pattem, inclination, height and distance of ribs), mechanical characteristics of both materials and type of applied load (monotonic, repeated or sustained), etc. also have an influence. Within this chapter the behaviour of steel-concrete interfaces together with the influencing parameters are discussed based on experimental data providing a possibility for comparison with the behaviour of those geomaterial interfaces such as different rock layers subjected to shear or soil and pile interaction where the interfacial characteristics have principle influence on the overall behaviour. Those parts of the behaviour of steel-concrete interfaces will be discussed in more details which may correlate with those of geomaterial interfaces. Specific questions of steel-concrete interaction concerning technological aspects of steel or concrete production (e.g. influence of additives) or special circumstances (e.g. fire or cryogenic conditions) will not be discussed herein.

2. BASIC BEHAVIOUR OF STEEL-CONCRETE INTERFACES A tensile or a compressive force on an embedded steel bar induces interactive forces in the surrounding concrete (Fig. 1). 2.1 Basics The component of the interaction force parallel to the bar is to be in equilibrium with the

256 applied force. Its value related to a unit of the interactional surface is referred to as bond stress (Tb) (in some literature as local bond stress). Its highest value is the bond strength (Zbu), and its average value over a given length (which is often used in design) is the average bond stress. The differences in local bond stress and average bond stress obtained on a bond length (gb) not greater than five bar diameters (50) are generally neglected owing to experimental difficulies with very short lengths.

interaction I force

applied force

steel bar concrete

hoop tensile stresses

Fig.1 Steel-concrete interaction (based on Tepfers [2])

The radial component of the interactional force induces hoop tensile stresses in the surrounding concrete which may lead to cracking parallel to the bar in case of inadequate concrete cover. Since the bond stress is a function of slip, the bond behaviour is generally characterized by the bond stress-slip (rb-S) relationship. 2.2 Phases of steel-concrete interaction

Relatively low loads may be transmitted by adhesion based on a physico-chemical contact of the two materials (Fig.2). Deformed reinforcing bars transfer, however, higher loads by mechanical interlock producing micro-cracking [3-5] and micro-crushing [6] in the close vicinity of bar deformations which lead to a measurable relative displacement of the concrete and the steel cross-sections. An increase in the applied force or deformation produces progressive micro-cracking and micro-crushing. (Cracks are considered to be micro-cracks which do not reach the concrete surface.) Based on slip measurements by the Moir6 technique, Gambarova and Giuriani [7] observed a higher contribution of micro-crushing than that of micro-cracking to the actual slip. For higher relative displacement than that at the bond strength, the concrete between the bar lugs will be sheared off producing strain softening. When the relative displacement becomes equal to the clear distance of bar lugs, a practically constant bond resistance remains. This resistance is often called as residual bond strength and is provided by friction over the cylindrical concrete surface at the tip of the bar lugs. Bond failure is theoretically possible due to failure of the ribs, however, this is not the normal situation for usual steel reinforcing bars in concrete.

257 B o n d stress

~eformed bar'~ bond strength

mechanical ,

shearing

.,

interlock

off

micro-cracking micro-crushing f~iction

residual bond strength

friction

r

.

.

.

.

[plain

.

-

.

barsl .

.

~ .

.

.

~ . .

friction - ~ . . friction

pull-out failure

_

adhesion J 9

|

slip at bond strength

r

slip

Fig.2 Phases of steel-concrete interaction for plain and deformed bars

Plain reinforcing bars develop much lower bond stresses than deformed bars because the mechanical interlock, which is the most important contribution in case of the bond of deformed bars, is provided only by surface irregularities rather than by pronotmced surface deformations (ribs). A very small slip, approximately 0.01 mm, is enough to reach the bond strength of plain bars. Then remains a frictional contact providing a constant bond stress which is the bond strength of plain bars. This bond strength is, however, only slightly higher than the adhesional bond stress. The adhesional bond stresses of plain and deformed bars are practically the same because it is only a function of the surface if otherwise equal conditions are considered (Fig.2). The ratio of bond strengths of deformed and plain bars depends mainly on the rib pattern, the concrete strength and the confining effects, and may even reach a factor of 10. Owing to the relatively poor bond properties, plain reinforcing bars are rarely used in construction. 2.3 Failure modes of steel-concrete interaction Two types of bond failures are distinguished considering the two different failure mechanisms of the concrete matrix (Fig.2). Pull-outfailure, where the concrete between the bar lugs is sheared off without a complete loss of the bond resistance (Fig.2). Visible cracks on the concrete surface are not necessarily produced. Splitting failure, where the hoop tensile stresses reach the tensile resistance of the surrounding concrete producing a longitudinal crack. In absence of transverse reinforcement or transverse pressure, it may lead to a complete loss of bond resistance.

258

2.4 Measurment technique The main concern of experimental bond studies is to detemine the bond stress-slip relationship that may be used for comparison of bond characteristics or for analytical studies. Slip over a length e b is the integral of the steel strains (esx) reduced by the concrete strains

(~x): eb

s = ~0 (esx'Ccx)dx

(1)

There are two different ways to determine slip: either to measure it directly e.g. with an LVDT on the bar end or by magnetic sensors in the interior of the concrete [39], or to measure the strains by resistance strain gauges [8-9] and to integrate them according to Eq.(1). If unloaded end slip also occurs, it is to be added to the result. Owing to the considerable work in preparing the steel bars for strain measurements and the difficulties in evaluating the results, simple specimens, the so called pull-out specimens [ 10] and beam specimens [11] were developed for direct measurement of slip. The slip is measured at the unloaded end of the specimen in both cases. The proposed bond length is 50 for pull-out tests and 100 for beam tests, respectively. Even if the shape of a beam specimen is more realistic, pull-out specimens provide the simplest way of testing and are more often used. The reinforcing bar is centrically embedded in a concrete prism of 100 (but maximum 200 mm) sides over the 50 bond length having an other 50 debonded length at the loaded end to eliminate stress concentrations. The bond stress-slip relationship is then determined from the registered tensile force-slip relationship/F(s)/relating the force to the steel-concrete interface:

% ( 0 - F(s) ~oe b

where s is the slip measured at the unloaded end of the specimen. The disadvantages of this method are that it neglects the changes of bond stress over the 50 bond length that may be considerable in case of larger diameters and uses a boundary condition for concrete (supporting block at the applied load) which is not typical. There are several proposals to improve the specimen or the evaluation method of measurements [12]. The bond stress is a function of the relative dispalcement, however, there are several parameters that influence the actual value of bond stress. The influence of rib pattern, concrete strength, confining effects, loading rate, long term loading and repeated loading are discussed in the following Sections. 3. INFLUENCE OF RIB PATTERN ON STEEL-CONCRETE INTERACTION The important parameter influencing the steel-concrete interaction is the shape of the interactional surface, i.e., the rib or deformation pattern of reinforcing bar. It consists of the shape, the heights, the inclination and distance of ribs. Different countries use different rib patterns. The rib pattern is not only intended to improve the bond characteristics but to distinguish bars of different strengths and manufacturers.

259 3.1 Influence of related rib area

To provide a possible comparison of bond characteristics of different bars, Rehm [1] introduced the related rib area (t~sb) meaning the axially projected front surface of transverse ribs related to the nominal interactional surface between two ribs. The deformation pattern not only influences the bond strength but also the failure mode. High ribs do not necessarily mean a more adequate bond behaviour because higher interactional stress increases the probability of longitudinal cracking.

rb/feU

0401

_ o.o

7#/

---

l/

0.10 .d'N.

0

0.25

0.50

0.75

1.00

1.25

slip [mm] Fig.3 Influence of the related rib area on the bond stress-slip relationship Rehm, Martin and Noakowski [13]

Fig.3 indicates experimentally derived bond stress-slip relationships as a function of the related rib area increasing from 0 (i.e. plain bars) up to 0.4. The increase in bond stress is less than proportional with the increase in related rib area. Twice of related rib area leads to less than twice of bond stress at a given slip. There should be an optimum in deformation pattern that is enough to transmit and anchor forces but still does not necessarily lead to longitudinal cracking. The optimal related rib area seems to be in the range of 0.05 < Otsb< 0.08 [ 14] which is less than that of some deformed bars used earlier. The related rib area is a practical tool for comparing bars of different rib patterns but the following analysis of single parameters of the rib pattern contributes to the characterization of bond properties.

260

3.2 Influence

of rib geometry

The experiments of Soretz und H61zenbein [15] have shown the increasing probability of splitting failure with increasing rib height on three series of reinforcing bars having the same relative rib area but different rib heights (0.10, 0.050 and 0.0250) and spacings (1.20, 0.60 and 0.30), respectively. With bars of different inclinations (30 to 90 ~ to the longitudinal axis, Soretz and HSlzenbein observed a slight improvement of bond characteristics (Fig.4). Varying the rib cross-section from a rectangle to a 45 ~ trapezoidal had no influence, and changing to a very flat triangle had only minor influence on the bond characteristics [15].

Rib patterns

Code

of rib pattern [] n

!Vl v w I

II ..... ! ......

III

1

EE

r if '~~

~_

,

e

!

IV

........ i:':/- ........,,-S;~ / --,

" tt

z'~ 2

,o I

,

V ...~~

w ~

't

I

I

, / I

l

I I(~176176176

slip

J I J ~(~bu) ~ / i/~ t ]-~

J~ r---~, i j I ,

I

o., m

',/'I i ~

i

~

0.01ram

VI

21

il

I

40 50 60 70 80 90(90E) r ' Rib inclination [~ Fig.4 Influence of rib inclination on bond stresses Soretz and H61zenbein [15]

Tepfers and Olsson [ 16] conducted pull-out tests on bars of 16 mm nominal diameter with 48 mm bond length of different rib heights and spacings providing different related rib areas. The radial component of the interactional force was determined by measuring the circumferential strain in a steel ring surrounding the concrete of the pull-out specimen. The force component which is parallel to the bar was obtained by supporting the specimen with a teflon covered circular support close to the bar. Tepfers and Olsson [ 16] observed that: - Increasing rib height improves the bond stress at a given slip. - There is an optimum rib spacing. If the spacing becomes too short, the bar starts to act as a plain bar with a diameter including the bar ribs. - The slip at maximum load decreases, when the related rib area increases.

261 The splitting tendency increases with increasing angle between the interactional force and the axis of the bar. This angle increases if the slip at maximum load increases otherwise it decreases if the related rib area increases. Kimura and Jirsa [17] compared experimentally various rib pattems of commercial and machined bars of 36 mm nominal diameter by pull-out tests on 150 mm bond length and using concretes with compressive strengths 40, 80 and 120 N/mm 2 and specimens which contained spiral reinforcement of 6 mm diameter and 40 mm pitch. They concluded that: - The average bond stress and the initial stiffness of the bond stress-slip curve increased as the rib spacing decreased for concrete strengths of 40 and 80 N/mm 2. - The average bond stress and the initial stiffness of the bond stress-slip curve increased as rib height increased. The increase was larger for higher strength concrete. - The bond stress at splitting was not strongly influenced by rib spacing or rib height, regardless of the concrete strength. - As the ratio of rib height to rib spacing increased, the average bond stress and the initial stiffness of bond stress-slip curve increased, regardless of concrete strength. However, for rib height to rib spacing ratios greater than 0.2, these values seemed to be constant. - The bond stress to splitting tended to decrease up to a rib height to rib spacing of 0.2. - Bars with rib face angles greater than or equal to 45 ~ exhibited almost the same behaviour. However, bars with a rib face angle of 30 ~ were initially less stiff.

4. I N F L U E N C E OF C O N C R E T E S T R E N G T H AND C O M P O S I T I O N ON S T E E L CONCRETE INTERACTION Not only the strength but also the composition and the consistence (i.e. cement content, water content, etc.) of concrete influences the bond characteristics.

4.1 Influence of concrete strength To develop bond stresses, contributions are derived from properties of concrete both in tension and in compression. Micro-cracking is controlled by tensile resistance, however, bearing stresses induce high compressive stresses in front of the ribs. Design equations generally relate the bond strength either directly or indirectly to the tensile strength of concrete. Considering the relationship of tensile and compressive strength of concrete as a known property, experimental studies were intended to determine the relationship between the bond stresses or the bond strength and the concrete compressive strength. The higher the concrete strength, the higher the bond stress, however, test results disagree in the actual ratios. Based on pull-out test results with concrete mixes of 16 to 50 N/mm 2 strength measured on cubes of 200 mm sides, Martin [18] concluded the following: - For the slip range of 0.01 to I. 0 ram, the bond stress of deformed bars is proportional to the concrete cube strength (Fig.5.b). - For very small slips, s < 0.01 ram, and for slips close to the bond strength, s > 1.0 ram,

262 the influence of concrete compressive strength is less important and proportional to the ~a power of the concrete cube strength (Fig.5.a). rb(s=O. 01 ) ['N/mm2] 2.5

1

1

1

rb(0.01)-~. 163 fc2P

a)

1.5

_ z ~ y w/c = 0.50 + 0.90 uu~ 9 grading curve B 7 days 28 days CI 0 9

0.5

C2 C3

A B

A 9

i

0

10

20

i

3O

4O

5O

fen ['N/mm~]

10

~(s~.Ol) t'N/mm~] consisl~ency

[~

c1.,~(01)-0.222 f~z

cl~'J

C2: %(0,1) = 0.197 fc_~ / C 2 : C3:xb(0,1)=0.145 f c ~ / /

/ ,,

C3/

b)

/////

(/y ~er legend see Fig. a)

f" 0

10

20

t

!

30

40

50

fed [N/mrn2] Fig.5 Influence of concrete strength on bond stresses; Martin [ 18] a) at 0.01 mm slip b) at 0.10 mm slip

Janovic [19] and Noakowski [20] also proposed a proportionality to the ~a power of the concrete strength. Based on pull-out test results with up to 0.25 mm slip with concrete mixes of 40, 80 and 120 N/mm 2 strength, Kimura and Jirsa [ 17] concluded that the bond stresses are proportional to the square root of the concrete strength.

263

4.2 I n f l u e n c e o f c o n c r e t e c o m p o s i t i o n

Applying various cement contents, grading curves of aggregates, water-cement ratios and consistencies, Martin [18] concluded from his test results the following" - The effect of water-cement ratio is eliminated if the bond stress is related to the concrete strength. - The grading curve of the aggregate and the consistency of fresh mix strongly influence the bond properties. The observed greatest difference was 1 to 5. - The highest bond stress belongs to the grading curve with the lowest amount of fine particles (A - Fig.6.a) and the lowest bond stress to the grading curve with considerably 100 80

a)

,~/ /I V" / / /

60

e~ o

=

40

~

20

fl,/. ~t,"1

0 0.25 0.5

1.0 2.0 4.0 8.0 16.0 31.5 sieve size [ram]

b) ~fc~

i IIII11

iw/e=0.75

0.8 -

0.5

I I

- consistencygrading c u r v e B IIII C I I 1 " 28 days ....

0.6

'

0.4

'-_/,,i,, " J~C2

/ ,/l!

rb(S = 0.2)/fc-~

~

,,"

""

0.4

/ N, . ~ . I r - ,

TM

_j

0.3

]

r

C3

9

i

f

0.2

J

0.2

/

0.1

/

I ,O 18 mm

def. bars 0 0.01

0 0.1

1

2

5

1

1.1

1.2

1.3

slip [mm]

1.4

1.5

consistency

Fig.6 Influence of concrete composition on the bond behaviour; Martin [18] a) grading of aggregates b) influence of grading curve on the bond stress at 0.2 mm slip c) influence of consistency (as degree of compaction acc. to Walz, DIN 1048)

264 higher amount of fine particles (C - Fig.6.c). The difference produced in bond stresses is more than 100 %. It means that coarser aggregates improve the bond capacity of embedded deformed bars (Fig.6.a and b). - The highest bond stress belongs to the stiff (C1) consistency and the lowest to the semifluid (C3) consistency. The produced difference in bond stresses is up to 100 % (Fig. 6. c). The reason for the influence of consistency is the composition of concrete since variation of consistency has been achieved by varying the cement content by about 30%.

5. INFLUENCE OF CONFINEMENT ON STEEL-CONCRETE INTERACTION Confinement concerns all kind of effects that control transverse deformations of concrete matrix surrounding the reinforcing bar induced by the interactional forces before or after splitting of the concrete cover. Confinement may be provided by transverse reinforcement, by transverse pressure or by improving the strength or thickness of concrete cover. By onset of longitudinal cracking, the interaction may completely vanish. The confinement provided by the reinforcement (Fig. 7.a) or by the transverse pressure (Fig. 7.b), however, controls the opening of the crack. 5.1 I n f l u e n c e o f t r a n s v e r s e r e i n f o r c e m e n t

Pull-out test results by Eligehausen, Popov and Bertero [23] (Fig. 7.a) indicate different bond stress-slip behaviours with and without transverse reinforcement. The specimen without transverse reinforcement (designated 1.4) failed by splitting of the concrete cover at rather small bond stress of 6 N/mm 2, then the bond resistance dropped rapidly. Specimens with transverse reinforcement failed by bar pull-out. The crack developed in the plane of the longitudinal axis of the bar. Therefore, transverse bars (called vertical bars) crossing the crack were effective in restraining the concrete, while the influence of bars parallel to the axis (called stirrups) was negligible. This can be seen comparing series 1.2 to 1.5 in Fig.7.a. The influence of the area of vertical bars was rather small in the applied range. The initial stiffness o f the bond stress-slip curves was almost identical for all test series, which indicate that there exists an upper limit for an effective restraining reinforcement beyond which the bond behaviour cannot be further improved, because the main role of this reinforcement is to prevent the opening of splitting cracks [23]. 5.2 I n f l u e n c e o f t r a n s v e r s e p r e s s u r e

Untrauer and Henry [21 ] studied experimentally the influence of transverse pressure on the bond behaviour on pull-out specimens. The normal pressure was applied to two parallel faces of the specimens up to 16.3 N/mm:. Slip was measured at the loaded end using a dial micrometer gauge system. Untrauer and Henry [21 ] concluded that:

- The bond strength increases with increased normal pressure. The increase is approximately proportional to the product of the square root of the normal pressure and the square root of the compressive strength of concrete. - The incerase in bond strength is higher than the increase of bond stress at lower slip values.

265 The bond strength is larger for a 28.6 mm bar than for a 19.0 mm bar when normal pressure is applied. They interpreted the larger increase in bond stresses at larger slips with the increasing contribution of the friction and bearing on the lugs which are influenced by the transverse pressure.

16 ,,

E

1

,,.

,

,~

a)

~

I//OJ

'N

)

@ l

It

r ~ , , , , ~ , ~ , , r s , .... o, r_

I

~,,

I 2~,

I

il J.2 I

|

~

~5"-~1

A,31

I~z,

! ,o

12 7

I tZ't

I

i.O

63~

I,z~

!

o z5

~

.~

0 L_

1

l

0

2

4

~E

~

1

I

6

8

Longitudinal

,

1

_--

10 12 Slip [ram]

p 0 : 0 N/mm ~ pl" 6 9 N/ram~ p2:13 8 N/rnm:

15

b)

.

)IS.,..,} Otm~} IO[~}! zA'*/As!

o~ o~ I-i

10

5

1

\

0 ,~

o

[

2.s

iii

/t

A

so

7.s

~oo

Slip [mm] Fig.7 Bond stress-slip behaviour with or without confinement leading to complete splitting failure or pull-out failure after splitting of the cover a) Confinement by transverse reinforcement; Eligehausen, Popov and Bertero [23] b) Confinement by transverse pressure; Malvar [29]

266 D/Srr [22] carried out tensile tests on cylindrical specimens with a centrally embedded steel bar of 16 mm diameter in order to obtain the bond stress-slip relationship for four different values of transverse pressure (0, 5, 10 and 15 N/minE). The specimen was 600 mm long with an embedment length of 500 mm. The specimens were notched at the middle section to localize transverse cracking. The concrete cube strength varied between 29 and 42.5 N / r a m 2 with an average of 35.9 N/ram 2. Test results by D6rr (Fig. 8) gave a 100 % increase in bond stress at 0.1 mm slip when the transverse pressure increases from 0 to 15 N/ram 2. Slip was derived from the sum of measured strain differences of steel and concrete and the bond stress was derived from the change of forces from section to section.

r

--'15 E E

~10

B

g.a

o

m:l

t/)

D

Q

5

D

rNmml

[31 p = 10 [N/mrn ~] p = 15 [N/mm 2] 0.1

t).3

0.2 Slip [ram]

Fig.8 Influence of transverse pressure on the bond stress-slip relationship, fc[]--36 N / m m 2 D6rr [22]

From a comparison of the experimental results [21-26] relating to the influence of transverse pressure on bond strength, Eligehausen, Popov and Bertero [23] have shown a 2.5 to 8.5 N/mm 2 increase for a pressure of 10 N/mm. This large scatter is partly a consequence of the different failure modes. Twelve specimens consisting of 19 mm reinforcing bar embedded in a 76 mm diameter concrete cylinder were tested by Malvar [28, 29] under controlled confinement. The bond length was a 5 lug spacing of approximately 70 mm. Concrete cylinder strength varied between 38.4 and 44.2 N/mm 2. The support of the specimen was provided by shear stresses on the surface of the specimen. Concerning the bond stress-slip behaviour he concluded that: Preceding the onset of longitudinal cracking, the influence of confinement stress could not be properly established. - After longitudinal cracking, bond stress increased significantly with applied confining -

267

20

El5 E o

10

0 m

5

o'

TEST TEST TEST TEST TEST

1:3.4 N/mm 2:10.3 " 3:17.2 " 4:24.1 " 5:31.0 "

~

16

Slip [mm] Fig.9 Bond stress-slip diagrams with various levels of transverse pressure; Malvar [29]

stress. The maximum bond stress was increased by almost 200 % by increasing the confinement stress from 3.4 to 31 N/mm 2 at the bar level (Fig.9 and Fig. 7.b). The effect of confinement on bond behaviour appeared less pronounced for the higher confining stresses. - After longitudinal cracking, radial deformation measured on the concrete surface showed an increase up to a limit value dependent on the confinement level, then a decrease due to interface deterioration. - Bars with normal ribs (at 90 ~ to the longitudinal axis) exhibited better bond characteristics than bars with inclined ribs. Bars with normal ribs also produced more severe radial cracking. - Increasing radial pressure generated more severe radial cracking. Pull-out tests on modified specimens were carried out by Nagatomo and Kaku [27] in order to investigate the influence of transverse compressive and tensile stresses on the bond behaviour of deformed bars. The major parameters were the transverse stress and the concrete cover. The following transverse compressive stress as: 0, 3.92, 7.85, 9.81 and 11.77 N/mm 2 and transverse tensile stress as: 0, 0.49, 0.98 and 1.47 N/mm 2, were applied. Different covers were intended to influence the type of bond failure (side splitting, V-notch splitting, comer splitting or pull-out). The bond length was 70=154 mm using a bar of 22 mm diamter. The concrete compressive strength varied between 15.5 and 18.8 N/mm 2. Nagatomo and Kaku [27] concluded the following:

- The bond strength increases linearly with increasing transverse compressive stress up to approximately 30 % of the concrete compressive strength, and tends to level off for the transverse compressive stress beyond that value (Fig. 10). The relative increase rate with transverse compressive stress decreases linearly with increasing concrete cover up to 2.5 times the test

268

2.0f[~~

2.0[

~b.(p~) Xb.(P=0)

l:bu(pr ~:bo(P=O)

"tb.(Pr Tb,,(p=O)

2.0-

o

~.~%o

~.o

o'.s

9

I

1 .o

ol

o'.s

p/f~

I

1 .o

o

I

0.5

p/fr

I

1.0

p/fr

Fig.10 Influence of transverse compressive strength on bond strength Nagatomo and Kaku [27] a) Cb/O:l.08 %/O=1.98 C) %/0=3.10

b)

bar diameter and becomes zero for the concrete cover beyond that value (Fig. 10). - The effect of transverse compressive stress on bond strength is less for the failure modes by a side splitting crack parallel to the direction of transverse compressive and by shearing of concrete passing through the top of bar ribs. - The bond strength decreases rapidly and parabolically with increasing transverse tensile stress and becomes zero at the transverse tensile stress equal to the tensile strength of concrete (Fig. 11). Based on their own experimental results and those of others' experimental results, Nagatomo and Kaku [27] developed a linear and a parabolic relationship to take into account transverse stresses. The normalized bond strength in case of transverse compression (p):

Xb"(P/fCvO) = %.(p/fr

k'p + 1

(2)

and the normalized bond strength in case of transverse tension (Pt, Fig. 1 I):

Xbu(Pt/ft~0) -- Xbu(Pe/ft=0)'( 90"755"Pt - 1)'( Pt _ 1) fc where l:bu(P/fcr

ft

) and factor k are functions of the concrete cover [27].

(3)

269

%.(p=O)~

O : %/0=1.08 A : cb/0=-1.98 !"1 : %/0=3.10

1.0~.~,, F"I ~,~ ~ , , ~

0.5- ~ 0

/Eq.(3) 0.5

1.0 P/fc

Fig.11 Influence of transverse tensile stress on bond strength; Nagatomo and Kaku [27]

6. INFLUENCE OF LOADING RATE ON STEEL-CONCRETE INTERACTION Vos and Reinhardt [30-31 ] studied experimentally the influence of loading rate on pull-out specimens of 30 bond length using concrete mixes with average cube strength of 22, 45 and 55 N/ram 2. 10 mm diameter plain and deformed bars of 0.076 related rib area were tested in addition to 3/8" diameter prestressing strands of 1730 N/mm 2 0.1% proof stress. Test results indicated that the bond resistance of plain bars and strands is hardly influenced by the loading rate (even increasing the loading rate by 100 000 times). It appears that the adhesion, the friction and the lack of fit (by prestressing strands) are insensitive to a variation in loading rate [30]. In case of deformed bars, the bond resistance increases with increasing loading rate, however, the concrete grade also has an influence. The higher the concrete strength, the less insensitive is the bond resistance to variations in loading rate (Fig.12). The influence of the loading rate and concrete strength is most pronounced for a small slip of the order of 0.01 mm. Vos and Reinhardt [30] developed the following relationship for the rate dependent bond stresses valid up to a slip (s) of 0.2 mm: Tb Tb0

_( X.b )q Tb0

0<s<0.2mm

(4)

where Tb and Tb0 are average bond stresses to a given slip and '~b and Xb0 are the respective loading rates. %o and Zb0 can be regarded as "static" reference values [30]. The power q of Eq.(4) was determined using the method of least squares for all results with deformed bars as a function of the concrete cube strength (fcc in N/mm 2) and the slip (s in ram):

270 rb/rbO 2.6 2.4 2.2 2.0

]fcz= 20 N / m m 2 [ / ~ /

1.8 1.6

/ "

./"

40.~

1.4 1.2

10

102

103

104

10'

106

,V;bO

2.0 1.8

[fc.=:O./mm

1.6 1.4 1.2 1

1

10

102

10'

104

10'

10~

fiCr

Fig.12 Influence of the loading rate on the bond resistance of deformed bars Vos and Reinhardt [30] a) 0.01 mm slip b) 0.20 mm slip 0.7(1-2.5s) fctJ 8 Eq.(4) is plotted in Fig.12 for three concrete strengths and two slip values 0.01 and 0.02 mm in double logarithmic scale. Fig.8 indicates that for a small slip as 0.01 mm, the influence of loading rate and concrete strength is more pronounced than for a higher slip as 0.2 mm. The bond resistance associated with impact loading can be twice the bond resistance of static loading. For a higher strength concrete (here 60 N/mm 2) this ratio is only 1.5. For higher displacement, the influence of the loading rate decreases to about 1.4 and 1.2, respectively.

271 7. INFLUENCE OF REPEATED LOAD ON STEEL-CONCRETE INTERACTION Repeated load produces an increase of slip as a result of progressive micro-cracking and micro-crushing [23,32-38] and can lead to failure at a cyclic stress level lower than the ultimate stress under monotonic loading. Repeated loading may be subdivided into low-cycle and high-cycle loading according to the number of load cycles up to failure. Low-cycle loading usually contains only a few cycles but at a high stress level. Low-cycle loading generally arise under seismic conditions. Highcycle loading contains (thousands or millions), but on a relatively low stress level. Bridge elements, offshore structures and supporting members of vibrating machinery are typically subjected to high-cycle loads. Another subdivision of cyclic loading, which is followed also herein, is to consider the type of applied stress as either having or not having stress alternations. Cyclic load will imply load cycles without change of sign of the applied load, and reversed cyclic load implies load reversals including both tension and compression in one cycle. 7.1 Influence of cyclic loads Test results with constant and variable amplitudes are discussed separately. 7.1.1 Constant amplitude cyclic loads According to pull-out test resuls by Rehm and Eligehausen [34] with constant amplitude cyclic loading, the slip versus number of load cycles diagram can be approximated by straight lines in double logarithmic scale. For loads below the fatigue bond strength these results give straight lines which are parallel to each other (Fig. 13). The gradient of lines were higher for specimens which failed by fatigue. 1.0 E

& 05 g:

t~ d~

O

o.o

._~ r,~

0.01 |.0

101

10 2

10 3

1r

10 5

]0 6

number of load cycles, n Fig. 13 Increase of slip at the free bar end as a function of the number of load cycles fc~=23,5 N/mm 2, O=14 mm, eb=30 Rehm and Eligehausen [34]

272 Rehm and Eligehausen [34] proposed a relationship to predict the slip after n load cycles (s,) as a function of the initial slip before applying the cyclic load (So) and the number of load cycles: sn -- s0(1+k.) where k. = (1 +n) 0"107-1 The slip coefficient, 1%, was not greatly infuenced by parameter variations of loads below the fatigue bond strength. Considering the total bond faitigue process, Baldzs [36] distinguished between three different phases in case of constant amplitude cyclic loads producing slip over the entire bond length (Fig. 14). Phase 1 Phase 2 Phase 3

: slip increase with decreasing rate : slip increase with a constant rate : slip increase with an increasing rate

Nevertheless, the slip versus number of load cycles diagram in linear scale contains, in turn, concave downwards, linear and concave upwards portions. The length (i.e. the corresponding number of load cycles) of all three portions are functions of the bond parameters and the cyclic load level. The slip rate during the secondary linear phase was also found to be dependent on the load level. In case of a relatively high load level, the second linear portion of the slip versus number of load cycles diagram can transform into a point of inflexion. To predict fatigue failure, the initial point of the failure branch (Phase 3) is of particular inter-est. It was found [36] to be equal to the slip, S(Zbu) at the monotonic bond strength (Fig. 14.b and c). Once the slip exceeds S('~bu), a pull-out failure occurs after additional load cycles.

7.1.2 Cyclic loads with variable amplitude Pull-out test results [37] with variable amplitude cyclic loading indicated that the development of slip is strongly dependent on the load history. Based on the load spectrum analyses of bridges and cranes, parabolically and logarithmically varying load histories were applied in addition to linear ones both with increasing and decreasing tendencies. These load histories were set up in stepwise fashion using shorter sub-blocks of constant amplitude loadings. Six typical test results obtained by using these load histories with a maximum cyclic load level of 40 % of the average bond strength applied in two blocks till 2.106 load cycles are presented in Fig. 15. The highest slip values of Fig. 15 are in the serviceability range far below the slip at bond strength. With increasing sequence of amplitudes, the first complete block of loading (here up to 1 million of load cycles) produces a pronounced slip increase with decreasing slip rate. Cycling in the next block at lower load levels, however, does not contribute to the slip increase (Fig.15.a, c and e).

273

a) -

I

I

I

I

!

!

y

T

r

i

7N

2 Slip I . i t i x~. t I. l pull-out ] [N/ram ] --.--...~decreasmg4.. i constant .L..4 lncreaslng..~failure ir - " >0 " 1 60 [ 1 rate ' n = 10r~- - 20 i. 30 40,,', 8c,,cles

I(-' r~r4 - tittiHJ,litttilttli#i"rhititttttttti:..~ ..... ~" i-t-

-"7

'oj. i ~ i f

,t

' ,llttttttttltl;1t!ttt,Itlllttl'ltttt I.-i~ : '~ltlttttNitttllt Itt/ltltN..... r,___~

.

L1.0cycles.'

~

-u-~ _

~

I

I

t 0__

i

+

0

!l]lJtl:l[ttt '-:i" ]ttiItt}]/tt]]tlt]tttttttt i ]l~t i ' i , I Slip i

p I 1o.2,5 I o.5o i

F I<%

! t

i I l:OOl

}0.75

Load history

t 1.25

I 1.50

[mm]

.

S"

1,, r,"l!lIIl!!!111111l lilii!lili!lll~llll'llil!Sll'Jlilllilllillllli !W,ti i t~~l /. ~%,~NNttil~lll[llllllllllJtliilli]lJ. , llJ~lli,,llllt,lJlillll,Ii~illtll,/ "15~0 0 N10u~b2e0o?~oad40yclSe~ 6~0 n;

Slip [mm]# . . 2 5-[

,

.

.

b ) /2/.!0 ~!.I ~ ,- l V! l /

I

a st

, ~/ 1 "U

I

'

I

!

: !

M~e~176176

Pull-out t fa!lure ]]

3.0 ~ /

-t / ~

20~

I/

~

)

|

"/

! ' !~;-~,i.~,,,._J

/

!

I I '

+tl+lll

10 20

:

4.01

i

;~~----~,f C " i!

i jt"Pt%uJl---] I! l 10

30 40 50 60 70 n Number of load cycles

c~

5

;

10

Bond stress [N/mm 2]

Fig.14 Bond fatigue process under cyclic loads,fern=20 N/mm 2, 0=8 mm, eb=120; Balhzs [36] a) bond stress-slip diagram b) slip vs. number of load cycles diagram c) monotonic bond stress vs. slip diagram

274

With decreasing sequence of amplitudes, significant slip increase is produced only by the highest cyclic load levels both in the first and in the later blocks (Fig. 15.b, d and J). Low cyclic load levels after higher load levels have hardly any influence on the slip increase. Slip

a)

Tb

[mm] t'N/mm~]

Slip

~~ 0 10 ~ 0

Slip ~ % [ram] [N/ram 2]

c) .., Shp

"t 0

~

Load

0.5 1.0 1.5 2.0 Number of load cycles [million]

Slip

Tb

e)

[mm]~tWmm~1 ,

0.5 1.0 1.5 2.0 Number of load cycles [million]

d) Slip

0 - 1 ~ ' - - ' ~ o a _ _ d 06

Slip

0.5 1'.0 1".5 2'.0 = Number of load cycles [million] xb

f)

0.2tlO '

!

[mm] I [N/mm:]

0.2+ 10

0

' r--W"-'-~- L~_ooad

Slip xb [mm]! [N/mm2]

[mm]l t'N/mm~] 0.1~5___ ~

0

---

c

0.2~ 0.1 ~ ~ ~

0.5 1.0 1.5 2.0 Number of load cycles [million]

0 2 10

Slip

-

~iLoad_~___ .__ _ ~ `

Slip. Tb

b)

'

i

~

015 1.0 115 2.0 Number of load cycles [million]

0

Slip Load!

015

~.0

~15

2.0

Number of load cycles [million]

Fig.15 Slip increase as a functon of the load history, fr N/ram 2, O=16 mm, eb=50 a)-b) parabolically c)-d) linearly e)-f) logarithmically variable amplitude Balfizs and Koch [37] 7.2 Reversed cyclic loads

The most common case of reversed cyclic loading is seismic loading. Cycles with reversed loading produce a more severe degradation of bond strength and bond stiffness than the same number of load cycles with cyclic loading without change of sign. Degradation primiraly depends on the number of load cycles and the peak slip in either direction between the bar is cyclically loaded [34, 36]. Under otherwise constant conditions the largest deterioration will occur for full reversals of slip. Whenever the load cycles are limited to produce slip only in one direction, there is no significant degradation of the bond strength [38].

275 Slip controlled load reversals produce deterioration both of the peak bond stress at the applied maximum slips and that of the frictional bond stress which is the necessary bond stress from the other direction to withdraw the previous slip (Fig. 16)[36]. Five cycles between +0.2 mm slip caused the peak bond stress to decrease by 35 percent (Fig.16).

Load history Decrease of peak bond stress

Ivvvvr I 2 3 ~ 5-

----q ---i

Xb(n) 100l -

i

--~-~.,....~

i

9

i

9

1

] ' ] 2"23+ "

Slip i [mm]_~

"

9

3

.

,

1

4" 5" 4

n

5

Decrease of frictional bond stress

15 ~ ~bf(n))

-

!

"l:bf(n- 8910 ~

[%] 5 ~ - -

_+__,_ 1+ 1. 2§

4 +

4

5+

5

n

Fig.16 Bond behaviour under slip controlled load reversals, fen=20 N/ram 2, O=16 mm, eb=20 (The ordinate axis of loaded side curves is skew to correct the elongation of reinforcing bar between the loaded end section and the point of attachment of the LVDT) Bal~s [36]

Force controlled load reversals produce a remarkable slip increase which indicates an increasing damage in the concrete matrix around the steel bar (Fig.17) [36]. The higher the

276 load, the higher the slip increment. The slip rate during force controlled load reversals was approximately four times higher in comparison to that of a repeated loading.

Load history FI "l:b

,,!j, !pJll!JJ!llJ!l!lI .,

:

.,..,

. . . . . . .

9

p

_ _ Slip at peak stress

0.4+~

. ~

[N/xb2]mm 13.3 11.2

.... 9.4 9 | 2 345-n

Fig.17 Bond behaviour under force controlled load reversals, fen=30 N/mm 2, O=16 mm, eb=20; Balfizs [36]

8. S U M M A R Y

The behaviour of steel-concrete interfaces is discussed in reference to experimental data. Tensile or compressive forces on an embedded steel bar induce interactional forces in the surrounding concrete matrix. Only relatively low loads can be transmitted by physicochemical adhesion, however, higher loads are transmitted by mechanical interlock and friction leading to relative displacement of the sections. Bond failure may be reached by pull out of the bar or by splitting of the concrete cover. Shape and dimension of ribs, strength and composition of concrete, confinement provided by transverse reinforcement or by transverse pressure, type and rate of loading all have a strong influence on the bond behaviour.

277 NOTATIONS Cb fc[] fc ft n p p, s So Sn S(Xbu) eb

C~sb ~cx ~s• zb Zbu "l~bx %-s 0

bottom concrete cover average concrete strength measured on cubes of 200 mm sides average concrete cylinder strength tensile strength of concrete number of load cycles transverse pressure transverse tension slip (relative displacement) initial slip before cyclic or sustained loading slip after n load cycles slip at bond strength bond length related rib area concrete strain distribution steel strain distribution bond stress bond strength bond stress distribution bond stress-slip relationship nominal bar diameter

mm N/mm 2 N/mm 2 N/rnrn 2 N/mrn 2

N/mm: mm mm mm mm mm % % N/mm 2 N/mm 2 N/mm 2

N/mm2-mm mm

REFERENCES 1. G. Rehm, 13ber die Grundlagen des Verbundes zwischen Stahl und Beton, Deutscher Ausschuss f'tir Stahlbeton, Heft 138 (1961) 2. R. Tepfers, Cracking of concrete cover along anchored deformed reinforcing bars, Magazine of Concrete Research, Vol.31, N-~ (1979) 3. 3. B.B. Broms, Technique for investigation of internal cracks in reinforced concrete members, ACI Joumal, Vol.62, N~ (1965) 35. 4. Y. Goto, Cracks formed in concrete around deformed tension bars, ACI Journal, Vol.68. (1971) N-~ 244. 5. Y. Goto and K. Otsuka, Studies on internal cracks formed in concrete around deformed tension bars, Transaction of of the Japan Concrete Institute 1980, 159. 6. E. Giuriani, Experimental investigation on the bond-slip law of deformed bars in concrete", IABSE Colloquium Delft 1981, IABSE Proceedings, Zfirich 1981, 121. 7. P. Gambarova and E. Giuriani, Fracture mechanics of bond in reinforced concrete, Discussion, Journal of Structural Engineering ASCE, Vol. 111. N~ (1985) 1161. 8. R.H. Scott and R.A. Gill, Short-term distributions of strain and bond stress along tension reinforcement, The Structural Engineer, Vol.65B/N~ (1987) 39. 9. S.M. Mirza and J. Houde, Study of bond stress-slip relationships in reinforced concrete", ACI Journal Vol.76. (1979) N~ 19.

278 10. RILEM/CEB/FIP, Recommendations on reinforcement steel for reinforced concrete. Revised edition of: RC6 Bond test for reinforcement steel: (2) Pull-out test, CEB News No-73, Lausanne May 1983 11. RILEM/CEB/FIP, Recommendations on reinforcement steel for reinforced concrete. Revised edition of: RC5 Bond test for reinforcement steel: (1) Beam test. CEB News No-61, Paris April 1982 12. A. Windisch, A modified pull-out test and new evaluation methods for a more real local bond-slip relationship, RILEM Materials and Structures, Vol.18, No-105(1985) 181. 13. G. Rehm, H. Martin and P. Noakowski, Einfluss der Profilierung und des Betons auf die Verbundqualit~it von Stahl in Beton- Ausziehversuche an gefr~ten St~.hlen, Report Nr.2203/1970, Lehrstuhl und Institut far Massivbau, Technische Universit~it Mfinchen 14. CEB, Bond Action and Bond Behaviour of Reinforcement, CEB Bulletin d'Information No-151, Paris, Dec. 1981 15. S. Soretz and H. H61zenbein, Influence of rib dimensions of reinforcing bars on bond and bendability, ACI Journal, Vol.76, No-I, (1979) 111. 16. R. Tepfers and P.-A. Olsson, Ring test for evaluation of bond properties of reinforcing bars, Proceedings, Int. Conf on Bond in Concrete, Riga Oct. 1992, Vol. 1. 1-89. 17. H. Kimura and J.O. Jirsa, Effects of bar deformation and concrete strength on bond of reinforcing steel to concrete, Proceedings, Int. Conf. on Bond in Concrete, Riga Oct. 1992. Vol.1. 1-100. 18. H. Martin, Bond performace of ribbed bars (pull-out-ests): Influence of concrete composition and cosistency, Bond in Concrete, Proceedings, (Ed. P. Bartos) Applied Science Publishers London, 1982, 289. 19. K. Janovic, Bericht fiber den neuen konsolenfOrmigen Ausziehk6rper als Vorschlag far ein allgemeingfiltiges Verbundprfifverfahren, Report, Massivbau TU Mfinchen, 1979 20. P. Noakowski, Verbundorientierte, kontinuierliche Theorie zur Ermittlung der Rissbreite, Beton- und Stahlbetonbau 7(1985) 185. + 8(1985) 221. 21. R.E. Untauer and R.L Henry, Influence of normal pressure on bond strength, ACI Journal, Proceedings, Vol.62, N-~ (1965) 577. 22. K. D6rr, Bond behaviour of ribbed reinforcement under transversal pressure. Nonlinear behaviour of reinforced concrete spatial structures, Preliminary Reports, IASS Symposium 1978, Vol. 1, Werner-Verlag, Dtisseldorf, 1978 23. R. Eligehausen, E.P. Popov and V.V. Bertero, Local bond stress-slip relationships of deformed bars under generalized excitations, Report N~ UCB/EERC 82-23, Earthquake Engineering Research Center, University of California, Berkeley, California, Oct. 1983 24. T.P. Tassios, Properties of bond between concrete and steel under load cycles idealising seismic actions", CEB Bulletin d'Information No-131, Vol.1, 1979 25. S. Viwathanatepa, E.P. Popov and V.V. Bertero, Effects of generalized loadings on bond of reinforcing bars embedded in confined concrete blocks, Research Report, Earthquake Engineering Research Center, No.UCB/EERC-79/22, University of California, Berkeley, August 1979 26. A.D. Cowell, V.V. Bertero and E.P. Popov, An investigation on local bond slip under variation of specimen parameters, Research Report, Earthquake Engineering Research Center, No.UCB/EERC 82/23, University of California, Berkeley, 1982

279 27. K Nagatomo and T. Kaku, Bond behaviour of deformed bars under lateral compressive and tensile stress, Proceedings of the Bond in Concrete Conference Riga 1992, 1-69. 28. L.J. Malvar, Bond of reinforcement under controlled confinement, ACI Joumal, Vol.89, N~ 1992, 593. 29. L.J. Malvar, Confinement stress dependent behavior, Part I: Experimental Investigation, Proceedings of the Bond in Concrete Conference Riga 1992, 1-79. 30. E. Vos and H.W. Reinhardt, Bond stress-slip behaviour of deformed bars, plain bars and strands under impact loading, Bond in Concrete (Ed. P.Bartos), Proceedings, Applied Science Publishers London, 1982, 173. 31. E. Vos and H.W. Reinhardt, Influence of loading rate on bond behaviour of reinforcing steel and prestressing strands, Materiaux et Constructions, 15 (1982), N-~ 3 32. B. Bresler and V. Bertero, Behaviour of reinforced concrete under repeated load, ASCE Journal of Structural Division, Vol.94. ST6 (1968), 1567. 33. A.D. Edwards and P.J. Yannopoulos, Local bond stress-slip relationship under repeated loading, Magazine of Concrete Research, Vol.30, N~103 (1978) 62. 34. G. Rehm and R. Eligehausen, Bond of ribbed bars under high-cycle repeated loads, ACI Journal, Vol.76, N~ (1979) 297. 35. P. Plaines, T. Tassios and E. Vintz61eo,,, Bond relaxation and bond-slip creep under monotonic and cyclic actions, Proceedb:gs, Bond in Concrete (Ed. P. Bartos), Applied Science Publishers London, 1982, 193. 36. G.L. Balfizs, Fatigue of bond, ACI Materials Journal Vol.88, N% (1991) 620. 37. G.L. Balfizs and R. Koch, Influence of load history on bond behaviour, Proceedings of the Bond in Concrete Conference Riga 1992, 7-1. 38. M.N. Hawkins, I.J. Lin and F.L. Jeang, Local bond strength of conrete for cyclic reversed actions, Proceedings, Bond in Conrete, Applied Science Publishers London, 1982, 151. 39. W. Nies, Ein neues Verfahren zur Messung der 6rtlichen Relativverschiebungen zwischen Stahl und Beton. Diss. Darmstadt 1979

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rigl-.tsreserved.

281

Steel-concrete interfaces: Damage and Plasticity computations J.M. Reynouard a, O. Merabet a and J.L. Cl6mentb a Laboratoire B6tons et Structures, Greco G6omateriaux-CNRS, Institut National des Sciences Appliqu6es de Lyon, 20 Avenue Albert Einstein, 69621 Villeurbanne Cedex, France b Laboratoire de Mecanique et de Technologie, Greco G6omateriaux-CNRS, Ecole Nationale Sup6rieure de Cachan, Paris 6, 61 Avenue du pdt Wilson, 94235 Cachan Cedex, France

INTRODUCTION Reinforced concrete is one of the most widely used construction materials. Composite by nature, its mechanical characteristics result from the combined action of two complementary materials: steel and concrete. Its structural behaviour is derived from the transfer of the stresses that can no longer be resisted by the cracked concrete, to the steel. This transfer is then made possible by the very nature of the interface between these two materials. Usually fibbed, the steel bars are intimately imbricated in the concrete, thus creating a transition zone essentially composed of concrete, and which has a minimum thickness equal to that of the fibs. As a result, reinforced concrete is a composite material, whose modelling needs to consider not only the separate behaviours of both materials: steel and concrete with their distinctive characteristics, but also their bond. Therefore, it is of prime importance to define clearly the class of problems for which the consideration of this bond is crucial on one hand, and the scale level that the modelling calls for, on the other hand. This is the problem which the present chapter addresses. It is divided into three parts. The first part is dedicated to the phenomenological aspects of the behaviour of the steel-concrete interface, the second attempts to show the main directions to follow towards its modelling, and finally, the last part focuses on reinforced concrete structures and aims at evaluating the importance of the bond in various real-life situations. 1. INTERFACE PHENOMENOLOGICAL ASPECTS

1.1. Physical parameters The quantification of the interface parameters is a difficult problem for composite structures. In order to model the interface behaviour, it is necessary to obtain experimental and physical information. All this information is local and related to the interface zone between reinforcing steel and concrete. In RC-elements tests, generally, local informations are not obtained directly : for example, the external load, the deflection, the crack opening, etc.., are measured. The measurement of local strains along a rebar and the concrete strains near the reinforcing bar is difficult and very sensitive to experimental conditions, and hence there is a significant scatter in test results. We do not have any information concerning the real

282 interfaces. Only the effects of the bond degradation can be observed and accurately measured. The steel-concrete interface problem is complex, as shown with Figure 1.

I

crack

..~ ,

I

.

t I

cover c~

_ _ concrete

- - -

9

~9 Lb

~

reinforcing

steel

i

" damaged interfacezone

Figure 1. Damaged interface zone in bending If a crack occurs in RC-beam bending test, the steel-concrete interface zone near the crack is damaged. It is difficult to obtain the relation between the physical measurements (applied load F, deflection, position, length and opening of cracks,...) and the mechanical local parameters (values and distribution of the bond stress x, relative slip U between steel and concrete, length of damaged zone Lb... ) . The bond stress 1: is calculated from strain measurements, with some hypothesis, as shown in paragraph 1.2. Experimentally, only the influence of the physical paremeters can be shown. The bond stress x is related to : the external confinement [ 1-3], the compressive and tensile strengths of concrete 6cand 6t, the age of concrete [4], the strength of the reinforcing steel 6s(F ) and the yield strength 6e. The global behaviour of a RC sample is influenced by the geometry of the reinforcement (with or without fibs), the bar diameter r [5], the concrete cover c [5-6], the reinforcement ratio p, the existence of stirrups, the geometry of the samples, the nature of the loading (monotonic or cyclic) [7-9]... The bond mechanism under monotonic loading, issued from a study of Eligehausen et al. [ 10] is represented Figure 2.

i

I

i

i

BOND

i CRAC - - t

I

i GAP

li

s A

9

* ~

i

CONCRZrE!-OND ~

9

9

0

S

v $

Figure 2. Bond mechanism under monotonic loading (from [ 10]) For RC slabs subjected to shear forces, the shear force is mainly transfered in the tension zone of the specimen by friction along the crack surfaces and by dowel action of longitudinal reinforcement [ 11 ]. A number of physical parameters have a significant influence on the interface local bond behaviour of real RC-structural elements: theoretical or analytical studies are only performed on small samples, like beam tests, pull-out tests or tie-members tests.

283 With the hypotheses H1 and H3, equation (4) can be written : I;-I

Esr 1 d2U =0 4 l + n p dx 2

(6)

1.2.4. S o l u t i o n s o f the differential b o n d e q u a t i o n

The equations (5) or (6) must be solved using the boundary conditions of the real experiment These relations can be applied for pull-out tests, push-out tests or tie-member tensile tests. The interface variables are the local bond stress x, the local slip I3 and the geometrical position variable x Assuming elastic steel behaviour and using the boundary conditions, we obtain (x, 13) [15] or (x,x) [16] bond laws. In the case of a plastic behaviour of the bar, the equations (4), (5) and (6) are modified [ 17-18] or empirical bond laws are used [19]. It was found that the interface behaviour depends on the position of a steel section over the embedment length of an axially loaded rebar. Examples of bond models [20] are shown Figure 5.

i 13.5

/+~J.O 6.0

'..~ +:/ ....

~.o

. 7 t 4.(I

/ , "~.o ,~.o ~ O.J

1.0 J.O

x/r

Q.O

* [mml

a. Unconfined model

i

~

~

9 I~ I ' X l / ~

IXl _V~-\-% .... -+ +.-'.+~--"" o..+t., :J.+

:..f+t ~,o,O ....+

+'+~- l~_-_"~162 ..........' ~ o +-~ i-"/+~o zo.,

to 9[ = = I

b. Confined model

Figure 5. Bond models (from [20]) With the H1 hypothesis, the confinement stress c~, does not appear in the bond equation, despite its great influence on the bond behaviour. It means that experimentally, the equation (4) can be solved only partially. The analytical bond relations are not universal, they are always approximate. From these results, the main conclusion is that the interface problem is physically complex, and the theoretical aspects are limited. According to Nilson [12] the location of the element interface area is an important variable and the relationship between bond stress and local slip varies at different sections of the specimen. Moreover the bond laws are strongly dependent on the geometrical and physical parameters. Consequently in finite element analysis of reinforced concrete structures these models have to describe the main physical aspects of bond: interface

284 1.2. Local bond equations In order to obtain local bond behaviour laws, which will be used in interface finite elements, the hypothesis for the bond analytical analysis are illustrated with the following example. 1.2.1. Tie-member tensile test We consider a prismatic tie-member shown in Figure 3. The reinforcement is a single steel bar of diameter ~. The ends of the bar are subjected to tensile load F. Strain gauges are glued on the surface of the bar embedded in concrete and on the concrete specimen.

'EC. ~'[A

I

'

BY

',

I

. strain gauges C]

~

- reinforcing steel cross section

Figure 3. Tie-member test The possible measurements are the applied load F, the displacements at the ends A and C, the crack width, the concrete strains arround the specimen e c and the strains of the steel bar e s. The strain measurements are generally limited. It should be noted that the steel gauges modify the interface between the steel and the surrounding concrete. It is possible to glue steel gauges in an hole made in the section of the bar. In this case, the interface behaviour is affected because the section of the steel is reduced. Hence, the steel strains distribution is modified [ 12]. This technique can be used only for high diameter bars and is very expensive. Experimentaly, E, can be measured by strain gauges, LVDT transducers [13] or piezoelectric sensor [ 14]. 1.2.2. Definition of the local bond parameters In order to write the differential bond equation of steel-concrete interface expressed by a relation between X,Ec,Es,Gn, we have to define : - the local bond stress x, through the equilibrium of a portion of bar (Figure 4)

J

I ....

! v A

i dx

-X'

v A

Figure 4. Equilibrium of a reinforcement part

285 The equilibrium relation on the X-axis leads to 9 dos 't = - - ~ 4 dx

(1)

To obtain this relation, we assume only that the longitudinal stresses in the X-direction, o, and % are uniformly distributed respectively on the bar section and on the bar perimeter. - G, cannot be directly measured. In order to obtain a relation between c, and o s we assume that the steel behaviour is an elastic and isotropic one. Considering that the stresses in the transverse direction perpendicular to X are equal to o n 9

O, 2V, 8, = ~ - - ~ O n E, E,

(2)

where E s is the steel Young modulus and v s the Poisson ratio. The relation 2 is applicable only in the case of an elastic behaviour of the bar. - we note U the local slip between steel and concrete, at the interface. U is defined through the following relation, from the axial steel strain 8s and the axial surrounding concrete int strain e c 9

dU

=8~ nt - 8 s

(3)

dx 1.2.3. Differential

bond equation

Using now the equations (1), (2) and (3), we obtain the differential bond equation : EsO d2U "1:+~~ 4 dx 2

EsO d8~nt 4

dx

Vs@ do n 2

dx

=0

(4)

The equation (4) is only correct in the case of an elastic behaviour of the steel (relation 2). All the parameters cannot be directly measured in experiments. To solve this equation, in order to obtain for example bond stress-slip (x,U) or ('r,,%,U) laws it is necessary to choose some other hypothesis 9 - the local confinement stress o n is unknown. On can be neglected in comparison with the values of o, (HI). It means that the result will be correct for high (elastic) values o f o,. - eomt is unknown, e cmt can be neglected in front of es (H2), or we can assume that ec~ is equal to the external measurement of the concrete strain ~ (H3). In this case we have int er = 8~ = - n p e , , where n is the modulus ratio E s / Er and p is the reinforcement ratio. Hypotheses H1 and H2 lead to the reduced bond equation : Es(~ d2U ~ + ~ ~ =0 4 dx 2

(5)

286 bond damage, influence of the confinement on the bond stress, and the expected numerical results will be a rough approximation. 2. M O D E L L I N G OF REINFORCED CONCRETE STRUCTURAL E L E M E N T Due to the significant scatter of the test results, a correct experimental determination of bond laws is not evident. Whether or not to include bond behaviour in finite element computations depends on the objectives of the structural calculation. In order to perform calculations on simple RC-structural elements, to obtain local informations on bond stress distribution, local slips, steel and concrete strains, two approaches are possible : these involve either the presence or absence of interface finite elements.

2.1. Interface finite element (I.F.E.) A simple test as tie-member tensile test can be analysed in two dimensions using interface finite elements or using continuum mechanics without numerical slip. These two approaches are shown Figure 6. I

i

I . concrete concrete- - _ steel-

I

~

. - interface element -- steel

1

a. without interface element

b. with interface element

Figure 6. Tie-member plane discretization

To use interface finite elements we need to define the tangential and normal constitutive response consistent with the experimental results. The main interest of this approach is to concentrate all the interface behaviour in a layer of elements which have an appropriate stiffness. The concrete and the steel are discretized by standard elements and their behaviour can be taken to be nonlinear (plastic or damaged materials). The interface finite element formulation is based on the equilibrium of a prism of concrete near the reinforcing steel. This prism has an unknown thickness [3t~. The possible prism displacements are u and v (Figure 7).

287

i

i

Y

- - ] - - -concrete

i

-~v . . . . 0

x)

IX u X

r .

.

.

.

i

-j .

.

.

interface layer

- - ] - - El interfaceYoungmodulus steel

i

Figure 7. Interface displacement field [21 ] u is the tangential slip between steel and concrete, related to the bond stress x, and v the transverse displacement issued from the confinement stresses 6.. The relation b e t w e e n , and o. is: 1: = -tgO. o.

(7)

with tg0 the so-called fib-factor. The law for interface behaviour can be expressed as :

{'1:}.. =[:

00]{U}v+E'c~k ~

-k]{:}l

(8)

where S is the elastic modulus of the x - u law, and k = tg0. This law depends on the choice of 0. If 0 = ~ is chosen, k is equal to zero. The equations (8) are reduced in 9x = S. u 2 With 0 = 0 equation (8) becomes :

It [So~ t:

t~n

=

El

~

(9)

V

The strains in the interface layer ex, ey and ex=0

v

r:y=~

y~

u

I~

"f~y are given by : (10)

We obtain an interface finite element composed of two independant springs, without a coupling between u and v. For example, the bond link element of Ngo and Scordelis [22] is based on this assumption (Figure 8).

288

{o:t:[o Figure 8. Bond link element ([22]) 2.1.1. Incremental interface behaviour laws

According to the experimental results from D6rr [ 1], for interface finite element, the yield function is often a Coulomb model function :

f(x,o.) = x-lo.ltg0-c

(11)

where c is the cohesion, 0 the friction angle before failure. A non-associated flow rule is often used: g(X,On) -- x-lOnltgtp- c

(12)

where rp is the friction angle after failure. The incremental law is derived from the following elastic relations :

AJ[ e; }

(13)

,14, with A = (Atg0 + X)tg{p + Xtg0 + C The relations (14) with 0 = {p (associated flow rule) give the equations obtained for example by Dragosavic [23]. Heuze and Barbour [24] use X=0 and 0 # 9 to study the failure of rocks. The experimental identification of the coupling parameter X is difficult. This invariably leads to the simplification X=0. 2.1.2. The CASTEM interface element

The interface element of the CASTEM finite element code [25], used to carry out tiemember calculation, has the following characteristics : - the failure criterion is shown Figure 9.

289 "l;zx Coulomb

C

n
(~n

Figure 9. Failure criteria If the transverse displacement, perpendicular to the interface surface, is an extensional one, the element cannot transmit bond stress. - the idealized x-)'xy laws are given Figure 10 and compared with experimental results [ 1].

/ iiiiiii

....

model Experimental results [ 1]

~n-O

C -

Figure 10. x-Tx~ laws - the x - u and c , - v relations are shown Figure 11. t~

EvO s

v

17 '

up

Figure 11. x - u and t ~ , - v relations

E1

290

2.1.3. Tie-member computation According to experimental results [1,12,26-28], the different parameters of the bond interface element are given by the following relations : f

2Mpa < c < 8Mpa 0 -- 10-15 ~

(15)

200N / mm 3 < S < 1000N / mm3 Parametric analysis, in comparison with experimental results derived from tie-members tests give

c = 2Mpa 0 = 10~ ~ = 5 ~ S = 200N / mm3 E 1 = 480N / mm 3 E2 =1000N/mm3

Ev0 =0,01

An example of simulation (applied tensile load versus end of specimen displacement) is given Figure 12.

3~-I LOAD [m~ 25-

28

CALCULATION TEST 1

15" - ' 4 - lEST 2 TEST )

~a

DISPLAC~

~

0 .eee .ese

~ .foe

.15e

.2e9

.25e

.3ee

.35a

.4ee

.4$0

.see

.sse

.~e

Figure 12. Comparison between computation and tie-member tensile test This approach gives us some local information, such as local slip and bond stress evolution. The predicted displacements at the ends of the specimen are in good agreement with the test results. Using interface finite elements, we need to identify a great number of parameters, which are not directly derived from test measurements. These parameters are only obtained for one kind of particular specimen.

291 2.2. Joint interface element 9contact approach

The bond interface element developed by El-ha[jar [29] for 2D soil-structure interaction problems and adapted by Merabet [30] to the steel-concrete interface, is based on a contact approach without bond stress-slip relationships [31-33]. The interface element shown in Figure 13 is a line element with a zero-length, composed of three nodes (i,j,1). The nodes i and j, connected to the solids A et B, have two degrees of freedom u x and uy. The contact forces ( ~ , ~ ) are defined on the third node 1 in the local coordinate system (n,t) having an angle 0 with the global X,Y axes. The equations of equilibrium are given in a matrix form, where the stiffness matrix is obtained by the standard procedure of minimizing the total potential energy

(16)

L

k

= a/

where [K] is the stiffness matrix of the continuum elements, [L] is the stiffness matrix of the contact element (2x2), {u} is the displacement vector, {~,} is the vector of contact forces treated as unknown variable vector (multiplying Lagrangian factor), {F} is the applied external load {a}={ a~at }T, is the relative displacements between the nodepairs (i j), in the local co-ordinate system, with an = Unjnl"U 9= An ; at = utjt'-u i = At The nonlinear behaviour of the joint-interface element is characterized by slip and separation taking place at the joint plane. Table 1 gives the various contact conditions (stick, sliding and open conditions). The iterations (i) correspond to the verification of all contact conditions at each loading stages (k). Using this method it is only necessary to solve a part of the simultaneous equations related to the contact surface and the calculations become simplified. Table 1 : Contact conditions

it,r. ( i -

l)".~

Stick

Stick

Sliding

Opening

k ~. .< F:

~.: .< Ft x) ) F,'

~.tq >

F tq

A., ~ F,

x: ~ F:

) , tJl >

F'~

A,. F,' < 0

A,.F; i>0

let

k: < F' t

Sliding Opening

a:
~:~0

The conditions are classified into the following three cases : Open condition : gap remains open Stick condition 9gap remains closed and no sliding occurs in shear (fixed condition) Sliding condition : gap remains closed and sliding occurs in shear

292

i

u,

i

I Figure 13. Joint interface element Separation of joint planes will occur when the tension normal to the joint plane becomes greater than Fnk. The yield function for this case is 9 Fnk = t n = c/tgq~ (t n is a tensile stress)

(17)

If the shear strength of the joint is exceeded irreversible slip occurs. The frictionnal force acting at the contact surface follows the Coulomb type criterion : Numerical simulation is proposed in paragraph 3.3 on reinforced concrete beam, with the following characteristics of the joint 9c = 4 MPa and tp = 10~ and hence Fnk = 22.7 MPa (obtained from a parametrical study [30])

2.3. Bond interface element for cyclic reversed loads If we consider that a law for bond between steel and concrete is written only in terms of quantities that are assumed to have a homogeneous distribution along the perimeter of the reinforcement, then this law will strongly depend on the topology of the problem as shown in Figure 14 Consequently, every test result will be associated to the type of test itself, and it will be difficult to extrapolate the results (especially the quantitative results) to too different types of problems. The steel-concrete bond problem is particularly important in a beam-column joint subjected to cyclic reversed loads, similar to those encountered during a seismic excitation. Therefore, it is of major interest to calibrate a bond law for this specific ease.

