Delamination Behaviour of Composites

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i

Delamination behaviour of composites

© 2008, Woodhead Publishing Limited

ii

Related titles: Multi-scale modelling of composite material systems (ISBN 978-1-85573-936-9) Predictive modelling provides the opportunity both to understand better how composites behave in different conditions and to develop materials with enhanced performance for particular industrial applications. This important book focuses on the fundamental understanding of composite materials at the microscopic scale, from designing micro-structural features, to the predictive equations of the functional behaviour of the structure for a specific end-application. Chapters discuss stress- and temperature-related behavioural phenomena based on knowledge of physics of microstructure and microstructural change over time. Impact behaviour of fibre-reinforced composite materials and structures (ISBN 978-978-1-8573-423-4) This study covers impact response, damage tolerance and failure of fibrereinforced composite materials and structures. Materials development, analysis and prediction of structural behaviour and cost-effective design all have a bearing on the impact response of composites and this book brings together for the first time the most comprehensive and up-to-date research work from leading international experts. Mechanical testing of advanced fibre composites (ISBN 978-1-85573-312-1) Testing of composite materials can present complex problems but is essential in order to ensure the reliable, safe and cost-effective performance of any engineering structure. Mechanical testing of advanced fibre composites describes a wide range of test methods which can be applied to various types of advanced fibre composites. The book focuses on high modulus, high strength fibre/plastic composites and also covers highly anisotropic materials such as carbon, aramid and glass. Details of these and other Woodhead Publishing materials books, as well as materials books from Maney Publishing, can be obtained by: • visiting our web site at www.woodheadpublishing.com • contacting Customer Services (e-mail: [email protected]; fax: +44 (0) 1223 893694; tel: +44 (0) 1223 891358 ext: 130; address: Woodhead Publishing Ltd, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, England) If you would like to receive information on forthcoming titles, please send your address details to: Francis Dodds (address, tel. and fax as above; e-mail: [email protected]). Please confirm which subject areas you are interested in. Maney currently publishes 16 peer-reviewed materials science and engineering journals. For further information visit www.maney.co.uk/journals © 2008, Woodhead Publishing Limited

iii

Delamination behaviour of composites Edited by Srinivasan Sridharan

Published by Woodhead Publishing and Maney Publishing on behalf of The Institute of Materials, Minerals & Mining

CRC Press Boca Raton Boston New York Washington, DC

WOODHEAD

PUBLISHING LIMITED

Cambridge England © 2008, Woodhead Publishing Limited

iv Woodhead Publishing Limited and Maney Publishing Limited on behalf of The Institute of Materials, Minerals & Mining Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, England www.woodheadpublishing.com Published in North America by CRC Press LLC, 6000 Broken Sound Parkway, NW, Suite 300, Boca Raton, FL 33487, USA First published 2008, Woodhead Publishing Limited and CRC Press LLC © 2008, Woodhead Publishing Limited The authors have asserted their moral rights. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. Reasonable efforts have been made to publish reliable data and information, but the author and the publishers cannot assume responsibility for the validity of all materials. Neither the author nor the publishers, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited. The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from Woodhead Publishing Limited for such copying. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging in Publication Data A catalog record for this book is available from the Library of Congress. Woodhead Publishing ISBN 978-1-84569-244-5 (book) Woodhead Publishing ISBN 978-1-84569-482-1 (e-book) CRC Press ISBN 978-1-4200-7967-8 CRC Press order number: WP7967 The publishers’ policy is to use permanent paper from mills that operate a sustainable forestry policy, and which has been manufactured from pulp which is processed using acid-free and elementary chlorine-free practices. Furthermore, the publishers ensure that the text paper and cover board used have met acceptable environmental accreditation standards. Project managed by Macfarlane Book Production Services, Dunstable, Bedfordshire, England (e-mail: [email protected]) Typeset by Replika Press Pvt Ltd, India Printed by T J International Limited, Padstow, Cornwall, England


v

Contents

Contributor contact details Introduction

xv xxi

S SRIDHARAN, Washington University in St. Louis USA

Part I Delamination as a mode of failure and testing of delamination resistance 1

Fracture mechanics concepts, stress fields, strain energy release rates, delamination initiation and growth criteria

3

I S RAJU and T K O’BRIEN, NASA-Langley Research Center, USA

1.1 1.2 1.3 1.4 1.5 1.6

Introduction Fracture mechanics concepts Delaminations Future trends Concluding remarks References

3 4 10 23 24 25

2

Delamination in the context of composite structural design

28

A RICCIO, CIRA (Centro Italiano Ricerche Aerospaziali – Italian Aerospace Research Centre), Italy

2.1 2.2 2.3 2.4 2.5

Introduction Physical phenomena behind delamination onset Physical phenomena behind delamination growth Introduction to delamination management in composites design Impact-induced delamination resistance in composites preliminary design


28 30 36 39 41

vi

Contents

2.6 2.7 2.8

Delamination tolerance in composites preliminary design Cost-effective delamination management References

46 55 60

3

Review of standard procedures for delamination resistance testing

65

P DAVIES, IFREMER Centre de Brest, France

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 4

Introduction Historical background Mode I Mode II Mode III Mixed mode I/II Conclusion on fracture mechanics tests to measure delamination resistance Future trends Conclusion Sources of information and advice Acknowledgements References Testing methods for dynamic interlaminar fracture toughness of polymeric composites

65 66 67 70 74 76 79 80 81 81 81 81

87

C T SUN, Purdue University, USA

4.1 4.2 4.3 4.4 4.5 4.6 4.7 5

Introduction Dynamic loading and crack propagation Mode I loading with double cantilever beam (DCB) for low crack velocity High crack velocity with modified double cantilever beam (DCB) and end notch flexure (ENF) Mode I by wedge loading with Hopkinson bar Acknowledgment References

87 90

95 104 115 115

Experimental characterization of interlaminar shear strength

117

93

R GANESAN, Concordia University, Canada

5.1 5.2 5.3 5.4

Introduction Short beam shear test Double-notch shear test Arcan Test


117 118 125 133

Contents

5.5 5.6 5.7

Conclusion References Appendix: Nomenclature

vii

134 135 136

Part II Delamination: detection and characterization 6

Integrated and discontinuous piezoelectric sensor/actuator for delamination detection

141

P TAN, Defence Science and Technology Organisation, Australia and L TONG, University of Sydney, Australia

6.1 6.2 6.3

6.4 6.5 6.6 6.7 6.8 6.9 7

Introduction Typical patterns for piezoelectric (PZT) or piezoelectric fiber reinforced composite (PFRC) sensor/actuator Constitutive equations and modelling development for a laminated beam with a single delamination and surface-bonded with an integrated piezoelectric sensor/actuator (IPSA) Parametric study Experimental verification Conclusions Acknowledgments References Appendix

141

146 149 157 165 165 165 167

Lamb wave-based quantitative identification of delamination in composite laminates

169

143

Z SU, The Hong Kong Polytechnic University, Hong Kong and L YE, The University of Sydney, Australia

7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11

Introduction Lamb waves in composite laminates Lamb wave scattering by delamination Lamb wave-based damage identification for composite structures Design of a diagnostic Lamb wave signal Digital signal processing (DSP) Signal pre-processing and de-noising Digital damage fingerprints (DDF) Data fusion Sensor network for delamination identification Case studies: evaluation of delamination in composite laminates


169 170 177 180 181 182 186 187 193 198 202

viii

Contents

7.12 7.13 7.14

Conclusion Acknowledgements References

211 211 212

8

Acoustic emission in delamination investigation

217

J BOHSE, BAM-Federal Institute for Materials Research and Testing, Germany and A J BRUNNER, Empa-Swiss Federal Laboratories for Materials Testing and Research, Switzerland

8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

Introduction Acoustic emission (AE) analysis Acoustic emission analysis applied to investigation of delaminations in fiber-reinforced, polymer-matrix (FRP) Acoustic emission monitoring of delaminations in fiber-reinforced, polymer matrix composite specimens Acoustic emission investigation of delaminations in structural elements and structures Advantages and limitations for acoustic emission delamination investigations Related nondestructive acoustic methods for delamination investigations Summary and outlook Acknowledgments References

217 218 222 223 253 267 272 272 273 273

Part III Analysis of delamination behaviour from tests 9

Experimental study of delamination in cross-ply laminates

281

A J BRUNNER, Empa-Swiss Federal Laboratories for Materials Testing and Research, Switzerland

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Introduction Summary of current state Experimental methods for studying delaminations Fracture mechanics study of delamination in cross-ply laminates Discussion and interpretation Structural elements or parts with cross-ply laminates Summary and outlook Acknowledgments References


281 282 285 286 300 304 305 305 305

Contents

10

Interlaminar mode II fracture characterization

ix

310

M F S F DE MOURA, Faculdade de Engenharia da Universidade do Porto, Portugal

10.1 10.2 10.3 10.4 10.5 10.6

Introduction Static mode II fracture characterization Dynamic mode II fracture characterization Conclusions Acknowledgements References

310 311 321 324 324 325

11

Interaction of matrix cracking and delamination

327

M F S F DE MOURA, Faculdade de Engenharia da Universidade do Porto, Portugal

11.1 11.2 11.3 11.4 11.5

Introduction Mixed-mode cohesive damage model Continuum damage mechanics Conclusions References and further reading

327 332 338 341 342

12

Experimental studies of compression failure of sandwich specimens with face/core debond

344

F AVILÉS, Centro de Investigación Científica de Yucatán, A C, México and L A CARLSSON, Florida Atlantic University, USA

12.1 12.2 12.3 12.4 12.5 12.6

Introduction Compression failure mechanism of debonded structures Compression failure of debonded sandwich columns Compression failure of debonded sandwich panels Acknowledgments References

344 344 346 353 362 362

Part IV Modelling delamination 13

Predicting progressive delamination via interface elements

367

S HALLETT, University of Bristol, UK

13.1 13.2 13.3 13.4 13.5

Introduction Background to the development of interface elements Numerical formulation of interface elements Applications Enhanced formulations


367 367 368 373 380

x

Contents

13.6 13.7 13.8

Conclusions Acknowledgements References

382 382 382

14

Competing cohesive layer models for prediction of delamination growth

387

S SRIDHARAN, Washington University in St. Louis, USA and Y LI, Intel Corporation, USA

14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9

Introduction User material model User supplied element model Double cantilever problem UMAT model: details of the study and discussion of results UEL model: details of the study and discussion of results Delamination of composite laminates under impact Conclusion References

387 388 391 394 394 403 407 427 427

15

Modeling of delamination fracture in composites: a review

429

R C YU, Universidad de Castilla-La Mancha, Spain and A PANDOLFI, Politecnico di Milano Italy

15.1 15.2 15.3 15.4 15.5 15.6 15.7

Introduction The cohesive approach Delamination failure in fiber reinforced composites Delamination failure in layered structures Summary and conclusions Acknowledgements References

429 431 432 440 450 451 452

16

Delamination in adhesively bonded joints

458

B R K BLACKMAN, Imperial College London, UK

16.1 16.2 16.3 16.4 16.5 16.6

Introduction Adhesive bonding of composites Fracture of adhesively bonded composite joints Future trends Sources of further information and advice References


458 458 460 479 480 481

Contents

17

Delamination propagation under cyclic loading

xi

485

P P CAMANHO, Universidade do Porto, Portugal and A TURON and J COSTA, University of Girona, Spain

17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 17.10 17.11 17.12 17.13 17.14 18

Introduction and motivation Experimental data Damage mechanics models Simulation of delamination growth under fatigue loading using cohesive elements: cohesive zone model approach Numerical representation of the cohesive zone model Constitutive model for high-cycle fatigue Examples Mode I loading Mode II loading Mixed-mode I and II loading Fatigue delamination on a skin-stiffener structure Conclusions Acknowledgments References and further reading

485 486 488 490 491 493 498 498 502 504 505 510 510 511

Single and multiple delamination in the presence of nonlinear crack face mechanisms

514

R MASSABÒ, University of Genova, Italy

18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8

Introduction The cohesive- and bridged-crack models Characteristic length scales in delamination fracture Derivation of bridging traction laws Single and multiple delamination fracture Final remarks Acknowledgement References

514 515 528 535 539 553 555 555

Part V Analysis of structural performance in the presence of delamination, and prevention/mitigation of delamination 19

Determination of delamination damage in composites under impact loads

561

A F JOHNSON and N TOSO-PENTECÔTE, German Aerospace Centre (DLR), Germany

19.1 19.2

Introduction Composites failure modelling


561 563

xii

Contents

19.3 19.4 19.5 19.6

Delamination damage in low velocity impact Delamination damage in high velocity impact Conclusions and future outlook References

570 576 583 584

20

Delamination buckling of composite cylindrical shells

586

A TAFRESHI, The University of Manchester, UK

20.1 20.2 20.3 20.4

20.5 20.6 21

Introduction Finite element analysis Validation study Results and discussion: analysis of delaminated composite cylindrical shells under different types of loadings Conclusion References

586 588 597

597 614 616

Delamination failure under compression of composite laminates and sandwich structures

618

S SRIDHARAN, Washington University in St. Louis, USA, Y LI, Intel Corporation, USA and S EL-SAYED, Caterpillar Inc., USA

21.1 21.2

21.5 21.6 21.7

Introduction Case study (1): composite laminate under longitudinal compression Case study (2): dynamic delamination of an axially compressed sandwich column Case study (3): two-dimensional delamination of laminated plates Results and discussion Conclusion References

635 644 647 648

22

Self-healing composites

650

21.3 21.4

618 619 628

M R KESSLER, Iowa State University, USA

22.1 22.2 22.3 22.4 22.5 22.6

Introduction Self-healing concept Healing-agent development Application to healing of delamination damage in FRPs Conclusions References and further reading


650 652 657 661 670 671

Contents

23

Z-pin bridging in composite delamination

xiii

674

H Y LIU, The University of Sydney, Australia and W YAN, Monash University, Australia

23.1 23.2 23.3 23.4 23.5 23.6 23.7

Introduction Z-pin bridging law Effect of z-pin bridging on composite delamination Z-pin bridging under high loading rate Fatigue degradation on z-pin bridging force Future trends References

674 675 677 693 699 703 704

24

Delamination suppression at ply drops by ply chamfering

706

M R WISNOM and B KHAN, University of Bristol, UK

24.1 24.2

706

24.3 24.4 24.5 24.6

Introduction Behaviour of tapered composites with ply drops Methods of chamfering plies Results of ply chamfering Summary and conclusions References

25

Influence of resin on delamination

721

707 711 713 719 720

S MALL, Air Force Institute of Technology, USA

25.1 25.2 25.3 25.4 25.5 25.6 25.7 25.8

Introduction Resin toughness versus composite toughness Resin toughness effects on different modes Resin effects on cyclic delamination behaviour Temperature considerations Effects of interleafing and other methods Summary References


721 722 725 729 733 735 737 739

xv

Contributor contact details

(* = main contact)

Introduction

Chapter 3

S. Sridharan* Department of Mechanical, Aerospace and Structural Engineering Washington University in St. Louis St. Louis, MO 63130 USA

P. Davies Materials and Structures Group (ERT/MS) IFREMER Centre de Brest BP70 29280 Plouzané France

E-mail: [email protected]


Chapter 1

Chapter 4

I. S. Raju* and T. K. O’Brien NASA-Langley Research Center Hampton, VA 23861 USA

C. Sun School of Aeronautics and Astronautics Purdue University West Lafayette, IN 47907 USA


Chapter 2 A. Riccio CIRA (Centro Italiano Ricerche Aerospaziali – Italian Aerospace Research Centre) Via Maiorise, S/N 81043 Capua (Caserta) Italy E-mail: [email protected] © 2008, Woodhead Publishing Limited


xvi


Chapter 5

Chapter 7

R. Ganesan Concordia Centre for Composites Department of Mechanical and Industrial Engineering Concordia University Room EV 4 - 211, 1515 St Catherine West Montreal Quebec H3G 2W1 Canada

Z. Su The Department of Mechanical Engineering The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong E-mail: [email protected] [email protected];


Chapter 6 P. Tan1 Human Protection and Performance Division Defense Science and Technology Organisation 506 Lorimer Street Fishermans Bend VIC 3207 Australia E-mail: [email protected]

L. Ye* Centre for Advanced Materials Technology (CAMT) School of Aerospace Mechanical and Mechatronic Engineering (AMME) The University of Sydney NSW 2006 Australia E-mail: [email protected]

Chapter 8

L. Tong* School of Aerospace Mechanical and Mechatronic Engineering University of Sydney NSW 2006 Australia

J. Bohse* BAM – Federal Institute for Materials Research and Testing Division V.6 Mechanical Behaviour of Polymers Unter den Eichen 87 D-12205 Berlin Germany



1

Ping Tan is currently working as a research scientist at the Australian Defence Science and Technology Organisation



A. J. Brunner Laboratory for Mechanical Systems Engineering Empa – Swiss Federal Laboratories for Materials Testing and Research Ueberlandstrasse 129 CH-8600 Duebendorf Switzerland E-mail: [email protected]

Chapter 9 A. J. Brunner Laboratory for Mechanical Systems Engineering Empa – Swiss Federal Laboratories for Materials Testing and Research Ueberlandstrasse 129 CH-8600 Duebendorf Switzerland

xvii

Chapter 12 Francis Avilés-Cetina* Centro de Investigación Científica de Yucatán, A.C. Unidad de Materiales Calle 43 # 103 Col. Chuburná de Hidalgo C.P. 97200. Mérida Yucatán México Leif A. Carlsson Department of Mechanical Engineering Florida Atlantic University Boca Raton, FL 33431 USA E-mail: [email protected] [email protected]

Chapter 13 E-mail: [email protected]

Chapters 10 and 11 Marcelo Francisco de Sousa Ferreira de Moura Departamento de Engenharia Mecânica e Gestão Industrial Faculdade de Engenharia da Universidade do Porto Rua Dr. Roberto Frias s/n, 4200465 Porto Portugal E-mail: [email protected]


S. Hallett Advance Composites Centre for Innovation and Science University of Bristol Queens Building University Walk Bristol BS8 1TR UK E-mail: [email protected]

xviii


Chapters 14 and 21

Chapter 16

S. Sridharan* Department of Mechanical, Aerospace and Structural Engineering Washington University in St. Louis St. Louis, MO 63130 USA

B. Blackman Department of Mechanical Engineering Imperial College London London SW7 2AZ UK E-mail: [email protected]

E-mail : [email protected]

Y. Li Intel Corporation Chandler, AZ 85224 USA E-mail: [email protected]

Chapter 15 R. Yu* E.T.S Ingenieros de Caminos, Canales y Puertos Universidad de Castilla-La Mancha 13071 Ciudad Real Spain E-mail: [email protected]

A. Pandolfi Dipartimento di Ingegneria Strutturale Politecnico di Milano Piazza Leonardo da Vinci 32 20133 Milano Italy E-mail: [email protected]

Chapter 17 P. Camanho* DEMEGI Faculdade de Engenharia Universidade do Porto Rua Dr. Roberto Frias 4200-465 Porto Portugal E-mail: [email protected]

A. Turon and J. Costa AMADE Polytechnic School University of Girona Campus Montilivii s/n 17071 Girona Spain

Chapter 18 Roberta Massabò Department of Civil, Environmental and Architectural Engineering University of Genova Via Montallegro, 1 16145, Genova Italy E-mail: [email protected]



Chapter 19

Chapter 23

Alastair F. Johnson and Nathalie Toso-Pentecôte German Aerospace Center (DLR) Institute of Structures and Design Pfaffenwaldring 38-40 70569 Stuttgart Germany

H. Liu* Centre for Advanced Materials Technology School of Aerospace Mechanical and Mechatronic Engineering The University of Sydney Sydney NSW 2006 Australia

E-mail: [email protected] [email protected]

xix

Chapter 20


A. Tafreshi School of Mechanical Aerospace and Civil Engineering The University of Manchester P.O. Box 88 Sackville Street Manchester M60 1QD UK

W. Yan Department of Mechanical Engineering Monash University Melbourne Victoria 3800 Australia



Chapter 24 Chapter 22 M. Kessler Iowa State University Materials Science and Engineering 2220 Hoover Hall Ames, IA 50011-2300 USA E-mail: [email protected]

M. R. Wisnom University of Bristol Advanced Composites Centre for Innovation and Science Queens Building University Walk Bristol BS8 1TR UK E-mail: M. [email protected]


xx


Chapter 25 S. Mall Air Force Institute of Technology AFIT/ENY Bldg. 640 2950 Hobson Way Wright-Patterson AFB, OH 45433 USA E-mail: [email protected]


xxi

Introduction S S R I D H A R A N, Washington University in St. Louis, USA

Laminated composites are becoming the preferred material system in a variety of industrial applications, such as aeronautical and aerospace structures, ship hulls in naval engineering, automotive structural parts, micro-electromechanical systems and also civil structures for strengthening concrete members. The increased strength and stiffness for a given weight, increased toughness, increased mechanical damping, increased chemical and corrosion resistance in comparison to conventional metallic materials and potential for structural tailoring are some of the factors that have contributed to the advancement of laminated composites. Their increased use has underlined the need for understanding their modes of failure and evolving technologies for the continual enhancement of their performance. The principal mode of failure of layered composites is the separation along the interfaces of the layers, viz. delamination. This type of failure is induced by interlaminar tension and shear that develop due to a variety of factors such as: Free edge effects, structural discontinuities, localized disturbances during manufacture and in working condition, such as impact of falling objects, drilling during manufacture, moisture and temperature variations and internal failure mechanisms such as matrix cracking. Hidden from superficial visual inspection, delamination often lies buried between the layers, and can begin to grow in response to an appropriate mode of loading, drastically reducing the stiffness of the structure and thus the life of the structure. The delamination growth often occurs in conjunction with other modes of failure, particularly matrix cracking. A study of composite delamination, as with any technological discipline, has two complementary aspects: an in depth understanding of the phenomenon by analysis and experimentation and the development of strategies for effectively dealing with the problem. These in turn lead to a number of specific topics that we need to consider in the present context. These comprise: 1. An understanding of the basic principles that govern the initiation of delamination, its growth and its potential interaction with other modes of failure of composites. This is the theme of the first chapter, but several authors return to this theme in their own respective contributions. © 2008, Woodhead Publishing Limited

xxii

Introduction

2. The determination of material parameters that govern delamination initiation and growth by appropriate testing. These must necessarily be interfacial strength parameters which govern interlaminar fracture initiation and interlaminar fracture toughness parameters, viz. critical strain energy release rates that must govern interlaminar crack growth. The book contains several valuable contributions from leading international authorities in the field of testing of composites. 3. Development of analytical tools : what are the methodologies one may employ to assess the possibility of delamination onset and growth under typical loading scenarios? This may be approached from the points of view of fracture mechanics, damage mechanics, cohesive modeling approach and approaches which draw from and combine these. In particular, the cohesive modeling approach has proven to be a powerful and versatile tool in that when embedded in a nonlinear finite element analysis, it can trace the two-dimensional delamination growth without user interference, it is robust from the point of view of numerical convergence, and can potentially account for a variety of interfacial failure mechanisms. This subject is discussed thoroughly in several authoritative contributions. 4. Detection of delamination: ability to diagnose the presence of delamination and to be able to capture in graphical terms the extent of delamination damage is a desideratum towards which the composite industry is continuing to make progress. Several nondestructive evaluation tools are available and have been used with varying degrees of success. Acoustic emission, Lamb-wave and Piezo-electric technologies are discussed in the context of delamination detection in the present work. 5. Prevention of delamination: several techniques of either inhibiting delamination or altogether suppressing it are available. The book contains a section treating the following techniques of delamination prevention/ inhibition: ‘Self-healing’ composites which internally exude adhesive material as soon as the crack advances thus effectively arresting the crack; Z-pin bridging in which fibers are introduced across the interlaminar surfaces, liable to delaminate, artfully tapering off discontinuities which are sources of potential delamination and the use of toughened epoxies. 6. Delamination-driven structural failure: certain loading scenarios can cause delamination growth if there is some pre-existing delamination in the structural component which in turn can lead to structural failure. Typically these are: impact, cyclic loading (delamination due to fatigue), compressive loading causing localized buckling in the vicinity of delamination and dynamic loading in the presence of in-plane compression. Impact loading and any form of dynamic loading in the presence of significant compressive stress in sandwich structures are known to trigger delamination failure which is abrupt and total. These aspects have been discussed in several contributions. © 2008, Woodhead Publishing Limited

Introduction

xxiii

The book has been divided into several sections to address the issues mentioned in the foregoing. It has been a pleasure to work with a number of authors of international standing and reputation who have spent a great deal of effort in developing their respective chapters. The references cited at the end of each chapter should supplement and corroborate the concepts developed in the chapter. We hope that researchers and engineers who are concerned to apply state of the art technologies to composite structural analysis, design and evaluation of risk of failure will find this book useful and a valuable source of insight.


Part I Delamination as a mode of failure and testing of delamination resistance

1 © 2008, Woodhead Publishing Limited

1 Fracture mechanics concepts, stress fields, strain energy release rates, delamination initiation and growth criteria I S R A J U and T K O ’ B R I E N, NASA-Langley Research Center, USA

1.1

Introduction

A complete understanding of composite delamination requires an appreciation for the fundamental principles of fracture mechanics and how these principles have been extended from the original concepts developed for isotropic materials to include the anisotropy typically present in composite materials. These extensions include the complexities of oscillatory singularities that occur for interface cracks in anisotropic media, and how these singularities are resolved for delamination growth prediction. Furthermore, full implementation of Interlaminar Fracture Mechanics (ILFM) in design requires development of composite delamination codes to calculate strain energy release rates and advancements in delamination growth criteria under mixed mode conditions for residual strength and life prediction. The chapter is organized as follows. First, fracture mechanics concepts for isotropic materials are presented. The stress field near a crack tip and the concept of the stress-intensity factor are introduced. Next, the evaluation of the strain energy release rate for self-similar crack growth, which is a measure of the crack driving force, through Irwin’s crack closure concept and the near-tip stress and displacement fields is presented. Cracks in orthotropic and anisotropic materials are considered next. A bi-material problem with an interface crack is considered as a precursor to cracks in layered media. In Section 1.3, the problem of delaminations in composite laminates is discussed. Mixed-mode behavior, determination of interlaminar fracture toughness, fatigue characterization, delamination onset are treated next. The process of evaluation of strain energy release rates in two- and three-dimensional finite element analyses is discussed. Two examples of delamination prediction and their validation with test data are presented next. Finally, future work needed to achieve a fully mature methodology for use in design certification of composite structures is outlined.


4


1.2

Fracture mechanics concepts

Consider a crack in a homogeneous isotropic linear elastic infinite plate as shown in Fig. 1.1(a). The crack lies on the y = 0 line and in the region x = ±a. This line discontinuity with zero thickness and with sharp ends is defined as a crack. A crack can also be thought of as a limiting case of an elliptical hole with a major axis of 2a and minor axis approaching a zero value. Under external loading the crack faces at θ = ± π in Fig. 1.1(a) can displace relative to each other. Figure 1.1(b) shows a crack in an infinite solid. The two- and three-dimensional stress states are also shown in Fig. 1.1. Any complex deformation of the crack faces can be described by a combination of three fracture modes, Mode-I, Mode-II, and Mode-III as shown in Fig. 1.2. Mode-I represents the opening mode of the crack faces, Mode-II represents the sliding mode, and Mode-III represents the tearing mode (out-of-plane shear mode) deformation.

σy τxy

Y, v

Y, v σx

σx

σy

τxy

r θ

τyz

σy

X, u

Z, w

2a

X, u

τzy σz

τyx τxz

τzx

σx τxy

a

(a) Crack in a plate (two dimensions)

(b) Crack in a solid (three dimensions)

1.1 Cracks in plates and solids.

Mode-I (opening)

Mode-II (sliding shear)

1.2 The three fracture modes.


Mode-III (tearing shear)


1.2.1

5

Crack-tip stress field

The elastic stress field around a crack tip has been well characterized and documented in research monographs and reference books (see Paris and Sih, 1965; Parker, 1981; Broek, 1982; Ewalds and Wanhill, 1984; Tada et al., 2000; Sanford, 2003; Anderson 2005). The stress field in the immediate vicinity of a crack tip can be written as (see Fig. 1.1(a))

θ KI cos    2 2π r

σx =

 θ   3θ   1 – sin  2  sin  2    

θ  θ 3θ  K II sin    2 + cos   cos    2  2 2      2π r 

–

θ KI cos    2 2π r

σy =

 θ   3θ   1 + sin  2  sin  2    

θ θ 3θ K II sin   cos   cos    2  2  2  2π r

+

3θ θ θ KI cos   sin   cos   2 2 2       2π r

τ xy =

θ K II cos    2 2π r

+

 θ   3θ   1 – sin  2  sin  2    

σz = 0 for plane stress and σz = ν(σx + σy) for plane strain conditions, 1.1 and

τ yz =

θ K III cos    2 2π r

τ zx = –

θ K III sin    2 2π r

1.2

Clearly, from Eqs 1.1 and 1.2, the stresses are singular at the crack tip (r = 0) and the stresses have a square-root singularity. The constants KI, KII, and KIII are termed as the Mode-I, Mode-II, and Mode-III stress-intensity factors, respectively. The stress-intensity factors describe the intensity of the stress field and are a measure of the severity of the crack. The displacements (u,v) that correspond to the stresses in Eq. 1.1 can be written as © 2008, Woodhead Publishing Limited

6


u=

+

v=

r cos  θ  2π  2

KI 2µ

–

r sin  θ  2π  2

K II 2µ

r sin  θ  2π  2

KI 2µ

 2θ  κ – 1 + 2 sin  2    

K II 2µ

 2θ  κ + 1 + 2 cos  2    

 2θ  κ + 1 – 2 cos  2    

r cos  θ  2π  2

 2θ  κ – 1 – 2 sin  2    

1.3

and the out-of-plane displacement (w) corresponding to the tearing mode in Eq. 1.2 is w=

r sin  θ  2π  2

2 K III µ

1.4

where µ is the shear modulus, κ = (3 – ν)/(1 + ν) for plane stress, κ = (3 – 4ν) for plane strain, and ν is the Poisson’s ratio of the material.

1.2.2

Strain energy release rate, G.

Utilizing the near tip stress and displacement fields, Irwin (1957) calculated the work required to close a crack of length a + ∆ a to a length a. Irwin argued that in a brittle material all the energy that is supplied externally is utilized in creating new crack surfaces as these materials undergo little or no plastic deformations. Thus, the work required to extend the crack from a to a + ∆ a will be the same as the work required to close the crack from a + ∆ a to a. As the crack increments are small, the crack opening displacement behind a new crack tip at a + ∆ a will be same as those behind the original crack tip, at a. Thus the work required to extend the crack from a to a + ∆ a is (see Fig. 1.3)

W= 1 2

∫

∆a

σ y ( ∆ a – r ) ⋅ v( r ) dr

1.5

0

Irwin obtained the strain energy release rate, G, as

G = lim W = lim 1 ∆ a→0 ∆ a ∆ a→0 2 ∆ a

∫

∆a

σ y ( ∆ a – r ) ⋅ v( r ) dr

1.6

0

Substituting the stresses in Eqs. 1.1 and 1.2, the displacements in Eqs 1.3 and 1.4, and integrating one obtains



7

Crack shape after Y growth of ∆a

r

Original crack

X

vR

∆a–r ∆a

a

1.3 Irwin’s crack closure concept.

G = G I + G II + G III =

K I2 K2 K2 + II + (1 + ν ) III E E′ E′

1.7

where GI, GII, and GIII are the Mode-I, Mode-II, and Mode-III strain energy release rates, respectively, G is the total strain energy release rate, E′ = E, in plane stress, E′ = E/(1–ν2), in plane strain, and E is the Young’s modulus of the material.

1.2.3

Orthotropic and anisotropic materials

For cracks in orthotropic or anisotropic materials, a similar square root singularity exists at the crack tips. The stress distributions are complicated and involve material properties (see Paris and Sih, 1965; Tada et al., 2000; Wang, 1984). The stress-intensity factors and strain energy release rates are defined in a manner similar to the isotropic case. The relationship between the K and G are more complicated and are as shown in Table 1.1 (Note: Table 6 in page 60 of Paris and Sih, 1965 has errors and they are corrected in Table 1.1. See also Tada et al., 2000.)

1.2.4

Interface crack problem

Consider the problem of an interface crack between two isotropic (orthotropic or anisotropic) materials as shown in Fig. 1.4. Williams (1959) analyzed the problem of a crack along the interface between two dissimilar isotropic materials and observed that the singularity at the tips of the crack is of the form:

σ ~ r –1/2±iγ or σ ~ where i =

– 1 and


r ±iγ r

1.8

8


Table 1.1 G to K conversion for orthotropic and anisotropic materials G i = c K i2 Values of c are given below for plane strain conditions Mode

Orthotropic case

Anisotropic case

2 A12 + A 66  A11A 22  A 22 +   2 2 A11 A 11  

I

II

2 A12 + A 66 A11  A 22 +  2 A11 2  A11

III

1 ( A A ) (1/ 2) 44 55 2

  

(1/ 2)

–

 ξ1 + ξ 2  A11 Im   2  ξ 1ξ 2 

+

A11 Im (ξ 1 + ξ 2 ) 2

(1/ 2)

2 (3/ 2) 1 ( A 44 A 55 – A 45 ) 2 A 44 A 55

Notes: In this table the ξ1 and ξ2 are the roots of A11ξ 4 – 2A16ξ 3 + (2A12 + A66) ξ 2 – 2A26ξ + A22 = 0, where Aij are coefficients of the A-matrix that relates strains, {ε}, to stresses, {σ}, through {ε} = [A] {σ}, and Im (·) denotes the imaginary part of the complex number in the parentheses.

