230223468 analysis of concrete structures by fracture mechanics by elfgreen and shah

Analysis of Concrete Structures by Fracture Mechanics

Other RILEM Proceedings available from Chapman and Hall1 Adhesion between Polymers and Concrete. ISAP 86Aix-en-Provence, France, 1986Edited by H.R.Sasse

2 From Materials Science to Construction Materials EngineeringProceedings of the First International RILEM CongressVersailles, France, 1987Edited by J.C.Maso

3 Durability of GeotextilesSt Rmy-ls-Chevreuse, France, 1986

4 Demolition and Reuse of Concrete and MasonryTokyo, Japan, 1988Edited by Y.Kasai

5 Admixtures for ConcreteImprovement of PropertiesBarcelona, Spain, 1990Edited by E.Vzquez

6 Analysis of Concrete Structures by Fracture MechanicsAbisko, Sweden, 1989Edited by L.Elfgren and S.P.Shah

7 Vegetable Plants and their Fibres as Building MaterialsSalvador, Bahia, Brazil, 1990Edited by H.S.Sobral

8 Mechanical Tests for Bituminous MixesBudapest, Hungary, 1990Edited by H.W.Fritz and E.Eustacchio

9 Test Quality for Construction, Materials and StructuresSt Rmy-ls-Chevreuse, France, 1990Edited by M.Fickelson

Publishers Note

This book has been produced from camera ready copy provided by the individual contributors, whosecooperation is gratefully acknowledged.

Analysis of Concrete Structures byFracture Mechanics

Proceedings of the International RILEM Workshop dedicated to Professor Arne Hillerborg, sponsored byRILEM (The International Union of Testing and Research Laboratories for Materials and Structures) and

organized by RILEM Technical Committee 90FM A Fracture Mechanics of Concrete StructuresApplications.

Abisko, SwedenJune 2830, 1989

EDITED BY

L.Elfgren and S.P.Shah

CHAPMAN AND HALLLONDON NEWYORK TOKYO MELBOURNE MADRAS

UK Chapman and Hall, 26 Boundary Row, London SE1 8HN

USA Van Nostrand Reinhold, 115 5th Avenue, New York NY10003

JAPAN Chapman and Hall Japan, Thomson Publishing Japan,Hirakawacho Nemoto Building, 7F, 1711 Hirakawa-cho,

Chiyoda-ku, Tokyo 102

AUSTRALIA Chapman and Hall Australia, Thomas Nelson Australia,102 Dodds Street, South Melbourne, Victoria 3205

INDIA Chapman and Hall India, R.Seshadri, 32 Second Main Road,CIT East, Madras 600035

First edition 1991

This edition published in the Taylor & Francis e-Library, 2005.

To purchase your own copy of this or any of Taylor & Francis or Routledges collectionof thousands of eBooks please go to www.eBookstore.tandf.co.uk.

1991 RILEM

ISBN 0-203-62676-1 Master e-book ISBN

ISBN 0-203-63058-0 (Adobe eReader Format)ISBN 0 412 36980 X (Print Edition) 0 442 31264 4 (Print Edition) (USA)

All rights reserved. No part of this publication may bereproduced or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording or otherwise, orstored in any retrieval system of any nature, without the writtenpermission of the copyright holder and the publisher, application

for which shall be made to the publisher.

The publisher makes no representation, express or implied, withregard to the accuracy of the information contained in this book andcannot accept any legal responsibility or liability for any errors or

omissions that may be made.

British Library Cataloguing in Publication DataAvailable

Library of Congress Cataloging-in-Publication DataAvailable

Dedication

This volume is dedicated to Professor Arne Hillerborg in recognition of his many outstanding contributions to thedevelopment of fracture mechanics for concrete structures.

Participants in the RILEM Workshop on Fracture Mechanics of Concrete Structures dedicated to Professor ArneHillerborg. The workshop took place in Abisko National Park in Northern Sweden in June 1989. From left to right:Lennart Elfgren (S), Herbert Linsbauer (A), Rune Sandstrm (S), Jan van Mier (NL), Ulf Ohlsson (S), Yu-Ting Zhu(PRC), Bjrn Tljsten (S). Surendra P.Shah (US), Ben Barr (UK), Anna Zolland (S), Carina Hannu (S), Herbert Duda(FRG), Jaime Planas (ES), Manuel Elices (ES), Ingegerd Hillerborg (S), Marianne Grauers (S), Per Anders Daerga (S),Crescentino Bosco (I) and Arne Hillerborg (S). Not present when the photo was taken were Mats Emborg (S), Jan-ErikJonasson (S) and Gottfried Sawade (FRG).

vi

Contents

Participants and contributing authors ix

Preface xi

1 Arne Hillerborg and fracture mechanicsL.ELFGREN

1

PART ONE BEHAVIOUR OF CONCRETE 15

2 Mode I behaviour of concrete: Influence of the rotational stiffness outside thecrack-zoneJ.G.M. van MIER

16

3 Experimental analysis of mixed mode I and II behaviour of concreteJ.G.M. van MIER, M.B.NOORU-MOHAMED, E.SCHLANGEN

26

4 Considerations regarding fracture zone response to simultaneous normal andshear displacementM.HASSANZADEH

36

5 Mixed mode fracture in compressionA.K.MAJIS.P.SHAH

49

6 Thermal stresses in concrete at early agesM.EMBORG

63

7 Grain-model for the determination of the stress-crack-width-relationH.DUDA

79

PART TWO STRUCTURAL MODELLING 89

8 Size effect and experimental validation of fracture modelsM.ELICES, J.PLANAS

90

9 General method for stability analysis of structures with growing interactingcracksZ.P.BAANT

117

10 Use of the brittleness number as a rational approach to minimumreinforcement designC.BOSCO, A.CARPINTERI, P.G.DEBERNARDI

121

11 Fracture mechanics analyses using ABAQUSK.GYLLTOFT

138

12 Design and construction of concrete dams under consideration of fracturemechanics aspectsH.N.LINSBAUER

144

PART THREE BENDING 152

13 Size dependency of the stress-strain curve in compressionA.HILLERBORG

153

14 Influence of the beam depth on the rotational capacity of beamsK.CEDERWALL, W.SOBKO, M.GRAUERS, M.PLOS

161

15 New failure criterion for concrete in the compression zone of a beamL.VANDEWALLE, F.MORTELMANS

166

PART FOUR SHEAR, BOND AND PUNCHING 178

16 Bond between new and old concreteYU-TINGZHU

179

17 Strengthening of existing concrete structures with glued steel platesB.TLJSTEN

187

18 Modelling, testing and strength analysis of adhesive bonds in pure shearP.J.GUSTAFSSON, H.WERNERSSON

197

19 Concrete surface loaded by a steel punchH.W.REINHARDT

211

PART FIVE ANCHORAGE 220

20 Fracture mechanics based analyses of pull-out tests and anchor boltsR.BALLARINI, S.P.SHAH

221

21 Anchor bolts in concrete structures. Two dimensional modellingU.OHLSSON, L.ELFGREN

254

Index 272

viii

Participants and contributing authors

Robert Ballarini, Case Western Reserve University, Department of Civil Engineering, Cleveland, Ohio44106, USA.Ben I.G.Barr, University of Wales, Division of Civil Engineering, P.O. Box 917, Cardiff CF2 1XH, UK.Zdenek P.Baant, Northwestern University, The Technological Institute, Evanston, Illinois 60208/3109,USA.Crescentino Bosco, Politecnico di Torino, Dipartimento di Ingegneria Strutturale, Corso Duca degliAbruzzi 24, 110129 Torino, Italy.Alberto Carpinteri, Politecnico di Torino, Dipartimento di Ingegneria Strutturale, Corso Duca degliAbruzzi 24, 110129 Torino, Italy.Krister Cederwall, Chalmers University of Technology, Division of Concrete Structures, S-412 96Gteborg, Sweden.Per Anders Daerga, Lule University of Technology, Department of Civil Engineering, S-951 87 Lule,Sweden.P.G.Debernardi, Politecnico di Torino, Dipartimento di Ingegneria Strutturale, Corso Duca degli Abruzzi24, 110129 Torino, Italy.Herbert Duda, Technische Hochschule Darmstadt, Institut fr Massivbau, Alexanderstrasse 5, D-6100Darmstadt, Germany.Lennart Elfgren, Lule University of Technology, Department of Civil Engineering, S-951 87 Lule,Sweden.Manuel Elices, Universidad Politecnica de Madrid, Departamento de Ciencia de Materiales, CiudadUniversitaria, E-280 40 Madrid, Spain.Mats Emborg, Lule University of Technology, Department of Civil Engineering, S-951 87 Lule,Sweden.Per Johan Gustafsson, Lund Institute of Technology, Department of Structural Mechanics, Box 118,S-22100 Lund, Sweden.Kent Gylltoft, National Swedish Testing Institute, Box 857, S-501 15 Bors, Sweden.Manoucheher Hassanzadeh, Lund Institute of Technology, Department of Building Materials, Box 118,S-221 00 Lund, Sweden.