Figure 14. Different boundary conditions

2.3.1. Bond stress-slip relationship We choose to use the bond law proposed by Eligehausen, Popov and Benero [10], calibrated after a series of tests representing the confined portion of a beam-column joint. Altogether, 120 specimens were tested, varying confining reinforcement, bar diameter, concrete strength, bar distance, transverse pressure and loading history. The specimen was designed such that an assumption could be made that the bond stresses were evenly distributed

293 along the bonded length, and that the measured slip represented with sufficient accuracy the slip at every point along that length. Therefore, the stress-slip relationships obtained can be considered to be local ones. The law is based on a theory of the mechanics governing the resistance and degradation of the bond, and is thus relatively general for application to typical civil-engineering problems. It is briefly presented herein, and the reader will find a complete description in [ 10]. The law consists in an envelope curve for each direction of loading, an elastic unloading modulus, followed by a perfectly plastic plateau, and functions representing the evolution of two damage variables that affect the envelope curve on one hand, and the friction branches on the other. The shape of the envelope curve is shown in Figure 15. The first part (path OA) has for equation:

(18)

1: = '~ma x "

It represents the mechanical interlocking between the steel lugs and the surrounding concrete, with local crushing of concrete in front of the lugs. This part is followed by a plateau (x = x=~=, path AB) which relates to advanced micro-cracking and initiation of shear cracks in concrete between the lugs. The descending branch (path BC) describes the loss of resistance due to partial shearing off of the concrete keys. The final plateau ( path CD, "c = xa) represents the residual adherence once the concrete keys are completely sheared off, leaving only concrete to concrete friction. The elastic unloading modulus (path EG) is independant of the slip value, while the friction plateau that follows (path GI) is a function of maximum attained slip and of "ca. It describes friction between steel and concrete. If unloading in the opposite direction is continued further, the face of the steel lug comes in contact again with concrete, and the envelope curve corresponding to loading in the opposite direction is followed (path IA'), but reduced to take damage into account. The damage variables change with the energy dissipated through cycling, thus the reduction of the envelope curves is a function of the total dissipated energy, while the degradation of the steel-concrete friction (path GI & LM) depends only on the energy dissipated by friction alone. 1", "" ~ = ~ : ~ , % _ ~r S=,,, ! ~ A'

r;r ;-s3

8'

,

"sztl's, o, s,l J s='r:.

'.s

~ J

J_~ ~-"~~

li ", : . j ~

'~r

'J~'---T r]--.,- ~ '

~ ~ -'~

s',/

.~ON_O'rON,C---" " ~ LOADING

M ON 0 T 0 N I C LOADING

,

I

r~

TI1,~ ss I fu ~--~J--'-

*

A;,r /

Auf- ~ ~ " l

' -To

Figure 15. Proposed analytical bond model for reversed loads (taken from [ 10])

294 A similar theory supports the extension of this law to the unconfined regions [34]. Being based on these physical phenomena, this model can be applied to all problems of bond between concrete and deformed bars. On the other hand, as was mentioned earlier, identification of the parameters must be carried out through tests representing the specific problem one wishes to address, unless a procedure is available in order to use standard-test results for a given situation. Six parameters are needed to define the model for zones of confined concrete, and twelve when the two directions of loading are unsymmetric, as is the case for unconfined regions. These parameters are 7:1,x3, ~ , ~ , S ~ , the definitions of which are given in Figure 15. The sixth parameter o~ allows adjustment of the shape of the first part of the envelope curve. Many authors have proposed values, tables or relations to access these parameters considering concrete strength, transverse reinforcement, steel geometry, shape of anchorage, transverse pressure, bar spacing, degree of confinement, etc... They cover an important class of problems usually encountered in civil engineering.

2.3.2 Application example This model was first used with a particular numerical scheme to solve the differential equations of bond along a reinforcing bar, considering concrete as being elastic [35]. Here is an example of its application which uses the finite element method. The element used here is a membrane element, and the stiffness matrix of the elastic material is taken to be a diagonal matrix (orthotropic with v = 0). In the longitudinal direction (with respect to the reinforcement axis), the material is assumed to be elastic, with a low Young's modulus. In the transverse direction, the behaviour is unilateral: elastic in compression, it has the same stiffness as the surrounding concrete, but the material has no resistance in tension, which allows the reinforcement to shrink freely inside its concrete sheath. The resistance to shear is governed by the bond-slip laws presented previously. The application example concerns tests No. 3 & 13 of the test series presented in [34]. It is a different boundary problem than the one used to calibrate the law, even if it also describes the pull-out/push-in response of a steel bar embedded in a concrete block, representing the conditions present in a beam-column joint (Figure 16). Only one end of the bar is pulled during test No. 3, whereas the reinforcement is pushed on one side and pulled on the other during test No. 13.

I

t_

25"

=

'~111 ll'H

111

s" [ l l

Ill

m I~aut~r I ~ I F ~ Axuu. .~t ~ LOa~NG l.,3 30UN,

~

lit

"li

~

~i',~i

TIE-DOW~

~

HORIZONTJ~../

LOAD CELL "

T

!

"

ill TEST

t

"

I

Figure 16. Reinforcing cage and test set-up (taken from [34])

I- -

--

295

2.3.3 Numerical results Only one half of the structure is discretized for reasons of symmetry. The biaxial material law used for concrete, developed by Nahas & Lepretre in Castem 2000 [36-37] is elastoplastic with linear softening and cracking is of the smeared fixed crack type, with a unilateral treatment of its opening and closing (more details are given in the next paragraphs). Strainhardening is assumed for steel, the uniaxial law of which is described point by point. Plots of slip versus applied force shown in Figures 17 and 18 show good agreement with experimental results, at a global level: beginning of yielding and maximum load are in excellent agreement, and the softening part of test No. 13 is correctly predicted. Because the measured slip in these tests represents the fixed-end rotation of the beam-column interface, the modelling of a beam-column joint using this law for bond will capture this phenomenon correctly. FORCE ( xl.E5 N ) 3.50 3.00 2.50

o

FIRST YEALDING OF STEEL

2.00 1.50

Prediction

........

1.00 0.50

Experiment 0.00

SUP (ram) 0.00

500

10.00

15.00

2000

2500

Figure 17. Force-Slip at pulled end (test N~ FORCE

( xl.E5 N )

3.00

2.50 r/oo.

2.00

9

if-

1.50

Prediction 1.00

Experiment

0.50

SLIP (mm) 0.00 0.00

5.00

10.00

15.00

20.00

Figure 18. Force-Slip at pulled end (test N~ 3)

25.00

296

Figure 19 shows the distribution of stress in steel after yielding across the specimen, for test No. 3. It can be seen that correct local results are also obtained. However, it must be noticed that this kind of model will not be able to reproduce the stress concentrations due to the steel lugs, which bring the strong disturbance of the bond distribution along the bar length evidenced in the tests. The crack pattern obtained analytically for test No. 3 (tension only) does not reproduce the real situation characterised by the formation of a traction cone in concrete at the face of the specimen. This is due to the method used in [ 10] to identify the parameters of the model for unconfined regions. Indeed, during the tests used for the calibration, the measured slip is the relative diplacement between the reinforcement and a point in concrete relatively far from it. This is equivalent to assuming that strain in concrete is negligible. This hypothesis is no longer valid when a concrete traction cone appears, and leads to assigning the deformation due to this macro-crack to the slip considered in the bond law. The results presented in Figure 20 concern the modelling of test No. 3, with only the constitutive relationship for concrete to represent the behaviour of the bond and without the use of interface elements. Although the quantitative results are not satisfactory, the formation of the traction cone is correctly predicted (Figure 21). Thus, it should not be necessary to integrate this particular phenomenon in the bond law as it was the case in these calculations, since the interface element is used in conjunction with a good concrete model. 2.3.4. Conclusion This example shows the relevance of using a bond law together with a nonlinear model for concrete to predict the behaviour of structures strongly influenced by bond. The limits of integrating either macro-cracks in the interface law or modelling typical bond mechanisms (local degradation induced by the pressure of the lugs) with a coarse mesh of concrete alone with the assumption of perfect bond are illustrated.

STRESS (MPa) P ~ I ~ 600

// "/

~

( xI.ES N )

3.00

EXlNmrlment

,500

..' ,/ / / /

2.50

6OO

2.00 ,300

Prediction tOO

,i100

. J

~o.

DISTANCE (mum)

5~o.

~o.

~o.

26o

t6o

o

Figure 19. Steel stress distribution, test N ~ 3

Experiment SLIP tram)

0.50

0.00 O.00

" '1'(30

'

'

Figure 20. Force-Slip at pulled end, without interface elements, test N ~ 13

297

i

Numerical

Experimental

Figure 21. Numerical (without interface elements) and experimental crack pattern 2.4. Interface finite element calculation using continuum mechanics The second procedure for modelling gC-samples consists of the discretization of the interface zone with a great number of small elements, without interface finite elements. In this case, there are no discontinuities of displacements across the interface. We consider that all the interface phenomena are derived from the surrounding concrete behaviour. Under this assumption, damage due to shear stresses near the reinforcing bar is due to progressive microcracking of" concrete. A reliable non linear concrete model is needed and all the effects of bond damage are concentrated in a zone which has the thickness of one or several elements. For small interface slip, the interface element approach and the continuum approach seems to be equivalent. In this second approach, there is no real slip between the two materials (Figure 22). The behaviour of the interface is taken into account through the concrete behaviour based on, for example, plasticity or damage mechanics [35] The same model is used for the concrete and for the interface layer of concrete

concrete

1

u' u x ]

concrete

/

interface layer concrete steel

rate.ace finite clement

steel

Figure 22. Discretization with and without bond interface elements

298 2.4.1. Damage model for concrete In order to calculate a tie-member subjected to tensile load, we define first the isotropic (scalar) damage model [39] based on the continuous damage theory. A strain softening behaviour is assumed both in tension and in compression. The stress-strain relationship is 9

<sij = (1 - D)Cijldekl

(19)

where <sij and Eij are the stress and strain tcnsors respectively, C~ju the initial stiffness matrix, D the damage variable. The concrete is assumed to remain isotropic, up to and beyond failure. The coefficient D=D(~ ) ranges from 0 for the virgin material to 1 at asymptotic failure. The variable ~ called equivalent strain controls the evolution of D [39]" an example of a damage evolution law is

=

.=

ei

(20)

where (~:i)=0 if ~:i < 0, (~:i)=ei if ~:i > 0, and ~:i are the principal strains. The non-local formulation can be used in order to eliminate localization modes and mesh sensitivity [40-41 ]. Then D(~: )=D(g), where the non-local variable g represents the average of over the representative volume surrounding each Gauss point of the concrete elements. 2.4.2. Tie-member computation A similation of a tie-member test without bond interface elements is shown Figure 23, in comparison with experimental results [42].

ae I LOAD[m,q 25

,

IS"

C~TION

10-

TF.~TI

~

TF~T2 S"

"4- TEST3

DISPLA~ ~-I

;

+

Oee 9 .OSe .lOe

[M~

i

I

!

I

;

1

!

i

1

.I$0

.2e9

.250

.3e0

.3sO

.400

.450

.$8II

.$s8

.eSg

Figure 23. Tie-member calculation - Damage model (without bond interface elements).

299 2.4.3. Pull-out-test calculation

A pull-out test has been simulated by Pijaudier et al. [41] using the non-local damage model. The predicted and the measured [43] strains distribution in the bar at various loading stages are plotted Figure 24.

/

l

14kN:

9 \

Damage frotll

11:

2kN~

,,%

"/'

:!:i- o.~ ' 'computation ~:~i~ 0 . 4 2 9 '~ - -e--experiments ~ o..~ 0.714

M o.~

0 A

II 52.$

C

r~ ,l,/mmy

D

r17.$ t

to

A

B

C

D

!~

F

z

0.143

i":~ ~ 0.7:4

'i',.,il .':. - .2

0al~ 1.0

2~ Ir

Figure 24. Non-local calculation of a pull-out test, without bond interface elements (from [41 ]) 2.5. A r e a s of application

In order to describe the bond-slip mechanisms in samples or structural elements, an efficient interface model as well as a reliable concrete model are needed. Due to the inherent variability in the experimental results, it is difficult to define accurately the bond law parameters. However, for sample tests the numerical results obtained with the different bond models are in good agreement with the experimental results. On the other hand, continuum mechanics modelling without the use of interface elements appears to be an efficient tool for capturing the bond-slip effects in simple tests (pull-out, tiemember). The main advantage of this second approach is that all standard concrete laws (damage, plasticity) can be used. However, a fine discretization of the concrete layers surrounding the reinforcement is needed and the calculations are strongly complicated in real concrete structures. Moreover these calculations are often dependent on the element size, because of the strain-softening laws needed for concrete in tension. In the case of structural elements, the local bond damage obviously influences the crack opening and the crack distribution, but has no systematically significant effect on the global behaviour. Another procedure for including the bond-slip effects in finite element analysis consits of modifying appropriately the concrete and steel laws, but, up to now, the problem is still unsolved especially for the cyclic reversed loads. Indeed, when a structure is subjected to a strong earthquake, the loss of its rigidity and the energy dissipated through the loading cycles are partly due to the slippage between steel and concrete. This phenomenon is generally neglected in numerical analysis. It is of prime importance now to define the class of problems for which the consideration of the slippage between steel and concrete is crucial, and to specify alternatively the different levels of structural modelling which allow a correct representation of the bond-slip effects.

300 3. STRUCTURAL MODELLING 3.1. P r o b l e m s t a t e m e n t

Many civil-engineering structures present either plane geometries (walls, slabs), or uniaxial ones (beams, columns). Towards efficiency, the modelling techniques for these structures must take advantage of these characteristics as much as possible. The reduction of a real threedimensional problem to an equivalent two-dimensional or one-dimensional one uses simplified mathematical models, as the beam or shell theory (Figure 25). This leads to the implementation of simple elements having the lowest computational costs, and being simple to use and to implement. Nevertheless, it is not always easy to predict the validity of the equivalence of the formulations. Certain problems show ambiguity in this respect, and one needs to conduct experiments to quantify the influence of the simplification on the quality of the results. At the local level (Figure 25), the slippage between steel and concrete can be analysed using bond interface elements, but actually only 2D calculations are reasonably possible. At the global level and an intermediate semi-global level, a correct representation of the bond mechanisms by the modification of the material laws is not evident. load

!11I

&

~ 3D

A

9.reinforcement

. concrete . steel

2D local level

global level

r

v semi-global level

steel t i b e t . . . concrete layer

. o

- steel layer

concrete fiber" I

3D multifiber beam element

3D layered shell element

Figure 25. Different levels of structural modellings

301 We consider a priori that the steel-concrete interface problem needs a three-dimensional approach in order to take into account all the parameters which can affect the behaviour of the interface : concrete cover, anchorage length, spacing of transverse reinforcement, spacing of the spliced bars on the splice length, influence of transverse pressure on bond behaviour, etc... [19]. It is however of interest to test 2-D and 1-D models and to improve them so they may include the salient features of a problem before considering a truly 3-D approach. When details in a concrete structure are evaluated, it may be essential to consider the slippage between steel and concrete. According to Ottosen [44] and Desayi [45], slip is of major importance for the crack widths and crack spacing developing in reinforced structures. However, a correct representation of problem geometry, regarding concrete, steel and steelconcrete interface, leads to a great number of elements. This complexity arises three difficulties: the implementation of the model, the volume of the nonlinear calculations and the post-processing of the results. Moreover, the variety of the parameters influencing the bond behaviour, especially when the structure is loaded in a cyclic manner, complicates significantly the finite element calculations. Thus, it is not surprising that bond-slip, in practice, is only included in the analysis of special details of a concrete structure in which the interaction of concrete and steel would play an essential part. In most current applications (frames, slabs and shells), the interaction between steel and concrete is assumed to occur via perfect bond. Our objective is to obtain, via numerical simulation, local and global information regarding a given beam, frame or shell subjected to monotonic and cyclic loads (the load carrying capacity, the entire load-deflection diagrams, the energy dissipated during several cycles as well as the crack patterns and steel strains distribution). The calculations presented here serves to evaluate

the influences of omission of the bond-slip behaviour. 3.2. Material laws Under the assumption of perfect bond between steel and concrete, accurate constitutive laws are required to predict the response of the material especially to pure shear. The nonlinear finite element procedure used in this work is shown to offer a reliable tool for estimating the strength, the permanent strains, the cracking process as well as the unilateral behaviour of the cracked concrete during the quasi-static cyclic loading reversals [30,46]. A standard elastoplastic model and a nonlocal damage model are used to describe the dissipative behaviour of the concrete subjected to monotonic and cyclic loadings. The elasto-plastic strain hardening model developed by Merabet and Reynouard [47] for concrete in compression is based on a Nadai yield criterion (Drucker-Prager type) with isotropic hardening and associated flow rule, coupled with a correct description of the unilateral opening and closing of cracks throughout the loading cycles. In tension, the concrete is modelled as a brittle or as a sottening material and a smeared fixed crack approach is considered. A characteristic feature of this approach [48] is that the effect of a crack on deformation is smeared over an area which, depending on the element type, may correspond to an integration (Gauss) point. With this representation concrete is treated as an orthotropic material with principal axes normal and parallel to the crack direction. Rough crack behaviour is incorporated in the model transmitting shear forces across the crack (the cracked concrete can still transfer 20% to 50% of the total shear in a beam or a shell by aggregate interlock [4849]). If overall load-deflection behaviour is desired, without regard to completely realistic local stresses, dowel effects and failure modes, the smeared crack representation is probably the best choice. A common criticism is that such an approach yields predictions which are dependent on the element size of a finite element discretization [50].

302 The uniaxial version [46,51-52], developed for frames subjected to dynamic loadings, is based on experimental observations and considers : - reduction of the elastic modulus due to damage and the reduction of the tensile resistance with increase in the inelastic compressive deformations; - friction between the surfaces of a crack. Then a layered finite element formulation is used for the computation of the integral expressions at Gauss points across the thickness and along the length of the beam element. The non-local damage model proposed by Pijaudier-Cabot et al. [40-41] is an isotropic (scalar) model based on irreversible thermodynamics processes (paragraph 2.4.1.). The concrete is considered as a strain softening material both in tension and in compression. In the presence of strain softening, spurious localization modes and mesh dependency may occur. The non-local formulation can be used to avoid these undesirable features. The steel is modelled as an elastic-perfect plastic or strain hardening material. The way in which reinforcement is incorporated into the finite element model depends upon whether either a discrete, or embedded, or distributed representation of the steel are adopted [48]. For our calculations, the reinforcing bars are discretized either by a set of uni-dimensional elements or by 2-D membrane elements. The load-step algorithm is based on the initial elastic stiffness matrix (plasticity model) and on the secant stiffness matrix (damage model). To be able to calculate the post-peak behaviour of the structure, a displacement control is used. 3.3. Structures under monotonic loading Nonlinear finite element analysis have been carried out on reinforced concrete structures by a number of researchers [48,53]. These structures have been studied from various viewpoints at different levels of complexity, either with or without bond interface elements. On the basis of available information (geometry, boundary conditions, conventional material characterization tests), it is possible to apply these models in order to obtain results of a predictive nature. As an example, we consider a simply supported beam [54], shown in Figure 26, analysed by Merabet et al. [55], with the following characteristics: standard cylindrical compression strength of the concrete 41MPa, tensile strength (bending test) 3.9MPa, initial Young modulus 37600MPa, Poisson's ratio 0.22, conventional steel yield strength 368MPa. Due to the symmetry, only half of the beam is analysed.

3.3.1. Global and local results (perfect bond) The predicted load-deflection results at midspan and the crack patterns at various stages are presented in Figures 27 and 28. Good fit of the experimental data has been achieved with a correct prediction of the yielding and the failure loads as well as the cracking process. The predicted stress distribution in the main longitudinal reinforcement at various load stages is in good agreement with the experimental results and indicates a correct redistribution of tensile stresses from concrete to steel. However, it should be noted that if a small number of elements is used in the 2D computations, the bar elements with linear displacement distribution, and hence a constant strain distribution along the element, do not give good results for the shearstress distribution. Incidently, discontinuities occur at the edge of the bar elements and only an average value can be calculated. A quartic or cubic displacement distribution would significantly improve the shear response especially for the calculations carried out with bond interface elements [56].

303

stirrups 08 e = 10

208

i

i 2T32 270

i

Figure 26. B e a m tested by Tuset [54]

9

250

t

200

EXPERIENCE

,..o.,~176176 i ....................................

,.,,o"""" -"""~

~ e~176

i

i ii

__'~

'

----.--m-.--PLASTICITY-2D - - +- - PLASTICITY-2D with contact elements

\q

DAMAGE 2D

100

~

\.,

.....

BEAM-2 elements BEAM-5 elements

50 . . . .

DISPLACEMENT (mm) 0,

0

0

t

I

I

20

40

6O

8O

BEAM- 10 elements

Figure 27. C o m p a r i s o n with calculated load-deflection diagrams

P=72kN

F .......

P- ll5kN

P= 80kN

~

PEAK LOAD: P = 204 kN

POST PEAK" P = 73 kN / f M ~ = 21mm DAMAGE MODEL

P - 120kN

P = 200 kN

,,crushed concrete

PEAK LOAD: P = 230 kN PLASTICITY MODEL E X P E R I E N C E : P PEAK LOAD = 236 kN

Figure 28. E v o l u t i o n o f damage and cracking at various loading stages

304 The layered beam model proves to be an efficient tool for the parametric studies of reinforced concrete structures composed of line elements. However, perfect bond is always assumed and due to its uniaxial formulation, the model neglects the transverse strains as well as the transverse reinforcement, and cannot correctly describe the crack orientation. The cracking zone first develops near the reinforcing bar and expands along the beam. At the peak load, the cracking evolution is localized at midspan. In the softening part of the loaddeflection diagram, the cracking does not expand in the axial direction while expanding further in the transverse direction (which corresponds to the formation of the softening plastic hinge). The peak load is mainly influenced by the element size [57] and the calculations which do not take into account the size effect overestimate the failure load for larger elements (Figure 27).

3.3.2. Modes of failure The beam exhibits ductile behaviour since it fails in flexure. This failure mechanism is not captured by the damage model because of its formulation (isotropic scalar damage model). Furthermore the experimental crack pattern at the failure load indicates the formation of a flexural-shear crack and the bursting of the compressed concrete at midspan. This shear failure is associated with the development of tensile stresses within the compressive zone of the beam due mainly to the complete degradation of the bond between concrete and tensile reinforcement [58]. The pronounced discrete crack observed in the test can be also captured by the 2D elasto-plastic model by selecting only the cracks with large normal crack strains (Elm > 0.007). This proves the ability of the smeared crack approach to perform qualitative prediction of crack localization and the evolution of discrete cracks. Another phenomenon which is often neglected in finite element analysis arises at the failure of the beam, namely, the plastic buckling of the longitudinal bars in the compressive zone. This type of buckling initiates flexural failure of a majority of the beams but it is impossible to predict exactly the flexural displacement that will be required to cause buckling of longitudinal reinforcement. Several investigations indicate that this type of buckling is influenced by the size and the spacing of the stirrups, the axial load, the shape of the stress-strain steel curve in the inelastic range and the amplitude of the flexural displacement of the beam [59-61]. Consequently the post-peak behaviour of reinforced concrete beams predicted by finite element models merits closer attention and should be interpreted with care. 3.3.3. Modelling of bond-slip In this analysis, the main longitudinal reinforcement could be reasonably modelled as having a perfect bond due to its good anchorage. The calculation carried out with the joint- interface element, based on a contact approach (presented in 2.2.) indicates a very similar global response (Figure 27), with a lower bound effect which does not exceed 5%, a maximum slip of l mm at the peak load, and small disturbances on the distribution of strains in the steel. Only sliding between steel and concrete has been predicted by the model without opening situations at the joint plane. However the investigation of Stevens et al. [56] on a similar beam with imperfectly anchored reinforcement, including bond interface elements, has shown a very significant deviation from the perfect bond response as soon as the diagonal cracks open, with an ultimate load reduction of about 27%. Ahmad and Bangash [62] carried out 3D finite element analysis on a scale model of an octagonal prestressed concrete slab with bonded tendons subjected to an increasing load up to failure. Three calculations have been done with partially bonded, perfectly bonded or unbonded

305 tendons. The simulations indicated a fairly close agreement between numerical and experimental values for the two first cases with a difference on the ultimate strength of 1.4%. The third case of the unbonded slab has shown a significant lower bo6nd phenomenon but a correct prediction of the failure load. Clearly, in numerical studies on reinforced concrete beams or slabs subjected to quasistatic monotonic loading up to failure, it is not crucial to include bond-slip effects when the main reinforcing bars are correctly anchored However, for structures submitted to general alternate cyclic loadings like those induced by earthquake, the deformation mechanisms are extremly complicated Indeed, after several cycles the loss of rigidity of the structure is due m part to the slippage of the reinforcement and the steel-concrete bond can be completely reduced to a simple frictional contact. 3.4. Structures subjected to repeated reversed loads When a structure is subjected to a strong earthquake, and goes through several cycles (1015 cycles) with high strains, a good estimation of the response should be necessary for the prediction of the importance of this damage and of its consequences on the stability of the structure. Due to the nature of the seismic motion, we need to consider the behaviour of the materials and that of the member connections under alternate cyclic loads. Much of the damage is the consequence of inadequate detailing that results in the absence of sufficient ductility capacity to withstand the imposed inelastic displacements, shear failure, and anchorage failure of longitudinal reinforcement in the plastic hinge regions. A beam-column subassemblage, for example, must satisfy two basic requirements: 1) avoid brittle failure due to diagonal shear and 2) maintain ductility of the connections. Hence, the structure should be able to undergo large inelastic deformations without substantial loss of strength. 3.4.1. Constitutive cyclic law The constitutive law for cyclic effects in concrete developed by Merabet et al. [46,51] is shown in Figure 29. This model allows a correct description of the nonlinear behaviour of concrete, the loss of stiffness, the permanent strains and the stiffness restitution during reversed loading. As soon as a crack starts to close, the concrete develops some compression, according to the imperfect contact of the crack surfaces.

1- Elastic tension 2,7- Crack opening 3,8- Crack closing 4- Non-linear compression 5- Damaged unloading (modulus E2) 6- Damaged unloading (modulus E I} 9- Linear compression (modulus E2) ~ 9

.'".71('12 ''~1

" 3 ' -ft

Figure 29. Cyclic law for concrete

l

tc/E0

306 The unilateral behaviour of the cracked concrete is one of the most important mechanism which has to be modelled if a good estimation of the energy dissipated throughout the loading cycles is expected. The modelling of steel can be described with the real stress-strain computed point by point. The hardening has up to now been assumed to be isotropic. 3.4.2. General input data

The numerical investigation presented here concerns a beam-column subassemblage subjected to reversed cyclic loads, tested by Deltoro-Rivera [63]. The specimen considered is an interior joint of a 4 m bay-2 m storey flame. The geometric characteristics and the reinforcement layout are described Figure 30. Basic concrete parameters are : E=37000MPa, v=0.2, compression strength 40MPa, tensile strength 4MPa. The steel characteristics are : E=200000MPa, yielding limit: 490MPa ((I)12), 440MPa ((I)14), 550MPa ((I)20). The boundary conditions are defined so that the test can reproduce the response of such a joint in a real building (Figure 31). First, a vertical prestressing (200kN) of the column is applied and an initial flexure is introduced by loading vertically (22kN) the end of the beams, in order to simulate the dead weight of the building and of the service loads on the floor. The vertical displacements of the end of the beams are prevented and an alternating horizontal displacement is imposed to the base of the column simulating the seismic action. One cycle (+33mm/-33mm) is carried out first to estimate the load-carrying capacity of the structure in order to define the cyclic loading procedure. The loading consists of a series of cycles of increasing amplitudes, a set of five cycles is carried out in order to check the stability of the structure properties under repeated loading. The first set of cycles, having an amplitude of d = 10 mm represents a service load, the following sets (d- 13, d- 26 mm) bring the structure into the post-elastic domain.

Shear ~| 1 reinforcement

2HA 0 20 * 1HA 0 12

2 H A 0 1 2 # 1HA0 14

60

..

South

North

....... 60

iJ H I

Beam section

.......

~

Column section

Figure 30. Beam-column subassemblage [63]

Figure 31. Boundary conditions

307 3.4.3. Numerical results

Since numerical simulation of cyclic loading is very computationally extensive, we limit the study to one cycle for the two first amplitudes (d= 10 and 13 mm) and a maximum of three cycles at the third amplitude (d=26mm). Three calculations are performed [64] using first the beam model and subsequently the membrane elasto-plastic model with two assumptions : with or without bond-interface elements between steel and concrete. The bond-interface elements are introduced only in the main horizontal reinforcing bars with the following characteristics (Figure 5.b and paragraph 2.3.) for the joint 9xl=lS.7MPa, x3=5.SMPa, S]=lmm, S2=3mm, S3=10.5rmn, for the beams: "r,]=SMPa, x3=0.1MPa, Sl=0.3mm, S2--0.31mm, S3=lmm. The global response of the subassembly concerns the history of the horizontal displacement d at the base of the column, versus the horizontal load H. The load-deformation characteristics of the structure under reversal of loading are shown in Figures 32 and 33. The predicted yield displacement (13 mm) and the global responses for the two first cycles (d=10 and 13mm) obtained by both three models are in good agreement with test results with no significant bondslip effects. Note that the test results plotted Figure 32 are related to the fifth cycle at the level d=13mm and the energy-dissipating capacities of the structure are function of the area of the hysteresis loops. At the third amplitude d=26mm (Figure 33), the slopes of the curves are considerably affected under the repeated loading. The stiffness degradation shown by the pinching of the hysteresis loops (characterised by reversed curvatures) and the loss of strength are mainly due to the bond deterioration under repeated and reversed loading [65]. These phenomena are correctly captured by the bond model while the calculations without bond-slip effects leads to stabilized cycles after the second cycle with the same amplitude. A maximum calculated slip of 1 mm is obtained within the joint. 9

TEST

LOAD (kN) 150

BEAM MODEL .....

PLASTICITY-2D with bond elements

1 O0

PLASTICITY-2D perfect

50 ~

ortd

~

"

-'5

-10

~

, ~ i

i

9

,

I 5

I

I

10 15 DISPLACEMENT (mm)

-loo -150

Figure 32. Global response at the amplitude d=13mm (predicted and test results) Figure 34 shows the evolution of cracking and the importance of the distorsion in the beamcolumn joint region obtained with the 2D plasticity models. Cracking results from both flexure in the beams and the columns, particularly in the vincinity of the joint, and of shear within the joint. However the test results only indicate a limited number of flexural cracks in columns immediatly above and below the joint. With the bond model the total dissipated energy through

308 cycling is partly due to the friction between the steel and the concrete and consequently this leads to reduced cracking (Figure 34). With the layered beam model, the joint behaves as a rigid bloc (conservation of the fight angle between the beams and the columns), and the global displacement obtained is entirely related to the overall flexure. Figure 35 indicates reasonable trends between measured and predicted stresses along the length of top and bottom longitudinal bars at d=+l 9mm especially with the bond model. LOAD (kN)

LOAD (kN) K , w

PLASTh ;/TY-2D

I'50

..^

//

, w

50

-:o ~ o

30

.z.~~

~3 20 DISPLACEMENT(ram)

-15C,

LOAD (KN)

LOAD (kN) L W

PLASTICITY-;tD ~" perf ect bo, ,d ' ~

BEAM I~fODEL

/,'~

i~., i

/

w

l-^

9

~" .

/

o /; -:o

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~

~

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-: 0

.,

DISP! ACEMENr (ram)

,~~.~

~

DI,c PLACEMI NT (mm)

....

--

- v , ~ J

~

Figure 33. Global response at the amplitude d=26mm (predicted and test results)

..................... : %:. ~ B o n d Elements .f

/ 1--/

Figure 34. Predicted crack patterns (with (a) and without (b) bond interface elements)

.

309 STRESS(MPa) ------o .

.

.

.

TEST

400

PLASTICITY (W'ah Bond Elements)

.

~ o~

/

PLASTICITY

(a) ....

..

i ~...~,',~:_'._.. ,~_r -800~

"~n.--~

~

~

i

0

-200

I

I

200

400

I

I

600 800 BEAMS X-axis (ram)

-100

-200 STRESS(MPo) o

TEST PLASTICITY (With Bond Elements)

t \

BEAMS X-Ioxis(ram) ;

"

-800

-600

(b)

PLASTICITY

\lO(t t,. I

I

-400

-200

'~.1

"

,"~._

i

-1O0

I

400 ~ ' - -

-200 -300

"\',- -

-400

\/

\

.

/

.

\"

I

/

I

800

I-

\i V

-5

Figure 3 5. Stress distribution along flexural reinforcement in top (a) and bottom (b).

CONCLUSION The work presented in this study concerns the behaviour and modelling of the steelconcrete interface, with an attempt to evaluate its influence in a number of real-life situations. Following a description of the phenomenology of the steel-concrete interface behaviour, three modelling techniques are presented. The first aims to implement a classical approach using bond-interface elements in conjunction with a monotonic bond-slip law. The second shows how a "contact element" type of solution, simply equipped with a failure criterion for slippage and loss of contact could improve the modelling process. Finally, the third describes the very significant contribution of a cyclic interface law integrated within a membrane element, and customized to a class of problems. Numerical analyses of typical reinforced concrete structures are then performed in order to evaluate the relevance of these models. The results of computational models are compared to tests performed on tensile members, beams subjected to bending and beam-column subassemblage.

310 With the objective of the accurate modelling of the dissipation due to the degradation of bond between steel and concrete, these models take into account the distributed characteristics of the interface laws and adapt in the best possible way to the specific features of the different problems. The first two models proposed are supported by general laws, the parameters of which need to be identified by standard tests. The third is completely associated to the beamcolumn joint. The latter option seems both efficient and reliable. Classes of problems can then be defined, making clear distinctions between the ones related to monotonic loading and those related to cyclic loading. For monotonic analysis up to failure, if the anchorage zone is correctly designed, the influence of the steel-concrete bond is negligible. Indeed, slippage is recorded from the very beginning of cracking, but it practically does not influence neither the local nor the global behaviour. Near failure, a number of phenomena concomittant to cracking (buckling of the reinforcing bars, dowel action) and usually neglected begin to dominate. Their influence is ot~en of the same order of magnitude than that of the loss of bond. In these circumstances it is difficult to separate the small influences of each aspect. When cyclic analysis is performed, particularly for beam-column joints, calculations have shown a small influence of the effect of bond during a half cycle (loading and unloading). On the other hand, when the steel-concrete interface is modelled, the analysis of a complete cycle allows a better estimation of the dissipated energy and a more accurate description of the cycle's shape (pinching of the hysteresis loops). In the same way, only this modelling is able to capture the drop of the peak and the losses of rigidity through cycling. Nevertheless, an increase of about 30% of the computation time is observed. It can be concluded that when performing cyclic or dynamic analysis of reinforced concrete structures, it seems absolutely necessary to take into account the steel-concrete bond in order to obtain reliable results.

REFERENCES

1.

2. 3.

4. 5. 6.

7. 8.

K. DOrr, Krat~ und Dehnungsverlauf von in Betoncylindem zentrisch einbetonierten Bewehrungsstfiben unter Querdruck, Forschungberichte aus dem Institut ftir Massivbau de Technischen Hochschule Darmstadt, n~ (1975). L.J. Malvar, Bond of reinforcement under controlled confinement, ACI Materials Journal, 89 (1992) 593-597. C. Laborderie and G. Pijaudier-Cabot, Influence of the state of stress in concrete on the behaviour of the steel concrete interface, proc. of First International Conference on Fracture Mechanics of Concrete Structures, Z.P. Bazant ed., Elsevier (1992). B.P. Hughes and C. Videla, Design criteria for early-age bond strength in reinforced concrete, Materials and Structures, 25 (1992) 445-463. H.M. Abrishami and D. Mitchell, Simulation of uniform bond stress, ACI Materials J., 89 (1992) 161-168. J.A. Den Uijl, Background of the CEB-FIP Model Code 90 clauses on anchorage and transverse tensile actions in the anchorage zone of prestressed concrete members, Bull. CEB 212 (1992) 74-94. M.A.F. Ismail and J.O. Jirsa, Bond determination in reinforced concrete subjected to low cycle loads, ACI Journal, 69 (1972) 334-343. A.D. Cowell, E.P. Popov and V.V. Bertero, Effects of concrete types and loading conditions on local bond slip relatioships, Earthquake Eng. Research Center, UCB/EERC82/17, Univ. of California, Berkeley, (1982).

311 9. 10. 11. 12. 13.

14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

24. 25.

26. 27. 28.

29. 30.

D.Z. Yankelevsky, M.A. Adin and D.N. Farhey, Mathematical model for bond-slip behaviour under cyclic loading, ACI Struct. Journal, 89 (1992) 692-698. R.Eligehausen, E.P. Popov and V.V. Bertero, Local bond stress-slip relationships of deformed bars under generalized excitations, Earthquake Eng. Research Center, Report UCB/EERC-83/23, Univ. of California, Berkeley (1983). K.H. Reineck, Ultimate shear force of structural concrete members without transverse reinforcement derived from a mechanical model, ACI Struct. J., 28-5 (1991) 592-602 H. Nilson, Internal measurement of bond slip, ACI Journal, 69-41 (1972) 439-441 G. Utescher and H. Herrmann, Experimental evaluation of the ultimate capacity of smooth round dowels embedded in concrete and made of austenitic stainless steel (in german), report 346, DAfSt, Berlin (1983) 49-104. M.D. Prisc and A. Gandelli, A new experimental approach to the investigation of contact forces at an interface, Materials and structures, 26 (1993) 214-225. D. Wastein, Distribution of bond stress in concrete pull-out specimens, ACI Journal, 18-9 (1947) 1041-1052. M. Wicke, CEB-FIP Model Code- Serviceability Limit States, Bull. CEB 217, Selected justification notes,(1993) 111-136. T.P. Tassios and P.J. Yannopoulos, Analytical studies on reinforced concrete members under cyclic loading based on bond stress-slip relationships, ACI Journal 3 (1981) 206-216 S. Somayaji and S.P. Shah, Bond stress versus slip relationship and cracking response of tension members, ACI Journal 3 (1981) 217-225 R. Eligehausen and G.L. Balzas, Bond and detailing, Bull. CEB 217, Selected justification notes (1993) 173-226. E. Cosenza, C. Greco and G. Manfredi, The concept of"equivalent steel", Bull. CEB 128, Ductility-Reinforcement (1993) 163-198. A.K. De Groot, G.M.A. Kusters and T. Monnier, Numerical modelling of bond slip behaviour, Heron publication 26, Deltt, (1981). D. Ngo and A.C. Scordelis, Finite element analysis of reinforced concrete beams, ACI Journal, 64-3 (1967) 153-163. M. Dragosavic and H. Groenveld, Bond model for concrete structures, Int. Conf. on Computer Aided Analysis and Design of Concrete Structures, F. Damjanic et al., ed., Pineridge Press, (1984) 203-214. B.E. Heuze and T.G. Barbour, New models for rocks joints and interface, ASCE Proc., 108 (1982), 757-776. A. Millard, G. Nahas and C. Douat, Stabilit6 des galeries de stockage pour drchets radioactifs- Amrliorations apportres aux 616ments joints du programme INCA, Report CEA-DEMT 85/417, Saclay, France (1985) R. Tepfers, Cracking of concrete cover along anchored deformed reinforcing bars, Magazine of Concrete research, 31-106 (1979) 3-12. D.H. Jiang, S.P. Shah and A.T. Andonian, Study of the transfer of tensile stress by bond, ACI Journal, 81-24 (1984) 251-259. M. Lorrain and P. Bravi, Etude de rinfluence de la nature du brton sur les dimensions de l'eprouvette pour ressai d'arrachement direct a moule cylindrique permanent, Report GRECO, F. Darve ed., IMG Press, Grenoble, France (1989) Z. EI-Hajjar, Contribution h la moddisation numerique des phrnomrnes de couplages entre milieux 61astiques ou 61astoplastiques, Doctoral Dissert. INSA, Lyon, France (1988) O. Merabet, Modrlisation des structures planes en brton arm6 sous chargements monotone et cyclique, Doctoral Dissertation, INSA, Lyon, France (1990).

312 31. N. Okamoto and M. Nakazawa, Finite element incremental contact analysis with various frictional conditions, Inter. J. Numer. Engrg., 14 (1979) 337-357 32. T.D. Sachdeva and C.V. Ramakrishnan, A finite element solution for the two-dimensional elastic contact problems with friction, Inter. J. Numer. Engrg., 17 (1981) 1257-1271 33. G. Beer, An isoparametric joint/interface element for finite element analysis, Inter. J. Numer. Engrg., 21 ( 1985) 585-600 34. S. Viwathanatepa, E.P. Popov and V.V. Bertero, Effects of generalized loadings on bond of reinforcing bars embedded in confined concrete blocks. Report No. UCB/EERC-79/22, University of California, Berkeley, California, U.S.A. (1979) 35. V. Ciampi, R. Eligehausen, V.V. Bertero and E.P. Popov, Analytical model for concrete anchorages of reinforcing bars under generalized excitations. Report No. UCB/EERC82/23, University of California, Berkeley, California, U.S.A. (1982) 36. G. Nahas, Calcul b, la mine des structures en b6ton-arm6, Doctoral Dissertation Universit6 Paris 6, France (1986) 37. C. Lepretre, Calcul/l la mine des structures en b6ton-arm6, Report DEMT 88/330, CEA Saclay, France (1988) 38. J.L. Cl6ment, J. Mazars and A. Zaborski, A damage model for concrete reinforcement bond in composite concrete structures, Struct. and Crack propagation in Brittle Matrix Composites, A.M. Brandt, ed., Elsevier, New York (1985)443-454. 39. J. Mazars and G. Pijaudier-Cabot, Continuum damage theory: application to concrete, J. Engrg. Mech., ASCE 115-2 (1989) 345-365. 40. G. Pijaudier-Cabot and Z.P. Bazant, Non local damage theory, J. Engrg. Mech., ASCE, 113-10 (1987) 1512-1513. 41. G. Pijaudier-Cabot, J. Mazars, and J. Pulikowski, Steel-concrete bond analysis with non local continuous damage, J. Struct. Engng ASCE 117-3 (1991) 862-882. 42. J.L. Cl6ment, Interface acier b6ton et comportement des structures en b6ton arm6. Caracterisation, mod61isation, Doctoral Dissertation, University of Paris 6, France, (1987). 43. K. Adrouche, M. Lorrain, Influence des param6tres constitutifs de l'association acier b6ton sur la r6sistance de l'adh6rence aux chargements cycliques lents, Materials and structures, France, 20 (1987), 315-320. 44. N.S. Ottosen, Material model for concrete, steel and concrete-steel interaction, Bull. CEB 194, Modelling of structural reinforced and prestressed concrete in computer programs, (1990) 47-67. 45. P. Desayi, Determination of the maximum crack width in reinforced concrete members, ACI Journal, 73 (1979) 473-447. 46. O. Merabet, M. Djerroud, I. Chahrour and J.M. Reynouard, D6veloppement d'un mod61e semi-global pour le calcul des systemes de poutres en b6ton arm6 sous chargements altem6s cycl6s, Report GRECO, J.M. Reynouard ed., INSA Press, Lyon, France (1991) 47. O. Merabet and J.M. Reynouard, Mod61isation des structures planes en b6ton arm6 sous chargements monotone et cyclique, Annales des Ponts et Chauss6es, Paris, France, 4 (1990) 21-41. 48. Z.P. Bazant et al., Finite Element Analysis of Reinforced Concrete, American Society of Civil Engineers (ed.), New York, 1982, 545 pp. 49. H.T. Hu and W.C. Schnobrich, Nonlinear analysis of cracked reinforced concrete, ACI Struct. J. 87 (1990) 199-207. 50. Z.P. Bazant and B.H. Oh, Crack-band theory for fracture of concrete, Materials and Structures, RILEM, Paris, France, 16-32 (1983) 155-177.

313 51. F. Fleury, O. Merabet and J.M. Reynouard, Nonlinear analysis of damage in reinforced concrete structures subjected to seismic loading, Proc. Fracture and Damage of Concrete and Rock, FDCR-2, 9-13 nov. 1992, Vienna, Austria, H.P. Rossmanith ed., E&FN Spon (1993) 472-482. 52. F. Fleury, N. Ile, O. Merabet and J.M. Reynouard, A R/C element for nonlinear structural seismic analysis, Proc. Structural Dynamics, EURODYN 93, 21-23 june 93, Trondheim, Norway, T. Moan et al. (eds.), Balkema, Rotterdam (1993) 161-167. 53. N. Bicanic and H. Mang (eds.), Computer Aided Analysis and Design of Concrete Structures, Proc. ofSCI-C, Pineridge Press, U.K. (1990) 1336 pp. 54. J. Tuset, Mod61es de structures en micro b6ton arm6 - Poutres isostatiques et hyperstatiques, Doctoral Dissertation, Univ. Claude Bernard, Lyon, France (1973). 55. O. Merabet, J.M. Reynouard, D. Breysse, Simulation du comportement des structures planes en b6ton arm6 sous chargement monotone, Annales I.T.B.T.P., Paris, France, 504 (1992) 95-111. 56. N.J. Stevens, S.M. Uzumeri, M.P. Collins and G.T. Will, Constituve model for reinforced concrete finite element analysis, ACI Struct. J., 88 (1991) 49-59. 57. Z.P. Bazant, J. Pan, G. Pijaudier-Cabot, Sot~ening in reinforced concrete beams and frames, J. Struct. Engrg., ASCE, 133-12 (1987) 2333-2347. 58. M.D. Kotsovos, Behaviour of beams with shear span-to-depth ratios greater than 2.5, ACI Journal, 83 (1986) 1026-1034. 59. C.F. Scribner, Reinforcement buckling in reinforced concrete flexural members, ACI Journal, 83 (1986) 966-973. 60. S.T. Mau, Effect of tie spacing on inelastic buckling of reinforcing bars, ACI Struct. J., 87 (1990) 671-677. 61. M. Papia, G. Russo and G. Zingone, Instability of longitudinal bars in reinforced concrete columns, J. Struct. Engrg., ASCE, 114 (1988) 445-461. 62. M. Ahmad and Y. Bangash, A three-dimensional bond analysis using finite elements, Computers and Stuctures, 25 (1987) 281-296. 63. R. Deltoro-Rivera, Comportement des noeuds d'ossatures en b6ton arm6 sous sollicitations altem6es, Ecole Nationale des Ponts et Chauss6es, Paris, France (1988). 64. O. Merabet, M. Pinto-Barbosa and J.M. Reynouard, Simulation of reinforced concrete beam-column subassemblage subjected to alternate loading, Proc. of European Workshop on Semi-Rigid Behaviour of Civil Engrg. Structural Connections, COST-C1, 28-30 oct. 1992, Ecole Nationale des Arts et Industries, Strasbourg, France (1992) 308-318. 65. M. Seckin and H.C. Fu, Beam-column connections in precast reinforced concrete construction, ACI Struct. J., 87 (1990) 252-261.

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MECHANICS OF ROCK AND CONCRETE JOINTS

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 1995 Elsevier Science B.V.

317

Mechanics of Rock Joints: Experimental Aspects L. Jing and O. Stephansson Royal Institute of Technology (KTH), S-100 44 Stockholm, Sweden

The laboratory test techniques and major results about the mechanical, hydraulic and thermal properties of rock joints are reviewed in this paper. Starting with an introduction to the joint roughness, the normal loading-unloading tests, direct shear tests under constant normal stiffness or stresses and the rotary shear tests with hollow cylindrical samples are reviewed in detail, followed by a brief introduction to the coupled thermo-hydro-mechanical behaviour of rock joint fluid flow during laboratory tests. Comparison is made between different test techniques and the needs for future study are discussed at the end.

1. INTRODUCTION The determination of physical properties of rock joints is of utmost importance for the characterization of jointed rock masses, formulation of constitutive models for rock joints and jointed rock masses, mathematical simulation of practical problems and rock engineering design. The chief method to obtain first hand information about joint behaviour is the laboratory tests. Field tests are performed to obtain knowledge about the joint behaviour at specific sites, but laboratory tests are most commonly used in practice. The joint behaviour is tested on rock samples containing a single joint. The most widely used laboratory test techniques are the roughness measurement and characterization, normal loading-unloading test with uniaxial or triaxial material test machines, direct shear test under constant normal stresses or normal stiffness, rotary shear test with hollow cylinder samples, and heated normal/shear tests with or without fluid flow for the coupled hydro-thermo-mechanical behaviour of rock joints. The loading condition could be static or dynamic, but mostly static in which the load-ing rate is maintained low and steady to avoid dynamic effects. In this article, following a brief discussion on the joint roughness characterization, the commonly used techniques and chief results of normal loading-unloading, direct shear under constant normal stresses and stiffness, and rotary shear tests are reviewed in detail, followed by a brief introduction to the heated direct shear test and heated normal stress-flow tests. The stress-flow tests without heating for studying conductivity of deformable joints and the dynamic effects are presented elsewhere in this book and will not be repeated here. Field experiments are important but have to be excluded because of the space limit. Comments on the merits and shortcomings of each testing method are given and the needs for further research are discussed. Due to the ever-growing literature and interest in the experimental study of rock joints, the review has to be very selective.

318 2. MEASUREMENT AND CHARACTERIZATION OF JOINT ROUGHNESS A rock joint is composed of two opposite surfaces of very complex topography due to numerous asperities of different dimensions. This morphological feature is called the roughness of the rock joint and is the major reason behind the complexity in mechanical, hydraulic and thermal behaviour of the rock joints. The characterization of the surface roughness remains one of the most important and challenging aspects in the study of rock joints. In practice, the roughness of rock joints can be measured by means of line profilometry, compass and disc clinometry and photogrammetry in laboratory and/or in situ scales [ 1]. In line profilometry, the roughness of rock joints is represented by an angle i (Figure 1), denoting the slope angle of the representative asperities in appropriate scales. The roughness of rock joints is scale-dependent, i.e. the value of the angle i is different at different scales. The low frequency "waviness" is more and more dominant with the increase of the joint size. Roughness of rock joints may also be anisotropic, i.e. different values of angle i may be found in different directions in the plane of the joint surface (Figure 2). Scale-dependency and anisotropy of the roughness are the major contributors to scale effects and anisotropy of the mechanical properties of rock joints, e.g. the shear strength and friction. A convenient laboratory technique to measure the anisotropic roughness of rock joints is the topographic recording of joint surfaces by mechanical or optical methods used commonly in mechanical engineering and the line profiling in different directions (Figure 3). The most widely used measure to quantitatively characterize the joint roughness is perhaps the JRC (Joint Roughness Coefficient) with special consideration for scale effect [2]. The JRC of a rock joint is usually determined by laboratory tilting test, given in the form ( JRC) ~ =

r - ~r log,o[(JCS)o/ aO ]

\

1 L a b o r a t o r y test 2 In situ test /

1

~-~ ~ . ,

"-

2

(1)

RoughnessProfile :

~~Q~mu,

"" ,j'~

i

".k

' ~O'~ y~ ,,,~

"i.

'

Figure 1. Different joint roughness measured with different scales. Waviness can be characterized by the slope angle i ([1]).

319

dip,. reaalng

C l a r c o m p a s s a l l o y disc ~ / ~

'\

I

i

-

5c 10cm ............... 20 cm -

-

-

....

20 ~....~ ~

,o

~03o4'0 ~

plate diameter cm

-

direction of 40 cm potential sliding -40

Figure 2. A method to measure and present the anisotropic roughness of rock joints in threedimensions with Clar compass and clinometer fixed on discs of different diameters ([ 1]).

Figure 3. A mechanical surface recording device (a), a measured three-dimensional surface of a rock joint by the device (b) and one of the profiles (c).

320 where (JRC)0 and

(JCS)o are the JRC of the joint sample at laboratory scale and the Wall

Strength of rock joints, respectively. ~ 0 is the normal stress exerted on the joint by the self weight of the upper half of the joint during tilting tests and r and r are the inclination angle of the upper half of the joint when slipping occurs during the tilting test and the residual friction angle of the joint sample, respectively. The scale effect of the roughness is considered by an empirical model

(JRC),, = ( ]RC)o[L,, I Lo]--0.02(JRc)0

(2)

where (JRC),, is the roughness of the joint at field scale of length L~ and L0 is the length of the joint sample at laboratory scale. The concept of JRC has the advantages of a simple form, easy to perform the necessary tilting tests in laboratory and explicit consideration of the scale effect. However, some different joints may turn out to have the same JRC value by tilting tests, but may have other different mechanical or hydraulic properties as a result of subtle surface differences. Secondly, the empirical relations for scale effect are valid only at scales which can be covered in laboratory tests. The JRCs for joints at field scales beyond the laboratory limit have to be extrapolated and the validity of the empirical constant in the model cannot be guaranteed. JRC is an experimental measure based on laboratory tilting tests; other analytical measures have also been invented to characterize the roughness of rock joints by using spectral analysis ([3, 4]) or fractal dimension (D) ([5, 6, 7, 8, 9]), based on the statistical, spectral or fractal analysis of the line profiles without mechanical tests. Some of these analytical models can be correlated to the JRC through empirical relations as listed below: 1

1) Source [3]"

JRC=32.2+32.471Oglo(Z2),

Z2 =

N(Ax) 21

N-li=l~.,(zi+l_zi)2] ~

(3)

where zi is the height of asperity at sampling point i along a profile, ~ is the length of horizontal intervals for the sampling and N is the number of sampling points.

2, Source

D-I

- -0.878 -,-37.784( 0.0,51_ ,6.930~.0.015) ~ 2

(4)

1

3) Source[8]" JRC=[ 4) Source [9]:

D-1

4.413x 10-5

~

JRC = 85.2671(D- 1) 0.5679

5) Source [5]" i = arccos(L1-~ where L is the length of the prof'fle.

(5) (6) (7)

321 The precondition to use fractal dimension is that the topographical pattern of the rough surface is self-affine. This means that the topographical patterns of rock joints are the same at all scales (the rule of self-similarity). This condition may not be satisfied by rock joints at all levels and conflicts with the observed scale-dependency of the roughness of rock joints. A common problem for above mentioned analytical measures is also the uniqueness. Due to the randomness of the asperity distributions and the uncertainties in the measurement, a value of z 2 or D may correspond to different joint surfaces whose morphological patterns are statistically equivalent, but may have different physical properties. The empirical constants in these correlation models were derived from laboratory measurements and may not be valid for rock joints at different field scales. Besides the uniqueness of the measures, all the above methods have been basically developed for two-dimensional problems based on line profilometry. To characterize roughness anisotropy, three-dimensional techniques for measurement, analysis and representation need to be developed. The roughness of rock joints also varies during a deformation process because of the accumulated damage on the asperities. Therefore, the roughness depends also on the stresses and the history ofjoint deformation. An exponential law was proposed ([ 10]) and experimentally verified ([ 11 ]) to describe the degradation of the roughness of rock joints. However, more experimental work is needed to further validate this law with consideration of the anisotropy and scale effects.

3. NORMAL LOADING-UNLOADING TEST The normal loading-unloading test is usually performed on a rock sample containing a single joint. The test is performed using an uniaxial test machine with a core sample (Figure 4a) or a direct shear test machine under normal loading with samples grouted into concrete blocks befitting the sample holder of specific dimensions (Figure 4b). A triaxial test chamber is used if very high normal load is required.

~

Normalforce

~ , Normalforce

Joint

Normalforce (a)

~

~

_Concret _ e block

Normalforce (b)

Figure 4. Principles for uniaxial normal loading - unloading tests of rock joints, a) Rock cores with a single joint tested with common material test machines, b) Rock joint samples grouted into two cement blocks and tested with direct shear machines.

322 Assuming the joint surface is nominally a flat surface of area s, the apparent normal stress o-. across the joint surface is calculated as cr, = F. / S where F. is the applied normal force. The net normal displacement (closure), u,, J of the joint is then the difference between the total displacement, u',,, and the displacement of the rock matrix, u,," i.e. u,J = u.-u. ' " (Figure 5a). Experimental results (Figure 5b) demonstrated that the normal displacement (closure) of the rock joints increases non-linearly with the increase of the normal stress magnitude and quickly approaches a limit value (Maximum closure). The unloading curve has a different rate of variation from that of the loading curve for repeated loading-unloading cycles. Each loadingunloading cycle results in a permanent residual normal displacement, indicating permanent damage on the joint asperities. An empirical model was proposed ([12]) for the relation between the normal displacement and normal stress of rock joints, given in a hyperbolic form, cr~-~_A -

( '. u,,1.j t Un--

(8)

n

where A and t are material parameters, ~ is called the initial setting normal stress and u~' is the maximum closure of the rock joints. Another hyperbolic relation was also proposed later in a simpler form ([13]), J un

(9)

o',, - a - b u ~

where a and b are material constants.

40'f~ "! 3 0 -

Solid rock

Interlocked joint

F~q

B

r n

J Un

35

,:

t ,'i" Un,,,,, sS S S t

30 o 20 -~ 15

20

o Z 0

At

25

10 5 I

0

0.04 0.08 0.10 Normal deformation, mm (a)

0.16

0

100 200 300 Closure (lam)

400

(b)

Figure 5. Behaviour of rock joints under uniaxial normal loading - unloading tests ([ 14]). a) Conceptual behaviour; b) experimental results (joint displacements only).

323 The hyperbolic relations presented above are widely accepted as proper descriptions of the joint behaviour under testing conditions in the laboratories. The geometrical and mechanical properties of the asperities are recognized as the reason for the non-linearity of the joint deformation. Considering the scale-dependency of the roughness, the sample size should also have some effect on the normal stress - normal displacement relation of rock joints, as reported in an experimental study on scale effects of rock joints[ 15]. However, the experimental data accumulated so far is still too limited to define any quantitative relation.

4. DIRECT SHEAR TESTS

The direct shear test is the most common laboratory technique to determine the shear behaviour of rock joints. The tests are performed on a direct shear machine which can supply both a normal force and a shear force (or velocity) to the joint sample (Figure 6).

\ Figure 6. Design of a direct shear test machine which can perform both monotonic and cyclic shear tests under constant normal stresses ([ 16]). 0) Steel frame; 1) Upper sample holder; 2) Lower sample holder, 3) bucket up; 4) carriage beam; 5) hydrostatic bearing; 6) ball bearing; 7) & 8) hydraulic actuator, 9) joint sample; 10) concrete grout.

Dilatancy is a particular phenomena of rough joint behaviour. It denotes the opening-up of a joint (so the increase of its normal displacement) when the asperities on the two opposite surfaces of the joint ride up over each other during shear (Figure 7a). The opposite occurs when the shear direction is reversed in cyclic shear and the normal displacement of the joint will decrease (compaction). The rate of dilatancy, d - u,J /u~, depends both on the geometry and strength of the asperities, the magnitude of the normal stress, and the history of the shear deformation. To eliminate the induced moments and maintain a constant nominal contact area during shear, the moving half of the joint sample is usually made smaller than that of the fixed half (Figure 7b and 7c).

324 The joint samples can be sheared either under constant normal forces or constant normal stiffness (Figures 8a and 8c).The constant normal stress condition represents the loading condition of a sliding block along a joint surface of a rock slope (Figure 8b). The normal dilatancy of the joints causes a lifting up of the sliding block whose self weight supplies a constant normal stress to the sliding surface (the joint). The constant normal stiffness condition represents the confinement of the rock mass surrounding a joint, such as a joint in a pillar between two rooms, Figure 8d. Neither the normal stress nor the normal deformation of the joint can remain constant due to the finite deformability of the pillar. The deformability of the pillar, however, may be taken as a constant measure.

-~Fn

~

J5F~

z ~ n

(a)

(b)

(c)

Figure 7. Dilatancy and changes of contact area during shear testing of joints, a) Definition of the dilatancy; b) Identical size for both upper and lower halves of a sample; c) Upper half with smaller cross-sectional area than the lower half.

~

Normal force J

Shear velocity i

V

i

G

~

N N - normal comp~>~onentof G

zx

(a)

A

17

Normal stiffness iii~i!~i~i~i~iiii~i~.~ii~iiiiiiiiii~iiiii~i~``..~ii~i~.~ii~ii~ii!] v iei!ii~i:.ii..".-iii::i~......................................... iii~..:;:Jiii~.:~!i!i~.-'.:~.-:.i~Shear i velocity

(b) -'~"~r~:'~'~'%-":~'~ ~ . ~ 1 ! surrounding rocks ~ii~~-~i!i~:,'.'i ~ +~ ~..~~~.~..'..::~~.'.~.'.:::::.~~:

Joint

...........:..:~.~.i.~.~.i.i.~.~.i.~.~.~.~.~.~.~.~.~.i..~.~.~.~.i.;.~.i.~........... .~.i.~.i.i.i.~.~:i.i.i.i.i.!.~.i.~.~.:

(c)

'.-'iiiii~..,,

~~"-:-~|

~,.-.,.....~ .,-~-,~t.,-~...-! 9

"~

:::-':~:::

~!

~ ~~,~,~~-'..-'.~-.'.'.-.':~.'~.,~ .~ ~.$~,~:::

":i:~$'Y"

.'::

% ~i:'~-~ .-~' ~'.,'~L~i~~ .'~ .::-4?.:~.~~s.~..::'.::.~i?~:....~ ..~'~.-.' :::..'.,~.," .,'~.:.~.i&.'

.:.'!

:.'i$:::.':::::."!:::-:::::.-:-:.-::::'.::.':i~ 1

li$i:~-~:.-':-::~::..ii:'.~..::-~~!:i~ ~!:-.'.i:i$.-'::'.'i!!:?.::?..-'.i:i:i:.%.~

~~.~:.-.-.:::::...--:::::.-.:::.:~, ~~~i~~:.,.:~i.~-"..--~~

(d)

Figure 8. Direct shear tests and natural loading conditions, a) Constant normal stresses condition; b) Constant normal stiffness condition; c) A rock slope; d) A joint in a rock pillar.

325 4.1. S h e a r s t r e s s a n d n o r m a l d i l a t a n c y u n d e r n o r m a l s t r e s s e s c o n s t r a i n t The experimental results of a flesh joint subjected to a monotonic shear under a constant normal stress is characterized by the peak and the residual shear stresses, and the shear stiffness (defined as the slope of'the shear stress curve before the peak), see Figure 9a ([2]) The peak shear stress is also called the shear strength of the rock joint The ratio between the residual shear stress and the applied normal stress is called the residual friction angle It represents the ultimate frictional strength of the joint in a monotonic shear test The maximum rate of dilatancy occurs at a shear displacement at which the peak shear stress appears After this peak point, the rate of dilatancy decreases and become zero when the residual shear stress is reached.

c1

2 3

Residual A

T I I

T I I I

I l

Dilation Contraction

w

-

12

A

~

~-'- 4 ~ - - - - ~ - - - ~

P'-

yI 9 It- U~ I i

4

~" U

Shear displacement

(a)

(b)

Figure 9. a) Shear stress and normal dilatancy versus the shear displacement for a rock joint under constant normal stress ([2]); b) Decrease of peak shear stress with increasing sample sizes ([ 13 ]).

The apparent peak shear stress may not always occur. It sometimes coincides with the residual shear stress. Based on experimental data obtained with different sizes of the joint sample of the same material, the peak shear stress is found to be scale dependent ([ 13, 15]), i.e. with the increase of the sample sizes, the apparent peak shear stress decreases and ultimately coincides with the residual shear stress (Figure 9b). While this argument may reflect the effect of the scale-dependency of the joint roughness, there may also be other reasons for the lack of an apparent peak shear stress: namely, the different normal confinement conditions (see the next section) and the initial state of the asperities, as demonstrated by results of cyclic shear tests (Figure 10). During cyclic shear tests under constant normal stresses, the shear stress curve occupies all four quadrants of the r-u, coordinate plane, compared with only the first quadrant for monotonic shear test. The shear stress curves in the quadrants I and III represent the forward

326 St (MPa) II

i~

3 I

Cycle 2 x" x.Cycle 1

2

Cycle 2

Ut (mm) -12

12

-- - - C y c l e 3

..a ~o

2,0i

mental

.