P Q Y, v Y

r

Mat–A θ

r θ

X, u X

2a

Interface

Mat–B

a Q P

1.4 Interface crack at a bi-material interface.

µ + κ AµB  1 – β γ = 1 ln  A = 1 ln   2π 2π  µB + κ BµA  1 + β

1.9

with κj = 3 – 4νj for plane strain conditions, µj is the shear modulus, and νj is the Poisson’s ratio for the jth material, and j = A, B. In Eq. 1.9 β is known as the Dundurs parameter and is defined in terms of µj and κj as

β=

µ A (κ B – 1) – µ B (κ A – 1) µ A (κ B + 1) – µ B (κ A + 1)


1.10


9

For orthotropic and anisotropic materials, the singularity is of similar form and the expressions for γ are much more complicated than the isotropic case (Wang and Choi, 1983a; 1983b; 1983c; Wang 1984; Aminpour, 1986). The stresses near field, t, along the interface (on the θ = 0 line in Fig. 1.4) can be written as (Erdogan, 1965; England, 1965) t = (σ y + iτ x y )θ = 0 =

Kr iγ 2π r

1.11

and the relative displacements uR at a distance r behind the crack tip at x = a as uR = (v + iu)θ=π – (v + iu)θ=–π Kr iγ 1 – νA 1 – νB  =2 + µ B  (1 + 2 iγ ) cosh ( πγ )  µA

r 2π

1.12

In Eqs 1.11 and 1.12, K is the complex stress-intensity factor defined as (Erdogan, 1965) K = (k1 + ik2)

π cosh (πγ)

1.13

The near-field solution in Eqs 1.8 and 1.11 suggests that the stresses very near the crack tip (r → 0), show oscillations (r±iγ can be written as e±iγ ln (r) and as r → 0 the e±iγ ln (r) term approaches ±i∞) and the crack faces interpenetrate each other in a small region near the crack tip. Similar behavior is observed for the orthotropic and anisotropic cases (Wang, 1984; Aminpour, 1986; Sun and Jih 1987; Raju et al., 1988]. The strain energy release rate, G, can be calculated using Eq. (1.11), Eq. (1.12), and Irwin’s crack closure integral as

G = lim

∆ a→0

1 2∆a

∫

∆a

t ( r ) u R ( ∆ a – r ) dr

0

KK 1 – νA 1 – νB  =4 + µ B  16 cosh 2 ( πγ )  µA

1.14

where K is the complex conjugate of K. The strain energy release rate in Eq. (1.14) will be denoted as the total strain energy release rate, G or GTotal. Raju et al. (1988) showed that for an interface crack between two isotropic materials, the individual strain energy release rates take the form GI = lim Re [C + D(∆a)iγ] ∆a→0

GII = lim Re [C – D(∆a)iγ] and ∆a→0


10


GTotal = GI + GII = lim Re [2C] ∆a→0

1.15

where ∆a is the virtual crack closure length, C and D are complex constants and Re denotes the real part of the complex function within the brackets. Equation (1.15) suggests that the individual strain energy release rates depend on ∆a and have no well-defined limits, while GTotal is independent of ∆a and has a very well defined limit. Raju et al. (1988) also showed that if the materials were chosen such that the Dundurs parameter β ≡ 0, then the individual as well as the total G values show independence of ∆a and have well-defined limits.

1.3

Delaminations

When two or more materials are combined to form a material with improved functionality, such a material is termed as a composite material. Composite materials usually exhibit the best qualities of their constituents and qualities that neither of the constituents possesses. Modern composite materials use high-strength fibers in a resin matrix. The fibers and matrix are combined to form a single ply. Composite laminates are formed by stacking plies of different orientations in a predetermined manner and curing the laminate under high temperature and pressure. Because the properties in the fiber direction are different from the other two orthogonal directions, a unidirectional (0-degree) ply can be modeled as an orthotropic material. The material properties of an angle ply (for example, a 45-degree ply) can be obtained using the unidirectional properties, the fiber angle, and the appropriate transformations. The resulting material properties of an angle ply shows anisotropic behavior (Jones, 1975). As such, a composite laminate with plies of different orientations can be treated as layered anisotropic medium. One of the most common failure modes of composite structures is delamination between plies. Delamination is a crack that forms between adjacent plies. The plies on either side of the delamination can have different fiber orientations. As such, a delamination can be viewed as an interface crack between two anisotropic materials. The stress fields and strain energy release rate analyses presented in Section 1.1 are applicable to these delaminations. The most common sources of delamination are the material and structural discontinuities that give rise to interlaminar stresses. Some of these sources are shown in Fig. 1.5. Delaminations occur at stress-free edges due to mismatch in properties of the individual layers, at ply drops (both internal and external) where thicknesses must be reduced, and at regions subjected to out-of-plane bending, such as bending of curved beams. When delaminations occur, all three failure modes, modes-I, II, and III, are usually present.



11

Curved free edge External ply drop Straight free edge

Z X

Internal ply drop

σz

τxz Corner

Interlaminar stesses

Skin stiffener interaction Solid-sandwich transition

1.5 Sources of delaminations at geometric and material discontinuities.

Over the past 25 years, significant progress has been made in research efforts to utilize fracture mechanics principles to characterize and predict delamination fatigue failures in composite laminates (O’Brien, 1990). Although these studies have demonstrated the promise of this approach, they have also highlighted some of the difficulties and differences relative to the wellestablished use of fracture mechanics for damage tolerance assessment of metallic structures. One notable difference is the propensity for cracks in composites to propagate in a mixed-mode fashion. This aspect can be attributed to the fact that delaminations are constrained to grow between composite layers. Delamination cracks do not immediately turn toward the opening mode direction as typically occurs in metals. Of primary concern is the need to characterize and analyze mixed-mode fracture involving the three fundamental fracture modes shown in Fig. 1.2. The interlaminar fracture toughness associated with each of the fracture modes must be characterized and the corresponding strain energy release rates for each mode (GI,GII,GIII) associated with the configuration and loading of interest must be calculated to predict delamination onset and growth. In addition, damage mechanisms that occur in the standardized test methods on unidirectional composite beams, such as the bridging of fibers above and below the crack plane in the opening mode (Johnson and Mangalgiri, 1987) and micro-cracking of the resin between fibers that coalesce to form hackles in the shearing mode (O’Brien, 1998), complicate attempts to achieve a generic characterization of delamination growth. © 2008, Woodhead Publishing Limited

12


1.3.1

Delamination characterization

The state of the art for using fracture mechanics to calculate interlaminar fracture toughness and delamination onset and growth has recently been outlined in two new sections developed for inclusion in Composite Materials Handbook 17 (formerly referred to as Mil Handbook 17) (Reeder, 2002; Paris, 2002). As noted in these documents, American Society for Testing of Materials (ASTM) standards have been developed for Mode I (double cantilever beam, DCB) (ASTM, 2001a) and mixed-Mode I and II (mixed-mode bending, MMB) [ASTM, 2001b] interlaminar fracture toughness. Although there are still no standard methods for pure Modes II and III, two promising test methods have been developed; the End Notched Flexure test (ENF) (Russell, 1982; Davidson and Sun, 2006) for Mode II and the Edge Cracked Torsion (ECT) test for Mode III (Lee, 1993; Li et al., 1997; Ratcliffe, 2004). Hence, ASTM standards for interlaminar fracture toughness for all three fracture modes should be in place shortly. A typical mixed-Mode I and II delamination failure criterion is shown in Fig. 1.6, and the inserts in this figure show the DCB, ENF, and MMB test configurations. The interlaminar fracture toughness is determined as a critical value of the strain energy release rate, Gc, plotted as a function of the mixed-mode ratio, GII/G. For the pure Mode I opening case, GII/G is equal to zero, whereas for the pure Mode II case, GII/G is equal

P

2.5

P2

IM7/E7T1-2 Graphite epoxy

Q2

ENF–Mode II

2.0 P

Gc, kJ/m2

c 2h

1.5

La

MMB–Mode I & II 1.0 DCB–Mode I 0.5

0 0

0.2

0.4

0.6

0.8

GII /G

1.6 Mixed-Mode I and II delamination criterion.


1


13

to unity. The apparent toughness increases monotonically as GII/G increases from zero to unity, i.e from the pure opening Mode I case to the pure shear Mode II case. Furthermore, due to the complex micro-mechanisms involved, the scatter is very large for the Mode II case (O’Brien, 1998). Delamination propagation procedures require a propagation criterion that is based on a mixed-mode failure criterion. One criterion is a power-law criterion developed by Wu and Reuter (1965) and given by: n p m  G I  +  G II  +  G III  ≥ 1      G Ic   G IIc   G IIIc 

1.16

where m, n, and p are empirically determined exponents for the mixed-mode criterion. For two-dimensional problems, only the first two terms in Eq. 1.16 are evaluated. If the left-hand side of Eq. 1.16 evaluates greater than or equal to unity then delamination growth will occur. Wu and Reuter (1965) found that for two-dimensional orthotropic plates, this empirical form of the general fracture criterion is applicable when m = 1 and n = 2. Another criterion developed by Benzeggagh and Kenane (1996) for twodimensional problems is expressed as: G I + G II G Ic + ( G IIc

G II  – G Ic )   G I + G II 

m

≥1

1.17

Reeder (2006) extended this mixed-mode criterion to three dimensions developed for the case when the critical strain energy release rates for Mode II and Mode III (GIIc and GIIIc, respectively) have different values. If they have the same value, then it reduces to the form given by Benzeggagh and Kenane (1996). Reeder’s criterion (2006) is expressed as:

G I + G II + G III m  G II + G III  G Ic + ( G IIc – G Ic )   G I + G II + G III  m  G III   G II + G III  + ( G IIIc – G IIc )  ≥1   G II + G III   G I + G II + G III 

1.18

or with some re-arranging of terms as: G I + G II + G III ≥1 G II ( G IIc – G Ic ) + G III ( G IIIc – G Ic )   G II + G III  m –1  G Ic + G I + G II + G III    G I + G II + G III 

1.19 © 2008, Woodhead Publishing Limited

14


Again when the left-hand side of the equation evaluates greater than or equal to unity, then crack growth will occur. In these expressions, the critical strain energy release rates for each mode of fracture are denoted by GIc, GIIc, and GIIIc, respectively. Gc is the total critical strain energy release rate for a given mixed-mode condition, and G is the sum of the three strain energy release rate components. The critical values are determined through fracture toughness testing. To be able to characterize delamination fully, fatigue characterization methods are also needed. Currently, the only ASTM standard is for Mode-I (DCB) delamination onset (ASTM, 1998). No standards exist for delamination onset in Modes II, III and mixed-modes. Furthermore, there are currently no standards for characterization of delamination growth. The rate of delamination growth with fatigue cycles can be characterized as a function of the maximum applied cyclic strain energy release rate, Gmax. This is usually described as a plot of da/dN vs. Gmax (Prel et al., 1989; Kageyama et al., 1995). The delamination growth rate can therefore be expressed as a power-law function:

da = A( G ) n max dN

1.20

Equation 1.20 is similar to the Paris law for metallic materials. For Mode-I delamination growth, the exponent n for composites may vary between 6 and 10, which is high compared to that for metallic materials. This exponent for Mode-II delamination growth and for toughened resin composites may be lower (between 3 and 5). Because of the large values of the exponent n, very small changes in Gmax can result in large changes in the delamination growth rate, which makes it difficult to establish reasonable inspection intervals for implementing the classical damage tolerance, slow crack growth methodology used for metals. Hence, a no-growth threshold approach is often proposed instead (O’Brien, 1990; Martin and Murri, 1990; O’Brien et al., 1989; Murri and Martin, 1993). Furthermore, for Mode I fatigue, fiber bridging typically develops in the unidirectional DCB specimens (Martin and Murri, 1990). Fiber bridging can cause a growing crack to arrest artificially early yielding a non-conservative threshold value. Therefore, as shown in Fig. 1.7, an alternate G versus N onset curve is typically generated to achieve a threshold characterization for delamination onset (Martin and Murri, 1990; O’Brien et al., 1989; Murri and Martin, 1993; ASTM, 1998). For some applications, this delamination threshold approach may be too conservative. Hence, several modifications to the classical Paris law have been suggested, including normalization by the static resistance curve (R-curve) (see Shivakumar et al., 2006) and adding additional terms to account for stress ratio (R-ratio) effects and near-threshold non-linearity (Paris, 2002).



Log da/dN

Delamination arrest Start of fatigue test Fiber bridging

15

Delamination onset

Static date

GImax

GIth GIth Log GImax

106 Cycles to delamination onset

1.7 Experimental technique to obtain a G-threshold (Glth) for delamination onset.

1.3.2

Strain energy release rate analysis

The strain energy release rate, G, associated with onset and growth is a measure of the delamination driving force. This G-value must be determined and compared to the measured fracture toughness, Gc, to predict delamination growth. Typically, a plot of the G components due to the three unique fracture modes (GI, GII, GIII) and the total G = GI + GII + GIII are calculated as a function of delamination length, a. The virtual crack closure technique (VCCT) (Rybicki and Kanninen, 1977; Raju, 1987) in conjunction with finite element analysis is widely used for evaluating the strain energy release rates. The VCCT is usually implemented using nodal displacements and nodal forces at the local elements in the vicinity of the delamination front in separate, individually developed, post-processing routines. The VCCT is now available in general purpose programs such as ABAQUS (ABAQUS, 2007) and MSC NASTRAN SOL 400 and SOL 600 (MSC NASTRAN, 2007). An example evaluation of these strain energy release rate in two and three-dimensional finite element analysis is presented next. Figure 1.8 shows local modeling near a delamination tip in two-dimensional finite element analysis modeled with 4-noded quadrilateral elements. The VCCT utilizes the nodal forces at the nodes at the delamination tip (Yi, Xi) and the relative displacements at nodes behind the delamination tip (∆vk,j, ∆uk,j). The strain energy release rates are computed easily as (Rybicki and Kanninen, 1977; Raju, 1987): GI =

1 Y ⋅ ∆v k,j 2∆a i


16

Delamination behaviour of composites y

J I k

i x

j

J+1

I +1

∆a

∆a

1.8 VCCT scheme for 4-noded quadrilateral elements in twodimensional analysis.

G II =

1 X ⋅ ∆u k,j 2∆a i

1.21

∆uk,j = uk – uj, and ∆vk,j = vk – vj where Xi and Yi are the nodal forces at node i in the x- and y-directions, respectively, evaluated from elements I and J, and ∆uk,j and ∆vk,j are the relative u and v displacements between the nodes k and j, respectively (see Fig. 1.8). Figure 1.9 shows local modeling on the delamination plane when the laminate is modeled with 8-noded hexahedron (brick) elements. The delamination front is represented by rectilinear segments (j – 1, j) and (j, j + 1). The nodes on the delamination front are labelled j – 1, j, j + 1. The elements ahead of delamination front and above the delamination plane are labelled I and I + 1. The elements behind the delamination front are labelled J and J + 1 above the delamination plane and K and K + 1 below the delamination plane. For clarity, the brick elements are not shown in the figure, but rather the element labels point to the corresponding face on the delamination plane. As the strain energy release rates, G, can vary along the delamination front, the G-values need to be computed at each of the nodes j – 1, j, j +1 on the delamination front. Raju et al. [1988] suggested evaluating the G-values at the nodes using the nodal forces at the delamination-front nodes and the relative displacements at the appropriate nodes behind the delamination front as:


Fracture mechanics concepts I +1

17

z, w

y, v x, u

J+1

j+1 b I+1

k +1 l+1

j ∆A

k

K +1

l

k–1

I bI

J

j–1

K l–1

∆a

∆a

1.9 VCCT scheme for 8-noded hexahedral element in threedimensional analysis.

GI =

Z j ∆w k , l 2∆ Aj

G II =

X j ∆u k , l 2∆ Aj

G III =

Y j ∆v k , l 2∆ Aj

1.22

where Zj, Xj, and Yj are the nodal forces at node j evaluated using elements I, I + 1, J, and J + 1. The relative displacements at the nodes behind the delamination front are computed as: ∆wk,l = wk – wl, ∆uk,l = uk – ul, and ∆vk,l = vk – vl

1.23

The ∆Aj is the area attributed to the node j and is the shaded region in Fig. 1.9. This area can be computed easily as:

∆A j =

( b I + b I +1 ) ⋅ ∆a 2

1.24

where bI and bI+1 are the widths of the elements I and I +1, respectively, as shown in Fig. 1.9. For the nodes at the ends of the delamination front, for the first layer, ∆A1 = (b1/2) · ∆a and for the last layer, ∆AN = (bN/2) · ∆a, where b1 and bN are the widths of the first and the last layers of elements on the delamination front. The G-calculations involved in Eq. 1.22 are repeated at © 2008, Woodhead Publishing Limited

18


each of the nodes on the delamination front to obtain the strain energy release rates along the delamination front. The total strain energy release rate can be evaluated using G = GI + GII + GIII

1.25

A comprehensive review of the state-of-the-art of VCCT was recently presented by Krueger (2004). The state-of-the-art using fracture mechanics based analyses of strain energy release rates (G) for composite delamination has recently been outlined in a new section drafted for Composite Materials Handbook (CMH-17) (Schaff, 2002). Several investigators (Sun and Jih, 1987; Raju et al., 1988; Krueger, 2004]) have shown that as the element size in terms of ∆a/h (where h is the ply thickness in a composite laminate) near the delamination tip is reduced, VCCT calculations for the individual modes do not show convergence while the total strain energy release rate remains constant. This non-convergence is attributed to the oscillatory part of the singularity (γ) discussed in Section 1.2.4 (see Eq. (1.15)). Krueger (2004) reviewed the recent literature on this issue and points out that for certain values of (∆a/h) the variation in the individual-mode strain energy release rates is small. After a careful study of this behavior, a recommended range of (∆a/h) is proposed. A value of (∆a/ h) >1 requires smearing ply properties for plies of different orientations, which is not acceptable if individual plies need to be modeled. At the other extreme, a value of (∆a/h) < (1/20), violates the assumption of a homogeneous continuum since this value borders on where micromechanics should be used. As such, the recommended element size range is (1/20) ≤ (∆a/h) ≤ 1. In this range of values of (∆a/h), the variation in the individual mode strain energy release rate is small. Most analysts in the literature tend to use element sizes in the range (∆a/h = 1/8 to 1/4 ) and hence the issue of the nonconvergence of the individual modes for delamination problems in composite laminates appears to be resolved. This issue is much more difficult to resolve for general interface cracks along a bi-material interface, such as coatings on substrates in the electrical chip industry.

1.3.3

Delamination prediction

Flexbeam fatigue life prediction Composite rotor hubs contain tapered flexbeams (see Fig. 1.10) with large numbers of ply terminations, or ply drops, to taper the beam thickness (see Fig. 1.11a). These ply drops cause delamination in the flexbeam under high combined axial tension and cyclic bending loads. Murri et al. (1998) tested 1/6 sized flexbeams under combined tension and cyclic bending loads (loads P and V in Fig. 1.10) and analyzed these flexbeams to determine the fatigue


Fracture mechanics concepts Full size flexbeam

19

P

V Load cell Top grip

Thin Strain gages Taper

Thick 1/6 section of flexbeam

Bottom grip

Flexbeam specimen in ATB load frame

1.10 Tension and bending testing of composite flexbeams.

life. Strain energy release rate, G, distributions associated with delaminations growing in either direction along a and a′ and b in Fig. 1.11(a) from ply drops were calculated from two-dimensional finite element analysis using VCCT. The peak G value of the highest distribution possible (see Fig. 1.11b) was compared to mode I delamination fatigue onset threshold characterization data (see Fig. 1.11c) to predict the number of cycles to onset of unstable delamination from ply drops in these tapered laminates (see Fig. 1.11d). These results are shown in Fig. 1.11 (Murri et al., 2001). Excellent agreement was obtained between the predictions and test data represented by the mean S-N test data, plus and minus one standard deviation. Skin/stiffener pull-off strength and life A simple specimen was developed for studying skin/stiffener debonding. This configuration, shown in Fig. 1.12, consists of a skin bonded to a tapered flange laminate (Minguet and O’Brien, 1996; Krueger et al., 2000). A generalized fatigue methodology based on the approach presented by Murri et al. (1998) was developed and used to predict the delamination onset life for skin/stiffener debonding under cyclic loading (see Krueger et al., 2002). © 2008, Woodhead Publishing Limited

20

Delamination behaviour of composites Y a′ a

b

14.5 plies 4.1 plies

X mid plane

Thin

Taper

Thick

(a) Delaminations modeled at ply drop Delamination growth along b with delaminations along a and a′

140 Delamination growth along a′

120

G, J/m2

100 80 60 40

P = 35.58 kN V = 4.45 kN

20 0 –40

–20

a, a′

b 0 20 40 Delamination length, mm

60

80

(b) FE G-analysis using VCCT

400 350

GImax = cNd

GImax, J/m2

300 250 200 150 100

DCB fatigue data Curve fit to data Curve fit ± one standard deviation

50 0 100

101

102 103 104 105 106 N, cycles to delamination onset

107

(c) Delamination characterization data

1.11 Flexbeam fatigue life prediction methodology.



21

0.015

0.010 εmax Flexbeam test data Calculated fatigue life Calculation ± one standard deviation

0.005

0.000 103

104 105 106 N, cycles to delamination failure

107

(d) Flexbeam life prediction

1.11 (Continued) Qxy

Frame or stiffener

Nxx Flange

Failure initiation

Mxx

Skin

Bondline

Tip of flange Flange Strain gauges

25.4 mm 203.2 mm Skin Simple specimen configuration

1.12 Composite skin-stiffener debonding characterization.

Delaminations from an initial matrix crack are modeled at the 45/-45 degree ply interface as shown in Fig. 1.13(a). For an applied load P and a delamination length a the strain energy release rates are evaluated using finite element analysis and VCCT. The resulting total strain energy release rate curve is shown in Fig. 1.13(b). This curve shows that the GT reaches a peak value and drops as the delamination length increases. The GII/GT ratio at the peak value is evaluated from the VCCT calculations. Owing to small variations in the © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites P

P

Initial matrix crack Adhesive pocket

Delamination ± 45° 0°/90° ±45° Adhesive 45° –45° 0° 26.8 mm

y

26 mm x (a) Skin/stringer debond failure

2000

GT, J/m2

1500

Flange tip

1000 ∆a 500

0 26

26.5

27 27.5 x coordinate, mm

28

(b) FE G-analysis using VCCT

350 300

GT, J/m2

250 200

GI /GT = 0.18

Gi

150 100 50 0 100

Ni 101

102

103 104 105 106 Fatigue life, N, cycles

107

108

(c) Delamination characterization data

1.13 Skin/Stringer debonding fatigue life prediction methodology.


Fracture mechanics concepts P

23

P

25

Applied load P, kN

20

Onset life = total + crack onset Mean

15

10

5 Predicted mean onset curve 0 100

101

102

103

104

105

106

107

108

Number of cycles, Nb, to delamination onset

(d) Skin/Stringer life prediction

1.13 (Continued)

computed values of the mixed mode ratio, an average value for GII/GT = 0.18 was used in the following step. From a delamination onset surface established from DCB, ENF and MMB fatigue tests (see Figure 27 of Krueger et al., 2002, page 75) an appropriate fatigue life curve was then extracted for a ratio GII/GT = 0.18 as shown in Fig. 1.13(c). With the information available, that links the total energy release rate GT, to the applied cyclic load, P, and the total energy release rate GT to the fatigue life, N, it was possible to create a relationship between the externally applied load, P, and the fatigue life for delamination onset, N.. The resulting plot yielded the prediction onset curve as shown in Fig. 1.13(d). Fatigue tests were conducted to determine the fatigue life and these P-N data are shown as symbols in Fig. 1.13(d). The PN data corresponds to the total life to delamination onset minus the life to matrix cracking onset. Predicted lives (solid curve in Fig. 1.13(d)) are in excellent agreement with test data. The details of this methodology are available in Krueger et al. (2002). The entire methodology is explained in a future publication CMH-17 (2009).

1.4

Future trends

The previous sections outline the basic theoretical underpinnings and the current state of the art for characterizing, analyzing, and predicting delamination. However, more work is needed in order to achieve a fully mature methodology for use in design and certification of composite structures. First, simple closed-form strain energy release rate solutions for typical material and geometric discontinuities in composite structural components © 2008, Woodhead Publishing Limited

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need to be developed for use in preliminary design tools, similar to the stress-intensity solutions available in handbooks for metallic structures. Second, for detailed design and certification, techniques for calculating strain energy release rates need to be incorporated in commercial finite element codes. A first attempt at this is the current inclusion of a VCCT element in ABAQUS (2007) and MSC NASTRAN (2007). Similar tools are needed in other codes commonly used in industry for detailed structural design. In addition, further refinements to resolve numerical issues are needed to achieve accurate, repeatable, and robust solutions. Furthermore, although delamination onset has been successfully predicted in laboratory coupons and sub-elements (Section 1.3.3), delamination onset and growth predictions obtained using these codes need to be validated by comparison with experimental data in structural elements. Finally, methodologies that address the unique aspects of composite delamination behavior must be developed and implemented. These may involve a combination of the threshold concepts noted earlier and the classical slow growth approach used for metallic materials (O’Brien, 2007). Implementation of these approaches will require standardization of all three pure modes and mixed-mode characterization test methods for delamination onset threshold and growth. All these attempts lead to development of accurate prediction methodologies and efficient composite structures and structural components.

1.5

Concluding remarks

This chapter presents delamination analyses, onset, growth, and prediction methods. First, fracture mechanics concepts for homogeneous, isotropic linear elastic materials are presented. Next, the evaluation of strain energy release rates from near-tip stress and displacement fields is discussed. Similar concepts for orthotropic and anisotropic materials with cracks are presented. As delaminations in composite laminates are a special case of cracks between two different anisotropic materials, the problem of an interface crack in a bimaterial plate is discussed in detail. The efforts in the literature for characterization of delaminations, the application of fracture mechanics principles to delaminations, and to characterize delamination onset are discussed. The methods available to evaluate the strain energy release rates in two- and three-dimensional analysis are presented. Guidance on finite element modeling near delamination tips is given. Two examples of predictions of life of delaminated composite components are presented. Finally, future directions on the work needed to achieve a fully mature methodology for use in design and certification of composite structure are outlined.



1.6

25

References

ABAQUS (2007), VCCT for ABAQUS User’s Guide, Version 1.3-1, Dassault Systèmes. Aminpour, M A (1986), ‘Finite Element Analysis of Propagating Interface Cracks in Composites’, PhD. Dissertation, University of Washington, Seattle, WA. (Available from University of Microfilms, Ann Arbor, MI) Anderson, T L (2005), Fracture Mechanics – Fundamentals and Applications, 3rd Edition, CRC Press, Boca Raton. ASTM (1998), ASTM Standard Test Method D6115–97, Standard Test Method for Mode I Fatigue Delamination Growth Onset of Unidirectional Fiber Reinforced Composite materials, in Annual Book of ASTM Standards, Vol. 15.03, ASTM International, West Conshohocken, PA. ASTM (2001a), ASTM Standard Test Method D5528-01, Standard Test Method for Mode I Interlaminar Fracture Toughness of Unidirectional Continuous Fiber Reinforced Composite materials, in Annual Book of ASTM Standards, Vol. 15.03, ASTM International, West Conshohocken, PA. ASTM (2001b), ASTM Standard Test Method D6671-01, Standard Test Method for Mixed Mode I-Mode II Interlaminar Fracture Toughness of Unidirectional Fiber Reinforced Polymers, in Annual Book of ASTM Standards, Vol. 15.03, ASTM International, West Conshohocken, PA. Benzeggagh, M L and Kenane, M (1996), ‘Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites with Mixed-Mode Bending Apparatus’, Composite Science and Technology, 56, 439–449. Broek, D (1982) Elementary Engineering Fracture Mechanics, Martinus Nijhoff Publishers, Kluwer Boston Inc., Boston, MA. CMH-17 (2009), Composite Materials Handbook, Volume 3, Chapter 12.6.10 on Composite Fatigue Damage Onset, To be published 2009. Davidson, B and Sun, X (2006), ‘Geometry and Data Reduction Recommendations for a Standardized End Notched Flexure Test for Unidirectional Composites’, Journal of ASTM International, 3, (9). England, A H (1965), ‘A Crack Between Dissimilar Media’, Jnl. Applied Mechanics, Trans. ASME, 32, (2), 400–402. Erdogan, F (1965) ‘Stress Distribution in Bonded Dissimilar Materials with Cracks’, Jnl. Applied Mechanics, Trans. ASME, 32, 403–410. Ewalds, H L and Wanhill, R J H (1984), Fracture Mechanics, Edwards Arnold Publishers, London. Irwin, G R (1957), ‘Analysis of Stresses and Strains Near the End of a Crack Traversing in a Plate’, Jnl. of Applied Mechanics, Trans. of ASME, 24, 351–369. Johnson, S J and Mangalgiri, P D (1987), ‘Investigation of Fiber Bridging in Double Cantilever Beam Specimens’, ASTM Journal of Composites Technology and Research, 2, 10–13. Jones, R M (1975), Mechanics of Composite Materials, McGraw-Hill Book Company, Washington, DC. Kageyama, K Kimpara, I Ohsawa, I Hojo, M and Kabashima, S (1995), ‘Mode I and Mode II Delamination Growth of Interlayer Toughened Carbon/Epoxy Composite System’, in Composite Materials: Fatigue and Fracture, Fifth Volume, STP 1230, American Society for Testing and Materials, 19–37. Krueger, R (2004), ‘The Virtual Crack Closure Technique: History, Approach and Applications’, Applied Mechanics Reviews, 57, (2), 109–143.