Arne Hillerborg, Lund Institute of Technology, Department of Building Materials, Box 118, S-221 00Lund, Sweden. Jan-Erik Jonasson, Lule University of Technology, Department of Civil Engineering, S-951 87 Lule,Sweden.Herbert N.Linsbauer, Technische Universitt Wien, Institut fr Konstruktiven Wasserbau, Karlsplatz 13/222, A-1040 Wien, Austria.Arup K.Maji, The University of New Mexico, Department of Civil Engineering, Albuquerque, NewMexico 87131, USA.Jan G.M.van Mier, Delft University of Technology, Department of Civil Engineering, Stevin Laboratory,P.O. Box 5048, 2600 GA Delft, The Netherlands.Fernand Mortelmans, Katholieke Universiteit te Leuven, Departement Bouwkunde, Park van Arenbergde Croylaan 2, B-3030 Heverlee, Belgium.M.B.Nooru-Mohamed, Delft University of Technology, Department of Civil Engineering, StevinLaboratory, P.O. Box 5048, 2600 GA Delft, The Netherlands.Ulf Ohlsson, Lule University of Technology, Department of Civil Engineering, S-951 87 Lule, Sweden.Jaime Planas, Universidad Politecnica de Madrid, Departamento de Ciencia-de Materiales, CiudadUniversitaria, E-280 40 Madrid, Spain.Mario Plos, Chalmers University of Technology, Division of Concrete Structures, S-41296 Gteborg,Sweden.Hans W.Reinhardt, Stuttgart University, Pfaffenwaldring 4, D7000 Stuttgart 80, West Germany (formerlyat Darmstadt University of Technology, Institut fr Massivbau, Alexanderstrasse 5, D-6100 Darmstadt).Rune Sandstrm, Lule University of Technology, Department of Civil Engineering, S-951 87 Lule,Sweden.Gottfried Sawade, Stuttgart University, Institut fr Werkstoffe im Bauwesen, Pfaffenwaldring 4, D-7000Stuttgart 80, Germany.Erik Schlangen, Delft University of Technology, Department of Civil Engineering, Stevin Laboratory,P.O. Box 5048, 2600 GA Delft, The Netherlands.Surendra P.Shah, Northwestern University, NSF Center for Advanced Cement-Based Materials (ACBM),The Technological Institute, Evanston, Illinois 602 083109, USA.ke Skarendal, Swedish Cement and Concrete Institute (CBI), S-100 44 Stockholm, Sweden.Wanda Sobko, Chalmers University of Technolgy, Division of Concrete Structures, S-412 96 Gteborg,Sweden. Angelo di Tommaso, Universit degli studi di Bologna, Facolt di Ingegneria, Viale Risorgimento 2,I-40136 Bologna, Italy.Bjrn Tljsten, Lule University of Technology, Department of Civil Engineering, S-951 87 Lule,Sweden.Lucie Vandewalle, Katholieke Universiteit te Leuven, Department Bouwkunde, Park van Arenberg deCroylaan 2, B-3030 Heverlee, Belgium.Hans Wernersson, Lund Institute of Technology, Department of Structural Mechanics, Box 118, S-22100 Lund, Sweden.Yu-Ting Zhu, Royal Institute of Technology, Department of Structural Mechanics, S-100 44 Stockholm,Sweden.

x

Preface

This volume contains the proceedings from an international workshop on the analysis of concrete structuresby fracture mechanics. The workshop was dedicated to Professor Arne Hillerborg in recognition of hismany outstanding contributions to this field. The workshop was organized by RILEM Technical Committee90-FMA Fracture Mechanics of Concrete StructuresApplications.

In addition to the presentations and discussions during the workshop, as summarized by the authors, thevolume also contains some papers contributed by colleagues and friends of Arne Hillerborg.

We would like to thank Arne Hillerborg and the other authors for their cooperation, the staff at theDivision of Structural Engineering at Lule University of Technology for the practical arrangements of theworkshop, and the personnel at Chapman and Hall for the publication of this volume.

L.Elfgren, S.P.ShahLule and Evanston

February 1990

1ARNE HILLERBORG AND FRACTURE MECHANICS

L.ELFGRENDepartment of Civil Engineering, Lule University of Technology, Lule, Sweden

1INTRODUCTION

Arne Hillerborg has played an important part in the development of fracture mechanics for concretestructures. His aim has always been to analyze problems in such a way that conclusions can be drawn whichare of value to practising desing engineers. In this paper a short outline is given of some of his contributions.

2CAREER

Arne Hillerborg was born in Stockholm, Sweden, on the 4th of January 1923. He graduated as a civilengineer (M.Sc.) from the Royal Institute of Technology in Stockholm in 1945. He subsequently worked asa research assistent in Structural Engineering at the Royal Institute of Technology, where he presented hisPhD thesis Dynamic influences of smoothly running loads on simply supported girders in 1951. After ashort period as design engineer he became involved in work for the Swedish concrete code, particularlyregarding design of two-way slabs. During a five year period 195560 he was a lecturer at the TechnicalCollege in Stockholm, teaching structural engineering. Then, 196068 he was head of Siporex CentralLaboratory, in charge of research and development for autoclaved aerated concrete. In 1968 he becameassociate professor in Structural Mechanics and in 1973 full professor in Building Materials at LundInstitute of Technology. Since January 1989 he is professor emeritus and in this function he is still veryactive.

Arne Hillerborg married Ingegerd in 1944. They have one daughter.

3THE STRIP METHOD

The strip method for reinforced concrete slabs was first proposed by Hillerborg (1956). It is a designmethod based on the lower bound theory of plasticity. It was further developed (1959). These first paperswere written in Swedish. A short English paper was published by Hillerborg (1960). The presentation to the

English-speaking world is mainly due to Crawford (1962), Blakey (1964), Wood (1968) and Wood andArmer (1968). Later Hillerborg has summarized and extended his work, first in Swedish (1974), English

2 ANALYSIS OF CONCRETE STRUCTURES

translation (1975), and In (1982a). An example of the simple and straight forward analysis is given inFigure 1.

Fig. 1. Example of an analysis with the Hillerborg strip method. In (a) dimensions and support condition are given for aslab with a uniform load of 8 kN/m2 . The slab is supported by a column, two sides are built in and two sides are simplysupported. In (b) and (c) chosen moment curves for the main strips are shown. In (d) resulting design moments aregiven. The load carried by the column is R=86.16.05=296 kN. From Hillerborg (1982a).

ARNE HILLERBORG AND FRACTURE MECHANICS 3

4FRACTURE MECHANICS

Arne Hillerborg first got interested in fracture mechanics of concrete when he taught Building Materials atLund Institue of Technology in the mid 70ies. He initiated a work for two of his students Petersson andModeer (1976) and they together later in the same year published a paper on it, Hillerborg et al (1976). Inthat paper the model which has later become known as the fictitious crack model was introduced, seeFigure 2. With the model it became obvious that linear elastic fracture mechanics (LEFM) could only beapplied to very large concrete structures and not to concrete elements of normal size as was earlier done,Mindess (1983a, b).

In 1979 and 1981 Matz Moder and Per-Erik Petersson published their PhD theses on fracture mechanicsof concrete. Their work was inspired by Hillerborg and separately or together they published many paperson related subjects. Some examples of results are given in Figures 3 to 5.

In order to illustrate the size dependance in a simple and dimensionless way Arne Hillerborg earlyintroduced the concept of a characteristic length 1ch of a material. The characteristic length 1ch is defined as

where E=the modulus of elasticity, GF=the fracture energy and ft= the tensile strength of the material.

Fig. 2. The fictitious crack model as it was first proposed by Hillerborg et al (1976), and by Hillerborg (1978a).


Size dependance can now be plotted as a function of d/1ch where d is 2 representative dimension of thestudied structural element (e.g. the depth of a beam), see examples in Figures 3, 7, 8 and 9. The ratio d/1ch=dt2/EGF=(d3 ft2/E)/(d2 GF) can also be interpreted as a brittleness number giving the ratio of the storedelastic strain energy (d3 ft2 /E) to the fracture energy needed to break the specimen d2 GF), see e.g.DiTommaso and Bache (1989).

When the first RILEM technical committee on fracture mechanics of concrete was formed in 1979 (TC50FMC) with Professor Folker H Wittman as chairman, Arne Hillerborg was one of the members (otherprominent members were H.K.Hilsdorf, M.Lorrain, H.Mihashi, S.Mindess, A. Rsli, R.N.Swamy,S.Ziegelsdorf and A.Di Tommaso).

In the RILEM work Hillerborg had the main responsibility for a series of round robin tests on fractureenergy of concrete according to a method proposed by Petersson. This resulted in a RILEMrecommendation (1985) for a three-point beam method to determine the fracture energy of concrete, seeFigures 6 and 7. This method is now used world-wide and has started much additional work on the testingmethods of fracture mechanics properties.

After the work of RILEM TC 50 FMC was finished in 1985, Wittmann (1983, 1986), two new RILEMtechnical committees were set up for the continuation of the work on fracture mechanics of concrete, onefor further work on test methods (TC 89 FMT, chaired by S.P.Shah) and one for application (TC 90 FMA,chaired by L.Elfgren). The latter committee was proposed by Hillerborg, who has also taken an active partin its work, see Elfgren (1989a, b).

In 1985 another of Hillerborgs students presented his PhD thesis, Gustafsson (1985). This thesiscomprised among other things many comparisons between tests and analytical results by means of the

Fig. 3. Theoretical relation between splitting strength fs and tensile strength ft for a concrete cube according to Moderas function of the brittleness number w/1ch. (w is the height of the cube, 1ch is a characteristic length=EGF/ft2, Emodulus of elasticity, and GF=fracture energy) From Hillerborg (1979b).


fictitious crack model and application of the model to some practical design problems, e.g. shear fracture ofreinforced beams. Some results from the thesis and related papers are given in Figures 8 and 9.

Ongoing work by Hillerborg and his doctor students includes mixed mode properties of concrete andstability problems in fracture mechanics testning, see Figure 10 and 11. Recently the possibility of a formalapplication of the fictitious crack model to the failure of concrete in the compression zone of a bent beamhas been studied by Hillerborg, and some preliminary conclusions of this work have been published (1988c,1989a).

A bibliography of the works presented by Hillerborg and his group is presented in the next section.

5

Fig. 4. The fracture zone and the stress distribution in front of the notch tip at the maximum load for different beamdepths. The figure is relevant for three-point bending with a ratio of notch depth a to beam depth d of a/d=0.25. Thematerial properties are ft=3 MPa, GF=75 N/m, E=30 GPa, 1ch=0.25 m. From Petersson (1981).


BIBLIOGRAPHY

In this section a list of references are given to the works published by Arne Hillerborg and his group ofstudents and collaborators. A few additional works of general interest are also cited.

Avd fr Byggnadsmateriallra (1988) Byggnadamateriallara LTH 1973 1988. Tillgnad Arne Hillerborg vid hansavgng frn professuren i december 1988 (Building Materials 19731988. A report dedicated to ArneHillerborg). Lund Institute of Technology. Report TVBM-3038, 96 pp.

Blakey, F.A. (1964) Strip method for slabs on columns, L-shaped plates etc. Translation of Hillerborg (1959).Melbourne, CSIRO, D.B.R. Translation No. 2.

Crawford, R.E. (1962) Limit design of reinforced ete slabs. Thesis submitted to the University of Illinois for thedegree of PhD. Urbana.