I-,~

A

i -30 -20 -10

i# | ||

10

20

B

'30U

t

-~.0. g..--

c "~2.0

III Cycle 1-3

Tangential Displacement, Ut (mm)

Un(mm) 1.0 0.8 0.6 0.4

Cycle 1

~

Cycle 2

E

,,~4.

.~ EE E~E

-0.2] -

(a)

, 6

, 9

i Ut(mm) 12

Experimental B

2. -40

, \\x~n/zy -9 -6 -3 " ~ o ~ 3 !

D

-30

-20

- l()'-'* ~"~'~20

C-2

30

D

Tangential Displacement, Ut (mm) (b)

Figure 10. Shear stress and normal dilatancy versus the shear displacement during cyclic shear tests under constant normal stresses, a) from [ 16]" b) from [ 17].

stages and the curves in the quadrants II and IV represent the reverse stages. It has been shown that the peak shear stress occurs only for the fin'st shear cycle and only during the forward stages ([ 16]). For the subsequent shear cycles, no apparent peak shear stress appears and the curves of normal dilatancy become smoother with much less small scale oscillations. It indicates that the disappearance of an apparent peak shear stress is a result of the damage of the asperities on the joint surface due to previous shear deformation during the first cycle. Another notable feature for cyclic shear tests is that the magnitude of shear stress at the reverse stages (shear stresses in quadrants II and IV) is almost constant and of less magnitude than that of the residual shear stress during forward shear. This is explained as the effects of different states of damage on the asperities for forward and reverse stages ([ 16]). The constant normal stress constraint is the most widely used test condition for rock joints. However, it represents only the loading conditions of rock joints at shallow surface environments and cannot consider the normal stress variation due to joint dilatancy due to f'mite deformability of the rock mass. This can only be achieved through shear tests under constant normal stiffness.

327 4.2. S h e a r stress a n d n o r m a l d e f o r m a t i o n u n d e r n o r m a l stiffness c o n s t r a i n t

In this test, the constraint to the normal deformation of a joint is, in most cases, imposed by the stiffness of springs installed to the test machine in the direction normal to the joint plane (Figure 11 a). The stiffness value, K, of the springs is preset so that it represents the deformability of the rock mass at different depth or under different stress states. The constant normal stress condition corresponds to K = 0 and the constant normal displacement condition corresponds to K = oo. The true stiffness of the rock mass surrounding a joint is of a finite value ranging from very low near the ground surface to very high at great depth. With an increment of dilatancy, an increment of normal stress is induced, resulting in an increment of shear stress. This is called displacement strengthening of the rock joints, a dilatancy - system stiffness- normal and shear stresses interaction mechanism. The experimental study on the effect of system stiffness on rock joints can be found in [ 10, 18, 19, 20, 21, 22, 23, 24].

(b) Figure 11. Direct shear machines with constant normal stiffness: a) a system with additional springs ([23]); b) a system with computer controlled normal loads - stiffness system without additional springs ([24]). 1 - normal load reaction frame; 2 -joint sample; 3 - load cell; 4 horizontal load actuator; 5 - horizontal load reaction frame; 6 - structural floor; 7 - bottom roller support system; 8 - horizontal load reaction frame; 9 - top support plate; 10 - load cell; 11 - normal load actuator.

During tests, an initial normal load is first applied to the sample and then the shear starts with the normal stiffness maintained constant throughout. The normal stress of the joint

328 increases significantly with the increase of the shear displacement and dilatancy, resulting in a significant increase of the shear stress (Figures 12 and 13). With an increase of the stiffness K, both the normal and shear stress increases proportionally, but the dilatancy decreases, due to the accumulated damage on the joint surface. An apparent peak shear stress occurs only when K = 0 or very low value, corresponding to a constant normal stress path. The initial normal stress affects the shear stress in the same fashion as in tests under constant normal stress conditions. Theoretically, this approach of system stiffness provides a unified basis on which the constraint conditions can be treated. However, it has received less attention in the past than might be expected, perhaps due to the special in situ tests needed to determine the deformability of a jointed rock mass required to quantify the system stiffness K.

7.0

7.0 (a)

6.0

K(kN/mm) 200

6.0 (C)

a, 5.0

:~5.0

4.0

4.0

3.0

--c~3.0

50

I,-i

K=200 kN/mm

on=l.O MPa 5.0 ~

15.5

o 2.0

,= 2.0 li'5

1.0

1.0

1.0

d,

b)

i( ~

.,,.~

0.5

- 5f 0 ~i1 5 " t I/ I

o Z

0 0

7.0 n~cl'~ . . . K--0 kN/mm 6.0 ~_\'J ---- K=50 kN/mm

_~ /

m I

m I 5 I

[ ' : ) ' - . . , - , . . . . . . , . ' - , 2.0 MPa 4"0 ~[ on=5.0 MPa ~ - ' "

!

.~ 3.0 ~

5 10 15 20 25 30 Shear Displacement (mm)

c~ 2 1.0 ~r 0 r-

:-" :.." : - ' - - ' - - ' . -'--'--" :-" :-" :.:" :.:" :.'." :-'.:.--'i'-'.:'-~-::--;:-.;:

L

cm

.o

k

:'t

.L

~ .

M 0

P

a

~

15"l~,IPa . . . . I I i I I I 5 10 15 20 25 30 Shear Displacement (mm)

Figure 12. Direct shear test of rock joints under different system stiffness ([23]): a) Shear stress vs. shear displacement; b) Normal displacement vs. shear displacement; c) Normal stress vs. shear displacement and d) The effect of initial normal stress on shear stress.

329 6 5

4 -

3 2 ,-

6 5 4

- (a) Forward

--

o 2 ~o 1

:~

1

.....

o-1 .t=

~d

/]

-

~0

-(c)

~-2 -3

I

~/2

I

1

~

0.07 _- (b) 0.06 9=-- 0.05 >,0.04 Reverse = 0.03 _~ 0.02 2 ~

1

I

'

'

1

r~_ 2 -3 _(d) ~, 0.060"07

Forward

0.05 /Jg

..,~0"040"03 Forward

-

'

o_ ,t=

|

1/~2//

~50.Ol

0 -0.01 -0.8

o

......................

'

'

'

'

-0.4 0 0.4 0.8 Shear displacement (in)

-0.01

//

everse

~ 0.02 0.01 0

/

1

/

/

.....

-0.8

..... i ...... , .....

-0.4 0 0.4 0.8 Shear displacement (in)

Figure 13. Shear load and normal dilatancy under constant normal load (F, =2.95 KPa) and normal stiffness (K=147.70 MPa/m) conditions ([24]). The initial normal load for the latter is also F,=2.95 KPa. a) Shear load vs. shear displacement under constant normal force; b) Dilatancy vs. shear displacement under constant normal force; c) Shear stress vs. shear displacement under constant normal stiffness and d) Dilatancy vs. shear displacement under constant normal stiffness.

4.3. Three-dimensional

effects

and stress-dependency

of joint

properties

The uniaxial normal loading-unloading test is a one-dimensional test and the direct shear test is two-dimensional. However, rock joints are located in a three-dimensional space and under usually general three-dimensional stress states. A rock joint in three-dimensions has 6 degrees of freedom to move or deform (Figure 14). Using a coordinate system defined on the joint plane (xz-plane in Figure 14a), the joint has three degrees of translational displacements (two in-plane displacements in x and z-directions and one normal dilatancy in n-direction, respectively, see Figures 14b and 14c) and three degrees of rotations. The moments of rotations can be resolved into a rotational friction moment in the joint plane about the n-axis (Mn in Figure 14f), and two other bending moments acting on either half of the joint about axes x and z (Figures 14d and 14e). The bending moments cause deformation of rock material in both halves of the joint, and the rotational friction moments causes rotational displacement (or deformation) of one half of the joint against another and induce a shear resistance moment on the joint surface accordingly.

330

n

I J~

J

L~

(a)

(b)

(c)

n

z

z

(d)

(e)

(f)

Figure 14 Degrees of freedom of a rock joint in three-dimensions, a) Initial state; b) Translations in x and z directions; c) Translation in n-direction (normal dilatancy); d) Rotation about x-axis (moment in zn-plane); e) Rotation about z-axis (moment in xn-plane); f) Rotation about n-axis (frictional moment in xz-plane).

One of the three-dimensional effects of rock joint properties is the anisotropy in its shear strength and shear deformability in the joint plane. The experimental data ([ 16, 25]) show that the shear strength and shear stiffness of rough joints are both anisotropic in the joint plane and dependent on the magnitude of normal stress (Figure 15).

z 120 ~

8p

A 90~

o---o T i l t i n g t e s t

aao

\ _.o--'~

7"

9__t 0.. = 3 MPa

~"

6 MPa

150 ~,,,,,;,,,Ji~.... r

30~ "--" 0".=9 MPa

=;/'~

"

"

1 8_.4 0 ~ .I.g

""

0~ v .~..}..u T~,72--r~A 50

- . '~, 1/ / . / - " ' "

--,5 0 ~v~/,...... : ~ , T ~ . . ` " 210~ .-

~; ,.,

240 ~ /

/

/

[

\\

\

"~1,."~3 30 ~ .;;....

.;,

7 6

~ 5 ~

,

o 0 =330 ~

3

O 0= 240 ~

~~ 2

1 0 ~~ ."0x=0 = 2180 0= 150 ~ 0= 120 ~ 9 0= 90 ~ a0 60 ~

9

. l

=

o

" "-"""

270~r - -

"" 300 ~

(a)

0

I

2

3

4

5

6

7

8

9

N o r m a l stress o ( M P a )

(b)

Figure 15. Experimental results about anisotropy and stress-dependency of joint properties ([ 16]). a) A polar diagram showing the anisotropy in the shear strength of a joint; b) Shear stiffness vs. normal stress magnitude.

331 The test was performed on concrete replicas of rock joints under constant normal stresses so that the initial surface state of joint samples can be kept the same for repeated tests. For the shear strength of joints, the degree of anisotropy decreases with increase of normal stress magnitude. For shear stiffness, both the degree of anisotropy and the stiffness value increase with increase of normal stress magnitude. This is also a manifestation of the anisotropy and stress-dependency of the joint roughness.

5. ROTARY SHEAR TESTS Direct shear tests are most widely used because it is relatively easy to build up the test equipment and perform the tests. However, there are some limitations with them: a) the contact area of the joint during shear changes, though the value of the nominal contact area can be constant, b) stress concentration always occurs at the front and back edges of the moving half of the joint sample during shear and may exert some unfavorable effects on the test results, c) the shear stress is not likely to be uniform over the joint surface during shear. d) the shear displacement is very limited (usually under 30 - 50 mm) which is undesirable if large shear displacement is required (e.g. to study the roughness damage). These limitations can be, at least partially, reduced by using rotary shear tests. Torsional or hollow cylinder shear tests have been performed on soil since early 30's. The first rotary shear tests on rock joints appeared 40 years later ([26, 27]). It has recently obtained a renewed interest for study of rock joints ( [28, 29, 30, 31, 32]). A sample of a rock joint is made into a hollow cylinder with the joint oriented perpendicularly to the axis of the cylinder (Figure 16a). The specimen is loaded in the normal direction by a constant normal load and a torque is applied to rotate one half of the specimen (the other half is fixed), thus to mobilize shear resistance on the joint surface (Figure 16b).

~N

IkN

T - Torque N - Normal force Pe, P i - Confining pressure

(b) Figure 16.(a) Hollow cylinder specimen for rotary shear tests ([30]); (b) Test principle.

The advantages of the rotary shear tests are: a) the shear displacement can be infinitely large; b) the normal stress across the joint can be much higher than that used for direct shear stress because the contact area is much smaller; c) the actual contact area is the same

332

throughout the test; and d) it can be combined with confining pressure (applied both inside and outside of the thin-walled cylinder) so that a general stress state can be applied to the rock sample. The direction of the shear stress on the joint surface is circumferential, so it varies at every point. The magnitude of the shear stress is assumed constant in the radial direction (because the thickness of the cylinder is usually small) and is given by ([31 ]) r=

3T

(lO)

27t(b 3 - a 3)

where T is the torque, a and b are the inner and outer radius of the cylinder, respectively. Figure 17 illustrates two systems for rotary shear tests of rock joints under constant normal stress, one without confining pressure (Figure 17a) and another with confining pressure (Figure 17b).

|

(a)

(b)

Figure 17. Rotary shear test system, a) Without confining pressure ([30]); (b) With confining pressure ([32]). A - normal load control (hydraulic jack); B - centralizing plate; C - vertical displacement transducer; D - upper sample holder; E - water bath; F - lower sample holder; I - rotational displacement transducer; J Thrust race; K - gear box shatt; L - Axis of machine; M - base plate; N - lower half of the sample; O -joint surface and infilling material; P - upper half of the sample; Q - torque arm; R - load beating plate; 1 -load cell; 2 - air relief; 3 - upper sample holder; 4 - cell ring; 5 - pore pressure; 6 - cell drain; 7 - bushing; 8 - 3" diameter shatt; 9 - fluid containment; 10 - MTS linear/rotary actuator; 11 - pore pressure and inst. blocks; 12 - lower sample holder; 13 - acrylic tube; 14 - tie rod; 15 -joint sample; 16 cell cap; 17 - knob; 18 - joint; 19 - O-ring. -

333 Figure 18a shows a typical recording of shear stress versus shear displacement (ranging from 0 to 860 mm) of a rotary shear test with an artificially profiled sample of uniform asperities subjected to a constant normal stress of 500 KPa. The regular undulation of the curve is caused by the regular waviness of the asperities. The gradual decrease of the undulations indicates the degradation of the surface roughness. Another example is given in Figure 18b in which mate-rial between the two opposite surfaces of the artificial joint was losing continuously during test so that the normal contraction continues with increase of circumferential shear distance. 800 400 ~ 0 0 i ~ 800

A

20

40

60

80

, 100 120 140 160 180 200 220 240 260 280 300 Shear displacement - mm

~ A ~ 40~0

tC

300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600 800 C 4 0

0

~

~

~

x

L

End of the test

0

1

I

1

I

I

600 620 640 660 680 700 720 740 760 780 800 820 840 860 (a) 2000

~

0

I

[

I

!

I

1

I

1

I

1

1

100 Shear displacement - mm

e~

"~::~ ~-~ o,,.~

I

,

I

"4~

I

I

I

I

I

I

I

I

I

I

I

I

I

1

I

I

I

I

I

I

I

I

I

200 I

I

I

I

o

z

-1.6 (b)

Figure 18. Behaviour of artificial sawcut joints during rotary shear tests ([30]). a) Shear stress versus shear displacement; b) Shear stress and normal displacement versus shear displacement.

For rotary shear tests, the roughness of the rock joints ought to be isotropic or axialsymmetric. This may not apply for real rock joints. Special techniques are also required to prepare the hollow cylindrical samples and maintain their integrity during tests. The test is, in essence, a three-dimensional one with axial symmetry.

334 All rotary tests so far were performed under constant normal stress conditions. It will be beneficial to conduct similar tests under constant normal stiffness with different initial normal loads, similar to what has been performed in direct shear tests. The special geometry of the sample allows a thermal coil to be installed around the sample so that a heating test with uniform temperature over the whole sample can be performed with or without confining pressure ([33 ]).

6. HYDRO-THERMO-MECHANICAL TESTING OF ROCK JOINTS The introduced test results so far represent only the mechanical behaviour of rock joints observed during laboratory tests under static loading conditions, in room temperature and without fluid. The behaviour of rock joints will be different when fluid and temperature gradient are present. The interactions between processes of mechanical deformation (M), fluid flow (H) and thermal transport (T) are called coupled processes of rock joints (Figure 19).

fO

HEATFLOW ~ f / ~,,~.THERMAL STRESSANDEXPANSION BUOYANCE CONVECTION f j / FRICTIONALHEAT , FLUIDPRESSURE

FLUIDFLOW (H)

L

STORAT,VITY CHANGE ~1~, JOINTDEFORMATION ] (M)

Figure 19. Coupled T-H-M processes associated with rock joints

The normal stress (deformation) - fluid flow interactions in a single rock joint have been experimentally studied since 1970's (see [34, 35, 36, 37, 38, 39]) and are the most thoroughly studied coupled H-M processes for rock joints. The basic aim of the test is to examine the fluid flow behaviour through rock joints under different normal stress conditions and develop flow models which can take the roughness of rock joints into account. The tests are concentrated on coupled normal stress- radial flow tests, but the coupled shear deformation-fluid flow tests under constant normal stresses were also reported recently ([40, 41 ]). No coupled tests of shear deformation -fluid flow tests under constant normal stiffness condition has been reported. Details about conductivity of deformable rock joints are given elsewhere in this book and will not be repeated here. The experimental studies of the mechanical behaviour of rock joints with heating are rather limited. In recently reported heated direct shear tests of both sawcut or tension-splited rock cores without fluid flow ([42]), the joint samples are heated up to a certain temperature and sheared under constant normal stresses (Figure 20). The aim of the tests is to observe the shear strength variations of the rock joints under different temperature. The inverse effect of

335

@ @ @

2.0

1.5

JRC= 8

N

JRC=

~tl

JRC = 3 JRC=0

5

t_

1.0 e-

0.5

i

0

(a)

I

I

I

100 200 300 400 Temperature (~ (b)

Figure 20. Heated shear tests of rock joints ([42]). a) Test system; b) Shear strength versus temperature for rock joints with different JRCs. 1 - normal load control; 2 - normal displacement reading; 3 - cylindrical upper heater; 4 - shear displacement reading; 5 thermocouples; 6 - temperature reading and control; 7 - lower shear box; 8 - annular space; 9 - cylindrical lower heater; 10 - shear load control; 11 - Annular space; 12 - normal load piston - base and cooling block; 13 - upper shear box; 14 - joint sample; 15 - concrete mold.

the frictional heating to the temperature field is ignored. The temperature was set to 20~

100~ 200~ 300~ and 400~ respectively, and kept constant during direct shear tests. The results indicate that the shear strength of the rock joints will increase with increase of the temperature to a peak value at a critical temperature (200 ~ in the test) and then decrease (Figure 20b). The initial shear stiffness is reported to decrease monotonically with the increase of temperature. The absolute difference in shear strength is, however, not great (about 0.2 - 0.3 MPa over a temperature span of 200 ~ ). This effect of "temperature strengthening" for shear strength and "temperature weakening" for shear stiffness was explained in [42] as a combined result of water content reduction, thermal expansion of the rock matrix and the thermally induced microcracks in the rock matrix. An earlier report on the temperature effect on the frictional property of rock joints can be found in [43 ]. An experimental study on the coupled thermo-hydro-mechanical behaviour of rock joints is reported in [44, 45, 46] in which both natural joints and extension induced cracks are subjected to normal stress loading with fluid flow and heating. The convective heat transfer through fluid flow in joints, relation between the mechanical and hydraulic apertures, joint thermal expansion and the effects of temperature on the fluid flow were investigated. The test was performed on a triaxial test system with heating facilities (Figure 21). The joint is so oriented that its two surfaces are parallel with the long axis of the sample core (102 mm in length and 51 mm in diameter) and across its diameter. The sample is subjected to hydrostatic pressure

336 and heated by raising the temperature of the confining oil. The fluid flow is kept laminar by simultaneous monitoring of the upstream and downstream flow rates under different effective normal stresses of the joint (see Figure 22a).

"~i

p

o

rl

"

I

Figure 21. Schematic arrangement of the coupled T-H-M test of rock joints ([46]). 1 - transducer; 2 - high pressure gas reservoir; 3 - upstream accumulator; 4 - cantilever device; 5 stainless steel platen; 6 - heating coil; 7 - sample sleeve (Secan rubber and head-shrink plastic jacket); 8 -joint sample; 9 - wiring; 10 - stainless steel platen; 11 - downstream accumulator; 12 - pressure transducer; 13 - wiring; 14 - adhesive type; 15 - differential pressure transducer; 16 - stand -off; 17 - joint; 18 - stainless steel net; 19 - PTFE disc.

1.6 ,_.=

"~ 1.2

c~ ~; 0.8 9

ei

20 9

,-==i

40

g

Q

9

.~.

0.4

i

0

0.0 0 0.5 1.0 1.5 2.0 2.5 Hydraulic head gradient, i (MPa/m) (a)

I

I

I

I

~I

I

0 10 20 30 40 50 Joint mechanical closure, (lam) (b)

Figure 22 Hydro-mechanical behaviour of rock joints ([46]). a) Transition from laminar to turbulent flow in joints; b) Linear relation between hydraulic aperture and mechanical closure.

337 The cubic law for fluid flow through single rock joints is found to be valid under the test conditions (up to 40 MPa of the effective normal stress, 6 MPa of the differential fluid pressure and 200 ~ of the temperature). The test results suggest that the hydraulic aperture (which was back-calculated using the cubic law) decreases linearly with increase of the normal deformation (mechanical closure) under low normal stress (< 8 MPa). By extrapolation of the test results (the solid line in Figure 22b), it was estimated that the hydraulic aperture would become zero when the mechanical closure equals the initial aperture E~. This, however, conflicts with observations from other coupled stress-flow tests (e.g. results in [34, 35, 36]) that a stress-independent residual flow rate exists even under very high normal stresses due m the tortuosity of the rough joints. The initial aperture and the normal displacement of joints are found m increase with increased temperature by this test, for both natural joints and artificially induced tension cracks (Figure 23). This increase was explained by thermal expansion of rock samples. The heat convection by fluid flow through joint is found to be significant and the heat exchange rate depends on the fluid velocity by a linear relation (Figure 24). The other findings from this test confirmed the importance of the joint roughness on the fluid flow and joint deformation, and the significance of the initial matching state of the two opposite surfaces of the rock joints. I

On, (MPa) 2 4 6

o Extension Fracture 9 Natural Joints _~.q

I

_o.8 f

,~

E

~0.4

>:

I

8

I

I

NJ2

40

at 180~

80

<120 20

60

100

140

Increase in sample temperatur, AT, (~ (a)

at 200~

180

160 (b)

Figure 23. Effect of temperature on the initial aperture (a) and the normal closure AV (b) of the rock joints ([46]).

The coupled behaviour of the rock joints is the most complex, but least experimentally investigated joint behaviour, especially under heated conditions. It is one of the keys to understand the coupled processes in jointed rock masses, fundamental for many engineering problems such as nuclear waste isolation and geothermal energy utilization. The laboratory studies so far have been dominated by the stress-flow problems, and are concentrated on the normal stress-radial flow tests. The results from the above mentioned heated tests provide us with an initial understanding of the joint behaviour under heated conditions, with or without

338 1200 - Sample temperature = 100~ Aperture, gm 91 5 . 8 5 z.j/O .,u o 16.44 _ a ~ ~o G,~ 800 9 16.96 ~ O r A21.16j~~

Sample temperature = 140~

1200 "-: e-.~

Aperture, Bm 91 4 . 5 0 r ~ o 15.50 .)7~ 9 16.54 A 18.09

~ , ~

~ ~ 800 ~~c~ ~.

91 9 . 7 ~

400

Gradient = 96.87

~

400

Gradient = 101.63

!

o

;

Flow velocity, ~, (mm/s)

o

!

1

4 6 8 Flow velocity, ~ (ram/s)

Figure 24. Linear relationship between the convective heat transfer coefficient, h, and fluid velocity in the joint at different temperature ([45]).

the presence of fluid flow. However, the amount of data may still be very limited and many influencing factors still need to be looked into: scale effect, dynamic effect, different loading conditions (a heated direct shear under constant normal stiffness constraint with fluid flow, for example), etc..

7. SUMMARY AND CONCLUSIONS The experimental study of rock joints has always been an active field of rock mechanics. The interest is expanding more and more into studies of the general physical behaviour of rock joints, not only the mechanical properties. This expansion of interest reflects the directions of the rock mechanics research towards general physics of the rock masses for which the joint is a key component. The major progress in laboratory experiment on rock joints is made in four types of testing: normal loading-unloading testing, direct shear testing ( under both constant normal stresses and normal stiffness), rotary shear testing under constant normal stress, and normal stress fluid flow interactions. Recent progress in the direct shear tests under constant normal stiffness provides a better understanding of the joint's behaviour under underground situations. Displacement strengthening is the major feature of rock joints under this loading condition, contrary to the displacement weakening under constant normal stresses at surface environments. The hyperbolic relation can be taken as a sound description between the normal stress and normal deformation of rock joints. The shear strength of the joint may be temperature dependent and the heat transfer between the rock matrix and fluid flow in the joints depends on the fluid velocity which may, in turn, depend on the mechanical deformation process. Based on the experimental data collected so far, we can be confident, at least qualitatively, that these general conclusions are valid. However, a number of uncertainties still remain about rock joints:

339 1) Roughness: All experimental findings clearly indicate that the roughness is the decisive factor for all aspects of mechanical, hydraulic and thermal behaviour of rock joints. The characterization of the roughness remains a great challenge. The strong correlation between roughness and other aspects of joint behaviour (e.g. scale effects, anisotropy, stress (path)-dependency and conductivity) demonstrates the strong requirement to represent, quantitatively and uniquely, the roughness of rock joints in both two and threedimensions. 2) Scale effect: Scale effects certainly exist in joint behaviour and is most likely a manifestation of the scale dependency of the roughness. Most of the experiments on scale effects have been performed in direct shear tests under constant normal stresses. Investigations about scale effects under other test conditions (normal compression tests, direct shear tests under constant normal stiffness, and other coupled stress - flow tests) are also needed. 3) Gouge material: The production of gouge material during tests on rock joints certainly affects the joint behaviour, especially the friction, shear strength and fluid conductivity, but research on its effect has not been successful. The difficulties lie in measuring the rate of gouge production, its distribution on the joint surface and the distribution of the actual contact area during tests. Some experimental results have shown that the gouge materials reduce the friction for rough surfaces and increase the friction for smooth surfaces. However, more experimental data are needed for more specific and quantitative conclusions regarding the effect of gouge material. 4) Three-dimensional effects: Almost all experimental studies performed so far are onedimensional (normal loading-unloading ), two-dimensional (direct shear) or pseudo-threedimensional (rotary shear) tests. The orientation of the joint surface in space, its f'mite dimension and the general stress state in situ, however, hardly justify such simplifications. The anisotropy of the joint roughness calls for the true three-dimensional experiment performed under combined shear and normal loading, under constant normal stiffness and normal stresses and other environmental conditions (e.g. temperature and fluid presence). 5) Dynarnir effects: Joint behaviour under general dynamic loading conditions cannot be included in this article due to space limitation, but its importance cannot be neglected. The experimental results demonstrated the existence of time and rate effects on friction of rock joints under constant normal stress conditions. Similar tests under constant normal stiffness with different shear velocities may be of great interest to study dynamic instabilities for underground excavations (e.g. rockbursts and mining induced seismicity). 6) Coupled processes: Tests on coupled processes with rock joints have been dominated by coupled normal stress - flow tests. Although heated shear tests and heated normal stress flow tests have also been reported, the number of tests and the data collected are not sufficient to draw quantitative conclusions or can be used to develop more general constitutive models for rock joints. More sophisticated tests on coupled behaviour of rock joints are needed to further our understanding of coupled processes of rock joints.

340 8. ACKNOWLEDGMENT The authors acknowledge that many excellent contributions to the study of rock joints cannot be included in this review, simply because of limited space. The authors are grateful to the Swedish Natural Science Research Council for financial support and to S. Bandis and M. H. de Freitas for the supply of two photographs about experimental equipment. Professor J. A. Hudson, Imperial College, London, reviewed the content of this article and upgraded the language. His kindness is gratefully appreciated.

REFERENCES 1. ISRM, Suggested methods for the quantitative description of joints in rock masses. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 15(1977) 319-368. 2. N. Barton, Rock Mechanics Review: The shear strength of rock and rock joints. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 13 (1976) 255-279. 3. R. Tse and D.M. Cruden, Estimating joint roughness coefficients. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 16(1979) 303-307. 4. S.R. Brown and C.H. Scholz, Broad bandwidth study of the topography of natural rock surfaces. J. Geophy. Res., 90 0314)(1985) 12575-12582. 5. N. Turk, M.J. Greig, M.R. Dearman and F. Amin, Characterization of rock joint surfaces by fractal dimension. Proc. 28th US Symp. on Rock Mechanics, Rotterdam, Balkema, 1987. (pp. 1223-1236). 6. J.R. Carr, Fractal characterization of joint surface roughness in welded tuff at Yucca Mountain, Nevada. Rock mechanics as a guide for efficient utilization of natural resources, Khair (ed.). 1989, (pp. 193-200). 7. Y.H. Lee, J.R. Carr, D.J. Barr and C.J. Haas, The fractal dimension as a measure of the roughness of rock discontinuity profiles. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 27(6) (1990) 453-464. 8. N. Wakabayashi and I. Fukushige, Experimental study on the relation between fractal dimension and shear strength. Proc. of the ISRM Conf. on fractured and jointed rock masses, Lake Tahoe, California, June 3-5, 1992. (in press). 9. H. Xie and W. C. Pariseau, Fractal estimation of joint roughness coefficients. Proc. of the ISRM Conf. on fractured and jointed rock masses, Lake Tahoe, California, June 3-5, 1992. (in press). 10. M.E. Plesha, Constitutive models for rock discontinuities with dilatancy and surface degradation. Int. J. Numer. Anal. Methods in Geomech., 11(1987) 345-362. 11. R.W. Hutson, Preparation of duplicate rock joints and their changing dilatancy under cyclic shear. Ph.D thesis, Northwestern Univ. Evanston, Illinois, 1987. 12. R.E. Goodman, Methods of geological engineering in discontinuous rocks. West Publishing Company, San Francisco, 1976. 13. S. Bandis, Experimental studies of scale effects on shear strength and deformation of rock joints. Ph.D thesis, Univ. of Leeds, Dept. of Earth Sciences, 1980. 14. N. Barton, S. Bandis and K. Bakhtar, Strength, deformation and conductivity coupling of rock joints. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 22(3)(1985) 121-140.

341 15. A.P. da Cunha and J. Muralha, Scale effects in the determination of mechanical properties of jointed rock masses. Proc. the ISRM Conf. on fractured and jointed rock masses, Lake Tahoe, California, June 3-5, 1992. (in press). (pp. 497-503) 16. L. Jing, Numerical modelling of jointed rock masses by distinct element methods for two and three-dimensional problems. Ph.D Thesis, 1990:90D, Lule~ University of Technology, Lule~., Sweden, 1990. 17. H.K. Kutter and G. Weissbach, Der einflues von verformungs-und elastungsgeschichte auf den scherwiderstand von gesteinskluften unter besonderer berucksichtigung der mylonitbildung, final report. DEG Research Project, Ku361/2/4, 1980. 18. L. Obert, B.T. Brady and F.W. Schmechel, The effect of normal stiffness on the shear resistance of rock. Rock Mechanics, 8(1976) 57-72. 19. W. Leichniz, Mechanical properties of rock joints. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 22(5) (1985) 313-321. 20. B. Benmokrane and G. Ballivy, Laboratory study of shear behaviour of rock joints under constant normal stiffness conditions. Proc. 30th US Symp. on Rock Mech., West Virginia Univ., 1989 ( pp. 899-906). 21. G. Archambault, M. Fortin, D.E. Gill, M. Aubertin and B. Ladanyi, Experimental investigation for an algorithm simulating the effect of variable normal stiffness on discontinuities shear strength. Rock joints, Barton and Stephansson (eds.) Balkema, Rotterdam, 1990. (pp. 141-148). 22. Y. Onishi and P.G. Dharmaratne, Shear behaviour of physical models of rock joints under constant normal stiffness conditions. Rock Joints, Barton & Stephansson (eds.), Balkema, Rotterdam, 1990. (pp. 267-273). 23. C.A. Skinas, S.C. Bandis and C.A. Demiris, Experimental investigations and modelling of rock joint behaviour under constant stiffness. Rock Joints, Barton & Stephansson (eds.), Balkema, Rotterdam, 1990. (pp. 301-308). 24. J.T. Wibowo, B. Amadei, S. Sture, A.B. Robertson and R. Price, Shear response of a rock joint under different boundary conditions: an experimental study. Proc. the ISRM Conf. on fractured and jointed rock masses, Lake Tahoe, California, June 3-5, 1992. (in press). 25. T.H. Huang and Y.S. Doong, Anisotropic shear strength of rock joints. Rock Joints, Barton & Stephansson (eds.), Balkema, Rotterdam, 1990. (pp. 211-218). 26. H.K. Kutter, Rotary shear testing of rock joints. Proc. 3rd ISRM Cong. Denver, 1974. ( pp. 254-262). 27. R.J. Christensen, S.R. Swanson and W.S. Brown, Torsional shear measurements of the frictional properties of westerly granite. Proc. 3rd ISRM Cong. Denver, 1974. (pp. 221225). 28. W.L. Power, T.E. Tullis and J.D. Weeks, Roughness and wear during brittle faulting, J. Geophy. Res., 93(B12)(1988) 15268-15278. 29. S. Xu and M.H. de Freitas, Use of rotary box for testing the shear strength of rock joints. Geotechnique, 38(2)(1988) 301-309. 30. S. Xu and M.H. de Freitas, the complete shear stress-vs-shear displacement behaviour of clean and infilled rough joints. Rock Joints. Barton & Stephansson (eds.), Balkema, Rotterdam, 1990. (pp. 341-348). 31. N. Yoshida and C.H. Scholz, Elastic properties of contacting surfaces under normal and shear loads. 2. Comparison of theory with experiments. J. Geophy. Res. 94(B 12)(1989)

342 17691-17700. 32. T.B. Reardon, E.C. Drumm and D. Lange-Kornbak, Comparison of direct shear and hollow cylinder tests on rock joints. Rock Mechanics as a Multidisplinary Science. Rogeirs (ed.), Balkema, Rotterdam, 1991. (pp. 1115-1123). 33. Z.P. Bazant, S. Prasannan, M. Hagen, S. Meiri, R. Vaitys, R. Klima and J.D. Hess, Large triaxial-torsional testing machine with hydrothermal control. Materiaux et Constructions, 19(112)(1986) 285-294. 34. J.E. Gale, A numerical, field and laboratory study of flow in rocks with deformable fractures. Ph.D thesis, Univ. of California, Berkeley, 225 pp, 1975. 35. K. Iwai, Fundamental studies of fluid flow through a single fracture. Ph.D thesis, Univ. of California, Berkeley. 208 pp, 1976. 36. P.A. Witherspoon, J.S.Y. Wang, K. Iwai and J.E. Gale, Validity of cubic law for fluid flow in a deformable rock fracture. Water Resources Research, 16(6)(1980) 1016-1024. 37. K.G. Raven and J.E. Gale, Water flow in a natural rock fracture as a function of stress and sample size. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 22(4)(1985) 251-161. 38. L.J. Pyrak-Nolte, L. Myer, N.G.W. Cook and P.A. Witherspoon, P. A, Hydraulic and mechanical properties of natural fractures in low permeability rock. Proc. 6th ISRM Cong., Herget and Vongpaisal (eds.), Montreal, Canada, 1987. (pp. 225-231). 39. A. Makurat, N. Barton, L. Tunbridge and G. Vik, The measurement of the mechanical and hydraulic properties of rock joints at different scales in the Stripa project. Rock Joints. Barton & Stephansson (eds.), Balkema, Rotterdam, 1990. (pp. 541-548). 40. A. Makurat, N. Barton and N.S. Rad, Joint conductivity variation due to normal and shear deformation. Rock Joints. Barton & Stephansson (eds.), Balkema, Rotterdam, 1990.(pp. 535-540). 41. T. Esaki, H. Hojo, T. Kimura and N. Kameda, Shear-flow coupling test on rock joints. Proc. 7th ISRM Cong. Aachen, Sep. 1991.( pp. 389-392). 42. H.A. Bilgin and A.G. Pasamehmetoglu, Shear behaviour ofjoints under heat in direct shear. Rock Joints. Barton & Stephansson (eds.), Balkema, Rotterdam, 1990 (pp. 179183). 43. R.M. Stesky, Rock Friction effect of confining pressure, temperature and pore pressure. Pure and Appl. Geophys. 116(1978) 690-704. 44. J. Zhao, Experimental studies of the hydro-thermo-mechanical behaviour of joints in granite. Ph.D thesis, Imperial College, London, 1987. 45. J. Zhao, Analytical and experimental studies of heated convection of water flow in rock fractures. Proc. 33rd US. Symp. on Rock Mechanics, Tillerson & Wawerisk (eds.), Santa Fe, New Mexico, 8- l 0 June, 1992. (pp. 591-596). 46. J. Zhao and E.T. Brown, Hydro-thermo-mechanical properties ofjoints in the Carnmenellis granite. Quart. J. Engng Geol., 25(1992) 279-290.

Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 1995 Elsevier Science B.V.

Rock Joints

-

343

BEM Computations

G. Beer Institute for Structural Analysis, Technical University Graz, GRAZ, AUSTRIA and B.A. Poulsen Division of Exploration and Mining, Commonwealth Scientific and Industrial Research Organisation, Center for Advanced Technologies, P.O. Box 883, Kenmore 4069 AUSTRALIA In this chapter boundary element methods which allow the modelling of a continuum which contains joints and faults will be discussed. Two main methods of analysis will be dealt with here: the Displacement Discontinuity method and the multiregion boundary element method. Efficient algorithms which can deal with slip and separation along joints, faults and material interfaces are presented. Details of implementation and programming will be discussed and examples of application in mining and geological engineering are shown. The chapter concludes with a summary of current research and of issues which have to be addressed in the future. 1. I N T R O D U C T I O N There are distinct advantages to using the Boundary Element Method (B.E.M.) for solving problems in Rock Mechanics. Here we are dealing with infinite or semi-infinite domains and sometimes complex excavation surfaces. As the B.E.M. requires the discretisation of surfaces only, both requirements are handled well. The classical Boundary Element Method considers that the continuum to be analysed is elastic and homogenous and that it contains no joints and faults, i.e. it is assumed that the displacements are continuous. To be able to model the discontinuous behavior along predefined joint planes two methods can be employed. Method 1: Special Displacement Discontinuity (D.D.)[1] boundary elements are developed which contain two opposing surfaces (instead of one surface as in the conventional B.E.M.). These D.D. elements are used to describe joint or fault surfaces. Method 2: Two or more boundary element regions, whose boundaries are described by conventional boundary elements, are connected[2],[3],[4]. Slip and separation is allowed to take place at these interfaces between regions. In the following we will deal with each method separately and discuss the theoretical background, implementation and examples of application. The advantages and disadvantages of each method will be highlighted.

344 2. B O U N D A R Y

ELEMENT

METHODS

There are two Boundary Element methods which can be used to solve boundary value problems: The indirect B.E.M. and the direct B.E.M. The Displacement Discontinuity Method belongs to the indirect B.E.M.

~ p

j

Figure 1. A boundary value problem where tractions are known

2.1. Indirect Methods, Displacement Discontinuity Method Consider the boundary value problem in Figure 1 where the tractions t(Q) are known over a surface S. In the indirect B.E.M. we seek a solution for fictitious force intensities /z(P) such that[5]

P~(Q) + fs TT(p' Q)p~(P)dSp - t(Q)

(i)

In the above equation T is a matrix containing fundamental solutions for the traction at point Q due to a unit force at a point P. In three dimensions the matrix T is of size 3 x 3:

T-

Tyx T~ Ty~ T=T~uT~

(2)

Txu(P,Q) is, for example, the Kelvin solution for the traction at Q in x-direction due to a unit load in y-direction located at point P. To be able to solve Eq. (1) it has to be discretised, i.e. we write the equation on a limited number of points Qi. The integrals are evaluated on a number of simple surfaces, the Boundary Elements, and summed. In the simplest case we assume a constant value of force intensity/z over a triangular element. The discretised form of Eq. (1) is written as: -~p~(Q,) + ~ e--1

T(P,

Q)dS(P)p~(Qe)- t(Q,)

(3)

e

where Es is the number of boundary elements describing the surface. Eq. (3) may be written in matrix form as: [ C ] { / z } - {t}

(4)

345 Here [C] is a coefficient matrix obtained by integrating the fundamental solutions T over boundary elements. Once the fictitious load intensities are known, the displacements at any point, including boundary points, can be computed by

Is uT(p' Q)lz( P )dS( P )

u(Q) -

(5)

In the above equation, U(P, Q) is a matrix of fundamental solutions for the displacements at Q due to unit loads at P. If displacement discontinuities are present in the rock mass (for example, if there are faults and joints) then Eq. (1) has to be expanded to (Figure 2):

/So T(P, Q')Tp(P)dS(P) + Is1Z(Q',P)6(P)dS(P) - t(Q') where

(6)

Z(Q',P)= N(Q)DBT(Q',P)

f

~"

S-

t ( Q - ) ~

st Figure 2. A boundary value problem with a displacement discontinuity For 3-D problems N(Q)-

n. 0

0 n~

0

0

I

0 n~ 0 T/,x n~ 0

0

0

(7)

nz

0

ny

nx

D is the elasticity matrix i.e.

1 D - C1

C2 C2

C2 1 C2 C2 C2 1 0 0 0

0 0 0

0 0 0

0 0 0

0 0 0 0

0 0 0 0 0

G/C1 0 G/C1 0 0 v/ci

with C1 - E

(i-u) (I + u)(l - 2u)

(8)

346 /2

C,.= (1-~) G=

E

2(1 +,)

and B is a differential operator matrix, i.e.

B-

O/Ox o o o 0/0y o o o 0/0z a/ay a/a~ o o

O/Oz

(9)

O / O z O/Oy o O/Oy

5(P) are defined as 8(P) = u ( P +) - u(P-)

The displacement discontinuities

where P+ denotes a point on S + and P - a point on S-. These displacement components are usually defined in a local coordinate system normal and tangential to the crack surface:

$(P)- dsl ds2

RIDE 1 RIDE 2

(10)

The displacement discontinuity has tractions acting on surfaces S + and S - which are equal in magnitude but opposite in direction. Consider a point O' (Figure 2) which approaches point O at a displacement discontinuity; then we have for the case of applied tractions t(O +) = - t ( O - ) lim

Q'~Q-

(Is

T(P,

Q')•(P)dS(P)+ Is Z(Q',P)$(P)dS(P)) -

o

t(~-)

(11)

i

It is shown in Reference [6] how this limit can be evaluated. The integral equation to be solved for the unknown fictitious forces ju and displacement discontinuities 6 is written as

Z(Q)/z(Q) + / s 0 T(P,

Q')lz(P)dS(P)+Is1Z(Q',P)6(P)dS(P)- [(Q)

(12)

Here Z(Q) is the "free term" computed by taking a limiting value of Eq. (11) as Q' approaches QIn a numerical implementation, the integral equation (12) is usually written for a number of discrete points Qi and the integrals are evaluated over a number of simple surfaces or boundary elements which are used to describe the boundary of the problem and the surfaces of displacement discontinuities. In the simplest case we may assume a constant value of force intensity/~ and/or displacement discontinuity 6 over a line element in 2-D or an element of triangular or quadrilateral shape in 3-D.

347 The discretised form of Eq. (12) is

,,,, T(P,Q')p~(P)dS(P)] +

Z(Q~)/~(Qi) + e--1

9

Z(Q',P)6(P)dS(P) e=l

-{(Q)(13)

9

where E0 is the number of elements on surface So and E1 is the number of elements on surface S1. For a constant distribution of bt, 6 over a boundary element, there is only one "nodal point" which we may assume at the center of the element. This node has 3 degrees of freedom. Note that in contrast to the F.E.M. there is no requirement of continuity of It, 6 across elements and this allows greater flexibility in generating boundary element meshes. After discretisation Eq. (13) can be written in matrix form as [C~]{V} + [C2]{6} = {i}

(14)

where [C~] and [C~] are matrices whose coefficients are obtained by integrating the fundamental solutions T and Z over boundary elements. {t} is a vector of known boundary tractions at the "collocation points" Q,. Eq. (15) can be rewritten into [C]{X} = {t} where [C] =

(15) [C~_]

To be able to solve Eq. (15) there must be enough collocation points so that [C] is a square matrix. In a typical problem we may have unknown intensities ~ at the centroid of each element on surface So and unknown displacement discontinuities $ at the centroid of each element on surface $1. If we assume that the centroids of boundary elements are also the collocation points, then [C 1 will be a square matrix and a solution of Eq. (15) can be obtained. 2.2. M o d e l l i n g of j o i n t s a n d f a u l t s The boundary element method just discussed lends itself easily to the modelling of a jointed or faulted rock mass. In the joint plane we may define a relationship between joint opening/closing , ride and the components of traction normal and tangential to the joint plane:

t ' = K~' where U a n d / i ' are vectors of tractions and displacements, i.e.

t' -

tsl ts2

~' -

6sl 6s2

The matrix K, which may be nonlinear, relates the relative displacements to the tractions and is equivalent to a spring which connects surfaces So and 5'1 at point Q. This matrix is the same as used for joint element formulations in the Finite Element method.

348 In a typical rock mechanics application K may initially contain high normal and shear stiffnesses until failure occurs. The condition for failure in shear for example is given by: ts _> Co + tntan r where ts - v/t2sl + t2s2, Co is the cohesion and r is the friction angle. The condition of failure in tension maybe that its tensile strength is exceeded. At failure the corresponding stiffnesses are reduced to very small values. The process of solution is iterative and failure may progress along a joint plane during iterations. In an iterative solution Eq. (14) is solved subject to constraint conditions

2.3. Example of Application An example of application of the indirect boundary element method / displacement discontinuity method is shown here. The program MAP3D is used to solve a mining problem[7]. In the mining example illustrated in Figure 3, an inclined fault intersects a simple rectangular shaped stope. This example is presented to illustrate the far reaching influence that the presence of a fault can have on the rockmass response around an excavation.

Figure 3. Inclined fault

In this problem, the far field stress state has been specified as 30 MPa in both horizontal directions, and 15MPa in the vertical direction. The elastic properties of the host rockmass are Young's modulus of 60GPa and Poisson's ratio of 0.25. The stope is 2m wide, 4m high and 4m long (along strike). The fault dips at 45 ~ and strikes parallel to the stope. The fault has a friction angle of 20 ~ with no cohesion. At the far field stress state, the fault has a strength of 8.2MPa and an active shear stress of 7.5MPa, giving a safety factor of 1.1, a stable condition. In the adjacent figures, analysis results are presented on a transverse section taken at the mid-point of the stope. Owing to slip on the fault, it can be observed that (Figure 4) the major principal stress is distributed non-uniformly around the stope. The presence

349 of the fault has significantly reduced the stress concentration directly over the top of the stope, and redistributed the stresses adjacent to the fault slip area.

Figure 4. Major principal stress

Figure 5 illustrates that the minor principal stresses are even more intensely redistributed, showing a reduction in the stress concentration over the upper right corner of the stope, and the development of some tensile stresses.

Figure 5. Minor principal stress

The displacements (Figure 6) show reduced closure of the stope in the area above and to the right of the fault. During the analysis, the fault slips a m a x i m u m of 0.14mm at the side of the stope. This occurs over a length of almost 2m. In the area directly above the stope, the fault slips

350

Figure 6. Displacement

a maximum of 0.6mm. This slip occurs over a length of more than 5m. As the stope is only 2m wide, this represents a relatively far reaching influence. 3. D I R E C T M E T H O D In the direct B.E.M., Betti's theorem is used to obtain a relationship between applied tractions and displacements. c ( P ) u ( P ) - _ rjs T(P,

Q)u(Q)dS + fs U(P, Q)t(Q)dS

(17)

where c ( P ) i s the free term, t(Q) and u(Q) are vectors containing the values of traction and displacement at boundary point Q, and T(P, Q) and U(P, Q) are the fundamental solutions for the traction and displacement at point Q due to orthogonal unit loads at P in the coordinate directions. Using isoparametric boundary elements [8] the displacement and traction can be approximated by

u(r

J

~-~ Nj(~)u#

(18)

j=l J

t(~)

~-~ N3(~)tj

(19)

j=l

where uj and tj are values of u and t at node j and Nj(~) are shape functions of intrinsic coordinates ~ and 77. Substituting Eqs. (18) and (19)into Eq. (17) the following matrix expression relating the displacements {u} and tractions {t} at all the nodes of the boundary of a region can be obtained using the point collocation method[8]

[A]{u} - [B]{t}

(20)

If faults and joints are present one may now combine two or more regions to analyse problems with joints or faults. Figure 7 shows an example of the combination of two semi-infinite regions to analyse an excavation with one fault.

351

=

+

Figure 7. Boundary element regions with a fault at the interface

For this example we can rewrite Eq. (:20) for region I

[A]'{u}' = [B]X{t} '

(21)

and for region II (22)

[A]H{u}H = [B]H{t} H

At the nodes of each region, except for the interfaces nodes, either tractions or the displacements are known. On the interface nodes both displacements and tractions are unknown. Let us re-arrange Eq. (21) as follows: [a r

B i ] I { HI } / -

[

] { t] } z

-

U t

(23)

Where u s and t s are vectors containing the displacements and tractions of nodes not on the interface and u ~ and t i are corresponding vectors for the nodes on the interface. Assuming that all tractions t r (i.e. the loads applied to the free surface) are known we can write: t'

= {b}'-[A']'

{u'}'

(24)

where { b } I = [B/]/{tf} / and [ A ] ' = [A 1, - S ' ] ' By treating each column of [A'] z as right hand side, the system of Eq. (24) can be solved to give t'

-

t '~

+

K'

{u'}z

(25)

The first part of the solution represents the displacement of the "free" nodes u f~ and the traction of the interface nodes t i~ assuming the displacements at the interface are zero. Each column of the second part of the solution represents the corresponding displacement and traction values for unit displacements of the interface nodes. Thus [K i] represents a "stiffness" matrix which relates the traction of the interface with the displacements at the

352 interface. For B.E. region I we can write the following relationship between the interface displacements and interface tractions: {ti} I -

{ti~ I + [K'] z {ui} I

(26)

For the case where the fault/joint does not slip or dilate, the conditions on the interface are: {u'}'

=

{u'}"

(27)

and provided no tractions are applied at the interface {ti} I + {ti} n = 0

(28)

which yields the following system of equations for the displacements at the interface ([Ki]I + [K'] I1) { u ' } - {t'~

+ {t'~ 11

(29)

The interface stresses t i may be obtained by substituting the interface displacements u i into Eq. (26). The displacements at the free nodes u f can be obtained by substituting into Eq. (25). For the case where the joint slips or dilates, Eqs. (27) and (28) no longer hold and an iterative scheme has to be employed. This is discussed in detail in the following section. 3.1. M o d e l l i n g of joints and faults There are basically two main approaches for the modelling of slip and dilation on fault planes. 1. Connection of the nodes at the interface by nonlinear springs. Initially a high stiffness is assumed for the spring. This stiffness is reduced in the direction normal to the interface if the yield condition for joint opening is reached and in the direction tangential to the interface if the yield condition for joint slip is reached. 2. Connection of the nodes at the interface by rigid links in the direction perpendicular and tangential to the interface. If the yield condition in tension is reached then the link is broken. If the joint yields in shear then the link in the tangential direction is broken. Figure 8 illustrates the two approaches. In method 1, the pair of interface nodes is connected by springs. Initially the springs are assigned high values of stiffness in normal direction, /c~ and shear direction, k~ (Figure 8a). If the yield condition for joint slip is reached then the shear stiffness is reduced to a small but nonzero value k~ for any subsequent load step (Figure 8b). If the yield condition for tension is reached then both the shear and normal stiffness are reduced (Figure 8c). In method 2, the nodes are connected by rigid links which prevent slip and separation (Figure 8d). If the yield condition in shear is reached the link which prevents slip is removed (Figure 8e). If the yield condition in tension has been reached then all links are broken (Figure 8f). The advantages of method 2 is twofold:

353

(a)

(d)

i

kP n

(e)

(b)

P

,

P

/!

(c)

(f)

Figure 8. Methods 1 and 2 for non-linear interface behavior

1. There is no need to specify initial and residual values of joint stiffness, quantities which are not easily obtained in the field 2. Convergence of the method is significantly faster since we do not have to deal with large differences in stiffness between iterations. For these reasons method :2 has been used in the present work and this will be explained in more detail here.

3.2. Local interface coordinate s y s t e m To be able to implement the method, the stresses at the interface have to be expressed in a local system of coordinates with one direction pointing normal to the joint and the other one or two directions being perpendicular to this direction (Figure 9).

354

Figure 9. Local coordinate system at the interface The transformation from global traction/displacement t,, uj to local tractions/displacements ti, ui at a node j is given by U;

=

ti

=

T,u, T,t,

where for 2-D problems

;}

u;={

I

For 3-D problems we have (Figure 9) u;=(

q

t; =

;

{ z}

'11.2

(34)

ts2

T,

=

[ z yi 21 s2.2

s2y

(35)

S2r

Eq. (26) can now be rewritten in terms of local components as:

{t'} = [T]{t'}' = [T]{t'o}'+ [T][K]'[T]T{~'}' or

{t'}'

=

{to'}' + [K']'{u'}'

(37)

where

{to'}' = [T](tao}' [K']' = [T][K']'[T]T

(38) (39)

355 The matrix [T] is given, for example, for 4 interface nodes, by T1

0

0

0

0 0 0

Tu 0 0

0 Tz 0

0 0 T4

[T]-

(40)

The transformation matrix Ts is computed with the outward normal ff,j to the node j. For three dimensional problems vectors s'l, and s'2 are computed using the following vector • product (Figure 9). Sx

-

ffx~7

(41)

s~

-

s-'i x a

(42)

where ~ is a unit vector in y direction i.e. r

{~

-

1

(43)

0

If ~ points exactly in y direction then Eq. (41) is replaced by s~ - g

x s

(44)

where s is a unit vector in x direction s -

0 0

(45)

For smooth interfaces without kinks there is a unique value for ff,j at each node. For interfaces with kinks an average value of ~,j can be assumed. If there are initial stresses tr0 present, then we have to work out the corresponding initial tractions in the local interface coordinate system. To convert the initial stress to tractions at the interface we have G,o

-

a=,on= + T=y,ony + T=~,on~

(46)

ty,o

-

ay,ony + T=~,on= + T,y,on~

(47)

G,o

-

a , , o n , + T=,,on= + Ty,,0n~

(48)

The tractions in the direction normal and tangential to the interface are given by to

-

Tto

(49)

These initial interface tractions have to be added to the vector of traction due to a rigid interface {to'} /

356

3.3. Onset of slip and separation - yield conditions The condition for the onset of slip and separation (yield conditions) may now be written in terms of the values of tnj and t,lj , t,2j at interface node j as

Slip F, = It,j] + t,~a tan Ca - ca = 0

Separation

Ft=t,~a-Ta=O where the tension positive sign convention is used and Ca is the angle of friction, ca is the cohesion and Ta is the tensile strength at node j. For three dimensional problems

I t , i t - v/t]~j + t ,2.

(50)

3.4. Iteration (time step) procedure An iterative (time step) scheme has to be used for the solution of problems which involve slip and separation. For the first iteration (time step = 0) we assume rigid links between all the nodes on the interface. For the two region problem in Figure 7 we can write

{u'},:0 = {u'}' = { u ' } " and assuming no external forces are applied at the joint {t'},=o - { t ' } ' + {t'} x1 = 0

(52)

This leads to the following system of equations ([K'] z + [K'] zI) {u'},=0 - {t'~ z + {t'~ z1

(53)

This system can be solved to give the displacements {u'}t=0 in the joint plane. The tractions at the joint plane in the local (joint) coordinate system can be obtained by substituting {u'},=0 for {u'} I in Eq. (37) {t'}tI=0- {t'~ I + [K']{u'}t=0

(54)

where {t'~ I includes the initial joint tractions mentioned above. Once we know these tractions we can check the yield conditions and determine if the joint either slips or opens. The action taken for the next iteration depends on the value of Ft or F,. If both Ft and F~ at an interface node are zero, then no action is taken for that node. If F, is greater than, or equal to, zero, then the condition for joint slip is satisfied. If F, is greater than zero, then at one stage during the increment the yield condition has been reached and the tractions at the end of the time step violate the yield condition. The component of the shear traction which is above F, = 0 is also called "excessive" shear traction. The procedure for joint yielding in shear is as follows (Figure 10, 11) 1. The link(s) in the direction perpendicular to the vector normal to the joint plane are broken 2. Any 'excessive' shear tractions are applied to the node in the opposite direction

357

tf = - F,. It,

Its1 >

-t,~j tan Cj + cj

(a)

(b)

Figure 10. Conditions for joint slip satisfied, 2-D problem: the figure shows (a) tractions and state of rigid links at the end of time step 0 and (b) applied tractions and state of rigid links (shear link broken) for next time step

For yielding in tension the procedure is as follows (Figure 12, 13): 1. All links are broken 2. 'Excessive' shear and normal tractions are applied in the opposite direction 4. P R O G R A M M I N G

CONSIDERATIONS

The implementation of the algorithm for connecting and disconnecting degrees of freedom is explained here in more detail. The iteration time steps involve the displacements on joint planes only via the "region stiffness matrices" [K']', [K'] H, etc. The system of equations is solved using a Frontal Solution method[9]. In this method each variable is assigned a 'destination' which determines where its stiffness coemcients are to be assembled. If two elements are connected by rigid links at a node then the stiffness coefficients for that node are assembled in the same location (that is they are added). If links are broken, then the stiffness coel~cients corresponding to the direction of the broken link (degree of freedom) are simply assembled in different locations. In the software implementation the allocation of 'destination' is handled as follows: Each node is assigned a 'restraint code' for each degree of freedom which is set to 1 for each connected degree of freedom and to 2 for each disconnected degree of freedom. Initially all 'restraint codes' are set to 1. If a link is broken then the 'restraint code' for the corresponding D.o.F. is set to 2. 4.1. Joint opening/closing, slip/stick monitor During the iteration (time stepping) it is important to keep track of joints which are closing after being open and joints where the direction of slip is reversed. This is done by a device known as opening/closing, slip/stick monitor.

358

j

tt, I > - t , v tan oj +%

.

Is 2

ts I -...... -,_ .,!

(b)

(al

§ Figure 11 C.~na,..:'~ns/or joint slip satisfied 3-D problem: the figure shows (a) tractions and state of rigid link~ :t the end of time step 0 and (b) applied tractions and state of rigid links (shear link broken) for next time step 9

~

~

The joint accumulated relative displacements (i.e. slip and opening) at time step t at node j can be computed from the absolute displacements at the interface by Opening Ar, j(t)

--

" u.,,(t)u.'j(t)

(55)

u,lj(t)-u~lj(t )

(56)

u u..(t)~,;,,(t)

(57)

Slip 1

Slip 2 As2j(t)

-

(a)

(b)

Figure 12. Conditions for joint separation satisfied, 2-D problem 9 the figure shows (a) tractions and state of rigid links at the end of time step 0 and (b) applied tractions and state of rigid links (all links broken) for next time step

359

t3 2

%1 .... -.~

.":;"...

t ~ ~ --ts i

;~: .~" ... ...... :~,,,

(a)

(hi

Figure 13. Conditions for joint separation satisfied, 3-D problem: the figure shows (a) tractions and state of rigid links at the end of time step 0 and (b) applied tractions and state of rigid links (all links broken) for next time step

Figure 14 shows the sign convention used for the relative joint displacement.

Figure 14. Sign convention for joint displacement

The monitor is implemented in the software in such a way that a rigid link is reestablished for the case where the joint closes again or the slip is reversed. Joint closing is detected when Anj(t) changes sign and slip reversal occurs if either Aslj(t) or As2j(t) changes sign. 5. A P P L I C A T I O N S The numerical method presented above has been implemented in program BEFE[10] and extensively tested. Comparisons were made with theoretical solutions or solutions obtained with other methods or software. Test examples are reported in [6] and will not be repeated here. Instead we will show applications in mining and geological engineering.

360 5.1. Mine excavations in faulted rock This application relates to the modelling of the excavation of ore by open stoping and caving at the Hellyer Mine in Tasmania, Australia.

l/-.

i

i

i

\ L,-.- / \

--

I

"

I

! ....

I I"

\ r

I :I ~,

\ ;oel~.

i

soe,=~l

------

\ '"

,

-

,,~,|

#

"--,JLZ,~

-TL

I,.-12

%'

"\l.

.,

Figure 15. Stope section, Hellyer Mine, Australia. From Aberfoyle Resources Ltd.

The region modelled is in the northern area of the mine. Figure 15 shows a typical section through the caved stope. The geological feature modelled is the Jack fault, which is sub-parallel to the caved stope and limits it in the upper part of the stope. Figure 16 shows a perspective view of the boundary element mesh for modelling the

361 caved stope, the 87A stope, the 400-91 stope and the Jack fault. The mesh was prepared from horizontal and vertical sections using the preprocessor FEMCAD[ll]. Four node boundary elements with a linear shape function were used. Although the Jack fault consists of two surfaces with boundary elements on each side of the fault, only one surface had to be specified with the program automatically generating the other side.

Figure 16. Boundary element mesh of the Hellyer mine. Also shown is the insitu stress tensors

Although there are about 4 different materials only one material was assumed for the purposes of the analysis. The elastic properties of this material are: E = 30 000 MPa v= 0.3 These values were derived from uniaxial compression tests on drill core. The properties of the Jack fault were assumed to be the following.

362

Cohesion (c) = 0.0 M P a Angle of friction (r = 20 ~ Dilation angle (~o) 0~ These properties were derived from observation of the fault surface where it is exposed, and from experience. Virgin stress m e a s u r e m e n t results from only one site were available, and these have been used. T h e stress field was assumed to vary linearly from the surface and have the following values: At surface: all stress zero At d e p t h of 280m: al = 1 6 . 3 M P a Dip = 1~ Bearing = 159 ~ Dip = 71 ~ a2 = 9.9 M P a Bearing = 66 ~ aa = 3.8 M P a Dip = 19 ~ Bearing = 249 ~ These stress tensors are shown in Figure 16. T h e following figures show some results of the analysis.

Figure 17. Contours of normal stress on Jack fault

Figure 17 shows contours of the stress normal to Jack fault. One can clearly see the destressed zone adjacent to the excavation and the stress concentration at the corners where the Jack fault surface is exposed. Figures 18 and 19 show contours of shear stress along strike and up dip. T h e contours of slip along strike after 2 iterations (time steps) are shown in Figure 20. T h e m a x i m u m values of slip of 1 5 m m occurs to the south of the caved area. Figure 21 shows contours

363

Figure 18. Shear stress along strike

Figure 19. Shear stress up/down dip

364 of slip up/down dip after 2 iterations. The maximum value of slip (14ram) occurs where the Jack fault surface is exposed to the caved area.

Figure 20. Contours of slip along strike after two iterations

Figure 21. Contours of slip up/clown dip after two iterations

365 The predicted directions and values of slip were found to agree well with observations although actual magnitudes were found to be larger than those computed.

.i

% Figure 22. Stress vectors at 360 level, no slip on Jack fault

X ~\

Figure 23. Stress vectors at 360 level, slip on Jack fault after two iterations

366 Figures 22 and 23 show the effect of fault movement on the stress distribution at 360 level. Figure 22 shows the stress tensors before any movement on the fault occurs, where as Figure 23 shows the stress tensors after the Jack fault has slipped by 15ram. The main effect is to turn the stress tensors so that they are more perpendicular to the fault in the area to the south of the caved zone. Consequently, the 87 Pillar, which was in a stress shadow before, has become more highly stressed.

5.2. Geological Modelling This application relates to gaining a better understanding of the development of ore deposits. Ore deposits usually occur along contacts between dissimilar materials. In the particular case presented here, a gold deposit was found along the contact between the basalt and ultramafic rock. Figure 24 shows structure contours of the top of the granite/ultramafic lithological contact whereas Figure 25 shows the position of the Lucky and Golden faults.