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Krueger, R Cvitkovich, M K O’Brien, T K and Minguet, P J (2000), ‘Testing and Analysis of Composite Skin/Stringer Debonding under Multi-Axial Loading’, Jnl. Composite Materials, 34, (15), 1264–1300. Krueger, R Paris, I L O’Brien, T K and Minguet, P J (2002), ‘Fatigue Life Methodology for Bonded Composite Skin/Stringer Configurations’, Jnl. Comp. Tech. and Research, 24, 56–79. Lee, S M (1993), ‘An Edge Crack Torsion Method for Mode III Delamination Fracture Testing’, Jnl. Comp. Tech. and Research, 15, (3), 193–201. Li, J, Lee, S M Lee, E W and O’Brien, T K (1997), ‘Evaluation of the Edge Crack Torsion (ECT) Test for Mode III Interlaminar Fracture Toughness of Laminated Composites’, Jnl. Comp. Tech. and Research, 19, (3), 174–183. Martin, R H and Murri, G B (1990), ‘Characterization of Mode I and II Delamination Growth and Thresholds in AS4/PEEK Composites’, in Composite Materials: Testing and Design, Ninth Volume, STP 1059, American Society for Testing and Materials, 251–270. Minguet, P J and O’Brien, T K (1996), ‘Analysis of Test Methods for Characterizing Skin/Stringer Debonding Failures in Reinforced Composite Panels’, in Composite Materials: Testing and Design, Twelfth Volume, STP 1274, American Society for Testing and Materials, 105–124. MSC NASTRAN (2007), Implicit Nonlinear (SOL 600) User’s Guide, MSC Software. Murri, G B and Martin, R H (1993), ‘Effect of Initial Delamination on Mode I and Mode II Interlaminar Fracture Toughness and Fatigue Fracture Thresholds’, in Composite Materials: Fatigue and Fracture, Fourth volume, STP 1156, American Society for Testing and Materials, 239–256. Murri, G B O’Brien, T K and Rousseau, C Q (1998), ‘Fatigue Life Methodology for Tapered Composite Flexbeam Laminates,’ Jnl. of the Amer. Helic. Soc., 43, (2), 146– 155. Murri, G Schaff, J R and Dobyns, A L (2001), ‘Fatigue and Damage Tolerance Analysis of a Hybrid Composite Tapered Flexbeam’, Proceedings of the AHS International 57th Annual Forum and Technology Display, Washington, DC. O’Brien, T K (1990), ‘Towards a Damage Tolerance Philosophy for Composite Materials and Structures’, Composite Materials: Testing and Design, Vol. 9, ASTM STP 1059, S P Garbo, Ed., ASTM, Philadelphia, 7–33. O’Brien, T K (1998), ‘Composite Interlaminar Shear Fracture Toughness, GIIc: Shear Measurement or Sheer Myth?’, Composite Materials: Fatigue and Fracture, Seventh Volume, STP 1330, American Society for Testing and Materials, 3–18. O’Brien, T K (2007), ‘Towards a Delamination Fatigue Methodology for Composite Materials’, Proceedings of the 16th International Conference on Composite Materials, Kyoto, Japan. O’Brien, T K Murri, G B and Salpekar, S A (1989), ‘Interlaminar Shear Fracture Toughness and Fatigue Thresholds for Composite Materials’, in Composite Materials: Fatigue and Fracture, Second Volume, STP 1012, American Society for Testing and Materials, 222–250. Parker, A P (1981), The Mechanics of Fracture and Fatigue – An Introduction, E & F N Spon Ltd., London. Paris, I L (2002), Composite Materials Handbook (CMH-17-1F), Vol. 1, Chapter 6, section 6.8.7.1. Paris, P C and Sih, G C (1965), Stress Analysis of Cracks, ASTM Special Publication No. 381, ASTM. © 2008, Woodhead Publishing Limited


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Prel, Y J Davies, P Benzeggagh, M and de Charentenay, F X (1989), ‘Mode I and Mode II Delamination of Thermosetting and Thermoplastic Composites’, in Composite Materials: Fatigue and Fracture, Second Volume, STP 1012, American Society for Testing and Materials, 251–269. Raju, I S (1987), ‘Calculation of Strain-Energy Release Rates with Higher Order And Singular Finite Elements’, Eng. Fract. Mech., 28, 251–274. Raju, I S Crews J H and Aminpour, M A (1988), ‘Convergence of Strain Energy Release Rates Components for Edge-delaminated Composite Laminates’, Eng. Fract. Mech., 30, 383–396. Ratcliffe, J G (2004), Characterization of the Edge Crack Torsion (ECT) Test for Mode III Fracture Toughness Measurement of Laminated Composites, NASA TM-2004213-269, September. Reeder, J R (2002), Composite Materials Handbook (CMH-17-1F), Vol. 1, Chapter 6, section 6.8.6. Reeder, J R (2006), ‘3D Mixed-Mode Delamination Fracture Criteria–An Experimentalist’s Perspective’, Paper #110, Proceedings of the 21st American Society for Composites Technical Conference, Dearborn, MI. Russell, A J (1982), ‘On Measurement of Mode II Interlaminar Fracture Energies’, Defence Research Establishment Pacific, Victoria, British Columbia, Canada, Materials DREP Report, 82–0. Rybicki, E F and Kanninen, M F (1977), ‘A Finite Element Calculation of Stress Intensity Factors by a Modified Crack Closure Integral’, Eng. Fract. Mech., 9, 931–938. Sanford, R J (2003), Principles of Fracture Mechanics, Pearson Education, Inc., Upper Saddle River, NJ. Schaff, J R (2002), Composite Materials Handbook (CMH-17), Vol. 3, Chapter 5, Debonding and Delamination Guidelines Task Group proposed new section on Energy Release Rate Analysis techniques for Composite Delamination , section 5.4.5. Shivakumar, K.N Chen, H Abali, F and Davies, C (2006), ‘A Total Fatigue Life Model for Mode I Delaminated Composite Laminates’, Int. J. of Fatigue, 28, (1), 33–42. Sun, C T and Jih, C J (1987), ‘On Strain Energy Release Rates or Interfacial Cracks in Bimaterial Media’, Engng. Fract. Mech., 28, 13–20. Tada, H Paris, P C and Irwin, G R (2000), The Stress Analysis of Cracks Handbook, 3rd Edition, ASME, New York, NY. Wang, S S and Choi, I (1983a), ‘The Interface Crack Behavior in Dissimilar Anisotropic Composites under Mixed-mode Loading’, Jnl. Applied Mechanics, Trans. ASME, 50, (1), 179–183. Wang, S S and Choi, I (1983b), ‘The Mechanics of Delamination in Fiber-Reinforced Composite Laminates, Part I: Stress Singularities and Solution Structure’, NASA Contractor Report, NASA CR 172269, November 1983. Wang, S S and Choi, I (1983c), ‘The Mechanics of Delamination in Fiber-Reinforced Composite Laminates, Part II: Delamination Behavior and Fracture Mechanics Parameters’, NASA Contractor Report, NASA CR 172270, November 1983. Wang, S S (1984), ‘Edge Delamination in Angle-ply Laminates’, AIAA Jnl., 22, (2) 256– 264. Williams, M. L. (1959), ‘The Stresses Around a Fault or Crack in Dissimilar Media’, Bull. of the Seismol. Soc. America, 49, (2), 199–204. Wu, E M and Reuter, R C Jr., (1965), ‘Crack Extension in Fiberglass Reinforced Plastics,’ T&AM Report No. 275, Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, IL, February 1965. © 2008, Woodhead Publishing Limited

2 Delamination in the context of composite structural design A R I C C I O, CIRA (Centro Italiano Ricerche Aerospaziali – Italian Aerospace Research Centre), Italy

2.1

Introduction

Advanced composite materials, due to their high potential in terms of stiffness/ weight ratio, are very attractive for structural applications where low weight and high stiffness conditions have to be met. Indeed, the inherent heterogeneity of composites facilitates the design of structures with optimised stiffness and strength in particular locations and directions according to specific requirements. Hence, when designing composite structural elements, additional design variables related to the constituent composites’ internal configuration (such as stacking sequence) must be taken into account. However, the heterogeneity of composites is also their main source of weakness, irrespective of the nature of the constituents. As an example, in laminated carbon fibre reinforced polymers, the interface between reinforced fibres and the matrix is critical for damage onset and development and the damage mechanisms themselves are numerous and closely connected. In many situations, the most critical damage mechanism for composites design is the delamination between adjacent layers. The delaminations can occur during the manufacturing process of the composite elements or they can arise as a consequence of impact with foreign objects. Whatever the cause of the delaminations, they can be very dangerous and can easily lead to a premature collapse of the structures. In Fig. 2.1, a delamination in a laminated carbon fibre reinforced polymer is shown. The picture is taken from a microscopy analysis of a damaged composite joint element performed as part of the European project BOJCAS (Bolted Joints in Composite Aircraft Structures – Contract N°: G4RD-CT-1999-00036). Nowadays, owing to the relative lack of knowledge about the physical phenomena which lead to the onset of delamination, its development and its interaction with other damage mechanisms, and the lack of robust models for predicting these phenomena, the possible presence of delaminations and damage in general is taken into account in composite design by introducing additional safety margins. These additional safety margins come from an 28 © 2008, Woodhead Publishing Limited


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2.1 Delamination between plies in a laminated composite (courtesy of the European project BOJCAS consortium Contract No: G4RD-CT1999-00036).

indiscriminate reduction of the allowables for composite materials which leads to over-conservative designs, not fully realising the expected economical benefits of composites. In order to avoid over-dimensioning in composite design, it is necessary to fully understand the physics behind composite damage mechanisms and to develop theories and analytical/numerical tools that can take into account the onset and growth of delamination from the earliest phases of design. The trend in composites-related research is to investigate the feasibility of virtual design environments comprising numerical tools that are able to simulate damage evolution in composites and that can allow for a reduction in design costs by reducing the amount of the experimental activity needed for the certification of composite materials and composite components. With such numerical tools, capable of simulating the behaviour of composite elements under realistic loading conditions, it is possible to design the composite items which are critical regarding the appearance of delaminations, limiting the over-dimensioning resulting from the indiscriminate reduction of composite materials allowables. The right type of virtual design environment enables the application of innovative design philosophies oriented to the definition of light-weight and low-cost composite structures obtained through effective damage management. In this chapter, delamination is investigated in the context of composite design, providing all the basic information needed to understand how the possibility of delamination is taken into account when designing with composite materials. In Sections 2.2 and 2.3, phenomenological considerations, respectively, on the onset and growth of delamination are described in order to explain the influence of loads and geometrical/material/process parameters on delamination formation and evolution. An attempt is made to classify the different sources of delamination onset and growth, providing all the basic information needed for an insight into the physical aspects behind delamination © 2008, Woodhead Publishing Limited

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phenomena. In these sections the delamination onset and the delamination growth are treated separately; however, they are physically strictly related to each other and the separation makes sense only from a design methodology perspective. Appropriate references are provided for all the physical phenomena cited in Sections 2.2 and 2.3, for which an in-depth description is outside the purposes of this chapter. The management of delamination in composite design is the focus of Sections 2.4, 2.5, 2.6 and 2.7. Examples of methodologies for the preliminary design of composite panels, considering impact-induced delamination onset and growth, are described and critically assessed. Finally, an example of the procedures for cost-effective delamination management in composite design is introduced and the influence of delamination management design philosophies on the lifecycle cost of composite components is quantified in order to prove their importance in ensuring the safety and affordability of composite materials in structural design.

2.2

Physical phenomena behind delamination onset

In this sub-section, an overview is given of the delamination onset phenomenon, focusing on the causes that can lead to delamination initiation in composite laminates. An attempt to classify the causes of delamination onset is made in Fig. 2.2. Manufacturing, hygrothermal and thermal conditions, machining, geometrical configurations, discontinuities and mechanical loads are identified as the main categories of delamination onset drivers. Each one of these categories is briefly described here.

2.2.1

Manufacturing and environmental effects

The manufacturing process of composite materials generally requires that the composite laminates are placed in a high-temperature environment, in order to ensure resin curing, and then cooled to ambient temperature. It is a known fact that thermal stresses can arise due to temperature gradients in restrained structures: even if the composite structure is globally unrestrained, the deformation of a single ply is highly influenced by the presence of adjacent plies that can be made of different materials and so characterised by a different value of CTE (thermal expansion coefficient). Therefore residual stress can be induced in the composite laminate and the onset of delamination is more likely [1]. The hygrothermal environment can significantly affect the laminate’s resistance to delamination because of the dependence of material properties on temperature and moisture [1–2].


Manufacturing

Environmental effects

Machining

Geometry

Mechanical load

Drilling

Impact

Residual stressinduced delaminations

Hydrothermal

Thermal

Skin-stringer debonding

2.2 Classification of the main delamination onset causes.

Free edge interlaminar stress

Tapered structures

Joints and bonding


Delamination onset

31


32

2.2.2


Machining

Machining of composites can also induce defects in laminates. A normal drilling operation [3–5] can create delaminations (see Fig. 2.3). The drill’s penetration into the lamination produces peel-off of the uppermost plies, and heat generation should be controlled in order to avoid altering the matrix properties. If the back surface of the laminate is not adequately supported, delamination can be induced on the other side of the laminate (see Fig. 2.3). The angle of penetration, the drill shape and also the fibre orientation and lay-up sequence affect the extent of drilling-induced delaminations.

2.2.3

Critical geometrical configurations

It is possible to recognise some geometrical configurations that can be considered critical for the initiation of delamination such as curved segments, transitions, sudden changes of section and inclusions. Laminated composite structures with tapered thicknesses, which are designed to produce parts for specific performance requirements, can be particularly critical. The thickness of the laminates is reduced by dropping plies internally (see Fig. 2.4). However, these ply-drop locations are sources for delamination initiation under bending and tension loads [6]. The low delamination durability of such critical configurations can result in high repair and part-replacement costs. A lot of studies have been dedicated to the analysis of damage development in composite laminates in the presence of inclusions. In this case, the failure occurs due to a process of damage accumulation caused by characteristic stress concentration

2.3 Delamination caused by drilling (on entry and exit respectively).



33

2.4 Tapered laminates.

2.5 Damage onset in bolted composite joints – bearing failure mode (courtesy of the European project BOJCAS consortium Contract No: G4RD-CT-1999-00036).

at the interface between the inclusion (such as a bolt [7–9], holes [10] and notches [11]), and the composite. In particular, in the case of bolted composite joints, delamination onset is likely to occur when the joints experience bearing failure conditions. An example of damage onset in bolted composite joints is shown in Fig. 2.5. In this figure, taken from the results of the European research project BOJCAS, and representing the damage status at failure for a single lap-bolted composite joint tested under tension, the delamination and shear matrix cracks in the bearing plane, perpendicular to the loading direction, are clearly visible. Smaller delamination effects and the rupture of fibres at the hole interface, due to drilling operations, are also important. In order to replace traditional mechanical fastening methods, co-curing, co-bonding, and secondary bonding have been used. In this case other problems such as the skin-stringer debonding of reinforced panels related to the stress © 2008, Woodhead Publishing Limited

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state during post-buckling [12] and the fatigue loading condition [13–15] can arise. The failure usually starts with the plies near the skin/stiffener interface and in particular at the flange tip where, due to the local stress state, matrix cracks can be induced and consequently delaminations can appear and grow. A schematic representation of skin-stringer debonding is given in Fig. 2.6. Free edge inter-laminar stresses can also be considered as one of the most frequent causes for delamination onset in laminated composites. Classical plate theory [16] assumes that it is possible to neglect the stresses acting through the laminate thickness (plane stress hypothesis) but this assumption is accurate only in regions sufficiently far from edges or holes and cut-out boundaries. This leads to the conclusion that in the presence of free edges, inter-laminar normal and shear stresses are not negligible (the stress state is three-dimensional). The presence of inter-laminar stress is principally due to the mismatch of engineering properties between adjacent plies [17] and the inter-laminar stress level at the edge is strongly dependent on the laminate stacking sequence [18–19]. In Fig. 2.7, an example of this dependency is given by showing the inter-laminar normal stress distributions along the thickness of a panel, at its edge, for two laminate stacking sequences. From the previous figure, it can be noted that the same plies can be rearranged in order to reduce the inter-laminar stresses: the plies positioning through the thickness, in fact, has a strong influence on the bending stiffness matrix [D] while the extensional stiffness matrix [A] is not affected by this

Stringer Delamination Skin

2.6 Skin-stringer debonding. © 2008, Woodhead Publishing Limited


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4

2

σz

z / ho 0

–2 [15°/-15°/45°/-45°]s [15°/45°/-45°/-15°]s

–4

2.7 Inter-laminar normal stress distribution at the edge of a composite plate for two different lay ups.

Interleaves

Ply termination

Free edge Edge cap

Tapering

2.8 Free edge delamination suppression concepts.

distribution. Failures can arise due to inter-laminar stresses exceeding the matrix strength and can cause delamination under both static and cyclic loading. In order to prevent the free edge effect, the angle between adjacent plies should be reduced as much as possible by rearranging the plies. In addition, edge reinforcement (such as edge cap, stitching, interleaves [20– 21]) or edge modification (ply termination or tapering) can be adopted (see Fig. 2.8) [17].

2.2.4

Low velocity impact

Low velocity impacts (due to dropped tools or caused by runway debris) generally happen accidentally during the manufacturing operation or the maintenance of the composite structure [22]. © 2008, Woodhead Publishing Limited

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Experimental studies have shown that delamination can appear between plies of different orientation when the threshold kinetic energy of the impactor has been reached. The formation of many matrix cracks as a result of impacts can induce delamination at ply interfaces. For thick laminates, matrix cracks are first induced at the top of the laminate due to the high contact stresses that arise where the impactor has dropped on the structure. In this case, the damage spreads from the top to the bottom of the laminate, inducing a typical damage configuration called ‘pine tree’ (see Fig. 2.9(a)). Thin laminates, on the contrary, can absorb the impact energy as bending deformation. In this case, the highest stresses are induced at the bottom edge of the laminate where matrix cracks can appear. Therefore for thin laminates, the characteristic damage pattern is a ‘reversed pine tree’ (see Fig. 2.9(b)). The delaminated area resulting from low velocity impact has an oblong shape (‘peanut shape’) inclined towards the direction of the fibres, in the lower plies at the interface.

2.3

Physical phenomena behind delamination growth

Whatever the cause of the onset of delaminations, the application of a load (compressive, fatigue or bending, and also a tensile load in joints) can induce a growth of the delamination. The identification of a threshold level for the growth of delaminations is of fundamental importance in understanding the real mechanical behaviour of delaminated composite structures. Starting from the fact that an interlaminate fracture, such as delamination, can be considered as a crack propagation [23], fracture mechanics is used for the analysis of the delamination growth and the ERR (energy release rate) is compared with the rate of fracture work in the delamination process. Three different fracture modes must be considered for composite materials (see Fig. 2.10) and the total ERR (GT) will be given as the sum of GI (Mode I due to interlaminar tension), GII (Mode II due to inter-laminar sliding shear) Impactor

Delaminations

(a) Thin laminates

(b) Thick laminates

2.9 Damage pattern for thin (a) and thick (b) laminates.



Mode I

Mode II

37

Mode III

2.10 Delamination fracture modes.

Delamination growth

Compressive load (buckling)

Fatigue

2.11 Classification of most critical loading conditions for delamination growth.

Unbuckled

Global buckling

Local buckling

Mixed buckling

2.12 Delamination growth scenarios under compressive loading conditions.

and GIII (due to inter-laminar scissoring shear) [24]. The prediction of the onset and growth of delamination can be obtained by comparing these ERR values (evaluated numerically) with the inter-laminar fracture toughness properties evaluated experimentally. In Fig. 2.11, a classification of the loading conditions which can induce the delamination growth is given. Compression and fatigue are identified as the most critical loading conditions for delamination growth. Delamination under compressive loading conditions can become critical and can lead to a premature collapse of the structures according to the scenarios shown in Fig. 2.12 [25–26]. Indeed, in most situations, after the local buckling of sub© 2008, Woodhead Publishing Limited

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laminates, the delamination growth can occur under compression and this phenomenon can quickly lead to sudden loss of load-carrying capability of the composite structures. In Fig. 2.13, the steps leading to failure of delaminated composite panels under compression are schematically shown: condition of un-buckled structure (Figure 2.13(a)), local buckling (Fig. 2.13(b)), delamination growth (Fig. 2.13(c)) and structural collapse (Fig. 2.13(d)). For particular structural details, such as the bolted joints, the tensile loading condition can induce a local compressive stress state, critical for delamination growth, near the hole. In fact, when joints fail following the bearing failure mode, as already pointed out in the previous subsection, delamination onset takes place in the bearing plane. These delaminations tend to growth under tensile loading condition or under tensile-compression fatigue conditions. Composites, in general, have been proven to have a better fatigue behaviour than metals for a tension-tension loading condition [27] but they have been

(a) Unbuckled

(b) Local buckling

(c) Delamination growth after buckling

(d)

Global buckling/structural collapse

2.13 Compression behaviour of delaminated composite panels.



39

found weak for tension-compression or multi-axial fatigue loading conditions due to the consequent degradation of the matrix and its fracture toughness properties (GIC, GIIC and GIIIC). The delamination growth can be induced by fatigue loadings (see Fig. 2.14) and usually the crack growth is related to the number of cycles for a given ratio between the minimum and the maximum applied stress [28–29]. Transverse reinforcement such as z pins and stitching are usually considered for limiting the delamination growth process [30] caused by static and fatigue loading conditions. However, they can facilitate the onset of the delamination due to the micro-damages and discontinuities introduced in the manufacturing process.

2.4

Introduction to delamination management in composites design

In the previous sections, an insight is given into the physical phenomena behind delamination formation and growth, and the main causes associated with the delamination failure condition are briefly illustrated. When designing with composites, these causes must be taken into account and delamination formation and growth must be adequately predicted and controlled in order to effectively exploit the potential of composite materials in structural applications. Recent research has shown how designing with composites needs a suitable virtual environment, based on effective analytical and numerical methodologies,

Initial delaminaton area

Propagation

2.14 Delamination growth under fatigue.


40


capable of simulating delamination-related physical phenomena at different levels of detail and with different degrees of accuracy, interacting with each other by means of multi-scale approaches. The need to predict delamination phenomena with different degrees of accuracy and consequent different computational effort, is directly related to the need for delamination prediction capabilities throughout the different design phases (conceptual design, preliminary design, optimisation and detailed design). Several methodologies for delamination onset and growth simulation can be found in the literature. Most of them are based on non-linear finite element analysis which can be adopted during the detailed design phase when the dimensions and the shape of composite components need to be refined. Only a few methodologies are fast enough to be effectively adopted in the preliminary design and optimisation phases. The availability of such fast analytical and numerical tools makes it possible to take the delamination phenomena into account from the earliest phases of the design, allowing effective delamination management in composite design. Depending on the design requirements, alternative and complementary delamination management philosophies are expected to be employed more often. Interesting examples are the delamination resistance concept, which is the ability of a structure not to develop delamination due to manufacturing processes, service loading conditions or external loading from unpredictable events (impacts, etc.), and the delamination tolerance concept, which is the ability of a delaminated composite structure to withstand the in-service loading conditions. Delamination management approaches can not only have an influence on the performance of composite structures, but they can also strongly influence their cost. In particular, the lifecycle cost, which nowadays is one of the driving parameters in composite structure design, has been proven to be dependent on delamination management approaches through the maintenance cost. In the following sections, some simplified methodologies for delamination onset and propagation prediction are introduced, which can be applied to delamination management in the preliminary structural design of composites. First, in Section 2.5, a simplified analytical/numerical approach for impactinduced delamination onset in stiffened composite panels is described. Then, in Section 2.6, a linearised numerical method for delamination growth initiation simulation, suitable for the prediction of the damage tolerance of composite panels under compression, is introduced. Finally, in Section 2.7, an example of a multi-objective optimisation procedure intended for the cost-effective design of stiffened composite panels is presented. This procedure can handle several performance constraints, including the ones related to delamination management. The application of this procedure to the design of cost-effective stiffened composite panels, with buckling and impact-induced delamination resistance constraints, is briefly described in order to show the benefits in © 2008, Woodhead Publishing Limited


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terms of lifecycle cost reduction that can be derived from a delamination management-based composite design.

2.5

Impact-induced delamination resistance in composites preliminary design

Among the different causes that can lead to delamination onset, attention is focused in this subsection on delamination onset induced by low-velocity impacts, in order to illustrate an example of methodologies for damage resistance design. As already shown in the previous section, matrix cracking, delaminations and superficial indentations are usually the result of lowvelocity impacts on composite structures. In order to take into account the complicated phenomena behind impactinduced damage, different methodologies have been developed over the last ten years to help predict the extent of the impact-induced damage [31–40]. Most of the earlier works are focused on the experimental investigations. Nevertheless, in recent years, some efficient numerical models have also been set up to study this problem. Some researchers [35–37] adopted a 1D finite element method (FEM) model based on a beam-like discretisation. This approach is computationally very efficient due to its simplicity and can be used to study some basic impact problems. However, this numerical model may fail to provide more detailed and comprehensive information, such as the in-plane delamination shape in the laminated plates. To perform more detailed stress analyses on the laminates, many authors [31, 32, 38] introduced 3D FEM models. However, there are some particular difficulties with 3D FEM, such as the exorbitant computational cost and the need for an automatic mesh generation following the extension of delamination. To overcome these difficulties, some researchers [33, 34, 39] employed the 2D plate model to study impact problems. In general, the research has focused mainly on the following aspects: 1. 2. 3. 4.

mechanisms of damage formation, especially the onset of various damages; dynamic fracture toughness for delamination extension; impact energy and velocity threshold for damage onset; relationship between the damage size and the impact parameters.

Very few works have been found on the full and direct numerical simulation of the entire impact damage process. Actually, most of the analysed numerical approaches give only a rough evaluation of the delamination sizes resulting from the impact parameters by using empirical formulas or simplified models, with the aid of experimental information based on the assumption that the impact-induced damage, as a first approximation and in certain conditions, can be represented by an equivalent delamination. This assumption is the basis for the majority of the fast methods adopted to predict the delamination © 2008, Woodhead Publishing Limited

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onset or, in other words, the damage resistance of composite structures. In this subsection, one of these methods (in particular as presented in [32, 41]) is introduced which seems to be particularly suitable for the preliminary design of stiffened composite panels required to sustain low-velocity impacts without delamination onset. The impact-induced delamination resistance method presented here is based on the assumption that delamination onset, due to an impact, depends on the maximum force exerted during the impact. As shown in [35], the threshold force needed for the delamination onset depends on the geometry and the material properties. This can be represented by Equation 2.1 where t is the thickness of the panel at the impact location, E and ν are the equivalent Young’s flexural modulus (usually, when dealing with laminates, the orthotropic equivalent engineering constants are adopted and starting from the flexural matrix [D], introduced in Classical Lamination Theory, it is possible to define an equivalent Young’s flexural modulus [42]) and the Poisson’s ratio of the laminate, and GIIC is the Mode II critical ERR of the adopted material.

Fcr = 1 ⋅ π ⋅ t 3/2 ⋅ 3

8 ⋅ G IIc ⋅ E /(1 – v 2 )

2.1

In order to check the impact damage resistance of a stiffened composite panel, the procedure introduced in [31, 41] can be adopted. The procedure is organised in the following three steps: 1. Calculate the threshold force by using the force-driven model proposed by Davies and Zhang [35]. If we assume the force-deflection curve to be a straight line, the impact energy is the triangular area below the curve and we can calculate the equivalent critical displacement, for a given impact location (see Fig. 2.15), impact energyEcr, geometry and material properties, as shown in Equation (2.2). d cr = 2 ⋅ E cr / Fcr = 2 ⋅ E cr 1 ⋅ π ⋅ t 3/2 ⋅ 3

8 ⋅ G IIc ⋅ E /(1 – ν 2 )

2.2

2. The next step is to determine the static linear structural response of the panel, undergoing a normal unit force in the impact location (see Fig. 2.16). This linear response is then linearly extrapolated in order to obtain a triangle characterised by an area equal to the impact energy Ecr. 3. Comparing this triangle to the one coming from the critical values, as shown in Fig. 2.17, it is possible to determine whether the designed panel is delamination resistant or not. In a panel resistant to delamination onset after an impact with energy Ecr, the extrapolated displacement relative to a triangle having area equal to Ecr, must be greater than or equal to dcr and the extrapolated force must be lower than the critical



43

Impact point

2.15 Impact location on the stiffened composite panel.

2.16 Linear static analysis of impact – out of plane displacements.

force. According to this criterion, an increment in the local stiffness of the panel reduces the capability of the panel to absorb the impact energy without delamination onset due to the increase in the impact force peak for a given impact energy. Indeed, higher values of the impact forces (closer to threshold force Fcr) for a given impact energy, are associated © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites F Panel not resistant to impact energy Ecr

Ecr Damage-resistant panel

Fcr

dcr

Uz

2.17 Damage resistance criterion.

with a reduced impact damage resistance, the impact force being the main parameter governing the onset of delamination induced by impact events. The impact-induced delamination resistance procedure presented here involves analytical expressions and linear finite element analyses and, due to its low computational cost, is suitable for the preliminary design of a complex composite structure with impact-induced delamination resistance requirements. In [41], this procedure has been applied to investigate the influence of compressive loads on the impact-induced delamination resistance of stiffened composite panels; while in [32, 43, 44] it has been applied to the design and optimisation of stiffened panels with impact-induced delamination resistance requirements. As already mentioned, it should be noted that this procedure is applicable only in certain conditions. In particular if the threshold impact energy Ecr, causes significant internal damage besides delaminations in a laminate, the presented procedure will be invalid, being based on the assumption that most of the impact energy is absorbed by elastic deformation of the structure. Indeed, a consistent internal damage formation absorbs a consistent amount of the impact energy leading to a decrease of the impact energy absorbed by elastic deformation. If the rate of impact energy absorbed elastically decreases, © 2008, Woodhead Publishing Limited


45

the maximum impact force decreases too, and Equation 2.1 will overestimate the real threshold impact force of the structure. It has been found numerically and further confirmed experimentally that, among the damage caused by transverse impact, the fibre breakage, which appears as micro-buckling on the impacted surface and as tensile failure on the back surface, is mainly responsible for the decrease of the rate of impact energy absorbed elastically and hence for the ‘attenuation of the maximum impact force’. On the contrary, the matrix cracking and the induced delaminations that can take place at very low load level due to the inherent weakness of the unidirectional plies in the transverse direction, were found to have an insignificant influence on the maximum impact force if fibre breakage does not take place. Hence a constraint for the applicability of Equation 2.1 to prediction of the threshold impact energy can be introduced. This constraint can be stated:  If Fcr < Pfb   If Fcr ≥ Pfb

Eq. applicable Eq. not applicable

2.3

where Pfb is the critical force at which the fibre breakage (in the form of fibre micro-buckling on the impacted surface or fibre tensile failure on the back surface) occurs. These two conditions can be easily checked by the linear static force/displacement analysis in step 2, which also provides the stresses in the laminates. These stresses can be substituted into a failure criterion to check whether fibre breakage takes place. Therefore, the constraint conditions are converted to: e ft ( Fcr ) ≥ 1  or  Eq. not applicable  e fc ( Fcr ) ≥ 1

2.4

where eft (Fcr) and efc (Fcr) are the tensile and compressive failure indices corresponding to applied load Fcr. From similar considerations of delamination shape and shear stress [31], the circular damaged area of the equivalent delamination, in case of impacts capable of overcoming the threshold force for the onset of delaminations, can also be predicted as: A=

2 9  P 16 π t 2  τ 

2.5

where t is the thickness of the laminate, P is the impact force and τ is the allowable inter-laminar shear stress.


46

2.6


Delamination tolerance in composites preliminary design

Whatever the origin, it is relevant to know whether a delamination affects the capability of the composite structure to sustain the in-service loads, that is, its integrity. This task is not trivial because the residual stiffness and strength of delaminated structures depend on the geometrical characteristics of delaminations and on the nature of the loads acting on the structure. Tolerance to delamination is an important issue when designing with composites because the need for lighter composite structures drives the design towards structural solutions which are more susceptible to delaminations and hence need to be delamination-tolerant in order to ensure safety. In this subsection, as one of the different causes that can lead to delamination growth, attention is focused on delamination growth in composite panels subjected to compressive loads, in order to illustrate an example of methodologies for damage tolerance design. Many authors have investigated the mechanical behaviour of delaminated composite structures under compression by modelling the delaminations with different degrees of complexity and by performing ad-hoc experimental tests. Various two-dimensional analytical approaches, suitable for the throughthe-width delaminations, have been introduced in [45–47] while threedimensional approaches both analytical [48–51] and numerical [52–57] have been adopted to simulate the mechanical behaviour of composite plates with embedded delaminations. The experimental activity on through-the-width [58–60] and embedded [61, 62] delaminations has been aimed to enhance the knowledge about these defects and to provide a useful database for the validation of brand new numerical tools. Relevant steps towards the realistic modelling of delamination have been made in [52] and [53], where the influence of the geometrical parameters of embedded delaminations on the buckling modes of delaminated composite panels have been investigated by means of numerical analyses. The importance of a proper model for the contact phenomena between the sub-laminates is highlighted in [54, 55] where an approach based on unilaterally constrained, finite, rectangular plates is employed. In [56, 57] the influence of contact phenomena, between the sub-laminates on the strain ERR distribution along the embedded delaminations front, is investigated in detail for composite panels with a single delamination. In [63–71], more complex models, taking the delamination growth into account, are introduced. In particular, in [63, 64], a two-dimensional approach for the simulation of the delamination growth is presented, for composite panels with through-the-width delaminations. This approach is based on the Modified Virtual Crack Closure Technique (MVCCT) for the calculation of the ERR and on the Penalty Method (PM) for the simulation of the contact phenomena. In [65], a detailed investigation on embedded delaminations’



47

growth is carried out, but only panels with a single delamination are investigated without giving information on the contributions of the single opening crack modes on the total strain ERR. The contribution of each opening crack mode on the overall ERR distribution is investigated in [66–68] where the approach adopted in [63, 64] has been extended and validated for single [66, 67] and multiple [68] embedded delaminations. An interface element, capable of addressing the problems of initiation and propagation of delaminations in laminated composites, is developed in [69], while cohesive surface approaches, based on mesomechanical models, are introduced in [70, 71]. Both the Virtual Crack Closure Technique (VCCT) and the Cohesive Zone-based Approach are progressively being implemented into commercial finite element codes, enabling the detailed simulation of delamination initiation and growth [72]. All the cited techniques for delamination growth simulation in composite structures under compressive loads, because of the strong relationship between delamination propagation and the buckling phenomenon, are based on geometrically non-linear models, which are suitable for detailed verification analyses. However, in a careful design where both safety and optimal design are properly addressed, it is essential to take into account the effects of delamination and its evolution, starting from the earlier stages of the design process. This goal can only be achieved by fast, numerical procedures capable of representing the main phenomena governing the structural behaviour of damage-tolerant composite structures, such as delamination growth. Very few examples of simplified and fast approaches for the preliminary design and optimisation of delamination-tolerant composite structures can be found in the literature [73, 74]. The simplified approach introduced in [74] is presented in this section. This approach seems to be particularly suitable for the preliminary design of delamination-tolerant composite panels under compression loads. The method provides a simulation of the delamination growth using four linear analyses. The first linearised buckling analysis is needed to determine the delamination buckling shape and load. Another linearised buckling analysis, together with two linear static analyses, is employed to evaluate the amount of energy released by delamination propagation using the elastic energy balance application. Under certain hypotheses, which can be considered acceptable in the preliminary design and optimisation phases, this approach can provide the delamination growth load and location in delaminated composite structures. Before presenting the method, it is important to review some phenomenological aspects on delamination, in order to clearly indicate the hypotheses and the approximations made. It is well known [75] that under compression load, delaminated structures can exhibit local forms of buckling. More precisely, for certain delamination sizes and locations along the laminate thickness, two main conditions can be singled out: © 2008, Woodhead Publishing Limited

48

• •


the thinner sub-laminate buckles locally while the thicker sub-laminate remains unaffected (Fig. 2.18(a)); both the sub-laminates buckle exhibiting a sort of mixed buckling mode, involving a combination of overall and local modes. (Fig. 2.18(b)).

The buckling of the thinner sub-laminate occurs first during the loading process, while the buckling in the thicker sub-laminate is in some way dictated by the geometrical arrangements of the structure, that is by parameters such as the delamination size and depth. As previously pointed out, in general, under compressive load, the first scenario may evolve, due to delamination growth, and may lead to a situation like the second scenario, in which the damage is much more pronounced and more dangerous forms of buckling/failure phenomena may commence. Thus, in order to design damage-tolerant structures, it is important to establish whether, starting from a delamination scenario like the first one that is induced, for example by accidental sources of damage such as impacts or manufacturing defects, the delaminations may propagate, leading to a delamination scenario like the second one. This is the reason why attention is focused on the first delamination scenario (Fig. 2.18(a)) and on the related delamination growth. According to the principles of linear elastic fracture mechanics, the propagation of delamination, schematically shown in Fig. 2.19, can be considered to be governed by the expression (Equation 2.6) defining the ERR as the variation of the potential energy ∆E associated to an increase ∆A of the crack size A. When this Released Energy reaches a critical value Gc, which depends only on the material properties, the delamination is supposed to grow.