DiTommaso, A. and Bache, H. (1989) Size effects and brittleness. Chapter 7 in Fracture Mechanics of ConcreteStructures. From theory to applications (ed. L.Elfgren) Chapman & Hall, London, 191207

Fig. 5. Experimentally determined stress-deformation-curves for concrete in tension. From Petersson (1981).

Fig. 6. Proposed standard beam for test of fracture energy GF Hillerborg (1985c).


Elfgren, L. editor (1989a), Fracture Mechanics of Concrete Structures. From theory to applications. A RILEMReport by Technical Committee 90-FMA. Chapman & Hall, London, 407 pp (ISBN 0412 306808).

Elfgren, L. (1989b) Applications of fracture mechanics to concrette structures. Fracture Toughness and FractureEnergy (eds. H Mishashi, H.Takahashi and F.H.Wittmann). Balkema, Rotterdam, pp 575590.

Gustafsson, P.J. (1983) Oarmerade betongrrs bjbrottlast. Teoretiska berkningsmetoder. (Unreinforced concretepipes.) Lund Institute of Technology, Rapport TVBM-3012.

Gustafsson, P.J. (1985) Fracture mechanics studies of non-yielding materials like concrete: modelling of tensilefracture and applied strength. Doctor thesis. Lund Institute of Technology. Report TVBM-1007. 422 pp.

Gustafsson, P.J. and Hillerborg, A. (1984) Improvements in concrete design achieved through the application of fracturemechanics. Application of fracture mechanics to cementitious composites, NATO Advanced ResearchWorkshop, September 47, Northwestern University.

Gustafsson, P.J. and Hillerborg, A. (1988) Sensitivity in shear strength of longitydinally reinforced concrete beams tofracture energy of concrete. ACI Structural Journal, Vol. 55, No 3, May-June, 286294.

Hassanzadeh, M. (1988) Determination of fracture zone properties in mixed mode I and II. Int. Conf. on Fracture ofConcrete and Rock, Vienna July 46, 1988. Abstract published in Engineering Fracture Mechanics, Vol 35, No1/2/3, 1990, p 614.

Hassanzadeh, M. and Hillerborg, A. (1989a) Theoretical Analysis of test methods. Fracture of Concrete and Rock,(eds S.P.Shah and S.E.Swartz), Springer, New York, pp 388395.

Hassanzadeh, M. and Hillerborg, A. (1989b) Concrete properties in mixed mode fracture. Fracture Toughness andFracture Energy Test Methods for Concrete and Rock, (eds H.Mihashi et al) Balkema, Rotterdam.pp 565568.

Hassanzadeh, M., Hillerborg, A. and Zhou, F.P. (1987) : Tests of material properties in mixed mode I and II. SEMRILEM International Conference on Fracture of Concrete and Rock. (eds. S.P.Shah and S.E.Swartz). Societyfor Experimental Mechanics, Bethel, CT, pp 353358 (ISBN 0912053135).

Fig. 7. Theoretical flexural strength of notched and unnotched concrete beams. Hillerborg (1985c).


Helmerson, H. (1978) Materialbrott fr olika byggnadsmaterial. (Material Fracture for various building materials.)Diploma work Lund Institute of Technology, Division of Building Materials.

Hillerborg, A. (1951) Dynamic influences of smoothly running loads on simply supported girders. Doctor Thesis,Department of Bridge Engineering, Royal Institute of Technology, Stockholm, 126 pp.

Hillerborg, A. (1956) Jmviksteori fr armerade betongplattor. (Equilibrium theory for concrete slabs). Betong(Stockholm). Vol. 41, No 4, 171182.

Hillerborg, A. (1959) Strimlemetoden (The strip method). Riksbyggen, Stockholm.Hillerborg, A. (1960) A plastic theory for the design of reinforced concrete slabs. IABSE Sixth Congress. Stockholm.

Preliminary publication. pp 177186. Hillerborg, A. (1974) Strimlemetoden (The strip method), Almqvist & Wiksell, Stockholm, 327 pp (ISBN 9120

039122).Hillerborg, A. (1975) Strip Method of design. English translation of Hillerborg (1974). Cement and Concrete

Association, Wexham Springs, (Viewpoint publication), E &FN Spon, London, 256 pp, (ISBN 0721010121).Hillerborg, A. (1978a) A model for fracture analysis. Lund Institute of Technology. Report TVBM-3005, 8 pp.Hillerborg, A. (1978b) Brottmekanik tillmpad p betong (Fracture mechanics applied to concrete) Nordisk Betong

(Stockholm) Nr 678, pp 512.Hillerborg, A. (1979a) The fictitious crack model and its use in numerical analysis. International Conference on

Fracture Mechanics in Engineering Applications, Bangalore, March 2630.

Fig. 8. Theoretical shear strength of reinforced concrete beams without shear reinforcement, is the percentage of rein-forcement. Gustafsson & Hillerborg (1984).


Hillerborg, A. (1979b) Some practical conclusions from the application of fracture mechanics to concrete. Studies onconcrete technology. Dedicated to Professor Sven G.Bergstrm on his 60th anniversary, Swedish Cement andConcrete Institute, Stockholm, December 14, 4354.

Hillerborg, A. (1980a) Brott i betong (Fracture of concrete). CBI:s informationsdag, Swedish Cement and ConcreteInstitute, Stockholm 1980.

Hillerborg, A. (1980b) Analysis of fracture by means of the fictitious crack model, particularly for fibre reinforcedconcrete. International Journal of Cement Composites, November, 177184.

Hillerborg, A. (1981) The application of fracture mechanics to concrete. Contemporary European ConcreteResearch, Stockholm June 911.

Hillerborg, A. (1982a) The advanced strip methoda simple design tool. Magazine of Concrete Research, Vol 34, No121, December 1982, pp 175181.

Hillerborg, A. (1982b) The influence of the tensile toughness of concrete on the behaviour of reinforced concretestructures. The Ninth International Congress of the FIP, Stockholm June 610.

Hillerborg, A. (1983a) Theoretical analysis of the double torsion test. Cement and Concrete Research, Vol 13, 6980.Hillerborg, A. (1983b) Analysis of one single crack. Fracture mechanics of Concrete. Editor F.H.Wittmann. Elsevier.

223 250.Hillerborg, A. (1983c) Examples of practical results achieved by means of the fictitious crack model. William Prager

Symposium on Machanics of Geomaterials: Rocks, Concrete, Soils. Northwestem University, September1115.

Hillerborg, A. (1983d) Concrete fracture energy tests performed by 9 laboratories according to a draft RILEMRecommendation. Report to RILEM TC 50-FMC. Lund Institute of Technology, Report TVBM-3015.

Hillerborg, A. (1983e) The fracture energy GF as a material property and its significance in structuralengineering. An outline of an introductory chapter of a report from RILEM TC 50-FMC.

Hillerborg, A. (1984a) Additional concrete fracture energy tests performed by 6 laboratories according to a draftRILEM Recommendation. Report to RILEM TC 50-FMC. Lund Institute of Technology, Report TVBM-3017.

Fig. 9. Theoretical strength of unreinforced concrete pipes, Gustafsson & Hillerborg (1984).


Hillerborg, A. (1984b) Numerical methods to simulate softening and fracture of concrete. Fracture Mechanics ofConcrete: Structural Application and numerical calculation. (Eds. G.Sih and A.DiTommaso). MartinusNijhoff. 141170.

Hillerborg, A. (1985a) Predictions of nonlinear fracture process zone in concrete. J. of Engineering Mechanics,January. Discussion of paper by Wecharatana and Shah, October 1983.

Hillerborg, A. (1985b) Influence of beam size on concrete fracture energy determined according to a draft RILEMRecommendation. Report to RILEM TC 50-FMC. Lund Institute of Technology, Report TVBM-3021.

Hillerborg, A. (1985c) The theoretical basis of a method to determine the fracture energy GF of concrete. Materials andStructures. No 106, 291296.

Hillerborg, A. (1985d) Results of three comparative test series for determining the fracture energy GF of concrete.Materials and Structures, No 107, 407413.

Hillerborg, A. (1985e) Determination and significance of fracture toughness of steel fibre concrete. Steel fiberconcrete, US-Sweden joint seminar, Stockholm June 35.

Hillerborg, A. (1985f) A comparison between the size effect law and the fictitious crack model. A Festschrift for theseventieth birthday of professor Sandro Dei Poli, Milano.

Hillerborg, A. (1986a) Dimensionless presentation and sensitivity analysis in fracture mechanics. Fracture Toughnessand Fracture Energy of Concrete, (ed. F.H.Wittmann), Elsevier, Amsterdam, pp 413421.

Hillerborg, A. (1986b) Fracture aspects of concrete. The 6th European Conference on Fracture, June 1520. Hillerborg, A. (1988a) Application of fracture mechanics to concrete. Summary of a series of lectures 1988. Lund

Institute of Technology, Report TVBM-3030.Hillerborg, A. (1988b) Fracture mechanics and the concrete codes. Fracture mechanics: Application to Concrete,

SP-118, American Concrete Institute, Dec 1989, pp 157170.

Fig. 10. Displacements and stresses in a cohesive zone where unstable conditions arize. From Hillerborg (1989c).


Hillerborg, A. (1988c) Rotational capacity of reinforced concrete beams, Nordic Concrete Research (Oslo) No 7,pp 121134.

Hillerborg, A. (1989a) Compression stress-strain curve for design of reinforced concrete beams. Fracture Mechanics:Application to Concrete, SP-118, American Concrete Institute, Dec 1989, pp 281294.

Fig. 11. Theoretical stress-deformation curves compared to correct curves (dashed lines) for different values ofrotational stiffness k, compare Figure 10. From Hillerborg (1989c).


Hillerborg, A. (1989b) Mixed mode fracture In concrete. Seventh international Conference on Fracture, Houston,Texas, March 2024, 1989.

Hillerborg, A. (1989c) Stability problems in fracture mechanics testing. Fracture of concrete and rock. RecentDevelopments (eds. S.P.Shah, S.E.Swartz and B.Barr). Elsevier, Amsterdam, pp 369 378.

Hillerborg, A. (1989d) Existing methods to determine and evaluate fracture toughness of aggregative materialsRILEM recommendation on concrete. Fracture Toughness and Fracture EnergyTest Methods for Concreteand Rock, (eds. H.Mihashi, H.Takahashi & F.H.Wittmann), Balkema, Rotterdam, pp 145151.