/ r

_

Figure 24. Structure contours of Granite, Ultramafic contact in plan view. Elevation given in metres

The analysis was performed to test a theory of geologists about what occurred in the region during the Precambrian period (the inferred age of the deposit is 830 million years.) The geologists are fairly certain that the region was shortened in an approximate east/west direction. It is assumed that the deposit was formed at approximately 5km depth below the surface. On this basis, it was suggested that uniform stress boundaries should be applied to a volume or rock of 2000 x 2000 x 1000m which contains the lithological contact and the Lucky and Golden faults. Figure 26 shows the boundary element mesh used for the analysis. The mesh consists of 4 boundary element regions which are shown in Figure 27. Note that the mesh does not have any internal elements, only surface elements thus making the mesh generation

367

-/>,?//// /r,i

Figure 25. S t r u c t u r e contours of the Lucky and Golden faults in plan view. Elevation given in m e t r e s

easier. T h e following m a t e r i a l properties are used: Rock Mass E = 75 G P a u = 0.25 Lithological Contact r = 30 ~ c = 5 MPa Fault Zones r = 20 ~ c = 0 MPa T h e model was restrained on one surface and loaded on the opposing surface. The b o u n d a r y stresses were as follows: O'1 = 145 M P a Horizontal, East West a2 = 132 M P a Horizontal, North South a3 = 50 M P a Vertical Figure 28 shows a view from the top of the displaced shape of the mesh after 2 iterations. Figure 29 shows a view from below the mesh. It can be clearly seen t h a t slip occurs on the Lucky and Golden faults. Figure 30 shows the distribution of stress normal to the lithological contact before and after slip has occurred. Figure 31 shows contours of slip u p / d o w n dip on the Lucky and Golden faults. (The values indicated are in metres.) Finally, Figure 32 shows contours of slip on the lithological contact. The dark blue area seems to be near the area where the actual deposit was found.

368

FEtlVUE

SCALE L. J 20~~g

Figure 26. Boundary element mesh used for the geological modelling example

FEMVUE

x~y

(a)

(b)

(c)

(d)

Figure 27. The four boundary element regions of the geological modelling example

369

FEtlVUE

36~,L SC A.LE 4 ~45 $ 2 A,.LE

Figure 28. View from above, note slip of Lucky and Golden faults

FEtlVUE

2-

!

i

J Figure 29. View from below, note slip of faults

~ %

.-\l;f~

D=S~L SC J,LE ~ J 4 545 SC~LE

370

Figure 30. Normal stress distribution on lithological contact after two iterations

Figure 31. Contours

of

slip up/down dip

on

Lucky and Golden faults

371

Figure 32. Contours of slip on lithological contact

6. S U M M A R Y

AND CONCLUSIONS

Efficient numerical models based on the B.E.M. have been presented which are able to simulate large amounts of fault movement in a rock mass with very few iterations. The advantages of these numerical models are: 1. A boundary discretisation only is needed to analyse the problem. This essentially reduces the dimensionality of the problem. For example, it would have been extremely difficult to analyse the first two problems with a reasonable amount of human and computer resources using a domain type method, i.e. finite element method, finite difference method or distinct element method. An analysis of the third example problem presented in this paper had previously been attempted, unsuccessfully, using a finite difference code. 2. If the second method is used there is no need to specify stiffness coefficients for the fault planes. Very often these are difficult to obtain in the field and unless the fault has infill the model is not really required to be able to simulate elastic joint deformation in compression. 3. In the second method, very few iterations are needed to obtain large values of slip and separation. For example, slip in the order of l m was modelled after only two iterations in the geological example. 4. Boundary Element models are very e f c i e n t in terms of computer resources as long as few faults are modelled. As the number of faults/joints increases, the model

372 becomes less efficient and, for a large number of faults, a distinct element code such as 3DEC would probably be a more efficient model to apply. The models are applicable to a large range of problems involving joints/faults some of which have been presented here. The initiation and propagation of cracks along material interfaces for example may also be modelled[12] by the multiregion boundary element method. 7. A C K N O W L E D G M E N T The authors would like to thank Dan Sullivan for drafting many of the figures used in this paper. Thanks are also due to Terry Wiles of Mine Modelling LTD for supplying the example for the Displacement Discontinuity Method. Thanks also to Aberfoyle Resources, Hellyer Division for permitting the presentation of the second example of the application. Greg Marshall and Kevin Rossengren have assisted in this analysis. The data for the third example has been supplied by Bill Power of the CSIRO Rock Mechanics Research Center, Perth. REFERENCES

1. Crouch S.L. and Starfield A.M. (1983) Boundary Element Methods in Solid Mechanics, George Allen and Urwin, London. 2. Cruse T.A. and Myers G.J., (1977) 'Three-dimensional fracture mechanics analysis' J. Struct. Div. Am. Soc. Cir. Eng. 103, 309-20 3. Gerstle W.H., Ingraffea A.R. and Perucchio R., (1988) 'Three-dimensional fatigue crack propagation analysis using the boundary element method' Int. J. Fatigue 10, No 3, pp189-192 4. Crotty J.M. and Wardle L.J., (1985) 'Boundary integral analysis of piecewise homogeneous media with structural discontinuities' Int. J. Rock Mech. Min. Sci. ~ Geomech. Abstr. Vol. 22 (6), pp419-427 5. Wiles T.D. and Nicholls D. (1993) 'Modelling Discontinuous Rockmasses in Three Dimensions Using MAP3D' First Canadian Symposium on Numerical Applications in Mining and Geomechanics, Montreal, Canada 6. Beer G. (1993) An efficient numerical method for modelling initiation and propagation of cracks along material interfaces, Int. Jnl. Num. Meth. in Eng., Vol. 36 7. Wiles T.D. and Curran J.H. (1982) 'The Use of 3-D Displacement Discontinuity Elements for Modelling', Proc. $th Int. Conf. Num. Methods in Geomech., Edmonton, Alberta, 1982 pp 103-112 8. Beer G. and Watson J.O. (1992) Introduction to Finite and Boundary Element Methods for Engineers, J.Wiley. 9. Irons B. and Abroad S. (1980) Techniques of Finite Elements, Ellis Horwood Ltd, Chichester. 10. B E F E - Users and Reference Manual, CSS International, P.O. Box 177 St Lucia, Qld., 4067, Australia

373 11. Beer G. and Mertz W. (1990) A user-friendly interface for computer aided analysis and design in mining Int. J. Rock. Mech. Sci. & Geomech. Abstr. Vol. 27, No. 6 pp 541-552 12. Beer G., Bunker K. and Poulsen B.A. Modelling of crack propagation due to a mechanical rock ezcavator (in preparation)

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Mechanics of GeomaterialInterfaces A.P.S. Selvaduraiand M.J. Boulon(Editors) 9 1995 ElsevierScience B.V. All rights reserved.

375

Rock Joints: Theory, Constitutive Equations Michael E. Plesha Dept. of Engineering Mechanics and Astronautics, University of Wisconsin, Madison, Wisconsin, 53706 USA This chapter describes constitutive equations for rock joints t h a t include phenomena such as friction, dilatancy, damage of surface roughness, and the ability to t r e a t arbitrary small deformation histories. Comparisons between theory and cyclic shear experiments on joints in artificial and natural rock are described. The equations that are developed are amenable to implementation in analysis software based on techniques such as the finite element method, the discrete element (rigid block) method, and the boundary element method. 1. I N T R O D U C T I O N At the shallow depths associated with civil excavation and construction, the mechanical and fluid transport response of a typical rock mass is controlled primarily by the behavior of joints. Because of the natural processes leading to their creation, rock discontinuities are rough and usually have a very high degree of initial mating, or correlation, as shown in Figure la. An individual joint's behavior is rich in phenomena and includes surface separation, nonlinear-elastic deformation, frictional sliding, dilatant deformation during slip, various forms of surface damage, and strong coupling between mechanical and hydraulic response that can produce dramatic changes in hydraulic conductivity. Despite the fact t h a t joints display very complex behavior, numerical simulations of civil and strategic facilities in jointed rock subject to load events such as construction sequences, earthquakes and detonations, have often been carried out using grossly simple models for joint behavior, the main reason being the lack of comprehensive, accurate models for joints that are amenable to numerical implementation and the arbitrary deformation histories t h a t are generally encountered. Dilatant deformations, or the propensity for these to occur, give rise to a joint's complex mechanical and hydraulic response. Dilatant deformations are not a purely kinematic phenomenon since the asperity surfaces that are engaged in sliding undergo various forms of damage, or destruction, that are stress and deformation history dependent. Laboratory tests described in reference [1] have shown that damage occurs in one or a combination of two forms: gradual damage mechanisms such as wear due to frictional sliding, and catastrophic damage mechanisms such as gross asperity shearing due to overstressing and/or repeated loading. Also, an important factor is the influence of damaged asperity debris,

376 which introduces a new material phase in the contact region that affects surface shape and subsequent response. A network of dense jointing, as is present in most rock masses, provides highconductivity flow paths for fluid transport. For joints that have low roughness and are essentially unfilled (i.e., little joint gouge or other material deposits), flow rate has been analytically and experimentally related to the cube of the joint aperture, or nominal joint separation. For higher degrees of roughness, or for very small apertures, the fluid flow behavior is more complex but nonetheless, is still very strongly dependent on joint aperture. Thus, small changes of aperture, such as will result from dilatant deformations during sliding, can produce enormous changes in joint conductivity and flow rate. Numerous empirical models relating aperture to normal stress have been put forth, but none of these adequately account for change of aperture due to dilation. It appears t h a t an essential and currently missing ingredient in the development of a model for coupling between hydraulic conductivity and mechanical deformation is an appropriate mechanical model that correctly predicts dilatant deformations as a function of joint stress and displacement history. A variety of constitutive models for the mechanical behavior of rock joints have been proposed in the literature, and many of these are reviewed in reference [2]. Some popular models are empirical while several others employ realistic micromechanisms in their derivations. In most cases however, these models only describe certain aspects of joint behavior such as peak shear strength as a function of compressive stress. More sophisticated models that are capable of predicting joint behavior under unidirectional sliding conditions have been developed for rock joints, and independently for predicting crack surface shear stresses in concrete. However, many of these are ad hoc in nature and were developed without reference to any specific deformation micromechanisms. Because the loading history that a joint experiences can be complex, leading to possible cyclic behavior (i.e., one or more reversals of sliding direction), constitutive equations to describe and quantify joint behavior should be of the incremental or integral type so that history-dependence can be appropriately accounted for. Some significant steps in this direction were taken in references [3-5] in which elastic-plastic deformations are considered with underlying micromechanisms for behavior. The constitutive models described in this chapter are developed in the spirit of this original work. 2. M A C R O S C O P I C CONSIDERATIONS The c o n s t i t u t i v e t h e o r y described h e r e i n d i s t i n g u i s h e s b e t w e e n m a c r o s t r u c t u r a l and m i c r o s t r u c t u r a l f e a t u r e s of a d i s c o n t i n u i t y . Macrostructural assumptions lead to the overall framework of the theory while microstructural assumptions and idealizations, which are discussed in Section 3, allow specialization of the general framework for application to rock joints. Throughout this chapter, two-dimensional contact is assumed. We begin by considering the presence of any surface roughness to be a microstructural feature and thus, we focus attention on contact between two

377 macroscopically smooth surfaces as shown in Figure lb. The significant assumption here is that there is some length scale at which the discontinuity has well-defined normal and tangent directions t and n. The appropriate kinematic variables are the relative surface displacements g = [gt , gn ] r where g,

(1) g.

-

UA and Us are the displacement vectors for points p+ and p-, respectively, shown in Figure lc, ~ and /~ are unit vectors in the tangent and normal directions, respectively, and superscript T denotes transposition. The kinetic variables are the joint stresses, or surface traction, ~ =[~t,~n] T where ~n is negative in compression. These stresses are macroscopic, or average stresses, as opposed to the far more complex and difficult to determine microscopic, or actual, stresses.

B

b)

p.... ~

!

.................

A

'"""

....

!

9

. ....:.:...

,dh V

B

"O . . . . . . . . 9

"....... -

:.:......

C)

I ..:::::.....:...

v

>t _

P

Figure 1. Surfaces in contact; a) closely mated rough contact surfaces, b) macroscopic contact surfaces with roughness not shown; point p identifies a contact point-pair consisting of two initially adjacent points on the macroscopic contact surfaces, c) close-up view of two contacting surfaces and the contact point-pair; the surfaces are shown separated for clarity; macroscopic tangent and normal directions are signified by t and n, respectively. We next assume that the relative surface displacements admit the additive decomposition g = ge + gS

(2)

378 where superscripts e and s denote the elastic (recoverable) and sliding (nonrecoverable) p a r t s of the deformation, respectively. The sliding displacements implicitly include dilatant deformations. There is considerable e x p e r i m e n t a l evidence supporting this decomposition for rock joints. Interestingly, similar evidence supporting such a decomposition exists for almost every other class of contact problem that has been carefully studied (e.g., metalmetal contact problems, soil-structure contact problems, etc.). Because stresses vanish upon unloading, they can be related to only the elastic part of Eq. (2) which also vanishes upon unloading. Thus, a general nonlinear elastic relation between stress and deformation is dO = E d g e

(3)

where the elastic joint stiffnesses are (4)

E LE~ E,,,,

and the symbol d preceding a quantity implies an increment in that quantity. In general, the elastic stiffnesses E are nonlinear functions of stress level, and there is considerable experimental data showing this, especially for large changes of stress level. However, we have found that it is usually satisfactory to let E be constant, which we do henceforth in this chapter. It is also common to take Etn = Ent = O, which we also assume. We assume that the sliding displacements are given by a flow rule, or sliding rule, of the form 0

if F
or dF
S

(5)

d~

0

where F is a scalar-valued function, G is the slip potential whose gradient gives the direction of sliding (including dilatant deformations), and ;t is a scalar that gives the magnitude of the slip. F < 0 corresponds to nonsliding (elastic) states of stress while F = 0 corresponds to sliding states of stress and F > 0 is not possible. If F and G are the same, then the friction is said to be associated while when F and G are different, which is the usual case, the friction is said to be nonassociated. While the slip function F clearly depends on stress, it may also depend on other parameters such as, for example, temperature. Nonetheless, we assume in this work that it is simply a function of stress where the specific form of F is to be determined later when microstructural considerations are addressed.

379 We also allow the contact surface geometry to change as damage of surface r o u g h n e s s evolves. While the exact m a n n e r in which this occurs will be determined through microstructural considerations, for the present we assume only t h a t damage is a function of sliding work per unit volume, W s, where the increment of this quantity is dW s = (~Tdg~. Later, we elaborate at greater length on the implications and limitations of such an assumption, and offer some enhanced damage models. But for the present, the rationale is as follows. For high compressive stress, high tangential stress is required to produce slip and rapid asperity degradation can occur for small tangential displacements. Under low compressive stress, low tangential stress will produce slip, yet if the a m o u n t of sliding is large, asperity damage will also occur. Regardless of the sliding history, w h e t h e r it be unidirectional or cyclic, the sliding work W~ is a nondecreasing function of time because of the dissipative n a t u r e of friction. F u r t h e r m o r e , given the same sliding displacement, W s clearly increases more rapidly for high stress levels than for low stress levels. If at a given i n s t a n t of time the interface is slipping or is about to slip (i.e., F = 0 ), and if at the next instant of time the interface is still slipping (i.e., dF = 0), we obtain

d(~ + ~--~

=0

(6)

Combining Eqs. (1) - (6) and eliminating ~t yields the following incremental constitutive equation:

d(~ = Eesdg

(7)

where if F
E E es =

~JL~J E E I-~3F~

E{3-~~)-H

(8)

ifF=dF=O

I is the identity matrix and the hardening-softening parameter is H ~

~

~Ws

Once the specific forms of F and G are determined, Eq. (8) is easy to evaluate and provides a clear relation between stress and deformation t h a t is valid for

380 arbitrary sliding histories. For this reason, the constitutive model is ideal for implementation in analysis software; some numerical and semi-analytical schemes that implement Eq. (7) are given in references [6-8]. Note that because friction is in general nonassociated, Ees is unsymmetric. 3. MICROSCOPIC CONSIDERATIONS

In this section, simple idealizations for the geometry of surface roughness are postulated in conjunction with assumptions for how this geometry changes with damage. Shown in Figure 2 are several possible idealizations for the most predominant roughness features of a discontinuity (this is commonly referred to as "first-order roughness"). Throughout this chapter, we assume small deformations, meaning that the tangential displacement gt is small enough so that one asperity does not override another (e.g., with reference to Figure 2a or 2b, we require - L l < gt < Lr). If gt exceeds these limits, then realistically speaking, most natural rock joints would be highly uncorrelated and simple nondilatant Coulomb friction, which has been very well studied, is probably adequate. Most joints however displace from an initially correlated position as shown in Figure la, and it is for tangential displacements in the range - L t < gt < L,. that a joint's response is especially complicated.

a)

. :::::::::...'~

~

~!!::::~~.i:. 9 ...

~

h

.

b) ~ - : . . . .'...:.:::::::.:

.i

.

~

T

9 9:--!.:~

9 ...

h

.., : . : . : ' : : : : - ' .

9

: ,..... ::..... .........

h

c)

d)

:::!::::; i;-:!::;;!~!i::;i:!: ::::::::::::::::::::: ....... ::~ii~!::i~!~

i "":

I

]

"'"

~

L

"':':'.

:,-

::'"

-,

1

e)

/7

l:

Figure 2. Various surface idealizations for the roughness shown in Figure la; a) sawtooth surface, b) sinusoidal surface, c) general periodic shape, d) displaced geometry for the surface of (c), e) geometry at tangent point of contact for the surface of (d).

381 Referring to the general surface idealization shown in Figure 2d, Figure 2e shows g e o m e t r y at the actual point of contact where a is the positive counterclockwise angle made by the microscopic tangent direction with respect to the macroscopic tangent direction. If we assume t h a t Coulomb friction governs the behavior at the actual point of contact, then Ifl[<-pf2 where p is the coefficient of friction and ]'1 and f2 are the microscopic tangential and normal forces, respectively. This friction constraint gives the slip function and slip potentials F=[fl[+llf2 and G=lfl[. Dividing F and G by the gross, or macroscopic contact area and transforming the resulting stress quantities to the macroscopic directions t,n gives [2] F = }0n sin a + o"l cos al + P(o'n cos a - o"t sin a) G = [o n sin a + at cos a[

(10)

In Eq. (10), a is a function of the sliding displacement gt- For the sawtooth surface idealization of Figure 2a, a = ar when gt > 0 while a = at when gt < 0 (assuming t h a t the sliding displacements are small enough so t h a t asperity overriding does not occur). For the sinusoidal surface idealization of Figure 2b, it can be shown t h a t a = 2--~sin

1+

where Lk takes the value L~ or Ll

depending on whether gt is positive or negative, respectively. Similar relations can be derived for more general periodic surface idealizations such as shown in Figure 2c. Because of damage due to stressing of asperities and sliding, contact surface geometry changes as the loading and sliding history evolve. The exact m a n n e r in which this occurs is certainly very complex and difficult to quantify. Here, we a t t e m p t to make some reasonable assumptions so t h a t some features of surface damage can be accounted for. The most significant assumption, made in the previous section, is t h a t damage is a function only of sliding work W s. Now we also a s s u m e t h a t the surface does not change shape as damage evolves, b u t changes only in amplitude. Thus, angle a changes with the history of the deformation process. An early model to characterize such damage, developed in reference [2], was

a = aoexp(-cW s)

(11)

where ao is the initial value of the asperity angle and c is an experimentallyd e t e r m i n e d positive number, called the asperity degradation constant, t h a t determines how rapidly the asperity surfaces degrade. Note t h a t unless c = 0, in which case the asperities experience no damage, as W~ becomes large, a -~ 0 and the macroscopic sliding surface becomes smooth. The exact interpretation of this is described more thoroughly later.

382

example The constitutive model given by Eqs. (7) - (11) will be compared to a laboratory cyclic direct shear test reported in reference [9]. The joint used in the test was artificially produced in sandstone by lineloading. The specimen had a nominal surface area of 495 cm 2 and was subjected to a constant compressive stress of 2.5 MPa. The experimental results are shown in Figure 3. The material parameters used in the simulation were obtained directly from the experimental results shown in Figure 3 (see reference [2]) and are given in Table 1. The constitutive model used the sawtooth surface shown in Figure 2a with (a r)o = 9~ and (at) o = - 7 ~ The simulation results are shown in Figure 3 with the following observations: 1) The shear stress vs. shear displacement results compare reasonably well, except during the initial loading in which the experimental results show much higher stresses. This is probably due to the presence of higher order roughness in the joint that is not accounted for in the constitutive model. If replicating such phenomena is important, it may be possible to enhance Eq. (11) to include damage of roughness according to a formula of the type . . . .

(12)

where subscripts 1, 2, etc., refer to parameters associated with the various orders of roughness. 2) The shear stress vs. shear displacement response for the analytical model is more abrupt near the transition between sliding down one asperity surface to sliding up another asperity surface. This is due to the abrupt trough between asperities in the sawtooth surface idealization (i.e., a changes discontinuously when gt goes from positive to negative). Later in this chapter, we investigate a variant of the sinusoidal surface idealization shown in Figure 2b which removes this deficiency. 3) The normal displacement vs. shear displacement response of the constitutive model shows some anomalous behavior in which the interface in the simulation substantially thickens as the sliding progresses, while the test shows very little change in net thickness. This points out a serious deficiency in the model that is related to how damage is treated. In the next section, damage modeling is discussed more thoroughly to help remove this deficiency. Other comparisons between the constitutive theory and experiments have been performed, but are not reported here (see reference [2]). We note however t h a t many of these comparisons include simulations of tests at different compressive stress levels, using the s a m e set of constitutive law material p a r a m e t e r s . Except for some of the deficiencies cited above, the agreement is generally good.

383

9

cycle 1

9

cycle 2

f " \ _ ,-xper, " - - - ' mental

3.0

9 - cycle 3

2.0

-

cycle I

-

cycle

c.c,.,

o,.

~, s

-30

-20

-I0

mi .J

10

20+J

ri ) 3 0

ml

o

I I

!i

'~

O)

"

Analytic

2

C:

-I

; f______]Ti

'///I/'

/j_2/)

i

_J

~ -/.,-+.,-.+..]/H///...

F

-3_so

tangentlal displacement, gt

-30

i

I

-10

10

i.

I

30

(mm)

tangential displacement, gt ( r a m )

E

s

Experlmental

s

-

~

._

"~ E

-40

,~-._-.~ -30

-20

~ _

~---~:~"~

20

30

-2'

c

.:

s

~ E 9 o

3

~

(mm)

2

0

._~ -,~

-2

"~

-3

E

tangential dlsplacement, gt

Analytlc

-S0

----.~~~Q, i

I

i

-30

I -10

,

I

30

10

o

tangentlal

displacement, gt

(mm)

Figure 3. Experimental and analytical results for a cyclic direct shear test in sandstone. The experimental results are taken from reference [9] while the analytical results are obtained using Eqs. (7) - (11). Table 1 Material parameters for simulation of Kutter and Weissbach's direct shear test.

E. =0.42 GN/m3

E,~= 1.0 GN/m 3

p =0.6

c=10-6 m2/Joule

4. D A M A G E M O D E L I N G energy-based wear theory In this sub-section, we more thoroughly discuss the implications of assuming that wear is dependent upon the energy dissipated during sliding, Ws. This discussion is based on the extensive treatment given in reference [10], which should be consulted by interested readers. Regardless of the materials involved in contact, wear on a microstructural level is a result of one or a combination of

384

several different and very complicated physical processes including subsurface cracking and delamination, adhesion, plowing, and thermal, environmental and chemical effects. Most of these damage processes rely on energy supplied by sliding. Our approach to wear modeling does not explicitly account for a p a r t i c u l a r m i c r o s t r u c t u r a l wear mechanism but r a t h e r uses a more global approach in which the actual wear mechanism is included under the general umbrella of an energy-related process. This is also a weakness of the theory in t h a t p a r a m e t e r s such as the asperity degradation constant c have no clear relationship, if indeed they should, with more established material properties of the contacting materials, and hence m u s t be considered as new m a t e r i a l properties to be determined by appropriate experiments. In reference [10], a general energy-based wear theory is developed. It was found t h a t the exponential wear model given by Eq. (11) is obtained using the following assumptions: 9 Destruction of a unit volume of contact surface material due to wear requires a specific amount of energy which is a material parameter. 9 The debris produced by damage can become firmly reattached to one of the contact surfaces; this requires a specific energy per unit volume which is a material parameter. 9 The sliding work is completely consumed by material removal due to wear and debris reattachment. 9 The two contacting surface shapes are dominated by their first Fourier components (i.e., they are sinusoidal). 9 Material loss always occurs in the neighborhood of asperity peaks and material reattachment always occurs near the asperity troughs. 9 The contact areas associated with material loss and r e a t t a c h m e n t are always located on the same side of a particular asperity. All of these assumptions are very plausible and realistic. However, to yield a wear theory in the form of Eq. (11), there are also a few assumptions t h a t m u s t be made which, while plausible, are not as easy to interpret or fully justify as those described above. These assumptions relate to how the sliding energy is divided among m a t e r i a l damage and r e a t t a c h m e n t m e c h a n i s m s , and some details regarding relationships between asperity heights and microscopic contact areas associated with regions undergoing material loss and reattachment. The important results of this work are: 1) Most of the assumptions used for the theory are very realistic. Those assumptions t h a t are more difficult to interpret are plausible, but otherwise c a n n o t be fully supported. F u r t h e r s c r u t i n y of these q u e s t i o n a b l e assumptions is perhaps a key to developing more accurate energy-based wear theories. 2) The forgoing assumptions lead to Eq. (11) except where it is clear t h a t c depends on the histories of (~, and g,.

385 This later remark is also supported by experiments that have been reported in the literature. In reference [11], surface shape changes are modeled by A = Aoexp(-CW s) where A and Ao are secant asperity angles and W is the total work whose increment is dW = qTdg. Based on numerous experiments using precision-sawn joints in limestone and granite, the authors report that parameter C has dependence on compressive stress and they suggest that C = 0.141 Ao(~n / Qu where Qu is the unconfined compressive strength of the rock, Ao is measured in degrees and C has units of cm2/Joule. Although the damage model used in reference [11] is not identical to Eq. (11), the strong functional similarity between these suggests that values of c and C should be close [12]. Note that to employ a damage parameter that has normal-stress or other types of dependencies, Eq. (11) m u s t be modified as follows to ensure that damage is irreversible (i.e., a should be a nondecreasing function of time)

(;s /

a = a oexp -

cdW

(13)

Experiments are also reported in reference [1] where normal stress dependence is also observed. Numerical simulations of laboratory tests at different compressive stress levels are also reported in which piecewise constant values of c are used in Eq. (13) with very good results.

surface approach during damage Closer study of Eq. (11) shows that while it does appear to capture m any important features of surface damage, its deficiency is that it does not correctly allow the two contact surfaces to approach one another as the roughness (i.e., asperities) degrades. For this reason, an interface undergoing damage usually thickens, sometimes considerably, as shown in the analytical results of Figure 3. In Figure 4, we more carefully consider kinematics during an increment of sliding motion; increments of elastic and sliding deformation, dg e and dgs are as described before, plus an increment of damage deformation dg d is included which allows the surfaces to approach one another as the asperities degrade. In what follows, we assume that the damage deformations have no tangential component so that gd = 0. Thus, in our enhanced damage model, Eq. (2) is replaced by

g = ge + gS + ga

(14)

where now it is necessary to specify how gd evolves. In the following explanation, we assume t h a t the contact surfaces fully conform before deformation so t ha t there is no initial void space. Once the asperities are fully destroyed, all dilatant deformations should be recovered regardless of the final tangential position gt. If the volume of the rubblized asperity material, which of course remains between the contact surfaces, has the

386

~gn

dg~,~dg~

> gl Figure 4. Relative surface displacement decomposition consisting of elastic, sliding and damage portions. same volume as when it was in the initial intact state, then the net normal deformation, neglecting ge which is small, should be zero. If the rubblized asperity material occupies greater volume than in its initial state, which would seem to be the normal situation, then the net normal deformation should be positive, while if the rubblized material has lower volume than initially, which would effectively be the case if asperity debris were lost from the sides of a laboratory specimen during testing, then the net normal deformation should be negative. An equation that characterizes this behavior is

where h is some measure of the surface roughness geometry as shown in Figures la-c, ho is the initial value of h, and ), is a bulking parameter that represents the fraction of ho that the interface increases or decreases in thickness after the asperities are fully degraded (i.e., when h ~ 0). We now assume that h degrades in the same way that the asperity angle a degrades, namely h = ho a / a o where a is given by Eq. (11), so that Eq. (15) becomes gnd :(Th o -gSn)[1-exp(-cWS)]

(16)

Increments of g a are dg d = -dgSn[1- exp(-cWS )] + ( ~ho - gS )c exp(-cWS )dW s

(17)

Using matrix notation, Eq. (17) becomes dg d = Pdg s where

(18)

387

0

0 P = (Yho - g S ) c e x p ( - c W S ) e t

e x p ( - c W s) - 1 + (yh o - gS)c exp(-cWS)crn

]

(19)

Combining Eqs. (3)-(6), (14) and (19) yields dO = E esa dg

(20)

where E

if F
E esd =

E I-

~

(21)

(I+e)vT ifF =

dF =0

and vector v is

V=

T

(22)

Note t h a t if damage deformations are neglected (i.e., P =0 and hence dg d = 0), Eq. (21) reduces to Eq. (8). In the next section, we demonstrate the improvement in predicting bulking displacements using this enhancement. In the remainder of this section, we offer a physical interpretation of the surface damage evolution predicted by this model. Shown in Figure 5 is a sequence of diagrams demonstrating the evolution of a sinusoidal surface shape. The first remark is that the surface shape that evolves is dependent only on the sliding work, W s, and not on the relative position of the top surface with respect to the bottom surface. The second observation is that the net sliding surface shape remains sinusoidal with the same period as the original surface; only the amplitude decreases. In Figure 5, the volume of rubblized material is equal to its initial intact volume (i.e., ~' = 0) so t h a t the interface has no net change of thickness. Making ?' a small positive number (e.g., 0 < ~' < 0.05) will give a small increase in thickness while making ), a small negative number (e.g., -0.05 < ), < 0) will give a small decrease in thickness, which is oi~en called s e a t i n g .

388

a)

b) 9

c)

.,

~176

9

.

, 9

9 .~

,,<.:.'..','"

d)

~

.

""

.

..,-

"" " ' 3 : " "

".'~"::i.:i:::i:.::::.:i::::' .. ,...~: "~, ":""'.::\ ": :-,::::!i.:.:~::?:::i:::i::'~..:.'"'"- :"": :"'-':,'.~:"::'i.:i:.'.!.':i'..,:i:"/~..,- :..' 9 rubblized a s p e r i t y material

Figure 5. Shape evolution for a sinusoidal surface predicted by the constitutive equations of Section 4; a) shows the initial surface shape, b) and c) show intermediate surface shapes as sliding work Ws accumulates in which the net sliding surface consists of regions of intact rock and regions of deposited asperity debris, and d) shows the final net shape of the sliding surface where a = 0 when Ws -+oo.

5. a - T M O D E L In reference [ 13] a new model, called the a-7 model, was developed using all of the ideas described in previous sections of this chapter, plus some new ideas that further increase accuracy and ease of use. In this section, the assumptions used in the theory are discussed and the features of the resulting constitutive equations are reviewed along with several example problems. Equations (3)-(6), (10), (11) or (13), and (14)-(22) are still key ingredients. A new assumption is that the two contacting surfaces, which have sinusoidal shape and the same period, can have slightly different amplitudes. A new p a r a m e t e r measuring the ratio of initial asperity amplitudes is defined as b = hBO/hAO where hAO and hBO are the initial asperity amplitudes, as shown in Figure 2b, for surfaces A and B, which are shown in Figure 1. By virtue of this assumption, the transition from sliding on one side of the asperity surfaces to sliding on the other side is smooth. Data analysis techniques are also discussed in reference [13] for determining necessary material parameters using the results of a minimum of one laboratory cyclic direct shear test at constant compressive stress. example In reference [13], a cyclic direct shear test on an artificial joint molded in hydrostone is reported. The material has an unconfined compressive strength of

389

Qu=38 MPa and simulates soft natural rock The joint had an initial sawtooth shape as shown in Figure 2a with asperities oriented at l0 ~ with respect to the horizontal (i.e., a r =10 ~ and at=-lO ~ and an asperity period of about 28mm (i.e., Lr=Ll=_14mm). The specimen had size 150mm x 200mm and was subjected to constant compressive stresses of 1.0 MPa for the first five cycles of shearing, followed by 1.5 MPa for the next five cycles, followed by 2.0 MPa for the last five cycles. Each cycle of shearing consisted of sliding the upper surface of material from the initially fully-seated position gt=O, to the right to gt =6mm, back to the left, passed the initially fully-seated position to gt =-6mm, and then back to the initial position gt=O. Experimental results for cycles 1-5, 6-10, and 11-15 are given in Figures 6-8, respectively. The numerical simulation employed the a-T model with a sinusoidal surface. The m a t e r i a l p a r a m e t e r s are given in Table 2 in which piecewise constant asperity degradation values were used with Eq. (13). Except for the asperity degradation values, all of these parameters were calculated from the first cycle test results in which the compressive stress was 1.0 MPa [13]. Using j u s t the first cycle test results, an asperity degradation p a r a m e t e r of c=0.5337(10) -2 cm2/Joule is computed, which yields the first entry in Table 2. However, simulation of the test using this single value of c, while it does give reasonably accurate results, does not permit the asperity surfaces to damage rapidly enough during the cycles at higher compressive stresses. Better agreement is obtained by using a value of c based on the average compressive stress during the entire test (reference [13] shows good simulation results for c=7.5(10) -3 cm2/Joule). In this chapter, the simulation is performed using the piecewise constant values of c reported in Table 2, with the results shown in Figures 6-8. While the value of c for r MPa was obtained according to the protocol described in reference [13], the remaining two values were obtained in an ad hoc m a n n e r based on our experience using this constitutive model. The simulation results are very good, b u t more research m u s t be done to establish the appropriate r e l a t i o n s h i p between c and or,, including necessary experimental protocols.

Table 2 Material p a r a m e t e r s for simulation of a fifteen-cycle direct shear test on an artificial joint molded in hydrostone [ 13].

Ett =6.5 GN/m 3

E,~=7.5GN/m3

p =0.8345

b=1.086

0.5337(10) 2 cm z / Joule

for a n =-1.0 MPa

c = ~ 0.5689(10) -2 cm z / Joule

for a n =-1.5 MPa

~1.8340(10)-2 cm 2/,Joule

for a n = -2.0 MPa

7=-0.06

390

(a)

--- 0.9 E

(c)

0.7

"~ o s 0.3 ~0.1 o _

J

._ -0.1 E-O.3 0

Z

-0.5

7'

-s'

3 ' -i

i

' 5 ' ~ ' 7

-7

-5

-3

-1

1

3

5

7

1.6

(b)

1.2

(d)

0.8

(-

0.4

-

l

,

t

E o.o

J

,_ -0.4

f

-0.8

.

.

J

-1.2 -1.6

. . . .

-7

-5

-3

-1

f

J

1

3

S h e a r displacement (mm)

5

7

-7

I

-5

. . . .

i

-3

. . . .

I

-1

. . . .

i'

1

9 ,''

i

. . . .

3

l , l l ,

5

7

Shear displacement (mm)

Figure 6. Results for first five cycles of a direct shear test (cycles 1-5) with G, =-1 MPa; (a) and (b) are experimental results, (c) and (d) are numerical results. Interestingly, if the relation C=O.141AoG,,/Qu, which was developed in reference [11] and is described in Section 4 of this chapter, is used with Ao and C replaced by ao and c, respectively, we obtain c values of 3.71(10) -2 cm2/Joule, 5.57(10) -2 cm2/Joule, and 7.42(10) -2 cm2/Joule, for compressive stresses of 1.0 MPa, 1.5 MPa and 2.0 MPa, respectively. While these values are higher t h a n those which we found to be most accurate, the a g r e e m e n t is n o n e t h e l e s s remarkable considering that the relation developed in reference [11] was based on tests using limestone and granite samples, which were certainly quite different t h a n the model material (hydrostone) employed in the laboratory test being simulated here. Note t h a t the a-T model employs a sinusoidal surface whereas the laboratory test used joints with a sawtooth shape. Despite this difference, the simulation and test results compare quite well. In fact, the numerical simulations were repeated, except in which a sawtooth surface was incorporated in the constitutive model and the comparison is, perhaps surprisingly, not as good. The probable reason is t h a t even though the laboratory specimen begins with sawtooth shaped

391

asperities, due to damage which is primarily concentrated near the peaks of asperities, the worn surface profile is better characterized as being sinusoidal.

O.g

(a) E 0.7-

(c)

0.7

o.s:

0.5

0.3~

0.3

to 0.1-

0.1

E ~ 0-

J

- - -0.1 E ,- -0.3 O

2:

-%,/-

0.1 0.3 .0.5

-0.5 7

-5

-3

-1

1

3

7

5

. . . .

7

. . . .

i

. . . .

-3

-5

-3 Shear

!

-1

. . . .

!

. . . .

1

i

,,

3

9 "1

. . . .

5

7

5

7

(d)

S

(b)

!

-5

to to

o

J

to L t~ qJ

r-

r

-1

7

-5

-3

-1

1

3

Shear displacement (mm)

5

7

-7

-1 1 displacement

3 (ram)

Figure 7. Results for second five cycles of a direct shear test (cycles 6-10) with Gn=-l.5 MPa; (a) and (b) are experimental results, (c) and (d) are numerical results. 6. C O N C L U S I O N S The major features of the constitutive theories described in this chapter are that they are based on specific microstructural mechanisms t h a t are easy to interpret, and the resulting constitutive equations are incremental and hence, easy to implement in analysis software based on techniques such as the finite element method, the discrete element (rigid block) method, and the boundary element method. Because the equations are incremental, they are applicable to arbitrary small displacement sliding motions, which are likely to occur in a numerical simulation of a realistic problem. Also, because the underlying microstructural mechanisms are easy to interpret, it is possible to develop simple and effective experimental protocols to determine the necessary constitutive law parameters as described in reference [13].

392

.~, 0.9

(a)

E E 0.7

(c) 3.7

"~ o.5

0.5

~ 0.3 ~J

0.3

~0.1

0.1

J

-0.1 E -0.3 o Z -0.5 t_

:).1 :).3 :).5

7

-5

-3

-1

1

3

5

7

3

:

9

7

i

9

-5

i

9

-3

i

9

-1

i

9

1

1

9

3

i

"

''

5

7

3

(d)

(b)

2

1

P.,,

J Y

0 L_

J

-2

-3

.

-7

-5

-3

.

.

.

-1

.

/ i

J .

.

.

.

1

3

Shear displacement

(mm)

.

9

5

7

7

I

-5

9

I

-3

9

'I

-1

9

I

1

,

Shear displacement

I

3

9

i

5

9

7

(mm)

Figure 8. Results for third five cycles of a direct shear test (cycles 11-15) with o',=-2 MPa; (a) and (b) are experimental results, (c) and (d) are numerical results. For the most part, the constitutive theories described in this chapter do a fairly good job of modeling the type of damage we refer to as wear, which is a more or less gradual destruction of asperities during sliding. However, when asperities are very highly stressed, damage in the form of catastrophic asperity shearing or fracturing may occur independent of whether there is sliding and the theories described in this chapter do not account for this. Furthermore, reference [1] reports results for cyclic direct shear experiments under constant compressive stress in which there is sometimes a transition between wear damage and shearing damage which appears to be fatigue-related. Incorporation of effects such as these requires further work. REFERENCES Huang, X., Haimson, B.C., Plesha, M.E., and Qiu, X., An Investigation of the Mechanics of Rock Joints--Part I, Laboratory Investigation, Int'l. J. for Rock

393

2. 3. 4. 5. 6. 7. 8.

9.

10. 11. 12.

13.

Mechanics, Mining Science and Geomechanics Abstracts, Vol. 30, pp. 257269, 1993. Plesha, M. E., Constitutive Models for Rock Joints with Dilatancy and Surface Degradation, Int'l. J. for Numerical and Analytical Methods in Geomechanics, Vol. 11, pp. 345-362, 1987. Ghaboussi, J., Wilson, E. L., and Isenberg, J., Finite Element for Rock Joints and Interfaces, J. Soil Mech. and Found. Div. ASCE, vol. 99, pp. 833-848, 1973. Hsu-Jun, K., Nonlinear Analysis of the Mechanical Properties of Joint and Weak Intercalation in Rock, Proc. 3rd Int'l. Conf. on Numerical Methods in Geomechanics, Aachen, W. Germany, pp. 523-532, 1979. Roberds, W. J., and Einstein, H. H., Comprehensive Model for Rock Discontinuities, J. Geotechnical Engr., ASCE, vol. 104, pp. 553-569, 1978. Ballarini, R., and Plesha, M.E., The Effects of Crack Surface Friction and Roughness on Crack Tip Stress Fields, Int'l. J. Fracture, Vol. 34, pp. 195-207, 1987. Plesha, M.E., Ballarini, R., and Parulekar, A., Constitutive Model and Finite Element Solution Procedure for Dilatant Contact Problems, Journal of Engineering Mechanics, ASCE, Vol. 115, pp. 2649-2668, 1989. O'Connor, K., Zubelewicz, A., Dowding, C.H., Belytschko, T., and Plesha, M.E., Cavern Response to Earthquake Shaking With and Without Dilation, Proceedings of the 27th U.S. Rock Mechanics Symposium, H.L. Hartman editor, pp. 891-896, 1986. Kutter, H. K., and Weissbach, G., Der Einflues von Verformungs- and Belastungsgeschichte auf den Scherwiderstand von Gesteinskluften unter besonderer Berucksichtigung der Mylonitbildung, Final Report, DFG Research Project Ku, 361/2/4, 1980. Qiu, X., and Plesha, M.E., A Theory for Dry Wear Based on Energy, ASME Journal of Tribology, Vol. 113, pp. 442-451, 1991. Hutson, R. W., and Dowding, C. H., Joint Asperity Degradation During Cyclic Shear, Int'l. J. for Rock Mechanics, Mining Science and Geomechanics Abstracts, Vol. 27, pp. 109-119, 1990. Plesha, M.E., Hutson, R.W., and Dowding, C.H., Determination of Asperity Damage Parameters for Constitutive Models of Rock Discontinuities, Int'l. Journal for Numerical and Analytical Methods in Geomechanics, Vol. 15, pp. 289-294, 1991. Qiu, X., Plesha, M.E., Haimson, B.C., and Huang, X., An Investigation of the Mechanics of Rock Joints--Part II, Analytic Investigation, Int'l. J. for Rock Mechanics, Mining Science and Geomechanics Abstracts, Vol. 30, pp. 271287, 1993.

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

395

Rock joints: F E M implementation and applications A. Gens, I. Carol and E.E. Alonso Departamento de Ingenierfa del Terreno, Technical University of Catalunya Gran Capitb. s/n. Edifici D-2. 08034 Barcelona, Spain

In order to incorporate the behaviour of rock joints in FEM analysis it is often useful to employ interface (or joint) elements especially devised to represent discontinuities. In this contribution some aspects of the implementation of this type of elements are described. A unified formulation for a family of isoparametric joint elements is presented first. Then the specific procedures required in the algorithm to integrate the constitutive law are indicated. Special reference is made to a simplified constitutive relationship that permits analytical integration. The relationship between the numerical integration rule and the performance of the element is discussed in detail. Finally two FEM analyses using joint elements are presented as examples of application. The analyses refer to a large rock slide and to the stability evaluation of an arch dam foundation.

1. I N T R O D U C T I O N In finite element (FEM) analysis of engineering problems involving fractured rock discontinuities in the rock mass are often represented by means of joint or interface elements. Their use is justified by the small thickness of rock discontinuities compared to the other dimensions of the problem and by the special characteristics of the mechanical behaviour of rock joints. The first interface element specifically developed for modelling rock joints was proposed in [ 1]. Afterwards, many more elements to represent discontinuities in rock and other related problems have been developed. They can be classified in the following categories: - Link elements in which only the interaction between opposite nodes is considered ([2-4]). - Continuum finite elements of small but finite thickness, sometimes referred to as 'thin' elements ([5-9]). - Zero thickness joint or interface elements in which relative displacements between opposite nodes are the primary deformation variables ([1], [10-13]). As demonstrated in [ 14] finite and zero thickness elements are not fundamentally different from a numerical point of view and similar performances can be obtained from both types of elements. Zero-thickness elements can be considered as a limiting case of finite thickness elements. A full literature review of the various interface or joint elements developed to date is presented in [ 15]. In this contribution a number of selected topics related to the implementation of joint elements in FEM codes will be described and discussed. In order to center the discussion reference will

396 be made to a class of zero-thickness isoparametric joint elements the formulation of which is presented first. The implications that the special features of the mechanical behaviour of rock joints have for the procedures included in the stress point algorithm special are then discussed. Again, in order to be more specific, reference will be made to a particular elastoplastic constitutive model although the discussion is also valid for other constitutive laws describing similar types of behaviour. A simplified model that permits the analytical integration of the constitutive relationship and is still able to describe basic features of rock joint behaviour is also presented. Afterwards the question concerning the interaction between element performance and the type of numerical integration scheme adopted is described in detail with particular emphasis on the behaviour of the element when modelling sliding. Finally the usefulness of this type of element in the analysis of engineering problems is demonstrated by a brief description of two cases: a 2-D analysis concerning the stability of a large rock slide and a 3-D analysis related to the safety evaluation of a large arch dam founded on fractured rock. A brief indication on the notation convention used: small bold letters (n, f) indicate vectors and capital bold letters (D, K) denote matrices. Normal typeface letters (o~,a, A) represent scalar variables.

2. FORMULATION OF A FAMILY OF ISOPARAMETRIC ZERO-THICKNESS JOINT ELEMENTS The basic variables used in the formulation are stresses, divided in normal and tangential components, and relative displacements, divided similarly in normal and tangential displacements (Fig. 1). Therefore, in this type of joint elements relative displacements play a role equivalent to that of strains in conventional continuum elements. Examples of joint elements are shown in Fig. 2: a quadratic 16-node surface (2-D) element for three-dimensional analysis and a quadratic 6-node line (l-D) element for two-dimensional analysis. Other interpolation orders and other element shapes (e.g. triangles) are of course possible. The similarity in the formulation for the different members of this family of isoparametric zerothickness joint element permits a unified description. The adoption of isoparametric interface elements facilitates their introduction in conventional finite element packages where the use of isoparametric continuum elements is very widespread. 2.1 G e o m e t r y and interpolation

As Fig. 2 shows, the position of the joint is defined by a number of midplane points occupying an intermediate position between two opposite nodes belonging to adjacent continuum elements. This of course does not preclude the possibility of the element having zero thickness. Relative displacement at those points are postulated to be equal to the difference between displacements of the corresponding adjacent nodes. Interpolation of geometry and relative displacements throughout the joint is obtained from standard shape functions, Ni, defined in a curvilinear coordinate system on the contact surface. Thus the global coordinates r and the relative displacements, u, of a point on the contact surface can be expressed as: r = N rm

(1)

u = N Um

(2)

397

Figure 1. Basic variables of the joint element.

8

a) 16

b)

~ 1 5

t

&

PM3

7

PH

1

5

3

//~~~ _/~ 9

S 2

Figure 2. a) Surface (2-D) quadratic joint element, b) Line (l-D) quadratic joint element. where r,,, and U m are the coordinates and the relative displacements of the mid-plane points. Typical expressions for comer and midside nodes for the quadratic line element are: N1 = 1/2 s(s - 1) ; N2 = 1 -

(3a)

s 2

and for the quadratic surface element: N[ = 1 / 4 ( 1 - s )

(1-t)(-1-s-t);

N5 = 1 / 2 ( 1 - s 2 ) ( l - t )

(3b)

where s and t are local curvilinear coordinates. In the line element the local reference system is constituted by simply two unit vectors: the tangent and the normal to the contact surface. In the surface element the local reference system is defined by three unit vectors: the normal to

398 the surface and two tangential vectors (Fig 3). The tangential plane is defined by vectors Or/Os and Or/Ot and the normal by n =

--l(0_sr A

0~)

J

(4)

where A indicates vectorial product. The expression for the jacobian is J=

Or Or ~ss A ~-~

(5)

The local reference unit vectors are grouped in a matrix, G.

Figure 3. Local reference coordinate system for surface joint element. 2.2. L i n e a r f o r m u l a t i o n

To obtain the basic element stiffness for linear analysis it is assumed that, in the local reference system, the relationship between stresses, trj, and relative displacements, uj, can be expressed as:

oj = D(uj - u ~ ) + tr]

(6)

where D is the constitutive law stiffness matrix, tr~ an initial stress vector and u] an initial strain vector. Local relative displacements, uj, and nodal displacements, 8 are related by the geometry matrix B: uj = B 8

(7)

For a joint element with 2n nodal degrees of freedom, B is given by B = ( - B 1 , - B 2 . . . . . - B , , , B l , B2 . . . . . B,,) = ( - B r , Br)

(8a)

399 and Bi -" Ni

G

(8b)

Prescribing equilibrium the element nodal force vector fe is fe = K e ~ + ~

(9)

where

Ke = fe Bt D B ds

(10a)

= fe Bt (o'~ + Du~)ds

(10b)

The structure of the resulting stiffness matrix K e is K e = ( _KeoKe -Ke~ e

(11)

where K e = Btr

D Br

(12)

The partial matrix K e is the element stiffness matrix in terms of relative displacements of the midplane points, urn, fe = Keo Um

(13)

The formulation for elements of different interpolation order (linear or higher than quadratic) follows the same steps, the only modifications required are the change of the number of nodes and of the shape function expressions. 2.3. Non linear analysis In general the behaviour of rock joints will be described by a non-linear constitutive law. The resulting analysis will also be nonlinear and an iterative procedure will be required. In that case equation (9) is expressed incrementally as: &~ = K~ A~ k + Affok

(14)

and

Kek =

f = f

B T Dk B ds

+ Dk U)~

(15a)

ds

(15b)

400 where Dk is the constitutive stiffness matrix used in a particular iteration and tr~ and u)~ play the role of 'initial' stresses and strains. They depend on the specific iterative scheme used. The matrix Dk can also take different forms (initial, tangent, secant or others) depending on the solution strategy adopted.

3. CONSTITUTIVE LAWS AND STRESS POINT A L G O R I T H M The special characteristics of the mechanical behaviour of rock joints imposes certain constraints on the algorithms used in the integration of the corresponding constitutive laws. Although a general discussion on suitable constitutive laws for rock joints is outside the scope of this contribution, in order to illustrate some important points concerning the stress point algorithm reference will be made to an elastoplastic constitutive law proposed in [13] and fully described in [ 16-17]. Basic points of this constitutive law are: a) yield surfaces (and hence the failure surface) are described by a non-linear hyperbolic equation: F = z 2 + z2 _ tan 2 tp (tr 2 + 2act) = 0

(16)

where a and tan q9 are hardening parameters (Figure 4a). b) variation of hardening parameters with plastic strain according to the law indicated in Fig. 4b, where the internal variable is d~ -" ~/(dvlP) 2 + (dvP) 2

(17)

and dv p and d v ( are the plastic components of the incremental shear relative displacements. c) dilatancy varies with stress level and plastic strain and is controlled by the plastic potential. In the tangential plane the plastic potential is a circle. d) nonlinear elastic normal stress/normal relative displacement relationship including locking behaviour (Fig. 4c). No stresses are allowed when the joint is open. As demonstrated in [ 16-17] this constitutive law is able to reproduce satisfactorily a wide range of rock joint behaviour observed experimentally.

3.1. Integration algorithm Different cases must be distinguished when integrating the constitutive law for rock joints. In the first place the joint may be open or closed. If the joint is open it must be ensured that no stresses are transmitted across it. In case of a closed joint, it is necessary to distinguish between a closing and a opening joint. A form of 'initial stress' iterative scheme will be adequate if the joint is opening but may not be applicable if the joint is closing due to the 'locking' shape of the normal stress-normal displacement curve (Fig. 4a). However in the latter case a pure 'initial strain' type of integration is not appropriate either due to the shape of the tangential stress/tangential relative displacements (z - v) curves. For the case of a closing joint the following 'mixed' initial stress initial strain procedure is advocated. It is assumed that stress and strain increments are related by a constitutive matrix D. do'j = D d u j

(18)

401

(7"

2nd degree parabola ap

3rd degree polynomial

b)

c)

I

ar

ao

v

~p

~r

Umc

v

u

Figure 4. Elastoplastic model for rock joints, a) Yield surface, b) Hardening law. c) Normal stress/relative normal displacement relationship. In the 'initial stress' type of algorithm, equation (18) is used directly (with a suitable subincrementation scheme). In the mixed procedure, dcr, dvl and dr2 are considered known and du, dr1 and dr2 must be computed. Partitioning D as D._IDnn

Dtn

Dnt] Dtt

(19)

the following equation is obtained:

[,uI [, drl dr2

= Dnnl

Dtn

ont 1[, 1

DnnDtt- Dnt

dr1 do2

that can be used directly in the 'mixed' integration procedure.

(20)

402 Finally if the constitutive model is an elasto-plastic one, distinction between loading and unloading relative to the yield surface is also required. Fig. 5 shows a computational scheme for the integration of the rock joint constitutive law in which all the different cases are explicitly considered. i,,,,,~u~ ~r~ a;..t',.,,,r, u,, rt] ~-os~ ~,Em ~.a;,' /

['NnCE'DpAoaax~ " :~ t~'.n.,an~~ ] FIND du.~Z~~na .~1~

[ELASTICDETERMINATION ~'

[I~T~RI~I~T~ POINT~:"

]

[I E3.AST~: u.~Os ~ ~',

!

[

i

OF

i tl.~

,b,~ . ~ 1

t

[

~s~

*mp0sto sr~.~ wuEsl ~, ,r... ~, /

ASSlG~ZEROSI'RESS~I.UES l 9 ~, ,o l

AS~C~CX~4FqJ~WCJ~ u. ~, .t~ .u~, g , ~

1".~,N PIn,POSED,UES ] J~ttlB O~ L u.t, .t~ .u,,.W,~.

9 '1

w

. s s . PACY~)SL~) .~S l ASSIGN~ ~ VlI.UIES ~ .0

Figure 5. Computational scheme for the integration of a rock joint constitutive law.

3.2. Simplified constitutive law and analytical integration In the analysis of real geotechnical structures the number of rock joints may be high and, therefore, the CPU time spent integrating the constitutive law according to the above procedure may be a large proportion of the total computational effort. In problems in which an exact reproduction of rock joint behaviour is not crucial, the use of a simplified law permitting analytical integration can be very advantageous as it results in a significant reduction of the computational resources required. Such a model can be derived assuming that the rock joint mechanical behaviour has the following characteristics: - elastic-perfectly plastic - hyperbolic yield surface according to equation (16) - linear normal stress-normal relative displacement behaviour - no dilatancy Figure 6 summarizes the type of behaviour described by this simplified model. The integration problem can be stated as that of obtaining the new values of the stresses (or, zl and "!:2) given an arbitrary imposed relative displacement increment (Au, Avl and Av2). The initial stress values are denoted as Cro, (rl)o and (r2)o. As shown in Fig 7, the angle of the

403 direction of the incremental relative displacement, Av, with the Avl-axis is called ft. Similarly, the angle of the tangential stress vector with the rl-axis is called 0.

e

O"n

O'n

v

c)

Figure 6. Simplified constitutive law permitting analytical integration.

. . . .

__~

a)

z~TPl

v~l

/ 'A

v2)o

7"1 v

//~_~

J

b)

_ _ ..t _

Vl

(vl)o

v

Figure 7. a) Yield surface in the r l - r 2 space, b) Variables in strain space. The normal stress, a, is related to u by the elastic relationship (21)

a = Knu

since dilatancy is assumed zero. The basic expressions for the incremental tangential stress are d'cl -- K t d v ~ = K t ( d v l

-dv

p)

"

dr2--

Ktdv~ = Kt(dv2-dv;)

(22)

Taking into account the flow rule in the rt - r2 plane, the plastic part of the tangential relative displacement increment can be written as dv p = dv p

cos0

" d v p -- d v p sin0

(23)

Also, the increments of rl and r2 are related to the increments of r and 0 according to drl = dr cosO - r sinOdO

9 d r 2 = d r sinO + r c o s O d O

(24)

404

From (22), (23) and (24) the following differential equation is obtained: Ktdv

dO

r

sin (/~ - O)

(25)

This expression can be integrated because it is possible to reduce the left hand side to an expression with a single variable since d v is related to du as d o = zXu Av d u , u is related to cr as indicated in (21) and r depends on ~r according to the hyperbolic expression (16). Substituting in (25) and integrating, the equation KI Av

tan ( / ~ - 0 2

15 ) =tan(,

0(o) [ __.+ ~/_.f2__-F.a2_tan2_~ ]-x,A, tan~ %

) r(o) + ~r

+ a2 tan2 qb

(26)

is obtained. From (26) the new value of 0 at the end of the relative displacement increment is determined. Then, the complete new stress state can be determined since cr is obtained from (21), ~ is derived from (26) and rl=r cos0; r2 = T sin0. If desired, the plastic relative displacement increment can also be easily computed. To illustrate the saving in computational time associated with the use of this simplified model, an example of application simulating a pull-out test of a geotextile reinforcement has been performed. It is a 3-D finite element analysis with 906 degrees of freedom in which both soil and reinforcement are assumed elastic and the behaviour of the soil/geotextile interface is described by means of the simplified model outlined above. More details of this analysis are given in the next section. Figure 8a shows the CPU times spent in the various runs performed. When the standard method of solution is used (i.e. no line search) the use of analytical integration results in a moderate reduction in computer time. However, the reduction is much larger when the 'line search' option is used since, in this case, many more constitutive law integrations need to be carried out. The same trends are even more noticeable in Figure 8b where only the CPU time spent during the integration of the constitutive law is plotted. It is interesting to note that the differences between the various runs is small during the first increments of loading. However, as failure is approached and more iterations are required for convergence, the computational advantages of using analytical integration become more apparent.

4. Q U A D R A T U R E

SCHEMES AND ELEMENT PERFORMANCE

As sliding is by far the most usual failure mode in rock joints, it is necessary to assess the performance of this family of elements in situations of this type. It has been observed (e.g. [ 18-20]) that the use of a conventional Gauss quadrature scheme in conjunction with a high stiffness joint moduli leads to spurious oscillatory stress profiles especially in the zones where high relative displacement (strain) gradients occur. Strain gradients reach a maximum during the passing of a sliding front across an element. The same type of behaviour has been reported in finite thickness elements when equivalent integration schemes and stiffness values are used [14]. Because of the difficulty of the element in reproducing the high strain gradients associated with a sliding front, the bad performance of the element may be attenuated by using a larger

405 number of elements and accepting the ensuing higher computational costs. As an alternative, the performance of the elements can be improved drastically by changing the type of numerical integration scheme. The change of the quadrature scheme not only affects the accuracy of integration but changes the structure of the stiffness matrix as well. This structure change can, in some cases, modify significantly the kinematic constraints of the element.

1o$

NOU,N__EE SEARC_____~H .....-~---. ,ncr~nt,, i~

~)

...... ,~----- Analytical int. 12

6

NO LINE SEARCH

~

.....~ - - - - - t t , ~ t i,,. . . . . ~9- - - - - Anolyticol

9

/o

a/

/ /

int.

o

5

o / a

Incremental

Anolyticol int.

/ j /

LINE SEARCH int.

/

v

r

Analytical int.

7

/ 3

.7.9 6 5

/"

4

/

..~

d':..-""

3

0

/

LINE SEARCH

I nertm~mtol int.

11 10

o. r

/ /

Z

0

1

2

3

4

5 6 7 Increment

8

9

10

11

12

0

0

1

2

3

4

5

6

7

8

9

110

11

12

Increment

Figure 8. CPU time used in the 3-D analysis of a pull-out test. a) Total CPU time. b) CPU time spent integrating the constitutive law. A convenient way to explore the kinematic behaviour of the element as a sliding front moves across it is by observing the eigenvalues and eigenvectors of the stiffness matrices resulting from different quadrature schemes. Because of space limitations only the results for full Gauss and full Newton-Cotes integration schemes for the quadratic elements will be shown. In fact, it is not necessary to carry out an eigenvalue analysis of the full stiffness matrix, K e, but it is sufficient to compute the eigenvalues and eigenvectors of the partial matrix K e. It can be shown that the non-zero eigenvalues of K e are just twice those of K e. Moreover the eigenvectors of the full matrix K e corresponding to the additional zero eigenvalues are modes of deformation implying equal movement of adjacent nodes which, of course, are zero energy modes given the formulation of the element. Therefore, the non-trivial eigenvalues and eigenmodes of the element are contained in the partial matrix K e. To simplify the analysis of the results, it is assumed that the matrix D representing the constitutive law is uncoupled with equal elastic shear and normal stiffness. When the sliding front reaches an integration point shear strain stiffness becomes zero. In the case of the quadratic line joint element (Fig. 9), the application of the Newton-Cotes scheme (which in this case coincides with Lobatto's) leads to a diagonal K e stiffness matrix. Therefore the eigenmodes are unit vectors corresponding to unit relative displacements of each

406 2'

3'

1'

2'

~b

~c

=•

i,b

1

2

l'

2

1

Newton

~, --"

-

3 Cotes

3'

~.c_ 3"

Gouss

1,67/0/0

1.67

6.67 I 6,6710

|

b)

1,66

0,tK)

O--~---~--o---O 0,80

c)

d~

,~

=

- o. oo~

1,17

0,~0

1,66

0,80

o0-"

1,66

~ --- -o.oo3 ~

S, 53

I.&6

=,','0

5,53

5,0&

5,S

0,00

~

0

3

1,77

~

5,53

Figure 9. Eigenvalues and eigenmodes for the line joint element: a) Newton-Cotes integration. b) Gauss integration with no sliding, c) Gauss integration with partial sliding affecting integration point a. d) Gauss integration with partial sliding affecting integration point b.

407 node (Fig. 9a). If the sliding front reaches the element, the shear stiffness of integration point a) will become zero and the deformation mode corresponding to the tangential displacement becomes a zero energy mode and can proceed unrestrained independently of the rest of the element. In the same way, when the sliding front reaches point b the eigenmode associated with the relative displacement of node 2 becomes a zero-energy mode. Therefore the element can reproduce, without spurious deformation couplings, the passing of a sliding front across the element. In the case of Gauss integration the element can not accommodate the deformation associated with a sliding front so easily. As Figures 9c and 9d show, the zero or near zero energy eigenmodes involves coupling of zones of the element that are undergoing sliding with parts that are not. Independent sliding of nodes is not therefore possible and spurious results are a strong possibility. Similar conclusions are obtained even when using a reduced 2x2 Gauss quadrature scheme. Another way of explaining the beneficial effect of employing a quadrature schemes with integration points at the nodes is to realize that in this case the resulting stiffness matrix is independent of the shape of the interpolation functions, N, used. This is due because the derivatives of N do not enter in the formulation of zero-thickness joint elements. As a consequence the stiffness matrix will accommodate any particular distribution of relative displacements inside the element without being constrained by the particular shape of the interpolation functions used. To illustrate the difference in performance caused by different integration rules an example simulating a pull-out test of a reinforcement strip is presented (Fig. 10). Naturally in this type of problem the sliding mode of deformation is dominant. A normal stress of 10 kPa is initially applied to soil and reinforcement. Afterwards, the reinforcement is pulled out, under displacement control, from a soil mass which is restrained at the boundaries. Both soil and reinforcement are considered as linear elastic materials. The simple elastic-perfectly plastic model of Fig. 6 is adopted for the interface. The following parameter values were used in the case shown here: a = 0 ; 99 = 30~ ; Kn = 106kPa/m ; Kt 106kPa/m, where Kn and Kt are the normal and tangential elastic stiffnesses respectively. =

Soil

Interface

Reinforcement

41,. ! t

l Figure 10. Finite element mesh for the 2-D analysis of a pull-out test.