(a)

(b)

2.18 Delamination scenarios.



G = – ∆E = Gc ∆A

49

2.6

For the first delamination scenario, when the thickness of the buckled sub-laminate is much smaller than the total laminate thickness, the following assumptions can be made: (a) according to the three basic failure modes defined in fracture mechanics, the contributions of fracture Mode II and Mode III can be neglected because the first opening mode clearly prevails over the remaining two; (b) the ERR distribution is mostly influenced by the out-of-plane displacements’ distribution along the delamination front. Based on these assumptions, the approach for the preliminary design of damage-tolerant composite structures is organised in the following steps: 1. calculation of the local buckled shape (mode) of the thinner sub-laminate (delamination buckling) by means of a linearised buckling analysis; 2. calculation of the energy globally released by the structure during delamination propagation by means of a linearised buckling analysis together with two linear static analyses; 3. redistribution of the overall released energy along the delamination front; 4. evaluation of the load at which the local critical ERR is attained. Thus the first analysis is meant to calculate the first buckling load and mode. In Fig. 2.20 a schematic representation of the analysed delamination scenario is presented. This representation, used to simplify the description of the proposed methodology, describes the adopted FEM modelling very well. Considering the overall structure subjected to an external compressive load F, by means of a linearised buckling analysis, it is possible to find the buckling load (eigenvalue) and mode (eigenvector) of the thinner sub-laminate. Hence, at each position i along the delamination front, it is possible to unequivocally determine the normalized out of plane displacements as ∆ui/

A

A + ∆A

2.19 Schematic representation of delamination growth.


50

Delamination behaviour of composites 1

0

0 2 (a) 1

Beam

F 0

F

0 2 Plate

(b)

0

Pristine part

1

Thinner sub-laminate

2

Thicker sub-laminate

2.20 Schematic representation of the analysed delamination scenario.

Location i along the delamination front

Buckled shape

1 ∆u i

F

F 0

0 2

A

2.21 Analysed delamination scenario – buckling shape and location along the delamination front.

∆u1 (i = 1…N), where N is the total number of considered locations along the delamination front (see Fig. 2.21). After the delamination buckling event, characterised by the critical load Fcr, the residual non-linear contribution of the buckled thinner sub-laminate to the global stiffness can be neglected. © 2008, Woodhead Publishing Limited


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Thus the stiffness K A of the overall post-buckled structure with a delamination size A, can be approximately calculated by removing the sublaminate labelled as (1) in Fig. 2.21. The same exercise can be repeated when the delamination size is A + ∆A. Hence, a new delamination buckling load Fcr′ and a new stiffness of the post-buckled structure KA+∆A can be defined. Figure 2.22 shows a graphical representation of the global structural stiffness evolution for both the structures, with different delamination sizes, during the loading process. Starting from the pre-buckling global stiffness value K0, the delamination buckling for the two analysed delamination sizes reduces the global stiffness to KA and KA+∆A respectively so that two distinct equilibrium paths can be defined. With reference to Fig. 2.22, the static propagation of the delamination size from A to A + ∆A, in a structure loaded by applied compressive displacements, can be reduced to a jump from the former to the latter equilibrium path [66]. With this assumption, for each value of the compressive displacement u*, the overall energy loss ∆E(u*) (area (A′ACB) in Fig. 2.22), related to the delamination size increment ∆Α, is given by:

KO F

A

C

KA

A + ∆A KA+∆A

Compressive force

B Delamination buckling

F cr F cr ′

A A′

ucr′ ucr

O O′

| ∆E (u*) |

u*

u

Applied compressive displacement

2.22 Graphical representation of the evolution of structural stiffness.


52


∆E(u*) = EA(u*) – EA+∆A(u*) = area (A′ACO′) – area (A′BO′) = area (ACO) + area (A′AOO′) – area (A′BO′)

2.7

= 1 K A ( u* – u cr ) 2 + 1 K 0 (2 u* – u cr ′ – u cr )( u cr – u cr ′ ) 2 2

– 1 K A+∆ A ( u* – u cr ′ ) 2 2 where EA(u*) and EA+∆A(u*) are the elastic energies absorbed as a consequence of the applied displacement u*, by the structure with the delamination size A and by the one with the delamination size A + ∆A respectively. Equation 2.7 gives us an idea of the total amount of energy released during the delamination propagation. However, the propagation of delamination is a local phenomenon which involves only some parts of the delamination front. Thus it is necessary to define a local ERR for each location i along the delamination front: G i (u*) = –

∆E i ( u * ) ∆A N

2.8

In Equation (2.8), ∆Ei(u*) is the local energy loss associated to the location i along the delamination front and ∆A/N is the fraction of delamination increment associated with the same location. Taking into account the balance of energy loss during the delamination propagation, one can write:

Σ ∆E i ( u * ) = ∆ E ( u *) ⇒ Σ G i = – N

N

∆E ( u * ) ∆A N

2.9

According to the assumption associated to the chosen delamination scenario, the ERR can be assumed to have the same distribution as the square of the buckling out-of-plane displacements along the delamination front: 2

 ∆ui  i –1 G =   G  ∆ ui –1  i

2.10

By combining Equations 2.9 and 2.10 we can write the local ERR associated with the first position along the delamination front: G i ( u* ) = –

∆E ( u *) ∆A Σ  ∆ui  N N  ∆ u1 

2

2.11

All the other values of Gi(u*) can be obtained by applying Equation 2.10. By progressively increasing the applied compressive displacement u* in © 2008, Woodhead Publishing Limited


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Equations 2.7, 2.10 and 2.11, it is possible to find the first location i whose ERR Gi(u*) exceeds the critical ERR GIC. The delamination propagation is presumed to take place in this location and the relative value of u* is considered to be the critical displacement udel at which initiation of delamination growth is expected. The external compressive load needed to initiate the delamination growth is then given by: Fdel = Fcr+ KA(udel – ucr)

2.12

In order to show the effectiveness of the simplified method presented here, a brief description, taken from [74], is given for its application to a composite plate with an embedded circular delamination under compressive load. From this application, it is possible to evaluate the influence of the linearisation of structural stiffness on the determination of the delamination growth initiation load and ERR distribution along the delamination front. The geometrical description of the square plate with an embedded delamination considered for this application and the adopted FEM model, built of eight noded, layered shell elements and rigid-links, are presented in Fig. 2.23. The material properties of the composite laminate used are summarised in Table 2.1. The stacking sequence of the composite laminate considered for this application is [90°, 0°, 90°]16. The results obtained with the simplified approach presented here are compared to the non-linear results obtained by an equivalent 3D model taken from the literature [67]. In order to verify the suitability of the proposed simplified approach to determine the ERR distribution along the delamination front, Fig. 2.24 shows the Gi values along the delamination front, obtained with the simplified linear model by linear, quadratic and cubic distributions, are compared to the 3D nonlinear numerical results of [67]. This figure presents the ERR values along a quarter of the delamination front and clearly shows how the quadratic distribution, adopted by the present formulation, gives reasonable results, while the linear and cubic distributions respectively overestimate and underestimate the local values of energy. The quadratic distribution lets us predict the correct maximum local ERR at θ = 0° (and θ = 180° for symmetry): hence, as found with the 3D non-linear model [67], the delamination propagation direction is perpendicular to the compressive load direction. In Table 2.2, the growth initiation values udel (= 0.240 mm) and Fdel (=79 821 N) calculated by means of the proposed novel approach, are compared to the corresponding values found in [67]. A maximum error of 12% is found, between the two sets of results, in the prediction of the force and the displacement required to initiate the growth of the delamination. This error, which is a measure of the influence of the linearisation of structural stiffness on the results in terms of initiation growth load, under the assumptions and hypotheses made, illustrates the applicability of the method for preliminary design of delamination tolerant composite panels. © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites h

H

h (mm) H (mm) W D (mm) W (mm)

D

0.375 6 80 150

W (a) Shell elements

Rigid links

A –A

1

0

0

2

A

Z X

A (b)

2.23 Geometrical description (a) and FEM model (b) of the delaminated square composite panel considered for the presented application.

Table 2.1 Material properties of composite plies Longitudinal Young’s modulus, EL (GPa) Transverse Young’s modulus, ET (GPa) Poisson’s ratio, νLT In-plane shear modulus, GLT (GPa) Critical ERR-Mode I, GIC (J/m2) Ply thickness (mm)


146 10.5 0.3 5.25 200 0.125


55

0.25 Simplified linear analysis (linear distribution) 3D non-linear analysis [23] Simplified linear analysis (quadratic distribution) Simplified linear analysis (cubic distribution)

0.2

G1 (N/mm)

0.15

θ

0.1

0.05

0 0

10

20

30

40

50 θ (deg)

60

70

80

90

2.24 ERR distributions along the delamination front – comparison between linearised and non linear approach.

Table 2.2 Comparison between results obtained by the linearised and the nonlinear approaches udel (displacement at growth in initiation) (mm)

Model

Fdel (growth initiation load) (N)

Simplified linear model (quadratic distribution)

0.240

79 821

3D non-linear model [23]

0.272

90 730

Err % with respect to 3D non-linear results

11.76%

2.7

12.02%

Cost-effective delamination management

The delamination-tolerant and the delamination-resistant design concepts applied to composite materials can strongly influence not only the performance, but also the cost of composite components. The reduction in cost that can be achieved by adopting these design concepts, apart from being a simple consequence of weight reduction (increase in payload), also involves aspects of manufacturing and maintenance (including repair). This is the reason why it is necessary to consider the influence of damage management design approaches on the lifecycle costs of composite structures. © 2008, Woodhead Publishing Limited

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Hence considerations on costs cannot be neglected even in the preliminary phases of the design when suitable structural configurations are searched for by adopting optimisation procedures based on simplified delamination management methodologies, like those introduced in the previous two subsections. In this subsection, in order to illustrate an example of procedures for cost-effective design of composite structures, attention is focused on the influence of resistance to impact-induced delamination on the lifecycle costs of stiffened composite panels. The literature lacks studies dealing with the optimisation procedures for composite stiffened panels. A few examples can be found relating to just the structural criteria for unstiffened panels [76–78] with some attention paid to weight saving [79]. In [43], an optimisation is performed on T-stiffened and I-stiffened composite panels in order to satisfy the requirements of minimum buckling load and minimum resistance to impact-induced delaminations. A posteriori analyses show that the delamination resistance configurations, though slightly heavier than the ones without delamination resistance constraints, are the most cost-effective ones. A similar result is found in [44], where a multi-objective performance cost optimisation is performed on stiffened composite panels, with buckling and delamination resistance constraints. The optimisation procedure introduced in [44] is briefly summarised here in order to demonstrate how a cost-effective design procedure can be realised for composite panels, with constraints on resistance to impactinduced delamination. Furthermore, some comparisons of the stiffened panels’ configurations, resulting from the optimisation analysis, are shown in order to give an idea of the order of magnitude of the influence of damage management on life-cycle costs in composite design. The proposed optimisation procedure can be split into the following steps: 1. setting up of a finite element parametric model; 2. definition of a lifecycle cost model; 3. implementation of methods for performance (including damage management) evaluation; 4. performing the optimisation cycles. The first step is the setting up of a finite element parametric model of stiffened composite panels where all the design variables should be appear as parameters. The next step is the introduction of a lifecycle cost (LLC) model for the evaluation of the lifecycle costs of the analysed configurations. The LCC of a stiffened skin panel is evaluated by calculating the manufacturing costs (MC), and the expected repair costs resulting from impact events during the service life: LCC = MC + Pr R · RC

2.13

Pr R being the probability of a repair during the service life, and RC the costs associated with repair consequence of a single impact. © 2008, Woodhead Publishing Limited


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The total manufacturing costs depend on material costs, labour costs, and non-recurring costs (such as tooling) written off over a certain number of components: MC = nonrece + mtc + lbrc

2.14

Costs of tooling and labour depend strongly on the manufacturing technology and panel topology. According to a multiple impacts formulation, both repair costs and the probability of a repair during the service life depend on the subpart involved (bay, stringer) and the impact location on the panel. Consequently, Equation 2.13 for LCC has to be altered slightly: n

LCC = MC + Σ Pr Ri · RCi i =1

2.15

where n is the number of impact locations considered in the analysis. The probability of a repair at a certain location on the panel is a function of the probability of an impact of given energy at the same location, and the associated damage resistance: Pr Ri = f(Pr Ii, DRi)

2.16

Since a variable number of locations is considered for an impact: each centre bay and stringer, the probability of impact occurrence at these locations is calculated on the basis of the ratio of the exposed surface to the full panel surface. Using the probability of impact occurrence at a given location and the related damage resistance, the number of repairs in the life of the component can be finally calculated. With the above-illustrated method, the damage resistance is implicitly related to the repair costs. As a consequence, a damageresistant panel with low (or nil) repair costs is probably expected to be relatively heavy and expensive to manufacture (due to the material costs). On the other hand, when a light design is considered (with reduced manufacturing costs), the damage resistance decreases, thereby increasing the probability of a repair occurrence with related repair costs. The third step is the implementation of the methods for performance evaluation. By adopting a linearised buckling procedure, it is possible to determine the critical load related to the onset of elastic instability of the panel (buckling load), while the resistance to impact-induced delamination can be easily determined by adopting the procedure based on the threshold force for impact-induced delamination onset previously described in this chapter. The last step involves performing optimisation cycles in order to find the best solution which satisfies the imposed constraints. Various optimisation algorithms can be adopted. In [44], a genetic algorithm developed in [80, 81] © 2008, Woodhead Publishing Limited

58


is used. A flow chart describing the logic behind the optimisation procedure is shown in Fig. 2.25. In order to show the effectiveness of the proposed multi-objective optimisation procedure, its application to composite stiffened panels with buckling (1500 N/mm) and resistance to impact-induced delamination (resistance to impacts up to 15 J) constraints is shown. The objectives are the minimisation of both structural mass and the lifecycle cost (LCC) of the panel. Omitting detailed information on design variables and on limitations in topologic configurations – this information can be found in [44] – the output of the optimisation procedure as the resulting Pareto set is shown in Fig. 2.26. By comparing the different Pareto solutions from Fig. 2.26, one may notice that the different optimum configurations can essentially be traced back to a minimum weight design (n.1) or a minimum cost design (n.4), plus two ‘compromise’ solutions (namely n.2 and n.3). All these equally ranked solutions give the designer some choice in selecting a suitable panel configuration for specific purposes. All the optimal solutions are buckling resistant. Solution 1 is lighter than 2, 3 or 4 but it is not damage resistant. As a consequence, a high LCC is experienced during its operative life due to a large number of repair costs. In Fig. 2.27, the analytical cost allocation for each Pareto solution is shown (MTC = Manufacturing Costs, NRC = Non-recurring costs, LBR= Labour Costs and RC = Repair Costs). Note that for partially compliant solutions 2, 3 and 4, a saving cost slice is also reported.

Project variables

Coding

Initial configurations

Genetic algorithm Crossover mutation

Selection No Convergence

FEM analysis (linearised buckling + impact-induced delamination resistance method) FEM environment

2.25 Representation of the optimisation procedure.


Yes

Optimal configuration

Pareto set 2

1

Lifecycle cost [k7]

1.9

1.8

1.7

1.6

1.5

1.4

2 3

1.3 4.4

4.5

4.6

4.7

4.8

4.9 Weight [kg]

5

5.1


5.3

4 5.4

59

2.26 Mass-lifecycle cost multi-objective optimisation – resulting Pareto set.

5.2


2.1

60

Delamination behaviour of composites MTC [k7]; 0.6252146; 31%

1

2

NRC [k7]; 0.0256; 1%

LBR [k7); 0.410480652; 21%

LBR [k7); 0.31057074; 16%

SAVE; 0.655815687; 33%

RC [k7]; 0.15609375; RC [k7]; 8% NRC [k7]; 1.0395; 52% 0.0128; 1%

SAVE; 0,0%

LBR [k7); 0.398706696; 20%

3 SAVE; 0.642183698; 32%

RC [k7]; 0.200625; 10%

4

MTC [k7]; 0.765695251; 37%

LBR [k7); 0.375611847; 19%

SAVE; 0.601378217; 30%

NRC [k7]; 0.0128; 1%

MTC [k7]; 0.746569946; 37%

RC [k7]; 0.31303125; 16%

NRC [k7]; 0.0128; 1%

MTC [k7]; 0.698064026; 34%

2.27 Mass-lifecycle cost multi-objective optimisation – cost allocation for each Pareto solution.

2.8

References

1. B. Pradhan, S.K. Panda, ‘Effect of material anisotropy and curing stresses on interface delamination propagation characteristics in multiply laminated FRP composites’, Journal of Engineering Materials and Technology, 128 (3), pp. 383–392, 2006. 2. R.Y. Kim, S.L. Donaldson ‘Experimental and analytical studies on the damage initiation in composite laminates at cryogenic temperatures’, Composite Structures 76 (1–2), pp. 62–66, 2006. 3. E. Persson, I. Eriksson, L. Zackrisson ‘Effects of hole machining defects on strength and fatigue life of composite laminates’ Composites Part A, 28 (2), pp 141–151, 1997. 4. S.K. Mazumdar, Composites Manufacturing, Materials, Product, and Process Engineering, CRC Press, 2002. 5. J.P., Davim, P., Reis, ‘Study of delamination in drilling carbon fiber reinforced plastics (CFRP) using design experiments’, Composite Structures, 59 (4), 2003, pp. 481–487.



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6. G.B. Murri, S.A. Salpekar, T.K. O’Brien, ‘Fatigue delamination onset prediction in tapered composite laminates’ NASA-TM-I01673. 7. T. Ireman, T. Ranvkin, I. Eriksson, ‘On damage development in mechanically fastened composite laminates’, Composite Structures, 19 (2), pp. 151–171, 2000. 8. P.P. Camanho, S. Bowron, F.L. Matthews, ‘Failure mechanisms in bolted CFRP’, Journal of Reinforced Plastics and Composites 17 (3), pp. 205–233, 1998. 9. Chen, Wen-Hwa, Lee, Shyh-Shiaw, Yeh, Jyi-Tyan, ‘Three-dimensional contact stress analysis of a composite laminate with a bolted joint’, Composite Structures, 20 (3), pp. 287–297, 1995. 10. Ko, Chu-Cheng, Lin, Chien-Chang, Chin, Hsiang ‘Prediction for delamination initiation around holes in symmetric laminates’, Composite Structures, 22 (4), pp. 187–191, 1992. 11. M.H.R. Jen, Y.S. Kau, J.M. Hsu, ‘Initiation and propagation of delamination in a centrally notched composite laminate’, Journal of Composite Materials 27 (3), pp. 272–302, 1993. 12. C. Meeks, E. Greenhalgh, B.G. Falzon, ‘Stiffener debonding mechanisms in postbuckled CFRP aerospace panels’, Composites Part A: Applied Science and Manufacturing, 36 (7), pp. 934–946, 2005. 13. R. Krueger, M. Cvitkovich, K. O’Brien, P. Minguet, ‘Testing and analysis of composite skin/stringer debonding under multi-axial loading’, Journal of Composite Materials, 34 (15), pp. 1263–1300, 2000. 14. P.J. Minguet, T.K. O’Brien, ‘Analysis of test methods for characterizing skin/stringer debonding failures in reinforced composite panels’, ASTM Special Technical Publication 1274, pp. 105–124, 1996. 15. R. Krueger, I.L. Paris, T.K. O’Brien, P.J. Minguet, ‘Fatigue life methodology for bonded composite skin/stringer configurations’, Journal of Composites Technology and Research, 24 (2), pp. 308–331, 2002. 16. J.N. Reddy, ‘Mechanics of Laminated Composite Plates, Theory and Analysis’, CRC Press 1997. 17. R. Jones, ‘Mechanics of Composite Materials’ Second Edition, Taylor and Francis, 1998. 18. N.J., Pagano, R.B. Pipes, ‘Influence of stacking sequence on laminate strength’, Journal of Composite Materials, 1971, pp. 50–57. 19. N.J., Pagano, R.B. Pipes, ‘Some observations on the interlaminar strength of composite laminates’, International Journal of Mechanical Sciences 15 (8), pp. 679–692, 1973. 20. L.A. Carlsson, ‘Fracture of laminated composites with interleaves’, Key Engineering Materials 120–121, pp. 489–520, 1996. 21. S. Subramanian, W.S. Chan, ‘Effect of loading frequency and interleaf on the delamination characteristics of laminated composites’, Journal of Composites Technology and Research 18 (3), pp. 179–193, 1996. 22. S. Abrate, ‘Impact on Composite Structures’, Cambridge University Press, 1998. 23. L.M. Kachanov, ‘Delamination Buckling of Composite Material’, Kluwer Academic Publishers , August, 1988. 24. R. Krueger, ‘Virtual crack closure technique: History, approach, and applications’, Applied Mechanics Reviews, 57 (2), pp. 109–143, March 2004. 25. G.A. Kardomates, D.W. Schmueser, ‘Buckling and post-buckling of delaminated composites under compressive loads including transverse shear effects’, AIAA Journal, 27, pp. 337–343, 1988.


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26. G.A. Kardomateas, ‘Large deformation effects in the postbuckling behavior of composites with thin delaminations’, AIAA Journal 27 (5), pp. 624–631, 1989. 27. M. Niu, Composite Airframe Structures, Adaso Adastra Engineering Center, 1992. 28. H.W. Bergmann, R. Prinz, ‘Fatigue life estimation of graphite/epoxy laminates under consideration of delamination growth, International Journal for Numerical Methods in Engineering, 27, pp. 323–341, September 1989. 29. B.C. Foos, W.E. Wolfe, Damage Initiation and Growth in Composite Materials Subjected to Low Velocity Impact, American Society of Mechanical Engineers, Materials Division (Publication) MD 69–1, pp. 391–400, 1995. 30. L.C. Dickinson, G.L Farley, M.K. Hinders, ‘Translaminar reinforced composites: A Review’, Journal of Composites Technology and Research 21 (1), pp. 3–15, 1999. 31. G.A.O. Davies, X. Zhang, ‘Impact Damage Prediction in Carbon Composite Structures’, Int. Journal of Impact Engineering 16(1); pp. 149–170, 1995. 32. J.F.M. Wiggenraad, E.S. Greenhalg, R. Olsson, Design and Analysis of Stiffened Composite Panels for Damage Resistance and Tolerance, NLR-report, NLR-TP2002–193, 1998. 33. G. Dorey, ‘Impact damage in composites – development, consequences and prevention’, Proc of ICCM 6 and ECCM 2, Elsevier Applied Science, 1987. 34. Z.L. Gu, C.T. Sun, ‘Prediction of impact damage region in SMC composites’, Composite Structures 7, pp. 179–190, 1987. 35. G.A.O. Davies, D. Hitchings, J. Wang, ‘Prediction of threshold impact energy for onset of delamination in quasi-isotropic carbon/epoxy composite laminates under low-velocity impact’, Composite Science and Technology 60, pp. 1–7, 2000. 36. O. Ishai A. Shragai, ‘Effect of impact loading on damage and residual compressive strength of CFRP laminated beams’, Composite Structure 14, pp. 319–337, 1990. 37. H.Y. Choi F.K. Chang, ‘A model for predicting damage in graphite/epoxy laminated composites resulting from low-velocity impact: Part I - Experiments’, J. Composite Materials 25, pp. 992–1011, 1991. 38. H.Y. Choi, F.K. Chang, ‘A model for predicting damage in graphite/epoxy laminated composites resulting from low–velocity impact: Part II - Analysis’, J. Composite Materials 25, pp. 1012–1038, 1991. 39. P.P. Camanho, F.L. Matthews, ‘Delamination onset prediction in mechanically fastened joints in composite laminates’, J. of Composite Materials 33 (10), pp. 906–927, 1999. 40. C.F. Li, N. Hu, Y.J. Yin, H. Semine, H. Fukunaga, ‘Low-velocity impact-induced damage of continuous fiber-reinforced composite laminates. Part I. An FEM numerical model’, Int. Composites Part A (33), pp. 1055–1062, 2002. 41. A. Riccio, N. Tessitore, ‘Influence of loading conditions on the impact damage resistance of composite panels’, Computers & Structures, 83, pp.. 2306–2317, 2005. 42. A.K. Kaw, G. Willenbring, ‘A software tool for mechanics of composite materials’, Int. J. Engng, 13(6), pp. 433–441, 1997. 43. A. Riccio, M. Gigliotti, L. Iuspa, L. Mormile, ‘Cost and Performance Optimisation of Composite Stiffened Panels’, presented at the XVIII Congresso nazionale AIDAA (Associazione Italiana di Aeronautica ed Astronautica). Volterra, 19–22 September 2005. 44. M. Corvino, L. Iuspa, A. Riccio, F. Scaramuzzino, ‘Multi-objective (weight and costs oriented) optimisation of impact damage resistant stiffened composite panels’, presented at The Eighth International Conference on Computational Structures Technology – Las Palmas de Gran Canaria, Spain 12–15 September 2006. © 2008, Woodhead Publishing Limited


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45. J.D. Whitcomb, ‘Approximate analysis of postbuckled through-the-width delaminations’, Composite Technol. Review, 4(3), pp. 71–77, 1982. 46. J.D. Whitcomb, ‘Parametric analytical study of instability-related delamination growth’, Composite Sci. & Technol, 25(1), pp. 18–48,1986. 47. H. Chai, C.D. Babcock, W.G. Knauss, ‘One delamination modelling of failure in laminated plates by delamination buckling’, Int. Journal of Solids & Struct, 17(1), pp. 1069–1083, 1981. 48. D.Y. Konishi, W.R. Johnston, ‘Fatigue effects on delaminations and strength degradation in graphite-epoxy laminates’, Proceedings of the Composite Materials: Testing and Design (Fifth Conference) S.W. Tsai (ed.) ASTM STP 674, American Soc. For Testing & Materials, pp. 597–619, 1979. 49. Z. Fei W.L. Yin, ‘Postbuckling growth of a circular delamination in a laminate under compression and bending’, Proceedings of the Twelfth South-eastern Conference on Theoretical and Applied Mechanics, Georgia Inst. Of Technology, Pine Mountain, Georgia, 10–11 May, 1984. 50. K.N. Shivakumar, J.D. Whitcomb, ‘Buckling of a sublaminate in a quasi-isotropic composite laminate’, Int. Journal of Composite Materials, 19, pp. 2–18, 1985 51. J.D. Whitcomb, K.N. Shivakumar, ‘Strain-energy release rate analysis of a laminate with a postbuckled delamination’, Numerical Methods in Fracture Mechanics, NASA TM–89091, 1987. 52. H.J. Kim C.S. Hong, ‘Buckling and postbuckling behaviour of composite laminates with an embedded delamination’, Procedings of ICCM-10, Whistler, BC, Canada, 1995. 53. K.L. Singh, B. Dattaguru, T.S. Ramamurthy, P.D. Mangalgiri, ‘Delamination tolerance studies in laminated composite panels’, Sadhana (printed in India), 25, part 4, pp. 409–422, 2000. 54. K. Shahwan, A.M. Waas, ‘Unilateral buckling of rectangular plates’ Int. J. of Solids & Structures, 31(1), pp. 75–89, 1994. 55. K. Shahwan, A.M. Waas, ‘Buckling of unilaterally constrained plates: application to the study of delaminations in layered structures’, J. Franklin Institute , 335B(6), pp. 1009–1039, 1998. 56. J.D. Whitcomb, ‘Analysis of a laminate with a postbuckled embedded delamination, including contact effects’, Int. Journal of Composite Materials, 26 (10), pp. 1523– 1535, 1992. 57. J.D. Whitcomb, ‘Three dimensional analysis of a postbuckled embedded delamination’, Int. Journal of Composite Materials, 23, pp. 862–889, 1989. 58. M. Ashizawa, ‘Fast Interlaminar Fracture of a Compressively Loaded Composite Containing a Defect’, Proceedings of the Fifth DOD/NASA Conference on Fibrous Composites in Structural Design. NASA Ames Research Center, pp. 1–269, 1981. 59. R.L. Ramkumar, ‘Fatigue degradation in compressively loaded composite laminates’, NASA CR–165681, 1981. 60. R.L. Ramkumar, ‘Performance of a quantitative study of instability-related delamination growth’, NASA CR-166046, 1983. 61. B.A. Byers, ‘Behaviour of damaged graphite/epoxy laminates under compression loading’, NASA CR-159293, 1980. 62. H. Chai, W.G. Knauss, C.D. Babcock, ‘Observation of damage growth in compressively loaded laminates’, Journal of Experimental Mechanics, 23, (3), pp. 329–337, 1983. 63. P. Perugini, A. Riccio F. Scaramuzzino, ‘Influence of delamination growth and contact phenomena on the compressive behaviour of composite panels’, Int. Journal of Composite Materials, 33 (15), pp. 1433–1456, 1999. © 2008, Woodhead Publishing Limited

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64. A. Riccio, P. Perugini, F. Scaramuzzino, ‘Modelling compression behaviour of delaminated composite panels’, Computers & Structures, 78, pp. 73–81, 2000. 65. K.-F. Nilsson, J.C. Thesken, P. Sindelar, A.E. Giannakopoulos, B. Storakers, ‘A theoretical and experimental investigation of buckling induced delamination growth’, Journal of Mech. Phys. Solids., 41(4), pp. 749–782, 1993. 66. P. Gaudenzi, P. Perugini A. Riccio, ‘Post-buckling behaviour of composite panels in the presence of unstable delaminations’, Composite Structures, 51(3), pp. 301–309, March 2001. 67. A. Riccio, P. Perugini, F. Scaramuzzino, ‘Embedded delamination growth in composite panels under compressive load’, Composites Part B: Engineering, 32(3), pp. 209– 218, April 2001. 68. A. Riccio, F. Scaramuzzino, P. Perugini, ‘Influence of contact phenomena on embedded delamination growth in composites’, AIAA Journal, 41(5), pp 933–940, May 2003. 69. G.A.O. Davies, D. Hitchings, J. Ankersen, ‘Predicting delamination and debonding in modern aerospace composite structures’, Composites Science and Technology, Vol 66, pp 846–854, 2006. 70. R. De Borst, Remmers J.J.C.,‘Computational modelling of Delamination’, Composites Science and Technology, 66, pp 713–722, 2006. 71. O. Allix, L. Blanchard, ‘Mesomodelling of delamination: towards industrial applications’, Composites Science and Technology, 66, pp 731–744, 2006. 72. ABAQUS MANUAL (revision 6.5–1): Theory. 73. M. Ashizawa, Fast Interlaminar Fracture of a Compressively Loaded Composite Containing a Defect. Proceedings of the Fifth DOD/NASA Conference on Fibrous Composites in Structural Design. NASA Ames Research Centre, pp. 1–269 – 1–292, 1981. 74. A. Riccio, M. Gigliotti, ‘A novel numerical delamination growth approach for the preliminary design of damage tolerant composite structures’, International Journal of Composite Materials, 41(16), pp 1938–1960, 2007. 75. P. Gaudenzi, P. Perugini, F. Spadaccia, ‘Post-buckling analysis of a delaminated composite plate under compression’, Composite Structures, 40(3), pp 231–238, 1998. 76. H. Fukunaga, G.N. Vanderplaats, ‘Stiffness optimization of orthotropic laminated composites lamination parameters’, AIAA Journal, 29, pp. 641–646, 1991. 77. S. Nagendra, R.T. Haftka, Z. Gurdal, ‘Stacking sequence optimization of simply supported laminates with stability and strain constraints’, AIAA Journal, 30, pp. 2132–2137, 1992. 78. R. LeRiche, R.T. Haftka, ‘Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm’, AIAA Journal, 31, pp. 951–956. 1993. 79. S. Venkataraman, L. Lamberti, R.T. Haftka, T.F. Johnson, ‘Challenges in comparing numerical solutions for optimum weights of stiffened shells’, Journal of Spacecraft and Rockets, 40, pp. 183–192, 2003. 80. L. Iuspa, F. Scaramuzzino, ‘A bit-masking oriented data structure for evolutionary operators implementation in genetic algorithms’, Soft Computing, 5, pp 58–68, 2001. 81. L. Iuspa, F. Scaramuzzino, P. Petrenga, ‘Optimal design of an aircraft engine mount via bit-masking oriented genetic algorithms’, Advances in Engineering Software, 34, pp. 707–720, 2003.


3 Review of standard procedures for delamination resistance testing P D AV I E S, IFREMER Centre de Brest, France

3.1

Introduction

The use of laminated materials is not new. Plywood is a traditional construction material, laminated glass fibre reinforced composites based on organic matrix resins have been used in boat construction for over 50 years, while carbon fibre composite aircraft applications have steadily increased since the early 1970s. It has been recognized for many years that the development of test standards is of major importance for composite materials [1] whose mechanical behaviour is largely determined by the manufacturing process. Changes in pressure, temperature or hygrometry can result in significant property variations, so coupon testing is essential, both to check fabrication quality and to validate design data. The anisotropic nature and multiple failure modes of composite materials have caused major difficulties in testing for strength, and there is still discussion over appropriate tests for in-plane shear and compression properties. As a result several different tests have been standardized for these loadings. The situation for out-of-plane properties is rather different. At the time of writing, the only delamination resistance test methods which are internationally accepted (by ISO, the International Standards Organization) are those for initiating a crack from an implanted defect under mode I loading, ISO 15024 [2], and the interlaminar shear strength test which promotes a delamination under the nominal out-of-plane shear loading of a short beam, ISO 14130 [3]. The former has only been in place since 2001, while the latter test has been used for many years but remains extremely controversial. Given the importance of the delamination failure mechanism it might appear surprising that so few tests are available. However, the lack of standard tests certainly does not mean that the subject has been neglected, even a rapid literature search will produce hundreds of papers and some have reviewed the subject [4–7]. Much of this work has been performed with the aim of developing standard tests, and a brief history of these efforts will be presented first in this chapter. Then the delamination tests based on fracture mechanics will be described in detail (mode I, mode 65 © 2008, Woodhead Publishing Limited

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II, mode III and mixed mode I/II), followed by a discussion of other tests for through-thickness properties. The aim of this chapter is to provide the reader with an overview of the test procedures currently being used to characterize interlaminar fracture behaviour and strength, together with an indication of the origins of the tests and the references necessary to understand the choices made during the standardization procedure. It should be emphasized that this is not an exhaustive review, that would require considerably more space than available in this volume, but rather a synthesis of the author’s experience, based on involvement in developing these test methods since the 1980s.