Hillerborg, A. (1990) Fracture mechanics concepts applied to moment capacity and rotational capacity of reinforcedconcrete beams. Engineering Fracture Mechanics, Vol 35, No 1/2/3. pp 233240.

Hillerborg, A. Moder, M. and Petersson, P-E. (1976) Analysis of crack formation and crack growth in concrete bymeans of fracture mechanics and finite elements. Cement and concrete research, Vol 6, 773782.

Hillerborg, A. Petersson, P-E. (1981) Fracture mechanical calculations, test methods and results for concrete and similarmaterials. Fifth international conference on fracture, Cannes March 29 April 3.

Horvath, R. and Persson, T. (1984) The influence of the size of the specimen on the fracture energy of concrete.Lund Institute of Technology. Report TVBM-5005.

Kaplan, F.M. (1961) Crack propagation and the fracture of concrete. J. American Concrete Institute, 58, 591610.Mindess, S. (1983a) The application of fracture mechanics to cement and concrete: A historical review, in Fracture

Mechanics of Concrete (ed. F.H.Wittman), Elsevier, Amsterdam, 130. Mindess, S. (1983b) The cracking and fracture of concrete: an annotated bibliography 19281981, in Fracture Mechanics

of Concrete, (ed. F.H. Wittman), Elsevier, Amsterdam, 539680.Moder, M. (1979a) Brottmekaniska analysmetoder fr betong (Fracture mechanics methods for concrete). Nordisk

Betong (Stockholm), Nr 179, pp 2429.Moder, M. (1979b) A fracture mechanics approach to failure analyses of concrete materials. Doctor thesis. Lund

Institute of Technology. Report TVBM-1001. 102+44 pp.Petersson, P-E. (1979) Betongs brottmekaniska egenskaper (Fracture mechanical properties of concrete). Nordisk

Betong (Stockholm) Nr 579, pp 3138.Petersson, P-E. (1980a) Fracture energy of concrete: Method of determination. Cement and Concrete Research, vol

10, 1980, pp 78 89.Petersson, P-E. (1980b) Fracture energy of concrete: Practical performance and experimental results. Cement and

Concrete Research, vol 10, 1980, pp 91101.Petersson, P-E. (1980c) Fracture mechanical calculations and tests for Fibre reinforced cementitious material.

Advances in Cementmatrix Composites. Materials Research Society, Annual meeting, Boston November 1718,1980, Proceeding, Symposium L, pp 95106.

Petersson, P-E. (1981) Crack growth and development of fracture zones in plain concrete and similar materials.Doctor Thesis. Lund Institute of Technology. Report TVBM-1006. 174+10 pp.

Petersson, P-E. (1982a) Determination of the fracture energy of mortar and concrete by means of three-pointbend tests on notched beams. Proposed RILEM Recommendation, 29th January 1982.

Petersson, P-E. (1982b) Comments on the method of determining the fracture energy of concrete by means ofthree-point bend tests on notched beams. Lund Institute of Technology. Report TVBM3011.

Petersson, P-E. and Gustafsson, P.J. (1980) A model for calculation of crack growth in concrete-like materials.Numerical Methods in Fracture Mechanics. (Proceedings of the Second International Conference held atUniversity College, Swansea, July 1980) Pineridge press, Swansea, pp 707719.

Petersson, P-E. and Moder, M. (1976) Brottmekanisk modell fr berkning av sprickutbredning i betong. (Afracture mechanics model for the calculation of crack development In concrete.) Lund Institute of Technology,Division of Building Materials, Report No 70, 47 pp.

RILEM Draft Recommendation (1985) Determination of the fracture energy of mortar and concrete by means of three-point bend tests on notched beams. Materials and Structures, vol 18, No 106, pp 285290.

Wittman, F.H. editor (1983) Fracture mechanics of concrete. Elsevier, Amsterdam, 8+680, (ISBN 0444421998).


Wittman, F.H. editor (1986) Fracture toughness and fracture energy of concrete. Elsevier, Amsterdam, 15+699,(ISBN 044442733 3).

Wood, R.H. (1968) The reinforcement of slabs in accordance with a predetermined field of moments. Concrete. vol 2,No 2. February 1968. pp 6976.

Wood, R.H and Armer, G.S.T. (1968) The theory of the strip method for design of slabs. Proceedings of theInstitution of Civil Engineers. Vol. 41, No. 10. October 1968. pp 285311.

Zhou, F. (1988) Some aspects of tensile fracture behaviour and structural response of cementitious materials.Lund Institute of Technology. Report TVBM-1008. 76 pp.


PART ONE

BEHAVIOUR OF CONCRETE

2MODE I BEHAVIOUR OF CONCRETE: INFLUENCE OFTHE ROTATIONAL STIFFNESS OUTSIDE THE CRACK-

ZONEJ.G.M. van MIER

Delft University of Technology, Department of Civil Engineering, Stevin Laboratory,Delft, The Netherlands

ABSTRACTIn the paper, the influence of the rotational stiffness of the specimen outside the crack-zone in

an uniaxial tensile test is discussed. Both the allowable rotations of the specimens loadingplatens as well as the flexural stiffness of the specimen itself determine the shape of the descendingbranch in a displacement controlled experiment. An LEFM based model is proposed, and it isshown that hardened cement paste fulfils the assumptions made in the model. In contrast,mortar and concrete show different behaviour, which may be explained by considering fracturein tension as a growth process in three dimensions.

INTRODUCTION

Since a number of years, displacement controlled uniaxial tensile tests are carried out for determining thefracture mechanics parameters for concrete and other cement-based materials under mode I loading. Thisspecific experiment was recomended with the introduction of the fictitious crack model [4]. Since then ithas become clear that some of the assumptions of this model are not fulfilled in an uniaxial tensile test,more specifically the development of a uniform process zone [14].

The rotational stiffness of the specimen outside the crack-zone determines the shape of the descendingbranch in an uniaxial tensile test. Testing a specimen between rotating end-platens will result in a gradualdescending branch as shown in [8] (see Fig. 1), whereas between fixed end-platens a typical bump arises inthe softening branch (see Fig. 2). The bump depends largely on stress redistributions during crack growthwithin the entire machine-specimen system. The stiffness of the specimen is crucial in this respect: thelength of a specimen [5], but also its shape [15] will determine if possible stress redistributions will occurwithin the system.

Utilising a simplified LEFM based analysis, the shape of the descending branch can be calculated fordifferent boundary conditions. The fracture process in hardened cement paste seems to be in agreement withsome of the peculiarities predicted with such an LEFM approach. Yet for concrete and mortar differentresponse is measured, which can be explained by considering the fracture process as a growth process inthree dimensions.

SIMPLIFIED LEFM ANALYSIS

In [14] a physical model was presented for explaining the fracture process in concrete. Let us assume thatthe assumptions made are valid, and that at peak (maximum load), a critical flaw has developed in thespecimen. The growth of this flaw results in a descending branch and can only be studied experimentally ina stable controlled testing machine (provided that the elastic energy release during crack growth is limitedand that no snap-backs occur). In Fig. 3, the crack growth beyond peak is shown using reflection photo-elasticity. In this particular experiment, the crack nucleated at the left notch of a Double-Edge-Notched(DEN) tensile specimen, and propagated gradually towards the other side while the load-displacementdiagram (displacement measured in the centre of the specimen, with a gauge length of 65 mm) described adescending branch. Note that In a DEN tensile specimen the crack will nucleate always from one of the

Fig. 1 Uniaxial tensile test between rotating end platens, after [8]. The numbers between brackets along the descendingbranch are the optically measured crack lengths at the specimens surface.

Fig.2 Results of uniaxial tensile tests between non-rotating end-platens, Single-edge-notched specimens, after [15].

MODE I BEHAVIOUR OF CONCRETE 17

notches as a direct result of the heterogeneity of the material under consideration. See also Fig. 1, where theoptically measured crack lengths at different stages along the descending branch are indicated.

Now assume that, due to pre-peak microcracking, at peak stress a critical flaw of size a0/W has developedin a Single-Edge-Notched (SEN) tensile specimen as indicated in Fig. 4. It is assumed that the crack frontremains straight during crack propagation. The stress intensity factor KI is equal to

(1)

where is the nominal externally applied stress, and Y is a geometrical factor, which depends on thespecimen geometry and boundary conditions. When a0 reaches the critical size, KI reaches its critical valueKI, crit. As mentioned before,it is assumed that a0 has reached the critical size at peak stress p:

(2)

When the macro-crack grows, KI=KI, crit=constant. Thus when the crack has extended to a length a1, thestress 1 that can be carried by the cracked specimen is equal to

Fig. 3a Crack growth in an uniaxial tensile test on a DEN specimen between fixed end-platens, detected with reflectionphoto-elasticity: load-displacement diagram, after [15].

Fig. 3b Crack growth at =13.5 m, after [15].


(3)

Thus the ratio 1/p can be calculated following

(4)

Eq. (4) determines the shape of the descending branch as function of the relative crack length a1/W. Whenthe geometrical factor Y is known, the exact shape of the softening branch can be calculated. The presentmodel is in fact similar to an R-curve type model, in which a gradual increase of KI is allowed for smallcrack lengths (in the current model when a1

Fixed end-platens

A similar result can be obtained for a SEN tensile specimen loaded between non-rotating end-platens(edge=0). A solution for the geometrical factor of the problem was recently published by Marchand et al.[9]. The solution is based on Tadas geometrical factors for bending and normal load respectively, and theexpression for the geometrical factor is:

(6)

where F1 and F2 are Tadas functions for normal load (eq.5) and pure bending respectively, and =a/W. TheCij are the dimensionless crack compliances that contain all information regarding the M, N, , relations.The details of the analysis can be found in [9]. Inherent to the solution is the development of aclosingbending moment as soon as the crack starts to grow. This phenomenon is shown in Fig. 6, where aplot of dimensionless stress 1/p versus relative crack length a1/W is given. The initial flaw was equal to a0/W=0.05. After an initial steep stress drop for crack lengths a1/W1. For longer cracks, the decay length isproportional to the specimen width, and a valid solution is obtained only when L/W is sufficiently large.