408 Fig. 11 shows the shear stresses in the interface for two values of reinforcement displacement. The different results obtained with the two types of interpolation schemes are readily apparent. The analysis using the Gauss integration scheme shows large oscillations which can not be part of a genuine solution to the problem. It has been observed that oscillations usually arise near the sliding front and propagate thereof. Fig. 11 also shows that good results are obtained with Newton-Cotes quadrature schemes. In fact the analysis using this scheme performed satisfactorily to the end of the analysis when full sliding of the reinforcement was achieved. Full details are given [ 18]. Similar type of results were obtained for the same problem using a wide range of realistic parameters for soil reinforcement and joint stiffness. For the line and surface linear joint elements the same type of dependence of performance on the quadrature scheme is predicted from eigenvalue analysis. The predictions have been confirmed by numerical experiments using the same pull-out test problem. 8 , 6

a~ J

8,1 -

0 N e w t o n l C o t e s , 3 points I - - 0 - - Gauss, 3 points I

,IX

.I\

l

j

l

~

~

~

1

n.t.

-e I -gO

'' ,I ~

~ I.

,-

-,

~.,l

'2'_ - & " .

~ " 0'2

I'

i

AL

.~_

-

'/1

Ill

.

~'

~4t

~

b)

~~---~-----~---e---~ . . . . . . . . . . . . . . . . . . . . . . . . . .

1

\

~

"

.

.

.

.

.

.

.

.

.

.

.

.

'

0'6

'

0'8

"

.

.

.

.

,,

,

''/~" ~""

~

I

I

,/,,"

I

I

I

0

~

] -41 0.0

.

i,"

~, " 0'4

,t

.

,

\

points

Neutron-Cotes, 3 - G a u s s , 3 points

10

length (m)

i/ . . 0.'2

. . 0.4

.

O.'S

0.'8

.

1.0

length (m) Er/Es=lO0. a) at 6=1 mm, b) at 5=3 mm.

Figure 11. Shear stress from a 2-D analysis with

The case of the quadratic surface joint element is somewhat different. The position and weights of the integration points of three quadrature schemes are shown in Fig. 12. It can be observed that the Newton-Cotes scheme (which again coincides with Lobatto's) has now an integration point that does not correspond to a node. In the lumped scheme integration points and nodes coincide but the weights associated with the comer nodes are negative. This leads to stiffness matrices with negative eigenvalues which are not acceptable in analysis. 0.077 X

0.~3 X b

0.077X c

0.123 X d

0.198 X e

0.123 X ,, f

0.077 X g

0.123 X h

0.077 N ~

1/36 ~a /

1/9 X b

1/36 ~ c

1/9

4/9 X e

1/9

d

g

h

i

g

1/9

1/36

-1/12

1/36

(a)

(b)

f

-1/12 I( a

1/3 X b

1/3 )I(

-1/12 " c

1/3

d

f

h 1/3

i -1/12

(c)

Figure 12. Integration schemes for the surface joint element, a) 3x3 Gauss b) 3x3 Newton-Cotes c) Lumped scheme.

409 - 0,003

O, O0

0,17

IZ x

b)

0,00

c)

,t=

d)

~9 - ' -

O, O0

- O, 004

- 0,009

- 0,004

O, O0

0,00

0,01

-0,001

0 O0

O, 04

0,006

O, OO&

Figure 13. Eigenvalues and eigenmodes corresponding to the lowest energy deformation modes: a) Newton-Cotes integration with partial sliding affecting integration points a, b and c. b) Newton-Cotes integration with partial sliding affecting integration points a, b, c, d, e and f . c) Gauss integration with partial sliding affecting integration points a, b and c. d) Gauss integration with partial sliding affecting integration points a, b, c, d, e and f .

410 Using the same constitutive assumption as in the line element, the lowest energy eigenmodes of the element for two assumptions of partial sliding in the y direction are presented in Fig. 13. Gauss and Newton-Cotes integration schemes are considered. It can be noted the Newton-Cotes integration rule does not show a complete uncoupling of the nodes undergoing sliding until the sliding front has reached the center of the element (Figs 13a and 13b) and that the Gauss scheme implies coupling for all partial sliding assumptions (Figs. 13c and 13d). In case of partial sliding involving only the first row of the integration points both integration schemes permit an uncoupled linear movement of the first row of nodes across the element. It is also noteworthy that the zero or near zero energy eigenmodes contain only relative displacements in the y direction and are not coupled to movements in the x coordinate direction. The performance of the element has been tested in a 3-D version of the pull-out test previously described (Fig. 14). The distribution of shear stresses on the central line of the interface between soil and reinforcement at a displacement S = 1 mm are plotted in Fig. 15a. Good results are obtained using Newton-Cotes integrations and moderate oscillations are observed in the analysis using Gauss quadrature. The analysis using the lumped integration scheme did not converge. Full results of the analysis using Newton-Cotes integration are described in [18]. A good element performance was observed throughout. In Fig. 15b the results for the same problem assuming an elastic interface are presented. Newton-Cotes and Gauss schemes provide results similar to the nonlinear case but large oscillations result from the lumped integration scheme case. It is likely that the good quality of the results in the Newton-Codes case are favoured by the fact that the case studied imposes an approximately uniform relative displacement across the reinforcement. As the eigenvalue analysis shows, this type of displacement can proceed uncoupled and a good element performance should be expected.

Soil

Reinforcement O.]6e

Interface

Figure 14. Finite element mesh for 3-D analysis of a pull-out test.

411 8 - - ~ - - Gauss.Newt~176

6 ~ ~,,

3x3

" ~ 4 \ \\

b)

o Newton-Cotes, 3x3 --O--Gauss. 3x3 - B- - Lumping scheme

,~ iv.\

'~/" A \

//

\\

b

\\\

-2

~t

-~.o

o.'2

0.'4 Zength

0.'6 (m)

0.'8

1.0

-~.o

'

o.'2

" o.'4 " o.'s " o.'e Zength (m)

///

"

~.o

Figure 15. Distribution of shear stress along the central line of the interface for different integration schemes, a) Elasto-plastic interface, 3=lmm. b) Elastic interface. When a more general sliding mode is applied the element performance is expected to deteriorate. However, because the coupling between nodes is introduced by a single integration point, it should be expected that an element integrated with the Newton-Cotes scheme will perform better then an element using Gauss quadrature where all integration points introduce coupling to some degree. The potential problems arising from the presence of the central integration point of the Newton-Cotes schemes would disappear if a 18-node lagrangian interface element is adopted. This strategy would not increase the size of the system of equations as the central node equation can be condensed out before assembling the global stiffness matrix. The main difficulty of this approach lies on the fact that lagrangian continuum elements are not usually implemented in conventional codes. Of course, linear 8-noded surface elements can be used in which all integration points of the 2x2 Newton-Cotes scheme are located at the nodes and a good element performance follows in all cases. Further studies have been made ([21-22]) on the interaction between numerical integration schemes and element performance for a similar range of interface elements. The conclusions reported are consistent with those outlined in this section.

5. EXAMPLES OF APPLICATION 5.1. Large rock slide at Cortes de Pallfis The first example of application concerns the analysis of a large rock slide immediately upstream of the Cortes II dam located on the Jt~car river in Eastern Spain. It was an old landslide reactivated by quarry excavations carried out in the lower part of the slope. A cross section of the slide is shown in Fig. 16a. Sliding was concentrated along a thin mudstone layer embedded in the general limestone formation. The volume of the sliding mass was of the order of five million cubic meters. Stabilization measures were can'ied out consisting basically of excavating the upper zone of the landslide and using this material to fill the lower part of the slope (Fig. 16b). A full description of the case history is given in [23]. Although the correction measures were designed on the basis of conventional limit

412 equilibrium analysis, their detailed effects were examined via a 2-D FEM analysis. Because the location of the sliding surface was perfectly identified, line joint elements were used for its representation. Additional joint elements traversing the upper part of the slide were included to provide kinematically admissible mechanisms that could simulate the observed sliding phenomenon. The mesh used and the initial stress distribution adopted are depicted in Fig. 17. _j.

EXCAVATIONZONE

1

6.60 ~0

t

t

~20 e 400

LOADING ZONE

NEUTRAL ZONE

P-6 S

"

t

OR G IN IALPROF L IE

zQ 380 t-. 360 uJ 36.0

_.1 IJ.J

6'/

....

~ / ~ ~ P Q R O F I L E

"...MAX.LRESEELR V 0,R

320 300 280

EXCAVATED f-ZONE

4.60 ~0

ORIGINALPROFILE OF THE QUARRY

_. ~20 s 4.00

LOADING

Z

o 380 I'--

..j 3Z+O

b)

. . . . . . . . . . . . . . . . . . . .,

t.LJ

320 300

0

50m

280

Figure 16. Representative section of the landslide, a) Geological conditions, b) Remedial measures scheme. The simulation of the quarry excavation and subsequent remedial measures was carried out in stages (Fig. 18). Stages 1 to 5 reproduced the progressive quarry excavation. Stage 6 simulated the excavation of the upper wedge of the slope and stages 7 and 8 modelled the loading of the lower part of the slide. From the stresses computed on the joint elements corresponding to the sliding surface it was possible to compute margins and factor of safety using more realistic stress distributions than those assumed in limit equilibrium analysis. The results are indicated in Fig. 18. More informative is the examination of the local safety margins along the sliding surface presented in Fig. 19. It can be seen that at the end of the quarry excavation the available safety margin is small and concentrated in the lower part of the slope. The increase in safety margin along the slide surface due to remedial measures can be observed in Fig. 19b. It is

413 noteworthy that although the global safety margin after applying the remedial measures is similar to that prevailing before quarry excavation, their actual distribution is in fact quite different. The resistance mechanism of the slide has therefore changed significantly. More details of the analysis are presented in [23-24].

Figure 17. Slope discretization and initial stress distribution. S.M.

STAGE

[~

Before

(x

10

F.S.

MNI

quarrying

!-~

J~l

3.00

1.285

1.290

1.16&.

tot,

1.157

Intermediate

Excavation of e

I I

t 1.51 ~

1.t,OS

II

Finat st.ate

I

T 2.98

1.505

Figure 18. Geometry changes and associated Margins (M.S.) and Factors (ES.) of Safety.

414 The finite element analysis carried out using joint elements not only allowed a better understanding of the sliding mechanism but it also increased confidence in the expected effects of the correction measures recommended.

......

~T 50 kPa

1

lal

3 0

/\ i

t

"..

.....

5

"Y''~

2i ..... I.. Figure 19. Local Margins of Safety along sliding surface, a) Before remedial measures, b) After remedial measures. 5.2. Stability assessment of an arch dam founded on fractured rock

Canelles dam is a 140 m high arch structure located on the Pyren6es, in northeastern Spain. It is founded in Cretaceous massive limestone which is visibly fractured by a main set of vertical joints (principal system) parallel to the valley. A second family of discontinuities is constituted by bedding planes dipping approximately 45 degrees upstream. Fig. 20 shows a plan view of the dam with some of the field observations of joint orientations. Because of its position at the end of the valley the dam has been subjected to an intense effort to evaluate its stability conditions. A significant element in the safety evaluation activities has been the performance of a numerical analysis of the problem. In the case of an arch-dam three-dimensional effects are probably predominant so a 3-D analysis was deemed necessary. The discretization of dam and foundation rock is depicted in Fig. 21. All the contacts between continuum elements are provided with surface joint element representing rock discontinuities. Joint elements are also included in all the contacts between dam and reinforcing wall with the rock. Linear elements are used throughout. The final mesh is composed of 1208 solid elements and 1505 joint elements with a total of 10758 degrees of freedom. Due to the complexity of

415 trying to approximate the real geometry and of including a sufficient number of discontinuities, several types of 3-D solid elements have been required" hexahedron, prism (triangular base), pyramid (quadrangular base) and tetrahedron. Consequently both quadrangular and triangular surface joint elements have been used.

J

\

\ HN

LEGEND I

STRATIFICATION (Strike and dip)

J

SHEAR JO4NTS (E-W and N-S Systems) VERTICAL JOINT (Princlpol system)

Figure 20. Plan view of the dam with rock joints. The analysis has involved four construction stages (1 for the foundation rock and 3 for

416

the dam) and a loading stage in which the hydrostatic load has been applied in increments. Solid elements have been considered elastic and the nonlinearity has been concentrated in the joint behaviour where the simplified model described in 3.2 has been used. Because of the uncertainties associated with estimating initial stresses two values for the far field Ko have been tried: 0.43 and 0.9. The results of the analysis indicated no instability under full hydrostatic load. Typical results of stresses acting on joints are shown in Fig. 22 in terms of contours of normal stress and available shear strength. A more detailed description of the analysis is given in [25-26].

Figure 21. Finite element mesh (hidden lines removed). Availability of stresses acting on the joints permitted the computation of factors of safety of a large number of possible slide mechanisms and the identification of the most critical ones. The minimum factor of safety in terms of strength parameters, Z~o, was obtained for the local abutment failure shown in Fig. 23. This result is consistent with failure mechanism observations made in model tests carded out as part of the safety evaluation study. The values of the factor of safety obtained were ~.~o=1.78 (Ko=0.43) and ~.~0=1.89 (Ko---0.9). Hydrostatic load was increased beyond its nominal value but instability was not observed even when it reached 300% of the maximum value. General stress redistribution and the increase in normal stresses acting on the joints meant that the most critical factor of safety reduced only to ~.~o= 1.30 (Ko=0.43). Again this example shows that the performance of a FEM analysis with a explicit modelling of rock discontinuities by means of joint elements provided a fuller appreciation of the resisting mechanisms of the dam foundation and a more sound basis for the assessment of the margin of safety available. Information from the analysis has contributed in a significant way to the final engineering evaluation of the degree of safety of the dam.

417

Fig. 22a ~ Normal stress contours on vertical joint.

Fig. 22d - - Available shear strength on vertical joint.

Fig. 22b - - Normal stress contours on bedding plane.

Fig. 22e - - Available shear strength on bedding plane.

Fig. 22c - - Normal stress contours on horizontal

Fig. 22f B Available shear strength on horizontal

discontinuity.

discontinuity.

418

Figure 23. Critical failure mechanism in the left abutment.

6. CONCLUDING REMARKS

Joint elements provide a powerful tool to model rock discontinuities in FEM analysis. A unified formulation for zero-thickness isoparametric joint elements for two- and threedimensional analysis has been presented. The mechanical behaviour observed experimentally in terms of relative displacements can be readily incorporated by means of appropriate constitutive relationships. Some features of this mechanical behaviour require special procedures in the stress point algorithm that have been described in section 3. It has been shown that the numerical integration scheme adopted influences strongly the performance of the element especially when high relative displacement gradients occur. The adoption of quadrature rules in which integration points coincide with the nodes uncouple the stiffness matrix. As a result a radically enhanced performance of the element is obtained. The joint elements described in this contribution can be profitably applied to the analysis of real engineering structures. As the two examples of application described demonstrate, FEM analyses using joint elements for the modelling of rock discontinuities can provide information of high engineering significance.

419 ACKNOWLEDGEMENTS

The support provided by the Direcci6n General de Polftica Cientffica y Trcnica through research grants PB90-0598, PB92-0702 and PB93-0955 is gratefully acknowledged. The Authors are thankful to their colleague P.C. Prat for his assistance during the preparation of this contribution.

REFERENCES

[ 1] Goodman, R.E., Taylor, R.L. and Brekke, R.L. (1968) A model for the mechanics ofjointed rock, J. Soil Mech. and Found. Div., Proc. A.S.C.E., 94, 637-659. [2] Anderson, H.W. and Dopp, R.L. (1966) FEM applied to rock mechanics, Proc. 1st ISRM Congr., Lisbon, 2, 317-321. [3] Frank, P., Guenot, A. and Humbert, P. (1982) Numerical analisis of contacts in Geomechanics, Proc. 4th. Int. Conf. Num. Meth. in Geomech. (Eisenstein, Z. Ed.), Edmonton, 1, 37-45. [4] Ahmad, M. and Bangash, Y. (1987) A 3-D bond analysis using FEs, Comput.& Struct, 25, 281-296. [5] Zienkiewicz, O.C., Best, B., Dullage, C. and Stagg, K.G. (1970) Analysis of nonlinear problems in rock mechanics with particular reference to jointed rock systems, Proc. 2nd. ISRM Congr., Beograd, 2, 501-509. [6] Pande G.N. and Sharma K.G. (1979) On joint/interface elements and associated problems of numerical ill-conditioning, Int. J. Num. Anal. Meth. Geomech. 3,293-300. [7] Desai, C.S., Zaman, M.M., Lightner, J.G. and Siriwardane, H.J. (1984) Thin-layer element for interfaces and joints, Int J. Num. Anal. Meth. Geomech., 8, 19-43. [8] Griffiths, D.V. (1985) Numerical modelling of interfaces using conventional finite elements, Proc. 5th. Int. Conf. Num. Meth. in Geomech. (Kawamoto, T. and Ichikawa, Y. Eds.), Nagoya, 2, 837-844. [9] Schweiger, H.E, Haas, W. and Handle, E. (1990) A thin-layer element for modelling joints and faults, Proc. Int. Conf. on Jointed and Faulted Rock (Rossmanith, H.P. Ed.), Vienna, 559-564. [ 10] Tardieu, B. and Pouyet, P. (1974) Proposition d'un modrle de joint tridimensionel courve, Proc. 3rd ISRM Congr., Denver, 2B, 833-836 [11] Carol, I. and Alonso, E.E (1983) A new joint element for the analysis of fractured rock, Proc. 5th. Int. Conf. Rock Mech., Melbourne, F147-F151. [12] Beer, G. (1985) An isoparametric joint/interface element for FE analysis, Int. J. Num. Meth. Engng. 21,585-600. [ 13] Carol, I., Gens, A. and Alonso, E.E. (1985) A three dimensional elastoplastic joint element, Proc. Int. Symp. Fundam. of Rock Joints (Stephansson, O. Ed.), Bjorkliden, 441-451. [ 14] Hohberg, J.M. and Schweiger, H.E (1992) On the penalty behavior of thin-layer elements, Proc. 4th Int. Symp. Numerical Models in Geomech. (Pande, G.N. and Pietruszczak, S. Eds.), Swansea, 1,241-248. [ 15] Hohberg, J.M. (1992) A Joint Element for the Nonlinear Dynamic Analysis of Arch Dams, Birkh~iuser Publishers, Basel.

420 [16] Gens, A., Carol, I. and Alonso, E.E. (1989) Elasto-plastic model for joints and interfaces, Proc. 2nd Int. Conf. on Computational Plasticity (Owen, D.R.J., Hinton, E. and Ofiate, E. Eds.), Barcelona, 2, 1251-1264. [17] Gens, A., Carol, I. and Alonso, E.E. (1990) A constitutive model for rock joints. Formulation and numerical implementation, Computers and Geotechnics, 9 (1990) 3-20. [18] Gens, A., Carol, I. and Alonso, E.E. (1988) An interface element formulation for the anlysis of soil-reinforcement interaction, Computers and Geotechnics, 7 (1988) 133-151. [ 19] Rots, J.G. (1988) Computational modelling of concrete fracture, Ph.D. Thesis, University of Delft. [20] Hohberg, J.M. (1990) A note on spurious oscillations in FEM joint elements Earthq. Engng. Struct. Dynamics, 19, 773-779. [21] Schellekens, J.C.J. (1992) Computational strategies for composite structures, Ph. D. Thesis, University of Delft. [22] Schellekens, J.C.J. and de Borst, R. (1993) On the numerical integration of interface elements Int. Jnl. Num. Meth. Engng, 36, 43-66. [23] Alonso, E.E., Gens, A. and Lloret, A. (1993) The landslide of Cortes de Pallas, Spain, G6otechnique, 43,507-521. [24] Alonso, E.E., Gens, A., Gili J.A. and Lloret, A. (1992) Analysis of the stabilization of a large landslide in Eastern Spain, Proc. 6th Int. Symp. on Landslides, Christchurch (Bell, D.H. Ed.), 1,637-641. [25] Gens, A., Alonso, E.E., Carol, I. and Prat, P.C. (1993) Safety evaluation of an arch dam founded on fractured rock, Proc. Int. Symp. Appl. Comp. Meth. in Rock Mech. and Engng. (Liu, H. Ed.), Xian, 2, 843-850. [26] Prat, P.C., Delhaye, C., Gens, A., Carol, I. and Alonso, E.E (1994) Safety evaluation of an arch dam founded on fractured rock based on a 3D nonlinear analysis with joint elements, Proc. Int. Work. on Dam Fracture and Damage (Bourdarot, E., Mazars, J. and Saouma, V. Eds), Chambery, 211-220.

Mechanics of Geomaterial Interfaces

A.P.S. Selvadurai and M.J. Boulon (Editors) 1995 Elsevier Science B.V.

421

C o n c r e t e Joints J.-M. Hohberg IUB Engineering Services Ltd. Thunstrasse 6 (P.O. Box), 3000 Berne 6, Switzerland

Concrete joints are in many respects similar to rock joints. Based on previous work from geomechanics, an isoparametric zero-thickness element for curved 3-D interfaces has been extended to mixed-mode fracturing of grouted concrete joints. It is based on an elastoplastic formulation of double-mechanism softening and includes non-frictional features such as interlock and free closure. Also addressed is the question of the correct stress integration in the penalty formulation.

1. I N T R O D U C T I O N Joint elements discretize the space between two mechanically interacting bodies. They have the character of a 'bidimensional continuum' with a virtual thickness, which may be interpreted as a derelict dimension inherited from degenerating a continuum element. Despite their vanishing thickness the behaviour of physical matter - such as deformable surface asperities, a soft filler material, or a fluid - can be accounted for through an appropriate constitutive relation. Alternatively, the same class of elements is derived from the regularization of contact constraints by means of penalty parameters, which are essentially very stiff springs, Figure 1. Such elements are quite common in numerical geotechnics. The parenthood is equally shared by the well-known Goodman element of 1968 [1] and the nodal spring boxes first advocated for concrete cracking and reinforcement pull-out by Ngo and Scordelis in 1967 [2]. Because of this common background the element family can be found over a wide range and scale of applications, from modelling slip mechanisms in seismology, over rock, concrete and masonry joints, soil-to-pile and soil-to-geotextile interfaces, to fictitious cracks in fracture mechanics and aggregate-matrix debonding in concrete micromodels. The present volume contains examples of similar formulations. The application in simple shearing has almost become standard practice, but the simultaneous opening/closing and the loss of an initial cohesion are tackled less often. Among the earliest applications in three dimensions was the modelling of interfaces in arch dams [3, 4]. The seismic behaviour of grouted contraction joints between dam monoliths motivated also the present investigation [5, 6]. The need for a coarse discretization and the dynamic loading rendered the problem particularly challenging, cf. [7, 8].

422

Figure 1. Degenerated continuum (left) versus constraint regularization (right)

2. E L E M E N T

FORMULATION

Despite their widespread use the theoretical foundation of zero-thickness joint elements seems a little obscure to many civil engineers. The penalty stiffness is argued to be but an arbitrary 'fudge' parameter and held responsible for various non-physical phenomena. A brief recapitulation is thus expedient. 2.1. T h e P e n a l t y C o n c e p t In presence of discontinuities the variational boundary problem to be solved is subject to unilateral constraints. For instance, two bodies may only separate but must not penetrate each other along their common contact surface Fc, gTui > 60i

for all nodes i on F c

(1)

where

ui : unknown displacement vector in Cartesian components z, y, z gi : connectivity vector between constrained degrees of freedom [9] 80i : amount of normal relative displacement representing full contact. The conventional finite element method approximates the potential energy IIk through a kinematically admissible displacement field, interpolated over the element domain from nodal displacement vectors. To minimize Ilk for the unknown state of the indeterminate

423 boundary condition, the functional need be augmented either by Lagrange multipliers or a fictitious energy term, growing with the amount of constraint violation [10]:

1

)T

6oi ( g ~ u i - 50,) --~ Min.

H~(u) = IIk(u) + ~ee E ( g / T u i -

(2)

i

Eq. (1) will be satisfied in a least-squares sense by virtue of the quadratic form of the additional term, representing the work done by 'push-back' forces, where e << 1 regularizes the unilateral constraint problem [11]: 1 Pi(0 = -s ( g T u i - 5oi)

(3)

An algebraic equation system is obtained from the first variation of the potential in Eq. (2), to which the constraint conditions are added in their 'perturbed' form [12]:

O]{U} {f}

+

GT

.

[ --eI

.

.

.

pC

.

(4)

$o

where K : system stiffness matrix assembled from the individual elements f : given vector of external nodal forces. The contact forces p~ can be condensed out owing to their proportionality to the amount of constraint violation: K u + Gp~ GGTu-

eGp~

=

f

=

G6o

/ =v

[K+aGG T]u=f+xG/5o

,

n=-

1 s

(5)

This penalty-augmented system matrix is assembled in the usual direct stiffness procedure and the equation system - of the same size as without constraints - is solved in conventional Newton-Raphson technique. A simple example is given in Table 1. Note that the displacements of double-nodes become identical to more and more digits, as the terms of the constraint matrix x G G T grow with respect to the stiffness contributions of the solid elements in K. The absolute accuracy, however, will deteriorate owing to truncation and round-off errors. This can be checked by computing the condition number of the system matrix or by the approximate criterion [13]:

II~ull Ilull

_
k,.,.

(6)

where kmin : smallest stiffness term overlaid by the penalty parameter e : unit round-off error (~ 10 -17 in 32-bit double precision) n : total number of degrees of freedom in the system. In finite-precision arithmetic there exists thus an optimum choice of the penalty parameter between constraint satisfaction and truncation error [14].

424

Table 1 Introductory Example Consider two square solid elements separated by an initial gap ~oi, which is to close under a load q. The connectivity vectors gi of all constraints form a matrix G (the angular brackets denoting column vectors) and have as many rows as interface degrees of freedom, here the nodes 3 and 4 of the upper and 5 and 6 of the lower block:

A G --

A

A]

A

g35x g35y g46x g46y V V V V

--

1 0 0 0 -1 0 0 0

0 1 0 0 0 -1 0 0

0 0 1 0 0 0 -1 0

0 0 0 1 0 0 0 -1

ly E

I'ol ,S~ 10 -z

(7)

: IOO0

I

t--J~L~-~~k-;4___.__. .L

II

,'o /

These 'relative' displacements take the place of strains found e.g. in thin-layer elements.

----.

E :~ooo

._o

~u

,o~

I

/

~o-----~

After deleting the rows and columns corresponding to fixed supports, the stiffness matrix K comprises only the nodes 3 to 6 from the two unconnected solids with Ec = 1000, coupled by the penalty constraint matrix (in italics), e.g. with ~ = 10 5.

(K+KGGT) =

(8)

100500. 125. -250. -125.

125. 100500. 125. 0.

-250. 125. 100500. -125.

-125. O. -125. 100500.

-100000. O. O. O.

O. -100000. O. O.

O. O. -100000. O.

O. O. O. -100000.

-100000. O. O. O.

O. -100000. O. O.

O. O. -100000. O.

O. O. O. -100000.

100500. - 125. -250. 125.

-125. 100500. -125. 0.

-250. - 125. 100500. 125.

125. 0. 125. 100500.

The interpenetration Sy and the block stresses converge with increasing penalty to their exact values:

K

~y

O.yOp

~bottom vy

103 104 105 106 107

1.8000.10 -2 2.2195.10 -3 2.2444.10 -4 2.2494.10 -5 2.2499-10 -6

6.400 5.610 5.511 5.501 5.500

-3.600 -4.390 -4.489 -4.499 -4.500

425 2.2. I s o p a r a m e t r i c I n t e r f a c e E l e m e n t s The isoparametric formulation of joint elements of zero thickness has been published several times before, beginning with [15, 16]. The derivation consists of four basic steps: 1. define 'relative displacements' {bn,58, bt} as the difference of top and bottom displacements and interpolate those to the stress sampling points p, ~v = H v { A u , } --

-H

ub~176 I

+H

(9)

utOV P

2. interpolate also the nodal coordinates xi to the sampling points p and erect in each a local frame of reference {n, s, t }, Figure 2, np=

Oxp

Ox v

O~ x Or/

o

np

, n p = in v[

o

, sp=ezxn

o

v

o_

, tp

o

npxsp

o

(10)

into which the relative displacements are transformed by a simple vector rotation with the direction cosines (the angular brackets indicating row vectors),

58 ~t

=

P

< sT >

Au~

< tT >

AUz

p

(11)

P

3. define a constitutive model - linking relative displacements to interface tractions {an, Ts, Tt} -- by penalty parameters (as long as the joint is bonded),

trp-

0 0

~8 0 0 ~t

5p

Figure 2. Geometry of curved joint element with 4 to 9 double-nodes

(12)

426

4. transform the tractions back to global coordinates and integrate them to nodal forces by means of suitable quadrature weights wp and 'tributary areas' dAp, which take the role of the Jacobian determinant:

dAp = In, I d~dr/=

OXp

OXp

(13)

The result of these four steps are nodal interface forces, computed from the contributions of nip quadrature points,

{f,}=

f,o,

=~'5

[AAA]

+HT

n

s

t

r~

v

v

v

rt

P

wvlnpl

(14)

P

and an interface stiffness matrix: +H T P

p=l

n s t V V V

P

0 0

~s 0 0 ~t

P

< sT > < tT>

-H

wplnpl (15)

+H

P

P

It is worth noting that all interpolation, transformation and integration is performed on a single (curvilinear) plane, either the midplane as in Figure 2 or simply the b o t t o m face of the joint element. The fact that the element actually possesses twice the number of nodes comes in only at the beginning and the end of the four steps. The same stiffness matrix can be derived as the limiting case of a degenerate solid element whose thickness t ~ is reduced at constant element length L ~, thereby decreasing the aspect ratio e = t~/L ~. For a four-node stretching element, for instance, one obtains:

k-g -Is Is

K:--*~ =

2G 1

k

= symm.

where 0 2E

G 0 2a

(16) 0 E 0 2E

2~, . Le T symm.

0 2x,,

xs 0

0 xn 0 2xn

The degeneration, which is discussed at some length in [6], is illustrated by Figure 3: As the aspect ratio decreases, the element reactions in the direction normal and tangential to the element base L ~ - the designated interface- start to uncouple, the effective stretching stiffness diminishes, and the 2:1 ratio of nodal forces given by Eq. (16) appears. Note that the joint element stiffness matrix with ~n, x~ depends on the element size L ~, whereas the continuum stiffness matrix with moduli E, G depends only on the element shape as expressed by e. The reason for the size-dependence of K~_ 0 is that the 'modulus' x is a stiffness per unit area [17]: E 1 G xnLe=-lE ~ ~,~=-and x~L e = - G ~ x ~ = - - t~ (17) e t~ e Obviously the penalty parameters can be viewed as local joint moduli and calibrated to represent asperity or filler properties. From Eq. (17) a standard choice is xs = xn/2 corresponding to u = 0, but any other ratio would be feasible as well.

427 0

500

~ 125

125 ...

250

t 125

A

~.t25 +

L= 2 t=2

125

,,,=o

...t25

~ 16,666.3

/

-..~125

,:o.o2

-

~t25

- - ' t .. ..G---

8,333.7 ~25

t 125 ++

t6,67t.3

J.gv---'----"

~25 ,~666.3

153,351.2

~Tkn'L ~

8,328.7

~ ,25

tt6,668.8

+

250

33,333.7

,,.

-i-

I I

E = ,,000

+ t25

500

A

_~kn'L

ks'L

:3 -

,,

t=

-+-

,I

0

l kn. L

3

ks'L

6 ---~=

~"

----~

ks'L

ks-L

6

3

k n = 50,000

k s = 25,000

Figure 3. Degeneration of a continuum element for imposed unit displacements [5]

2.3. A n a l y s i s o f J o i n t E l e m e n t P e r f o r m a n c e

Although the joint element stresses are interpolated by smooth shape functions, they are evaluated only at discrete sampling points for the numerical integration by means of the Eqns. (14) and (15). The latter yields a constraint matrix, which is generally coupled because of 1. the local constitutive relations, as soon as off-diagonal terms appear in Eq. (12) due to friction or dilatancy 2. the spatial components due to coordinate rotation 3. the nodal contributions of the interpolation functions from different sampling points. Hence the constraint matrix derived from an isoparametric joint element is decoupled only if (1) the modulus matrix is diagonal, (2) the element is planar and aligned with a global coordinate axis, and (3) the contributions from different shape functions do not interact. If the sampling points, where the interface constraints are invoked, lie inside the joint element (Gauss points), the displacement shape functions overshoot at the nodes such that both the displacements and the interface stresses oscillate, unless a fairly soft joint stiffness can be used, e.g. in soil mechanics [16]. Similar problems with Gaussian quadrature in combination with high penalty values become noticeable in quadratic interpolation

428 as spurious debonding and 'erratic' slip directions. Suggested remedies were to simply discard the centre node output or to average the stresses over the element, cf. [18]. But although the stress output appears smoothed, the displacement oscillations even grow, Figure 4. These effects are not limited to zero-thickness joint elements but are equally found in thin-layer elements [19], ~ to be expected from Eq. (17).

A

,

i

I

i

!

i e-@ ,

I I

__.1 . . . . I

I

.

'

', ,

I

,,-o. '

i

, '

; I

,, ,

i

9

.

I

!

I

1

,I__L_ J__ .1_ a__l__~._ _

I , I !l

i

,

', I 1

~

I

I

,I',',I,1', S

"

I

: ' I

,

I

I

'

I

'

I

.

I

5 El./i lO

2 Gauss •n = t07

I

I I 10Eb,/

/

,ool ! I, I , I I I ! /

I

.

I

'

,

'

I

,

I

'

I

,

I

~

I

I

~

I

% / /~//)/~o// K s = 105 ,

K n = 10 "5... 409

'

I I I , I ~,,~

V~

2 Lob.,~

y/; 10,>,,/ t / 10 .,L lO9

of zonation

(b) influence

//,9. / ) /,~

5 Elements 2 Gauss

(a) influence of penalty

2Gauss-J t

t Gouss--..J

/ i

"

:

.

.

'

I I , ~, i

I I I i I

-

|

i

I

I

~

I

I

!

I

I

i

I

I

I I I I I

I I | I

,

I i

5 Elements ,,._ = qnt 'xn

',

I '

_I

''

I I

I

I

i

I

,

I

I

i

I

'

I

(c) influence of quadrature

Figure 4. Classical test for joint stress oscillation [20] The true remedy is found in sampling the joint stresses at the nodes: For the price of allowing 'aliasing' violation of the constraints between the nodes and somewhat stiffer behaviour, the oscillations disappear irrespective of the penalty size. The rea.son is that for nodal ('Lobatto') quadrature the penalty formulation converges to the Lagrange multiplier solution [21], satisfying the constraint equations exactly with c ~ 0 in Eq. (4). The resulting lumped constraint matrix is identical to the one ~sembled from nodal spring boxes; its superior behaviour under steep deformation gradients has been noticed before, e.g. for propagating slip fronts or crack tips [22, 23]. The good news is that such matrix can be gained from standard isoparametric interface elements, with automatic computation of direction cosines, quadrature weights etc. Since its publication in [5] this suggestion enjoys steadily growing support [24-26].

429 The differing constraint matrices can be investigated in an eigenvalue analysis of the submatrix K from Eq. (16), since the top and bottom face use identical interpolation: (K-

AI) r = O

(18)

where : the displacement eigenvectors of the individual eigenmodes : the eigenvalues representing the modal deformation energy. According to Eq. (15) in the elastically (bonded) state the interface must react in the same manner for each degree of freedom (n, s or t), even though this may sometimes not be apparent [27]. The zero-energy modes due to the zero stretching stiffness do not show up; the eigenvalues )~ are half those of the full matrix, since the deformation energy is measured with respect to the joint midplane.

3-pt. Gauss:

~. 6 . .§ . .

2-pt. Gauss:

T

K=~

s

Simpson:

4. 1

3

R

~2

1

:~

2 16

i

-12 22

K^ = - CI

28

k I= 0.1608

~kl= O.

~kI = 0.3333

),2= 0.3333

k2= 0.3333

)'2: 0.3333

~3= 1.

)'3 : 1.3333

k3= 1.1059

-.o . . . .

3

,00]

2

o 1 o o o 4

-'0

. . . . o. . . . --..._....~.....~

Figure 5. Eigenvalue analysis for line joints of unit length [28] As shown in Figure 5 for a line joint, with exact quadrature (three or more Gauss points) the midside nodes move in a direction opposite to that of the element vertices; underintegration with two Gauss points (the well-known 'Barlow points') removes the energy associated with the spurious motion of the midside nodes, which is now allowed to occur completely unrestrained. But if integrated at the nodes, using the Simpson rule, the element behaves perfectly well; this is evident from the fact that the multiple eigenvalues can be combined such as to allow the independent motion of each double-node. One may attempt to improve the 3-D surface joint element in the same manner: Starting with the consistent submatrix as obtained in exact quadrature, the diagonal counterpart of least spectral deviation [29] need be constructed, Table 2.

430

Table 2 Eigenvalue Analysis of 3-D Joint Elements [30] The consistent submatrix of Eq. (19) and has the following eigenmodes and eigenvalues:

AI=.0268 A2=.0268 A3=.0411 A4=.0556 A5=.0889 A6=.1843 A7=.1843 As=l.0811 If underintegrated by 2 x 2 Gauss points, they change to:

AI=O.

A~=O.

A3=O.

A4=O.

As=.0556

A6=.1667

A7=.1667

A8=.9444

If diagonalized according to the row-sum technique, some eigenvalues become negative:

AI =-.1667 A2=-.1667 A3=-.1667 A4=-.1667 A5=.6667

A6= .6667

A7= .6667

A8= .6667

In presence of a 9th centre node, however, the eigenvalues are all positive:

A1= .0556 As = .0556 A3= .0556 A4= .0556 A5= .2222 A6=.2222 A7=.2222 As = .2222 A9=.6667 Applying the product Simpson rule to the 8-noded element, the eigenvaJues are:

A1=.0556

A2=.0556

A3=.0556

A4=.0852

As=.2222

A6=.2222

A7=.2222 A8=1.3037

431

The standard technique is to add up the rows to the total stiffness per degree of freedom and to place the sum on the diagonal. Applying this rule to the Serendipity type joint element (8 double-nodes) of side lengths L~, Lb with the consistent submatrix,

L~Lb I ~ = a9--~-

30 10 15 10 -30 -40 -40 -30

10 30 10 15 -30 -30 -40 -40

15 10 30 10 -40 -30 -30 -40

10 15 10 30 -40 -40 -30 -30

-30 -30 -40 -40 160 100 80 100

-40 -30 -30 -40 100 160 100 80

-40 -40 -30 -30 80 100 160 100

-30 -40 -40 -30 100 80 100 160

(19)

gives, however, for the corner and midside nodes the terms -75/900 and 300/900, respectively, which is the familiar ratio - 1 / 1 2 : 1/3 of nodal forces equivalent to a unit surface pressure. The negative stiffness being acceptable only for very soft interfaces anyway, such element would be very fragile, entailing reversed-sign implementation of constitutive models, etc. [31]. Positive corner coefficients have been derived in mass lumping by redistributing the total mass of all nodes in proportion to the size of the consistent diagonal terms [32]. Since the stiffness matrix of a zero-thickness joint element also uses the shape functions undifferentiated, Eq. 15, this technique could be applied in principle. However, an even deformation field would not produce an even stress field, i.e. the resulting element is inconsistent and would only be practicable for stiffly bonded interfaces. Once again the best option is nodal quadrature, here the product Simpson rule. Although complete uncoupling of nodes would only be achieved in presence of a double-node in the element c e n t r e - i.e. for a joint element designed to connect to 27-node solid ele m e n t s - , the reduced interaction of shape functions is still beneficial in that the state determination in nonlinear computation is accelerated [24]. The stress output is smoothed because neighbouring joint elements share at their common nodes the same stress history (provided their penalty values are the same); in coarse discretization it has the advantage that stresses are sampled in the outermost fibres of a segmental cross-section, cf. the examples at the end of the paper. 3. J O I N T C O N S T I T U T I V E

MODEL

Having coped with the element formulation, the constitutive model need be discussed next. Complicated crack-dilatancy relations have been proposed to fit laboratory results of shear tests with controled crack width, cf. [33]. This approach is typical for concrete mechanics, postulating that a crack must first form in pure tension ('mode I' in terms of fracture mechanics) and that interlock, i.e. the transfer of shear forces between asperities of the opposite crack faces, can be neglected during the initial opening phase [34]. Rock mechanics, in contrast, permits discontinuities to exist a priori, and shear failure may occur in any state of compression or opening. Consequently, remaining material bridges in incomplete joints can be destroyed by shear ('mode II') and in arbitrary combination with tension or compression, which is referred to in the following as 'mixed-mode decohesion'.

432 The natural concept for shearing under compression is friction; not in the simple form of a straight Coulomb line [35], but in the form of a 'mobilized' friction coefficient [36, 37]: #mob =

tan(C, + r

= # + Umob

(20)

1 - - ]Al]rnob

where r

: the friction angle on a smooth surface # : the 'basic' friction coefficient (= tan r

r

: the stress-dependent angle of dilatancy

vmob : the coefficient of dilatancy (= tan ~ ) . From the observation that a body slipping on a smooth surface does not heave, perfect friction (r must be accompanied by zero dilatancy. Adding roughness to the surface results in a dilatant volume increase due to the riding up on the asperity flanks, or the built-up of compression if this is inhibited by confinement. The dependence of the mobilized friction coefficient on the normal stress an is conveniently modelled by a curved failure criterion, e.g. a parabolic or hyperbolic surface (Table 3). 3.1. F r i c t i o n as N o n - A s s o c i a t e d P l a s t i c i t y This surface is amenable to plasticity theory [38], treating the penalty stiffness terms of Eq. (12) as a diagonal elastic modulus matrix D e. The representation of frictional sliding by yielding in shear is fairly common, e.g. [39, 40]. Both slip and the concurrent dilatancy are regarded as plastic deformations. For a given total displacement, only the elastic portion results in a stress increment: dtr = Ded~ e = D e (d~ - d~ p) = d a e - Ded~ p =: DePd~

(21)

This is the familiar kinematic split of plasticity theory, applied to the relative joint displacements instead of strain; dtre is the elastic trial stress increment. Because of the

Table 3 Hyperboloid as Generalization of the Coulomb Friction Cone From the general formula of the hyperboloid opening to the a~-axis, with using b / a = c / d = t a n r and d - a = f, the following expression is derived for the failure surface" ~max

tan 2 r

(an - f)[(an

I Irl ~~.~tote

I tensi~cut-~

- f ) - 2a] = 0

In friction a controls the apparent cohesion at rma~[(~.=0) and r is the 'basic' friction angle. With a ---, 0 the approximation to the classical Coulomb cone can be made as close as desired.

1

"

433 non-dilatant character of perfect friction, a plastic potential surface G need be defined, which gives the flow direction as the partial derivative with respect to the stress vector: d6 p

(22)

= G,a (dA)

where (dA> : plastic multiplier existing for a trial stress outside the yield surface G,~ : vector normal to the plastic potential surface. The correct slip potential for perfect friction (repesented by a Drucker-Prager cone) is the von-Mises cylinder, Figure 6 (left). Its cross-section is sometimes addressed as 'slip circle', deforming to an ellipse if the friction properties were to be either initially orthotropic or subject to an orthotropic degradation law [41].

-or,

-on

-%

I/z friction cone

cone with tension cut-off

hyperboloid

Figure 6. Friction criteria and slip potentials [42]

As in conventional continuum plasticity, the rate-form elastoplastic modulus matrix can be computed as: D~P_D ~_Dp=D

~

D ~G,a| -

TD ~

7./~p

(23)

where ~,~r : vector normal to the failure surface ~ P 9hardening modulus (= ~-,~ D ~ G,a +~v). It is based on the vector product (e) of the surface normals modified by the metric D ~. A simple example of how to evaluate the pertinent terms is given in Table 4. When only symmetric solvers are available, an elegant way to make D ~p symmetric is to find an equivalent associated yield criterion for the current trial stress increment do"~

434 Table 4 Example of Surface Normals and Elastoplastic Modulus Matrix In the plasticity formulation the slip direction need not be collinear with the shear stress increment but depends primarily on the total shear stress vector when slip is imminent. This is given by the direction cosines/~s,t, i.e. tan ~ = rs/rt. In the simple case of a perfect isotropic friction cone the surface normals thus become: .T,o- =

,

~,o"=

~

and

7-/ep= #u~;n + ~

( ~ + ~ = 1)

(24)

~t Inserting these into Eq. (23), the elastoplastic modulus matrix of 3-D friction is derived: ~ D~p =

•s

I

-~,~

+

-~~ (25)

Note that it becomes grossly unsymmetric a.s u << #. For u = 0 the submatrix for the shear degrees of freedom gives the slip circle, on which only the tangential shear component will experience the shear stiffness ~ . The term -pua,~ accounts for the change in the current shear strength due to confinement.

[43]. Postulating that the same elastoplastic state must be reached by computing d)~ from the true (non-associated) model as from a substitute pseudo-associated model, denoted by the subscript (9), an equivalent hardening modulus is derived: ^ e 7"/~) = ~7-/~p - 7-/(0)

with

^ G,T d~r~ ~ = .T,Terd(r e

(26)

The construction of the equivalent associated surface at the contact stress point demonstrates the existance of the so-called 'wedge region' between potential surface and the more steeply inclined yield surface. This region would be entered if the normal compression in a joint were to be reduced at constant or slightly diminishing shear stress. By considering a 'directional stiffness modulus' it can be shown that loading into the wedge region is in fact characterized by a stiffness exceeding the elastic one, such that energy could be extracted in closed loading-unloading cycles [44]. Eq. (26) reflects this possibility by ~/~a) becoming negative even for ideal plasticity, when ( < 0. (This effect is, of course, not particular to joints but widely discussed for geotechnical plasticity models.) True softening - e.g. due to wear of asperities - is represented by a negative hardening modulus 7"(~. It is obviously limited by the condition ~',Ter D e G,er +Tff > 0, and the larger the angle between the normals of the yield and the potential surface (representing the degree of non-associatedness), the smaller is the permissible rate of softening [45, 46]. In contrast to continuum plasticity, however, it can be controlled by selecting sufficiently high penalties in the metric D r to avoid constitutive snap-back for a given rate of softening.

435

3.2. Multi-Mechanism Plasticity Besides the wear of asperities other degradation phenomena may be of interest, such as the breaking of an initial cementation or asperity crushing, cf. [47]. Attempts to incorporate these features date back to the Darmstadt school of concrete mechanics [48]. Their approach amounts to a cohesive interface concept, in which a 'back-bone curve' prescribes the current shear capacity in function of the contact pressure or joint closure, Figure 7. As soon as dilatancy is to be considered, however, the presumption of uncoupled constitutive behaviour does no longer hold, and the interaction of different yield surface segments must be accounted for. A proper cap formulation for crushing failure in a 2-D line joint element has been presented in [49]. The remainder of this paper focusses on the tension cap, which constitutes the apex of the hyperboloid. Most formulations blend the joint opening into the friction criterion by gradually changing the flow rule to associated behaviour near the apex [50]. If, however, the reversible loss of contact is to be distinguished from plastic dilatancy, the two mechanisms need be separately monitored. This is possible through the existance of an apex singularity in the potential surface, which results naturally from the normal G,tT lacking behind the normal of the yield surface by the constant angle r Eq. (20). The apex singularity can be interpreted as the contracted opposite corners of a tension cut-off plane, Table 3. By shifting this plane to a position close to or remote from the apex, any combination from a hyperboloid to a truncated friction cone can be effected. For a smooth interface with initial cementation, on the one hand, cohesion and tensile strength would be destroyed simultaneously; hence the surface should contract to a perfect

Figure 7. Combining friction with tension and compression caps

436 residual cone [51]. For a remaining asperity interlock, on the other hand, some or all of the cohesion might be preserved in absence of tensile strength, Figure 8 (left). Similarly for the hyperboloid the collapse to the asymptotic cone should be distinguished from a mere cap retraction [52, 53], Figure 8 (right).

Figure 8. Principal collapse mechanisms of truncated cone and hyperboloid Because of the coupling of tensile strength and cohesion, which belong to two different surface segments, the damage variables are ascribed to generalized damage modes [54]" loss of cementation (debonding) as 'mode 1' and wear of asperities as 'mode 2'. The mathematical formulation introduces for this purpose as damage variables

dxj-Zs

,

j,k=l,2

(27)

k which are a linear combination of damage functions s with the plastic multipliers d~k (j ~ k) stand for a 'latent' hardening/softening, as arguments. The coupling terms s whereas the s describe the direct effect ('self'hardening~softening) [55]. Since the plastic multipliers have the dimension of a displacement, dimensionless damage functions correspond to the hypothesis of effective plastic 'strain'; if they were given the dimension of a surface traction, their product with d)~ would stand for a fracture energy (plastic-work hypothesis). For the two mechanisms the hardening/softening modulus, as defined in Eq. (23), becomes a non-symmetric 2 • 2 matrix [56]"

7-l~ = 3vj,T D e Gk,cr - Z ~J ')Cm T

(28)

f--'mk

m

By invoking the consistency condition for either surface segment, the two plastic multipliers are solved from a coupled equation system (see the example in Table 5)"

d)~2

=

det[~ep]

_ ~;]

~

.~--1,T

d~re =:

&T

d6

(29)

437

Table 5 Example of Coupled Softening For a purely cementitious cohesion the softening of a truncated friction cone (Figure 8 left) when controlled by a single damage variable [57] is derived from the general case of Eq. (27). Assuming that both mechanisms are active, the damage variable is simply made a combination of interface separation and slip distance: d x =- &Sgnat' 4-

Id,SV,tl

(30)

= ~ , a = dA1 4- I ~ , r l dA2

With a single derivative of each failure surface segment, .T'l , X = - f t ,

(31)

:F2 , x = - C"

the hardening modulus matrix of Eq. (28) simplifies to: [?/~]-

#x,~x'~ #v~nv~'~+~s

['Hv]- [ c" f[ c" f[

'

(32)

To relate the decrement in cohesion c" to the tension softening, it is split into the decrement ~'" at constant f t (Fig. 8 left) plus the decrement due to f~: c " = ~" + #f[

(33)

This yields the following concise expression for the determinant:

(34)

d e t [ ~ ~p] = tCs(~n 4- f [ ) 4- tcn(1-v)e" The vectors ~1,2 are solved from Eq. (29):

~n C" ~ ( , ~ + f[) flt~,(x,,+ f[) --

~,1 = det[~Pl

From dAk = r

-~:~("'~

+ f[')

-fltr~(vrm4- f[')

,

~2-

det[7.l~v]

)

(3s)

d&~ + tCs(tCn + f [ ) I d & l det[~ ep]

(36)

the plastic scalars are evaluated to

dA1 - n,~(t% -4- ~")d&~ - tc~(vtr + f[')Id~• det[7-/ep]

and

dA2 = -tr

where Id~i• = {flsd,5~+fltd~t} is the displacement component normal to the slip circle (deviatoric plane). The resulting plastic displacement and the rate-form tangent modulus matrix could now be evaluated from Eq. (38). For the case of no-tension material, ie. ft = f[ = 0 for which ~" = c", this simplifies to: dA2 = - c " d & . , + ~s [diS• to, + ( l - v ) c"

and

dA1 = &Sn - u dA2

(37)

Note the term for loss of cohesion due to d~n, increasing the amount of slip in dA2. Moreover, the opening component governed by dA1 is diminished by the concurrent slip dilatancy (-vdA2). For certain paths the stress point will not be attracted by the corner singularity but could be driven up the mantle line due to confinement.

438 In generalization of Eqs. (22) and (23) the resulting plastic displacement increment and the rate-form tangent matrix become a linear combination of the mechanism contributions: d~ v = ~

,tr

T

,

D "p=D"

k-'l

I-

Gk,tr|

(38)

k=l

Two particularities are worth mentioning: First, the dilatancy is naturally limited by the asperity height, which itself need be linked to the accumulated damage to the joint roughness. The asperity height is therefore included from the onset in the damage function /~2k for mode 2 and the pertinent derivatives of the yield surface -T2,Xm. If the dilatancy was not to be limited in one way or another, overriding might proceed indefinitely without ever losing contact in the joint [59]. Second, the apex represents the position of asperity tips barely touching each other. Further separation of the joint faces must be reversible, as it entails loss of contact. Depending on the mating state of asperities, some stress-free closure may occur before they are fully disengaged. This effect is accounted for in an approximate manner by an additional 'interlock memory', allowing for free closure to the gap width as defined either by the end of tension softening or the last plastic shearing state. It seems physically plausible that plastic shearing in a half-mated position destroys the previous match of asperities and creates - together with rubble filling up the surface troughs - a new reference configuration for joint closure, Figure 9.

Figure 9. Idealized shear key model As drawback the model cannot accommodate contractancy, which may arise from shear reversal (without damage), crushing, ploughing, compaction or loss of rubble. A more elaborate memory of the mating state has been proposed for line joints [58]; in 3-D a solution might be sought in the framework of bounding surface plasticity.

3.3. Stress Point Algorithm Because the contact constraints are treated as a pseudo-constitutive problem, the load/time stepping does not need to respect the individual events of debonding or rebonding, onset of slip or return to stick (as would be necessary in Langrange multiplier

439 formulations). Instead, special attention must be paid to the elastoplastic stress integration, as the elastic trial stresses computed from the penalty parameters may become very large. Furthermore, the drastic stiffness changes in joints suggest a path-independent iteration strategy [60], which always evaluates the total stress path from the last equilibrium state: t+atui = tu + ZXtui

(39)

This is done by projecting the elastic trial stress state back onto the yield surface by /,

t+ato" = t+ato'e -- D~ ]o

G,a(~,• d~

(40)

In the present work the integral is evaluated by a two-stage Runge-Kutta formula,

i~t), G,r

J0

dA " (1 - 0) G,r I1

At~, + 0 G,~r [2 AtA2

(41)

with a similar integration for the damage parameters at~j. Although the elastoplastic tangent matrix is not involved, evaluating the predictor gradient G,cr I1 requires determination of the so-called 'contact stress' s t a t e t+At~, at which the stress vector penetrates the yield surface. For the choice of 0 = 1/2 the predictor step yields directly a first estimate for the endof-step value t+At~r [61], Table 6. This permits comparison of the difference between both estimates of the plastic deformation and to use it as input for an adaptive sub-stepping scheme [62]. No end-of-step correction is thus necessary. The sub-stepping algorithm not only controls the angular error in purely tangential steps, it also curbs the instability of the semi-explicit scheme in overlarge radial steps and takes care of all intermediate state changes such as from single to multiple mechanisms, from softening to residual capacity, including the switch between plastic and reversible joint behaviour. A rate-form tangent modulus matrix D ep is only computed at the end of the last substep to improve the search direction for the global Newton-Raphson iteration or - if converged - to enter the next load step. If the projected stress point lies on the frictional segment, the matrix is made symmetric using Eq. (26); for a corner stress state the offdiagonal terms are simply averaged. To avoid negative stiffness terms during softening, the modulus matrix is routinely checked for positive-definiteness, and the plastic correction D p to the penalty constraint matrix is scaled as necessary. The algorithm is robust and can easily accommodate trial implementations of various softening rules, etc. However, the sub-stepping tolerance need be a function of the current shear capacity, decreasing towards the apex, to avoid large fluctuations of the end-of-step stress state during successive iterates. An implicit projection method is expected to fare better under such circumstances. While a simple radial return within the plane of the slip circle is unsuited except for the simplest form of frictional behaviour [63], the closest-point projection method seems to have reached a standard that it could handle segmented yield surfaces and still allow the construction of the so-called consistent tangent, preserving the second-order convergence property of the global Newton-Raphson iteration.

440

Table 6 Flow Chart for the Plastic Integration Routine With

an

Euler

t+Atoe

forward

step an intermediate stage is computed for the instant t T At/(28) ('predictor'); this is subsequently repeated for the full step length with the intermediate gradient ('corrector'), and finally the two results are averaged as given by Eq. (41).

/ D"%? t+Ato

~ t+~-~'l~""~--~G'OI2

/s.,\

o..))

trial stress for subin~ernent

model

n

n

n

'

" .-9

~~" average ~

~ 7

n accumulate I I resulls J

9

-~

~[

J

return( ~i ~ mapping

tensile softening

/

--"--I sub- ~ "_'_"'2'_',':_"~ [increment n ~ y

~

"::_"."_"~

1

pi. multip4ier J J pLmuitiplier J pl. rnultiolier II I ~ J _ intension inshear

associated slipmodulus mc~chanism modulus

elastic ,. modulus reduce plastic par1

441 4. A P P L I C A T I O N S Tests were made with simple structures such as voussoir arches, investigating in particular the effect of joint discretization on the performance in opening and closing. As seen from Figure 10, the overall behaviour of a slender structure is not much changed by whether two layers of Gauss-integrated joints or a single layer with nodal quadrature (Simpson rule) is used; only the stress pattern in the solid blocks differs, as their shape functions cannot represent the local stress concentrations. Also investigated was the impact problem during closing in dynamic application, known as 'joint chattering', where local viscous damping was successfully implemented to mitigate the rebound of nodes [5, 281.

Figure 10. Discrete joint models of voussoir arch [64] The tractability of a full-blown dynamic analysis was tested with an arch dam under earthquake excitation. This type of analysis is notorious for spurious high-frequency noise [7, 8], but it could be mastered by means of first-order algorithmic damping. While earlier analyses always postulated unlimited shear strength of the vertical contraction joints between dam monoliths, the slip and transient eigenvibrations of the cantilever blocks, including their resynchronization by friction during closure, could be demonstrated for the first time, Figure 11. Upon presentation it was spontaneously nicknamed 'the piano effect'. There is evidence that the static solution of a structure with several joints opening at the same time poses an even greater challenge, because softening of one interface may

442

//)

/

~ ? , . - ' V \ ,4

I

/

.- - "/'X'//~

/

. - . --. " ~ , ' - ~ IA'

. ~ .-. ' Y ~ YI

. ' . I-)"

A VI

i

,1~

t = 2.55 s

vii

, KI ]

X/ I I

f= 2.65s

~ ~t, I If

,,

f= 2.75 s

I /il

',

,,

f=185 s

t= 2.95 s

Figure 11. Earthquake-induced motion sequence of a high arch dam with unkeyed joints at empty reservoir [30]

trigger the softening of others in a 'zip effect', before eventually a stable residual state is reached [65]. Tests are now underway with a more sophisticated solver based on an arc-length method, and also the question of more accurate tangent operators - possibly consistent by virtue of a fully implicit stress point algorithm - need be reconsidered. The Runge-Kutta integration routine will thereby serve as a reference standard for accuracy and efficiency. 5. C L O S I N G R E M A R K S

The formulation as double-mechanism softening may seem unduely complicated for ordinary crack-dilatancy computations, but only this enables a true mixed-mode 'fictitious crack' concept. As an example for the relevance of the developed constitutive model, a result from the testing phase of the stress point algorithm is reproduced in Figure 12, which involved a single joint element underneath a quasi-rigid block. Assuming a hyperbolic failure surface, a trial stress increment of a certain length was imposed in a single step, exploring six different directions from a plastic stress state on the tension cap: Paths 4 and 5 experience softening with the cap, while path 6 even moves around the apex to the other side (as evident from the reversal of the slip direction), demonstrating the robustness of the sub-stepping algorithm. The most. interesting path, however, is no. 3: The dilatancy evolving during softening in shear is sufficient to drive the stress point back into compression, which thus escapes the contracting cap. This behaviour was actually observed in recent laboratory experiments, in which concrete specimens were sheared after microcracks had formed in pure tension [66, 67].

443

t

1;11 s

lateral stabilisation:

2

3

II'~(t ell = IOMPa 4

30 ~

angles

5

1
applied loads

load paths

~n

Xs

1

3 2 6

5 v

,,,.._

5n

5s

Figure 12. Performance test of constitutive model in dilatant softening

In summing up it can be stated that rock joint models concentrate by and large on the frictional behaviour, stopping short of separation; crack-dilatancy models, in contrast, typically postulate shearless separation with subsequent interlock in function of the crack width. The present work attempts to combine both approaches and to reconcile friction with interlock. This results in an advanced joint element suited for all kinds of discontinuities as present in arch dams: from rock joints and concrete-to-rock interfaces, to construction and grouted contraction joints, as well as potential crack surfaces. 6. A C K N O W L E D G E M E N T S The work described in this paper was jointly supported by the Dam Safety Board of the Swiss Federal Office of Water Resources and the Swiss Federal Institute of Technology (ETH) Zurich. It was performed at the ETH Institute of Structural Engineering within a research group developing the programme DANAID for the 3-D seismic analysis of arch dams with full media interaction. The supervision, encouragement and support by Prof. Dr. H. Bachmann and former colleagues in Zurich is greatly appreciated. Special thanks go to Dr. Promper (Salzburg) and to Dr. Hilber (Stuttgart) for readily sharing their experience with the degeneration of continuum elements and stress point algorithms.

444 REFERENCES 1.

10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23.

R.E. Goodman, R.L. Taylor and T.L. Brekke: A model for the mechanics of jointed rock. Proc. ASCE (Soil Mech.) 94 (1968) 637-659. D. Ngo and A.C. Scordelis: Finite element analysis of reinforced concrete beams, ACI Jnl. 64 (1967) 152-163. B. Tardieu and P. Pouyet: Proposed three-dimensional curved joint model. Proc. 3rd ISRM Congr., Denver/CO 1974, 2D/833-836. J.P.F. O'Connor: The modelling of cracks, potential crack surfaces, and construction joints in arch dams by curved surface interface elements. Trans. 15th ICOLD Congr., Lausanne 1985, II/389-406. J.-M.Hohberg and H. Bachmann: A macro joint element for nonlinear arch dam analysis. Proc. 6th ICONMIG, Innsbruck 1988, II/829-834. J.-M. Hohberg: A Joint Element for the Nonlinear Dynamic Analysis of Arch Dams, IBK Report 186, Birkhs Basle 1992. D. Row and V. Schricker: Seismic analysis of structures with localized nonlinearities. Proc. 8th WCEE, San Francisco/CA 1984, IV/475-482. M.J. Dowling and J.F. Hall: Nonlinear seismis analysis of arch dams. J. Eng. Mech. ASCE 115 (1989) 768-789. E.L. Wilson and B. Parsons: Finite element analysis of elastic contact problems using differential displacements. Int. J. Num. Meth. Eng. 4 (1970) 387-395. P.D. Panagiotopoulos: Mechanics of Material Interfaces. Elsevier, Amsterdam 1986. O.C. Zienkiewicz: Constrained variational principles and penalty function methods in the finite element analysis. Lecture Notes Math. 363, Springer, Berlin 1974; 207-214. B. Nour-Omid and P. Wriggers: A note on the optimum choice of penalty parameters. Comm. Appl. Num. Meth. 3 (1987) 581-585. J.H. Wilkinson: Rounding Errors in Algebraic Processes. Prentice Hall, Englewood Cliffs/N.J. 1963. C.A. Felippa: Error analysis of penalty function techniques for constraint definition in linear algebraic systems. Int. J. Num. Meth. Eng. 11 (1977) 709-728. D.N. Buragohain and V.S. Shah: Curved isoparametric interface surface element. J. Struct. Div. ASCE 104 (1978) 205-209. C. Katz and H. Werner: Implementation of nonlinear boundary conditions in finite element analysis. Comput. Struct. 15 (1982) 299-304. O.C. Zienkiewicz, B.Best et al.: Analysis of nonlinear problems in rock mechanics with particular reference to jointed rock systems. Proc. 2nd ISRM Congr., Beograd 1970, III/501509. L.R. Herrmann: Finite element analysis of contact problems. Proc. Eng. Mech. Div. ASCE 104 (1978) 1043-1057. J.-M. Hohberg and H.F. Schweiger: On the penalty behaviour of thin-layer elements. Proc. NUMOG-4, Swansea 1992, 1/241-248. R.E. Goodman and C. St.John: Finite element analysis for discontinuous rock. Numerical Methods in Geotechnical Engineering, McGraw-Hill, New York/N.Y. 1977; 148-175. N. Kikuchi: A smoothing technique for reduced-integration penalty methods in contact problems. Int. J. Num. Meth. Eng. 18 (1982) 343-350. H.M. Hilber and R.L. Taylor: A finite element model of fluid flow in systems of deformable fractured rock. SESM Report 76-5, UC Berkeley/CA 1976. J.G. Rots: Computational Modelling of Concrete Fracture. Doct. diss., TH Delft 1988.