3.2

Historical background

Concerted efforts to develop standard test procedures for delamination resistance measurement started in the early 1980s. A NASA document published in 1982 described a mode I delamination test for toughened resin composites [8]. The ASTM D30.02.02 task group on Interlaminar Fracture Toughness, chaired by Dr Kevin O’Brien of NASA Langley, then started to examine these tests and launched several pre-standardization actions. In Japan a group developing test methods for Mechanical Properties of Carbon Composites set up a sub-committee on Fracture Toughness, chaired by Professor Kageyama of the University of Tokyo. This group subsequently proposed national standards for mode I (DCB, double cantilever beam) and mode II (ENF, end notched flexure) within a JIS (Japanese Industrial Standards) framework [9]. In Europe a technical committee of the European Group on Fracture, subsequently to become ESIS (European Structural Integrity Society), started to work on delamination tests in 1984. This work was one of the activities within Technical Committee 4, Polymers and Composites, under the chairmanship of Professor Gordon Williams, co-chaired by Professor Kausch and then Professor Pavan. The main mode of operation of this group, which meets twice a year, is drafting of test protocols, testing them by round robin tests (providing the same test protocol and specimens from the same batch for testing in different laboratories), and detailed discussion of the results, before drafting a revised protocol and re-testing. A summary of the work of this group appeared in 2001 [10]. The ASTM, ESIS and JIS groups participated at a joint meeting during an ASTM conference in Orlando, Florida in November 1989 and subsequently collaborated closely on mode I test standardization. The ESIS mode I test protocol (then at its third revision, following three round robin exercises) was written in ASTM format with modifications from JIS. It was then subjected to further round robin testing [11], and passed as an ASTM standard in 1994 [12]. Based on this document a mode I ISO new work item was proposed in 1994. This finally emerged as the ISO standard in 2001 [2].


Standard procedures for delamination resistance testing

67

In parallel, the VAMAS group (Versailles Project on Advanced Materials and Standards), an international collaboration aimed at drafting codes of practice and specifications for advanced materials, started a task group on Polymer Composites chaired first by Professor Bathias of the University of Compiègne, then Dr Sims of NPL. They also launched delamination test round robin activities, but did not pursue these when ASTM, JIS and ESIS started to work together, preferring to concentrate on fatigue tests. VAMAS did initiate subsequent mode II round robins, however, as will be described below. Alongside these activities a number of other standards were drafted for specific industrial applications, particularly in the aerospace industry. These included the CRAG (Composite Research Advisory Group) methods [13], and AECMA (European Association of Aerospace Industries) standards [14]. Several aerospace companies also have their own internal composite test procedures.

3.3

Mode I

As described above, the current ISO mode I delamination resistance test standard [2] is the result of several years of collaboration between the ESIS TC4, ASTM D30.06 and JIS groups. The main points from this work are described briefly below.

3.3.1

Specimen

The choice of specimen geometry was straightforward, as only the DCB (double cantilever beam), Fig. 3.1a, had been studied in detail. Some work had been performed by VAMAS on width tapered DCB specimens (WTDCB), following early work by Bascom [15], but this was

(a)

(b)

3.1 Mode I specimen geometry (a) DCB; (b) WTDCB.


68


not pursued as it was hoped that a single parallel-sided specimen geometry might be used for mode I, II and I/II tests. Load introduction was mainly by rectangular blocks in Europe and through hinges in the USA, so both were included in the standard.

3.3.2

Procedure

Defining a test procedure required a choice between continuous loading and load-unload cycles. The former was retained once the data analysis had been chosen. The main innovation in this test procedure is using initiation values of GIc. The first test procedures from ASTM used only propagation values but ESIS had always been concerned with measuring complete R-curves from initiation onwards, due to a strong influence of work by Benzeggagh and colleagues at the University of Compiègne [16]. The switch to measuring only initiation GIc was found to be necessary when several studies indicated that propagation values depended on the specimen geometry, and in particular on specimen stiffness [17, 18]. Higher propagation values could be obtained with stiffer specimens. The mean propagation value, while providing some useful information in comparative studies when specimen geometry is kept constant, is not an intrinsic material property. Once this decision had been made the questions remaining to be resolved were: • What should the starter defect be made of? • How should initiation be defined? It was these questions which occupied much of the time between the first documents and the final draft ISO version. The first question was the easier to deal with, several studies were run to examine the influence of starter films and pre-cracks on GIc values [19, 20]. Figure 3.2 shows an example of results, which indicated higher values and an unstable load drop when thick films were used, related to a resin pocket in front of the film. Below a certain thickness a plateau value was found for the materials tested. The thinnest films studied, 6 microns aluminium, are of similar dimensions to the carbon fibre diameter but are not easy to keep flat during manufacture. They also require a release agent, and there were concerns that this might migrate into the resin. A baking procedure was developed to avoid this, PTFE (poly(tetrafluoroethylene)) is preferred when high cure temperatures do not exclude it. It should be mentioned that one material was tested which showed lower propagation values than those at initiation, even when a sufficiently thin film was used. This highlights the difficulty in drafting a test procedure for all possible materials. This case was dealt with simply by stating in the standard that for the atypical case when an R-curve shows a decrease in apparent toughness when loading from the film then wedge precracking may be used. © 2008, Woodhead Publishing Limited


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Normalized initiation values 2 13 µm

C/PEEK C/EPOXY

1.5

1

0.5 0

20 40 Film thickness, microns

60

3.2 GIc initiation values versus starter film thickness for carbon/epoxy and carbon/PEEK unidirectional composites [19, 20], re-plotted normalized by the lowest value measured.

The second question to be addressed was how to define initiation. Crack initiation starts at the specimen centre and cannot be detected visually on the specimen edge until it has grown one or two millimetres. Work at EMPA, a materials research institute in Switzerland, with acoustic emission and microradiography revealed that the non-linearity on the load-displacement plot is a reasonable indication of crack initiation at the specimen centre [21]. This notion is accepted as the basis for detection of crack initiation. It is not completely satisfactory, defining a deviation from linearity is rather subjective, so compliance change values were also examined, similar to those used in the KC fracture test for metals ASTM E399 [22] and in the subsequent polymer fracture toughness test standard developed by ESIS [23]. A 5% change had been used in the ESIS protocols, studies to determine whether smaller changes (down to 1%) might give better results were examined but did not show any particular improvement and the 5% change was retained. A third initiation criterion, visual onset, was also included to be compatible with measured propagation values. These crack propagation values, necessary for the data analysis, are obtained by following the crack visually on the specimen edge.

3.3.3

Data analysis

The basic analysis is that of Williams et al. [24], corrected beam theory (CBT), employing a crack length correction determined experimentally. Various other analyses had been used previously, in particular a power law suggested by Berry [25], but the CBT expression was retained as it allows an effective modulus to be determined, which can provide a useful check on results. © 2008, Woodhead Publishing Limited

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Comparisons between Berry’s method and CBT gave identical results. Corrections for large displacements and end blocks are also included. A second slightly different method used in Japan, the modified compliance calibration (MCC), in which the width-normalized cube root of compliance is plotted as a function of thickness-normalized delamination, is also included in the standard. The drawback with both these methods which are based on experimental compliance calibrations is that crack length measurements are needed during propagation. In early ESIS testing, a procedure with no crack measurements during propagation was proposed, the crack length was recalculated from a compliance calibration based on testing several specimens with inserts of different lengths. This multi-specimen calibration was tested in one round robin and was shown to give similar results to single specimen calibrations for that material. It was subsequently abandoned, but when extensive fibre bridging occurs the multi-specimen method should give better results.

3.3.4

Current status

The mode I test is now an established ISO standard, and is widely used. It is not perfect but does allow materials to be compared on a common basis by avoiding the large differences due to specimen geometry effects during propagation. This should not be taken to imply that research is no longer being performed on mode I delamination. The development of a standard test procedure acceptable to all parties is an exercise in pragmatism and several of the underlying issues, particularly related to fibre bridging, are still being actively studied [26, 27]. New tougher materials are also requiring a re-evaluation of the limits of the test, this will be discussed further below.

3.4

Mode II

Mode I test development focused on details of the procedure and analysis, as there was already agreement on specimen geometry. For mode II (forward shear) even the choice of specimen has not been fixed, 20 years after the first pre-standardization exercises. There are four main contenders plus a number of more recent proposals, as will be described in the next section. As a result, only the Japanese group has proposed a standard. International collaboration has been pursued for mode II tests and two round robin exercises conducted by the author in 1996 and 1998 within the VAMAS framework [28, 29] enabled results on the same materials to be compared using the four candidate tests, but no agreement was reached to enable a single method to be proposed. However, this situation is evolving as will be discussed in Section 3.4.4.



3.4.1

71

Specimens

Figure 3.3 shows the four specimens proposed for standardization, together with two more recent suggestions. The ENF (end notch flexure) specimen, Fig. 3.3a, was proposed by Russell and Street for composites [30], based on its prior use to characterize wood [31]. It is simple to use and analyze, but has one major drawback; it does not allow stable propagation unless long crack lengths are used (a/L>0.7). In order to stabilize the test one possibility is to measure the relative shear displacement of the two specimen faces and use this in a control loop to adjust the loading rate, Fig. 3.3b. This was developed in Japan [32], but requires an appropriate test machine and some pre-calibration. A second, simpler, option, proposed more recently by Martin and Davidson [33, 34], is to load the specimen in four-point loading instead of three, the 4ENF test, Fig. 3.3c. This induces stable propagation between the central loading points. Another approach is to use the end loaded split (ELS) specimen, Fig. 3.3d, loaded in a sliding fixture. This was first proposed by Bradley and colleagues in the early 1980s [35] and has been extensively studied by the ESIS group. It allows stable propagation provided that a/L>0.55. Its main disadvantage is the need to clamp one end of the specimen, which introduces uncertainty in the boundary conditions. However a recently-proposed procedure [36] enables this to be calibrated and should provide more consistent data. Two other proposals are shown in Figs 3.3e and 3.3f, described in references [37, 38] respectively. The fact that new tests are still being developed emphasizes the difficulties with existing configurations. Two round robins were run. The first [28] compared ENF, SENF and ELS, with a few results for the 4ENF (which had just been proposed) included. The second focused on the 4ENF specimen, on the same carbon/epoxy material. Protocols were provided by ASTM, JIS, and ESIS respectively for the first three tests. Rod Martin of MERL (Materials Engineering Research Laboratory) drafted a 4ENF protocol [39]. Six laboratories participated, two American, two Japanese and two European. Based on these two exercises a comparison was made between results from the four specimens, as shown in Fig. 3.4. This shows that all three stable specimen configurations give initiation values lower than the ENF specimen. Scatter is lower for tests from mode II precracks. (It should be noted that one set of data gave very low ELS values from inserts, which explains the large scatter for this result.)

3.4.2

Procedures

Once a specimen geometry has been adopted, the main point of discussion is the choice of defect from which to measure toughness values. Given the


72


(a)

(b)

(c)

(d)

(e)

(f)

3.3 Mode II specimens; (a) ENF; (b) SENF; (c) 4ENF; (d) ELS; (e) ONF; (f) CENS.



73

800 Insert 700

II pre-crack

GIIc, J/m2

600 500 400 300 200 100 0 ENF

4ENF

SENF

ELS

3.4 Comparison between results from four mode II specimens, [28, 29].

quite extensive study of this parameter during the mode I standard development it might appear obvious that the same type of starter defect, an insert film less than 13 microns thick, should be adopted for mode II testing. While this has been the opinion of some, including the author, others have suggested that the only GIIc value of interest is the most conservative one; as values obtained from mode I and mode II pre-cracks are often lower than those from film inserts these should be adopted. Intuitively it is easy to imagine that a pre-cracking procedure which introduces a damage zone in front of the crack tip is very likely to lower the delamination resistance during a subsequent test. There is some evidence to support this, (e.g. Fig. 3.4), though much of the published data was obtained on ENF specimens for which the unstable nature of crack propagation does not allow the validity of the film to be checked. The disadvantage of pre-cracks is that they are rarely straight, and it is not easy to measure their length, whereas a starter film length can be measured accurately on the fracture surface after testing. Once the initial defect type has been chosen, the specimen is loaded to failure. There has been little discussion of loading rates, which are generally around 0.5 to 1 mm/minute.

3.4.3

Analyses

Data analysis for the ENF specimens uses either beam theory expressions or a compliance calibration, performed by sliding the specimen to change the crack length and applying load-unload cycles, before the test to failure. Carlsson and Gillespie have reviewed data reduction methods [40, 41]. For the SENF specimen a compliance calibration is used. For the 4ENF specimen beam theory expressions exist [33, 42, 43] but experimental compliance is © 2008, Woodhead Publishing Limited

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favoured, as the crack length can be measured during propagation, and this was used in the VAMAS round robins. Davidson and colleagues [44] have examined factors which affect the accuracy of the 4ENF test. They concluded that if compliance and crack length are measured accurately then the 4ENF and ENF will produce essentially the same toughness values. One way to reduce errors in crack length measurement is by inverse determination from the deformed specimen shape. This is now possible, using image analysis, and may lead to a simplification of the test procedure in the future [45]. For the ELS specimen both corrected beam theory [46] and compliance calibrations have been used [36, 47].

3.4.4

Current status

The VAMAS round robins were intended to allow a single specimen to be selected for an ISO standard. The 4ENF and ELS tests both appeared promising, but for various reasons the mode II activity did not appear as a New Work Item within the ISO TC61 committee, which is responsible for composites. The ASTM group favoured the 4ENF, ESIS preferred the ELS, but subsequent detailed work by Davidson and colleagues on the former revealed some anomalies. These are described in detail in [48, 49], but the most important point is that in order for the compliance calibration analysis to be accurate the fixture must be very stiff. The compliance of the upper bearing in the 4ENF fixture can introduce errors in the analysis. As a result Davidson advocated returning to the ENF test, perceived to be more accurate, and at the time of writing this chapter an ENF test protocol was being drafted for ASTM. In the author’s opinion this test, condemned previously due to its inherently unstable nature, is not the best way to obtain reliable GIIc values. The use of the ELS specimen or even the 4ENF with a fixture calibration (similar to that required for the MMB fixture in the ASTM standard) would be preferable alternatives. It should also be noted that there is still discussion over whether pure shear loading can be achieved in a test specimen (or in a structure). O’Brien discussed this [50] in a provocative paper entitled GIIc: shear measurement or sheer myth?

3.5

Mode III

The resistance of composite materials to out-of-plane shear (torsion) may be of interest in certain applications such as helicopter rotor hubs. Mode III testing was the subject of some interest in the late 1980s and has been revisited periodically since then. It is sometimes assumed that GIIc = GIIIc but this has never been proved. There have been several attempts to develop a mode III test, but there is no accepted test method.



3.5.1

75

Specimens

Several geometries have been proposed, Fig. 3.5, including split cantilever beams [51–53] and cracked rail shear specimens [54]. However, both of these have drawbacks and neither was pursued. Interest revived with the introduction of the ECT (edge cracked torsion) test by Lee [55] in 1993. This involves applying equal and opposite moments to the corners of rectangular specimens with a specially designed stacking sequence including 90/+45/– 45 plies and a mid-plane insert film (Fig. 3.5c). This test has been investigated in detail with respect to friction and mode II contributions, and both were found to yield negligible contributions [56]. It should be noted, however, that the need for a special stacking sequence for the ECT test is a significant disadvantage compared to the other delamination tests described in this chapter, which can all be run on unidirectional beam specimens.

(a)

(b)

(c)

3.5 Mode III specimens; (a) split cantilever beam; (b) cracked rail shear; (c) edge cracked torsion.


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3.5.2


Procedures

The mode III test which has been closest to reaching standardization is that performed by torsion loading of plate specimens. The load was initially applied at one corner only. An ASTM D30 round robin was organized to evaluate this test on carbon/epoxy samples but the results reported in 1997 indicated large scatter and considerable non-linearity. The test frame was then modified so that the load could be applied symmetrically by two pins, and a second round robin was organized. Results presented in 1999 indicated that delaminations did not always grow at the correct interface and a significant mode II component was indicated near the loading pins. Li et al. [57] applied the ECT specimen to measure GIIIc values for E-glass/epoxy and showed that intralaminar cracks in the 90° plies contributed significantly to GIIIc values during propagation. It is interesting to note that for mode III tests values are measured directly from starter films, without pre-cracking.

3.5.3

Analyses

Crack growth is not visible during the ECT test, so multi-specimen testing with different initial crack lengths, as discussed above for mode I, and posttest inspection are necessary to obtain the compliance calibration. Ratcliffe [58] recently ran tests and numerical analyses. He noted a strong effect of initiation values on insert length when maximum load was used to determine GIIIc values. When onset of non-linearity was used values obtained were lower than GIIc values for the same materials. He also underlined the need for detailed fracture surface analysis to validate the tests.

3.5.4

Current status

At present there is relatively little interest in GIIIc values, and this is not a priority for standardization.

3.6

Mixed mode I/II

In parallel with the actions to develop pure mode I, II and III tests there has also been extensive work on mixed mode loading. This is necessary if mixed mode criteria are to be validated, but also in order to test laminates under more realistic loading conditions: pure modes are rarely encountered in practice. We will concentrate on the I/II combination, I/III and II/III have received much less attention.

3.6.1

Specimens

Figure 3.6 shows the MMB (mixed mode bending) specimen and how it is loaded. It was introduced by Reeder and Crews [59–61] and standardized by © 2008, Woodhead Publishing Limited


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(a)

(b)

(c)

(d)

(e)

3.6 Mixed mode (I/II) specimens: (a) MMB; (b) MMF; (c) CLS, (d) SLFPB; (e) ADCB.

ASTM as D6671 [62]. Figure 3.6 also shows some of the other mixed mode specimens which have been proposed for unidirectional materials. Most are based on flexure. The advantage of the MMB set-up compared to these other fixtures is that different mode combinations can be applied simply by moving the moment arm. This makes construction of a failure envelope easier, though for fatigue studies a fixed ratio specimen may be easier to use. The MMF specimen was employed by Russell and Street [63], among others. Williams and colleagues proposed a similar variable ratio mixed mode specimen [64]. The cracked lap shear (CLS) configuration, loaded in tension, appeared promising and was tested in an early ASTM round robin [65]. However, analysis proved complicated and required non-linear FE analysis so the test was not developed further. A version loaded in flexure, the single leg four point bend (SLFPB) test was proposed recently [66]. Finally, the fixed ratio mixed mode (FRMM) specimen, often referred to as ADCB (asymmetric © 2008, Woodhead Publishing Limited

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double cantilever beam), developed by Bradley and colleagues in the early 1980s, was studied extensively by the ESIS TC4 group in the 1990s. Simple to use, it only provides a single ratio of 4:3 of mode I to mode II component [10, 67].

3.6.2

Procedures

The MMB test procedure, together with details of the loading fixture, can be found in ASTM D6671 [62]. The standard is based on values measured at initiation from the same starter film (thickness < 13 microns) as that recommended for the mode I test.

3.6.3

Analyses

The MMB data reduction is based on a corrected beam theory expression rather than compliance calibration, as the crack propagation is not always stable when there is a large mode II contribution, and shifting the specimen to measure compliance for different crack lengths (as performed for ENF specimens for example) is difficult due to the bonded hinges. Bhashyan and Davidson studied MMB data analyses in some detail and their results indicated that the modified beam theory agreed well with FE analyses [68].

3.6.4

Current status

The MMB test is now widely used. An error in the equation relating the mixed mode ratio to the lever arm length in the original document was corrected in the 2006 revision. The determination of lever length for a required mixed mode ratio in the standard is now determined by a more accurate expression than the original simple beam equations, as round robin test results showed that mode ratios were often very different from those expected. One consequence of the difficulties in harmonizing pure mode II tests is that the mixed mode test with a large mode II component is being used to characterize the influence of shear. Given that there is doubt as to whether a pure mode II state can be obtained in a test specimen, this sounds quite attractive. The problem, however, is that the mode II component in the MMB test is simply obtained from an ENF test, so propagation in mode II dominated tests is still unstable. Longer crack lengths are recommended to encourage more stable propagation. Another limitation, noted in recent work on very high modulus carbon fibre composites, is that the bending loads can lead to compression failure before the crack propagates. This leads to a limit to the type of materials which can be tested [69].



3.7

79

Conclusion on fracture mechanics tests to measure delamination resistance

The previous sections indicate that delamination resistance testing under mode I and mixed mode I/II loading are now well established. The situation for mode II and mode III is less clear. Developing fracture mechanics tests involving pure shear is no simpler than the development of standards for classical stress-based tests to measure in-plane shear strength, (which currently include tension on ±45° specimens, rail shear fixtures and the Iosipescu test). Various tests exist to provide GIIc and GIIIc values; whether they reach standard status will depend on whether the need for these values is sufficient to encourage the effort required.

3.7.1

Stress-based interlaminar tests

The preceding sections have focused on fracture mechanics tests, but there are also several stress-based delamination resistance tests. Descriptions of through-thickness test methods are available elsewhere [70, 71], here only a brief overview will be given. Figure 3.7 shows some of the tests available to obtain interlaminar strength properties. Among these only the ILSS specimen is recognized as an ISO test method [3], ASTM standards are available for the 90° bend and Iosipescu specimens.

(b)

(a)

(d)

(c)

(e) (f)

3.7 Interlaminar strength test specimens. Tension: (a) 90° bend; (b) Cspecimen; (c) ILTT. Shear: (d) ILSS; (e) Iosipescu. Tension and shear: (f) Arcan fixture.


80


The 90° bend specimen provides a means of obtaining interlaminar tensile strength from a simple test. A standard ASTM test procedure exists [72], which is based on work by Jackson and Martin [73] who analysed unidirectional samples with a 90° bend loaded in tension. A close alternative is to use a Cshaped specimen, this is particularly interesting for the study of tubular structures [74]. The choice of these tests on angular or curved structures reflects the difficulty in introducing loads directly into flat plates. Several attempts have been made to measure through thickness strength by bonding blocks onto both faces of a laminate. Local stress concentrations can be reduced by appropriate specimen and/or block profiling [70, 71], and an ASTM group is currently looking at this type of test configuration within D30.06. For interlaminar shear the loading of short beam specimens in three point flexure is probably the most widely used test performed on composite specimens. It is simple to run, requires little material and provides a failure strength. However, as the ISO standard [3] points out it provides apparent shear strength, and many authors have underlined the limited usefulness of these values for anything but quality control [75, 76]. The four point ILSS test [77] is preferable but is rarely used. Alternative shear tests such as the so-called Iosipescu configuration [78] allow interlaminar shear strengths to be measured, provided sufficiently thick samples are available. Finally, the Arcan fixture [79, 80] can be used to obtain interlaminar strengths under a range of loadings combining tension and shear. Fixtures allowing compression and shear can also be designed [81]. This is a very attractive fixture, in some ways the stress equivalent of the MMB specimen. Its main disadvantage is the stress concentrations at the specimen ends, but the use of ‘beaks’ on the blocks which hold the composite can reduce these. The modified fixture has been applied to adhesives and should be suitable for composite specimens.

3.8

Future trends

There are three areas in which pre-standardization studies of delamination resistance testing are continuing: First, the application of these tests to multidirectional laminates, more representative of most structural applications, is being examined by ESIS, among others. This was examined many years ago by Chai [82], but several recent papers have also treated this subject [83, 84]. Chapter 9 discusses this in more detail. A second area of interest is the extension of these tests to higher loading rates. This has been studied by many authors [85] and is discussed in Chapter 4, but no standard method exists yet. A third area of increasing importance is the adaptation of these methods to composites with through-thickness reinforcements. Techniques such as stitching and Z-pinning [86] enable large increases in toughness to be achieved, but their characterization poses considerable difficulties. Damage is no longer confined to a small crack tip © 2008, Woodhead Publishing Limited


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zone but involves the entire specimen thickness. It is important to clearly define the limits of the present standards and to develop alternative, more appropriate, methods to characterize these very tough composites. Again, the subject is discussed elsewhere in this volume, in Chapter 23.

3.9

Conclusion

Over the last 20 years, thanks to considerable efforts by ASTM, JIS, ESIS and VAMAS, fracture mechanics tests have been developed to characterize the delamination resistance of composite materials. GIc and GI/IIc values are now quoted by material suppliers. There has also been increasing interest in these values from designers, as damage modelling and damage tolerance concepts become more common. The challenge for those developing test methods is now to accommodate the greatly improved toughness of modern composite materials.

3.10

Sources of information and advice

ESIS website, http://www.esisweb.org/ ASTM website, http://www.astm.org NASA technical report server, http://ntrs.nasa.gov/search.jsp VAMAS website, http://www.vamas.org

3.11

Acknowledgements

The author is grateful to the many colleagues in the ESIS TC4 group and at IFREMER, who have contributed to the work described above.

3.12

References

1. Sims GD, Development of standards for advanced polymer-matrix composites, Composites, 22, 4, 267–274, July 1991. 2. Fibre-reinforced plastic composites – Determination of mode I interlaminar fracture toughness, GIC, for unidirectionally reinforced materials, 15024, International Standards Organisation, ISO., 2001. 3. Fibre-reinforced plastic composites – Determination of apparent interlaminar shear strength by short-beam method, 14130, International Organisation for Standardisation, ISO., 1997. 4. Sela N, Ishai O, Interlaminar fracture toughness and toughening of laminated composite materials: a review, Composites, 1989, 20(5), 423–435. 5. Davies P, Blackman BRK, Brunner AJ, Standard Test Methods for Delamination Resistance of Composite Materials: Current Status, Applied Composite Materials, 1998, 5(6), 345–364. 6. Tay TE, Characterization and analysis of delamination fracture in composites – a


82

7.

8. 9. 10. 11. 12.

13. 14.

15. 16. 17.

18. 19. 20. 21.

22. 23. 24.

25. 26.

Delamination behaviour of composites review of developments from 1990 to 2001, Applied Mechanics Reviews, 2003, 56(1), 1–23. Brunner AJ, Blackman BRK, Davies P, A Status Report on Delamination Resistance Testing of Polymer-Matrix Composites, Engineering Fracture Mechanics, 75, 2008, 2779–2794. NASA reference publication 1092, Standard Tests for Toughened Resin Composites, Langley Research Center, 1982. JIS 7086, Testing methods for interlaminar fracture toughness of carbon fiber reinforced plastics, 1993. Moore DR, Pavan A, Williams JG, (eds), Fracture Mechanics Test Methods for Polymers, Adhesives and Composites ESIS Publication No. 28, Elsevier. 2001. O’Brien TK, Martin RH, Results of ASTM Round Robin testing for mode I interlaminar fracture toughness of composite materials, NASA Tech Memo 104222, 1992. ASTM, Standard Test Method for Mode I Interlaminar Fracture Toughness of Unidirectional Fibre-Reinforced Polymer Matrix Composites, D5528, American Society for Testing and Materials International, ASTM 1994. Curtis PT, CRAG Test Methods for the Engineering Properties of Fibre Reinforced Composites, RAE Farnborough, February 1988. AECMA Aerospace series, Carbon Fibre Reinforced Plastics. Determination of Interlaminar Fracture Toughness Energy Mode I-GIc (prEN6033) and Mode II-GIIc (prEN6034), 12, 1995. Bascom WD, Bitner JL, Moulton RJ, Siebert AR, The interlaminar fracture of organicmatrix woven reinforcement composites, Composites, 11, 1980, 9–18. Davies P, Benzggagh ML, Interlaminar mode I fracture testing, in Application of Fracture Mechanics to Composite Materials, ed. K. Friedrich, Elsevier 1989. Prel YJ, Davies P, Benzeggagh ML, de Charentenay FX, Mode I and Mode II delamination of thermosetting and thermoplastic composites, ASTM STP 1012, p 251, 1989. Davies P, Cantwell WJ, Moulin C, Kausch HH, A study of delamination resistance of IM6/PEEK composites, Composites Science and Technology, 36, 1989, 153–166. Davies P, Cantwell WJ, Kausch HH, Measurement of Initiation Values of GIc in IM6/ PEEK Composites, Composites Science and Technology, 35, 1989, 301–313. Davies P, Moulin C, Kausch HH, Fisscher M, Measurement of GIc and GIIc in Carbon/epoxy composites, Comp Sci & Tech., 39, 1990, 193–205. de Kalbermatten T, Jaggi R, Flueler P, Kausch HH, Davies P, Microfocus radiography studies during mode I interlaminar fracture toughness tests on composites, J. Mat. Sci. Letters, 11(9), 1992, 543–546. ASTM 399 Standard test method for linear elastic plane strain fracture toughness KIc of metallic materials. ISO 13586:2000 Plastics – Determination of fracture toughness (GIC and KIC) – Linear elastic fracture mechanics (LEFM) approach. Hashemi S, Kinloch AJ, Williams JG, Corrections needed in double cantilever beam tests for the interlaminar failure of fibre composites, J. Mat. Sci. Letters, 8, 1989, 125–129. Berry JP, Determination of fracture surface energies by the cleavage technique, J. Appl Physics, 34, 1963, 62–68. Sorensen BF, Jacobson TK, Large-scale bridging in composites: R-curves and bridging laws, Composites Part: applied science and manufacturing, 1998, 29A(11), 1443– 1451.



83

27. Brunner AJ, Blackman BRK, Williams JG, Calculating a damage parameter and bridging stress from GIC delamination tests on fibre composites, Composites Science and Technology, 2006, 66(6), 785–795. 28. Davies P, Sims GD, Blackman BRK, Brunner AJ, Kageyama K, Hojo M, Tanaka K, Murri G, Rousseau C, Gieseke B, Martin RH, Comparison of test configurations for the determination of GIIc of unidirectional carbon/epoxy composites, an international round robin, Plastics, Rubber and Composites, 1999, 28(9), 432–437. 29. Davies P, Summary of results from second VAMAS mode II round robin test exercise using the 4ENF specimen. IFREMER report ref. TMSI/RED/MS 99.82, July 1999. 30. Russell AJ, Street KN, Factors affecting the interlaminar fracture of graphite/epoxy laminates, Proc ICCM4, Tokyo, 1982, 279–286. 31. Barrett JD, Foschi RO, Mode II stress intensity factors for cracked wood beams, Eng Fract Mech, 9, 1977, 371–378. 32. Kageyama K, Kikuchi M, Yanagisawa N, Stabilized end notched flexure test: Characterization of mode II interlaminar crack growth, ASTM STP 1110, 1991, 210–225. 33. Martin R, Davidson BD, Mode II fracture toughness evaluation using a four point bend end notch flexure test. Proceedings 4th International Conference on Deformation and Fracture of Composites, Institute of Materials, 1997, 243–252. 34. Martin RH, Elms T, Bowron S, Characterization of mode II delamination using the 4ENF. Proceedings 4th European Conference on Composites Testing & Standardization, 1998, 161–170. 35. Jordan WM, Bradley WL, Micromechanisms of fracture in toughened graphite/ epoxy laminates, ASTM STP 937, 1987, 95–114. 36. Blackman BRK, Brunner AJ, Williams JG, Mode II Fracture testing of composites: a new look at an old problem, Engineering Fracture Mechanics, 73, 2006 2443– 2455. 37. Fan C, Jar PYB, Cheng J-JR, A unified approach to quantify the role of friction in beam-type specimens for the measurement of mode II delamination resistance of fibre-reinforced polymers, Composite Science and Technology, 67, 6, 2007, 989– 995. 38. Caimmi F, Frassine R, Pavan A, A new jig for mode II interlaminar fracture testing of composite materials under quasi-static and moderately high rates of loading, Eng Fract Mechanics, 73, 16, 2006, 2277–2291. 39. Murri GB, Martin RH, Effect of initial delamination on mode I and mode II interlaminar fracture toughness and fatigue fracture threshold, ASTM STP 1156, 1991, 239–256. 40. Carlsson LA, Gillespie JW Jr, Pipes RB, On the analysis and design of the end notched flexure (ENF) specimen for mode II testing, Journal of Composite Materials, 1986, 20(6), 594–604. 41. Carlsson LA, Gillespie JW Jr, Mode II Interlaminar fracture of composites, in Application of Fracture Mechanics to Composite Materials, ed. K. Friedrich, Elsevier, 1989. 42. Kageyama K, Kimpara I, Suzuki T, Ohsawa I, Kanai M, Tsuno H, Effects of test conditions on interlaminar fracture toughness of four point ENF specimens, Proc, ICCM12, Paris 1999. 43. Cartié D, Davies P, Peleau M, Partridge IK, The influence of hydrostatic pressure on the interlaminar fracture toughness of carbon/epoxy composites, Composites Part B: Engineering, 37, 4–5, 2006, 292–300. 44. Schuecker C, Davidson BD, Evaluation of the accuracy of the four point bend end© 2008, Woodhead Publishing Limited

84

45. 46. 47. 48.

49. 50. 51. 52. 53.

54.

55. 56.

57.

58.

59. 60. 61. 62.

63. 64.

Delamination behaviour of composites notched flexure test for mode II delamination toughness determination, Composites Science and Technology, 60(11), 2000, 2137–2146. Davies P, Casari P, Carlsson LA, Characterization of Mode II delamination, Proc JNC15 (in French), 2007. Wang Y, Williams JG, Corrections for mode II fracture toughness specimens of composite materials, Composites Science and Technology, 43(3), 1992, 251–256. Davies P, Blackman BRK, Brunner AJ, Mode II delamination, in reference [10], pp. 307–333. Davidson BD, Sun XK, Effects of friction, geometry and fixture compliance on the perceived toughness from three- and four- point bend end-notched flexure tests, Journal of Reinforced Plastics and Composites, 24(15), 2005, 1611–1628. Davidson BD, Four point bend end-notched flexure (4ENF) test standardization update. Presentation at ASC/ASTM D30 Joint 19th Tech Conference, October 2004. O’Brien TK, GIIc: shear measurement or sheer myth? Tech Memo 110280 February 1997. Donaldson SL, Mode III interlaminar fracture characterization of composite materials, Composites Science and Technology, 1988, 32(3), 225–249. Martin RH, Evaluation of the split cantilever beam for mode III delamination testing, ASTM STP 1110, 1991, 243–266. Rizov V, Shindo Y, Horiguchi K, Narita F, Mode III interlaminar fracture behaviour of glass fiber reinforced polymer woven laminates at 293 to 4K, Appl Comp Mats, 13, 2006, 287–304. Becht G, Gillespie JW Jr. Design and analysis of crack rail shear specimen for mode III interlaminar fracture, Composites Science and Technology, 1988, 31(2), 143– 157. Lee SM, An edge crack torsion method for mode III delamination fracture toughness, Journal of Composite Technology & Research, 1993, 15(3), 193–201. Zhao D, Wang Y, Mode III fracture behaviour of laminated composite with edge crack in torsion, Theoretical and Applied Fracture Mechanics, 1998, 29(2), 109– 123. Li X, Carlsson LA, Davies P, Influence of fibre volume fraction on mode III interlaminar fracture toughness of glass/epoxy composites, Composites Science and Technology, 2004, 64(9), 1279–1286. Ratcliffe JG, Characterization of the Edge Crack Torsion (ECT) test for mode III fracture toughness measurement of laminated composites, NASA Technical Memorandum NASA/TM-2004-213269, September 2004. Crews JH, Reeder JR, A mixed mode bending apparatus for delamination testing, NASA Technical Memorandum 1000662, 1988. Reeder JR, Crews Jr JH, Non-linear analysis and redesign of the mixed mode bending delamination test. NASA Technical Memorandum 102777, 1991. Reeder JR, Refinements to the mixed mode bending test for delamination toughness, Journal of Composites Technology & Research, 25(4), 2003, 191–195. Standard test method for Mixed Mode I-Mode II interlaminar fracture toughness of unidirectional fiber reinforced polymer matrix composites, ASTM D6671, American Society for Testing and Materials. Russell AJ, Street KN, Moisture and temperature effects on the mixed mode delamination of unidirectional graphite/epoxy, ASTM STP 876, 1985, 349–370. Hashemi S, Kinloch AJ, Williams JG, Mixed mode fracture in fibre-polymer composite laminates, ASTM STP 1110, 1991, 143–168.



