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.


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


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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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-.." . 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.


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~

~

~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.


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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Laboratory Miniature Pile Test Apparatus (After Coyle and Sulaiman [4])

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.

132 To hyClratdic lack

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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.

Tylx;

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

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135

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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Simple Shear Apparatus (After Uesugi [15])

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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Figure 8.

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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Figure 9.

Details of Translational Test Box (After Desai [22])

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).


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(.'~ 9 s ' . ; . I ". x~,.. 'L._~'.x. ,.~,'.~. - Z:,'x~, ~ " ,~ ~" ~ "/.. ~', "~.I_, "-,,_.

.. t'-!~ .

F tq

A., ~ F,

x: ~ F:

) , tJl >

F'~

A,. F,' < 0

A,.F; i>0

let

k: < F' t

Sliding Opening

a: 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

-Je' ' ~

~

~

:Z~

3O

-: 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.


MECHANICS OF ROCK AND CONCRETE JOINTS


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

-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)


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 : 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 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 15qNlx1C [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

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