445 24. A. Gens, I. Carol and E.E. Alonso: An interface element formulation for the analysis of soil-reinforcement interaction. Comput. Geotechn. 7 (1989) 133-151. 25. J.G. Rots and J.C.J. Schellekens: Interface elements in concrete mechanics. Proc. SCI-C '90, Zell a/See 1990, II/909- 918. 26. Qiu X.-j., M.E. Plesha and D.W. Meyer: Stiffness matrix integration rules for contactfriction. Comp. Meth. Appl. Mech. Eng. 93 (1991) 385-399. 27. V.N. Kaliakin and J. Li: Insight into deficiencies associated with commonly used zerothickness interface elements. Comput. Geotechn. (to appear). 28. J.-M. Hohberg: A note on spurious oscillations in FEM joint elements. Earthq. Eng. Struct. Dyn. 19 (1990) 773-779. 29. R. Ruge: Spectral classification of Co problems. Comput. Struct. 30 (1988) 795-799. 30. J.-M. Hohberg: Seismic arch dam analysis with full joint nonlinearity. Proc. Dam Fracture (EPRI), Boulder/CO 1991, 59-75. 31. M. Coulthard: Testing and enhancement of a nodal tie element with Coulomb plasticity. Scand. J. Metallurgy 12 (1983) 289-292. 32. E. Hinton, T. Rock and O.C. Zienkiewicz: A note on mass lumping and related processes in the finite element method. Earthq. Eng. Struct. Dyn. 4 (1976) 245-249. 33. P.H. Feenstra, R. de Borst and J.G. Rots: Numerical study on crack dilatance - Part 1: Models and stability analysis; Part 2: Applications. J. Eng. Mech. 117 (1991) 733-753 + 754-769. 34. J.C. Walraven and W. Keuser: Fracture of plain concrete under mixed-mode conditions. Proc. Recent Developments in the Fracture of Concrete and Rock, Cardiff 1989, 625-634. 35. S.G. Millard and R.P. Johnson: Shear transfer across cracks in reinforced concrete due to aggregate interlock and dowel action. Mag. Concr. Res. 36 (1984) 126/9-21 + disc. 38 (1986) 134/47-51. 36. W. Leichnitz: Mechanical properties of rock joints. Int. J. Rock Mech. Min. Sci. 22 (1985) 313-321. 37. C. Gerrard: Shear failure of rock joints - Appropriate constraints for empirical relations. Int. J. Rock Mech. Min. Sci. 23 (1986) 421-429. 38. D.C. Drucker: Coulomb friction, plasticity and limit loads. J. Appl. Mech. ASME 21 (1954) 71-74. 39. I. Carol, A. Gens and E.E. Alonso: A three-dimensional elastoplastic joint element. Proc. Rock Joints, BjSrkliden 1985, 441-451. 40. T. Rodit and D.R.J. Owen: A plasticity theory for friction and joint elements. Proc. COMPLAS-2, Barcelona 1989, II/1043-1062. 41. R. Michalowski and Z. Mr6z: Associated and non-associated sliding rules in contact friction. Arch. Mech. Stos. 30 (1978) 259-276. 42. A. Curnier: A rather general theory of friction, inspired by the classical theory of plasticity, including contact impenetrability. Int. J. Solids Struct. 20 (1984) 637-647. 43. G.N. Pande and S. Pietruszczak: Symmetric tangential stiffness formulation for nonassociated plasticity. Comput. Geotechn. 2 (1986) 89-99. 44. Z. MrSz: Non-associated flow laws in plasticity. J. M~canique 2 (1963) 21-42. 45. G. Maier and T. Hueckel: Nonassociated and coupled flow rules of elastoplasticity for rocklike materials. Int. J. Rock Mech. Min. Sci. 16 (1979) 77-92. 46. J. Lubliner: On uniqueness of solutions in generalized plasticity at infinitesimal deformations. Int. J. Solids Struct. 23 (1987) 261-266. 47. G. Lacombe and M. Pommeret: The structural joints in structures of large prefabricated panels. Annales I.T.B.T.P. no. 314 (1974) 113-144 (in French).

446 48. W. Stauder: Contribution to the investigation of reinforced concrete diaphragms with finite elements, considering realistic material behaviour. Doct. diss., TH Darmstadt 1972 (in German). 49. P.B. Louren(;o, J.G. Rots and J. Blaauwendraad: Implementation of an interface cap model for the analysis of masonry structures. Proc. EURO-C '94, Innsbruck 1994, II/123-134. 50. J.-M. Hohberg: Multimechanism plasticity with coupled damage in tension and shear. Proc. COMPLAS-3, Barcelona 1992, II/1503-1514. 51. Z. Mr6z and M. Angelillo: Rate-dependent degradation model for concrete and rock. Proc. NUMOG-1, Zurich 1982, 208-217. 52. I. Carol and P.C. Prat: A statically constrained microplane model for the smeared analysis of concrete cracking. Proc. SCI-C '90, ZeH a/See 1990, II/919-930. 53. T. Stankowski: Numerical simulation of progressive failure in particle composits. PhD thesis, Univ. Colorado, Boulder/CO 1991. 54. J. Dubois, J.C. Bianchini and A. de Rouvray: Coupled damage modes plasticity models for the simulation of complex materials used in reactors. Proc. SMIRT-5, Berlin 1979, paper M2/5. 55. B. Loret: Geomechanical applications of the theory of multimechanisms. GeomaterialsConstitutive Equations and Modelling, Elsevier, London 1990; 187-211. 56. H.R. Riggs and G.H. Powell: Tangent constitutive matrices for inelastic finite element analysis. Int. J. Num. Meth. Eng. 29 (1990) 1193-1203 + disc. 32 (1991) 645. 57. R.A. Vonk, H.S. Rutten et al.: Micromechanical simulation of concrete softening. Proc. Fracture Processes in Concrete, Rock and Ceramics, Noordwijk 1991, 1/129-138. 58. M.F. Snyman and J.B. Martin: A consistent formulation of a dilatant interface element. CERECAM Rep. 168, Univ. of Cape Town, Rondebosch 1991. 59. H.R. Riggs and G.H. Powell: Rough crack model for analysis of concrete. J. Eng. Mech. ASCE 112 (1986) 448-464. 60. N. Bi(~ani(~ and Pankaj: Some computational aspects of tensile strain localization modelling in concrete. Eng. Fract. Mech. 35 (1990) 697-707. 61. H.M. Hilber and D. Raisch: Nonlinear two-dimensional finite element method for practical tunnel analyses. Proc. l l t h FE Congr., Baden-Baden 1982, 118-161 (in German). 62. W. Sloan: Substepping schemes for the numerical integration of elastoplastic stress-strain relations. Int. J. Num. Meth. Eng. 24 (1987) 893-911. 63. A.E. Giannakopoulos: The return mapping method for the integration of friction constitutive relations. Comput. Struct. 32 (1989) 157-167. 64. J.-M. Hohberg: Planes of weakness in finite element analysis. Coll. Remaining Structural Capacity, Copenhagen 1993; IABSE Rep. 67, 157-164. 65. J.-M. Hohberg: Peak-strength and path dependency in arch dam analysis. Proc. Dam Safety Evaluation, Grindelwald 1993, 1/175-186. 66. M. Hassanzadeh: Behaviour of fracture process zones in concrete, influenced by simultaneously applied normal and shear displacements. Doct. diss., Lund Inst. of Techn. 1992. 67. E. Schlangen: Experimental and numerical analysis of fracture processes in concrete. Doct. diss., TU Delft 1993.

INTERFACES IN DISCRETE SYSTEMS

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Meclaanlcs of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All fights reserved.

449

Pore Pressure Effects on Interface Behavior R.O. Davis Department of Civil Engineering, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

1. THE EFFECTIVE STRESS PRINCIPLE The fact that pore fluid pressures are important in understanding and predicting the behavior of many geologic materials has been known at least since the late nineteenth century. A paper by Clarke 1 in 1904 presented correlations of monthly rainfall with displacement of a large slow-moving landslide at Portland, Oregon. Clarke's work showed clearly that it was the pressure in the pore water, rather than its weight or any lubricating effect, that caused the motion of the landslide to accelerate or to slow down. Although Clarke did not have the analytical tools to fully understand the Portland landslide, his paper was a remarkably well documented case history of pore pressure effects. There are a great many similar case histories in the engineering and geologic literature in which the reduction of frictional strength on a potential surface of sliding due to pore fluid pressures has resulted in failure. Sometimes the failure is manifest in only small displacements accompanied by minor damage. Other times catastrophic results have ensued. The Malpasset dam which failed in 1959 in southern France took the lives of 300 people. The dam failed when a massive slip occurred on a pre-existing fault surface which lay beneath the dam foundation. This failure occurred despite the fact the dam had been in operation for more than four years. The failure was directly linked to pore pressures developing on the fault due to abnormally heavy rainfall during the three days preceding failure. The link between pore pressure and frictional strength of geologic materials lies in understanding the principle of effective stress discovered by Karl Terzaghi 2. Terzaghi realized that whenever a pore fluid is present, the stress it supports will contribute to the overall stress state in the material. On any geologic interface there will be areas of rock or soil particles in direct contact and other areas where no contact occurs but where pore fluid separates material on either side of the interface. Let us consider some area A of the interface. At the points of contact between soil particles or rock asperities, there will exist some average contact stress crc acting normal to the interface. If we let A c denote the area of contact, then the normal force supported by the soil particles or the rock will be ocAc. The remaining area, A-Ac, will support the hydrostatic pore fluid stress which we will denote by u. Thus the normal force carried by the pore fluid is u(A-Ac). The combination of these two normal forces must equilibrate the applied normal stress o acting over the whole area A. We have o A = or162 + u ( A -

450 or

a =a/+u

1---~

(1)

where o / is the effective stress given by o/_

~ A

(2)

We see from this expression that the effective stress represents the ratio of normal force supported by the solid particle contacts or rock asperities per unit area of the interface. In many practical situations, the ratio AriA in equation (1) will be small in comparison with 1. Because of this, (1) is commonly approximated by o

=

o ~ +u

(3)

This is the form of the effective stress principle put forward by Terzaghi in 1925. The total stress o is partitioned into the effective stress o ~ and the pore pressure u. The more exact form of the effective stress principle given in (1) was first noted by Skempton 3 in 1960. It should be noted that eq. (1) or the approximate eq. (3) are appropriate for dealing with geologic interfaces where frictional strength and slip are matters of primary interest. The frictional strength will depend upon o / and will be independent of u, a fact experimentally verified many times. Other forms for the effective stress principle may be required for other types of problems 4 but these will not concern us here. We can generalize the effective stress principle for three-dimensional stress states. Equation (3) becomes '

Oij "-- Oij + U6ij

(4)

where oii is the Cauchy total stress tensor, o/..tj is the effective stress tensor and 6~i is the Kronecker delta symbol. We see that eq. (4) decomposes the stress state into a hydrostatic part with magnitude u and an effective stress part supported by the rock or soil skeleton. It should also be noted that even though we employ the notation of continuum mechanics, we must still consider finite sized areas and volumes. The usual limit procedures cannot be applied to granular materials or rock interfaces. Thus we define stress as the ratio of force divided by a finite area large enough to encompass a representative number of grains or a representative region of rock surface. The interfaces we wish to consider here are surfaces on which slip may occur in a geologic mass. With regard to soils the interface may be a well defined slip surface or it may be a shear zone where shearing deformation is concentrated in a narrow band. In rock we are concerned with joints or faults and these also may be well defined surfaces or may be narrow gouge filled shear bands. In either case the resistance to slip or shear deformation will be characterized by the Coulomb friction law which states the maximum shear stress ~: which may be supported is a linear function of the effective stress o / . r = c +/~ o /

(5)

Here c is the cohesion term, independent of normal stress, and p is a coefficient of friction. According to (5), the actual shear stress on the interface must always be less than or equal to ~:. If the actual stress is equal to r, then slip will occur.

451 Equation (5) is an idealization of the complex processes involved in slipping on a geologic shear surface. According to (5) no deformation occurs until the actual shear stress equals ~:, and deformation will cease immediately if the stress drops below ~:. In fact the slip mechanism is much more complicated than this and deformation may accompany lesser stresses. Nevertheless, (5) is a useful representation of strength which may be fruitfully applied in most practical problems. For high normal stress (e.g. crY> 200 MPa) the coefficients in eq. (5) become relatively independent of the material being tested. In 1978 Byerlee 5 compiled test data for a wide range of rock types and found generally good agreement that p = 0.6. For normal stresses lower than 200 MPa, the data becomes more scattered. For modest stress levels in soils (on the order of 100's of kPa) the value of p may take on values as small as 0.1 in clays to values in excess of 1.0 in sands and gravels. The cohesion c may also change as a function of confining stress. At low stresses on a well developed shear surface, c will generally be zero. At high stress levels c will normally have some small positive value. The strength parameter/J may also depend upon the rate of shearing. Rate effects have been observed in both soils 6'7 and rocks 8'9. Increasing the rate of shearing may lead to weakening or strengthening depending upon the material involved. The importance of pore fluid stress is clear in eq. (5). Increasing or decreasing u, while holding the total stress o constant, may have a dramatic effect on the stability of a fault or landslide. Changes in u may occur either because of outside influences such as infiltration in a landslide during heavy rainfall, or because of the mechanical behavior of the material itself. Practically all geologic materials exhibit some volume change when subjected to shear. Both compaction (volume decrease) and dilatation (volume increase) have been observed in different materials. If the void space in the material is filled with pore fluid, any tendency toward volume change will be accompanied by changes in pore fluid pressure. These pore pressures are required to drive the flow of pore fluid which must accompany the volume change. Once sufficient flow has occurred the pore pressures may return to their normal values, although situations arise in which flow and pore pressure dissipation encourage further shearing deformation with its accompanying compaction or dilatation. We will investigate these effects in more detail below. In order to quantify changes in pore pressure which result from either compaction or dilatation, Skempton 1~ introduced two pore pressure coefficients called A and B. These relate the change of pore pressure to changes of deviatoric and mean stress in undrained deformation. Let p represent total mean stress and let q be the deviatoric stress 1 p = 3O'kk

,

1 [3 Crkk)2]1 q = -~ OijOij - (

(6)

Then A and B are given by A=__

du

aq

, B=--

Ou

ap

(7)

The coefficient B depends primarily on whether the void space is fully saturated or not. For full saturation with de-aired water, B will theoretically be exactly equal to 1.0, and careful experiments confirm this. If undissolved air is trapped in the pores, B will be less than 1.0. The range of possible values for A is quite broad. Negative values correspond to dilating materials and positive values to compacting materials. If A equals zero, then

452 the material maintains constant volume during shearing. The value of A will not in general remain constant during any loading procedure. Depending on loading details, some geologic materials may first compact, then dilate, and finally deform at constant volume in a typical test. Nevertheless A can be a useful measure of the overall pore pressure behavior expected for a particular material. In the remainder of this chapter we will consider a number of situations where pore pressures may have a significant effect on the behavior of a geomaterial interface. In the next section, the question of induced seismicity due to pore pressure is examined. This is followed by a section describing how pre- and post-earthquake pore pressures may affect the earthquake process. Section 4 briefly discusses frictional heating of pore fluids with implication for landslide stability. Finally, in section 5, recent experimental evidence of dynamic fluctuation of pore pressures in shearing materials is described.

2. PORE PRESSURE INDUCED SEISMICITY The Denver earthquakes of 1966 first suggested that pore pressures could induce seismic activity 11. Injection of fluid waste in a disposal well was linked to earthquake occurrence, and seismic activity ceased when injection was discontinued. This phenomena was confirmed in an elaborate and carefully documented experiment carried out at Rangely, Colorado between 1971 and 197312. The Rangely experiment utilized existing wells at the Rangely oilfield to increase pore pressures in the region of an active fault. Since the overall fault dimensions were small, there appeared to be no possibility of producing a damaging earthquake, and indeed the largest earthquake induced by the experiment had magnitude 3.1. Using hydraulic fracturing techniques the in-situ stress field near the fault was measured, and it was found that the shear and normal stresses acting on the fault were approximately 7 and 34 MPa respectively. Taking c - 0 and/~-0.81 in eq. (5), the value of effective stress required to produce failure with 1:-7 MPa was found to be about 8.6 MPa. This suggested that the critical value of u required to induce earthquakes was ur

- ~'=34

- 8.6=25.4 MPa

The experiment consisted of carefully raising the pore pressure by water injection to a value greater than ur and then lowering it below ur while monitoring seismic activity on the fault. As expected a strong correlation between number of earthquakes and pore pressure near the fault was discovered, and the value of critical pore pressure was experimentally confirmed. It is noteworthy that the value of ur was predicted before the complete cycle of increasing and decreasing pore pressure was carried out. The Rangely experiment provided conclusive evidence for the applicability of the effective stress principle to frictional stability of faults. Many laboratory experiments had shown the principle to be valid for small scale samples, but the Rangety data extended the experimental dimensions by four to five orders of magnitude and removed any lingering doubt that pore pressures and effective stress were not controlling factors. Increased seismicity associated with injection from wells is not the only form of induced earthquake activity. The most common occurrences of induced seismicity are associated with reservoir construction. There have been numerous instances of increased numbers of earthquakes resulting from reservoir filling 13. These may simply result from increasing

453 levels of deviatoric stress in the rock near the reservoir due to the additional weight of water, or may be due to pore pressure effects combined with increased stress levels. Two pore pressure effects are present in reservoir filling. The most obvious effect is that water from the reservoir may infiltrate into the ground below and directly affect pore pressures in the adjacent rocks. A second, more immediate effect is the possibility of enhanced pore pressures due to the weight of the reservoir. We can gain an intuitive understanding of both effects by considering the idealized situation depicted in Figure 1. In the figure the reservoir is idealized as having constant depth h and extending indefinitely in all horizontal directions. Uniaxial strain conditions exist and the increase in vertical and horizontal components of total stress due to reservoir filling will be

Figure 1. Reservoir induced seismicity

'

&~

v ]v.h

A~

(8)

l-v

Here v~ represents the value of Poisson's ratio which is applicable for undrained loading of the rock. The pore pressure increase Au will be given by eqs. (6) and (7). Using (8) we have 1

Ap = 3 ( A O , + 2 k , OH)

_

_

,

1[ 1 + v ]?w h

~q = ~o

=

- Ao H

1-2v.)

y~h

(9)

)] 1~,~h -v,

(10)

I -v

Thus the pore pressure increase is Au=BAp

+AAq=

-1B 3 (l+v)

+A(1-2v

If the degree of saturation of the rock is near total, then B will be close to 1.0. In most instances the value of A will be negative but its magnitude will be small. If we assume A(1-2v~) is near zero and set B equal to 1.0, then

454

,u

=

_

3 i-v

?w h

(11)

If v~ were 0.4 for example, we see that Au = 0.78Ywh. Thus reservoir filling will immediately increase the pore pressure in the surrounding rock. How this increase may affect the stability of a fault depends primarily on the fault surface orientation. If the fault surface is horizontal, the effective stress increases AOZv= A o , -

Au = -

2[ 1 - 2 v

] u yw h 1-v u

3

If the fault surface is vertical, the effective stress it supports decreases by this amount.

AOH:AOH AU:

-

3 ....1

-

v u

/

Y

wh

Thus vertical fault surfaces become less stable and horizontal surfaces more stable. Surfaces with intermediate orientation suffer this change Ao~ = [ 1 - 2 v ) ( 1 ) 1-vu c~

Ywh

where 1~ denotes the angle between the fault surface and the horizontal. We see that cos"11/v3 - 55 degrees marks the maximum value of 0 for which the effective stress will be increased. These effects occur immediately upon loading. As time passes two things may occur. First, the additional or excess pore pressure beneath the reservoir may dissipate as flow occurs into the surrounding rock. Second, infiltration from the reservoir may link the existing ground water with the reservoir water. The result of these two processes will be a return of the effective stress conditions to the situation which existed before reservoir filling. Both pore pressure and total stress will be increased, but effective stress will be unchanged, implying unaltered stability of existing faults. This too is an idealized situation. Infiltration may often not connect the reservoir to existing ground water due to the presence of impermeable materials. The dissipation of excess pore pressures will occur regardless, in which case the long term equilibrium pore pressure may be unchanged from pre-reservoir conditions. In that instance the reservoir induced changes in effective stress will be equal to the changes in total stress and stability may be enhanced. In general, the most likely time of occurrence for reservoir induced earthquakes is shortly after reservoir filling. The stress increase caused by the added weight of water will, in fully saturated rock, be initially largely carried by the pore fluid. As flow occurs away from the loaded region the stress increase will gradually be transferred from the pore fluid to the rock. This process is familiar in soil mechanics where it is called consolidation, and it was the discovery of the effective stress principle which led Terzaghi to an understanding of the consolidation problem. The same understanding is necessary for the reservoir induced seismicity problem.

455

3. PORE PRESSURE EFFECTS ASSOCIATED WITH EARTHQUAKES Changes in pore pressures accompany earthquakes and may affect the aftershocks which follow. Pore pressure changes may also precede an earthquake and, in the 1970's, much interest was generated by the possibility that precursor changes in pore pressure could be used as a prediction tool. The dilatancy-diffusion theory was put forward as an explanation of changes in the ratio of dilatational and shear wave velocity observed before several earthquakes. The basic idea which underlies the dilatancy-diffusion theory is simple 14as. Shearing deformation in the rock adjacent to a fault will generally be accompanied by dilatation which in turn will cause a decrease in pore pressure. The mechanism is embodied in eq. (7)1 whenever the pore pressure coefficientA is negative. Lowered pore pressures near the fault will result in increased effective stress and greater strength. The phenomena is called dilatancy hardening. Strength increases will only be temporary however, since flow will occur into the region due to the negative pore pressure gradient. Diffusion of pore fluid into the region increases the pore pressure, decreasing the effective stress, and decreasing the strength. In theory then, the increased shear stress which might have caused an earthquake, instead causes the fault to become temporarily stronger. The increased strength cannot persist, and after some time the pore pressure drops and rupture occurs. The amount of time needed for all this to occur depends on the level of dilatation, the volume of rock involved and the permeability in the region of the fault. One particular aspect of the dilatancy-diffusion theory made it especially attractive from the standpoint of earthquake prediction, namely that the whole process could, in theory, be monitored. It was speculated that dilatancy would manifest itself by changing the velocity of seismic waves which propagate through the affected rock. Early observations suggested that the ratio of speeds of dilatational and shear waves, Vp/Vs, decreased by as much as ten percent in response to dilatation. As dilatation developed, the Vp/Vs ratio decreased. Then as diffusion brought additional pore fluid into the focal region and pore pressures increased, the ratio Vp/V~also increased, supposedly returning to its normal value of about 1.75, roughly at the time the earthquake would occur. The optimistic view taken in the 1970's held that observation of the vp/v~ ratio was one key to earthquake prediction. Unfortunately, subsequent events do not totally conform with the theory 16. Careful measurements of seismic wave velocities were carried out annually in Japan using explosive generated waves in the Izu region. Three large earthquakes occurred in the region in 1974, 1978 and 1980 during the time the velocity measurements were being made. No significant changes in vp/v~ were observed, and the dilatancy-diffusion theory has since lost much of its credibility in regard to earthquake prediction 17. Failure of the dilatancy-diffusion theory as an earthquake prediction tool does not imply the phenomena of dilatant hardening accompanied by pore fluid flow may never occur. There is considerable geological evidence suggesting just the opposite. Sibson TM has summarized a number of cases of hydrothermal vein systems associated with faults which could have arisen from fracture dilatancy and pore fluid flow. Post rupture flow and redistribution of pore fluid stress has also been advanced to explain both the temporal and spatial distribution of aftershocks associated with large earthquakes. A simplified model connecting pore fluid motion with aftershock occurrence was put forward by Nur and Booker 19. Their model considered pore pressure variations

456 set up by a two-dimensional edge dislocation in an infinite elastic space. Regions of compression and dilation lie on either side of the dislocation and corresponding pore pressure changes are expected to be found near the ends of a fault immediately following rupture. Flow then occurs from the region of higher pressure to the region of lower pressure. Nur and Booker hypothesized that occurrence of aftershocks was directly related to the rate of increase of pore pressure in the dilatant quadrant. Their simple model explained clustering of aftershocks near fault ends as well as the absence of aftershocks following deep earthquakes or small earthquakes. A more sophisticated model for this process has since been advanced by Li, et al.20.

4. FRICTIONAL HEATING EFFECTS Whenever slip occurs on a geological interface, energy will be dissipated. If we let r denote shear stress and v slip velocity, then at any instant the rate of dissipation per unit area of interface is the product rv. Dissipated energy takes the form of heat which may be transported away from the interface by a combination of convection and conduction. Heat conduction occurs through rock or the solid particle matrix in soil and also through pore fluids. Convection is manifest in flow of pore fluids away from the interface. If steady state slip is occurring, a state of thermal equilibrium will exist. The frictional heat generated will be balanced by conduction alone as convection can occur only as a transient process. No excess pore pressure will accompany steady state slip. In contrast to the steady state, transient conditions give rise to a rich variety of possible responses depending upon the details of loading and the materials involved. Any increase in either r or v will result in an increase in temperature at the interface surface. In general, for a fully saturated geomaterial, increasing temperature will result in increasing pore pressure since the coefficient of thermal expansion for most common minerals is about an order of magnitude smaller than that for water 21. Increasing pore pressure of course implies decreasing effective stress and decreasing strength. This slip-weakening effect may have important ramifications for overall stability of a fault or landslide. The simplest situation of interest occurs when we assume the interface is of infinite extent with uniform conditions everywhere. In this case heat transport occurs perpendicular to the interface and only one spatial dimension is required. Letting x be the spatial coordinate normal to the interface, the field equations appropriate to the problem are 22

mfi- anO=-aV dx V=_

(12)

k du "tw Ox

6 = 6 . 020 aX 2

(13/ PwCw O(OV) pC

O~X

where u = u(x,t) = pore pressure 0 = 0(x,t) = pore fluid temperature V = V(x,t) -- pore fluid velocity

(14)

457 n -- porosity rn = compressibility of solid matrix = (1 + v)(1 - 2v) E(1 -

a k 6 p law c %

= = = = = = =

coefficient of thermal expansion of pore fluid coefficient of permeability of solid matrix thermal diffusivity of solid-pore fluid mixture mass density of solid-pore fluid mixture mass density of pore fluid specific heat capacity of solid-pore fluid mixture specific heat capacity of pore fluid.

Here eq. (12) represents conservation of pore fluid mass, eq. (13) is Darcy's law, and eq. (14) represents conservation of pore fluid energy. These three equations can be solved, given appropriate boundary conditions and a constitutive equation for the shear stress 1:. The boundary conditions for this problem depend on whether we are concerned with earthquakes or landslides. In the latter case, there will be an isothermal boundary (the ground surface) not too distant from the interface. In the former case we may assume the interface is contained within an infinite medium and require the solution to remain bounded as x -, oo. In either case, the boundary conditions at the interface are atx=0"

V=0

and

- K a0 ax

=

t:v

(15)

Here K denotes the coefficient of thermal conductivity of the solid-pore fluid mixture. The no-flow boundary condition (15)1 implies a symmetric pore pressure gradient on either side of the interface. Finally, we require a constitutive equation relating shear and normal effective stress on the slipping interface. The Coulomb equation (5) is appropriate, and would normally be applied with c equal to zero. The coefficient of friction/~ may be taken to be a function of the slip velocity v as discussed in Section 16'7'8'9. Consideration of frictional heating of pore fluids in regard to earthquakes were first treated by Lachenbruch 23. Without solving the field equations, he considered critical combinations of the material parameters in special cases. He concluded that if the permeability was sufficiently large, thermal effects would not be important; but, in sufficiently impermeable rock, high pore pressures could be thermally generated and these could affect the dynamics of faulting. Lachenbruch's work was pursued further by Mase and Smith 24. They solved the governing equations for the special case of a fault slipping with constant velocity. They identified a range of material parameters within which frictional heating affected the strength of the fault. Again permeability was the most interesting parameter. In cases where permeability was sufficiently high, any excess pore pressures which might be generated by frictional heating were quickly dissipated by flow away from the fault. For low permeability rock however, high pore pressures could be generated by their model. Effects of frictional heating in landslide behavior have been considered by several investigators. Habib 25 carried out a simplified analysis, omitting the effects of fluid flow, and concerned primarily with the possibility of vaporization of pore fluid. Gogue126 was also concerned with the possibility of vaporization, particularly in regard to very large rockslides which appear to exhibit behavior compatible with near zero friction coefficients.

458 The Vaiont rockslide 27 is a case which has generated particular interest in the rock mechanics community. Voight and Faust 2a'29 carried out numerical calculations for the Vaiont slide which incorporated conduction and convection of heat. Their findings suggested that vaporization of pore fluid was unlikely, but pore pressures could nevertheless be significantly enhanced by frictional heating, leading to loss of frictional strength and high slide velocities. Similar conclusions were given by Anderson 3~ Frictional heating of pore fluids has also been implicated as a possible mechanism for the gradual acceleration and loss of stability observed in many creeping landslides 22. In materials of low permeability, small changes in creep velocity may result in slightly increased pore pressures, and this effect may compound in time leading to complete loss of stability. The effect may have been observed in the East Abbotsford landslide 31. Frictional heating can also result in stick-slip behavior for a simple elastic slider such as illustrated in Figure 2. The slider is pulled by an elastic spring which is connected to a load point moving with constant velocity vo. Frictional dissipation and pore fluid heating occur at the slider-base interface affecting frictional resistance. For certain combinations of material parameters, the slider exhibits stationary periods interspersed with rapid jumps forward 32. The full role of pore fluid heating in interface behavior is not yet clear. While theoretical calculations suggest its importance in certain situations, these are specialized cases dependent on full saturation and (especially) low permeability. The exceptional mobility of some large rock slides remains both the impetus and the only experimental evidence for the theory. Figure 2. Elastic slider 5. DYNAMIC PORE PRESSURE FLUCTUATIONS An alternative explanation for landslide mobility has been advanced by Iverson and LaHusen 33. They hypothesize that while the overall motion of a landslide may appear steady, internally there exist isolated regions of high or low pore pressure induced by local compaction or dilatation of the solid matrix. At any point on the sliding surface, local compaction and dilatation would be expected to occur in some roughly periodic fashion. The rate at which changes occur would presumably depend upon the velocity of sliding v and some characteristic dimension A. The dimension A might be the mean particle size or might be larger if groups of particles are moving together as a relatively rigid mass. The characteristic time (period) associated with the dilatation-compaction fluctmtions would be A/v. Pore pressures will develop locally in response to the dilatation-compaction fluctuations. Any excess pore pressure, whether negative or positive, will result in flow, either toward or away from the region affected. Flow results in dissipation of pore pressure depending upon the permeability and compressibility of the solid matrix and the distance to regions of lower pressure. The characteristic time for pore pressure dissipation may be taken as

459 A2/c where A is the characteristic particle (or particle group) dimension, and c is the coefficient of consolidation, defined by k c--

Ywm

where k and m are the permeability and compressibility of the solid matrix [defined previously following eq. (4)] and y, is the unit weight of the pore fluid. The coefficient of consolidation c is a familiar parameter in soil mechanics. We now have two characteristic times: A/v, the period of fluctuation of compression and dilatation, and A2/c, the time for dissipation of excess pore pressure. The ratio of these defines a dimensionless number R34 A/v c t~2/c vA whose magnitude characterizes the tendency of a landslide to develop sustained excess pore pressures. For large values of R the dissipation time is short in comparison with the period of pore pressure generation, and sustained pore pressures are unlikely. Conversely, for small R, the pore pressure dissipates slowly and sustained pressures are likely. Iverson and LaHusen 33 carried out pore pressure measurements in controlled shearing of a carefully constructed array of fibreglass rods. For values of R between 10 and 50 they found pore pressure fluctuations on the slip surface characterized by plateaus of high pressure (slightly higher than the initial static pressure) separated by deep troughs of low pressure. During the high pressure plateaus the pore pressure was sufficient to reduce the effective stress to zero. The low pressure troughs were of shorter duration but during them the pore pressure was sufficiently reduced so that the mean pressure over the plateautrough cycle was equal to the initial hydrostatic pressure. They also performed measurements in a large scale simulated landslide which showed quite large pore pressure fluctuations for an R value of approximately 0.3. If significant local pore pressure fluctuations do accompany sliding on geomaterial interfaces, many regions of high and low excess pore pressure might exist simultaneously on the sliding surface. The regions of high pressure would be weakened while the low pressure regions would be hardened. The edges of the high pressure region might be expected to exhibit stress concentrations tending toward dilatation, while flow from high to low pressure regions would decrease strength in areas of dilatency hardening. Migration of pore pressure from one region to another might presumably result in an overall pattern of motion which is apparently uniform. 6. SUMMARY Pore pressures tend to complicate the picture of geomaterial interface strength. They may be generated by the material itself through compaction or dilatation or possibly frictional heating, and they will dissipate only as rapidly as the material will allow. They introduce the dimension of time to strength considerations which might otherwise exhibit no time dependence, and their effects on stability of faults or landslides may be catastrophic. An understanding of the effective stress principle together with the phenomena of compaction and dilatation and the laws governing flow through porous media is required for any consideration of strength of geomaterial interfaces where pore fluids are involved.

460 REFERENCES

1. Clarke, D.D., (1904) A phenomenal landslide, Tram. ASCE, vol. 53 (paper 984) pp. 332-397. 2. Terzaghi, K., (1925) Erdbaumechanik, Franz Deuticke, Wien, 399 p. 3. Skempton, A.W. (1960) Correspondence, Geotechnique, vol. 10, No. 4, p. 186. 4. Robin, P-Y.F. (1973) Note on effective pressure, Jour. Geophys. Res. vol. 78, No. 14, pp. 2434-2437. 5. Byerlee, J.D., (1978) Friction of rocks, Pure AppL Geophys., vol. 116, pp. 615-626. 6. Hvorslev, M.J. (1960) Physical components of the shear strength of saturated clays, Proc. Res. Conf. on Shear Strength of Cohesive Soils, ASCE, Boulder, pp. 169-273. 7. Salt, G. (1985) Aspects of landslide mobility, Proc. llth Int. Conf. Soil Mechanics and Foundation Engineetqng, Part 3/A/4, pp. 1167-1172. 8. Dieterich, J.H. (1979) Modelling rock friction: 1 Experimental results and constitutive relations, Jour. Geophys. Res., vol. 84, pp. 2161-2168. 9. Tullis, T.E. (1988). Rock friction constitutive behavior from laboratory experiments and its implication for earthquake prediction field monitoring program, Pure AppL Geophys., vol. 126, pp. 555-588. 10. Skempton, A.W. (1954) The pore pressure coefficients A and B, Geotechnique, vol. 4, pp. 143-147. 11. Healy, J.A., Rubey, W.W., Griggs, D.T., and Raleigh, C.B. (1968) The Denver earthquakes, Science, vol. 161, pp. 1301-1310. 12. Raleigh, C.B., Healy, J.H., and Bedehoeft, J.D. (1976) An experiment in earthquake control at Rangely, Colorado, Science, vol. 191, pp. 1230-1237. 13. O'Reilly, W. and Rastogi, B.K. (eds) (1986) Induced seismicity, Phys. Earth Planetary Interiors, vol. 44, pp. 73-199. 14. Nur, A. (1972) Dilatency, pore fluids, and premonitory variations in tJtp travel times, Bull. SeismoL Soc. Am., vol. 62, pp. 1217-1222. 15. Whitcomb, J.H., Garmony, J.D., and Anderson, D.L., (1973) Earthquake prediction: Variation of seismic velocities before the San Fernando earthquake, Science, vol. 180, pp. 632-641. 16. Mogi, K. (1985) Earthquake Prediction, Academic Press, Tokyo, 355 p. 17. Scholz, C.H. (1990) The Mechanics of Earthquakes and Faulting, Cambridge University Press, Cambridge, 439 p. 18. Sibson, R.H. (1981) Fluid flow accompanying faulting: Field evidence and models, in Earthquake Prediction, an International Review, ed. D. Simpson and P. Richards, Am. Geophys. Union, Washington, pp. 593-603. 19. Nur, A. and Booker, J.R. (1972) Aftershocks caused by pore fluid flow? Science, vol. 175, pp. 885-887. 20. Li, V.C., Seale, S.H., and Cao, T. (1987) Postseismic stress and pore pressure readjustment and aftershock distributions, Tectonophysics, vol. 144, pp. 37-54. 21. Campanella, R.G., and Mitchell, J.K. (1968) Influence of temperature variations on soil behavior, Jour. Soil Mech. Found. D&. ASCE, vol. 94, pp. 709-734. 22. Davis, R.O., Smith, N.R., and Salt, G. (1990) Pore fluid frictional heating and stability of creeping landslides, Int. Jour. Num. AnaL Methods Geomech., vol. 14, pp. 427-443.

461 23. Lachenbruch, A.H. (1980) Frictional heating, fluid pressure, and the resistance to fault motion, Jour. Geophys. Res., vol. 85(B11), pp. 6097-6112. 24. Mase, C.W., and Smith, L. (1985), Pore-fluid pressures and frictional heating on a fault surface, Pure AppL Geophys., vol. 122, pp. 583-607. 25. Habib, P. (1975) Production of gaseous pore pressure during rockslides, Rock Mech., vol. 7, pp. 193-197. 26. Goguel, J (1978) Scale dependent rock slide mechanisms, with emphasis on the role of pore fluid vaporization, in B. Voight (ed.), Rockslicles and Avalanches 1: Natural Phenomena, Elsevier, pp. 693-705. 27. MiJller, L. (1968), New considerations on the Vaiont slide, Rock Mech. Engrg. GeoL, vol. 6, pp. 1-91. 28. Voight, B. and Faust, C. (1982) Frictional heat and strength loss in some rapid landslides, Geotechnique, vol. 32, pp. 43-54. 29. Voight, B and Faust, C. (1992) Frictional heat and strength loss in some rapid landslides: error correction and affirmation of mechanism for Vaiont landslide, Geotechnique, vol. 42, pp. 641-643. 30. Anderson, D.L. (1980), An earthquake induced heat mechanism to explain the loss of strength of large rock or earth slides, Proc. Int. Conf. Eng. for Protection from Natural Disasters, Bangkok, pp. 569-580. 31. Smith, N.R. and Salt, G. (1988) Predicting landslide mobility: an application to the East Abbotsford Landslide, Proc. 5th New Zealand Geomech. Conf., Sydney, pp. 567-572. 32. Davis, R.O., and Mullenger, G. (1992) Frictional sliding in the presence of thermally induced pore pressures, in C.S. Desai, et aL (ed.) Constitutive Laws for Engineering Materials, ASME Press, New York, pp. 549-552. 33. Iverson, R.M., and LaHusen, R.G. (1989) Dynamic pore-pressure fluctuations in rapidly shearing granular materials, Science, vol. 246, pp. 796-799. 34. Rudnicki, J.W. (1984) Effects of dilatent hardening on the development of concentrated shear deformation in fissured rock masses, Jour. Geophys. Res., vol. 89, pp. 9259-9270.

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Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

463

F r i c t i o n a l c o n t a c t in c o l l e c t i o n s of rigid or d e f o r m a b l e b o d i e s : n u m e r i c a l s i m u l a t i o n of g e o m a t e r i a l m o t i o n s M. Jean Laboratoire de Mrcanique et Grnie Civil, Universit6 Montpellier II, CNRS, cc048, place E. Bataillon, 34000 Montpellier, France

The purpose of this paper is to present some general numerical methods for treating dynamical problems involving unilateral contact and dry friction. Some examples of applications related to the structural response of rigid or deformable geomaterials such as, rocks, soils, collections of blocks, granular materials, are given. Emphasis is put on Coulomb's dry friction law. This law is relevant for a large class of applications to geomaterials. It accounts for the main features of dry friction. It may be easily improved without drastic changes in the proposed methods. The frictional problems appear to be strongly non linear, and call for the techniques of nonsmooth mechanics. Convex Analysis is widely used to formulate friction equations and numerical algorithms. INTRODUCTION When modelling a mechanical problem, some mechanical variables are selected and the equations governing the system are formulated. Some equations derive from mechanical principles such as the equations of motion or the equations of continuum mechanics which are universally accepted. Other equations such as constitutive laws for a material, or frictional contact laws, are often complex and difficult to define and may be unreliable. Finally some numerical method is chosen to find approximate solutions to the system of equations. When the main interest is devoted to the description of physical phenomena, numerical results are of assistance. There is some inclinacy to take numerical results for granted, especially when graphical outputs are produced showing some similarity with observed phenomena. The numerical results should not be blindly accepted when sensitive mechanical phenomena are studied, and the influence of the modelling process and in particular the choice of frictional

464 contact laws should be examined. The influence of computational parameters, such as the time step and convergence criteria are also important. Many complicated phenomena are involved when contact and friction occur between rigid or deformable bodies. Reproducible behaviours may be experimentally difficult to obtain. Moreover in some cases reproducibility should not be expected due to either insufficient control of experimental circumstances or due to lack of information. Frictional contact laws are usually written as relations between the local stress at some point of the interface and the relative velocity. Formulating such laws and selecting particular expressions depend on the time and length scales of the investigation, the scope of applications, the expected results, and the methods used to solve the equations of the problem. For instance, time and length scales decide whether the materials should be considered as either rigid or deformable. Generally, the equations of motion govern the evolution, but if inertia effects are negligible with respect to external and internal forces, one may end with a quasi-static problem. When a collection of rigid bodies is under consideration, for instance a granular flow, or a wall made of blocks under seismic excitation, these problems have to be treated in a dynamic sense. In applications such that strain-stress experiments in granular material, or progressive loading of a wall, interest is focussed on the quasi-static behaviour, while supposedly negligible dynamical effects certainly occur. In such cases it is difficult to propose consistent equations governing the system, other than the equations of motion. In this paper, finite dimensional systems which may be collections of rigid bodies or models of continuous media will be considered. Finite dimensional models of continuous media are obtained through such methods as finite elements methods. The question of the choice of the finite elements is not within the scope of this paper, which is devoted to numerical treatment of contact and friction.

Frictional contact laws are presented in part 1. The equations governing the problems are written in part 2, and the discretization processes are presented in part 3. The method used to compute the frictional contact forces is presented in part 4. Some examples are given in part 5.

1. F R I C T I O N A L CONTACT LAWS Complicated phenomena, occuring at the so-called microscopic scale, take place between contacting bodies or edges of discontinuities. Surfaces in contact are found to be rough, and contact is established through asperities which suffer damage during compression and sliding, so fragments of material are generated. Molecular interactions may occur as well. When investigating the structural response of the system, such processes are omitted. A frictional contact law should appear as a relation between tangential and normal stresses and the sliding velocity at the point of contact. This law may be constructed from experimental data. It may also emerge from theoretical analysis based on multiple length-scale such as the

465 homogeneization technique. In such techniques a neighborhood of the contacting zone is considered, including parts of the two bodies, and sometimes also a third body interacting with the two bodies. When the thickness of this neighborhood vanishes the homogeneization technique yields a constitutive law at the interface, or a contact element in finite element methods, (Licht, [ 1]). This law or this contact element accounts for interfaces features and also for the response of the materials. When large sliding displacements or deformations are investigated, other parameters such as the sliding displacement have to be introduced to take into account wear or structural phenomena such as dilatancy in granular materials. When such a degree of complexity has to be attained, and when micro-scale variables are purposely ignored for the developpement of the model, a phenomenological model might prove valuable, (Selvadurai, this volum, Boulon,[2]). When developping a frictional contact law, it is first generally assumed that motions are smooth, i.e. the velocities are continuous functions of time and forces may be described as piecewise continuous functions of time. Nevertheless when frictional contact occurs, the velocities may have jumps and reactions appear as impulses. The proper mathematical tool to describe jumps of the velocity is the concept of a function U with bounded variation on the considered time interval. This secures the existence of the left limit U', i.e. the value before the impact, and the right limit U +, the value after the impact. Derivatives of such functions are measures, such as the Lebesgue measure and the Dirac measure at each point of discontinuities. The densities with respect to the Lebesgue measure describe the usual forces, and the densities with respect to the Dirac measure describe the impulses. Lebesgue forces are usually applied. Impulses are exerted when rigid bodies collide. In some applications, it seems relevant to consider the bodies to be rigid. When deformable bodies are subjected to impact, they are kept into contact during a certain interval of time, the stiffer the bodies, the shorter the interval. When performing a double time-scale analysis, the stiffness coefficients of the bodies are introduced as increasing parameters, so that the bodies tend to become rigid, the duration of the contact might vanish, and in most cases the contact locus reduces to isolated points. It may happen that kinematic variables, such as the relative velocities before the impact, and after the impact, and the time density of impulse at the contact point, are found to satisfy a relation, which is referred to as a shock law. For instance some elementary models, like contacting particles or rigid spheres, interacting through a frictionless thin elastic layer, yield the well known restitution shock law when the stiffness tends to infinity: the normal relative velocity e/N+ after the impact and the normal relative velocity qdN- before the impact, satisfy the requirement q./N+=-eC./N-, where ee [0,1] is Newton's classical coefficient of restitution. Furthermore, if the contact episode obeys Coulomb's law, the same restitution law still emerges while the sliding velocity U.r+ after the impact and the time density of impulse R, are found to satisfy Coulomb's law, (Jean, [3]). If a single contact is involved this frictional shock law proves to be dissipative. If other shocks occur the dissipativity cannot be proven, except for the inelastic case e=0. Moreau [4] has introduced the dissipation coefficient ~i: the linear combination (1-~)/2U'+ (l+5)/2rU+, and the time density of impulse R satisfy a relation similar to Coulomb's law. The dissipativity of such a law is proven. Generally, there is no evidence,

466 neither experimental, nor theoretical, arising from a double time-scale analysis for instance, that a shock law may be exhibited or even more be relevant. A number of authors have mentioned cases where such laws fail to be dissipative (Stronge, [5]). These remarks apply to collection of rocks or stone blocks. In this paper, only the inelastic shock law will be used as outlined below. In paragraphs 1.1 ..... 1.5, motions are assumed to be smooth. The extension to motions with discontinuities will be discussed at the paragraph 1.6. 1.1. Local variables For the sake of simplicity we consider the 2-dimensional case of a body O , candidate for contact with some opposing body O', at some time t. These bodies may be rigid or deformable ones. Some special particles P of ~O are defined as particles candidates for contact. For instance if O is described by a mesh, the nodes of the boundary may be defined as candidates for contact. Another criterion which may be used in general situations is that the particle P be a proximal point to O'. An opposing particle P' to a candidate P is defined as a particle of the boundary ~O' of O' which is an orthogonal projection of P. A local frame is defined by its origin P', by the unit normal vector N directed from P' toward P, and by a tangent vector T to ~90' at P', (figure 1). Such a frame might be easily constructed when the boundaries of O and O' are smooth. However, a candidate for contact P might have several projection P'. Bodies O and O' should be partitioned so that a candidate P for contact be associated with a unique pair of objects, namely a candidate for contact object, and an opposing object, allowing one to construct a unique opposing particle P' and a unique local frame. So it may happen, that a particle P be several times candidate for contact, defined by a single candidate object, but various opposing objects. The following local variables are defined:

Figure 1. Local frame.

467

qN R = (R T , R N )

U = ( U T , UN )

N coordinate of P in the local frame, or gap, components of the reaction from O' exerted on P, components of the relative velocity of P with respect to O', Vp-Vp' ; when contact occurs, q.t1. is the sliding velocity.

When two continuous deformable bodies O , O', are contacting, particles of some continuous subset of/90 are possibly candidates for contact. The reaction from O' exerted on each particle of ~O candidate for contact, is described by a line (2-dimensional case), surface (3-dimensional case) density of force, i.e. a stress vector. Since numerical methods are the subject of this paper, only a finite number of degrees of freedom is considered. When dealing with such finite models, a f'mite number of candidates for contact is selected, for instance nodes or Gauss points in a finite mesh, and reaction forces are exerted on these candidates, possibly affected with some weight coefficients. To derive the relations governing these forces from those frictional contact laws adopted for the contacting continuous media is a mathematical and numerical analysis problem, (Jean, [6]). Here, for the sake of simplicity, it will be assumed that frictional contact laws are applied to nodes of meshes, which proves to be a correct approximation for linear elements, and is considered to be correct when appropriate weight coefficients are used.

1.2. U n i l a t e r a l c o n t a c t The main features of unilateral contact are:

i Impenetrability: qN > 0 . ii Bodies are assumed not to attract themselves when contacting: qN = 0 =~ R N > 0 . iii If P is is not in contact with O , the reaction force is still mentioned, but with a null value: qN > 0 :::0 RN = 0 .

The relations i, ii, iii, may be summarized all together in one of the three equivalent relations, qN > 0 R N > 0 and qN > 0 and

RN > 0 VS N > 0 'q' PN > 0

qN RN = 0 , qN(SN-RN) > 0 , (PN" q N ) RN > 0 .

(1) (2) (3)

Another equivalent form is there exists P > 0 such that R N = proj R+ (R N - p qN ) '

(4)

(if this relation is true for some p > 0, it is actually true for any p > 0). The symbol proJE x denotes the orthogonal projection of x on a convex set E. The relation (1) is known as a complementarity condition, or Signorini's condition. The graph of this relation is shown on figure 2. The above relations are all equivalent forms to express the following

468

convex analysis property: let WE be the indicator function of the set E, i.e. WE(X) = 0 if x~ E, WE(X) = +oo if x~ E;

the variables qN and

-

RN are conjugate with respect to the pair of conjugate functions

W ' R + = WR- , W R + .

In this paper the mathematical details are not considered. This property is only mentioned to emphasize the fact that some useful mathematical properties, such as monotonicity, underlie Signorini's condition. For brevity, Signorini's condition will be referred to as follows: S (qN' RN) is true.

y

Figure 2. Signorini's condition graph.

1.3. Friction

law

Coulomb's law is first presented since it accounts for the main features of dry friction. In the 2-dimensional case, R T E [- I.tRN , ~ R N ] ,

I.l. friction coefficient,

r > 0 :=, R T =-I.tRN UT < 0 =:~ R T = I.tRN

, .

(7)

U T is the sliding velocity. The graph of this relation is displayed on figure 3. Another equivalent form is the so-called principle of maximal dissipation,

RT ~ C

V ST ~ C

r

RT ) > 0.

(8)

where C denotes the interval [- 12RN , 12RN ] . This relation is in turn equivalent to the following" there exists p > 0 such that R T = proj C (RT- p ~ I ' ) '

(9)

(if this relation is true for some p > 0, then it is true for any p > 0). Here again the above relations are equivalent ways of expressing the convex analysis property:

469 the variables -UT and R T are conjugate with respect to the pair of functions W ' C , WC.

The function W*C conjugate of Wc actually equals q'*C (-U,r ) = IUTIW*C may be viewed as a dissipation "pseudo-potential". The same holds in the 3-dimensional case. The convex set C then equals the disk, C = { R " IIRII< g }, g = la R N 9 For brevity, Coulomb's law will be referred to as follows: CRN (U T , R T) is true.

v

% Figure 3. Coulomb's law graph.

1.4. Regular forms of frictional contact laws The graphs of figures 2 and 3 are not the graphs of mappings, since '/hi is neither a function of R N n o r R N a function of qN 9Similarly, UT is neither a function of R T n o r R T a function of UT. Convex analysis allows one to deal with such graphs. Usual techniques of regular nonlinear analysis may be applied only to graphs of mappings. A classical example of such graphs of the latter sort is displayed on figures 4 and 5. The graph on figure 4, shows the normal reaction force opposing interpenetration as a linear function of the negative gap. The slope of the graph, namely a stiffness coefficient, is supposed to be large enough to restrict penetration at an acceptable level. Such a mechanical behaviour appears realistic if one figures out that the boundaries of the contacting bodies are coated with a thin elastic layer, or if the possible asperities are elastic. On the graph on figure 5, when the sliding velocity U T is vanishing, the friction force is proportional and opposite to this velocity. This is viscous damping, with a viscosity coefficient large enough to ensure a reasonably small sliding velocity. For numerical purposes, the sliding velocity is approximated as the ratio, ALT/At, where At is the time step and ALT is the increment of tangential displacement. Thus, the friction force appears as proportional to the displacement from a reference position, the end position of the previous time step. This is intepreted as the action from elastic layers or elastic asperities. Nevertheless, especially when contact involves high pressures and large sliding distances, more complicated phenomena should be expected, like plasticity and wear. This suggests that the graphs on figures 4 and 5 should be smoothed. Besides, using smooth graphs allows one

470 to apply smooth nonlinear analysis, (Oden, Martins, [7]). Nevertheless, reproducible experiments which could produce reliable values of tangential or normal elastic stiffness coefficients, tangential viscosity coefficients, or any physical value related to frictional contact, are still unavailable.

%

Figure 4. A regularized unilateral contact law.

%

Figure 5. A regularized friction law.

%

Figure 6. Static and dynamic friction coefficients.

With the elastic shear behaviour, it is more advisable to introduce one more variable, the shear elastic displacement AL T , together with Coulomb's law, to avoid any error when interpretating the graph on figure 5. Suppose a tangential loading causes an elastic shear ALT and a relative velocity UT with the same sign. When the loading is reversed, the relative velocity changes its sign while the elastic shear is still ALT . The graph on figure 5 does not make any difference between ALT and ~tT , and shows a friction force with the incorrect sign. Complicated phenomena such as wear, solid lubrication, existence of a joint lying between the contacting bodies, may produce a frictional behaviour rather discrepant from Coulomb's law. For instance the graph of the friction law displayed on figure 6, distinguishes a static friction coefficient and a dynamical one. In this example, the friction force R T may be considered as the sum of two terms: RT = ~r + 9~r ; 9~T is a friction force obeying Coulomb's law while 9 ~ = fl UT) is a smooth function of the sliding velocity '//T"

1.5. Relaxed contact and friction laws, thick graphs As it has been mentioned in paragraphs 1.2 and 1.3, Signorini's condition and Coulomb's law have interesting properties in the context of Convex Analysis. Many techniques such as quasi-variational inequalities, differential inclusion, piecewisc continuous mapping fixed point theories, may be used to deal with such laws. This is a reason to favour them when constructing numerical algorithms. It has been noticed that these laws are not adexluate to describe complex phenomena. A way to overcome the inherent uncertainty of the situation is to define relaxed frictional contact laws. A pair qN' ~N' is said to satisfy a relaxed Signorini's unilateral condition, up to some given gap margin AqN, and some given reaction force margin Ag~N, if there exists a pair ~qN' ~ N ' with 15qNl<_AqN, 15~l<_Ag~, such that the pair qN+SqN, 9~+~, satisfies Signorini's condition. The set of pairs qN' 9~, satisfying the relaxed

471 Signorini's condition is displayed on figure 7, as what may be called a thick graph. In the same way one may define a relaxed Coulomb's law, up to some given sliding velocity margin A and some given friction force margin ART. The corresponding thick graph is displayed on figure 8.

qdAqs

Figure 7. Thick graph of Signorini's condition.

Figure 8. Thick graph of Coulomb's law.

These relaxed frictional contact laws account for the lack of experimental information. They also account for numerical computation errors, since the computed values obtained from an algorithm satisfy the asserted laws up to certain margins only. For the applications of the paragraph 5, Signorini's condition and Coulomb's law up to certain margins are adopted.

1.6. Non smooth motions Only smooth motions have been considered in the foregoing. The variables used to describe a frictional contact law are, the gap qN' the sliding velocity UT , and the reaction force R. The frictional contact law is referred to under the form, S (qN' RN) is true,

Signorini's condition

CRN ( u T , R T) is true.

Coulomb's law

When non smooth motions occur, i.e. discontinuous velocities, the relation to be obtained should involve only those elements significant for bounded variation functions, i.e. the right and left limits, U +, U-, and the densities of impulse. Signorini's condition is positively homogeneous with respect to ~N' i.e. if a pair, qN' P'N' verifies S, the pair qN' ~'~N' verifies also S, where k is any positive number. The relation S may thus be readily extended to the case where ~N is a density of impulse. The relation C is also positively homogeneous with respect to R, i.e. if a pair, UT, R, verifies C, the pair u T, ~,R, verifies also C. The relation C may thus also be extended to the case where R is a density of impulse. But what interpretation should then be given to UT ? When one of the two contacting bodies is deformable, Coulomb's law is extended as CRN (UT+, R T) is true.

472 When two rigid bodies are contacting, a double time-scale analysis should be performed, to understand or model the short physical process producing the steep changes of velocity, and see if a shock law may finally be exhibited. This is not always the case. For the applications of the present paper, inelastic shocks are assumed namely: >0~

UN+ = O,

and Coulomb's law is extended as

CRN ( UT+, R T) is true. It may be proved that Signorini's condition together with the inelastic shock law is a complete law in the sense of Moreau [4]. This law is found equivalent to qN>O~ ~,~ = O, qN = 0 :=~ UN+ > 0

> 0

UN+ ~

= 0.

(lbis)

The proof is partly based on the property that when a contact occurs at some time t, a strictly negative value UN+ would yield strictly negative values of the gap qN in some interval ]t, t+X].

2. E Q U A T I O N S OF T H E F R I C T I O N A L C O N T A C T P R O B L E M

2.1. Kinematic equations A mechanical system with n degrees of freedom is to be described by a variable q~ R n. When the system is a discrete model of a continuous medium obtained through such a method as the finite element method, the variable q is for instance the vector of node displacements. If t ---) q(t) is a smooth motion, the first and second time derivative are denoted respectively, c], ~1". Let pa be a candidate for contact (upper Greek indices are numbering candidates for contact, or=1,..., X, where ~ is the number of candidates). The relative velocity ~ of the candidate Pa with respect to the antagonistic body may be written, U a = H*(q) et ~,

(9a)

where H*(q) ot is the transpose of a linear mapping H(q) from R n into R 2 (2-dimensional case) or R 3 (3-dimensional case). This relation is obtained from mere kinematic considerations. It involves the matrices giving the components of vectors in the local frame from those in the general frame. The dual relation is an expression of the representative R a of the local reaction force ~ , for the system of parameter q, R a = H(q) a Ra.

(10s)

473 The following property allows one to relate the gap to the normal relative velocity,

UNa =qNa ,

(lla)

Considering all candidates for contact, introducing the 2~ (or 3X)vectors U = (U 1..... U a .... U~), R = (R1..... ~ .... J~:), the above relations may be summarized as U - H*(q) ~l, R = H(q) R ,

(9) (10)

o

UN = qN 9

(11)

The linear mapping H(q) from R2X into R n is injective if and only if H*(q) from R n into R2~; is surjective. The mapping H(q) is injective, if there exists no system of local reaction forces R with a null representative R; the mapping H*(q) is surjective, if any system of relative velocity U is the image of some (]. In other words H(q) is an injective mapping or H*(q) a surjective mapping if constraints are kinematically independent. This is not the general case, for instance when two rigid bodies have severn contact points. Another example is when several nodes of the same element are contacting the same boundary element.

2.2. The equation of Dynamics Lagrange's equation governing the smooth motion of the mechanical system may be written as, M(q) ~l~ = F(t, q, ~t ) ,

(12)

where M(q) is the inertia matrix, F represents external forces, and quadratic inertia terms with respect to ~t- When the system is a discrete model of a continuous medium obtained through such a method as the finite element method, F stands for external or internal forces as well. When frictional contact occurs, the representative R of frictional contact forces R has to be written at the righthand side of the dynamical equations together with the other forces, M(q) [l* = F(t, q, ~t ) + R.

(13)

This equation has to be written together with the kinematic relations, (9), (10), (11), and the frictional contact law. So far smooth motions have been considered. When contact and friction occurs, the velocities are not generally smooth, but must be expected to be bounded variation functions, while the reaction forces R, R, are densities with respect to positive real measures as already discussed in paragraph 1.6. The equation (13) must be understood in the sense of distributions, or written as a differential equation in the sense of measures, (dt is the Lebesgue measure on the real line R, dO is a positive measure, standing for the Lebesgue measure as well as for the Dirac measure),

474 (13bis)

M(q) d~l = F(t, q, ~t )dt + Rd0.

The kinematic relations may be readily extended to the case where ~t and U are bounded variation functions, and R and R are densities of measures. The question to establish whether the frictional contact laws may be extended to the case of bounded variation functions and density of measures is not merely a mathematical matter and has already been discussed in 1.6. The equations of the frictional contact problem are as follows: the equation of Dynamics, M(q) d~t = F(t, q, ~t )dt + Rd0, the initial conditions, q(t0)=q0, the kinematic relations, U = H*(q) ~t , R = H(q) R , o

UN =qN ' the frictional contact laws, for or= 1..... X, S (qN~ , RN ~ is true,

Signorini's condition (or relaxed Signorini's condition)

CRN (q/Tu+, R T) is true.

Coulomb's law (or relaxed Coulomb's law)

RN u > 0 ~

inelastic shock when rigid bodies are contacting

UN~

= 0

,

f

o

Equation of D y n a m i c s - - ~ ~ ~

1~t}

{R}

H*

~

H

~ _ _Fricti~

c~

la__...ws..,. ")r

Figure 9. Equations of the frictional contact problem.

The question of the existence and uniqueness of the solutions of these equations is not within the scope of this paper. The literature provides theorems of existence dealing with the frictionless case or with regularized frictional contact laws, for the dynamical or the quasi-static case. If the friction coefficient is "small enough", there exists a unique solution. With

475 Coulomb's law, even when the constraints are kinematically independent, (H(q) is an injective mapping), the existence and uniqueness of a solution is not ensured, and a locking phenomenon may occur. This occurs a fortiori when the constraints are kinematically dependent, for example when collections of rigid bodies are concerned.

3. D I S C R E T I Z E D E Q U A T I O N S

3.1. The equation of Dynamics To begin with, the non contact case is considered. M(q) d(~ = F(t, q, t~ )dt . One considers a time step, ] ti, tI+l], h = tI+l-tI 9The symbol ~t(I) denotes an approximate value of ~t(tI) and ~t(I+1) an approximate value of ~t(tI+1); the meaning is similar using q(I) and q(tI), q(I+l) and q(tI§ Integrating both sides of the equation of Dynamics on the time interval and using the approximations, q(I+ 1)-q(I)

= h~(I+ 1) ,

~M(q) d~l

= M(q(I+ 1) )(~(I+ 1)-~t(I)) ,

IF(t, q, ~t )dt = hF(~,q(I+ 1),~l(I+l)),

i=tI+l ,

allows one to derive a discretized form of the equation of Dynamics, M(q(I+ 1))(~(I+ 1)-~(I)) = hF(~,q (I+ 1)),~(I+ 1))

(1)

The terms ~l(I+ 1), q(I+ 1), are obtained as limits of sequences qk, qk, defined by the following algorithm. The matrix M(q) is assumed to be a slowly varying function of q. The term F may be predicted using a first order expansion, as usually done when performing a Newton Raphson method. One sets Mk

= M(qk)

,

Vk =-()F(~, qk, qk ) / t)~] , Kk=-/)F (i , q k , q k* ) / / ) q

Fk=

F

,

,qk,qk),

= Mk + hVk + h2Kk, Wk =/~Ik "1 The matrices Mk, Vk, Kk, are supposed symetric positive definite so that 10k has a symetric positive definite inverse matrix. One chooses, qk - q(1) = h qk , q k + l -

q(I) = h (~k+l .