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65. Johnson S, Stress analysis of the cracked lap shear specimens An ASTM round robin, NASA-TM-89006, 1986. 66. Tracy GD, Feraboli P, Kedward KT, A new mixed mode test for carbon/epoxy composite systems, Composites Part A: A34(11), 2003, 1125–1131. 67. Hashemi S, Kinloch AJ, Williams JG, The analysis of interlaminar fracture in uniaxial fibre-polymer composites, Proc Roy Soc. Lond., A427, 1990, 173–199. 68. Bhashyan S, Davidson BD, Evaluation of data reduction methods for the mixed mode bending test, AIAA Journal, 35, 1997, 546–552. 69. Baral N, Davies P, Baley C, Bigourdan B, Delamination behaviour of very high modulus carbon/epoxy marine composites, Comp. Sci & Tech., 68, 3–4, 2008, 995– 1007. 70. Mespoulet S, Hodgkinson JM, Matthews FL, Design, development and implementation of test methods for determination of through thickness properties of laminated composites, Plast Rubber Compos, 29(9), 2000, 496–502. 71. Lodeiro MJ, Broughton WR, Sims GD, Understanding the limitations of through thickness test methods, Proc 4th European conf on Comp, Testing and Standardization, London, 1998, 80–90. 72. ASTM D6415/D6415M-06 Standard Test Method for Measuring the Curved Beam Strength of a Fiber-Reinforced Polymer-Matrix Composite. 73. Jackson WC, Martin RH, An interlaminar tensile strength specimen, ASTM STP 1206, 1993, 333–354. 74. Hiel CC, Sumich M, Chappell DP, A curved beam test specimen for determining the interlaminar tensile strength of a laminated composite, Proc. Eighth DOD NASA FAA Conference on Fibrous Composites in Structural Design, Part 1 pp. 247–261, 1990. 75. Berg CA, Tirosh J, Israeli M, Analysis of short beam bending of fiber reinforced composites, ASTM STP 497, 1972, 206–218. 76. Chaterjee S, Adams D, Oplinger DW, Test methods for composites, a status report Volume III: Shear test methods, DOT/FAA/CT-93/17, June 1993, 97–102. 77. Browning CL, Abrams FL, Whitney JM, A four point shear test for graphite epoxy composites, ASTM STP 797, 1983, 54–74. 78. ASTM D5379/D5379M-05 Standard Test Method for Shear Properties of Composite Materials by the V-Notched Beam Method. 79. Arcan M, Hashin Z, Voloshin A, A method to produce uniform plane stress states with applications to fiber reinforced materials, Experimental Mechanics, 18, 1978, 141–146. 80. Jurf RA and Pipes RB Interlaminar Fracture of Composite Materials, J. Comp Mats, 16(5), 1982, 386–394. 81. Cognard J-Y, Davies P, Gineste B, Sohier L, Development of an improved adhesive test method for composite assembly design, Composites Science and Technology, 65(3-4), 2005, 359–368. 82. Chai H, The characterization of Mode I delamination failure in non-woven, multidirectional laminates, Composites, 15(4), 1984, 277–290. 83. Brunner AJ, Blackman BRK, Delamination fracture in cross-ply laminates: What can be learned from experiment? Proceedings 3rd ESIS TC4 Conference in: Fracture of Polymers, Composites and Adhesives, BRK, Blackman, JG, Williams, A, Pavan (eds), ESIS-Publication No. 32, Elsevier 2003, 433–444. 84. de Morais AB, Analysis of mode II interlaminar fracture of multidirectional laminates, Composites Part A. applied science and manufacturing, A35(1), 2004, 51–57. © 2008, Woodhead Publishing Limited

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85. Blackman BRK, Kinloch AJ, Wang Y, Williams JG, The failure of fibre composites and adhesively bonded fibre composites under high rates of test, J. Mat Sci, Part 1 (30), 1985, 5885-5900, Part 2, 31, 1996, 4451–4466, Part 3, 31, 1996, 4467–4477. 86. Partridge IK, Cartié DDR, Delamination resistant laminates by Z-fiber® pinning. Part I manufacture and fracture performance, Composites Part A, 36(1), 2005, 55– 64.


4 Testing methods for dynamic interlaminar fracture toughness of polymeric composites C T S U N, Purdue University, USA

4.1

Introduction

One of the major failure modes in unidirectionally fiber-reinforced polymeric composites is delamination cracking. Delamination in composites is often produced by impact loads, and spreads dynamically. Thus, its dynamic propagation behavior and methods for characterizing the dynamic fracture properties are of interest to many researchers. There are several specimen configurations developed for dynamic fracture tests for metals and ceramics. Among them are the compact compression specimen (CCS) [1] for mode I loading and the side impact specimen [2] for mode II loading. Figures 4.1 and 4.2 show the sketches of these two types of specimens, respectively. Neither specimen produces a pure mode I or mode II state of stress at the crack tip. Common among most of the dynamic fracture tests is the use of dynamic loads either through a rigid body impact or the use of a Hopkinson bar. Using the measured impact force history in conjunction with the observed crack tip velocity as the input boundary conditions, the dynamic response of the specimen is modeled numerically, from which the fracture parameter such as energy release rate history is Crack gage

Strain gage

Striker

Instrumented bar

4.1 Compact compression specimen for mode I loading.


88


Striker

4.2 Side impact specimen for mode II loading.

calculated. In other words, dynamic fracture toughness is not measured directly. Rather, it is calculated using the solid mechanics model with the aid of some experimental data. Consequently, modeling errors may result in inaccurate ‘measured’ dynamic fracture toughness even if the experiment is carried out flawlessly. Traditional testing methods for dynamic fracture toughness usually require sizable specimens. It is well known that making thick-section composites is not easy and their mechanical properties may differ from those of the thinner counterparts. Since composites are often used in the form of thin laminates, to maintain the K-dominance region size similar to that in the realistic laminate, it is highly desirable to keep the thickness of the test specimen similar to that of the structure. In static delamination fracture tests, the double cantilever beam (DCB) and end notch flexure (ENF) specimens as shown in Figs 4.3 and 4.4 are most popular for mode I and mode II loadings, respectively. Although these specimens can maintain a realistic thin thickness of the composite structure, their large flexibilities will require very fast loading machines and may make the dynamic test quite challenging. By various testing methods, dynamic delamination fracture toughnesses of polymer composites have been obtained by a number of authors. However, there are discrepancies among these results. Some researchers reported that initiation fracture toughness increased with loading rate [3–5]. Some researchers found the opposite result [6]. Some claimed stepwise-drop of fracture toughness with increasing loading rate [7]. On the other hand, a few researchers reported that dynamic initiation fracture toughness was insensitive to loading rate [8, 9], and that dynamic fracture toughness was insensitive to crack propagation speed [9, 10]. It should be noted that most of these conclusions were made © 2008, Woodhead Publishing Limited

Testing methods for dynamic interlaminar fracture toughness

89

Hinges

4.3 Double cantilever beam specimen for mode I loading. P

4.4 End notch flexure (ENF) specimen for mode II fracture test.

based upon test data that exhibited significant scatter and for different material systems. Contributions from earlier researchers such as Takeda et al. [11], and Grady and Sun [12] have undoubtedly helped our understanding in dynamic delamination fracture of polymeric composites. Recent works by Lambros and Rosakis [13–14] and Coker and Rosakis [15] who used impact loads to create high speed mixed mode delamination fracture provided understanding of the dynamic mixed mode problem. Recently, Sun and Bing [16] used a cracked off-axis composite specimen to conduct mode II delamination experiments and showed that the presence of compressive transverse stresses ahead of the crack tip could greatly raise the mode II fracture toughness of the composite. This article reviews the technique developed by Guo and Sun [10] using a film adhesive embedded ahead of the pre-crack tip to delay the onset of crack propagation in DCB specimens in order to generate high speed mode I delamination crack propagation in composite laminates. This technique was later applied to ENF specimens by Tsai et al. [17] for dynamic mode II delamination fracture. Dynamic crack propagation in these modified DCB and ENF specimens was produced by quasi-static loading on an MTS machine. In addition, a wedge loaded compact-tension (WLCT) specimen [9] for mode I loading is also discussed. In [9] a Split Hopkinson Pressure Bar (SHPB) was used for loading. The finite element method was employed to simulate the dynamic delamination crack propagation in the fiber composite from which both initiation fracture toughness and dynamic fracture toughness were determined. © 2008, Woodhead Publishing Limited

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4.2

Dynamic loading and crack propagation

4.2.1

Loading with a Hopkinson bar

The Split Hopkinson Pressure Bar (SHPB) as depicted in Fig. 4.5 is often used as a instrumented loading device in dynamic fracture experiments. It works on the principle of one-dimensional wave propagation. This implies that a stress wave propagates non-dispersively in a long elastic bar at elastic bar velocity. Details of the classical SHPB set up can be found in the literature [18–19] and are not elaborated in this chapter. In conventional SHPB analysis, the deformation history of the specimen is extracted from the signals in the strain gages mounted on the incident and transmission bars. Analysis is accomplished by referencing the three signals (two in the incident bar, the incident and reflected wave; and one in the transmission bar, the transmitted wave) to the same instant of time. Usually, the chosen instant of time is either the arrival of the incident pulse at the incident bar-specimen interface or the time when the wave arrives at the center of the specimen, although the former seems to be more appropriate as the start of specimen deformation. The incident wave arrives at the specimen-incident bar interface (point B in Fig. 4.5) after a duration ∆tAB from the time it crosses the incident bar gage A, and the reflected pulse is recorded by the same set of gages after another time interval ∆tAB. Thus, the same set of gages on the incident bar records both the incident pulse εi(t) and the reflected pulse εr(t), but they are separated by a time period of 2∆tAB. Thus,

εi(t) = εI (t – ∆tAB)

4.1

εr(t) = εI (t + ∆tAB)

4.2

These strains ε i (t) and εr(t) determine the displacement of the incident barspecimen interface, u1(t). The strain εI (t) is recorded by the incident gages at any instant t. The strain in the transmission bar corresponding to this displacement is recorded after a time period of ∆tCD, which is the time taken by the elastic wave to travel the distance between the interface C and the pair of gages D shown in Fig. 4.5. Thus,

εt(t) = εT (t + ∆tCD)

4.3 Gage

Gage

Striker bar

A Incident bar

B

C

D Transmission bar

Specimen

4.5 Schematic of Split Hopkinson Pressure Bar.


Throw-off bar


91

where εT (t) is the strain measured by the transmission bar strain gages at any given instant of time. The time shifting assumes non-dispersive mechanical waves in the bars. Using the data recorded by the strain gages, the displacements of the bar interfaces are calculated as a function of time as follows.

u1 ( t ) = c 0

∫

t

(– ε i + ε r ) dτ

4.4

0

u2 ( t ) = – c 0

∫

t

ε t dτ

4.5

0

where c0 is the longitudinal wave velocity of the bar, and u2 is the displacement at the specimen/transmission bar interface. The average strain is then given by

εS =

u 2 – u1 c = 0 L L

∫

t

(– ε t + ε i – ε r ) dτ

4.6

0

where L is the original specimen length. The loads on each face of the specimen are given by: P1 = AE(εi + εr)

4.7

P2 = AEεt

4.8

where A is the bar cross-sectional area and E is the Young’s modulus. The average stress in the specimen is given by

σS = P AS

4.9

where As is the cross-sectional area of the specimen and P = (P1 + P2)/2.

4.2.2

Calculation of dynamic energy release rate

An important step in determining dynamic fracture toughness using the measured data is the numerical simulation of the dynamic response of the specimen. The stress and deformation field histories from the simulation can be used to calculate dynamic energy release rate history for both stationary and propagating cracks. The modified crack closure method is commonly used to calculate energy release rate in static fracture problems. By referring to Fig. 4.6 which depicts the 2D finite element mesh near the crack tip, the mode I and mode II energy release rates are calculated as GI =

1 f ( u – u ), G = 1 f ( u – u ) II 2b 1b 2 ∆a 2 c 2 a 2 ∆a 1 c 1 a


4.10

92


∆a a c

b

4.6 Elements and nodes used in the MCC method.

where f2c is the vertical nodal force component at crack tip node c, and u2a and u2b are the vertical displacements at nodes a and b, respectively. Similar meanings for f1c, u1a, and u1b are obvious. It was shown by Jih and Sun [20] that the modified crack closure method is valid for calculating the dynamic energy release rate. This method is especially efficient for calculating the energy release rate history for dynamically propagating cracks for 2D as well as 3D problems. The dynamic energy release rate can also be calculated from the energy balance consideration. This procedure is particularly simple if the boundary is fixed and no work is done during the time of interest. In that case, the energy released during a finite crack extension ∆a is equal to the decrease in the total elastic energy increment ∆ALLSE and total kinetic energy increment ∆ALLKE, i.e.,

G = – ∆ALLSE + ∆ALLKE 4.11 ∆a where G is the total energy release rate. This method allows one to use structural elements such as plate elements to model the dynamic response of a delaminated composite structure. These two methods (MCC and energy balance) produce very close results. However, the energy method can only be used to calculate the total energy release rate but not those of the individual fracture modes. In the static formulation of crack problems, there is a fixed relation between the stress intensity factor K and the strain energy release rate G. These two parameters are often interchangeably used to characterize fracture toughness of a material. However, in dynamic conditions, the relation between G and K depends on the crack speed. Thus, a constant (with respect to crack speed) Gc may not imply a constant Kc. © 2008, Woodhead Publishing Limited


93

For a moving crack in a homogeneous orthotropic material, the energy release rates can be related to the stress intensity factors KI and KII and the crack speed v as [21]

G I = 1 H 22 K I2 , G II = 1 H11 K II2 2 2

4.12

where H11 =

1 ρ α 2 [2 (1 + s )/ ξ ]1/2 C66 R 22 2

H 22 =

1 ρ [2 ξ (1 + s )]1/2 C66 R 22

R = ρ22 ( ρ22 ξ – 1 + α 22 ) – ρ122 α 22 / ξ

ρ11 =

α1 =

C11 C C , ρ22 = 22 , ρ12 = 12 C66 C66 C66

ρv 2   1 – C  , α 2 =  11 

ρv 2   1 – C   66 

ξ = α 1α 2 ρ11 / ρ22 s=

α 22 + ρ11 ρ22 α 12 – (1 + ρ12 ) 2 2 ρ11 ρ22 α 1α 2

and Cij are elastic constants. The material principal axes coincide with the coordinate axes. A state of plane strain parallel to x1–x2 plane is assumed and the crack is assumed to propagate in the x1 direction.

4.3

Mode I loading with double cantilever beam (DCB) for low crack velocity

In the experimental study of dynamic crack propagation in isotropic materials, the DCB specimen has been widely used because of its simplicity. Usually, the crack tip is blunted so that high crack propagation speed can be attained after crack initiation. Unfortunately, such practice does not work for composite materials. Experiments using conventional DCB specimens made of fiber composites loaded by servohydraulic loading machines can only produce stable crack propagation and the crack speeds obtained are limited by the high compliance of the DCB specimen and the crosshead speed of the loading machine. In order to achieve higher crack propagation speeds, Yaniv and Daniel [4] used a height-tapered DCB (HTDCB) specimen as shown in Fig. 4.7. In their tests, specimens were loaded on an Instron loading machine


94

Delamination behaviour of composites Conductive paint

VDC

To oscilloscope

4.7 Height-tapered DCB specimen with conductive paint circuit for monitoring crack propagation.

under constant displacement rate control. Limited by the crosshead speed of the loading machine, the maximum crack propagation speed obtained was 26 m/s. Because of its higher stiffness coupled with a lower crack propagation speed, the deformation of the HTDCB can be approximated as quasi-static and the dynamic (inertia) effect is neglected in the calculation of the deformation during loading. Moreover, since the resulting deflection is small, a linear beam theory that accounts for transverse shear can be used to model the quasi-static deformation of the specimen from which the mode I energy release rate is obtained as [4] 2 2   ˙ G I = 12 P2 3 k1 1 –  a  k1 k 2   c11 L  E11 b c  

4.13

where

k1 = 1 +

E11 2 –2/3 c a 10 G12

k 2–1 = 5 c 2 a –2/3 12

c=

h x 2/3


4.14 4.15 4.16


95

In Equations 4.14–4.16, E11 and G12 are the longitudinal Young’s modulus and shear modulus, respectively, of the composite; a is crack length, a˙ is the crack tip speed, h is the half height of the specimen at the distance x measured from the point of the applied load. Using the HTDCB specimen Yanif and Daniel [4] investigated AS4/35016 graphite/epoxy composite. Specimens were cut from 24-ply and 48-ply plates. Opening deflections were obtained directly from the Instron actuator piston and the crack tip position was monitored using lines of a conductive silver paint applied along the crack path as shown in Fig. 4.7. The spacing of the conductive lines was 6.3 mm. These conductive lines were connected to a battery. Stepwise variations of the voltage across the circuit occurred as the crack cut through these conductive lines yielding the instantaneous location as well as the speed of the crack tip. The dynamic delamination fracture toughness (energy release rate during crack propagation) of AS4/3501-6 was found to depend mildly on crack speed: it increases with crack speed up to a speed of 1 m/s and then decreases with increasing crack speed.

4.4

High crack velocity with modified double cantilever beam (DCB) and end notch flexure (ENF)

It is difficult to attain high crack velocities with the DCB specimen because the part of the specimen ahead of the crack tip is unstressed. For an ENF (end notch flexure) specimen loaded under three-point bending, the whole specimen is deformed during crack propagation, allowing locally stored strain energy to flow continuously to the crack tip and, as a result, to produce higher crack propagation speeds. In general, the conventional DCB and ENF specimens are not capable of producing high crack propagation velocities because of their high compliance and low toughness which prevent significant pre-cracking deformation in the specimen. Recently, Guo and Sun [10] developed a technique by toughening the tip of the pre-crack in the DCB specimen to produce high velocity mode I crack propagation in polymeric composites. The basic idea was to increase the crack initiation toughness and generate unstable crack propagation after crack initiation. This was accomplished by placing a strip of adhesive film at the tip of the pre-existing crack during laminate lay-up as shown in Fig. 4.8. The test results showed that the presence of the adhesive strip increased the crack initiation toughness significantly so that after initiation unstable crack propagation was generated. Later, Tsai et al. [16] used the same specimen design and used as an ENF specimen for mode II fracture tests producing even greater crack speeds.


96


Crack surface

Adhesive

Conductive lines

4.8 Specimen design of toughened specimen. Nickel print

P Copper fingers

Aluminum line 272 mm

4.9 Three-point bending set-up for the ENF specimen.

4.4.1

Specimen design

The basic configurations for the DCB and ENF specimens are identical. The only difference between the two is in how the load is applied. As depicted in Fig. 4.8, a layer of Myler release film was inserted at the midplane of the laminate to create the crack. A strip of FM-73 film adhesive is placed next to the Myler release film and cured together with the laminate. The Myler release film creates the pre-crack, and the tough FM-73 adhesive is used to delay the onset of cracking. Conductive lines are placed along the path of the propagating crack as shown in Fig. 4.8. In [16], 16 pure aluminum lines (each 0.6 mm thick) were deposited onto the edge side of the specimen by the vacuum vapor deposition technique using a template with precisely machined slits. Each slit was about 0.6 mm wide producing a 0.6 mm aluminum line. The spacing between two adjacent lines was 3 mm. For graphite/epoxy composites which are conductive, a thin layer of insulating varnish has to be first applied on the specimen before the deposition of thin aluminum lines. These aluminum lines were connected to an array of copper fingers by nickel print as shown in Fig. 4.9. © 2008, Woodhead Publishing Limited


97

These copper fingers were thin copper strips 1.6 mm wide and 0.1 mm thick, and were connected at the other end to a specially designed circuit [22] to detect the voltage change when an aluminum line is broken. Each line functions as a mechanical switch which has two conditions, either broken or connected. Because this switching system involves a mechanical fracture process which may cause sparking or unclear tearing, each line is connected to the input of a debounce circuit of two NAND gates. The debounce circuit prevents the generation of false signals which may result from either unclear tearing or momentary reconnection of the broken lines. The output of this device is a square wave-like signal with each rising or falling edge corresponding to the breakage of a conductive line. The output signals give the times of crack arrival and are recorded by an oscilloscope. Load, deflection, and crack extension can be continuously recorded.

4.4.2

Example for mode I delamination

Guo and Sun [10] studied mode I dynamic delamination fracture toughness of AS4/3501-6 graphite/epoxy using the toughened DCB specimen. DCB specimens were 30-ply unidirectional laminates. The specimen size was 260 mm (10 in) long, 20 mm (0.75 in) wide and 3.7 mm (0.174 in) thick with a 100 mm (3.9 in) pre-existing crack. Specimens were loaded at a displacement rate of 0.1 mm/s (0.004 in/s) on the MTS machine. At this stroke rate, the loading boundary condition can be considered as fixed boundary during dynamic crack propagation. A representative crack extension measurement is shown in Fig. 4.10 from which the crack extension history can be constructed. The least squares method was used to fit the obtained data into the crack extension history curve shown in Fig. 4.11. Since the crack velocity is the derivative of the crack extension curve, its value is quite sensitive to the way the crack extension curve is fitted. The process of curve fitting was completed in two steps. In the first step, the first several points, e.g. 5 points, of the experimental data were fitted by a second order curve, and then this curve was discretized into a large number of points, e.g. 1000 points. In the second step, these points combined with the rest of the experimental data were fitted by a higher order curve. This procedure gives a smoother fit of the data (up to crack extension of 0.015 m) which were not affected by reflected waves. The crack speed was then calculated by taking the derivatives of the crack extension curve. The result is shown in Fig. 4.12 as the solid line. The obtained crack extension history curve was then used as the input for the numerical simulation. For the quasi-static loading condition adopted in the experiment, the opening displacement of the loading end of the specimen was assumed to be fixed during crack propagation. The specimen was assumed © 2008, Woodhead Publishing Limited

98


3.5 3

Voltage (v)

2.5 2 1.5 1 0.5 0 –0.5 –1E–4

0E+0

1E–4

2E–4 3E–4 Time (s)

4E–4

5E–4

6E–4

4.10 Crack extension measurement. 0.05

Crack extension (m)

0.04

0.03

0.02

0.01

0 0E+0

1E–4

2E–4

3E–4 Time (s)

4E–4

5E–4

6E–4

4.11 Crack extension history curve.

to be in a static state until the displacement of the loading end reached the critical displacement obtained and then the crack was allowed to propagate according to the given crack extension history. Figure 4.12 shows a typical crack velocity history from which we can see that the crack speed is the highest at the beginning and drops quickly as it propagates. Such a sudden © 2008, Woodhead Publishing Limited


99

400 350

Crack speed (m/s)

300 250 200 150 100 50 0 0

0.01

0.02 0.03 Crack extension (m)

0.04

0.05

4.12 A typical crack speed history.

rise in crack speed could cause numerical oscillations in the transition from static state to dynamic state. Noting that the crack propagation should have initiated at a small distance from the first conductive line, and the actual crack speed history should start from zero, we modified the crack speed history by adding a transition stage represented by the dashed line shown in Fig. 4.12. In fact the crack speed was assumed to reach its initial value from zero at a constant acceleration with respect to crack extension. The dynamic energy release rate history corresponding to the crack speed history such as that given by Fig. 4.12 was calculated using the energy balance approach as described in Section 4.2.2 using a double plate model proposed by Zheng and Sun [23]. In this model, a composite laminate with a delamination crack is treated as two separate Mindlin plates. Outside the delamination region, tying constraints are imposed to ensure structural integrity. In this study, the double plate approach was extended to model mode I dynamic crack propagation in the DCB specimen. Because of symmetry, the tying constraints which ensure the continuity of the material ahead of the crack front become very simple: only the displacement in the direction perpendicular to the midplane of the specimen needs to be constrained, and this can be accomplished by using boundary constraint in the finite element formulation. The material constants for AS4/3501-6 are E1 = 142 GPa, E2 = E3 = 10.3 GPa, G12 = G13 = 7.2 GPa, υ12 = υ13 = 0.27, υ23 = 0.45, ρ = 1580 kg/m3. Figure 4.13 shows a typical calculated dynamic energy release rate. It should be noted that the initial portion (crack extension < 0.003 m) of the dynamic strain energy release rate should be viewed with doubt because of © 2008, Woodhead Publishing Limited

100


Dynamic energy release rate (N/1)

1200

1000 800

600

400

200

0

0

0.01


0.04

0.05

4.13 Dynamic energy release rate history of a DCB specimen.

the uncertain crack speeds in that region. Consequently, the data in this region were not taken. The oscillatory behavior of the energy release rate at the later stage (crack extension > 0.017 m) of propagation is caused by the waves reflected from the loading end. Because the hinges at the loading ends cannot be accurately modeled, the calculated wave reflections are not accurate. Therefore, the dynamic energy release rates were taken only from the range of crack extension between 0.003 m to 0.017 m. The results from three specimens are plotted in Fig. 4.14 against crack speed. The critical static energy release rate GIC obtained by using area method is also plotted as the dashed line. Since the actual movement of the crack tip was prescribed in the numerical simulation, the calculated dynamic energy release rate can be considered as fracture toughness. From Fig. 4.14, the critical dynamic energy release rate seems to be a constant for crack speed less than 200 m/s and it is very close to static fracture toughness GIC of the composite. For higher speeds, the critical dynamic energy release rate seems to rise slightly as crack speed increases.

4.4.3

Example for mode II delamination

Tsai et al. [16] used the toughened ENF specimen to study dynamic mode II delamination in Hercules S2/8553 glass/epoxy unidirectional composite. The specimen design is shown in Fig. 4.8 and three-point-bending test configuration is shown in Fig. 4.9. To produce crack propagation, the specimen was loaded at a displacement rate of 0.025 mm/s on an MTS machine. Crack extension was monitored by means of a conductive line circuit applied along the path © 2008, Woodhead Publishing Limited


101

Dynamic energy release rate (N/1)

500 450

GIC

400 350 300 250 200 150 100 50 0 50

100

150

200 250 Crack speed (m/s)

300

350

4.14 Critical dynamic energy release rate vs crack speed based on three specimens.

of the propagating crack as described in detail in Section 4.4.1. The dimensions were 278 mm long, 12.7 mm wide and 11.6 mm thick. The pre-crack length was 62 mm. The material constants for S2/8553 are E1 = 43.3 GPa, E2 = E3 = 12.7 GPa, G12 = G13 = 4.46 GPa, ν12 = ν13 = 0.29 and mass density ρ = 2100 kg/m3. For each specimen, the center deflection of the specimen at which crack propagation initiated was measured, and the signals generated by crack extension were recorded. Figure 4.15 shows recorded signals from the conductive lines circuitry on a specimen. The vertical lines indicate the times when the conductive lines were broken, which were then used to construct the time history of the position of the propagating crack as shown in Fig. 4.16. Subsequently, the least squares method was used to fit these data into the crack extension history curve which is also shown in Fig. 4.16 from which crack propagation velocities were calculated. The crack extension history curve was then used as the input for the finite element simulation. In a close examination of the recorded crack extension signals, such as those shown in Fig. 4.15, it is noted the first time passage of the crack through the first pair of conductive lines was much greater than the subsequent times. This indicates that the crack speed was very low initially. Such behavior was not observed in the mode I test using the DCB specimen as reported in Section 4.4.2. In the case of the ENF specimen, the film adhesive FM-73 placed ahead of the pre-crack tip fails in shearing mode that could exhibit a large failure strain. It is likely that the crack might have started to propagate from a location ahead of the film adhesive before the film adhesive was © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites 3.5 3 2.5

Voltage (v)

2 1.5 1 0.5 0 –0.5 –5.0E–5

0.0E+0

5.0E–5 1.0E–4 Time (s)

1.5E–4

2.0E–4

4.15 Recorded crack extension signals for specimen ES34. 4.5E–2 4.E–2

Crack extension (m)

3.5E–2 3.E–2 2.5E–2 2.E–2 1.5E–2 1.E–2 5.E–3 0.E+0 0.E+5

2.E–5

4.E–5 6.E–5 Time (s)

8.E–5

1.E–4

4.16 Crack extension history of specimen ES34.

completely severed. In view of the foregoing, the initial part of the interlaminar toughness was regarded as contaminated by the film adhesive. Data from three specimens were used for numerical analysis. Since the crack propagated very slowly between the first conductive line and the second, the first signals of all three specimens were neglected, and the crack was assumed to initiate from the position of the second conductive line. Since the © 2008, Woodhead Publishing Limited


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stroke rate (0.025 mm/s) was very low in the experiment and the fracture event took place in a very short time, the stroke loading was taken as stationary in the analysis. In contrast to impact loading, in this simulation the loading and boundary conditions during the period of crack propagation were precisely prescribed. A finite element simulation of mode II dynamic crack propagation was performed based on ABAQUS version 5.4. Crack surfaces to be generated by crack propagation during the analysis were modeled as bonded surfaces connected by interface contact elements INTER2 of zero thickness as shown in Fig. 4.3. The BOND SURFACE option was used so that these surfaces were perfectly bonded initially. By using DEBOND and CRACK GROWTH options, these surfaces could be separated during the analysis through the separation of the nodes on these surfaces according to the measured crack propagation history. Note that the interface contact element can be used only in conjunction with 2-D plane elements. The 4-node plane strain element was used in the analysis. The crack propagation was simulated by separating the nodes. In each node separating step, crack length increases by the size of one element ahead of the crack tip. Each step was completed in a number of increments defined in the input file. At the beginning of each step, the nodes at the current crack tip were released simultaneously by using DEBOND and CRACK GROWTH options. Then the nodal force at the crack tip was reduced to zero linearly by setting the parameter AMPLITUDE of the STEP option equal to RAMP. The energy release rate was calculated at the end of each step by either the modified crack closure method or the energy balance method as described in Section 4.2.2. Because crack surfaces are modeled as contact surfaces, for mode II problems, small numerical oscillations can incur contact and cause severe numerical problems of convergence. To avoid such a problem, a small amount of opening displacement was added to the cracked end of the specimen in the analysis to keep crack surfaces open. Figure 4.17 presents the dynamic interlaminar toughness obtained from three specimens plotted against crack speed. For the purpose of comparison, the static mode II fracture toughness of S2/8553 glass/epoxy obtained using the conventional ENF test is also shown in this figure. It is evident that lower than 300 m/s, the dynamic fracture toughness of this composite seems to be more or less the same as the static value for crack speeds from 300 m/s to 700 m/s. For crack speeds the dynamic toughness appears to be significantly higher than the static value. It should be noted that this range of low crack speed corresponds to the initial portion of crack propagation that might have been contaminated by the residual resistance from the adhesive film. For this reason, the toughness data below 300 m/s were not included in Fig. 4.17. Tsai et al. [16] also reported the result of testing AS4/3501-6 carbon/ epoxy composite ENF specimens for mode II dynamic toughness using the © 2008, Woodhead Publishing Limited

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GIIC

Critical energy release rate (J/m2)

3500 3000 2500 2000 1500 1000 500 0 300

400

500 Crack speed (m/s)

600

700

4.17 Mode II dynamic fracture toughness vs. crack speed for S2/8553.

same three-point-bending setup. They found that crack speeds reached over 1000 m/s. Because of the high crack speeds, one or two signals associated with breakage of the conductive lines were often missing during the test. This problem could be caused by the lack of opening of crack surfaces making breakage of conductive lines at high crack propagation speed more difficult to detect by the circuit. To alleviate this problem, they moved the pre-crack from the mid-plane to one third of the specimen thickness. This new ENF specimen would generate some crack surface opening due to the presence of mode I action. This design resulted in complete breakage of conductive lines and reduced the probability of missing signals. However, the new modified ENF specimen produces mixed mode fracture instead of pure mode II fracture.

4.5

Mode I by wedge loading with Hopkinson bar

Wedge loaded compact-tension (WLCT) specimens of unidirectional composite laminates were used by Sun and Han [9] to investigate dynamic delamination fracture behavior in polymeric composites. This method is similar to the wedge-insert-fracture (WIF) method employed by Kusaka et al. [7] except for the specimen design. Sun and Han [9] performed quasi-static tests on an MTS machine, and conducted dynamic fracture tests on a Split Hopkinson Pressure Bar (SHPB) apparatus for achieving high speed mode I delamination © 2008, Woodhead Publishing Limited


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Table 4.1 Material properties of composites Composite

S2/8552 IM7/977-3

E11 (GPa)

E22 (GPa)

E33 (GPa)

G12 (GPa)

ν12

ν23

ρ kg/m3

50 162.47

16 9.52

16 9.52

6.9 6.09

0.28 0.37

0.4 0.4

1980 1585

Crack gage

11

Pre-crack

45°

W

6~8 19.5

4.18 The WLCT specimen and its dimensions in mm.

crack propagation (up to 1000 m/s). The finite element method was used to simulate the dynamic delamination crack propagation in the unidirectional fiber composites from which both initiation fracture toughness and dynamic fracture toughness were determined. Two polymeric composite systems (Hexcel S2/8552 glass/epoxy and Cytec Fiberite IM7/977-3 carbon fiber/epoxy) were considered. The material constants are listed in Table 4.1.