The descending branch has some interesting characteristics. The plateau level increases with decreasingL/W ratio, which demonstrates the influence of the flexural stiffness of the specimen outside the crack-zone. Furthermore in a truly LEFM material, the external stress has to be increased near relative cracklengths of approximately 0.80.9, which implies that the load-excentricity and the associated closingbending moment have become so large that the axial load must be increased to facilitate further crackgrowth.

DISCUSSION AND COMPARISON WITH EXPERIMENTAL RESULTS

In tensile tests on concrete or mortar specimens between fixed end-platens, an increase of applied stress hasnever been observed in the descending branch, see for example [5], [15] (see Fig. 2), and [17]. For DEN

Fig. 5 Calculated shape of the descending branch for a SEN tensile specimen loaded between free rotating end-platens.


tensile specimens also a plateau is observed which depends on the actual specimen dimensions and notchdepths. However, the length of the plateau is smaller in terms of deformations ([15], see Fig.3).

In contrast to this, for hardened cement paste (hcp), the external stress must be increased during someinterval of the descending branch as shown in Fig. 7.

This may indicate that LEFM-concepts are applicable to hcp, and that the model assumptions in thepreceding paragraph are correct. Note however that experiments on hcp specimens are extremely difficult

Fig. 6 Calculated shape of the descending branch for a SEN tensile specimen loaded between fixed end-platens.

Fig. 7 a P- diagrams for single-edge-notched hcp specimens loaded in tension between fixed end-platens. The numbersbetween brackets indicate the number of days of under water curing. Between removal from the water basin and testingat an age of 3540 days, the specimens were kept at 50 % RH.


mainly because of the sensitivity to shrinkage cracking of the material. The results of Fig. 7a were obtainedon specimens that were cured under water for different periods, viz. 14 and 28 days as indicated. Shrinkagecracking was extreme in the specimen which was cured for 14 days only. As shown in Fig. 7b, the crackplane is in this case largely determined by the presence of shrinkage cracks which will extend primarilyfrom the specimens surface into the interior of the specimen [19].

It must be concluded that concrete and mortar do not follow the assumptions of the model, at least at thissize level. Yet the descending branches of both mortar and concrete specimens have the typical plateau in testswith fixed end-platens, and the different behaviour can be explained by considering the fracture of aspecimen as a three dimensional growth process.

In Fig. 8, the hypothesized process is shown schematically. Due to non-uniform drying, the surfaces of aspecimen will always be subjected to tensile eigen-stresses. Therefore it is quite likely that crack nucleationis from the outer parts of the specimen towards the centre of it. The stress redistributions in a SEN specimenloaded between fixed end-platens are limited in the case of concrete and mortar as demonstrated by the factthat no stress increase is measured during some interval of the descending branch. This can occur when thecrack front is not straight, but rather curved as indicated in Fig. 8 (see cross-section A-A). The crack hasextended further in parts of the specimen near the surfaces as compared to the specimens centre. Becausethe core is still intact, the developing load-excentricity during macro-crack propagation is considerable lessas compared to a growing crack in a truly LEFM material with a straight crack front. Consequently, thebending moment is relatively small and no increase of external stress is needed for facilitating further crackgrowth. Note that curved crack fronts are observed also in three-point bend tests by means of impregnationtechniques (see for example [1], [12]). When the crack front reaches the other side of the specimen (whichwill first occur near the surfaces), the specimen will unload rapidly as soon as the last part of the cross-section cracks. However, a small part of the specimens core may still be intact as will be discussed next.

Several reasons can be given for the development of an intact core in a specimen. First of all, the crackbranches which develop from the surfaces do not necessarily grow in the same cross-sectional plane, andmay avoid each other, [15]. In this way, flexural ligaments develop internally. Flexural failure of theligaments may be responsible for the long tail that is normally observed in the load-deformation curve intension (see Fig. 8, inset (a.)). This follows also from theoretical considerations, [2].

The second reason may be that compressive eigen-stresses which develop in the specimens core due tonon-uniform drying-out, will prevent the formation of a straight crack front (see Fig. 8, inset (b.)). In this

Fig. 7b Fracture surfaces for the two specimens of Fig. 7a.


case, the long tail in the load-deformation diagram is explained by the fact that the increased tensileresistance of the core must be overcome.

The third reason may be that the final separation of the crack planes is prevented by friction due to thepull-out of aggregates. This might also explain the long tail of the load-deformation curve (see Fig. 8, inset(c.)).

Currently It is not clear which of the mechanisms is responsible for the observed long tail in thedescending branch. Most likely a combination of the three factors occurs. The end of the plateau resultsnormally in a sudden unloading, which might be explained from the second mechanism. The explosiveunloading is facilitated by the presence of tensile eigen-stresses near the specimens edges.

One last remark should be made. The non-uniform opening and the associated plateau in the descendingbranch can be simulated numerically with a computational softening diagram, [14], [11], [18]. Thecomputational softening diagram is a simplified two-dimensional representation of the three dimensionalfracture process for plane-stress analysis. Crucial is that the first part of the softening curve is extremelysteep, and corresponds to almost purely brittle behaviour (see Fig. 9). If the first steep part of the bi-lineardiagram is exceeding the upper bound for non-uniform opening as indicated in Fig. 9, the non-uniformity ofthe fracture process cannot be simulated in a numerical analysis [14]. This is in agreement with the abovereasoning. Yet, if the shallow tail of the computational softening diagram is dependent of the specimendimensions (thickness) and possibly also of non-uniform drying-out, the size dependency of the Gf-approach might be explained [3]. Note that the largest contribution to the fracture energy Gf is due to theshallow second branch in the computational softening diagram.

CONCLUSIONS

The influence of the boundary conditions on the load-deformation response of Single-Edge-Notched tensilespecimens can be explained from a simple analysis based on linear fracture mechanics. Assumption in themodel is that a critical notch has developed in the specimen before the peak load is reached. Subsequently

Fig. 8 Three-dimensional fracture mechanism for concrete and mortar specimens loaded in tension.


this critical flaw grows under constant stress intensity factor. When a specimen is loaded between freerotating end-platens, a smooth descending branch will occur, whereas the analysis of a specimen loadedbetween fixed end-platens results in a typical plateau in the descending branch. The plateau is associatedwith severe stress redistributions in the specimen-machine system, which depend on the specimensgeometry (shape and size) and boundary conditions, and the machine stiffness. A material fulfils theassumptions made in the analysis if the external load must be increased during some interval of thedescending branch when a SEN specimen is loaded under fixed end displacement. Hardened cement pastespecimens fulfil this requirement, whereas concrete and mortar specimens give different results. Note thatthe model gives insight in the post-peak behaviour only. The maximum load depends on the micro-crackprocesses that take place before the maximum is reached [14].

The behaviour of concrete and mortar specimens can be explained by considering the fracture of aspecimen as a three-dimensional growth process. Primarily, the critical flaw will propagate with a curvedcrack front as observed also in three point bend tests [1], [12]: the crack has extended further along the edges,and is delayed in the centre of a specimen. When the macro-crack grows towards the free (unnotched) edgeof the specimen, sudden unloading will occur, which marks the end of the plateau. The residual carryingcapacity of a specimen, i.e. the long tail of the load-deformation diagram, can be explained from threedifferent (possibly interacting) mechanisms: the development of internal flexural ligaments, thecompressive eigen-stresses in the specimens core due to non-uniform drying-out and frictional pull-out ofaggregates. Future research should clarify the mechanism.

REFERENCES

1 [] Bascoul, A., Kharchi, F. and Maso, J.C., Concerning the Measurement of the Fracture Energy of a Micro-ConcreteAccording to the Crack Growth in a Three Points Bending Test on Notched Beams, in Fracture of Concrete andRock, Proceedings of the SEM-RILEM International Conference, Houston, June 1719, 1987, (S.P.Shah andS.E.Swartz, eds.), Springer Verlag 1989, pp.396408.

Fig. 9 Computational bi-linear softening diagrams for plane-stress analysis, after [16].


2 [] Bazant, Z.P., Snapback Instability at Crack Ligament Tearing and its implications for FractureMicromechanics, Cement & Concrete Research, 17(6) (1987), 951967.

3 [] Brameshuber, W., Bruchmechanische Eigenschaften von jungem Beton, Schriftenreihe des Instituts furMassivbau und Baustofftechnologie, Karlsruhe, 5 (1988), pp.233.

4 [] Hillerborg, A., Petersson, P.E. and Modeer, M., Analysis of Crack Formation and Crack Growth in Concreteby means of Fracture Mechanics and Finite Elements, Cement & Concrete Research, 6(6) (1976), 773782.

5 [] Hordijk, D.A., Reinhardt, H.W. and Cornelissen, H.A.W., Fracture Mechanics Parameters from UniaxialTensile Tests as influenced by Specimen Length, in Preprints SEM-RILEM Intern. Conf. on Fracture of Concreteand Rock, Houston, June 1719, 1987 (S.P.Shah and S.E.Swartz eds.), pp.138149.

6 [] Jenq, Y.S. and Shah, S.P., Two Parameter Fracture Model for Concrete , Journal of Eng. Mech., ASCE, 111(10)(1985), pp.12271241

7 [] Karihaloo, B.L., Do Plain and Fibre-Reinforced Concretes have an R-curve Behaviour?, in Fracture ofConcrete and Rock, Proceedings of the SEM-RILEM International Conference, Houston, June 1719, 1987(S.P.Shah and S.E.Swartz eds.), Springer Verlag, 1989, pp.96105.

8 [] Labuz, J.F., Shah, S.P. and Dowding, C.H., Experimental Analysis of Crack Propagation in Granite, Int.J.RockMech. Min. Sci. & Geomech. Abstr., 22(2), (1985), 8598.

9 [] Marchand, N., Parks, D.M., and Pelloux, R.M., KI-Solutions for Single Edge Notch Specimens under FixedEnd Displacements, Int J. Fracture, 31 (1986), 5365.

10 [] Peterson, P.E., Crack Growth and Development of Fracture Zones in Plain Concrete and Similar Materials,Report no. TVBM 1006, Lund Institute of Technology, Sweden, (1981), pp.174.

11 [] Rots, J.G., Computational Modelling of Concrete Fracture, PhD Dissertation, Delft University of Technology(1988).