476 An approximate value of F is

F(I, qk+l, ~k+l ) = F(~, qk + (qk+l - qk ), ~k + (~k+l-~k)). Fk+l _= Fk - hKk (qk+l-qk)- Vk (qk+l-qk) . The algorithm can be written as

qk+l= ~]k+ Wk(-Mk ( ~]k- ~l(I))+hFk ), o

qk+l - q(I) = h qk+ 1. A more general statement is, # ~k+l= ~tk + Wk(-Mk ( ~k- ~(I)) +hFk ) , qk+l - q(I) = h qk+ 1, o

#

where Wk is a symetric positive def'mite matrix.The usual initial values are, ~0 = ~ ( I ) , qO = q(I). If the sequences ~tk, qk, converge to some values ~t(I+ 1), q(I+ 1), they are found to satisfy the equation (1). If Mk = M is a constant matrix, and F(t, q, ~t ) = Kq + 0 ~t + P(t), #

where, K, 0, are symetric positive definite matrices, when the choice Wk = (M+hV+h2K) -1 is made, the algorithm converges within a single iteration. Remark 1" A first order time expansion has been used to approximate q, ~1- Higher order approximations are most often used in dynamical computations. It is certainly very valuable as far as smooth motions are concerned. But when contact and friction occurs, the velocity ~1 is not differentiable anymore in the usual sense. It is a function with discontinuities. At a point of discontinuity, the derivative ~1" in the sense of distributions is a Dirac measure. So, a second order approximation scheme has to manage with approximate values of ~*, which are quite perturbating large numbers. Approximations such as, q(I+ 1)-q(I) = h(1-0) ~l(I) +h0 ~(I+ 1), may also be used giving numerically stable algorithms provided 0e ]1/2,1]. # Remark 2: When the choice Wk = 1~Ik-1is made, the corresponding algorithna is implicit. Notice

that if the mass matrix Mr: is vanishing the algorithm is found to be the one generated by the Newton-Raphson #method for quasi-static formulations. When Mk = M is a constant diagonal matrix, choosing Wk = M-1 , and performing only a single iteration on the index k, one gets a fast explicit scheme. When large elasto-plastic deformations are involved, implicit methods may be favoured, since the equation of Dynamics is then satisfied with external and internal forces computed at the end of the time step, while when using explicit methods these forces are

477 computed at the end of the previous time step, with the risk of error accumulation. The implicit method is most costly since derivatives of F have to be computed and a large linear system has to be solved for each iteration k, while a straightforward computation is performed when using the explicit scheme. Nevertheless, the time step should be small enough to ensure numerical stability and sufficient accuracy, while the time step is usually larger when using implicit methods which prove to be unconditionally stable. Of course many other choices of correcting # matrices Wk are possible.

3.2. The discretized equations of the frictional contact problem When contact and friction occur, the representative of the reaction forces must be writen together with other forces. Approximate values of the relative velocities and gaps, at the end of the time step, are denoted, U(I+l), qN(I+l). The right limit of the relative velocity r involved in the writing of Coulomb's law and of the inelastic shock law, paragraph 1.6. The choice is made to consider U(I+ 1) as an approximation of U ~. The local reaction impulses and the representatives of these impulses, during the time step, are denoted, h ~ I + 1), hR(I+ 1). The approximated value R(I+ 1) is sought as the limit of a sequence Rk satisfying # ~k+l = ~k + W k ( - M k ( ~k- ~(I)) +hFk +hRk+l), qk+l - q(I) = h ~k+l .

Notice that the unknown is hR(I+l) and not hR(I). Notice also that Rk+l has been written at the second hand rather than Rk. The scheme is thus implicit with respect to the frictional contact variables. Approximate values ~ I + 1), U(I+ 1), qN(I+1), are sought as the limits of sequences, ~lc, Uk, qNk . The kinematic relatrions are written using the discretized form, Uk+l

= H*(qk+l) qk+l,

Rk+ I = H(qk+ I) Rk+ I , qNk+l = qN(1) + h UNk+I .

The last relation comes from 2.1 (11). The term qNa(I) is the gap for the contact a at the beginning of the time step. When the radius of curvature of contacting regions is large enough with respect to the distance run within a time step, the local frames do not change much, and an explicit computation of values of H and H* may be done once for all, for instance H(qk), H(qk), or merely H(q(I)), H*(q(I)). Such values are noted Ilk, Ilk*. The discretized contact and friction relations are S (qN~

1, RNOtk+1) is true,

Signorini's condition

CRNk+ 1 ( UTak+ 1, RTak+ 1) is true,

Coulomb's law

RNtXk+I ::~ UNtXk+I = 0 .

inelastic shock when rigid bodies are contacting

478 Due to the positive homogeneity of Signorini's condition and Coulomb's law, the impulse h ~ + l may be choosen as a variable as well as the density of impulse ~+1. To summarize, the discretized equations are

The equation of Dynamics # fik+l = (]k + Wk(-Mk ( qk- fi(I)) +hFk +hRk+l), o

(14)

o

qk+l - q(I) = h qk+l , the initial conditions, the kinematic relations Uk+l = Hk* qk+l,

(15)

Rk+l = Hk Rk+l ,

(16)

qNk+1 = qN(1) + hUNk+l

(17)

,

the frictional contact laws, for a= I .....;(,

S (qNak+ 1, RN~

1) is true,

CRNk+ 1 (UTak+ 1, RTak+ 1) is true, RNk+I

:::# UN~

= 0 .

Signorini's condition Coulomb's law inelastic shock when rigid bodies are contacting

4. SOLVING THE D I S C R E T I Z E D EQUATIONS 4.1. The system of equations with local variables as primary unknowns A number of methods may be used to solve the system of equations of paragraph 3, such as variational inequalities techniques or fixed point techniques, (see a review, Jean, [8]). The method presented here is the one used to compute the examples of paragraph 5. Using the kinematic relations, (15), (16), (17), and the equation of Dynamics (14), one obtains

Uk+l = 'b~ib k + '14,khRk+l , # r = Hk*WkHk, # '/}lib k = Hk*(qk + Wk(-Mk ( ~]k- ~(I)) +hFk)).

(18)

The term 'l,]ib k is the free relative velocity, i.e. the relative velocity of candidates when no reactions are applied, Rk = 0. The relation (18) shows the linear dependance between the unknowns, Uk+l, hRk+l, obtained through the linearized form of the equation of Dynamics and the kinematic equations. The equation (18) is written together with Signorini's condition and Coulomb's law. The unknowns of the system are thus ~ + 1 and Rk+l. Sub iterations are needed to solve this system. It is understood that quantities with indices k have provisional

479 values which are known. In the following, the indices k, k+l, shall be omitted to simplify. The system to be solved is then U = 'l'tib + W h R , qN = qN(I) +hU N , for a= 1..... X, S (qNa, RN a) is true,

Signorini's condition

CRN (UTa, RT a) is true,

Coulomb's law

RN a ~

UNa

inelastic shock when rigid bodies are contacting

= 0 .

the unknowns of which are the relative velocities U and the density of impulse R. 4.2. Rewriting Signorini's condition Signorini's condition may be written

UNa >_ 0 TN a >_ 0 UNa TNa= 0 where UNa = qNa(I)/h + UNa . When the candidate a defines a contact between two rigid bodies, the form (Ibis), paragraph 1.6, equivalent to Signorini's condition together with the inelastic shock law is used: if a contact is not expected, ~qa = 0 if a contact is expected, UNa > 0

RNa > 0

UNaP~Na= 0

where UNa = U N a . A criterion has to be chosen, in order to decide if a contact is expected. Two possible criteria ale:

i) A value of the flee gap, i.e. the predicted gap when reactions are not exerted is qNa = qNa(I) +h~ibN a . If this free gap is negative a contact is assumed to be expected. A prediction at half the time step may also be used. This criterion decides explicitly the status of contact. ii) A value of the flee gap, i.e. the predicted gap when the reaction on the candidate a is null, and when the reactions on other candidates f l ~ have provisional values, is qNa = qNa(I) +hVNa, 'I/Ix =

V l i b ot +

y'

13#-a

Wotl3

hRI3

.

480 If this free gap is negative a contact is assumed to be expected. This criterion, of the implicit kind, is examined each time values of R a are being computed. To summarize, the following form of Signorini's condition is written S (UNO~, RN ~ is true. According to the definition of UNa, in the case where one of the contacting body is deformable this relation accounts for the usual Signorini's condition S (qN ~ RN~ and when the two contacting bodies are rigid, it accounts for the usual Signorini's condition together with the inelastic shock law.

4.3. T h e s o l u t i o n for the c a s e o f a s i n g l e c o n t a c t

The case of a single contact is first discussed. The index a is omitted here. One definesU = (UT, U N ), U T = U T, UlibT = UlibT, and UN is defined as in 4.2: U N = qN/h = qN(I)/h + U N , Ulib N = qN(I)]h + UlibN when one of the contacting body is deformable'U N = U N , UlibN = UlibN when two rigid bodies are contacting. Using the forms (4) and (9) of Signorini's condition and Coulomb's law, the system may be written, U - Ulib - W h R

= O ,

R N - proj R+ (~,j - PUN) = 0, R T - proj C ( R T - p UT) = 0, C = [I.tRN,-I.tR N] in the 2-dimensional case, C is the disk with center 0 and radius laR N in the 3-dimensional case. This system has the form O(X)--0, where X = ( U ,90, and 9 is a piecewise continuous linear function. In the 2-dimensional case, the solution may be exhibited in a straightforward manner: one sets f

= '/4)'1 Ulib,

w=[

wNNNwT wTTWNT]'

a =-WNTIWNN

It is a s s u m e d -1 < laa < 1 . Then the system has a unique solution:

if UlibN > 0

then

h R T = 0 , hRN = 0 , i.e. no contact

if UlibN <__0 and fT + I'tfN- 0 i.e. forward sliding;

then h R T = -It h R N , h R N = -Ulib N / (l+l-ta) WNN ,

481 if UlibN < 0 and fT - lafN < 0 i.e backward sliding;

then hRT = 12 hR N , hRN = -UlibN / (1-11a) wNN ,

if UlibN < 0 and fT + I'tfN < 0 et fT- I'tfN > 0 then hRT = -fT ' hRN = "fN' i.e. contact without sliding. When the inequality -1 < laa < 1 is not satisfied, the solution is not unique. In the 3dimensional case, the solution is not straightforward and it is numerically computed using a generalized Newton method which yields the solution within a few iterations, (Jean, [9]). 4.4. The solution when several contacts are involved

The unknowns U t~, R~t , tx=l ..... ~, (actually Uak+l, 9(~ but the indices k+l, k, are omitted here) are sought as limits of sequences U~tp, RtXp (actually subsequences q./tXk+l,p, 9~k+l,p ), (Jean, Moreau, [ 10]). Suppose that provisional values of the local reactions ~ p are adopted for the candidates [ga. The equation (18) writes, UCtp+l = U lib a + E ff12ot13 hg(.13p + q42t~ct hRap+l , together with Signorini's condition and Coulomb's law written for the candidate a. A value of U~p+I , Rap+l, is readily obtained from the formulas of paragraph 4.2. All candidates for contact are successively examined, repeatedly, until some convergence criterion is satisfied. Namely the approximate values are prescribed to satisfy relaxed Signorini's condition and Coulomb's law up to certain given margins. The convergence of this relaxation algorithm has not been mathematically established. Nevertheless, the numerical convergence is satisfactory.

5. E X A M P L E S OF G E O M A T E R I A L MOTIONS 5.1. Compression of geologic layers

Two viscoelastic layers, 10 kms deep, 20 kms long, are subjected on the left edge to a constant velocity equal to v0=0.33 10 -10 m/s toward the fight edge, while only vertical displacements are allowed on the fight and left edges, and only horizontal displacements occur along the bottom edge. The layers are subjected to gravity. The viscoelatic layers obey a simplified behaviour law, Maxwell's law ~*=2~tD+Xtr(D)I-2"~dev(a), ~,=vE/(1 +v)(1-2v), ~=E/2(1 +v), rl=l/2"y, c is the Cauchy stress tensor and 6" is the Green-Naghdi derivative. The stiffness coefficient E is equal to 1010 Pa, the Poisson coefficient v=0.25. The upper layer has a viscosity coefficient 11 equal to 1023 Paxs, and the lower layer, quite less viscous, has a viscosity coefficient rl equal to 1021 Paxs. The experiment duration equals 106 years. A fault (crack) crosses the two layers. The layers are discretized with 1890 linear triangular elements T3, and 2062 degrees of freedom; 25 nodes are candidate for contact on each edge of the fault. Signorini's condition and Coulomb's law are assumed on the fault, with a friction coefficient equal to 0.2. An explicit scheme with respect to q, ~t, is used to integrate the equation of

482 Dynamics, and an implicit scheme is used to compute the reaction forces as described in paragraph 4. One is interested in the quasi-static evolution of the layers. Nodes are equipped with fictitious adaptive masses in order to allow one the use of large time steps, (about 3000 steps for the duration of the experience), the ratio of inertia forces to internal forces being controled during the computation. The final state is displayed on figures 10, 11, 12. The figure 10 displays the sliding velocity between the edges of the fault (divided by v0) versus the depth. It shows that no sliding occurs on the lower part of the fault, and the sliding velocity rapidly increases, from the lower layer, up to the surface of the upper layer. Large deviatoric stresses are developed in the neighborhood of the fault when crossing from a layer to the other one, figure 11. Large deviatoric strains are developed in the neighborhood of the fault in the lower layer, figure 12. This example has been computed by J. Chery, Laboratoire de G6ophysique, Montpellier.

=,,

depth (meter)

O

v0 9

9

sliding velocity/vO -20

I

0 Figure 10.

1

Figure 11. Second invariant of the deviatoric stress; scale from 20 to 220 MPa.

Figure 12. Second invariant of the deviatoric strain; scale from 0.01 to 0.09.

483 5.2. Bi-directional strain-stress experiment of a Schneebeli material The Schneebeli material consists of a collection of rigid disks or rolls. It is widely used as an approximate model of soil, either in experiments, or in numerical simulations. A number of authors have used the numerical program TRUBAL, or improved versions of this program,

originally developped by Cundall, [11], to perform numerical simulations of classical mechanical tests. In the TRUBAL program, the contact is established through a system of springs and dampers, and the tangential stress follows Coulomb's law. An explicit integration scheme is used for the equation of Dynamics. Numerical strain-stress experiments presented in this paper have been performed with the program LMGC developed by the author, using relaxed Signorini's condition, relaxed Coulomb's law and an implicit Euler integration scheme for the equation of Dynamics, as described in the previous section. The results obtained by Yemmas, [16], are compared with those obtained by Cambou, Mahboubi, Ecole Centrale de Lyon, France, using TRUBAL. This comparison was the object of a research program within the GRECO G6omat6riaux (a report is to be published). Using TRUBAL, the question of the influence of the regularizing coefficients, namely the tangential and normal stiffness and viscosity coefficients has to be investigated. This question has been widely surveyed by Kruyt, [ 12]. It seems that for the strain-stress experiment, the results do not depend too much on these coefficients. When using LMGC, the question of the permitted margins is also raised. To provide a comparison, an interpenetration less than 1/100 of the minimal disk radius was allowed. The facilities offered by the two programs are different and it is so far impossible to perform exactly the same numerical experiments. For instance, the samples of material are not generated in the same way, boundaries conditions are not applied in the same way, and the stress tensor is not computed in the same way. The sample used for the purpose by LMGC contains 256 disks (rolls with a unit lenght of 1 m), 48 disks with radius 1.6 mm, 80 disks with radius 1.05 ram, 1:28 disks with radius 0.65 ram, figure 13. Samples with 1024 disks have also been used. The upper frictionless wall, is submitted to a constant vertical force P=3.300 N, while the fight hand side frictionless wall, moves with a constant velocity V=10 cm/s and induces a resulting reaction force R. The two other walls are fixed and frictionless. The ratio of P to the box width L, is referred to as if2, and the ratio of R to the height box as ~1. The initial width box is L0. The response cl/o'2 versus the strain (L-L0)/L0 is rather erratic. The disks are moving discontinuously. A system of rigid disks may produce locking, and the deformation is possible only if some ease is allowed between disks, for instance allowing elastic deformation in TRUBAL, or allowing margins of interpenetration in LMGC. The figure 13 displays the internal or global angle of friction r defined by sin~ - (~1-o'2)/(ff1+o'2) versus the local angle of friction r between disks. The experimental results were obtained by Abriak, Ecole des Mines de Douai, (to be published). These angles are computed using "the maximum value" of ~1/c2. The figure shows that when the local angle q~becomes large, the global angle seems to reach a steady value. One may also note that when q~is large enough, the void ratio increases when the strain (L-L0)/L0 increases. Disks instead of sliding on each other, rather roll, the sample expands, and more ease is allowed between disks. Consequently r does not increase as much as r

484

~

L

P

I I

II

i

o i t

~', : ~

~_"L~-~----

I o

!

trajectories It--0.5

al/o'2 2.5

TRUBAL LMGC .

1.0

i

strain 50

"

16%

(L- Lo~/Lo

global angle of friction (degree) 0.3 I void rate

It" friction coefficient la=l It 0.5 g 0.375 0.25

/ 25

9 lb

9

n ~

9

9 *

,u

14

LMGC ECL ABRIAK 9

0.2 strain

local angle of friction (degree)

a 16%

(L- Lo)/L 0 Figure 13.

485

5.3. A temple_like building made of rigid blocks This example has been computed by Moreau (private communication), with a program using an algorithm as described in this paper. Blocks are rigid and inelastic shocks are assumed to occur between comers and edges. A horizontal deviation of the ground (1 meter) is applied for 1 second, figure 13. This is an example of numerical simulation performed in order to understand the behaviour of ancient buildings made of blocks without any joint when subjected to earthquakes. More complicated examples may be found in [ 10]. Examples with a single block are also interesting, since they allow one to make comparisons with rigid body models or finite element viscoelastic models, with analytical solutions obtained for special motions, (Sinopoli, [13]) or experiments, (Raous, [14]). The underlying question, already raised paragraph 1, is the existence of a relevant shock law for blocks. The computer program UDEC, originated by Cundall, [15], deals with such collections of blocks. As in TRUBAL, the contact is established through a system of springs and dampers, and the tangential stress follows Coulomb's law. Similarly one has to enquire which normal and tangential stiffness and viscosity coefficients are to be adopted in an analysis to obtain a realistic frictional contact model.

486 REFERENCES

1. C.Licht, Comportement asymptotique d'une bande dissipative mince de faible rigidit6, C. R. Acad. Sci. Pads, t. 317, S6rie I, p. 429-433,1993. 2. M. Boulon, Basic features of soil-structure interface behaviour, Computers and G6otechnics, 7, pp 115-131. 3. M. Jean, Dynamics of rigid bodies with dry friction and partially elastic collisions. International Series of Numerical Mathematics, vol 101, Birkh~iuser Verlag, Basel, 1991, pp 57-70. 4. J.J. Moreau, Unilateral Contact and Dry Friction in Finite Freedom Dynamics, in Non Smooth Mechanics and Applications (ed. J.J. Moreau & P.D. Panagiotopoulos), CISM Courses and Lectures, No 302, Springer-Verlag, Wien, New-York, 1988, pp 1-82. 5. W.J. Stronge Rigid body collisions with friction, Proc. R. Soc. Lond. A (1990) 431,169181. 6. M. Jean, Unilateral conctact with dry friction: time and space discrete variables formulation, Arch. of Mech., Vol. 40, 5-6, Warszawa, 1988, pp. 677-691. 7. J.T. Oden, J.A.C. Martins, Models and computational methods for dynamic friction phenomena. Computer Methods in Applied Mechanics Engineering, vol 52, 527-634, 1985. 8. M. Jean, Simulation num6rique des probl~mes de contact avec frottement, MatEriaux et Techniques, Tribologie et Mise en Forme, No 1-2-3 1993. 9. M. Jean, Numerical methods for three dimensional dynamical problems, Proceedings of the Conference Contact Mechanics 93, 13-15 July 1993, Southampton, Computational Mechanics Publications, Southampton Boston, ed. M.H. Aliabadi, C.A. Brebbia,1993. 10. M. Jean, J.J. Moreau, Unilaterality and dry friction in the Dynamics of rigid bodies collections. Proceedings of the Contact Mechanics International Symposium, ed. A. Curnier, Presses Polytechniques et Universitaires Romandes, Lausanne, 1992, pp. 31-48. 11. P.A. Cundall, O.D.L. Strack, A discrete numerical model for granular assemblies. G6otechnique 2, No. 1, pp. 47-65, 1979. 12. N.P. Kruyt, Towards Micro-Mechanical Constitutive Relations for Granular Materials. Modem Approaches to Plasticity, ed. D. Kolymbas, Elsevier Science Publishers B.V.1993. 13. A. Sinopoli, Dynamic analysis of a stone column excited by a sine wave ground motion. Appl. Mech. Rev., ASME, vol 44, 199, pp 246-255. 14. M. Raous, Experimental analysis of the rocking of a rigid block, 3rd Pan American Congress of Applied Mechanics (PACAM III), Sao Paolo, Brazil, january 1993. 15. P.A. Cundall, A computer model for simulating progressive large scale movements of blocky rock systems. Proceedings of the Symposium of the International Society of Rock Mechanics (Nancy, France, 1971), vol 1, pp 132-150. 16. R. Yemmas, Simulation numErique de mat6riaux granulaires. Th~se M6canique, GEnie MEcanique, GEnie Civil, Montpellier 20 dEcembre 1993.

ACKNOWLEDGMENTS The author is a member of the research group "MEcanique Non REguli~re" (Nonsmooth Mechanics) in the Laboratoire de MEcanique et GEnie Civil, Montpellier. Most of the ideas presented in this paper are developed by the group. Most of these ideas originated with J.J. Moreau who advised in the writing of this paper.

Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

487

Interfacial l o c a l i s a t i o n in s i m p l e shear tests on a g r a n u l a r m e d i u m m o d e l l e d as a Cosserat continuum I. Vardoulakis* and P. Unterreiner** *

National Technical University of Athens, Department of Engineering Science, Section of Mechanics, 5 Heroes of Polytechnion Avenue, 15773 Zografou Athens, Greece ** CERMES (ENPC-LCPC), Ecole Nationale des Ponts et Chaussres, La Courtine, 93167 Noisy-le-Grand Cedex, France

SUMMARY Interfacing of a granular material with a structural member involves the localisation of the deformations within a few grain diameter thick layer, which is called interface layer. This layer has been shown experimentally to depend mostly on the roughness characteristics and on the grain diameter but not on the geometrical dimensions of the sample or structural member. Among all the interface shear tests available for studying the formation of interfaces in a granular medium, we have selected the plane simple shear test and the ring simple shear test since they allow the sample to deform freely in its volume before interface localisation starts and since they are one dimensional tests (the stress and strain fields depend only on one space variable, the distance to the interface). Within a granular material, individual grains can translate and rotate. A classical continuum takes into account only the transitional degrees of freedom, while a Cosserat continuum considers the additional rotational degrees of freedom. The appropriate equilibrium equations and boundary conditions for such a continuum are derived beforehand by utilising the principle of virtual work. When modelling the plane simple shear test with a classical continuum, one cannot explain the formation of an interface layer unless ones introduces a strong heterogeneity with a softer material near the interface. The analysis of the ring simple shear test with a classical continuum and rigid plastic constitutive equations yields a boundary layer of plastic material near the interface which has either a zero thickness for an associated material or a thickness directly proportional to the radius of the interface cylinder for a non-associated material. Limitations of a classical continuum can be overcome within the framework of a Cosserat continuum. The plane and ring simple shear tests are analysed using linear elastic and rigid plastic constitutive equations respectively. Both types of constitutive equations introduce an internal material length which is shown to control the thickness of the interface layer independently of the geometric dimensions of the sample.

488

1. INTRODUCTION Most civil engineering constructions involve the interfacing of soil and structural members. For various materials such as, natural soils like sand and gravel [1-3] or model materials like Schneebeli rolls [4-5], it has been observed that very soon after sheafing starts across an interface, the shear deformations concentrate into a thin layer of soil, a few grains diameter thick, at the contact with the structural member. This thin layer can be called the interface layer and is characterised by high displacement gradients, strong local dilantacy and significantly high grain rotations [3, 6]. The interface thickness depends on the grain size and the roughness of the contact surface. At the scale where such phenomena occur, i.e., at the scale of a few grains, the interface surface of the structural member is viewed by the soil as long and large. The only remaining space variable of interest is thus the distance to the interface surface. distance d to the interface surface H -~ ......... top of layer zone with [ \ small l k _ _ _gradients .

.

interface surface

.

.

.

.

.

.

Interface layer (granular soils)

~_F- - strong displacement gradient ~ - significant grain rotation

~

- strong local dilatancy

.

~__ --~-tangential displacement

U1

Figure 1 : Tangential displacement profile in sheafing of a layer Among all interface shear tests, we may select the plane simple shear test [6] and the ring simple shear test [7] for two reasons (Figure 2). First, they induce homogeneous strain and stress fields in the sample, which depend only on the distance to the interface. Secondly, they allow the sample to deform freely in its volume before localisation of the deformations. i~

'

U2

! ~Xl

f

f

~

~3 Figure 2a : Plane simple shear apparatus

/

/

/

/

rigid interfacein translation

Figure 2b : Modelling of plane simple shear

x2 UO

.

-

/ interface

( rotation Figure 2c : Ring simple shear apparatus

~ x!--- ~ )

Figure 2d : Modelling of ring simple shear

489 The plane simple shear test simulates the shearing by a plane structural member of a long layer of soil, of finite thickness (Figures 2a and 2b). On the other hand, the ring simple shear test simulates the shearing of a thick wall cylinder in plane strain by an internal cylindrical inclusion submitted to torsion (Figures 2c and 2d). Both tests minimise, as much as possible, the undesirable comer effects which can be encountered in all types of direct shear tests : plane direct shear test (modified Casagrande box) [8-11], ring direct shear test (modified Bromhead shear apparatus) [12-13], and axisymmetric shear test (pull-out of a rod out of a cylindrical sample) [14]. In both simple shear tests, the geometry as well as the lateral boundary conditions which are applied, generate displacement and strain fields which do depend only on the distance to the interface. In the case of the plane simple shear test, this is possible because of the plane strain conditions which are generated by the two lateral rigid sides of the box and due to the system of piled up frames which ensures to some degree the invariance of the displacement field parallel to the interface surface (Figure 2a). In the case of the ring simple shear test, plane strain conditions are imposed by the top and bottom plates and the geometry has a symmetry of revolution (Figure 2c). Analysis of the two simple shear tests, using a "classical" continuum model does not yield any of the localisation phenomena actually observed. Is defined as a "classical" continuum, a continuum where only degrees of freedom in translation are taken into account for defining the kinematics. For such a continuum, the displacement vector at a given point M is determined experimentally by averaging the displacements of the grains over a sufficiently large surface or volume as compared to the grain size. The individual grain rotations are ignored for defining the kinematics. When modelling plane simple shear of an homogeneous sample using a classical continuum, it is impossible to explain the formation of an interface layer even when one assumes different boundary conditions (at the interface and at the top of the sample) and whatever the constitutive equations are. The only solution to this problem is to assume that the material is heterogeneous with a strong gradient of properties, which is not realistic as will be seen later. An analysis for the ring simple shear test, based on a classical continuum and using a MohrCoulomb rigid plastic non-associated behaviour, yields a solution with a zone of plastified soil in contact with the interface. As it will be demonstrated later, the thickness of this layer in contact with the interface is directly proportional to the internal diameter of the sample. However, experiments show evidence of an interface layer, where tangential displacements concentrate and whose thickness is independent of the diameter of the sample. The use of more complex constitutive equations does not alleviate these limitations of the classical continuum approach for modelling such phenomena and thus other approaches must be developed. Laboratory experiments, analytic computations and discrete numerical modelling of granular materials undergoing localisation tend to show consistently that strain localisation in shear bands [15-17] or at interfaces [18-20] is systematically accompanied by significant individual grain rotations. What should be understood by significant individual grain rotations is that the grains, on average, tend to rotate more than would be expected if they were embedded in a classical continuum medium and would follow its rotation f~ij defined by the antisymmetric part U~,j of the gradient of displacements Ui,j . Therefore, it seems natural when modelling granular media where, for the considered range of stresses, the grains behave almost as rigid bodies, to use a generalised continuum which takes into account not only the

490

translational degrees of freedom of the individual material points but also their rotational degrees of freedom [21-23]. In the case of simple interface shear tests, where the only space variable of interest is the distance to the interface, the experimental values of the new degrees of rotation in a given point M at a distance d = Y-Yint (plane shear) or r-rint (ring simple shear) of the interface surface, are determined by averaging the individual grain rotations over a sufficiently large length parallel to the interface (Figure 3). Similar averaging procedures have already been performed in discrete numerical simulations of an assembly of grains [17]. Experimental measurements of individual grain rotation has been reported only recently for polycrystalline metals [24] and for sand grains within an interface layer [20]. Within the experimental range of error, all these measures confirm that grains may rotate differently than the rotation ot the classical continuum medium in which they are assumed to be embedded.

c3

top of layer U2

(x2)=< %rain(d)>x1C [O,L]

c

H]~(~

(~}(2)Q - -

granular

~~@~0 ~~d~ 2 5 1 2 ~ ( i )

me d i u m

X

A 2

Cosserat continuum

3

U1

~Xl 9----

/

/

/

/

/

/

/

interface Figure 3 : Modelling of a granular medium with a Cosserat continuum A Cosserat continuum owns two main features which are of major interest in modelling interfacial localisation. First, the constitutive equations introduce an internal length, which can be related to the grain size based on micro-mechanical considerations [15,25] and which controls the interface thickness independently of any other geometrical length of the system. Secondly, corresponding to the extra degrees of freedom, additional boundary conditions stem naturally out of the principle of virtual work. Those boundary conditions have to be defined with respect to the surface of the interface and in particular its roughness as previously proposed within the framework of second gradient plasticity theories [26-27]. In this article, we will analyse the plane simple shear test and the ring simple shear test of a granular medium modelled as a Cosserat continuum. The emphasis will be on the features of the Cosserat continuum as compared to the classical continuum, rather than on the constitutive equations. Therefore, two limiting cases of constitutive equations will be studied, i. e., linear elastic and rigid plastic behaviour. It will be shown how a Cosserat continuum can be successful in reproducing the salient properties of an interface layer.

2. PRINCIPLE OF VIRTUAL WORK AND EQUILIBRIUM EQUATIONS Derivation of the equilibrium equations for a Cosserat continuum has been done by many authors [21, 28-30]. In a 1972 paper, Germain advocates the systematic use of the method of virtual work in continuum mechanics to derive the fundamental equations of a given continuum [31] and he applies it to the theory of second gradient [32] as well as to a continuum with microstructure [23].

491 In this article, we will summarise and apply the method proposed by Gerrnain [23] to a continuum with a rigid microstructure [33]. Such a continuum will be referred hereafter as a Cosserat continuum. A classical continuum is composed of a continuous distribution of panicles, each one being represented geometrically by a point M of co-ordinates x i in a Cartesian frame and being characterised kinematically by a displacement vector field U i . In a continuum with a rigid microstructure (Cosserat continuum), each particle is still represented by a point at the macroscopic level. However, its kinematics are defined in a more refined way at the microscopic level. At this level of observation, a point M appears itself as a particle, i.e., a continuum V(M) of small extent with M as its centre of gravity (Figure 4). Let's introduce a local frame x i attached to the particle, parallel to the Cartesian frame x i , and with M as its origin. The velocity U' i in the Cartesian frame, of a point M' of the panicle, is a function of the local co-ordinates xi of the point M'. Since the particle size is small compared to the other dimensions of the continuum, it is reasonable to develop the velocity U' i in a Taylor expansion of the local co-ordinates x'i and to limit it to the terms of degree one :

(1)

O'i "- U i + Zij x'j

"Ihe kinematical description of the continuum is thus completely defined if one knows the macro-displacements field U i and the micro-displacements gradient Xij 9 For a continuum with a rigid microstructure which is allowed only to rotate and not to deform, the second order tensor Xij is antisymmetric and will be called the micro-rotation or Cosserat tensor co~] (Figure 4). One can introduce the corresponding micro-rotation vector co~ defined with 9 c

c

{Oij = -eijk~ k

(2)

where eij k is the alternating tensor with

(

/

/

i

xI

I

x (:]O------~ l

~....

,artesiar, c~tesianff,"ame aae

)s

~

eij k = ejk i =

I ............. /

[ J

macroscopic level

/ !

ekij, eijj = 0 , el: 3 = 1 and

/ M'(x;)

t ~

,

e213 = - 1 .

x3i

..-

.- - ~ - - 4 ~ x '

,~.,,..,,,, U i (M') Ui(M>

,

1

/

/

/

~ ~ / / /

microscopic level

Figure 4 : Description of the kinematics of a continuum with rigid microstructure

492 The work of internal forces

Wo)

is assumed to be a linear function of the macro-

displacements vector U i , its gradient Ui, j , the micro-rotation vector e0~ and its gradient, also called the curvature tensor, ~:ij = r c ,

(3)

W(i) = - I v { a i g i + bijUi,j + Cijf'0~ + dijKij }dV

where a i , bij , cij and dij are statical variables associated in energy to the corresponding kinematical variables. Since the work of internal forces is an objective quantity, it must remain unchanged when computed in a different frame which yields a i equal to zero and allows the work of internal forces to be rewritten in the following form 9

j-{

a

a

c

W(i ) ------ 13';U~,j--I-O'ij(gi, j -(oij)-l-~l.ijl(ij v

}

(4)

dV

where the statical variables have been identified, cy~j is the symmetric stress tensor (Cauchy stress tensor), 6~ is the antisymmetric stress tensor, and ~t~j is the couple stress tensor (Figure 5). The stress tensor 6ij is, in the case of a Cosserat continuum, a general second order tensor with both a symmetric and an antisymmetric component 9

a

(rij = (r~j +(r~ i

(5) Boundary conditions classical Cosserat

Xl x3

/ / / J / /

~ 4r--.--~ ,2 ,% ~

tj = (Yij nj

mi = gij nj

statical

Ui

co~ or E~2

kinematical

interface

Figure 5 9Kinematics, statics and boundary conditions in a Cosserat continuum

s gija and ~tij, are respectively identified as e0s symmetric strain The strains, conjugated to G~, tensor, E~ antisymmetric strain tensor and ~:ij curvature tensor. The following relations are applicable 9 F--'-,ij =U.~1,j --

f~i~ = Ua 'J-

Ui,j + Uj,i 2

(6)

Ui'j-Uj'i

(7)

2

493 a

c

e~j =~ij -m~j

(8)

a

c

Eij =E~j +Eij = Ui, j -(oij

(9)

Kij = CO1,j c.

(10)

The physical interpretation for the various strains can be found in [21] with the corresponding terminology which is the most appropriate from a physical point of view. Ui is the macro-displacement. The symmetric part ei~ of the strain tensor eij, which is equal to the symmetric part of the displacements gradient, measures the macro-deformations of the continuum. The antisymmetric part U~,j of the displacements gradient measures the macrorotation of the medium and is noted as ~ j . The antisymmetric strain tensor e~ , which measures the difference between the macro-rotation of the medium ~ j and the micro-rotation c

coi~ of the microstructure, is referred to as the relative deformation or rotation tensor. The curvature ~:ij is the micro-deformation (rotation) gradient. Correspondingly 6~j , 6~ and Bij can be called macro-, relative, and micro-stresses, respectively. The work of internal forces W(i ) over a volume V can be written as follows : (11)

W(i) - -IV {~ij Ui,j - (Y~jo); + laijKij } dV and integrated by parts to obtain : w(i~

= Iv{ ij u i + (lLtij,j + 2 a) c}dV-I { ij

nj U i +Bij nj olC}dS

(12)

where n i is the vector normal to the surface S and cy~ the antisymmetric stress vector defined with the alternating tensor eijk : a _

(Yij

a

- eij k O k

(13)

The general form for the work of external forces W(e)in a Cosserat continuum is defined by introducing the volume forces fi and volume couples c i as well as contact forces t i and contact couples m i corresponding to the displacement and rotation fields: W(e) : IV { f i U i "k-Ci0) ~

}pdV +

I s { t i U i -k-

mir ~}dS

(14)

Thus, the application of the principle of virtual work, which states that the work of internal and external forces must be equal to zero for any arbitrary kinematically admissible displacement and rotation fields, yields the following equilibrium equations :

494 (Yij,j + P f i = 0

(15)

a

~ij,j + 2 Gi +PCi = 0

(16)

with the corresponding boundary conditions" (Yij nj = t i

(17)

~ij nj -- m i

(18)

The refinement of the kinematics of a Cosserat continuum introduces new statical variables, (~ and ~tij, with their conjugated kinematical variables, e~ and ~:ij, as well as corresponding extra boundary conditions which will be called Cosserat boundary conditions. The coupling between the equilibrium equations (15) and (16) occurs through the antisymmetric part of the stress tensor (cf. equation 5). Statical Cosserat boundary conditions can be formulated either in terms of the antisymmetric stress 6~ or in terms of the couple stress ~ij while the kinematical boundary conditions can be formulated in terms of the relative rotation E~ , the c Cosserat rotation c0ij or the curvature ~qj. The extra Cosserat boundary conditions in couple

stress ~j or in Cosserat rotation ~

can be called micro-boundary conditions since they

involve only microscopic quantities, while the boundary conditions in stresses 6ij can be called total boundary conditions since they combine both the macroscopic and the relative terms. The micro-boundary conditions will have to be defined in correspondence with the micro-features of the interface surface. For the following analyses, volume forces fi and volume couples c i will be not considered. In the case of the plane simple shear, modelled as the shearing of a long layer of soil of finite thickness by a rigid interface, the equilibrium equations (15) and (16) reduce to 9

=0

(19)

dx 2 d(Y~2

= o

(20)

dx 2 d~.32

~ -

a

2 012 = 0

(21)

dx 2

In the case of the ring simple shear, which has a symmetry of revolution and obeys plane strain conditions, the equilibrium equations (15) and (16) reduce to 9 _

dr

r

=0

(22)

495

+

+ 2 as~ - 0 dr dpzr dr

+

(23)

r

Pzr r

a - 2 arO

=0

(24)

When r tends toward infinity, these latter equations reduce to those for the plane simple shear with the correspondence (r, t3, z) = (2, 1, 3). After combination and partial integration, the two last equilibrium equations (23) and (23) can be rewritten as follows 9 A aS0r = ~ - -

1 ( p-zr r

dktzr) dr

1 .Pzr + dPzr ) dr

(25)

(26)

where A is a constant of integration. The Cosserat solution is the sum of the classical solution and some additional terms which are function of the couple stress laz~ and its first derivative. The case of a classical continuum is reached either by assuming a zero antisymmetric tensor ~ and at least one boundary condition in zero couple stress ,t/ijn J - m , or a zero couple stress tensor p,~ which automatically yields a zero antisymmetric stress tensor ~ . The strains and curvature for the plane simple shear of a Cosserat continuum are equal to: (27)

6"11 - - 0

dU~ 6"22 = dx~ J

(28)

_ l dU~ 6"12 - 2 dx~

(29)

a _ 1 d U i + co~ 6"12 - 2 dx~

(30)

K'32 = ~dx 2

(31)

The strains and curvature for the ring simple shear of a Cosserat continuum are equal to" dU r 6"rr = ~ dr

(32)

496

Ur r

%o = ~

1 ( dUO ~;Or = ~ dr

(33) UO) r

(34)

l(dUo UO) c = +, -m z ~:~r "~ dr r

(35)

K:zr =

(36)

dr

In the case of a classical continuum, the antisymmetric stresses and curvatures contained in (30), (31), (35) and (36) are equal to zero. 3. LINEAR ELASTIC CONSTITUTIVE EQUATIONS

The constitutive equations of a linear elastic isotropic Cosserat continuum can be written under the following form [21, 34-37] 9 {(y}- [L] {E}

(37)

K + GK - G

-G Lij =

K+G 2G

(38) 2Go 2N

where the vectors {c} and {E} are the normalised generalised stress and strain vectors respectively, which in the case of the simple shear test are equal to 9

{(3'}t = (Yll' ~22'0~2' 0 ~ 2 ' -['t32 ~ }

(39)

{E} t ={Ell,E22,E~2,E~2,RK32}

(40)

and in the case of the ring simple shear" {o'}t = (CYrr, 000, CYSt,CYSt,~ - )

(41)

{E} t = {Err, g00, E~r, g~r, R Kzr}

(42)

497 The constitutive equations introduce an elastic material length, noted by R, through the ratio 2N of the bending modulus namely - - ~ relating ~t32 (or ~ r ) with K32 (or Kzr) and the shear modulus G, where N and G both have dimensions of a stress. Those constitutive equations do not introduce any coupling between the microscopic, relative, and macroscopic variables. The necessary condition of stability of such an elastic material, in the sense of Hadamard, i. e., the elastic potential is a positive definite quadratic function, requires that : G > 0, 3 K - G > 0, G c > 0 and N > 0. Following the notations of Vardoulakis and co-workers, we can re-write all the moduli under the following forms : G = G / 2 (h 1 + h 2 )

(43)

G c = G / 2 (h 1 - h 2 )

(44)

2 N = G / h3

(45)

1 with the necessary conditions of stability h 3 > 0 and hi > - since 2 (h~ + h 2 ) = 1. 4 4. PLANE SIMPLE SHEAR OF A CLASSICAL LINEAR ELASTIC MATERIAL In the simple shear test of a classical continuum, the equilibrium equations imply that the stresses 0"12 and 0"22 are uncoupled and constant over the height of the layer. The equations in tangential displacement U 1 and shear deformation 1312 are respectively " 2 1312 (X 2 ) = dU1 dx 2

(46)

2 1312(X2) = O'12(constant) G(x2)

(47)

where the shear modulus may depend on the x 2 co-ordinate for a heterogeneous material. In the case of a homogeneous classical material, whatever the boundary conditions, the solution in tangential displacement U 1 will be linear in x 2 and the shear strain 1312will be constant over the height of the layer. Concentration of the shear strain near the interface can occur only if a strong heterogeneity of the shear modules G(x2) exists. Let's consider for example, a shear modulus G(x2) which varies exponentially between the interface (x 2 = 0) and the upper boundary (x 2 = H) with G(H) > G(0) according to the equations (Figure 6a) : G(x2) = G ( 0 ) e x p ( X ~ - )

(48)

498

(G(H)) ~, = In. G----~

(49)

Heterogeneity defined as the derivative c)G/c)x 2 , introduces a length scale to the problem which compares to the geometric dimensions : g-1 = 1 /)G _ ~ G/)x 2 H

(50)

The shear strain 1~12 and the tangential displacement U 1 decrease exponentially with the distance to the interface (Figures 6b and 6c) :

E12(X2) = 2 G(0 )

exo(- /

(51)

U l ( X 2 ) --- U I ( 0 )

1-

(52) 1-exp(-H)

0.1

0.1

0.05

0.1

Height x2 [m] 0.05

i

J 1

i

0~

a : Shear modulus log G(x 2) [mPa]

-4.0E-6

-2.0E-6

J 0.0t 0.0E+( 0

0.5

1

b : Shear deformation

c : Tangential displacement

a 1~12

U 1 (x 2) [mm]

Figure 6 : Plane simple shear of an heterogeneous classical linear elastic material The distance to the interface d~ (~12) over which the shear stress reaches a small dimensionless value of p % is given by :

(.__q di[E12 =

p] =

g In 2 p U I ( 0 )

7H

(53)

499 This thickness depends on both the thickness H of the sample through the length I and the intensity of shearing measured by the ratio H / U 1 (0). The distance d I to the interface over which the tangential displacement U 1 decreases and reaches a fraction p of its value at the interface U 1(0), is given by : di[U 1 = p U 1(0)] = - g In p

(54)

and is proportional to the height H of the sample through the length t. For example, for a sample with a height H = 10 cm, a distance d I of the order of 5 grains of 1 mm diameter, and a fraction p of 10%, the coefficient X is equal to 46 and the ratio G(H)/G(0) is about 1020. Therefore, for a linear elastic classical material, only a strong heterogeneity could explain the very rapid decrease of the tangential displacement and shear strain near the interface. Therefore, it is expected that simulations of interfaces with heterogeneous boundary layers will lead to highly ill-conditioned structures.

5. PLANE SIMPLE SHEAR OF A HOMOGENEOUS LINEAR ELASTIC COSSERAT MATERIAL In a Cosserat continuum, the two classical boundary conditions for the tangential displacement U 1 at both boundaries : U 1(0) = fl WI

(55)

UI(H) =0

(56)

have to be complemented by two extra boundary conditions, corresponding to the new degree of freedom in rotation [ 18] :

m~ (o)= a

f~ w~ R

e12(H) = 0

(57) (58)

where W I is the tangential displacement of the structural member whose interface is being tested, fl ~ [0,1] is the fraction of W I which is transmitted to the soil in translation (partial stick), and f2>0 is the fraction which is transmitted in rotation. The second boundary (58) condition dictates that the soil behaves "classically", at least near the upper boundary. This corresponds to the experimental observations, when a sample is sheared, of the formation of a "plug zone" where rotations are not predominant and the assumption of classical behaviour is reasonable.

500 Table 1 : Boundary conditions used for the analysis of the plane simple shear

Cosserat continuum

Top of layer H (~'22(H) < 0 normal stress UI(H) =0 zero tangential displacement 1~12(H) = ~"~3(H) - ~ (H) = 0 zero relative rotation "classical solution"

Interface x 2(0) U2(0) =0 rigid interface

Classical continuum boundary conditions Cosserat boundary conditions

UI(O)--

f 1 W I >0

tangential displacement R co; (0)= f2 W, > 0 Cosserat rotation

a

C

When solving the problem of the simple plane shear, it is interesting to use the relative a rotation el2 as the unknown variable. After combining the equations 9(19) to (21), (27) to (31), (37) to (40) we obtain the differential equation in e~2 " d2 (X2 dx22 (e~2)--R-~-e~2 = 0

(59)

where the coefficient c~ is a dimensionless constitutive coefficient : o~ = I

2 G Gc N(G+Gr

= I ~h3 h1

(60)

The necessary conditions of stability for the elastic material impose h 1 and h 3 to be positive. Therefore, oc2 is always positive and the solution of the differential equation is in exponential. In the case of the statical model, the h i coefficients are equal to (3/4, -1/4, 1) while for the kinematical one they are equal to (3/8, 1/8, 1/4) [15]. This yields oc equal to ~ and for respectively, the statical and kinematical models. The Cosserat solution is thus equal to the classical one plus two additional terms proportional to exp(+_ocx2/R ), which will be called Cosserat terms and decrease or increase very rapidly over a few internal lengths R (Figure 7). The solutions in tangential displacement U 1 and Cosserat rotation coc are : Ua(X2)=U1(0)+r~12(0)G x , . - - ~

r [ co (0)-U12H(0).](ch(ocx2-RH)-ch(-oc HR))

l[ c UlOl

r r (x 2 ) = coc ( 0 ) + ~ -

(0)-

2H

1 = ~~/4"hxh--~

X=-sh

o~- +~

X2

-H)) R

(61)

(62)

(63)

ch 0 ~ - -1

(64)

501 The solutions in symmetric and antisymmetric shear deformations are :

~,x~, o1~,o, 1 joe,o, ~,o,1 )~h / x~. 2G

4hlX

~ J

R

'ha-h~'I~ c ( 0 ) _ ~~1'0'1 / ot x2-" jsh R )

e; 2(x 2 ) = ~ 2 h l

(65)

(66)

It should be noticed that all the Cosserat terms are multiplied by the factor coc (0) - U 1(0)/2 H which measures the difference between the Cosserat rotation coc (0) at the interface and its value for the classical case : f~12(0) = U1 (0)/2 H . It is thus clear that if the extra boundary conditions do not prompt the degree of freedom in rotation, more than the classical ones do it, the Cosserat solution will degenerate to the classical one. For example if one changes the Cosserat boundary condition chosen at the interface R m~(0) = f2 W~ > 0 into either coe( 0 ) - U1 (0) = 0 or ~2 (0)= 0 , the classical solution will be obtained. 2H The internal length R is of the order of the grain radius and thus it is typically much smaller than the height of the layer H. The above exact formulas thus reduce to : UI(x2)=UI(O)+~'2(O)x:G + 2 1 3 R [ m C ( 0 ) - U )l ( 0 ) l (2e x p ( H -CZ~)-I

(67)

~ (x2) = f'~ (0) + [c~ ( 0 ) - UI(0) 2H 1 (exp(-c~-)-

(68)

~,x~,: hl-~ exp/-~/2hi E~c'~ ~2~~1

1)

(69)

Using these formulae, it is possible to calculate the distance dI to the interface over which the kinematical variables U 1 , coc and e12 decrease and reach only a small fraction p of their values at the interface. Those distances are given by : a

dI(U 1) = (1- p) 4

U1(0) R o~C(0)

(70)

di(03 c ) = ------~-di (E~2)= - h~31 ln p --------~-

(71)

hI

All these distances are independent of the height H of the sample. They are directly proportional to the internal length R with a coefficient which depends on the constitutive parameters h i and the fraction p. Only di(U1) depends on the value of the boundary conditions

502 and, precisely, the ratio between the displacement U~(0) and the Cosserat rotation R03c(0) which are imposed at the interface. Numerical applications with p = 1% and a ratio of UI(0) by R0~c(0) equal to 2, yield, for the statical model, d~(U~)=4R and d~ (co c ) = d~ (c~2) - 3 R , while for the kinematical model they are equal to 5.6 and 1.5 R, respectively. 10

10

10

Normalised height x2/R

5

00

0.2 0.4 0.6 0.8 1 a 9Tangential displacement U1 / R

O~

0.1 0.2 0.3 0.4 00 0.1 0.2 0.3 0.4 b 9Cosserat rotation c" Relative rotation c a CO E 3 12_ Figure 7 9Plane simple shear of an homogeneous Cosserat linear elastic material

6 RING SIMPLE SHEAR OF A RIGID PLASTIC CLASSICAL MATERIAL In the present paragraph, the ring simple shear will be analysed using a classical continuum with rigid plastic constitutive equations and a Mohr-Coulomb criterion [4]. In this case, out of the three equilibrium equations only (22) and (23) are not null. Moreover, (23) can be integrated directly. For the sake of simplicity, the statical solution will be given in terms of Mohr stress variables : p , q and ot : crrr = p + q cos 2 ct

(72)

O-0o = p - q cos 2 ct

(73)

Croc - O'r0 = q sin 2 a

(74)

The Mohr Coulomb criterion is selected as it is well suited for granular soils 9 F - q + p sin~b-c cos~b

(75)

with r being the angle of internal friction. The plastic potential G is chosen to be of the type 9 G - q + p sinfl

(76)

where 13 -< ~ is the angle of dilatancy. For 13 = r , the behaviour is associated. Within the framework of the J2 theory of rigid plasticity, the total plastic deformations are given b y

503

Err = 5L 0(3 = ~ (sin [3 + cos2 or)

(77)

%0 = 9v ~)___Q__G = ~, (sin ~ - c o s 2 o~) ~Coo 2

(78)

e r 0 = e0r =

~, ~ _ - - O_G~, sin2 a OOrO 2

(79)

It should be noticed that there are 3 statical variables (p, q, ~) plus 6 kinematical ones (er~, eoo, eOr, Ur, U o , ~,) while there are 3 equations containing only the statical variables and 6 which contain both statical and kinematical ones. The statical problem can be thus solved independently of the kinematical one as long as appropriate boundary conditions are chosen. Here, the set of boundary conditions, summarised in Table 2, will be chosen. It corresponds to the conditions of the laboratory test and allows the statical problem to be solved separately from the kinematical one. Table 2 9Boundary conditions for the statically determined ring simple shear problem External radius

Interface radius rint Classical continuum boundary conditions

U r =0 rigid interface OOr < 0 interface shear stress

rex t

Orr
The only kinematical boundary condition in U r can be transformed into a statical one knowing that the stress and strain tensors are co-axial and using the definition of eoo. This boundary condition is equivalent to o~ = -0t/4 - 13/2) at rint . For convenience, the angle o~ will be used as independent variable rather than r. The differential equation in o~ is : d~ - -

dr

1 =

sin2

-

(80)

r sinr

This equation can be integrated as follows 9 r- M f(~) f(c~) = [tanl~] sine/sin2 Ic~l]1/2 = sin Iotl

(81) sin 0-1 -sin 0-1 2 cosl(xl 2 /'42

(82)

where M is a constant of integration. Knowing the function r(~), the solutions in stresses ~r~(r) , oo0(r), pressure p(r) and deviatoric stress q(r) are immediate. The differential equation in U r and c~ is obtained by eliminating ~ from the equations (77) and (78) and can be integrated 9

504

U r (0~)= K gl (0~) g2 (0~)

(83)

gl (0~)= [2 sinlo~[l+sin 13coslo~ll-sin 13]1/2

(84)

g 2 (0r =

Isin 13- cos

2sin 13 l+sin 13 1-sin ~ 1 20~ I cos 213 sinloe I 1-sin 13coslc~l l+sin

sin ~--sin 13 ~

(85)

with K constant of integration. For associated material, g2 = 1. The function g~ is equal to zero only if o~ is equal to 0 or _+ n / 2. In the case of associated plasticity, the boundary condition in U r = 0 at rint implies that the constant K is equal to zero within the plastified zone at the interface. As a result the radial displacement U r is identically null. It follows that the interval for r is reduced to the point rint and the thickness of the plastified zone is zero. The solution is thus composed of two zones : a zero thickness plastified zone in contact with the interface and an external rigid zone extending from r+t to rext . In the case of non associativity, one can develop an additional solution composed of a plastified zone between rint and rp with rp > rint and an external rigid body zone between rp and rext . The limit rp between those two zones is calculated by solving the following equation in O~p which insures continuity of the radial stress C~r in r(O~p) = rp 9 f2(l~intl) (tanl0~p I)sin*

sin~-cos20~p ICOr (rin t)l sin~)

(Yrr (rext)

+

cos2 (Xp c ~ ~ =1 tan~) (Yrr (rext)

(86)

It follows that the radius rp is directly proportional to the radius of the interface cylinder rint " rp = Iint f((Xp )/f(0~in t )

(87)

Therefore, in a classical continuum, the layer of plastified soil in contact with the interface has either a zero thickness or a finite thickness which is directly proportional to the interface cylinder radius. These results which have been developed analytically for a rigid perfectly plastic material, are still valid for more general elasto-plastic constitutive equations. This dependency of the "interface layer" on the geometric dimensions of the sample comes from the classical continuum description which does not introduce any internal length to scale down the problem. In this case, the layer thickness is controlled by the geometric dimensions.

7 CONSTITUTIVE EQUATIONS FOR A RIGID PLASTIC COSSERAT MATERIAL While there has been a lot of research on elastic Cosserat continuum [34-37], relatively little has been done on developing yield criteria. Concerning plasticity, some previous work has been done for granular materials modelled as isotropic Cosserat continuum [4, 15, 24-25, 38], while in Rock Mechanics, some work has been done to develop anisotropic Cosserat type models for layered and blocky rocks [39-40].

505

Two approaches have been investigated for isotropic plastic Cosserat continuum. On one hand, one can start from the von Mises or Drucker-Praguer criterion and generalise it for a Cosserat continuum based on micro-mechanical considerations [15,25]. This leads to a plasticity theory with one yield criterion based on the definition of a generalised shear stress invariant ~c :

2 1"13 gc = 1"11Sij Sij + 132 Sij Sji +-~--~- ~ij ~ij where rll

+ TI 2 ---

(88)

1/2. On the other hand, one can start from the Mohr-Coulomb criterion and a

generalise it to a Cosserat continuum by including in it the antisymmetric stresses cYij [4]. However, this criterion has to be completed by a second one which is function at least of the couple stresses l,t0 . Such an approach has already been tried but yields an interface layer with a zero thickness despite the introduction of an intemal length through the second criterion [4]. Hereafter, we choose to keep the classical Mohr-Coulomb criterion as it is : (89)

F1 = q + p s i n e - c cos~) = 0

It is completed with a second criterion F 2 which is given by a function of only the antisymmetric and couple stresses. This is the simplest possible choice. Different types of such a criterion have been proposed [5, 38]. Similarly to the Mohr-Coulomb criterion which is a linear combination of the 1rst and 2 na invariants p and q of the symmetric stress tensor o~j , one can think of F 2 as a linear function of the two new invariants -~J2 ((Y~j) and 4J2

)

For

the ring simple shear, these two invariants reduce to IO~rl and II.tzrl/4~ respectively. This linear criterion can then be written as follows :

h3

Fz = he Io~l +--~- I~tzrl- N = 0

(90)

where h 2 , h 3 are dimensionless constants, N is a sort of "rotational cohesion", and R is the plastic internal length. Equation (90) introduces only 2 new constitutive parameters which are 9CYma x a = N/h2 and ~maXzr= N R/h 3 since F 2 can be rewritten as follows 9

F2=

a + (Ymax gzr

9 max

(91)

-1=0

An other type of criterion F 2 can be considered starting from the generalised shear stress invariant 1:c defined in equation (88). After separation of the classical terms from the Cosserat ones, 1:c can be written as a function of q, which is the second invariant of o~j , and of qc a

which is an invariant of cij and l.tij I:c =

s

qc

a,

(92)

506

(93)

Since q is already incorporated in F~ , it will not be introduced into F 2 which will be chosen simply as a function of qc 9For the ring simple shear, F 2 will be written as follows 9

+ ~5- ICtzr - N - 0

(94/

where h 2 , h 3 are dimensionless constants, N is a sort of "rotational cohesion", and R is the plastic internal length. With O'~max-N / ~ 2 -2 and ,//WznaX-N R/~ff3--3 , equation (94)can be rewritten as follows 9

F2 =

_.a ax

+

nax ,t/zr

-

I- 0

(95)

The two constants ~nax and ~tmax can be interpreted physically as the maximum (yield) antisymmetric and couple stresses allowable in the material with respect to the microscopic criterion. Their ratio, which has the dimension of a square length, introduces a plastic internal length which will be denoted by the symbol R but it should be mentioned that it is different from the elastic internal length R introduced earlier. With the present choice of criteria, the Mohr-Coulomb criterion in the (q, p) plane is not modified. Since graphical representation of the criteria in the 4 dimension space ( q , p , cra0r , gz~) is not intuitive, we will restrict ourselves to the 3 dimensions space ( q , oa0r , l.tz~) for a given value of the pressure p. In this space, the Miihlhaus and Vardoulakis criterion [15] is represented by an ellipsoid (figure 8a) while the linear and quadratic micro-criteria are represented by cylinders of axis parallel to the q axis and of section a parallelogram and an ellipse respectively (Figure 8b) .Therefore, the quadratic criterion F 2 can be considered as a simple approximation of the MiJhlhaus and Vardoulakis criterion.

(Y

a

...

Or

( gzr

Figure 8a 9MiJhlhaus and Vardoulakis criterion

~Lzr

-'--

(ya

Or

j,

Figure 8b 9Linear and quadratic criteria

507

8 RING SIMPLE SHEAR OF A COSSERAT RIGID PLASTIC MATERIAL We consider hereafter the ring simple shear of a Cosserat material with rigid plastic constitutive equations and two yield criteria : the Mohr Coulomb criterion F l , which is termed macroscopic criterion, and either the linear or quadratic criterion F 2 which are termed microscopic criteria. The first flow rule G 1 will be non-associated (~ < r while the second flow rule will be associated G 2 = F 2 . The calculations will be performed within the framework of the J2 t h e o r y of rigid plasticity. With such choices the problem becomes reasonably simple to be solved analytically. Four features are most important with that respect. The number of statical unknowns (c=, 600, g~r , C~r , l.t~) is equal to the number of equations containing only the statical unknowns (3 equilibrium equations plus 2 yield criteria). This allows to solve the statical problem before solving the kinematical one. Moreover, the uncoupling of the two yield criteria allow to solve for the antisymmetric and couple stresses before solving for the symmetric stresses. For the considered plastic potentials G 1 and G 2 , the symmetric stress and strain tensors remain co-axial and the plastic multipliers ~,l and k 2 are independent. Developing the solution for the complete loading curve of the interface would require to define an evolution law for the interface boundary conditions and therefore to introduce new parameters which are not at the present known. Therefore, we will develop the solution for the ring simple shear only at a given point of the loading for the set of boundary conditions summarised in Table 2 and completed by Table 3 for the Cosserat boundary conditions. When compared to the modelling with a classical continuum, only three new parameters are introduced : two for the microscopic yield criterion and one for the Cosserat boundary condition in couple stress. The couple stress imposed at rint by the interface surface measures its roughness with respect to the maximum allowable couple stress in the material ~tmax . For very smooth interface surfaces, the roughness defined by the ratio [Bzr(rint)[/B max will be equal to 0 while for very rough interface surfaces, it will be equal to 1. Table 3 : Cosserat Boundary conditions for the ring simple shear problem

Cosserat boundary conditions

Interface radius rin t

External radius rex t

Pzr>0 measure of interface roughness

~tzr = 0 "classical solution"

Four different types of zones with different behaviours can exist in the present case. Zone R1-R2 : None of the two yield criteria is reached (F 1 < 0 and F 2 < 0). The soil behaves as a classical rigid material. Zone P I - R 2 : Only the first yield criterion is reached (F 1 = 0 and F 2 < 0). The soil behaves as a classical perfectly plastic material. Zone R1-P2 : Only the second yield criterion is reached (F 1 < 0 and F 2 = 0). The macroscopic displacements are those of a classical rigid material : Ur (r) - 0 and U0 (r) = f~z r . However, the microstructure is free to rotate and its rotation is in a plastic flow state. Zone P1-P2 : Both yield criteria are reached (F~ = 0 and F 2 = 0). The soil plastifies both in translation and rotation. This is the zone where the behaviour is the closest to the experimental observations in an interface layer when this one is fully developed.

508 Therefore, we will develop a solution with a P1-P2 zone of thickness e near the interface (rin t < r < rin t + e). Earlier in the loading, other types of zones may exist near the interface surface but will not be considered presently. Starting from a P1-P2 zone near the interface, it can be shown mechanically that the next zones are a P1-R2 zone between rin t + e and rp and then a R l-R2 zone between rp and %xt.

8.1 Solution for the statical problem Within the interface layer, the second criterion F 2 is reached. Elimination of the antisymmetric s t r e s s (Y~r in F 2 using the equation (26) yields a non linear differential equation of first order in ~tz~. At the interface contact, the couple stress ~t~ (qnt) is known. We are looking for a transition between a Cosserat zone to a classical one where the couple stress g~ is identically zero. Therefore, ~tz~ is taken equal to zero in rint + e and is continuous. The boundary values in rint and rint + e are known and one can thus solve for the thickness e. In the case of the ring simple shear, this non linear equation cannot be solved analytically. Figures 9a and 9b shows numerical solutions of the couple and antisymmetric stresses in the case of a linear criterion with h z = 1.0, h 3 = 1/16, N = 8 000 kPa and R = 1 ram. 150 couple stress 100

10 anti.

['t zr / R 50

8

symmetric 6! stress

~ ~ (r int! = ~tmax/2

A

gzr(rint)=gmax/2

a

zr (rint) = g 2 ax

(Y0r

[kPa]

42 t ~ ~ . , ]

gzr (rant) = g m a x

[kPa] q5 16 17 18 19 20 21 22 23 24 2-5

q5 16 17 18 19 20 21 22 23 24 25

radius r / 10 R (internal length)

radius r / 10 R (internal length)

Figure9a: Variation of couple stress

Figure 9b: Variation of antisymmetric stress

The plane simple shear can be obtained as a limiting case of the ring simple shear when r tends toward infinity and the difference rext-rint stays constant and is equal to the sample thickness H. In that case and for the linear criterion, the equation in ~tz~can be integrated:

zr/r

[.tma-"-------~ = 1- 1-

Eh3rr t]

~max

exp 2

h2

(96)

R

a max where (h3/h 2 R) is equal to the ratio of Gma x and ~tzr . The interface thickness is given by 9

R h2 E 'rint' ] = l maxIn 1- 'rMt' 1

e. . . . In 1 2 h3

II max r~zr

2 O'ma a x

max [.tzr

(97)

and depends both on the plastic internal length R introduced by the second microscopic yield criterion and the value of the interface roughness.

509 For a smooth interface, the thickness is equal to zero. When the roughness increases, the thickness e increases and tends toward infinity for a very rough interface. This is a limitation of the linear criterion for the plane simple shear. In the case of the ring simple shear, the interface thickness e remains finite even for a roughness of 1. However, we are looking for a model where the interface layer is of finite thickness, whatever the geometry of the interface surface. Therefore, the linear criterion is not acceptable with respect to this point. In the case of the plane simple shear and the quadratic criterion, the differential equation in Bz~ can be integrated as follows :

9max gzr

rint)

max gzr

(98)

R

. max . The thickness e of the interface where ~/h2/h 3 R is equal to the ratio of ~ maa x and gzr layer is given b y :

e

=

P'Fa /'zr'rnt)) 'zmaX ' an/'zr'rint))

--

2 ~h 3

in

.