4.5.1

Specimen fabrication

The wedge loaded compact-tension (WLCT) specimen of unidirectional composite laminates is shown in Fig. 4.18. The small dimensions make it suitable for dynamic tests on the SHPB. The width of the specimens, w, ranges from 6 to 10 mm. A pre-crack is created by inserting a thin layer of Mylar release film (about 0.025 mm in thickness) or a TEF Teflon film (TFE-1, about 0.05 mm thick from Measurement Groups, Inc.) into the laminate before curing. In order to make the laminates about 9 mm thick, 136 plies of S2/8552 (68 plies + film insertion + 68 plies), and 90 plies of IM7/977-3 (45 plies + insertion + 45 plies) were used, respectively. The cured panels were cut by a water-jet cutter into strips of about 19.5 mm in width, and machined by a © 2008, Woodhead Publishing Limited

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double edge 45° V-cutter to create a V-notch behind the pre-crack tip. These strips were subsequently cut into blocks with almost the final dimensions of a WLCT specimen. The V-notch surfaces of the block specimen were polished and their opposite end surfaces were lapped and polished. Two crack propagation gages (TK-09_CPB02_005/DP by Measurements Group, Inc.) were mounted on the specimen as shown in Fig. 4.18 with MBond 600 adhesive to measure the speed of crack propagation. Each crack propagation gage consists of ten conductive strands or ten parallel resistors with the same resistance of 50 Ω for each strand. The distance between two adjacent strands is 0.25 mm. When a strand is broken by the advancing crack, the total resistance of the gage changes and so does the output of voltage. The crack speed is determined with the knowledge of the strand spacing and the times of breakage of two adjacent strands.

4.5.2

Experimental procedure

A specially designed wedge tip was used to generate the cleavage loading on the WLCT specimen in both quasi-static and dynamic tests. The wedge (see Fig. 4.19), which was made of heat-treated drill steel rod of 12.7 mm in diameter, has a cylindrical front tip of 0.77 mm in radius and a curved transition neck to ensure that only the wedge tip portion would be in contact with the V-notched surface of the specimen. A 10 kip MTS machine was used to perform the quasi-static test. The specimen was placed on a rotation-wise self-adjustable horizontal platform that was held by the lower crosshead of the MTS machine with the V-notch pointing up. Lubricant (petroleum jelly) was applied on the wedge tip to minimize friction. A stroke control at rate of 0.01 mm/s was used. Outputs of P Upper Xhead Wedge

Specimen CV1

OSC PC

Crack gage Platform CT

CV2

Lower Xhead CV1: crack gage converter CT: MTS controller PC: personal computer

CV2: MTS converter OSC: oscilloscope

4.19 Schematics of quasi-static test setup for MLCT specimen.



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both load and displacement from the MTS machine were sent to a PC and acquired by LabView (a software product of National Instrument). In the case that crack propagation information was acquired, the output from the crack gage was sent to a Tektronix® TD 420 oscilloscope, and was downloaded to a PC for further processing. The dynamic test was conducted on an SHPB apparatus (see Fig. 4.20). The setup consists of a launch system including a pressure reservoir and an air gun (not shown in Fig. 4.20), bars (strike bar, incident bar, and transmission bar), and data acquisition system (converters, Wheatstone bridges, amplifiers, oscilloscope and PC). The wedge tip placed against the V-notch of the specimen was sandwiched between the incident bar and transmission bar. Lubricant was applied on the contact surfaces of the wedge tip and specimen as well as the contact surface between the specimen and transmission bar. The striker bar, incident bar and transmission bar were made from solid steel. They are 12.7 mm in diameter, 102 mm long for the striker bar, 914 mm long for the incident bar, and 559 mm long for the transmission bar. The striking end of the striker bar is slightly rounded to reduce high frequency contents in the incident signal from a pair of general purpose strain gauges (gage factor 2.155, 350 Ω). Because the load level in the transmission bar is much lower than that in the incident bar, a pair of sensitive N-type semiconductor strain gages (gage factor –107.4, 350 Ω) were used on the transmission bar at a position 178 mm from the end that was in contact with

Wedge Specimen

Striker bar

Transmission bar

Incident bar

WB

AMP

Crack Trans. gage bar

CV1

OSC

WB

AMP

PC WB: Wheatstone bridge AMP: amplifier PC: personal computer

CV1: crack gage converter OSC: oscilloscope

4.20 Schematics of dynamic test setup for WLCT specimen.


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the specimen. The double strain gage scheme is used to remove the signal of any possible bending waves. The strain gauges on the bars were connected to Wheatstone bridges (WB) excited by about 6 DC voltages, and the outputs of WB were sent to amplifiers (Tektronix® AM 502). Proper gains of the amplifier (500 times for the incident channel, and 200 times for the transmission channel) were used, and outputs from the amplifier were sent to a fourchannel oscilloscope (Tektronix® TD 420), which was then connected to a PC. Output signals of the crack propagation gages from the conversion circuit suggested by the manufacturer were too weak to be accurately measured if no amplifier was used. Due to the bandwidth limit of the amplifier (Tektronix® AM 502) and the fast event of the delamination crack propagation (a few microseconds), the conversion circuit was modified as shown in Fig. 4.21 in which R1 through R10 represent the ten strands of the crack propagation gage. A dummy resistor of 50 Ω in parallel and another 275 Ω resistor (instead of the suggested 100 kΩ resistor) in serial plus a 15 V DC power supply were used. This modified circuit can provide much stronger signatures to be accurately recorded by the oscilloscope without an amplifier. However, this modification may cause a heating problem to the crack gage. To avoid the potential heating problem, the duration of the circuit with power on was always kept as short as possible. Shielding wires were also necessary for the wiring from the gauges to the Wheatstone bridges and to the oscilloscope in order to eliminate radio noises. In dynamic tests, signals from the two crack propagation gages, and from the transmission bar were recorded by the oscilloscope, which was set at 100 MS/s (the highest sampling rate of the oscilloscope) and with 60 000 sampling points in total (the full capacity of the oscilloscope). The channel connected

275Ω Ch 1, 2, 3, 4 15V DC 50Ω

R1 R2

R9R10

Crack gage

4.21 Modified crack gage conversion circuit.


Oscilloscope


109

to the first crack propagation gage was used as the trigger channel. The triggering level was set at 870 mV.

4.5.3

Experimental results

A typical signal recorded from the crack propagation gage in dynamic testing (specimen F09) on the SHPB is shown in Fig. 4.22 in which (a) and (b) are signals from the first and second crack gages, respectively. A rising signal step corresponds to the breakage of each strand of the gage. The time between two steps was used to compute the crack propagation speed. All the ten steps (corresponding to the ten strands) of the captured signals can be identified throughout the testing time. Figures 4.23 and 4.24 show typical recorded incident and transmission signals, respectively, obtained from the test. The incident and reflected wave signals in the incident bar are almost identical but are of opposite signs 4 F09_1

3

Amplitude (V)

Amplitude (V)

4

2 1

3 2 1 0

0 –2

–1

0 1 Time (µs)

2

3

F09_2

2

3

4 5 Time (µs)

4.22 Typical output of crack propagation signal. 1.5 1

F09_I

Output (V)

0.5 0 –160

–60

40

140

–0.5 –1 –1.5 Time (µs)

4.23 Typical output from the strain gage on the incident bar.


6

110

Delamination behaviour of composites 2 F09_T

Output (V)

1

0 –25

75

175

275

–1 –2 Time (µs)

4.24 Typical output from the strain gage on the transmission bar.

indicating a nearly complete reflection and, as a result, the transmission wave signal is very weak. This is why high-sensitive semi-conductor strain gages were needed for the strain measurement in the transmission bar. It also implies that signals from the incident bar are not suitable to be used for strain or stress calculations. Instead, signals from the transmission bar shown in Fig. 4.24 were used for further derivations and simulations.

4.5.4

Dynamic fracture toughness

The finite element analysis was used to investigate the quasi-static and dynamic fracture initiation toughness as well as dynamic fracture toughness (during crack propagation) of the two composites. Owing to symmetry, only one half of the specimen was modeled. Furthermore, the problem was modeled in 2D under a state of plane strain. Commercial package MARC/MENTAT was used as a preprocessor, and ABAQUS (Standard and Explicit) were used as the solver and partially as the post-processor. Figure 4.25 shows the finite element model used in modeling dynamic fractures. The transmission bar is represented by the portion with coarse meshes and with the semi-infinite plane strain element (INFPE4) on the right end (elements with one side open) of the model. The symmetry line of the specimen is aligned with the horizontal axis (coordinate 1). The wedge tip was modeled as a rigid body with a cylindrical shape (see Fig. 4.25) using a mass element (*MASS in ABAQUS/Explicit). The mass element was prescribed with a very small density (10–2 kg/m3). Contact pairs were prescribed for the wedge and V-notch surfaces, and for the flat specimen end and the transmission bar (heavy solid line in Fig. 4.25). All contacts were assumed to be elastic and frictionless. Fine square meshes (element length = 3.43 × 10–5 m) were used along the constrained horizontal boundary of the symmetry line to facilitate the energy release rate computation discussed in the following. © 2008, Woodhead Publishing Limited


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Transmission bar

Specimen Wedge

P (t ) 2

1 3

4.25 The FE model for fracture analysis. 3

350 F09_T F09_Crack1 F09_Crack2

250

2.5 2

200

1.5

150 1

100

0.5

50 0 0

10

20

30 Time (µs)

40

50

Cractk output (V)

Load, P(t) (N)

300

0 60

4.26 Loading history and time/position of crack propagation measurements.

Delamination crack propagation was simulated by sequentially releasing the constraints in the vertical direction on the boundary nodes along the crack propagation path (the symmetry line) according to the measured crack speed. The timing of node release must be synchronized with the loading history. The time-step for each nodal release at the crack tip was determined from the finite element size and the measured crack tip position history. Within each time-step, there were, on the average, more than a hundred time increments in the time integration used by the solver (ABAQUS). Figure 4.26 shows typical synchronized loading and crack propagation histories for IM7/977-3, in which curve ‘F09_T’ denotes the converted loading history (half of the total load in amplitude), ‘F09_crack1’ and ‘F09_crack2’ denote the first and second crack gage measurements, respectively. A typical crack speed history measured from the crack gages is shown in Fig. 4.27. A linear fit is used to capture the crack propagation velocity. Since © 2008, Woodhead Publishing Limited

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Crack speed (m/s)

800 600 400

Test (F09) Liner fit

200 0 0

2

4 6 Crack extension (mm)

8

4.27 Propagation speed of a delamination crack in IM7/977-3 composite. 2000 S2FG/8552 IM7/977–3

GIc (J/m2)

1500

1000

500

0 0

5 ˙ log GI (J/(m2s))

10

4.28 Effect of loading rate on crack initiation toughness.

no information is available for crack propagation from its initial tip position to the first strand of the first crack gage, the first segment (rising part) of the linear fit was assumed. The modified crack closure method [20] was used to compute both static and dynamic strain energy release rates, GI. The energy release rate for dynamic crack initiation, G Ii_dyn , was calculated according to the nodal force of the crack tip node and nodal displacement behind the crack tip at the time of onset of fracture. The dynamic energy release rate, GI_dyn, for a moving crack was calculated in the same manner for the entire crack propagation history. Owing to serious fiber bridging during the dynamic crack propagation in S2/8552 specimens, only its crack initiation fracture was investigated. Figure 4.28 shows the initiation fracture toughness data for both S2/8552 and IM7/ 977-3 specimens. In the figure, initiation fracture toughness GIc was plotted against G˙ I which is defined as © 2008, Woodhead Publishing Limited


Gi G˙ I = I ∆t

113

4.17

where G Ii is the energy release rate at crack initiation, and ∆t is the time for GI growing from zero to G Ii . This average time rate of GI can be used to represent the loading rate before onset of cracking [7]. Note that the dynamic initiation fracture toughness was obtained with dynamic loading rates that were six orders higher than those used in the quasi-static test. The data in Fig. 4.28 show that the initiation fracture toughness of IM7/977-3 does not exhibit appreciable load rate-dependency. Although the static value is slightly higher than the dynamic value, they are all in the data scatter range. The dynamic initiation fracture toughness of S2/8552 is clearly higher than its static counterpart. Such an increase in fracture toughness could be partially attributed to fiber bridging, which was noted to be more severe in the dynamic test than in the static one. Dynamic energy release rate during crack propagation in IM7/977-3 was also calculated based on the measured delamination crack speed history. Figure 4.29 shows the results of calculated dynamic energy release rate versus crack extension for two test specimens. It should be noted that the calculated values shown to the left of the vertical dashed line in Figure 4.29 may not be reliable due to the uncertainty of the initial crack speed. Figure 4.30 shows the computed dynamic energy release rate GI against propagation speed of the delamination crack. For comparison, the average dynamic initiation fracture toughness, G Ici_dyn which is about 458 J/m2, and static initiation fracture toughness, G Ici_static , which is about 495 J/m2, are also shown in Fig. 4.30. The computed dynamic energy release rate GI in Figs 4.29 and 4.30 are the dynamic fracture toughness, GIc-dyn, of IM7/9773 during crack propagation. Based upon these results, it can be concluded 700 600

GI (J/m2)

500 400 300

F09

200

F07

100 0

0

0.001


0.004

0.005

4.29 Dynamic energy release rate versus crack extension for IM7/9773 composite.


114


Dynamic GI (J/m2)

700 600 500 400 i GIc_dyn

300

i GIc_static

200 100 0 600

700

800 Crack speed (m/s)

900

4.30 Dynamic energy release rate GI versus crack speed for IM7/977-3 composite.

that the dynamic fracture toughness GIc-dyn of IM7/977-3 is basically insensitive to delamination crack propagation speeds up to 950 m/s, and that dynamic fracture toughness, GIc-dyn is approximately equal to the quasi-static initiation fracture toughness G Ici_static and to the dynamic initiation fracture toughness G Ici_dyn . This conclusion is similar to the result obtained by Guo and Sun [10] who studied fracture toughness for crack speeds up to 300 m/s in the AS4/3501-6 carbon/epoxy composite.

4.5.5

Comparison of wedge loaded compact-tension (WLCT) and double cantilever beam (DCB) specimens

The mode I fracture toughness obtained from the WLCT specimen was compared with the result obtained from the conventional double cantilever beam (DCB) fracture test for IM7/977-3 [9]. The IM7/977-3 DCB specimen was fabricated from a 20 ply unidirectional laminate panel with a Teflon film placed in the middle plane (10 plies + insertion + 10 plies) to form a prexisting delamination crack. The DCB specimen was 2 mm in thickness (2h), 18 mm in width (b), crack length a is 70 mm, and the total specimen length was 110 mm. The quasi-static DCB test was performed on a 1 kip MTS machine (444.8 N load cell, 63.5 mm stroke cell). The stroke control mode was used for loading with a rate of 0.127 mm/s. The measured peak load Pcr was used to compute the mode I fracture toughness GIc as

12 Pcr2 a 2 Eb 2 h 3 where E is Young’s modulus in fiber direction (E11 in Table 4.1). G Ic =


4.18


115

The averaged mode I fracture toughness GIc of IM7/977-3 from the DCB test was 460 J/m2 which is in fairly good agreement with G Ici_static (about 495 J/m2) from the WLCT test.

4.6

Acknowledgment

The majority of the work by this author reported here was supported by ONR grants.

4.7

References

1. Duffy, J., Suresh, S., Cho, K. and Bopp, E.R., A method for dynamic fracture initiation testing of ceramics, ASME J. Engng Mater. Tech., 1988, 110, 325–331. 2. Kalthoff, J.F., Shadowoptical analysis of dynamic fracture, Opt. Eng., 1988, 27, 835–840. 3. Aliyu, A.A. and Daniel, I.M., Effects of strain rate on delamination fracture toughness of graphite/epoxy. Delamination and Debonding of Materials, ASTM STP 876, W.S. Johnson (ed.), American Society for Testing and Materials, 1985, 336–348. 4. Yaniv, G. and Daniel, I.M., Height-tapered double cantilever beam specimen for study of rate effects on fracture toughness of composites. Composite Materials Testing and Design 8th Conference, ASTM STP 972, J.D. Whitcomb (ed.), American Society for Testing and Materials, 1988, 241–258. 5. Smiley, A.J. and Pipes, R.B., Rate effects on mode I interlaminar fracture toughness in composite materials, J Comp Mats, 21, 1987, 670–687. 6. Daniel, I.M., Shareef, I. and Aliyu, A.A., Rate effects on delamination fracture toughness of a toughened graphite/epoxy, Toughened Composites, ASTM STP 937, N.J. Johnston (ed.), 1987, 260–274. 7. Kusaka, T., Hojo, M., Mai, Y.-W., Kurokawa, T., Jojima, T. and Ochiai, S., Rate dependence of mode I fracture behavior in carbon fiber/epoxy laminates, Comp Sci & Tech., 1998, 59, 591–602. 8. Blackman, B.R.K., Dear, J.P., Kinloch, A.J., MacGillivray, H., Wang, Y., Williams, J.G. and Yayla, Y., The failure of fiber composites and adhesively bonded fiber composites under high rates of test: Mode I loading experimental studies, J Mat Sci, 1987, 30, 5885–5900. 9. Sun, C.T. and Han, C., A method for testing interlaminar dynamic fracture toughness of polymeric composites, Composites Part B, 35(6–8), 2004, 647–655. 10. Guo, C. and Sun, C.T., Dynamic mode I crack propagation in a carbon/epoxy composite, Composite Science and Technology, 58(9), 1998, 1405–1410. 11. Takeda, N., Sierakowski, R.L., Ross, C.A. and Malvern, L.E., Delamination-crack propagation in ballistically impacted glass/epoxy composite laminates, Experimental Mechanics, 22(1), 1982, 19–24. 12. Grady, J.E. and Sun, C.T., Dynamic delamination crack propagation in a graphite/ epoxy laminate. In: Composite Materials: Fatigue and Fracture, ASTM STP 907, American Society for Testing and Materials, Philadelphia, PA, 1986, 5–31. 13. Lambros, J. and Rosakis, A., Experimental investigation of dynamics delamination in thick polymeric composite laminates, Experimental Mechanics, 37(3), 1997, 360– 366.


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14. Lambros, J. and Rosakis, A.J., Dynamic crack initiation and growth in thick unidirectional graphite/epoxy plates, Composites Science and Technology, 57, 1997, 55–65. 15. Coker, D. and Rosakis, A.J., Experimental observations of intersonic crack growth in asymmetrically loaded unidirectional composite plates, Philosophical Magazine A, 81(3), 2001, 571–595. 16. Sun, C.T. and Bing, Q., Characterizing mode II fracture toughness in polymeric composites using off-axis specimens. Procedings of the 48th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, Hawaii, 23–26 April, 2007. 17. Tsai, J.L., Guo, C. and Sun, C.T., Dynamic delamination fracture toughness in unidirectional polymeric composites, Composites Science and Technology, 61, 2001, 87–94. 18. Kolsky, H., An investigation of the mechanical properties of materials at very high rates of loading, Proc Phys Soc B, 62, 1949, 676–701. 19. Davies, E.D.H. and Hunter, S.C., The dynamic compression testing of solids by the method of the split Hopkinson Pressure Bar, Journal of The Mechanics and Physics of Solids, 11, 1963, 155–179. 20. Jih, C.J. and Sun, C.T., Evaluation of a finite element based crack-closure method for calculating static and dynamic strain energy release rates, Eng Frac Mech, 37, 1990, 313–322. 21. Yang, W., Suo, Z. and Shih, S.F., Mechanics of dynamic debonding. Proc Roy Soc, A, 443, 1991, 679–697. 22. Chang, C., A digital dynamic-crack speed measuring device, Experimental Techniques, 13, 1989, 16–17. 23. Zheng, S. and Sun, C.T., A double plate finite element model for impact induced delamination problems, Composites Science & Technology, 53, 1995, 111–118.


5 Experimental characterization of interlaminar shear strength R G A N E S A N, Concordia University, Canada

5.1

Introduction

The stresses acting on the interface of two adjacent plies in composite laminates are called interlaminar stresses. There are three such stresses, which are, interlaminar normal stress σz and interlaminar shear stresses τxz and τyz. Here z direction is along the thickness of the laminate and further, x and y are in-plane directions. Relative deformations between adjacent plies are caused by these stresses. If these stresses exceed the corresponding strength values that are constants for a particular composite material, failure along the interface takes place. Interlaminar stresses cause delamination failure. Delamination is the most critical failure mode in laminated composite materials. It can cause unstable crack growth in composite structures that leads to a catastrophic failure condition as a result of its influence on a component or structure made of laminated composite material in weakening its resistance to subsequent failure modes. It may be noted here that the resistance to delamination of laminated composite material can be characterized using two different approaches, which are, the strength-of-materials approach and the linear elastic fracture mechanics approach. In the present chapter the interlaminar shear strength of composite material is considered. The interlaminar shear strength is used in the strength-ofmaterials approach as the controlling factor for the initiation of delamination due to interlaminar shear. Therefore, it is of considerable interest and importance to determine the interlaminar shear strength of composite materials through appropriate tests. These tests must be such that failure of the laminate specimen is initiated in a transverse shear mode. Based on this consideration, test methods have been devised to determine as accurately as possible the interlaminar shear strength of the laminated composite material. Interlaminar shear strength is a measure of the in situ shear strength of the matrix layer that is present between plies in a composite laminate. Even today, there is no method available for the exact determination of this strength property. However, approximate values of the interlaminar shear strength, 117 © 2008, Woodhead Publishing Limited

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called the apparent interlaminar shear strength, can be determined by using various test methods. In this regard, it is noted that an ideal test method for characterizing a basic material property is the one that can provide a known state of relatively uniform stress and strain in the specimen gage section while minimizing the stress concentration in the load introduction section. Researchers have been trying hard during the past to establish such ideal test methods based on analytical studies, numerical investigations, and, more importantly, engineering intuition. So far, three test methods: short beam shear test, double-notch shear test, and Arcan test have been very well developed and established as standard test methods for the determination of interlaminar shear strength. In the present chapter, these three experimental methods are presented. In addition to describing in full detail the test methods, the related modeling and analysis aspects and observations based on test experience are also presented in order to justify the approaches taken and to provide insight into the procedures used for data reduction. This will also help in understanding clearly the limitations and validity of these test methods and the care that must be taken in interpreting and using the test results in design and development tasks. In Section 5.2, the short beam shear test is considered. In Sections 5.3 and 5.4, the double-notch shear test and Arcan test are considered. The last section provides overall discussion and concluding remarks, wherein research trends and future research directions are discussed.

5.2

Short beam shear test

5.2.1

Test method

This test method is covered in the ASTM Standard D 2344 / D 2344M-00. In this method, a laminate specimen with a low value of L/h ratio, where L is the length of the support span and h is the specimen thickness, is subjected to three-point loading as shown in Fig. 5.1. The beam is machined from a relatively thick, at least 16 plies thick, unidirectional laminate with the fibers in the axial direction of the test specimen. Precise values of dimensions recommended for the test specimen and of the distance between support points can be obtained from ASTM Standard D 2344 / D 2344M-00. The test specimen is loaded in the direction normal to the plies, that is, direction-3.

5.2.2

Test specimen parameters

Appropriate values for the geometrical parameters of the test specimen have to be selected with the objective of eliminating unwanted failure modes in the specimen. When a beam is subjected to a three-point bending loading, both flexural stress and shear stress are induced in the beam. The shear stress



119

P

1

h

b L

5.1 Short beam shear test for measurement of interlaminar shear strength.

acts in the direction transverse to the beam axis and is significant in the case of relatively short beams. A large span-depth ratio in a flexural test increases the maximum normal stress without increasing the interlaminar shear stress. As a result, the influence of interlaminar shear on the beam deformation is less pronounced and the developed interlaminar shear stresses across the cross-section will be smaller compared to flexural stresses. Hence, the tendency for longitudinal failure due to flexure increases. Either tensile or compressive flexural failure may take place at the outer plies of the laminated beam before shear failure at the mid-plane takes place. On the other hand, if the span is short the failure is initiated and propagated by interlaminar shear. Such a short beam specimen can be used to determine the interlaminar shear strength of the composite material. The maximum bending stress σmax occurs at the top and bottom surfaces of the beam cross-section at the mid-span. Based on the classical beam theory, the maximum shear stress τmax can be related to the maximum bending stress σmax through the equation

τ max = h σ max 2L

5.1

for the case of three-point loading. This relationship can be used to determine the appropriate specimen geometry that can ensure the occurrence of interlaminar shear failure in the test specimen. From Equation 5.1, it can be observed that an interlaminar shear failure would be assured if

2L S p 1 S13 h

5.2

where S1 and S13 are respectively the tensile strength or compressive strength (whichever is lesser) and the interlaminar shear strength of the composite material. © 2008, Woodhead Publishing Limited

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To ensure that interlaminar shear failure occurs prior to flexural failure, the length-to-height ratio must satisfy Equation 5.2. For a typical laminated beam made of 15 to 20 plies of carbon-epoxy composite material, a span of 1 cm/0.4 in., width of 0.64 cm/0.25 in., and thickness of 1.9 to 2.5 mm (0.075 to 0.1 in.) can be used.

5.2.3

Data reduction

In accordance with the classical beam theory, the value of the transverse (interlaminar) shear stress τ13 at the mid-plane of the laminated beam can be calculated using the formula

τ 13 = 3 P 2 bh

5.3

where P is the applied transverse load on the beam, b denotes the width of the beam and h denotes the height of the beam, i.e. the specimen thickness. During the test, the maximum value of the applied load before failure of the specimen can be recorded. By substituting the value of the maximum load in Equation 5.3 the interlaminar shear strength of the composite material can be calculated.

5.2.4

Assumptions

Since Equation 5.3 is based on the assumption of a parabolic shear stress distribution across the thickness, the term ‘apparent’ is used in the title of the ASTM test method. Therefore, it is not appropriate to use this method for generating actual design information. It should be noted that the stress distributions near the loading point and at the support points are quite complicated, which cannot be predicted accurately using the classical beam theory and classical laminate theory. In addition, free-edge effects, i.e. development of three-dimensional stress state at and near the end surfaces of the beam in the width direction, are present and these have not been accounted for in the beam and classical laminate theories. Further, the stress distributions across the width have been assumed to be uniform in the beam theory. They may not be as uniform in the test specimen as assumed in the beam theory formula given by Equation 5.3.

5.2.5

Limitations

In the test specimen, if the fibers are oriented parallel to the longitudinal axis of the beam, for a unidirectional composite laminate, the flexural stress induced is the fiber-direction normal stress σ1 and the shear stress induced is the transverse shear stress τ13 which represents an interlaminar shear stress. © 2008, Woodhead Publishing Limited


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In addition, it is well known that the flexural stress σ1 is compressive on the surface of the beam where the loading is applied and is tensile on the surface where the support is provided. If the linear elastic material response is considered, the normal stress σ1 would vary linearly through the beam thickness, and further, for each beam cross-section the magnitudes of the stresses at top and bottom surfaces will be equal for symmetric beam cross-section subjected to bending in the plane of the beam. Correspondingly, the neutral plane coincides with the mid-plane of the beam. If a parabolic shear stress distribution over the cross-section with zero shear stress at both top and bottom surfaces of the beam is considered, the maximum shear stress is present on the midplane of the beam. Since the mid-plane is a neutral plane, the flexural stress at the mid-plane will be zero and therefore, the stress state at any point on the mid-plane is pure shear. If the laminated beam is made of plies with different fiber orientations, the interlaminar shear stress will be parabolic within each layer, and a discontinuity in the slope of the parabolic distribution will be present at each ply interface. The ply stacking sequence also influences the interlaminar shear stress distribution across the thickness of the beam. In this case, the test data should be interpreted in conjunction with the laminated beam theory. It is important to realize that the maximum shear stress in the cross-section of the laminated beam may not necessarily be the same as that calculated using the classical beam theory formula. Attempts have been made in the past to induce interlaminar shear failures in laminated beams with long spans and smaller depths. Experience gained through such investigations has shown that interlaminar shear failures are difficult to attain in laminated beams with higher L/h ratio values. This test is not useful unless interlaminar shear failure is the governing mode of failure. The interlaminar shear strength of advanced composite material systems such as graphite-epoxy composites is relatively high. In general it is very difficult to achieve interlaminar shear failure in the test specimens made of such materials. When thin unidirectional beams made of such material systems are tested, the test may not yield interlaminar shear failure for some L/h ratio values. Rather, local compressive failures near the loaded points would occur. In general, better results are obtained with thicker laminates that are approximately 50 plies thick. The test method is problematic also in the case of materials with low flexural to interlaminar shear strength ratio, such as Kevlar, textile and carbon-carbon composites. The test specimen could fail by fiber rupture, microbuckling or interlaminar shear cracking, or a combination of fracture modes. An apparent low strength value as measured by this test does not necessarily mean low interfacial strength. Also, the matrix properties are important. It has been noted in previous research that higher interlaminar shear strength can be observed in material systems that have higher matrix tensile strength and higher volume © 2008, Woodhead Publishing Limited

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fraction of the matrix. It has also been noted that the interlaminar shear strength is a function of the fiber volume fraction. However, these are not accounted for in the calculation of the short beam shear strength. Modifications of the test method for this case have been proposed recently in the works of Weisshaus and Ishai [1], Short [2], and Abali et al. [3]. Weisshaus and Ishai [1] and Short [2] proposed a short sandwich beam test in which the lamina or laminate to be tested is sandwiched between two strips made of steel or tougher composite material. The testing has to be conducted under threepoint or four-point bending loading. The modification proposed by Abali et al. [3] is to replace the direct concentrated loading by a distributed patch loading using an aluminum plate and a rubber pad. The short sandwich beam test proposed is illustrated in Fig. 5.2. The maximum flexural stress in the core, i.e. the actual lamina or laminate test specimen, has been determined using sandwich theory and is given by [4]

σc =

PLh1 4 Ie

5.4

where h1 is the half-thickness of the actual laminate and Ie is the equivalent moment of inertia of the beam cross-section given by [4] I e = 2 b [ h13 ( n – 1) + h23 ] 3n

5.5

In the above equation, n denotes the ratio of the axial moduli of the core and face sheet materials and h2 is the distance from the mid-plane to the outer surface of the top or bottom face sheet. Equation 5.3 for interlaminar shear stress at the mid-plane of the beam that is based on the classical beam bending theory has been modified through introducing a correction factor β as follows:

P σ2 2

h1

h2

1

h

2

b L

5.2 Short sandwich beam specimen proposed by Weisshaus and Ishai [1] for measurement of interlaminar shear strength.


σ1


τ = 3P β 2 bh

123

5.6

The factor β is given by Weisshaus and Ishai [1] in terms of k = h1/h2 as

β=

1 + k 2 ( n – 1) 1 + k 3 ( n – 1)

5.7

Typical test data for currently used high-performance composite material systems, both of brittle epoxy resin matrix composites and ductile matrix composites, indicate that [5], the apparent interlaminar shear strength as well as the occurrence of the interlaminar failure mode depend on several factors including the specimen geometry and the distance between points of load application and support. Elasto-plastic stress analysis of test specimens has shown that [6] the test configuration results in stress concentration effects at and near the points of load application and support which are not fully eliminated. This is due to the fact that for short highly orthotropic beam specimens, the St-Venant’s principle is not satisfied. Elasticity solutions for the short beam shear test specimen were determined in the works of Pagano and Wang [7], Kasano et al. [8], Gerhardt and Liu [9] and Whitney [10]. It has been shown that the stress distribution given by the classical beam theory is not completely realized in the case of three-point bend specimens for certain values of geometric parameters. Also in many cases, it has been observed that significantly skewed shear stress distributions are present near the support and load points. It appears that the specimen produces minimum interlaminar shear strength based on the observation that constant shear stress is present along a segment of the beam mid-plane. This, in conjunction with the presence of complex failure modes in test specimens makes it very difficult to compare the interlaminar shear strengths of different material systems based on this test method. Another aspect is the effect of the width of the beam specimen. In general, the width-to-thickness ratio, b/h, of commonly used test specimen geometries is usually large compared to that of beams for which the classical beam theory is appropriate. The width effect has been considered in the works of Sattar and Kellogg [11] and Kedward [12]. They have shown that the width effect is more pronounced for materials with relatively high values of ν12, G12/E1, and G12/G13. The ASTM standard specifies specimen width and maximum thickness. There is no minimum thickness required, and as a result, short beam shear specimens as thin as 1.5 mm have been tested. When L/h = 4, the load nose will cover the entire beam span thus creating a ‘punchout’ test rather than a flexure test. In order to avoid this problem, test specimens should be as thick as possible. Pipes et al. [13] investigated a very large beam for interlaminar shear test with the objective of avoiding the above-mentioned difficulties and the difficulty associated with the measurement of interlaminar shear


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stiffness. These beams were 13 mm/0.5 in. wide, 25 mm/1 in. thick and 102 mm/4 in. long. In this approach, the problem associated with excessively small values of length resulting from low L/h ratios has been alleviated. However, the problem of large shear stresses in excess of those predicted by classical beam theory is still present. Such large shear stresses in the vicinity of the applied transverse load at the mid-span have been determined by Berg et al. [14] using a finite element analysis. They have shown that the shear stress distribution is skewed near the load and reaction points and that the interlaminar shear stresses larger than those predicted by classical beam theory do exist in the specimen. Pipes et al. [13] have also obtained a similar result in their work. They have shown that at x/L = 0.25 shear strains predicted using classical beam theory are recovered. Experimental investigations using photoelastic coatings have been used to verify the locations and magnitudes of such large shear strains. Considering the limitations of conventional thin short beam shear test specimens, alternatives have been proposed by Browning et al. [15]. Thick specimens and specimens subjected to four-point bending at quarter points were proposed. The specimen configuration has been designed so as to induce an interlaminar failure. However, the modes of failure observed in these specimens were complex and further, the test data are dependent on specimen geometry. The above-mentioned research works point to the difficulty and uncertainty that are encountered in correctly interpreting the short beam shear test data for laminated composite materials. The test method is more applicable to polymeric and composite materials which can be treated as homogeneous or nearly homogeneous.