12 [] Swartz, S.E. and Refai, T., Cracked Surface Revealed by Dye and its Utility in Determining FractureParameters, in Preprints IntI. Workshop on Fracture Toughness and Fracture EnergyTest Methods forConcrete and Rock, Tohoku University, Sendai, Japan, Oct. 1214, 1988, pp.393405.

13 [] Tada, H., Paris, P.C. and Irwin, G.R., The Stress Analysis of Cracks Handbook, Del Research Corporation,Hellertown, USA (1973).

14 [] Van Mier, J.G.M., Fracture of Concrete under Complex Stress, HERON, 31 (3) (1986), pp.190.15 [] Van Mier, J.G.M. and Nooru-Mohamed, M.B., Geometrical and Structural Aspects of Concrete Fracture,

International Conference on Fracture and Damage of Concrete and Rock, Vienna, July 46, 1988 To appear inEng. Fract Mech. (1989).

16 [] Van Mier, J.G.M., Mixed-Mode Fracture of Concrete under Different Boundary Conditions, in ProceedingsSEM Spring Conference on Experimental Mechanics, Cambridge (Ma), May 28-June 1 (1989), SEM, Bethel,pp.5158.

17 [] Willam, K., Hurlbut, B. and Sture, S. (1985), Experimental and Constitutive Aspects of Concrete Failure, inProceedings US-Japan Seminar on Finite Element Analysis of Reinforced Concrete Structures, May 2124,(1985), ASCE Special Publ.(Ch.Meyer and H.Okamura eds.), pp.226254.

18 [] Wittmann, F.H., Roelfstra, P.E., Mihashi, H., Huang, Y.-Y., Zhang, X.-H. and Nomura, N., Influence of Ageof Loading, Water-Cement Ratio and Rate of Loading on Fracture Energy of Concrete, Materials and Structures,RILEM, 20 (1987), 103110.

19 [] Wittmann, F.H. and Roelfstra, P.E., Constitutive Relations for Transient Conditions, In Proceedings IABSEColloquium on Computational Mechanics of Concrete StructuresAdvances and Applications, Delft, 54 (1987),pp.239259.


3EXPERIMENTAL ANALYSIS OF MIXED MODE I AND II

BEHAVIOUR OF CONCRETEJ.G.M. van MIER, M.B.NOORU-MOHAMED, E.SCHLANGEN

Delft University of Technology, Department of Civil Engineering, Stevin Laboratory,Delft, The Netherlands

ABSTRACTResults are presented of combined tensile and shear tests on concrete, mortar and Lytag

lightweight concrete plates in the recently developed biaxial test rig of the Stevin Laboratory.The experimental procedure is explained in detail, and some preliminary results are presented.The behaviour of pre-cracked concrete specimens subjected to lateral (tensile or compressive)shear is, at small crack openings (smaller than 250 m), governed by diagonal cracking (tensileshear) or strut splitting (compressive shear) due to the rotation of the principal stresses. Slidingfailure was observed only for larger crack openings. The results suggest that assumptions madein conventional aggregate interlock theories are not valid for small crack openings. Rather, theassumptions made in recently developed so-called smeared rotating crack models are confirmed.

INTRODUCTION

Mixed mode I and II fracture of concrete and other brittle disordered materials is currently studiedextensively worldwide. The efforts can be subdivided between two basic approaches: the conventionalfracture mechanics approach, and an approach in which the constitutive relations of a fracture zone arestudied. In the first approach the primary goal is to study the growth of cracks under combined tensile andshear loading. Of main interest are the derivation of a crack growth mechanism and the evaluation of themixed mode fracture toughness of materials (e.g. [1], [2]).

Because concrete and mortar do not obey the assumptions of linear fracture mechanics, it is assumed thata fracture process zone preceeds the development of a macroscopic traction free crack. In the secondapproach it is tried to isolate a fracture process zone and to study its behaviour under combined tensile andshear loadings (e.g. [3], [7], [10]). As a result constitutive equations should be derived (normal and shearstress versus crack opening and crack sliding displacement, see pp. 9496 in [6]) for implementation infinite element codes. This last approach is in fact an extension of the fictitious crack model [4] and relatedmodels for mode I loading. The input parameters for the fictitious crack models must be derived by loadingsmall specimens in a very stiff loading apparatus. However, recently researchers became aware of the factthat besides the stiffness of the machine, the stiffness of the specimen itself has a considerable influence onthe measured response in tension.

In [13], It was argued that the fracture in uniaxial tension can also be regarded as a growth process inthree dimensions. Macroscopic cracks grow through the specimens cross-section with a curved crack front.The surfaces of a specimen are cracked first, and the propagation of the surface cracks complies toassumptions from linear fracture mechanics. Yet because different mechanisms take part in the fracturing ofthe central core of the specimen, a deviation from linear fracture mechanics occurs. These mechanismsmust be borne in mind when fracture processes under mixed mode I and II conditions are studied.

Currently, experiments are carried out in the Stevin Laboratory, where the fracture process of concretespecimens subjected to combined tension and shear is studied. The experiments should lead to thederivation of a mixed mode crack growth criterion for brittle disordered materials. In this paper, theexperimental method and some preliminary results are discussed.

MIXED MODE I AND II EXPERIMENTS

Description of biaxial test rig

The experiments were carried out in the biaxial test rig of the Stevin Laboratory [10]. Basically, theapparatus consists of two independent rigid frames that are fixed in an overall frame by means of platesprings. An exploded view of the apparatus is shown in Fig. 1. One of the two rigid frames consists of twocoupled frames, and the second frame can slide in between the coupled frames. The coupled frames canmove in the horizontal direction and are fixed to the overall frame in the vertical direction. The middleframe can move vertically, and is fixed in the horizontal direction. To both the middle frame and thecoupled outer frames a load-cell and a hydraulic actuator, with a capacity of 100 kN in tension orcompression, are connected as indicated. The maximum load is however restricted to 50 kN. Squareconcrete plates of size 20020050 mm are loaded in the apparatus.

The loading procedure is clarified in Fig. 2. In uniaxial tension, a concrete plate is glued between theupper side of the middle frame and the lower side of the coupled outer frames (Fig. 2a). When the glue hasset, the specimen is loaded by moving the middle frame upward. Because the coupled outer frames arefixed in the vertical direction, a tensile stress develops in the concrete specimen, which will eventuallyfracture the plate (Fig. 2b). The concrete plates are double or single-edge-notched at half height (asindicated) in order to generate crack growth at a known location. This facilitates deformation controlledtesting. The same procedure is used for applying a shear load to a pre-cracked specimen, as shown in Fig. 3.In these biaxial experiments not only the upper and loweredges of a specimen are glued to the frames, butalso parts of the left and right edges. After pre-cracking a specimen as described before, a lateral in-planeshear load can be applied over the cracked area. This can be done either in displacement-control (when thedifferential shear deformation between the two specimen halves is used as control variable), or in load-control.

Specimens and materials

Specimens were double-edge-notched square concrete plates of size 200*200 *50 mm. The notches were 25mm deep saw-cuts. Up to now, experiments are carried out using three materials: a 2 mm mortar, a 16 mmconcrete and Lytag high strength lightweight concrete with a maximum aggregate size of 12 mm. Thedetails of the various mixes as well as the average compressive strength and splitting tensile strength of thethree materials are given in [8] and [12]. The plates were cast in a vertical position in a battery mould(containing six plates of size 300*300*50 mm) . Two days after casting, the plates were demoulded and

EXPERIMENTAL ANALYSIS OF MIXED MODE 27

placed in a fresh water basin. After 14 days, the specimens were sawn to the required size and were placedin a fresh water basin again. At 28 days the specimens were removed from the basin, and were allowed todry in the laboratory (1718C and 50% R.H.). The age at loading varied a little for the different test-series,but was always larger than 35 days. The differences in treatment are reported in the previous publications [8],[12].

Load-paths investigated

Three different load-paths were studied so far: (1) shear at constant axial crack opening, (2) shear at zeronormal confinement and (3) shear at small constant compressive normal confinement. Always a specimenwas pre-cracked in tension (in displacement-control in the vertical direction of the test rig) to a prescribedaxial crack opening. Displacement control in the vertical direction was done by using the averagedeformation (meas. length 65 mm) measured with either two [12] or four [8] LVDTs as a feedback signal inthe servo-controlled system. Subsequently the axial load was decreased to zero or to the prescribed normalconfinement (-1000 N, corresponding to an average normal stress of -0.13 N/mm2). The axial opening wasin most experiments larger than 50 m, at which stage the crack process was uniform again (see [12]). Onlya small number of tests was carried out at smaller crack openings. In load-paths (1) and (2) the shear loadwas either compressive or tensile, whereas in load-path (3) compressive shear was applied. In the initialexperiments [12], the shear was applied in load-control, whereas in the more recent tests [8], shear loadswere applied in displacement-control via two LVDTs measuring the relative displacements of the twospecimen halves (that are separated partially by the axial crack).

The loading rates were as follows: 1.652.0 m/min in the vertical (tensile) direction and 1.60 m/min(when a test was carried out in displacement-control) or 20 N/s (in load-controlled testing) in the horizontal(shear) direction. When the prescribed axial crack opening was reached, the specimen was unloaded in the

Fig. 1 Exploded view of the biaxial test rig


vertical (tensile) direction by an immediate change of polarity of the axial loading. Therefore, the influenceof creep-effects can be neglected.

EXPERIMENTAL RESULTS

Depending on the actual boundary conditions (load-paths) in the experiments, rather distinct responses weremeasured. The fracturing of the specimens was recorded in detail, in some cases even using reflection photo-elasticity [11]. For all details, the reader is refered to [8], [11] and [12]. Here only some of the results willbe discussed.

Fig. 2 Sectional view of the biaxial apparatus: at the beginning of an experiment (a), and after cracking (b). Note thebending of the plate springs of the middle frame in Fig. b as a result of the upward movement of this frame with respectto the coupled frames.

Fig. 3 Double-edge-notched square plate for the mixed-mode tests.