~, ~tzr .max

=

a 20'ma x

max ~lzr

(99)

This thickness is finite whatever the value of the roughness. Its maximum value is : 7t h ~ 32 /t[.t max emax: ~ R = ~ ~ (Ymax

(100)

Knowing the variations of the couple stress Bz~ between rint and rint + e , one can deduce immediately the variations of C~r and 6Or 9The equation in angle cx is obtained by combining (22), (25) and (72) to (75). The value of o~ at the contact with the interface is given by the boundary condition in U r (rint) = 0 which yields o~ = -(rd4-13/2). Knowing the variations of (Y0r and o~, all the other statical variables can be easily calculated. At the limit between the P1-P2 and P1-R2 zones, only : Bzr , (~0r and 6rr have to be continuous. Outside the interface layer (zone P1-P2), it will be assumed that the material behaves classically with zero couple and antisymmetric stresses. At rin t + e , B~ is equal to zero and remains null for r > rint + e . The antisymmetric stress 6~r is not zero at rint + e- but one can bring it down to zero at rint + e § as long as continuity of 6Or is insured. Once the jumps of the statical variables are known, the classical solutions in the P1-R2 and R1-R2 zones are straightforward. The point rp at the limit between those two zones is calculated to insure the continuity of the radial stress Cry.

510

8.2 Solution for the kinematical problem Like most rigid plastic problems in classical continuum, construction of a solution kinematically admissible at the limit between the two classical P1-R2 and R1-R2 zones is not possible. The combination of the rigid plastic constitutive equations with the geometry is too stringent with regards to the radial displacement U r . This one is equal to zero in the rigid classical zone whatever the boundary conditions. Therefore, we will restrict ourselves to develop a solution in the P1-P2 and P1-R2 zones with U o = O and e0cz - 0 as boundary conditions in rp .To replace the boundary condition on Ur at rp, we consider a condition on the plastic multiplier ~,1 at the interface contact similarly to what is done in gradient theories of plasticity [27]. It tums out that this boundary condition is equivalent to specifying the derivative of the radial displacement U r at the interface contact. The solution in angle o~ is known from the statical solution developed previously. The solution in U r is obtained after integrating the differential equation in o~ .Knowing the function Ur(00, the calculation of the plastic multiplier k 1 is immediate with equations (77) and (78). The differential equation in U 0 over the P1-P2 and P1-R2 zones is solved starting from rp where U0 = 0 . Figure 10a presents the numerical simulations for U0 in the case of a linear criterion. One should notice the bent at the limit between the Cosserat plastified P1-P2 zone (interface layer) and the classical plastified P1-R2 zone which corresponds to a discontinuity of the gradient of the tangential displacement U 0 . Since the boundary conditions at the interface are specified in total tangential stress ~0~ and couple stress ~z~ while in rp the displacement Uo is blocked, the three solutions for a roughness of 0, 0.5 and 1.0 do not yield the same tangential displacements at the interface contact. The solution in Cosserat rotation o cz in the classical plastified P1-R2 zone is function of the tangential displacement U0 and its first order derivative since the Cosserat rotation coz is constrained to follow the rotation of the macroscopic continuum f~z 9In the interface layer, the differential equation in Cosserat rotation o~cz is obtained from the second flow rule by eliminating the plastic multiplier L2 . A good evidence of Cosserat effects in the interface layer is the relative rotation 8~2 = f~z -~ which is significantly high in that zone and equal to zero outside (Figure 10b). A classical continuum gives Cosserat rotations cocz = ~ z of the order of a few degrees near the interface. For a Cosserat rigid plastic continuum, the plastic flow in rotation which occurs in the interface layer yields Cosserat rotations of several hundreds of degrees.

,o[

U0

11i

relative 1,400 rotation 1,200

---~tzr(rint)= 0

tangential 25 displacement 20I, '

~ '

~.

=

s

max gzr

a

0r [degree]

[ ~tm]

" " " ~zr (rint)= 0

1,000

m 2 ~tzr (rint)=l.tzra~

800

ma

~

max

600 - - - ~tzr (rint)= ['tzr 400 200n . ~ ~5 20 25 .

q5

....

20 . . . .

....

30

]-tzr(rint)=gzr ~2

.

.

.

.

.

.

.

.

.

.

.

30

radius r / 10 R (internal length)

radius r / 10 R (internal length)

Figure 10a : Solution in tangential displacement

Figure 10b : Solution in relative rotation

511

9 CONCLUSION A classical continuum, whatever its degree of heterogeneity and constitutive equations is not appropriate for modelling interface layers since it does not introduce an internal length which can scale down the interface layer thickness. This phenomena exists also for volume localisation phenomena in shear bands [ 15]. A Cosserat continuum which describes both at a microscopic and macroscopic level the kinematics of a granular material with rigid grains, is able to overcome those limitations. The linear elastic constitutive equations for an isotropic Cosserat continuum introduce an elastic internal length R through the ratio of the bending modulus N by the shear modulus G. During shearing, the shear deformations concentrate in an interface layer of finite thickness which is a function of R and the Cosserat boundary condition. Although the decrease of the antisymmetric stress and couple stresses is very fast over a few internal length distance, they never reach exactly zero in the elastic case. In a Cosserat rigid plastic isotropic continuum, the constitutive equations introduce an internal length through the ratio of 0F2/0Bi j and /)F2//)~j which has the dimension of a square length. This plastic internal length controls directly the thickness of the interface layer where both the macroscopic and microscopic criteria are reached. When investigating the kind of second microscopic criterion which would be appropriate for modelling interfaces, it has been found that a quadratic one is preferable to a linear one.

THANKS The authors want to acknowledge the Commission of the European Communities who sponsored this research through the SCIENCE Project No. SC1"-CT91-0659. Part of the present research was developed within the framework of the French GRECO-G6omat6riaux.

REFERENCES 1. 2.

Vesic, A. (1977). Design of pile foundations; NCHRP report N ~ 42. Clichy, W., Boulon, M., and Desrues, J. (1987). Etude exp6rimentale st6r6ophotogramm6trique des interfaces. Proc. 4 th French-Polish Conf., Grenoble, France. 3. Boulon, M. (1989). Basic features of soil-structure interface behaviour. Comp. & Geo., Vol. 7. 4. Bogdaneva-Bontcheva, N., and Lippmann, H. (1975). Rotationssymmetrisches ebenes Fliessen eines granularen Modellmaterials. Acta Mechanica, Vol. 21. 5. LOffelmannn, G. (1989). Theorische und experimetelle Untersuchungen zur Schiittgut- Wand Wechslwirkung von Granulaten. Dr.-Ing. Dissertation, Universit~it Karlsruhe. 6. Kishida, H., and Uesugi, M. (1987). Test of the interface between sand and steel in the simple shear apparatus. Ggotechnique, Vol. 37, No. 1. 7. Unterreiner, P., Lerat, P., Vardoulakis, I., Schlosser, F., De Laure, E., and Belmont, G. (1993). Brevet sur l'Appareil de Cisaillement Simple Annulaire (ACSA) (patent). 8. Potyondi, J. G. (1961). Skin friction between various soils and construction materials. Ggotechnique, Vol. 11, No. 4. 9. Desai, C. S., Drumm, E. C.,and Zaman, M. M. (1985). Modeling of interfaces. ASCEE Journal Geotechnical Engineering, Vol. 111, No. 6. 10. Boulon, M., and Nova, R. (1990). Modelling of soil-structure interface behaviour. A comparison between elasto-plastic and rate-type laws. Int. J. Comp. & Geotech., Vol. 9. 11. Hoteit, N. (1990). Contribution ~ l'6tude du comportement d'interface sable-inclusion et application au frottement apparent. Ph. D. Thesis, I. N. P. G., Grenoble.

512 12. Yoshimi, Y., and Kishida, T. (1981). A ring torsion apparatus for evaluating friction between soil and metal surfaces. Geotechnical Testing Journal, Vol. 4, No. 4. 13. Boutrif, A. (1993). Mesure du comportement d'interface sol-structure ~. la bo~te de cisaillement annulaire et mod61isation. Ph.D. Thesis, Universit6 Joseph Fourier. 14. Brumunds W. F., and Leonards, G. A. (1973). Experimental study of statics and dynamic friction between sand and construction materials. J. Test. & Eval., Vol. 1. 15. Miihlhaus, H. B. and Vardoulakis, I. (1987). The thickness of shear bands in granular materials. Gb,otechnique, Vol. 37, N ~ 3. 16. Ord, A., Vardoulakis, I., and Kajewski, R. (1991). Shear band formation in Gosford sandstone. lnt. J. Rock Mech. Min. Science & Geomechanics Abstracts, Vol. 28, N ~ 5. 17. Bardet, J.P., and Proubet, J. (1991). The structure of persistent shear bands in idealized granular media. Computer Methods and Advances in Geomechanics, Balkema. 18. Teichman, J. (1989). Scherzonenbildung und Verspannungseffekte in Granulaten unter Beriicksichtigunug von Kerndrehungen. Dr.-lng. Dissertation, Universit~t Karlsruhe. 19. Kishida, H., and Uesugi, M. (1987). test of the interface between sand and steel in the simple shear apparatus. G~,otechnique 37, No. 1. 20. Boulon, M., and Hassan, H. (1993). Development of a visualisation of the movement of the grains within a soil-structure interi~,ce. CEC SCIENCE Program No. 659. Internal Report. 21. Mindlin, R. D. (1964). Microstructure in linear elasticity. Arch. Rat. Mech. Anal., Vol. 16. 22. Cowin, S. C., and Pennington, C. J. (1970). On the steady rotational motion of polar fluids. Rheol. Acta., Vol. 9. 23. Germain, P. (1973b). The method of virtual power in continuum mechanics. Part 2 : Microstructure. SIAMJ. Appl. Math., Voi. 25, No. 3. 24. Diepolder, W., Mannl, V., and Lippmann, H. (1991). The cosserat continuum, a model for gram rotations in metals. International Journal qfPlastici~, Vol. 7. 25. Kanatani, K. I. (1979). A micropolar continuum theory for the flow of granular materials. International Journal Engineering ,Science, Vol. 17. 26 Chen, Z., and Schreyer, H. L. (1987). Simulation of Soil-Concrete Interfaces with Nonlocal Constitutive Models. Journal Engineering Mechanics, Vol. 113, No. 11. 27. Vardoulakis, I., Shah, K. R., and Papanastasiou, P. (1992). Modelling of Tool-Rock Shear Interfaces using Gradient dependent Flow Theo~, of Plasticit3,. International. J. Rock Mech. Min. ,Science & Geomechanical Abstracts. Vol. 29, No. 6. 28. Eringen, A. C. (1966). Linear theory of micropolar elasticity. J. Math. Mech., Vol. 15. 29. Eringen, A. C. (1968). Mechanics of micromorphic continua. Mechanics of Generallized Continua. Proc. 1UTAM,Symposium, E. Krrner ed., Springer-Verlag. 30. Eringen, A. C. (1970). Balance laws of micromorphic mechanics. Int. J. Eng. Sc., Vol. 8. 31. Germain, P. (1972). Sur l'application de la mrthode des puissances virtuelles en mrcanique des milieux continus. CRAS, Paris, Srrie A, No. 274. 32. Germain, P. (1973a). La mrthode des puissances virtuelles en mrcanique des milieux contmus16re partie, Throrie du second gradient. Journal de MOcanique, Vol. 12. 33. Salenqon, J. (1987). Mrcanique des milieux continus, Tome I : Concepts grneraux. Cours de m~canique de l'Ecole Polytechnique, Presses de rEcole Polytechnique. 34. Koiter, W. T. (1963). Couple-stresses in theor3, of elasticity. Proceedings Koninklijke Nederlands Akademie Van Wetenshcq(fen, Series B, Vol. 67. 35. Neuber, H. (1966). l~lber Probleme der Spannungkonzentration im Cosserat-Krrper. Acta Mecanica, Vol. 2. 36. Schaeffer, H. (1967). Das Cosserat-Kontinuum. Zeitschrift .fi~r Angewandte Mathematik und Mechanik, Band 47, Het~ 8. 37. Selvadurai, A. P. S. (1973). Bending of an infinite beam resting on a porous medium. G~otechnique, Vol. 23, No. 3. 38 Lippmann, H. (1969). Eine Cosserat Theorie des plastischen Fliessens. Acta Mechanica 8. 39. Besdo, D. (1974). Ein Beitrag zur nichtlinearen Theorie des Cosserat-Kontinuums. Acta Mechanica 20. 40. Miihlhaus, H. B. (1993). Continuum models for lavered and blocky rock. in Comprehensive Rock Engineering. Vol. 2, Hudson Editor in Chief, Pergamon Press.

Mechanics of Geomaterial Interfaces A.P.S. Selvadurai and M.J. Boulon (Editors) 9 1995 Elsevier Science B.V. All rights reserved.

513

I N T E R F A C E S IN P A R T I C U L A T E M A T E R I A L S Anil Misra Department of Civil Engineering, University of Missouri-Kansas City, 600 W Mechanic, Independence, MO 64050, U.S.A.

The discrete, particulate nature of geomaterials has a significant influence on their mechanical behavior. To account for the effect of the discrete nature, there has been a growing interest, in the recent years, in developing models of particulate media from a particle interaction standpoint. The essence of these efforts has been to develop models that predict the bulk, overall mechanical behavior of particulate materials based upon the geometric arrangement of constituent particles, viz. fabric, and the particle mechanical and geometrical properties. In these modelling methodologies, the behavior of the interfaces between particles has a pivotal role. In this work, we will present an exposition of the interface models that describe the force-deformation behavior of interfaces between particles. The study of effects produced by mutual compression of non-conforming elastic bodies was pioneered by Hertz who considered the case in which forces were normal to a smooth interface surface. Since then a variety of studies have been published on contact mechanics of elastic bodies considering a variety of interface conditions (Johnson [1]). Among the primary factors that influence the mechanical behavior of the particle interfaces vis avis the study of geo-materials are the particle elastic properties, size, shape, friction, surface adhesion, surface topography, and presence of liquid at interface. The intent of this paper, is to present expressions for interface stiffness describing interface force-deformation relationship for a variety of interface conditions. These expressions will be presented in a form suitable for use in the modelling of particulate materials. In this work, therefore, we first present a brief discussion of the modelling methodologies in the mechanics of particulate media. In the subsequent discussion on interface models, we begin with the Hertzian contact theory of contacting non-conforming elastic bodies under normal forces. The extension of Hertzian theory to account for tangential forces, including the effect of partial slip and consequent irrecoverable deformations, is presented next. Further work on Hertzian theory to account for non-spherical particle shapes, surface roughness, and particle adhesion are described. Interface models for particles undergoing relative rotation, such as rolling and torsion, are also described. Finally, a compendium of measured inter-particle friction angles on various types of particle materials is presented from the data available in the literature.

514 1. MECHANICS OF PARTICULATE MATERIALS In the classical continuum mechanics of materials, the problem scale is naturally considered to be such that the effect of material micro-structure may be neglected. In addition, the classical continuum mechanics approach does not explicitly consider the material properties of the constituents and the micro-structure. However, for a large number of problems in particulate materials, it is well recognized that phenomena exhibited at 'macro' scales are governed by mechanisms which occur at the particle or 'micro' scale. In recent years several researchers have been motivated to explore this connection between the micromechanisms and the macro-phenomena. As a result, there has been a growing interest in modelling mechanical behavior of particulate media from a particle interaction standpoint. These models may be considered in two broad categories, namely the computer simulation models and the micro-mechanical stress-strain models (cf. Misra [2]). In the computer simulation model, the deformation behavior of a collection of particles is obtained by considering the equilibrium of each particle in terms of the forces developed at particle interfaces. The micro-mechanical stress-strain model, on the other hand, approximates the deformation behavior by representing the micro-structure in terms of appropriate statistical measures while accounting for the interaction at particle interfaces. In both these modelling methodologies, however, the interaction at the particle interface is accounted for in a similar fashion. 1.1 Particle Interface Consider, for example, the arbitrary deformation of an element of a particulate

material. The incremental relative displacement A8 ~" between two particles, m and n, is given by A8 ,"" = Au,.~- Au + eisk (Aoi~ rk" - Aco'r~) S

(1)

while the incremental relative rotation A0~" is given by

A0."" = A o.).'-A co." z I !

(2)

where ui = the particle displacement, cok = the particle rotation, r~ is the vector joining the centroid of a particle to the contact point, the superscripts, n and m, refer to the particles, and e;ik is the permutation symbol. The tensor summation convention is used for the subscripts throughout this paper. The relative movement between interacting particles leads to development of forces and moments at the particle interface. The contact force f~ and the relative displacement 8~ are conveniently related via the interface stiffness K;j as follows:

af~ = K~sASs+ A.f~

(3)

and the contact moment IX,.and relative rotation 0, are related via the interface rotational

515

stiffness Gij as follows: (4)

All i = GiiAOj + A~_li

where Af/ and A]~ i are relaxation force and moment caused by the sliding and separation at the particle interface. Both the computer simulation models as well as the micromechanical stress-strain models use this fundamental model of interaction at particle interfaces. Further, when the tangential force and torsional moment reach the shear strength and torsional strength, respectively, sliding occurs at the interface. For frictional contacts, the sliding conditions are expressed as: f, = f tanq)

and

11 =

3~:af tanq~

(5)

16

where q), is the interface friction angle and a is a measure of the size of contact area.

1.2 Computer Simulation Model As illustrated in Fig. 1, the computer simulation model considers the motion of each particle in the assembly under the forces developed at particle interfaces under imposed boundary conditions. Based on this basic premise, dynamic and quasi-static computer simulation schemes have been presented (Cundall and Strack [3], Kishino [4], Chang and Misra [5]). Under quasi-static conditions, for example, the inertial forces may be neglected and the equation of motion reduces to the following equilibrium conditions which, for the mth particle in the assembly, are written as A F ,m

-E A~

cx

= 0

(6)

ot

AM2- ~e eAg~rk

+ tai

= 0

(7)

ot

Force-DeformationBehavior i of ParticleAssembly

1

I

Force-DeformationBehavior 1 of Particle Interface

Figure 1. Computer simulation approach in particulate mechanics.

516 where F,." and M 7 are the externally imposed force and the moment on the m-th particle, f and ~t~ are the force and moment, respectively, acting at the o~-th contact of the m-th particle and the summation is performed over the contacts of the m-th particle. For the particles within the granular assembly which are not located at the boundary no external load is acting, therefore F~=O and M~m=o. Thus, for quasi-static conditions, the force-deformation relationship for the assembly may be obtained as

a vz: - E K'q LfauJ" - au?+%(aco

]

k r,

r;"~-aco

) +

:1/

= 0

(8)

and

~~-

E eijl K"~[Au; k

-

AUk+ ekm (At0"p r q' ~ - Ae~q

OL

+e/jlA ~r I +G"~[Aco"-Aco ij t J

;]+A0"'~= i

rl"~

(9)

0

which represents 6M equations in terms of 3M particle forces, 3M particle moments, 3M particle displacements, and 3M particle rotations for an assembly of M particles. A variety of methods can be used to solve the above force-deformation relationship under prescribed boundary conditions (cf. Chang and Acheampong [6]). 1.3 Micromechanical Model As illustrated in Fig. 2, in the micromechanical model, the overall stress-strain relationship for an element of particulate materials is developed by relating the overall stress and strain to the interface force-deformation relationship. Based on this basic framework, stress-strain relationships have been presented by considering the relationship between the overall strain and the interface deformation, the overall stress and the interface force, and statistical representation of micro-structure (cf. Misra and Chang [7]). In an approach that accounts for effects of random heterogeneity, the micro-structure of particulate materials is considered in terms of micro-elements composed of a particle and its immediate neighbors. Along this approach, a local stress-strain relationship is derived for the micro-element. The overall stress-strain relationship is then obtained by considering the volume averages of local stresses and strains (cf. Chang et al [8], Misra and Chang [7]). The local stress-strain relationship of the micro-element is written as A ~ q"

= C

?I

qtaA

"+Ac~"

~" kl

'1

(10)

where the superscript n refers to the micro-element and CijI"1u is the local stiffness tensor. The local stiffness tensor is derived in terms of the interface stiffnesses and the relative position of the neighboring particles as (Chang et al. [8]) ,, C ijkl

=

1 ~ 2V "

I,~,K,,,~(5 ,,~, , ) i jr rll k - Fr~

where li is the branch vector joining the centroids of particles in contact and

(11)

517

Stress-Strain Relationship of Particle Assembly

1

l

,,

Local Stross-Strain Relationship of a Microelement

~

I

- C o n t a ~ elationship/ . Strain-Contact Deformation] lationship~~,,J

1

Force-Deformation Behavior of Particle Interface Figure 2. Micromechanical approach in particulate mechanics.

Fj~ : ( ~ K~!~ -] ' ~ (z g"~l "= -'il "k

(12)

It is noted that the strain in this model is defined to include the effect of rotation as " eo~aco"k AE~ = Auq+

(13)

The conventional definition of strain tensor is recovered by taking the symmetric part of the distortion, r which is identical to the symmetric part of the displacement gradient, ua~). The non-symmetric part represents the net particle rotation in excess of rigid body rotation. Noting the heterogeneity with respect to the micro-element, the local strain a~ is

518 related to overall strain eu, as A e : ~m n

=

H n A "~kl mnkl n

(14)

i

where H,,~t is a concentration' tensor. Using the following volume averages

-r1

V"

Ao;,

and

(15) V

ij

the overall stress-strain relationship is obtained to be (16)

A ~ ij = CijuA'~kl + At~ q

where the effective stiffness tensor C iju is written as 1

(17)

Cijkl = V~n vnCi~nn:nk,

Methods for obtaining the 'concentration' tensor have been discussed by Misra [2] and Chang et al [9]. 2. INTERFACE STIFFNESS As seen from the above discussion, the interface stiffness tensors K,j and Gij, that connect the force and deformations at particle interface, have a central role in the mechanics of particulate media. The earliest study of force-deformation behavior of particle interface is traceable to Hertz, who studied the contact of frictionless non-conforming elastic bodies under normal loads (Johnson [ 1]). Hertz's pioneering work has proved to be the foundation for subsequent development of contact mechanics for a variety of surface conditions, material behavior and loading conditions. Among the subsequent work, perhaps the most significant is the extension of Hertzian contact to frictional non-conforming elastic bodies with smooth topography under combined normal and tangential loads by Mindlin [10] and Mindlin and Deresiewicz [11]. Beginning with the Hertz-Mindlin contact, we will present a discussion of a variety of interfaces of interest in particulate mechanics. It is convenient to consider the interface stiffness tensor in terms of stiffnesses that describe the behavior along the direction of the normal and the tangent to the interface, such that the interface stiffness tensor K;j is expressed as Kij = K n~n s + Ks(SiS j + t~tj) + K

(nis j + njs~ + n~L + nits)

(18)

where Kn and Ks are the interface stiffnesses along the normal and tangential direction of the interface surface as depicted in Fig. 3, and K,~ cross-links the normal and shear behavior.

519

Gn

Gs

Figure 3. Schematic representation of the interface stiffness. The unit vector n is normal to the interface and vectors s and t are arbitrarily chosen on the plane tangential to the interface, such that nst forms a local cartesian coordinate system, as illustrated in Fig. 4. For most cases the cross-link term K~ may be neglected, resulting in a simpler form of interface stiffness tensor given by (19)

Kij = K nin j + K( s,~ +tit i)

a ~

%

r

Y

Figure 4. Coordinate system at a particle contact.

520 The interface rotational stiffness Gij may be similarly expressed as (20)

Gii = Gnnin j + Gs(SiS j + tit j)

where Gn is the torsional stiffness and Gs is the rolling stiffness of the interface as depicted in Fig. 3.

2.1 Interface of Smooth Spherical Elastic Particles The interface of topographically smooth elastic particles is the simplest type of interface of interest in the mechanics of particulate materials. It is also important from a theoretical standpoint, since closed form analytical expressions of the interface stiffnesses are possible. Hence, these stiffnesses prm, ide a starting point for developing interface forcedeformation models for more complex interfaces. N o r m a l Stiffness K n The normal contact of topographically smooth elastic spheres was first studied by Hertz in 1882 (Mindlin [10]). Under zero load, the particles will be in contact at a single point. As the load is increased, the particles deform in the vicinity of the point of initial contact such that the contact area will be finite though small compared to particle size. Considering the contact area to be circular with a parabolic pressure distribution, the deformation of the particle in the vicinity of the point of contact is obtained using the linear elasticity theory for pressure load on elastic half space. Based on this solution, the relative approach of the particles in contact is obtained in terms of the radius of the contact area, a, the normal force at the interface, fn, and the elastic properties of the particles. The resultant normal stiffness K n is expressed by

(21)

K n = 2Ea

where I - v 1 + - - 1- ---v- 2 ,

1 --'--

E"

" - - " - - " - -

2G~

a

2G 2

i j,3

--1 - --1

3]:,0

----

8E

I

19

1 +. 1

--

2

~2

)

(22)

G~ and G 2 are the shear moduli, v t and v 2 are the Poisson's ratios, p~ and 132 are the radii of the particles in contact. In addition to the aforementioned assumptions on the nature of the contact and analysis method, it is assumed in the derivation that the particles are frictionless, isotropic and homogeneous. Under vanishing friction no shear forces are transmitted under the normal load and the cross-link stiffness K,~ vanishes. Johnson [1] reports that friction plays a role only if the elastic properties of the particles are distinct. Additionally, the effect of friction is reported to be small on the normal stiffness even for contacts of dissimilar particles. Particles with anisotropic elastic properties have been studied by Willis [12]. However, closed form analytical expression for stiffness are generally difficult to obtain. The subsequent discussion in this paper will be restricted to particles with isotropic elastic properties.

521

Tangential Stiffness Ks For the contact of two topographically smooth spheres, the tangential stiffness under monotonically increasing tangential force and constant normal force was first studied by Mindlin [10]. The theory was subsequently extended by Mindlin and Deresiewicz [11] to account for the effect of oscillating tangential force accompanied by varying normal force. Under oscillating tangential force, it is convenient to consider the following three loading conditions: (a) virgin loading; (b) unloading - the loading reversal in the direction opposite the virgin loading; and (c) reloading - the loading reversal that leads to loading in the direction of the virgin loading. In Mindlin's theory, the shear tractions, caused by the tangential force, have a parabolic distribution on the contact surface, such that shear tractions are larger near the edge of the contact area and decrease towards the center. The slip resistance, on the other hand, has the same distribution as normal force, with a smaller slip resistance near the edge, increasing towards the center of the contact area, forming an inverted parabola. Thus, in Mindlin's theory, the slip initiates at the edge of the contact area forming a slip annulus as illustrated in Fig. 5. The shear force distribution on the contact surface is depicted in Fig. 5 by the envelope PQRS. In portions PQ and RS, the shear forces equal the slip resistance. As the tangential force, f,, is increased, the radially symmetric slip annulus enlarges, progressing radially inward until the complete contact surface is slipping, which results in the sliding condition, f~=f, tan ~, where ~ is the frictional resistance of the interface. Considering that the tangential displacement over the adhered portion is uniform and the tangential force in the slip annulus equals the slip resistance, the inner radius of the slip annulus, b, is given by Q

R

Sheartraction ~11 ~ distribution I / ~ j / V ,~ -along / D section O ' " /i ~'~~~~'""/

~ W ~ _/" Positiveslip i~ /~-N~ / D"

I I I I I I I i I I I I I I I I

I i I I

I

I I I I I I I I I I I ' I

~

1

I I i I I I I I I I I I

I ! i I I I I I I I I

~ Negativeslip

I I I I II I I I i I I I I I

t A Contactarea -"/ /

Figure 5. Slip annulus at contact (after Chang et al [12]).

522

(23)

b:a/,_ __ 1'3 f tanq~

where a, is the radius of the contact area. Immediately on load reversal, the shear traction over the contact area becomes less than the slip resistance such that no slippage occurs at the contact. Continued load reversal results in an increase of the shear traction in the opposite direction which leads to counter-slip in the opposite direction to the initial slip. A new slip annulus develops starting from the edge of the contact area. As depicted in Fig. 5, the new shear traction distribution is given by the envelope PUVWXS with shear traction in portions PU and XS equal to shear resistance. The new slip annulus between circles A and B' develops with slip in the direction opposite to the original slip. The stick-slip phenomena is also effected by the variation of normal force which results in variation in shear resistance. A lower normal force will result in smaller slip resistance and, generally, a larger slip annulus. The stick-slip phenomena under increasing normal force is, however, more complex and discussed in detail by Mindlin and Deresiewicz [11]. A concise form of the resulting tangential stiffness are expressed as (Thornton and Randall [14], also see Chang et al [13])

/

1

Af.tan,

[o II

(24)

1

where the positive sign is invoked during unloading only, --- G/

I aL) L

03= 1 - f.tan"""~ + f.

(virgin loading)

2-v G1

2 -V 2

G~

, and

(25)

P

~ ,-I 2f tan~,

f,

2f tan~,

f~

(unloading)

(26)

(reloading)

(27)

where f" and f'* are the loading and unloading reversal points, respectively, as shown in Fig. 6. If the incremental tangential force is less that the shear resistance, i.e. Af,EAf,, tan 0,. Additionally, during unloading and reloading regimes when

523

r

it,'"

Figure 6. Interface tangential force-displacement relationship (after Chang et al [13]). the slip annulus exceeds the size of largest previous slip annulus, the slip is said to be fully reversed and the subsequent loading follows the virgin loading regime, i.e. Eq. 25.

Torsional Stiffness Gn The torsion of two spheres in contact is caused by the twisting moment ~tn about the common normal. The torsional behavior is qualitatively similar to tangential behavior in that it depends upon the normal force and is accompanied by slip along the slip annulus. If the slip is neglected the torsional stiffness Gn is expressed as (Johnson [1]) G = ~16Ga 3

(28)

3

where a is the radius of the contact area, _~ --

§

and G~ and 6;2 are the shear

moduli of particles in contact. Slip annulus during torsion of spheres in contact has been considered by Lubkin [15] and Deresiewicz [16]. Although no closed form relationship between the twist and twisting moment has been reported, it is feasible to derive an expression for torsional stiffness Gn, using the series approximations presented by Deresiewicz [16], as follows

G = 1--~6Ga3 (8af"tan@"-121a" ,),,2 "3 (_4af tand~ +61a )l/2_(32af tan~

(29)

Interface behavior under oscillating torsional load has also been studied by Deresiewicz [ 10].

524 The behavior is qualitatively similar to that under oscillating tangential force; however, analytical expressions are difficult to obtain. Rolling Stiffness G s

The rolling stiffness describes the resistance to relative rotation at particle interface. On relative rotation of particles in contact under normal load, normal pressure distribution is altered to a non-axisymmetrical distribution of normal pressure on the contact area as distinct from the axisymmetrical distribution used in the Hertzian theory of purely normal load. Although it is difficult to derive analytical closed form expressions of rolling stiffness, Q, for spherical particles, approximate expressions may be obtained, from the solutions of deformation field in elastic half space under linear pressure distribution equivalent to a rolling moment of kt,, as (30)

G = 3 Ea 3

where the radius of the contact area, a, and material property, E, have been defined earlier. It is noted, however, the linear pressure distribution will lead to tensile pressure at contact edge. Also, the above approximation neglects the possibility of partial separation of the contact. The approximate expression compares well with the rolling stiffness of a rigid punch on elastic half-space based on results of Muskhelishvili [17] written as G = ~Eb 2 4

(31)

where b is half width of the punch.

2.2 Interface of Smooth Non-Spherical Elastic Particles For non-spherical particles or particles with general surface profiles, a elliptical contact area may be assumed, although the contact shape is not known a priori (Johnson [1]). The subsequent analysis methodology is similar to that of spherical elastic particles; however closed form analytical solutions are not feasible. Normal Stiffness Kn

The normal stiffness, Kn, for non-spherical elastic particles is written as (32)

K n = 2 E a ' W ( e ) -~

where a'=

3f.pr '8E

Pe = 2 PiP2

)~/z '

P l and P2 are the radii of curvature of the two particles in contact, and

(33)

525

re(e) =

K(e)

E(e) - K(e)

(34)

K(e) - E(e) )

In Eq. 34, a and b are the major and the minor axes of the elliptical contact, E(e) and K(e) are complete elliptic integrals of argument e=(1-b2/a2) m, b
Tangential Stiffness K s The tangential stiffness of non-spherical particles differs from that for spherical particles by a corrective term denoted by ~, such that

/

1 Af~ tan, " K , = 8G'a _~.

[o

- 1

cI) (e)-'

(35)

where 0 is given by Eqs. 25, 26 and 27, and from Deresiewicz [18]

9 (e) -

t

[4a/71: b ( 2 - v ) ] [ ( 1 - v/e2)Kl(e) § v E,(e)/e2],

a
1, a=b [ [4/~: (2- v)][(1 - v + v / e 2) K(e)- v E(e)/e2],

a>b

(36)

where E(e) and K(e) are complete elliptic integrals of argument e=(1-b2/a2) ~/2,ba.

Torsional and Rolling Stiffness Although little information is available in the literature on torsion and rolling of nonspherical particles, it is noted that the behavior is qualitatively similar to that exhibited by spherical particles. The torsional as well as the rolling stiffness depend upon the contact size which, in turn, is related to the normal force. Equivalent contact size may be defined as discussed in Johnson [1] and Hills et al [19]. The approximation of contact size using Eq. 33 is found to give close estimates when compared to complete solutions. We note here, that for most purposes, the torsional and rolling stiffness based upon spherical particles may suffice. 2.3 Interface of S m o o t h Spherical Elastic Particles with Surface A d h e s i o n

Small clay-sized particles may have non-vanishing surface potentials which leads to attractive forces at the interface of such particles. Therefore, these particle interfaces can sustain tensile loads. Additionally, the surface potential can alter the interface behavior under compressive normal forces. Although the origin of surface potential is not completely understood, contact of spherical particles with surface energy has been studied (cf. Johnson

526 et al. [20], Derjaguin et al. [21]). Based on these studies, we present expressions for the normal and tangential interface stiffnesses. The torsional and rolling stiffness are not discussed due to lack of information.

Normal Stiffness K~ The extension of the Hertzian contact theory by Johnson et al. [20], suggests that surface energy alters the contact size and the pressure distribution. Based on their results, the normal stiffness is obtained to be 96 E 2a 4 (f,, + Zr K = " 40 E X,ca3 +32 E f,, a 3 - 3 p

(37)

where the radius of contact area a is given by

a

=

(3p[

2]1

fn + 2Zr 2r162 + Xr

(38)

and ~ is the interface adhesion force. When f~=-Xc (i.e. the interface tensile force equals contact adhesion force) the contact separates and the normal stiffness vanishes. Based on the work by Johnson et al. [20], the interface adhesion force, 7~, is given by 7~=3x~, where y is the interface surface energy. According to Derjaguin et al [21], however, 7~--4xTp. It is noted that Eqs. 37 and 38 reduce to those given by Hertzian theory for contact with no adhesion, i.e. X~--0.

Tangential Stiffness Ks Application of tangential force to interfaces with adhesion results in a reduction of the radius of contact area to

a =

3p ~

+2Zc+

2 2 E 4 f . z + 4 x - f ~ ~_~

(39)

The reduction is caused by a 'peeling' mechanism which continues until the tangential force

fs =~ --4[(f~7~+7~2)G/E] "2, (cf. Savkoor and Briggs [22]). Under the 'peeling' regime, no slip occurs at the contact and the tangential stiffness may be estimated as

K = 8Gla $

(40)

where the radius of contact area, a, is given by Eq. 38. Once the tangential force exceeds the critical peeling force, ~c, slip occurs at the contact. The tangential stiffness under contact slip can be estimated using Eqs. 24-27, noting that the sliding condition is given by fs = (fn+7~)tan~ (cf. Chang et al [13]). Thornton [23] has argued that a more realistic sliding condition is expressed by fs = (fn+2Xc)tan~ w

527

2.4 Interface of Rough Elastic Particles Although topographically smooth particles provide an important starting point in the study of interface behavior they seldom represent realistic situations. At the interface of rough contacts the actual contact is between the asperities on the particle surface. Thus the force-deformation behavior of the interface is determined by overall deformation of the particle in the Hertzian sense as well as the deformation of the individual asperties. In practice, the particle surface roughness, in terms of asperity heights and wavelength, is difficult to characterize and introduction of statistical measures are necessary to describe the roughness. By using gaussian distributions of surface heights, Greenwood and coworkers have numerically studied the behavior of rough interfaces under normal loads (cf. Hills et al [19], Chapter 14). However, expressions for stiffnesses have not been presented. In addition the possibility of sliding at inclined asperity contacts have been neglected in this analysis. In an alternative approach, which considers the deformation of individual asperity contacts only, the overall stiffness, Kii, of the rough interface may be obtained as the algebraic sum of individual asperity contact stiffness, written as Kij = ~ K~

(41)

where superscript ~ refers to the asperity contact in the o~-th direction. Introducing a density function, ~(f~), that describes the distribution of asperity contact orientation and assuming the asperity size to be uniform for simplicity, Eq. 41 can be rewritten as (42)

Kq = f K q ( ~ ) ~ ( ~ ) d ~

Using the following truncated spherical harmonic expansion to represent the distribution function, ~(f~) (cf. Chang and Misra [24])

[A

1 1+ (3cos27§ ~(f2) = 2~ "4"

]

(43)

the interface stiffnesses are obtained in terms of the single asperity normal stiffness, K, '~, and tangential stiffness, Ks~, as

K

~_.~[5 ( K ~ + 2 K '~) + 2 A ( K ~ - K s'~)

(44)

and K$ =

5 ( K~§

$

) - A ( K ~ - K $s)

(45)

In this derivation, the individual asperity normal stiffness, K, ", and tangential stiffness, K, '~, are assumed to be uniform for all the asperities. The individual asperity stiffness can be

528 taken to be the interface stiffness for smooth particles.

2.5 Interface of Spherical Particles with Viscous Bridge In particulate materials with liquid bridge at particle interface, such as in partially saturated soils or in particulate materials that use viscous liquids as a binder for agglomeration, the interface stiffness due to the capillary action is of interest. Based on the work of Adams and Perchard [25], the normal force, f,, due to the capillary action of a liquid bridge at particle interface, is found to be (Lian et al [26]) [. = ~ y , r 2

r2)

(46)

+--

r1

where y/is the surface tension of the air-liquid surface, rl and r 2 are the principal radii of a torroidal shaped liquid bridge shown in Fig. 7. The capillary action exerts attractive forces on particles upto a separation distance approximately the cube root of liquid bridge volume (Lian et al [26]). The time dependent viscous forces will also develop at the interface due to the relative approach or retreat of the particles. The normal force, fn, due to liquid viscosity is related to the relative velocity, vn, of the particles via coefficient, Cn, given by

C ?I

3~rlP 2 2h

(47)

where 11 is the fluid viscosity, and h is the particle separation along the common normal. Y

r

1

r t'M

Figure 7. Torroidal viscous bridge between spherical particles.

529 3. INTERFACE FRICTION The friction at an interface has been traditionally described using Amonton's friction law which states that at incipient sliding, the friction force f, =fn tan~,, where ~ is termed the interface friction angle. Although microscopic processes associated with interface friction are being explored, the interface friction phenomenon has not been completely understood (Singer and Pollock [27]). In light of this, Amonton's law and friction angle have served an important role in practice. Along the lines of Amonton's law, friction angle of a variety of materials of interest in geomechanics have been measured under a variety of conditions. A compilation of these measurements are given in the Appendix based on the data reported by Rowe [28] and Mitchell [29]. It is seen that the values of friction angle vary not only for mineral type, but also for test conditions and test apparatus.

REFERENCES Johnson, K.L. (1985). Contact Mechanics. Cambridge University Press, London. ,

,

,

Misra, A. (1991). Constitutive relationships for granular solids with particle slidings and fabric changes. Ph.D. dissertation, University of Massachusetts at Amherst. Cundall, P.A., and Strack, O.D.L. (1979). "A discrete numerical model for granular assemblies," Geotechnique, V ol. 29, 47-65. Kishino, Y. (1987). Discrete model analysis of granular media. Micromechanics of granular materials, Eds. M. Satake, and J.T. Jenkins, Elsevier, Amsterdam, The Netherlands, 143-152. Chang, C.S., and Misra, A. (1989). Computer simulation and modelling of mechanical properties of particulates. Computers and Geotechnics, Vol. 7, No. 4, 269-287. Chang, C.S., and Acheampong, K. (1993). Accuracy and stability for static analysis using dynamic formulation in discrete element methods. Proc. 2nd Intl. Conf. Discrete Element Methods, Eds. J.R. Williams and G.G.W. Mustoe, IES1, Massachusetts Institute of Technology, 379-390.

,

Misra, A. and Chang, C.S. (1993). Effective elastic moduli of heterogeneous granular solids. International Journal of Solids and Structures, Vol. 30, No. 18, 2547-2566. Chang, C.S., Misra, A. and Acheampong, K. (1992). Elasto-plastic deformation for particulates with frictional contacts. Journal of Engineering Mechanics, Vol. 118, No. 8, 1692-1707. Chang, C.S., Chang, Y. and Kabir, M.G. (1992). Micromechanics modeling for stress-strain behavior of granular soils. Journal of Geotechnical Engineering, ASCE, Vol. 118, No. 12, 1959-1974.

530 10.

Mindlin, R.D. (1949). Compliance of elastic bodies in contact. Mechanics, Vol. 16, 259.

11.

Mindlin, R.D., and Deresiewicz, H. (1953). Elastic spheres in contact under varying oblique forces. Journal of Applied Mechanics, Vol. 20, 327.

12.

Willis, J.R. (1966). Hertzian contact of anisotropic bodies. Journal of Mechanics and Physics of Solids, Vol. 14, 163.

13.

Chang, C.S., Misra, A. and Sundaram, S.S. (1990). Micromechanical modelling of cemented sands under low amplitude oscillations. Geotechnique, Vol. 40, No. 2, 251263.

14.

Thornton, C. and Randall, C.W. (1987). Applications of theoretical contact mechanics to solid particle system simulation. Micromechanics of granular materials, Eds. M. Satake, and J.T. Jenkins, Elsevier, Amsterdam, The Netherlands, 245-252.

15.

Lubkin, J.L. (1951). Torsion of elastic spheres in contact. Mechanics, Vol. 18, 183.

16.

Deresiewicz, H. (1954). Contact of elastic spheres under oscillating torsional couple. Journal of Applied Mechanics, Vol. 21, 52.

17.

Muskhelishvili, N.I. (1949). Some Basic Problems of the Mathematical Theory of Elasticity. (English translation by J.R.M. Radok, Noordhoff).

18.

Deresiewicz, H. (1957). Oblique contact of non-spherical bodies. Journal of Applied Mechanics, Vol. 24, 623-624.

19.

Hills, D.A., Nowell, D. and Sackfield, A. (1992). Mechanics of Elastic Contacts. B utterworth-Heinemann, London.

20.

Johnson, K.L., Kendall, K. and Roberts, A.D. (1971). Surface energy and the contact of elastic solids. Proc. R. Soc., A324, 301-313.

21.

Derjaguin, B.V., Muller, V.M. and Toporov, Y.P. (1975). Effect of contact deformations on the adhesion of particles. Journal of Colloid Interface Science, Vol. 53, 314-326.

22.

Savkoor, A.R. and Briggs, G.A.D. (1977). The effect of tangential force on the contact of elastic solids in adhesion. Proc. Roy. Soc., A356, 103-114.

23.

Thornton, C. (1991). Interparticle sliding in the presence of adhesion. Journal of Physics D: Applied Physics, Vol. 24, 1942-1946.

Journal of Applied

Journal of Applied

531 24.

Chang, C.S., and Misra, A. (1990). Packing structure and mechanical properties of granulates. Journal of Engineering Mechanics, ASCE, Vol. 116, No. 5, 1077-1093.

25.

Adams, M.J. and Perchard, V. (1985). The cohesive forces between particles with interstitial liquids. I. Chem. E. Symp. Ser., 91, 147-160.

26.

Lian, G., Thornton, C. and M.J. Adams (1993). Effect of liquid bridge forces on agglomerate collisions. Powders and Grains 93, Ed. C. Thornton, 59-64, A.A.Balkema, Rotterdam.

27.

Singer, I.L. and Pollock, H.M. (1992). Fundamental of Friction: Macroscopic and Microscopic Processes. Kluwer Academic, Dordrecht, The Netherlands.

28.

Rowe, P.W. (1971). Theoretical meaning and observed values of deformation parameters for soil. Stress-Strain Behavior of Soils, Ed. R.H.G. Parry, G.T.Foulis & Co., 143-194.

29.

Mitchell, J.K. (1992). York.

30.

Tombs, S.G. (1969). Strength and Deformation Characteristics of Rockfill. Ph. D. Thesis University of London.

31.

Horn, H.M. and Deere. D.V. (1962). Geotechnique 12 4, 319-335.

32.

Tschebotarioff, G.P. and Welch, J.D. (1948). Lateral earth pressures and friction between soil minerals. Proc. 2nd Int. Conf. Soil Mech. Rotterdam VII 135-138.

33.

Hafiz, M.S. (1950). Strength characteristics of sands and gravels in direct shear. PhD Thesis, University of London.

34.

Penman, A.D.M. (1953). Shear characteristics of saturated silt measured in triaxial compression. Geotechnique 3 4, 312-328.

35.

Bishop, A.W. (1954). Correspondence on shear characteristics of a saturated silt measured in triaxial compression. Geotechnique 4 1.43-45.

36.

Gray, J.E. (1960). The relationship between principal stress dilatancy and friction angle of a granular material. M.Sc. Thesis, University of Manchester.

37.

Rowe, P.W. (1962). The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc. Roy. Soc. A 269 500-527.

38.

Bromwell, L.G. (1966). The friction of quartz in high vacuum. M.I.T. Dept. of Civil Engineering, Research Report B66-18.

Fundamentals of Soil Behavior.

Wiley Interscience, New

Frictional characteristics of minerals.

532 39.

Lee, I.K. (1966). Stress dilatancy performance of feldspar. J. soil Mech. Fdns. Div. Am. Soc. civ. Engrs. 92 SM2. 79-103.

40.

Parikh, P.V. (1967). The shearing behavior of sand under axisymmetric loading. Ph.D. Thesis, University of Manchester.

41.

E1-Sohby, A.A.K. (1969). Deformation of sands under constant stress ratio. Proc. 7th Int. Conf. Soil Mech. Mexico 1 111-119.

42.

Skinner, A.E. (1969). A note on the influence of interparticle friction on the shearing strength of a random assembly of spherical particles. Geotechnique 19 1, 150-157.

43.

Tong, P.Y.L. (1970). Plane strain deformation of sands. Ph.D. Thesis, University of Manchester.

4.

Procter, D.C., and Barton, R.R. (1974). Measurements of the angle of interparticle friction. Geotechnique, Vol. 24, No. 4, pp. 581-604. Appendix - Laboratory measurements of interface friction angles.

Ref.

Type of test

Material

Conditions

Ir ~

Tombs [30]

Large particle on large particle

Chert

15

Mudstone

24

Granite

36

] i

Horn and Deere [31 ]

Block on block

Horn and Deere [31]

Direct shear box, fixed particles on flat surface

Feldspar

Quartz (clean)

Dry

6.8

Water-saturated

37.6

Dry Water-saturated

Quartz (milky)

Dry Water-saturated

Quartz (rose)

Quartz (clear)

27

Dry Water-saturated

Feldspar (microline)

23

24

Dry Water-saturated

37

Amylamine

31

Carbontetrachloride

11

533 Appendix - Laboratory measurements of interface friction angles. (continued)

Ref.

Type of test

Material

Conditions

1r

Tschebotarioff and Welch [32]

Direct shear box, fixed particles on flat surface

Quartz

Dry

.6

Hafiz [33]

Three fixed particles over block

Quartz

Penman [34]

Direct shear box, fixed particles on flat surface

Quartz

Bishop [35] Gray [36]

Three balls on glass plate

i Moist and watersaturated

I

25

Water-saturated

21-27

Dry

11

Water-saturated

33

High load

19

Low load

29

Quartz

Air dried

22

Glass

Dry, Low load 19

Water-saturated and low load After Redryin g

Gray [36]

Rods on rods

Steel

Dry, cleaned with carbontetrachloride

Gray [36]

Light load apparatus

Steel

Dry, polished

16 I

9-14

Dry, polished, and cleaned with carbontetrachloride

9.5-

12

i Gray [36]

Ball on ball

Steel

i Gray [36]

Free particles on plate

Steel

Water-saturated, dry, cleaned with carbontetrachloride

8.8

Glass

Dry

10-12

Water-saturated

17

Gray [36]

534 Appendix - Laboratory measurements of interface friction angles. (continued)

Ref.

Type of test

Material

Conditions

Rowe [37]

Direct shear box, free particles on flat surface

Steel

Air Water-saturated

Glass ballotini

Water-saturated

Quartz

Water-saturated Water-saturated Water-saturated Water-saturated Water-saturated

Bromwell [38]

Direct shear box, fixed particles on flat surface

Atmospheric High vacuum

Bromwell [38]

Block on block

Quartz

Dependent on surface condition

Lee [39]

Free particle

Feldspar

Water-saturated

Parikh [40]

Ball on ball

Phosphorbronze

Water-saturated

EI-Sohby [41]

Direct shear box, free particles on flat surface

Zircon

Water-saturated

Feldspar

Water-saturated

Ball on ball

Glass ballotini

Dry, low load

Skinner [42]

Dry, high load Flooded, low load

28

Flooded, high load

38

Flooded, low load

40

Flooded, high load

40

535 Appendix - Laboratory measurements of interface friction angles. (continued)

].ef Tong [43]

Tong [43]

Type of test

I Material

Particles fixed with wax after initial sliding

Glass ballotini

Direct shear box, free particles on flat surface

Quartz

]Conditions

]~,

Cleaned with soap, water and acetone

15

Cleaned with soap, water and acetone

14

Water-saturated

26

Moist and watersaturated

28

Dry, tested in dry nitrogen

15

Acetone cleaned, tested in dry nitrogen

16

Trichloroethylene, acetone, detergent rinses

21

Water-saturated

15

Cleaned with soap, water and acetone

15

Glass ballotini

Tschebotarioff and Welch [32]

Block over particle set in

Quartz

mortar

Dry Moist

24.25

Water-saturated

24.25

Dry

7.4

Water-saturated

24.2

Horn and Deere [31 ]

Block on block

Rowe [37]

Particles on polished block

Quartz

Water-saturated

22-31

Procter and Barton [44]

Particle-particle Particle-plane particle-plane

Quartz

Saturated

26

Saturated

22.2

Dry

17.4

Quartz

536 Appendix - Laboratory measurements of interface friction angles. (continued)

Ref.

I

Type of test

[Material

Conditions

Procter and Barton [44]

Particle-plane

Feldspar

saturated

Horn and Deere [31 ]

Block on block

Calcite

Dry

Hron and Deere [31]

Along cleavage faces

Water-saturated Muscovite

Dry Dry Saturated

i Horn and Deere [31 ]

Along cleavage faces

Phlogopite

Dry Dry Saturated

Horn and Deere [31 ]

Along cleavage faces

Biotite

Dry Dry Saturated

Horn and Deere [31 ]

Along cleavage faces

Chlorite

Dry Dry Saturated

Skinner [42]

Ball on plate

Glass ballotini

Dry, low load Dry, high load Flooded, low load Flooded, high load

Skinner [42]

Friction apparatus

Steel balls

Dry

28.9

537

AUTHOR INDEX Alonso, E.E. 395

Plesha, M.E. 375

Alvappillai, A. 127

Poulsen, B.A.

Bal~zs, G.L. 255

Reinhardt, H.W. 255

Beer, G. 343

Reynouard, J.M. 281

Boulon, M.

Rigby, D.B.

147

343

107

Carol, I. 395

Riska, K. 77

C16ment, J.L. 281

Selvadurai, A.P.S.

Davis, R.O. 449

Shah, S.P.

Desai, C.S.

107 147

Sodhi, D.S.

57

Stephansson, O. 317

Gens, A. 395 Hohberg, J.-M.

227

Shao, Y. 227

Frederking, R.M.W. 35 Garnica, P.

173

Timco, G.W. 35 421

Unterreiner, P. 487

Jean, M. 463

van Mier, J.G.M. 201

Jing, L.

Vardoulakis, I. 487

317

Ladanyi, B. 3

Vermeer, P.A.

Li, Z. 227

Vervuurt, A. 201

Merabet, O. 281

Zaman, M. 127

Misra, A. 513

147

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539

SUBJECT INDEX This index was compiled using keywords supplied by the authors and supplemented by the Editors: the numbers refer to the first page of the paper concerned Adaptive local step, 147 Adfreeze strength of interfaces, 3 Adhesion phenomena, 173 Aggregate-matrix interface, 201 Anchor cylindrical, 173 Anchor pull out experimental modelling, 201 Annular shear device, 127 Arch dams stability on fractured rock, 395 Bond concrete to reinforcement, 255 healing tests, 3 Bonded-debonded interfaces, 227 Bonded interface, 227 Bond interface element cyclic reversed loads, 281 Bond parameters differential, 281 local, 281 Bond-slip modelling, numerical results, 281 Bond stress-slip relationship, 281 Boundary element modelling of geomaterial interfaces, 173, 343 of non-linear interfaces, 173, 343 of rock joints, 343

540 Calibration chamber experiments in, 147 Circular anchor at precompressed geological interface, 173 Computer simulation of particulate materials, 513 Concrete composition influence on steel-concrete interface, 255 Concrete joints cracking of, 421 finite element modelling of, 421 Concrete-steel interface, 201 Concrete-steel fibre interface, 227 Confinement influence on steel-concrete interface, 255 Constitutive equations for rock joints, 375 Constitutive interpolation, 147 Constitutive laws elastoplasticity, 147, 173, 281, 375, 395, 421 rate type, 147 stress point algorithm, 395 Constitutive modelling, 107 slip based, 227 Constitutive models of interface behaviour, 3, 107, 147, 227, 281, 375, 463, 487 Contact zone in ice, 35 Cosserat continuum linear elastic, 487 rigid plastic, 487 Couette flow, 487 Coulomb friction, 173, 227, 343, 463

541 Crack growth in lattice model, 201 Cracked layer contact problem for, 173 wedge indentation of, 173 Cracking of concrete joints, 421

Creep at pile-soil interface, 3 in frozen soils, 3 Cyclic loading of interfaces, 107 Cyclic multi-degree-of-freedom device, 127 Cylindrical anchor, 173 Damage computations steel-concrete interfaces, 281 Dams, 421 Debonded interface frictional shear stresses, 227 Debonding fracture-based criteria, 227 Debonding criteria for steel-cement interface, 227 Deformable bodies frictional contact, 463 Differential bond parameters, 281 Dilatancy, 421 hardening and softening effects, 3 localized, 147 Dilatant interface, 173, 375 Direct boundary element method, 343 Direct shear device, 107, 127 Direct shear path, 147

542 Direct shear tests, 127 of rock joints, 317 Displacement discontinuity method, 343 Disturbed state concept, 107 Double-mechanism plasticity, 421 Dynamic loading interface testing, 127 Earthquakes pore pressure effects, 449 Effective stress Principle, 449 Elastoplasticity constitutive laws, 147, 173, 281, 375, 395, 421 Experimental investigations ice indentation, 35 rock joints, 317 soil-structure interfaces, 3, 107, 127, 147 steel-concrete interface, 201, 227 Experimental model of ice-structure interaction, 57 Experimental modelling of interfaces, 127 of steel-concrete interface, 227 Failure at pile-soil interface, 3 pull-out, 255 splitting, 255 Faults and joints boundary element modelling of, 343 Finite crushing depth, 77 Finite element method, 107, 147 for rock joints, 395

543

Finite element modelling concrete joints, 421 ice-structure contact, 77 of non-linear interfaces, 147, 281, 395, 421 Flaking ice models, 77 Floating ice, 77 Floating ice sheets indentation tests, 35 Flow rule non-associated, 147, 173, 281, 375, 421, 463 Fracture surfaces non-linear interfaces, 173 Fractured rock stability of arch dams on, 395 Friction at interface, 513 in concrete joints, 421 in rock joints, 375 Frictional contact deformable bodies, 463 laws, 463 rigid bodies, 463 Frictional heating effects, 449 Frictional interface plane crack at, 173 Frozen soil creep of, 3 strength of, 3 Frozen soil-structure interfaces numerical analysis, 3 Geological interface precompressed, circular anchor at, 173

544 Geomaterial interfaces boundary element modelling, 173 Geomaterial motions of geological layer, 463 Global loading, 147 Goodman element, 421 Grain crushing, 147 Granular medium, 487 Hardening effects, 3 High pressure ice crushing, 35 Hydro-thermo-mechanical testing of rock joints, 317 Ice content of frozen soil effect on strength, 3 Ice crushing, 77 Ice extrusion, 57 Ice flaking, 77 Ice friction, 35 Ice indentation, 57 experimental investigations, 35 Ice sheets indentation modelling, 77 Ice-structure contact, 57 finite element modelling, 77 Ice-structure interaction, 35 experimental modelling, 57 theoretical modelling, 57 Incremental laws interface behaviour, 281 Incremental paths, 147 Incremental plasticity, 421 Indentation modelling of ice sheets, 77

545 Indentation tests on floating ice sheets, 35 Indirect boundary element method, 343 Integration local, 147 Newton-Cotes, 147 spatial, 147 Interaction steel-concrete, 255 Interface aggregate-matrix, 201 bonded, 227 bonded-debonded, 227 constitutive models, 227 constitutive relationships, 3, 107, 147, 173, 227, 281, 375, 463, 487 dilatancy, 173, 375 finite elements, 281 ice-structure, 77 roughness, 487 soil-structure, 127, 147 steel-concrete, 201 steel fibre-concrete, 227 thin layer, 147 zero-thickness, 147 Interfaces adfreeze strength of, 3 coordinate system, 343 Coulomb, 173 cyclic loading, 107 damage, 281 dilatant, 173 elements, 147 experimental modelling of, 107, 127, 147, 201, 227, 317 friction, 513

546 frozen soil-structure, 3 in particulate materials, 513 modelling, 107 non-linear, BEM modelling, 173 shear tests, 3 steel-concrete, 255, 281 stiffness, at particle contact, 513 testing, 107 testing, dynamic loading, 127 testing, static loading, 127 reverse cyclic loading of, 255 Interface behaviour incremental laws, 281 piles in permafrost, 3 pore pressure effects, 449 Interface modelling applications to rock slides, 395 Interfacial localization in shear, 487 Joints and faults BE modelling of, 343 concrete, FE modelling of, 421 in rock, 375 Joint elements zero-thickness, 395 Joint interface element contact approach, 281 Joint roughness measurement and characterization, 317 Joint testing, 107 rock joints, 317 Korzhavin model, 77

547 Lagrange multiplier, 421 Lattice model crack growth in, 201 Linear elastic Cosserat continuum, 487 Load transfer, 147 Load transfer mechanism at frozen interfaces, 3 Loading global, 147 Loading rate influence on steel-concrete interface, 255 Local bond parameters, 281 Local convergency, 147 Local integration, 147 Local paths, 147 Localized dilatancy, 147 Localized iterative solution procedures, 173 Lumped stiffness, 421 Matrix shear lag, 227 Mechanical interlock in steel-concrete interface, 201 Micromechanical model of particulate interface, 513 Modelling bond slip, 281 concrete joints, 421 interfaces, 107 reinforced concrete structural element, 281 Moire interferometry technique, 227

Newton--Cotes integration, 147 Nodal springbox, 421

548 Non-associated flow rule, 147, 173, 281, 375, 421, 463 Non-linear interfaces boundary element modelling, 173, 343 finite element modelling, 147, 281, 395, 421 fracture surfaces, 173 Nominal ice pressure, 77 Normal loading-unloading test of rock joints, 317 Numerical analysis frozen soil-structure interfaces, 3 Numerical results for bond slip, 281 Numerical simulation of rock joints, 375 Oscillations stress, 147, 421 Particle contact surface adhesion, 513 viscous bridge, 513 Particles smooth, non-spherical, elastic, 513 smooth, spherical, elastic, 513 rough, elastic, 513 Particulate materials interfaces, 513 micromechanical model, 513 Particulate model of concrete, 201 Penalty parameter, 421 Permafrost interface behaviour of piles, 3

549 Piles 3, 147 in permafrost, interface behaviour, 3 Pile-soil interface creep and failure at, 3 Plane crack at frictional interface region, 173 Plane simple shear, 487 Plasticity double-mechanism, 421 incremental, 421 Plasticity computations steel-concrete interfaces, 281 Plasticity theories second gradient effects, 487 Pore pressure effects associated with earthquakes, 449 effects on interface behaviour, 449 fluctuations, 449 -induced seismicity, 449 Prescribed normal stiffness, 147 Pressure-area curve, 77 Pull-out of anchor in concrete, 201 experiments, 227 failure, 255 test, 147 Quadratic interpolation, 421 Rate type constitutive law, 147 Reinforced concrete modelling of structural element, 281 Repeated loading influence on steel-concrete interface, 255

550 Reverse cyclic loading of interfaces, 255 Rib pattern influence on steel-concrete interface, 255 Rik's method, 147 Rigid bodies frictional contact, 463 Rigid plastic material ring shear, 487 Rigid plasticity, 487 Ring shear of rigid plastic material, 487 Ring shear device, 127 Ring simple shear, 487 Ring torsion device, 127 Rock fractured, arch dams on, 395 Rock joints BEM computations, 343 constitutive equations, 375 experimental investigations, 317 FEM computations, 395 friction in, 375 _ mechanics of, experimental aspects, 317 numerical simulation of, 375 quadrature schemes for FEM, 395 shear in, 375 surface roughness of, 375 three-dimensional effects, 317 Rock slides applications of interface modelling, 395 Rotary shear tests of rock joints, 317 Roughness of interface, 487 surface, of rock joints, 375

551 Runge-Kutta algorithm, 421 Schneebeli material granular, 463 Second gradient plasticity theories, 487 Seismic analysis, 421 Seismicity pore pressure induced, 449 Shaking table tests, 127 Shear in rock joints, 375 Shear apparatus, 107 Shear device annular, 127 simple, 107, 127 Shear tests on interfaces, 3 Shear strength, 107 Signorini condition, 463 Simple sheardevice, 107, 127 Simple shear tests on granular media, 487 Slippage, 255 Slip and separation yield conditions, 343 Softening effects, 3 tension, 421 Soil-structure interfaces, 3, 107, 127, 147 Spatial integration, 147 Splitting failure, 255 Stability of arch dams on fractured rock, 395 Static loading interface testing, 127

552 Steel-concrete interaction, 255 Steel-concrete interfaces, 255 damage and plasticity computations, 281 experimental investigations, 201, 227 mechanical interlock, 201 Steel fibre-concrete interface, 227 Strain rate effect on strength of frozen soil, 3 Strength of frozen soils, 3 Stress oscillations, 147, 421 Stress point algorithm constitutive laws, 395 Slip relationship of bond interface, 281 Springbox nodal, 421 Structural model of interface behaviour, 227 Sub-incrementation technique, 147 Surface adhesion at particle contact, 513 Surface roughness of rock joints, 375 Tension softening, 421 Testing of interfaces, 107 of joints, 107 Theoretical model of ice-structure interaction, 57 Thin layer element, 107, 421 Thin layer interface, 147 Three-dimensional effects of rock joints, 317 Transverse pressure influence on steel-concrete interface, 255

553 Transverse tension influence on steel-concrete interface, 255 Triaxial loading, 127 Unilateral contact, 463 Undrained shear tests, 107 Viscous bridge at particle contact, 513 Wedge indentation of cracked layer, 173 Yield conditions at slip and separation, 343 Zero-thickness element, 421 interface, 147 joint elements, 395

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