Tensile shear at constant crack-opening (load-path 1)

A number of tests was carried out using reflection photo-elasticity as a crack detection technique. A verythin coating (0.25 mm) of photo-elastic material was glued to the concrete specimen. In Fig. 4 some of theresults are shown. After axial pre-cracking, in this case up to 11.6 and 23.4 m respectively, atensile shearload was applied in the horizontal direction. The axial P- diagram and load-path P- Ps are shown in Fig. 4aand b. The growth of macro-cracks in the specimen could be followed quite accurately by means of thephoto-elastic reflection technique. In Fig. 4c, the cracking at step 27 is shown. The corresponding point inthe P- diagram is indicated in Fig. 4a. At the moment of application of tensile shear, a diagonal cracknucleated from the left saw-cut, and propagated towardsthe upper edge of the specimen. This was observedalready during the first shear cycle at =11.2 m. Redistributions in the specimen could be followed closely,and at some moment even the closure of the axial crack was observed. In fact, the axial crack was shieldedby the propagating diagonal shear crack. Final failure of the specimen was through the development of a

Fig. 4 Tensile shear at constant crack opening: (a) P- diagram, (b) load-path P-Ps and (c) photo-elastic fringes at stage27.


second diagonal crack from the small crack branch near the lower right notch. For this boundary condition(viz. =const. during shear) the in-plane shear resistance of the axial crack was higher than the crackresistance of the intact specimen parts.

Tensile shear at zero axial load (load-path 2)

In addition to the constant crack opening experiments (load-path 1), several tests were carried out in whichtensile shear was applied to a specimen containing a partially developed axial crack, while maintaining azero load normal to the crack plane (viz. P=0=const. during shearing). In Fig. 5, the results are shown as P-and Ps- plots for a number of experiments with various axial crack openings.

A sudden decrease of shear capacity, or rather maximum applicable shear-load PS, was measured nearcrack openings of 250 m. Beyond this point, a more or less constant shear load (Ps=2 kN) was measured.When the post-mortem crack patterns were compared, it was found that failure was through sliding forcrack openings larger than 250 m, and that diagonal cracks developed when the initial crack width atshearing was smaller than 250 m (see Fig. 6). A parallel might be drawn with shear in beams, when afterstress redistributions a new set of diagonal cracks grows through a previously developed crack pattern.

Compressive shear with normal confinement (load-path 3)

Examples of load-path 3 experiments are shown in Fig. 7. In Fig. 7a, the P- diagrams for three tests withdifferent axial crack opening are given, in Fig. 7b the corresponding Ps-s diagrams. The replicability of theexperiments was good [8], so only single results are compared. Basically the fracture mechanism of theexperiments is of interest, and not (yet) the quantitative evaluation of the results. With increasing crack

Fig. 5 P- and Ps- diagrams for tensile shear tests at constant zero axial load.


opening at shearing, a decreasing shear stiffness was measured in the experiments. Also, the tests at largercrack opening showed a lower maximum shear-load. The typical transition as was observed in the tensileshear tests at zero axial load was not found. The failure pattern was completely different in the compressiveshear tests. Failure was not through the development of diagonal shear-cracks, but rather by tensilesplitting in the direction of the compressive strut that would develop in a specimen. This is shown in Fig. 8.Crack A-B-C developed due to the axial tensile loading. Under subsequent shear, the diagonal splittingcrack C-D developed parallel to the compressive strut direction. The residual carrying capacity could beassociated with the length of crack B-C, which was more or less perpendicular to the compressive loadingdirection. Clearly, the shear strength of the crack, in combination with the axial confinement was still solarge that no shear failure could occur. This was observed only In the tests without axial confinement.

Shear stiffness reduction

In Fig. 9, the initial shear stiffness Ps/s, which is defined as the secant modulus between 10 % and 30 % ofthe maximum shear load Ps, is shown as function of the axial crack opening a* (which is the true crackopening at the beginning of shearing, and is smaller than the unloading deformation in the axial direction,see Fig. 7a). The results of 2 mm mortar, 16 mm concrete and Lytag concrete are included in the samefigure. As expected, a decreasing shear stiffness was measured with increasing crack width. For crackopenings larger than 200 mm, there seemed to be a slight influence of the confining stress normal to thecrack plane.

In computational models, the decrease of shear stiffness of cracked concrete is expressed by means of ashear retention factor (see for example in [6], p.95). For normal-weight concrete with a maximumaggregate size of 16 mm, values of between 0.12 and 0.40 were calculated based on the new experimentalresults [8]. This is far in excess of values determined from conventional aggregate interlock experiments

Fig. 6 Post-mortem crack patterns of the experiments of Fig. 5.


(e.g. [9]), viz. 0.04 -0.09 for the same crack opening rangeas tested (200400 m for normal-weightconcrete, see [8]). As argued in [6], the differences might be explained from neglecting the couplingbetween shear stiffness and shear displacements. Furthermore, an assumption in conventional aggregateinterlock theories is that the principal stress axes are fixed and that shear is transfered in the crack plane.Rotation of principal stresses occurred in the experiments reported in this paper (Figs. 47), and it isbelieved that the current tests are suitable for testing so-called smeared rotating crack concepts (see in [5],pp.138146). The assumptions made in this type of modelling are in agreement with the currentexperimental results.

CONCLUSIONS

Results are presented of mixed mode I and II tests of concrete. A shear-like load was applied to a specimenafter it was pre-cracked to a prescribed axial crack opening. The results indicate that the response of thespecimens upon shearing is governed by the development of a rotated principal stress state, especially forvery small axial crack openings. Due to this rotation of principal stresses, diagonal cracking (under tensileshear) or splitting of compressive struts (under application of compressive shear) was observed in the

Fig. 7 P- (a) and Ps-s (b) diagrams for three compressive shear tests with compressive confinement normal to thecrack plane. The tests were carried out at different values for the axial crack opening.

Fig. 8 Crack pattern for one of the experiments of Fig. 7 (=150 m crack opening at shearing).


experiments. Sliding in the crack plane has been found for very large crack openings only (larger than 250m). This suggests that aggregate interlock models are not applicable for small crack openings (smaller than250 m). The experimental results seem to be in agreement with assumptions made in so-called smearedrotating crack models [5].

REFERENCES

1 [] Arrea, M. and Ingraffea, A.R., Mixed Mode Crack Propagation in Mortar and Concrete, Report No. 8113,Dept. Struct.Eng., Cornell University, Ithaca, N.Y., (1982), pp. 143.

2 [] Bazant, Z.P. and Pfeiffer, P.A., Shear Fracture Tests of Concrete, Materials and Structures, RILEM, 19(110)(1986), 111121.

3 [] Hassanzadeh, M., Hillerborg, A. and Zhou Fan Ping, Tests of Material Properties in Mixed Mode I and II, inPreprints SEM-RILEM Intern. Conf. on Fracture of Concrete and Rock, Houston, June 1719, (1987), (S.P.Shahand S.E.Swartz eds.), pp. 353358.

4 [] Hillerborg, A., Peterson, P.E. and Modeer, M., Analysis of Crack Formation and Crack Growth in Concrete bymeans of Fracture Mechanics and Finite Elements, Cement and Concrete Research, 6(6) (1976), 773782.

5 [] Hillerborg, A. and Rots, J.G., Crack Concepts and Numerical Modelling, Chapter 5 in Fracture Mechanics ofConcrete StructuresFrom Theory to Applications, (L.Elfgren ed.), Chapman & Hall (1989), pp.128146.

6 [] Hordijk, D.A., Van Mier, J.G.M. and Reinhardt, H.W., Material Properties, Chapter 4 in Fracture Mechanics ofConcrete StructuresFrom Theory to Applications, (L.EIfgren ed.), Chapman & Hall (1989), pp. 67127.

7 [] Keuser, W., Kornverzahnung bei Zugbeanspruchung, Forschunngskolloquium DAfStb, Darmstadt (1986),pp. 1318.

8 [] Nooru-Mohamed, M.B. and Van Mier, J.G.M., Fracture of Concrete under Mixed-Mode Loading, in Fractureof Concrete and Rock-Recent Developments, (S.P.Shah, S.E.Swartz, B.Barr, eds.), Elsevier Science Publishers,London/New York, 1989, pp.458467.

9 [] Paulay, T. and Loeber, P.J., Shear Transfer by Aggregate Interlock, ACISpecial Publication 42, Shear inReinforced Concrete, (1974), Vol. I, pp. 116.

Fig. 9 Decrease of shear stiffness with increasing axial crack opening.


10 [] Reinhardt, H.W., Cornelissen, H.A.W., and Hordijk, D.A., Mixed Mode Fracture Tests of Concrete, inFracture of Concrete and Rock, Proceedings of the SEM-RILEM International Conference, Houston, June1719, 1987 (S.P.Shah & S.E.Swartz eds.), Springer Verlag (1989), pp.117130.

11 [] Van Mier, J.G.M., Fracture Study of Concrete Specimens Subjected to Combined Tensile and Shear Loading,in Proceedings GAMAC IntI. Conf. on Measurement and Testing in Civil Engineering, (J.F.Jullien ed.), Lyon-Villeurbanne, Sept. 1316 (1988), Vol. 1, pp. 337347.

12 [] Van Mier, J.G.M. and Nooru-Mohamed, M.B., Fracture of Concrete under Tensile and Shear-like Loadings, inPreprints IntI. Workshop on Fracture Toughness and Fracture EnergyTest Methods for Concrete and Rock,Tohoku University, Sendai, Japan (H.Mihashi ed.), Oct. 1214, (1988), pp. 433447.

13 [] Van Mier, J.G.M., Model Behaviour of Concrete: Influence of the Rotational Stiffness outside the Crack-Zone,in Analysis of Concrete Structures by Fracture Mechanics, Proceedings of the RILEM workshop dedicated toProf. A.Hillerborg, Abisko, Sweden, June 1989 (L.Elfgren ed.), Chapman & Hall, (1990).


4CONSIDERATIONS REGARDING FRACTURE ZONE

RESPONSE TO SIMULTANEOUS NORMAL AND SHEARDISPLACEMENT

M.HASSANZADEH Lund Institute of Technology, Sweden

AbstractA testing arrangement has been developed in order to determine the fracture process zone

properties, affected by simultaneously applied normal and shear displacement. The testingarrangement makes it possible to perform stable displacement-controlled mixed-mode testsaccording to any arbitrary displacement path. The paper presents the testing arrangement anddiscusses a few aspects of the determination of the fracture zone properties.

1Introduction

The present project is part of a major fracture-mechanics project which has been going on since 1974 . Theproject has resulted in the development of the Fictitious Crack Model (FCM), Hillerborg (1989-a), whichhas become the principal model regarding fracture-mechanics analyses at the Division of BuildingMaterials, Lund Institute of Technology. The goal of this particular project has been to determine materialproperties which are necessary for mixed mode I and II fracture-mechanics applications of FCM.

2Mixed mode according to the fictitious crack model

According to the FCM, three phases may be distinguished in mixed-mode crack propagation (see figure 1).The first one is the initiation phase characterized by the formation of the fracture process zone or theformation of a zone in which the material starts to soften. In homogeneous material, the condition forinitiation is fulfilled when the first principal tensile stress reaches the tensile strength of the material. At theinstant the fracture zone is formed, it also assumes the final orientation which is perpendicular to the firstprincipal tensile stress direction. The second phase is the continuous softening of the fracture zone, which ischaracterized by the gradual weakening of the material inside the fracture zone. The subsequent state ofstress within the fracture zone depends on alterations in the first principal stress direction in the subsequentstages. If the direction remains unchanged during the complete softening process, the state of stress will bemode I, i.e. only normal stresses will occur inside the fracture zone despite global mixed-mode loading.

However, if the direction of the first principal tensile stress is changed, the state of stress inside the fracturezone will be mixed-mode, i.e. both normal and shear stresses will occur within the zone. The third phase isthe formation of a real crack characterized by stress-free crack surfaces.

The fracture zone properties which are needed for incremental mixed-mode applications of the FCM canbe formulated as follows:

and wn are stress and displacement increments normal to the fracture plane. r and Ws are shearstress and shear displacement increments inside the fracture process zone. The diagonal terms in thestiffness matrix (K11 and K22) are normal and shear crack stiffnesses, and the off-diagonal terms (K12 andK21) are the coupling functions. The terms in the stiffness matrix are not constants but are functions of bothtotal normal and total shear displacements.

The FCM differs from the linear elastic mixed-mode models in two aspects. FCM utilizes the strengthcriterion for the initiation of crack propagation, while the linear elastic models utilize the stress intensitycriterion (Hillerborg 1989). Furthermore, the linear elastic models neglect the effects of the fracture processzone which, leads to a completely different stress field in front of a propagating real crack.

3Testing arrangement

Figure 2 shows test equipment which, together with a closed-loop testing machine, forms an arrangement formixedmode I and II tests. The details, performance and design of the equipment are discussed in references

Figure 1. Mixed-mode crack propagation according to the fictitious crack model.

CONSIDERATIONS REGARDING FRACTURE ZONE 37

[2, 3, 4, 6, 7]. Prior to the test, a prismatic notched concrete specimen is glued in between the beams. Afterhardening of the adhesive, the equipment is put in position in the closed-loop machine. The tests aredisplacement-controlled and are run in a semi-automatic manner, which means that the closedloop machinecontinuously runs a programmed normal dis placement scheme and that the shear displacement is manuallyimposed by rotating the crank indicated in figure 2. By means of continuous adjustment of the sheardisplacement, it is possible to perform mixed-mode tests according to any arbitrary displacement path, orrelation between normal and shear displacements. The mixed mode procedures are such that they fulfil thecriterion outlined by the FCM, i.e. normal displacement is first imposed so that a tensile fracture zone startsto form. Shortly after the normal stress starts to decrease, shear displacement is also imposed. In thesubsequent stages, the test is continued in such a way that the normal and shear displacements follow apredefined displacement path.

4Mixed mode tests

A predefined displacement path is needed to conduct the mixed-mode tests by means of the test arrangementpresented here. At present there are no established sets of displacement paths which represent real structuralbehaviour. Therefore, arbitrary displacement paths have been used so far. Two types of displacement pathhave been tested, a linear path and a parabolic path which are defined by the following formulae.

Figure 2. Test equipment.


Wn and Ws are normal and shear displacements within the fracture process zone.Low a and values describe situations where the normal displacement is somehow suppressed, for instance

due to reinforcement. On the contrary, high a and values describe situations where the normaldisplacement is not considerably suppressed. The results of such tests are presented and discussed inreference [4]. Some additional tests will be presented in this paper.

Four additional tests were performed on ordinary concrete specimens with 50 MPa compressive strengthand 8 mm maximum aggregate size. The specimen geometry is demonstrated in figure 3. The first and thesecond tests were conducted on three-month-old specimens. The first test was conducted on a water-saturated specimen. The second test was conducted on a specimen which was water-saturated during thefirst month and conditioned for two months in a laboratory climate. The displacement path for both testswas parabola with =0.5 mm1/2 (see figure 4a) . The results are presented in figures 4b and 4c. The figuresalso show the mean value of three similar tests conducted on 28-day-old specimens.

The third and the fourth tests were conducted on specimens which were treated in the same way asabove, but the displacement path was different. The displacement path, a mixed-path, is shown in figure 5a,in which four parts may be distinguished. Part (a) is a straight line with =60, part (b) is a parabola with=0.4 mm1/2 , part (c) is a straight line with slope 76 and part (d) is a parabola with =0.6 mm . The testresults are demonstrated in figures 5b and 5c. Figure 6 shows the results of two parabola paths (=0.4 and=0.6) and a straight line path (=60) conducted on 28-day-old specimens (Hassanzadeh 1989). The resultsof the parabola paths are the mean values of three tests.

5Discussion

Due to the low number of samples, the statistic significance of the results presented in the previous sectionis dubious. Nevertheless the results are useful, to a certain extent, in studying the degree of influence of someparameters.

Figures 4 and 5 confirm that the moisture content has a striking influence on the tensile strength butalmost no influence on the mixed-mode behaviour of concrete. In figure 4, the influence of the age of theconcrete can be observed. It is low due to the fact that most of the hardening occurs during the first month.

In figures 5 and 6, the influence of the history of the displacement is studied. It is interesting to comparethe shear stressshear displacement curves at: points where the first straight line (=60) intersects theparabolas. The first intersection is with parabola =0.4 mm1/2 (point ws=0.05 mm wn=0.09 mm) where theshear stresses pertaining to the straight line displacement path are slightly higher than the correspondingstress pertaining to the parabola path. The second intersection is with parabola =0.6 mm1/2 (point ws=0.12mm wn=0.21 mm) where the shear stresses pertaining to the straight line displacement path are much higherthan the corresponding stress pertaining to the parabola path. Further, the linear path stresses descend whilethe parabola path stress rises. The same tendencies can be observed with the normal stressnormaldisplacement curves. It can be concluded that there are two, as in this case, or more stress states for onetotal displacement point.

As was mentioned above, the displacement paths have been chosen arbitrarily and it is not known to whatextent they cover the real displacement paths occurring in practice. The displacement paths which occur inpractice depend on geometry, loading and boundary conditions. Therefore, it is necessary that this type of


mixed-mode tests be combined with numerical applications in order to check the reasonableness of theassumed displacement path.

Figure 3. Specimen geometry and measuring points, in mm, Normal displacements are measured at points 1 1 and 2.Shear displacements are measured at points 3 and 4.


6Conclusion

Some degree of path dependency has been observed, which may cause difficulties in the derivation of asimple, general expression which describes the mixed-mode behaviour of a fracture process zone. Since thetests and the conclusions drown from the test results have been based on arbitrary deformation paths, furtherresearch is needed to verify the phenomena which have been observed. Such research should combine bothlaboratory and theoretical experiments.

REFERENCES

1 . Gustafsson, P.J. (1985), Fracture mechanics studies of non-yielding materials like concrete, reportTVBM-1007, thesis, Div. of Building Materials, Univ. of Lund, Sweden.

2 . Hassanzadeh, M., Hillerborg, A., Zhou, F.P. (1987), Tests of material properties in mixed mode I and II. procof the International Conference on Fracture of Concrete and Rock, pp 353358. Houston, U.S.A.

3 . Hassanzadeh, M., Hillerborg, A. (1989), Concrete prope rties in mixed mode fracture, Fracture toughness andfracture energy, Ed Mihashi, H., Takahashi, H., Wittmann, F.H., A.A. Balkema Publishers.

4 . Hassanzadeh, M. (1989), Determination of fracture zone properties in mode I and II. proc of the 1989 SEMSpring Conference on Experimental Mechanics, pp 521527. Cambridge, MA, U.S.A.

5 . Hillerborg, A. (1989-a), Discrete crack approach, chapter 5 in Fracture mechanics of concrete structures,RILEM report, Ed L. Elfgren, Chapman and Hall.

6 . Hillerborg, A. (1989-b), Mixed mode fracture in concrete, Seventh International Conference on Fracture,Houston, Texas, March 2024.

7 . Zhou, F.P. (1988), Some aspects of tensile fracture behaviour and structural response of cementitiousmaterials, report TVBM-1008, thesis, Div. of Building Materials, Univ. of Lund, Sweden.

Figure 4a. Displacement path, parabola =0.5 mm1/2.


Figure 4b. Normal stressnormal displacement curves pertaining to the parabola path =0.5 mm1/2.


Figure 4c. Shear stressshear displacement curves pertaining to the parabola path =0.5 mm1/2


Figure 5a. Different displacement paths.


Figure 5b. Normal stressnormal displacement curves pertaining to the mixed-path.


Figure 5c. Shear stressshear displacement curves pertaining to the mixed-path.


Figure 6a. Normal stressnormal displacement curves pertaining to the parabola path =0.4 mm1/2 =0.6 mm1/2 and=60.


Figure 6b. Shear stressshear displacement curves pertaining to the parabola path =0.4 mm =0.6 mm1/2 and =60.


5MIXED MODE FRACTURE IN COMPRESSION

A.K.MAJIDepartment of Civil Engineering, University of New Mexico, Albuquerque, USA

S.P.SHAHCenter for Advanced Cement-Based Materials, Northwestern University, USA

SUMMARY

Mechanical response of concrete under compressive loading is largely dependent on the development ofmicrofracture in the material. It is generally accepted that interface debonding and matrix cracking isresponsible for nonlinear behavior and eventual failure of concrete. A study of these phenomena is thereforecritical in understanding the material and for developing adequate constitutive models.

In order to study the influence of compressive loading, a phenomenological study was first undertakenusing model concrete specimen with prefabricated microstructure. Holographic Interf

230223468 analysis of concrete structures by fracture mechanics by elfgreen and shah

Documents