5.2.6

Concluding remarks

Despite all its limitations, this test method usually produces reasonable values of shear strength for many composite material systems. The simplicity of the test method and the inexpensive nature of the specimen are the main reasons for its wide popularity for materials characterization. The test specimen can be very small in size thereby requiring the least amount of material and minimum specimen preparation time and cost. The test fixture is relatively simple compared to that required for other material property tests, and further no strain or displacement measurements are taken in this test. Therefore, the test can be performed quickly and economically. Specimens made of materials that have intermediate or low interlaminar shear strength display the occurrence of a clear interlaminar shear failure in the test. The short beam shear test method is more appropriate for these materials. This method has also been used to determine the capabilities of new matrix resin materials relative to a base-line resin material, for instance an epoxy resin. If the shear strengths S13 and S23 are considered to be equal based on the transversely isotropic © 2008, Woodhead Publishing Limited


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assumption, this test method can also be used to determine the in-plane shear strength. This is often the case for unidirectional composite lamina. It can be said that the short beam shear test is used extensively as more of a materials screening and quality control test. It has been widely used as a qualityassurance tool for evaluating the quality of a manufactured composite and to determine the related matrix-dominated properties of the composite. Another important application of this method is in the determination of effectiveness of new fiber finishes and surface treatments, and to determine the fiber-resin compatibility and the ductility or brittleness of the resin.

5.3

Double-notch shear test

5.3.1

Test method and data reduction

This is another test method that can be used for determining the interlaminar shear strength of composite materials. This is also called as notched-plate test. In this test, two notches from the opposite surfaces are provided in the thickness direction of the test specimen, as shown in Fig. 5.3. The interlaminar shear strength is determined based on tension or compression tests on such double-notched specimens (see Figs 5.4 and 5.5). The distance between the notches is selected such that the failure load corresponding to the interlaminar shear failure between the notches is smaller than the failure load corresponding to tensile failure of the notched cross-sections in the test specimen. The shape of the grooved test specimen is chosen so as to satisfy the requirement that its cross-section should guarantee that only tangential stresses will be present in the specimen and that the failure of the specimen will be due to interlaminar shear. In the past, the specimen with two notches and the specimen with notches and a hole were used, and tension or compression load was applied [16]. These two specimen configurations are shown in Figs 5.3 and 5.4. In the test specimen with two notches, the cross-sectional area over which the shear loading acts, is located between the notches and is parallel to the longitudinal axis of the specimen. In the specimen with notches and a hole, this cross-section lies between the notches and the hole. In this case, the shearing action takes place over two planes. The dimensions of the test specimen as well as the test method must be chosen in consideration of the bending of the specimen and the stress concentration effect. In the specimen with asymmetrically located notches, under a tensile loading, a bending moment is developed in the specimen. This is analogous to the bending that takes place in an adhesive lap joint. The bending moment M will be equal to Fw/2, where F is the applied tensile load, and w is the width of the specimen at the weakened section. The effect of this bending moment on the measured interlaminar shear strength has been studied by researchers. It has been


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3 30

φ16

5 14

15

30 5 14

15

16

(a)

(b)

5.3 Notched specimens for interlaminar shear tests. (a) Specimen with grooves in two sections; (b) Specimen with grooves in one section and a central opening. All dimensions are in mm.

shown that [16] the maximum value of interlaminar shear strength in tension has been obtained in test set-ups in which the value of M/EI approaches the value of zero, where E is the elastic modulus of the material and I is the moment of inertia of the cross-section on which the bending moment acts. The plane on which the bottom surfaces of the two notches lie is the failure plane for the notched specimen. This condition was achieved by installing the specimen in guides that prevent specimen bending. The effect of the stress concentration is that the distribution of tangential stress along the length of the section between the notches is not uniform. The maximum © 2008, Woodhead Publishing Limited


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(a)

(b)

(c)

5.4 Specimen types and loading schemes for interlaminar shear tests under tension or compression. (a) Grooves in the lay-up plane; (b) Grooves in the transverse plane; (c) Grooves and an opening.

tangential stress in the specimen with asymmetric notches can be calculated using the formula [16]

τ max = F θ cot ( hθ ) ab

5.8

where F is the applied tensile or compressive load, a is the groove spacing, © 2008, Woodhead Publishing Limited

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b is the specimen width, and θ is equal to 2ak/h, where h is the specimen thickness and k = (Gxz/2Ex)0.5. Here Gxz and Ex correspond to the material of the specimen. Equation 5.8 indicates that the concentration of tangential stresses increases with the increasing notch spacing and shear modulus, and with decreasing specimen thickness and elastic modulus of the material. A careful study of the effect of stress concentration, which is a function of notch spacing, on the measured interlaminar shear strength indicated that for obtaining optimal results in the determination of the interlaminar shear strength, the specimens should be installed in guides which prevent bending and that the notch spacing should be no more than about 10 mm [16]. The conclusion reached was the same whether the bending effect was considered or not. The effect of bending is not present in specimens that have notches and a hole. Deformation occurs in two planes in this type of specimen. A thorough investigation of the stress concentration effect in this type of specimen was not conducted in existing work. However, the specimen that has notches and a hole has the additional advantage that in addition to the interlaminar shear strength, the shear modulus can also be determined. A variety of notched specimen shapes have been developed in the past [16–20] for the determination of interlaminar shear strength. Detailed investigations have been conducted on these specimens. It has been concluded that changes in the notch shape and in the specimen shape do not cause qualitative changes in the characteristics of the stress distribution over the gage section of the specimen. In spite of this, research on new specimen shapes is continuing. The precision of cutting of the notches has a significant effect on the measured interlaminar strength value. Undercutting, where the notch does not reach the midplane of the specimen, leads to an increase in the measured interlamimar strength value. On the other hand, overcutting, where the notch reaches beyond the midplane of the specimen, leads to the decrease in the measured strength value. High-quality cutting instruments such as diamond wheels have to be used to cut the notches in the specimen. The test method currently used is described [17] in ASTM Standard D 3846-02. The configuration of the test specimen is shown in Fig. 5.5. It is made of unidirectional composite material, 79.5 mm/3.13 in. long, 12.7 mm/ 0.50 in. wide, and 2.54 to 6.60 mm (0.100 to 0.260 in.) thick. Two parallel notches (grooves) are machined, one on each face of the test coupon, 6.4 mm/0.25 in. apart and of depth equal to half the coupon thickness. The specimen is subjected to a uniaxial tension or compression along its longitudinal direction. Interlaminar shear failure occurs along the mid-plane of the specimen between the notches. When compressive loading is applied, it is necessary to prevent the buckling failure of the specimen. For this purpose, a supporting fixture has been recommended in the ASTM specification. The interlaminar shear strength can be calculated using the formula © 2008, Woodhead Publishing Limited


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h F

F L

W F

F

5.5 Double-notch specimen for determination of interlaminar shear strength.

S13 =

Fmax wl

5.9

where Fmax is the failure load, l is the distance between the notches and w is the width of the specimen. The interlaminar shear stress along the mid-plane between the notches is actually non-uniform as the stress analysis results indicate [18]. However, as the distance between the notches decreases, a more uniform stress distribution is achieved. For interlaminar shear failure to take place between the notches, the specimen dimensions should satisfy the condition

S13 p h S1 t 2l

5.10

S13 p h S1 c 2l

5.11

or

where S1t and S1c are respectively the longitudinal tensile strength and compressive strength of the composite lamina.

5.3.2

Sensitivity to notch parameters

Markham and Dawson [19] have shown that the measured interlaminar shear strength is very sensitive to the ratio of notch space to thickness. They have conducted a detailed stress analysis of the test specimen and found that the interlaminar shear stress distribution on the mid-plane between the two notches significantly depends on this ratio and that this distribution is non-uniform. In the vicinity of the notches, stress concentration is also present. This stress concentration leads to a decrease in the measured interlaminar shear strength. © 2008, Woodhead Publishing Limited

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This stress concentration also has a dependence on the ratio of notch space to thickness. A series of unidirectional carbon/epoxy test specimens with different values of notch spacing have been tested. Based on the test results, they have concluded that the notch spacing is a critical parameter in the experimental determination of interlaminar shear strength. Chiao et al. [20] have shown that the measured value of interlaminar shear strength is also very sensitive to notch depth. They have noted the difficulty of cutting the notch to precisely half the specimen thickness. The variations in the notch depth have considerable influence on the interlaminar shear strength. The fiber tearing in the undercut specimen also affects the shear strength. However, Shokrieh et al. [21] found that when a supporting fixture is used as recommended by the ASTM standard to prevent the buckling of the specimen, slight variations in the notch depth have no influence on the measured interlaminar shear strength. It was also noted that whether the specimens are overcut or undercut, the delamination will still be initiated at the notch bottom surface rather than the notch free edges due to the stress concentration at the notch bottom. Notch width is another important parameter. In the test specimen with large notch width the bending deformation would be large when compressive loading is applied and as a result, higher stress concentrations at notches would be induced. However, with the use of supporting fixture as recommended in the ASTM specification, this effect is somewhat eliminated.

5.3.3

Effect of fiber orientation

The dependence of interlaminar shear strength determined using the doublenotch shear test on the difference between the fiber orientation angles of the two plies, one just above and another just below the mid-plane, has been investigated by Zhang et al. [22]. Experiments were conducted on laminate specimens that have interfaces around which plies are oriented with a fiber orientation angle difference of θd. The influence of this angle θd on the interlaminar shear strength has been investigated. With direction 1 oriented along one of the fiber orientations of the plies surrounding the interface, the interlaminar shear strengths S13 and S23 were measured. Two types of specimens, as shown in Fig. 5.6, were designed and used corresponding to these two interlaminar shear stress components. The following values of fiber orientation angle difference θd were used in the test program: 0, 10, 20, 30, 40, 50, 60, 70, 80 and 90 degrees. In order to have non-zero values for θd, test specimens with unsymmetric stacking sequences were used. The number of plies in the test specimen has been chosen considering the fact that thick specimens will result in more reliable experimental data for interlaminar shear strengths due to the development of more uniform shear stress distribution at the mid-plane of the specimen. In the ideal case, delamination would be developed along the mid-plane in the test specimen. © 2008, Woodhead Publishing Limited


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2

θd 1

Interface with fiber orientation angle difference θd

F

F

θd θd

Type I specimen (for S13)

Type II specimen for (S23)

5.6 Types of specimen used in Reference [22] for interlaminar shear tests.

However, owing to manufacturing tolerance, the notch depth variation will be present and therefore, a number of interfaces with fiber orientation angle difference of θd were preferred around the mid-plane of the laminate so that the delamination takes place always at one such interface. For the determination of S13 the stacking sequence [(0/θ)3/(θ/0)3/(θ/0)3/(0/θ)3] was used. For this type of specimen the analysis using the classical laminate theory had indicated that there was no bending-twisting coupling. The free edge effects were reduced due to the specific arrangement of 0° and θ° plies. For the determination of S23, the stacking sequence [(90/(90-θ))2/04/((90-θ)/90)2/((90-θ)/90)2/04/ (90/(90-θ))2] was used. In both cases, a notch space of 7 mm and a notch width of 1.8 mm were used. Further details about the particular designs of specimens are available in Ref. [22]. © 2008, Woodhead Publishing Limited

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The specimens with supporting fixture and subjected to compressive loading have been analyzed for three-dimensional stress distribution. The quadratic eight-node three-dimensional layered solid element has been used in the study. It has been shown that in both the specimen types the possibility of development of transverse matrix cracks is considerably reduced, since the two-direction normal stress and in-plane shear stress are much smaller compared to interlaminar shear stress. The interlaminar shear stress τxz has been shown to have a uniform distribution at the mid-plane between the notches, in both the specimens. The effect of notch depth on the stress distribution within the gage sections of the specimens has also been investigated using the threedimensional finite element analysis. Idealized specimens that have notch depth of exactly half the specimen thickness were analyzed. In addition, specimens with different notch depths and specimens with both undercut and overcut were also analyzed. It has been observed that in the overcut specimens the interlaminar shear stress τxz is more uniformly distributed at most of the interface regions between the notches. Also, the distribution of interlaminar shear stress τxz at the two interfaces that are adjacent to the mid-plane is the same as the distribution of τxz at the mid-plane of the specimen that was neither overcut nor undercut. In the undercut specimens, the interlaminar shear stress τxz tends to be smaller at relatively larger region of the interface under the notch bottom. Based on this, it has been concluded that the overcut would have relatively little effect on the measured interlaminar strength, and that it is necessary to avoid the undercut to obtain reliable interlaminar shear strength. It has also been observed that a specimen with a fillet has a uniform shear stress distribution at the mid-plane between the notches and further, the stress concentrations in the vicinity of the notches are reduced. In the specimen with rectangular notch, the interlaminar shear stresses are maximum at the tips of the notches and then they drop down to zero rapidly. However, in the specimen with fillet, the interlaminar shear stresses reach their maximum values within the fillet region and they rapidly reduce away from the fillet. The applied loading is transferred through the mid-planes in between the points of maximum stress, and hence the material inside the fillet regions seemed to be carrying a part of the loading. Considering these, the notch shape has been modified so as to have a round bottom with a fillet radius of 0.9 mm, as shown in Fig. 5.7, and the interlaminar shear stress has been calculated using the modified formula [22]

S13 =

Fmax w ( l + 2δ )

5.12

where δ is a correction parameter. A value of 0.5 mm has been used for specimen with circular fillet. Compared to the standard unidirectional laminate specimens, the specimens © 2008, Woodhead Publishing Limited


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Z b w X d

t

s L L = 80 mm, b = 12.7 mm, s = 7 mm, t = 4.44 mm, w = 1.8 mm d = 2.22 mm, fillet radius = 0.9 mm

5.7 Double-notch specimen with modified notch shape [22].

with plies of different fiber orientations will have a more non-uniform interlaminar shear stress distribution between the notches. In addition, at the free edges of the specimen, since the values of in-plane ply stiffness coefficients change through the thickness due to the change in ply orientation angles through thickness, concentration of interlaminar stresses is present.

5.4

Arcan test

The Arcan fixture and test specimen configuration were developed by Arcan et al. [23] in an attempt to produce uniform plane stress state in the test section of the specimen. The Arcan test fixture and test specimen are shown in Fig. 5.8. The test method is based on the fact that a shear force transmitted through a section between two edge notches produces a nearly uniform shear stress along the section. The test coupon is mounted on the Arcan fixture through a bolted specimen holder. The load can be applied at various orientations with respect to the section through the notches. This allows the application of any biaxial state of stresses from pure shear to transverse tension or any combination thereof. When the specimen is loaded in the ydirection by setting α = 0, a state of pure shear is produced at the test section of the specimen. The test fixture can also be used to determine the interlaminar shear strength. The average shear stress applied through the notched specimen is equal to PA/lnh, where ln is the specimen height at notch location, h is the specimen thickness, and PA is the applied load. The test specimen is a short coupon with two 90° notches. The faces of a unidirectional coupon are bonded to the specimen holder shown in Figure 5.8 and the failure load is recorded. The interlaminar shear strength is calculated using the expression S13 =

PA max w AlA

5.13

where PAmax is the failure load, lA is the length of the test coupon, and wA is the width of the test coupon. © 2008, Woodhead Publishing Limited

134

Delamination behaviour of composites Y

α = 0° Specimen holder

α

PA

203 mm diameter 13.2 mm thickness

α = 45°

(metal fixture)

Specimen region

α = 90°

x

Adhesive

α

PA

5.8 Specimen holder and loading fixture for Arcan test.

This test method has the advantage that the metal fixture (specimen holder) to which the composite laminate test specimen is bonded, is reusable. After testing is completed, the specimen can be removed by heating the specimen and the metal fixture beyond the glass transition temperature of the adhesive.

5.5

Conclusion

In this chapter, experimental characterization of interlaminar shear strength of fiber-reinforced laminated composite material is considered. The three test methods, which have been developed and established over the years, have been described. Aspects related to the theoretical background for these test methods, requirements for test samples and loadings, influences of various material and geometric parameters and test specimen parameters on the measured strength values, validity of test data, and limitations of the test methods have been discussed. Observations and recommendations regarding test samples, test procedures, etc., based on existing research works have been given. Considering all the above-mentioned aspects, it can be said that the short beam shear test and double-notch shear test are the most popular and widely used test methods. In particular, the short beam test is the most widely used © 2008, Woodhead Publishing Limited


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test method for all types of composite materials that are currently in use. It is emphasized that a short beam shear test becomes invalid if either the tensile failure of the ply precedes the transverse shear failure or if both the tensile and shear failures occur simultaneously in the test specimen. The current research activities focus on applications of these two test methods to newly developed composite materials such as textile composites, braided composites, smart composites, nano composites, green composites, and bio composites. In particular, modifications of test specimen configurations and loading fixtures so as to obtain reliable test data on interlaminar shear strengths of these materials are the main thrust of on-going research in the area of experimental mechanics of composites. In this regard, use of the state-ofthe-art computational methods, instrumentation, and experimental techniques is increasingly made.

5.6

References

1. H. Weisshaus and O. Ishai, ‘A Comparative Study of the Interlaminar Shear Strength of C/C Composites,’ Proceedings of 32nd International Sampe Technical Conference, 5–9 November 2000, pp. 854–872. 2. S. R. Short, ‘Characterization of Interlaminar Shear Failures of Graphite-Epoxy Composite Materials,’ Composites, Vol. 26, No. 6, 1995, pp. 431–449. 3. F. Abali, A. Pora and K. Shivakumar, ‘Modified Short Beam Test for Measurement of Interlaminar Shear Strength of Composites,’ Journal of Composite Materials, Vol. 37, No. 5, 2003, pp. 453–464. 4. I. M. Daniel and O. Ishai, Engineering Mechanics of Composite Materials, Second Edition, Oxford University Press, 2006. 5. J. M. Whitney, ‘Experimental Characterization of Delamination Fracture’, In: Interlaminar Response of Composite Materials, Edited by N. J. Pagano, Elsevier Science Publishers BV, 1989. 6. P. E. Sandorff, ‘Saint-Venant Effects in an Orthotropic Beam’, Journal of Composite Materials, Vol. 14, No. 3, 1980, pp. 199–212. 7. N. J. Pagano and A. S. D. Wang, ‘Further Study of Composite Laminates Under Cylindrical Bending’, Journal of Composite Materials, Vol. 5, No. 4, 1971, pp. 521– 528. 8. H. Kasano, K. Ogino, H. Matsumoto and I. Nakahara, ‘Theoretical Analysis of an Orthotropic Rectangular Plate Under In-Plane Loading’, Transactions of Japan Society of Composite Materials, Vol. 6, No. 1, 1980, pp. 33–42. 9. T. D. Gerhardt and J. Y. Liu, ‘Orthotropic Beams Under Normal Loading’, Journal of Engineering Mechanics, ASCE Engineering Mechanics Division, Vol. 109, No. 2, 1983, pp. 394–410. 10. J. M. Whitney, ‘Elasticity Analysis of Orthotropic Beams Under Concentrated Loads’, Composites Science and Technology, Vol. 22, No. 2, 1985, pp. 167–184. 11. S. A. Sattar and D. H. Kellogg, ‘The Effect of Geometry on the Mode of Failure of Composites in Short-Beam Shear Test’, In: Composite Materials: Testing and Design, ASTM-STP-460 (1969), pp. 62–71. 12. K. T. Kedward, ‘On the Short Beam Test Method’, Fibre Science and Technology, Vol. 5, No. 1, 1972, pp. 85–95. © 2008, Woodhead Publishing Limited

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13. R. B. Pipes, D. L. Reed and J. E. Ashton, ‘Experimental Determination of Interlaminar Shear Properties of Composite Materials’, SESA paper No. 1985 A, Presented at 1972 SESA Spring Meeting, Cleveland, OH, 23–26 May 1972. 14. C. A. Berg, J. Tirosh and M. Israeli, ‘Analysis of Short Beam Bending of Fiber Reinforced Composites’, In: Composite materials: Testing and Design, ASTM-STP497 (1972), pp. 206–218. 15. C. E. Browning, F. L. Abrams and J. M. Whitney, ‘A Four-Point Shear Test for Graphite/Epoxy Composites’, In: Composite Materials: Quality Assurance and Processing, Edited by C. E. Browning, ASTM-STP-797 (1983), pp. 54–74. 16. Y. M. Tarnopolskii and T. Kincis, Static Test Methods for Composites, Van Nostrand Reinhold Company, New York, 1985. 17. Standard Test Method for In-Plane Shear Strength of Reinforced Plastics, ASTM D 3846–02. 18. G. Menges and R. Kleinholz, ‘Comparison of Different Methods for the Determimnation of Interlaminar Shear Strength’, (in German), Kunststoffe, Vol. 59, 1969, pp. 959– 966. 19. M. F. Markham and D. Dawson, ‘Interlaminar Shear Strength of Fiber-Reinforced Composites’, Composites, Vol. 6, 1975, pp. 173–176. 20. C. C. Chiao, R. L. Moore and T. T. Chiao, ‘Measurement of Shear Properties of Fibre Composites: Part I – Evaluation of Test Methods’, Composites, Vol. 8, 1977, pp. 161–169. 21. M. M. Shokrieh, P. E. Olivia, P. Kotsioprifits and L. B. Lessard, ‘Determination of Interlaminar Shear Strength of Graphite/Epoxy Composite Materials in Static and Fatigue Loading’, Proceedings of ICCM-10, Vancouver, Canada, 1995, Vol. IV, pp. 81–88. 22. C. Zhang, S. V. Hoa and R. Ganesan, ‘Experimental Characterization of Interlaminar Shear Strengths of Graphite/Epoxy Laminated Composites,’ Journal of Composite Materials, Vol. 36, 2002, pp. 1615–1652. 23. M. Arcan, Z. Hashin and A. Voloshin, ‘A Method to Produce Uniform Plane-Stress States with Applications to Fiber-Reinforced Materials’, Experimental Mechanics, Vol. 18, 1978, p. 141.

5.7 L h b P σ1 σmax τ13 τxz τmax S1 S13 S23

Appendix: Nomenclature Length of the support span of the short beam test specimen Thickness of the short beam test specimen / double-notch test specimen Width of the short beam test specimen / double-notch test specimen Transverse load applied on the short beam test specimen Normal stress in the fiber direction Maximum bending stress Interlaminar shear stress in the principal material coordinate system Interlaminar shear stress in the Cartesian coordinate system Maximum shear stress Tensile strength or compressive strength (whichever is lesser) of the composite material Interlaminar shear strength of the composite material in 1-3 plane Interlaminar shear strength of the composite material in 2-3 plane



S1t S1c h1 Ie h2 k n

β w a l ln M F E G I θ θd δ PA lA wA

137

Longitudinal tensile strength of the composite lamina Longitudinal compressive strength of the composite lamina Half-thickness of the actual laminate in the short sandwich beam test specimen Equivalent moment of inertia of the cross-section of the short sandwich beam test specimen The distance from the mid-plane to the outer surface of the top or bottom face sheet of the short sandwich beam test specimen Ratio of h1 to h2 The ratio of the axial moduli of the core and face sheet materials of the short sandwich beam test specimen Correction factor for short sandwich beam test specimen Width of the double-notch test specimen Groove (notch) spacing of the double-notch test specimen Distance between the notches of the double-notch test specimen Specimen height at notch location in Arcan test Bending moment Axial load applied on the double-notch test specimen Elastic modulus Shear modulus Moment of inertia Fiber orientation angle of the ply Fiber orientation angle difference at ply interface Correction parameter for modified double-notch test specimen Applied load in Arcan test Length of the Arcan test coupon Width of the Arcan test coupon


Part II Delamination: detection and characterization


6 Integrated and discontinuous piezoelectric sensor/actuator for delamination detection P T A N, Defence Science and Technology Organisation, Australia and L T O N G, University of Sydney, Australia

6.1

Introduction

Laminated composites are being increasingly used in critical engineering structures due to their high specific stiffness and specific strength. The applications of laminated composite, however, have been limited by delaminations, which can be introduced during the fabrication process or later in the service life, for example under impact by a foreign object. The existence of delamination degrades the stiffness, strength and fatigue properties of laminated composite structures and affects the structures’ dynamic response, and it also has a potential to cause catastrophic failure of the structures. However, owing to the fact that delaminations often occur beneath the composite structure surfaces, they are usually invisible or difficult to detect by visual inspection. Thus, reliable and inexpensive delamination detection methods or technologies must be developed to improve safety and reliability of laminated composite structures in service. In order to effectively detect delaminations in laminated composites, various non-destructive techniques and methods have been proposed [1]. These methods possess clear advantages in some aspects but also have limitations in others. For example, the vibration-based methods and techniques [2–4] can usually be used to identify the presence of delaminations, but not the size and location of delaminations; the delamination monitoring electric resistance change method [5–6] can locate internal delaminations according to the electric resistance changes between equally spaced electrodes, and seems to limit to the graphite fiber reinforced plastics composites; and the techniques using fiber Bragg grating sensor system [7–9] have shown [10] that the embedded optical fibers significantly reduce the fatigue life of composite structures. All the delamination detection methods mentioned here are considered generally as passive. They can only be used for inferring the integrity and safety of a laminated composite structure under an external loading. The reliability and confidence of a delamination detection technique would be significantly improved if active, instead of passive, delamination detection techniques 141 © 2008, Woodhead Publishing Limited

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were used. This is due to the active techniques that are able to generate preselected diagnostic signals and transmit them to neighboring sensors whose responses can then be measured and interpreted in terms of delamination location and size within a laminated structure. There has recently been an increasing interest in a relatively new area of non-destructive delamination monitoring using active or smart materials. Piezoelectric material is one of the most commonly used smart materials for detecting delaminations in composites. This emerging non-destructive delamination monitoring method involves using piezoelectric (PZT) sensor/ actuators, bonded on laminated composite structure surfaces or embedded within the composite structures [1]. For example, Teboub and Hajela [11] presented a neural network based strategy for online damage (e.g., delamination) detection in unsymmetric composite laminate beams, in which a network of piezoelectric sensors/actuators have been embedded in a structure during manufacturing. Keilers and Chang [12–14] conducted an experimental and analytical investigation for detecting a delamination and estimating its size and location for laminated composite structures. The experimental test was conducted using piezoelectrics attached to a composite beam with and without an artificial delamination, and the structural analysis was developed for predicting the output voltages of the piezoelectric sensors when the delaminated beam was excited by the piezoelectric actuators. Saravanos et al. [15] proposed a detection technique for identification of the delamination in composite beams using piezoelectric sensors. By employing this technique, the presence and location of a delamination can be detected based on variations in voltage, which is generated in the piezoelectric layers when the beam is excited either externally or via piezoelectric actuators. Yin et al. [16] used a modified quasi three-dimensional finite element method to detect damage such as delamination in a laminated composite through piezoelectric film sensors. However, delamination detection at higher frequency was not discussed in all the methods mentioned previously. In addition, these methods only involve using discrete or sole sensor and actuator, i.e., the piezoelectric layer used as a sensor is separated from that used as an actuator, to deal with an idealized case of single delamination. In the present investigation, various dynamic analytical models are developed for the identification of single and triple delaminations embedded within a laminated composite beam bonded with integrated PZT sensors/actuators. Compared are the sensor charge output (SCO) distributions between the beams with single delamination and integrated and discontinuous PZT sensor/actuators, and those between the beams with single and triple delaminations but bonded with integrated PZT sensors/actuators. First, the typical patterns for piezoelectric (PZT) and piezoelectric fiber reinforced composite (PFRC) sensor/actuator employed for delamination detection in composite structures are outlined. Subsequently, a dynamic analytical model is developed for identifying single delamination embedded © 2008, Woodhead Publishing Limited

Piezoelectric sensor/actuator for delamination detection

143

in a cantilever laminated beam, which is surface-bonded with an integrated PZT or PFRC sensor/actuator. This model, which includes parameters characterizing the through-the-width delamination, is developed using the classical beam theory and the assumption of constant peel and shear strains through the bond line [17]. For simplicity, the delamination front lines are assumed to be straight and perpendicular to the longitudinal direction of a beam. The contact and friction between the upper and lower delaminated surfaces are not considered since for the cases considered in this investigation, only the open mode is considered. That means there is no stress transferring between the upper and lower delaminated surfaces. The PZT or PFRC actuator is employed to excite the beam and those PZT or PFRC sensors are used to measure the SCO along the beam surface. Hence, by monitoring the SCO distribution along the beam for the first three frequencies, the presence, size and axial location of single through-the-width delamination embedded in a laminated beam can be identified. The influence of the PZT actuator location on the predicted SCO distribution is also discussed. In order to compare the SCO distributions between the beams bonded with integrated PZT or PFRC sensor/actuator (IPSA) and discontinuous PZT or PFRC sensor/actuator (DPSA) and those between the beams with single delamination and triple delaminations. Two additional dynamic analytical models are proposed, one is for the beam with single delamination and bonded with DPSA, and the other is for the beam with triple delaminations and bonded with IPSA. Then, quantitative comparisons of the SCO between the beams bonded with IPSA and DPSA as well as that between the beams with single delamination and triple delaminations are conducted. Finally, an experimental testing program is conducted to verify the proposed model for a beam with single delamination.

6.2

Typical patterns for piezoelectric (PZT) or piezoelectric fiber reinforced composite (PFRC) sensor/actuator

For the sake of detecting delaminations embedded in a laminated composite beam using PZT or PFRC patches, two typical forms, namely integrated PZT or PFRC sensor/actuator (IPSA) and discontinuous PZT or PFRC sensor/ actuator (DPSA) (see Fig. 6.1), are considered in this investigation. For a PZT actuator, a laminar electrode is required and coated or sputtered on the actuator surface, and thus the required electric field can be applied to the actuator for exciting the beam system. For a PFRC actuator, it was revealed from the literature reviews [18–20] that there are two general ways to lay electrodes. One is referred to as laminar electrode (see Fig. 6.2(a) [21]), where the applied electric field in the y direction causes actuation strain in the piezoelectric fiber (or z) direction primarily due to the piezoelectric constants d31. In order to create larger strain actuation in the piezoelectric © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites PZT or PFRC patch

Delamination Composite beam (a) For the case of IPSA PZT or PFRC patch

Composite beam Delamination (b) For the case of DPSA

6.1 A schematic for a beam system with a single delamination and bonded with IPSA or DPSA. y

Laminar electrode +V

x z

PZT fiber

Matrix

Electric field (a) PZT fiber with laminar electrode

Interdigitated electrode +V

+V PZT fiber

Matrix Electric field

(a) PZT fiber with interdigitated electrode

6.2 Schematics for the laminar and interdigitated electrode [21].

fiber direction using the piezoelectric constants d33, an interdigitated electrode, i.e., IDE, was proposed by Hagood, et al. [18–19] (see Fig. 6.2(b) [21]). It is evident that the electric field gradient at the electrode edge may result in large stress concentration in piezoelectric fibers (see Fig. 6.2(b)). For the sake of reducing large stress concentration, the authors proposed a new type © 2008, Woodhead Publishing Limited


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of electrode, namely circular-linked interdigitated electrode, i.e., CLIDE (see Fig. 6.3 [21]). It is noted that the electric field distribution shown in Fig. 6.3(a) is more uniform than that in Fig. 6.2(b), which may lead to a reduced stress concentration around the CLIDE edge compared to that around the IDE edge. For the purpose of obtaining a continuous curve of SCO vs x location, following two methods were suggested by Yin et al. [16]. One approach is using a scanning probe referred to a ‘moving’ electrode to obtain a continuous curve of voltage distribution, and the other possible approach is to locate electrodes point by point to record voltage of each isolated point. In this investigation, the electrode strips, which are evenly distributed along the beam length (see Fig. 6.4), are proposed and employed for measuring the required SCO distribution, and thus identify the presence, size and axial location of delaminations embedded in a laminated beam. It is worth pointing out that the length of an electrode strip along the x direction le should be larger than the thickness of the piezoelectric sensor ts. The electrode strips and a laminar electrode are, respectively, coated on the top and bottom surfaces of the PZT or PFRC sensor surfaces via depositing conductive

+V +

Electric field

PZT fiber Matrix

Electrode (a) With matrix

PZT fiber Electrode (b) Without matrix

6.3 A schematic for the circular-link interdigitated electrode [21]. z Ie

y x

V

Electrode strip

Composite beam

Delamination

PZT of PFRC layer Laminar electrode

6.4 A schematic for the electrode strip distribution and voltage record.


146


metal or following the similar procedure for manufacturing integrated or printed circuit board.

6.3

Constitutive equations and modelling development for a laminated beam with a single delamination and surface-bonded with an integrated piezoelectric sensor/actuator (IPSA)

Figure 6.5 [22] shows a cantilever laminated composite beam with single delamination embedded in the beam system and bonded with an IPSA layer. For simplicity, we assume that the delamination runs throughout the width of the host composite beam and the delamination front lines are straight and perpendicular to the longitudinal direction of the beam. In order to investigate the effect of the IPSA actuator segment location on the SCO value, we divide the beam system into nine major span-wise regions, namely I to IX as shown in Fig. 6.5. In this beam system, the IPSA segments located in regions II, V and VIII are, respectively, considered as actuators. The electric field E is applied along the actuator segment thickness direction. For example, for the case of electric field E38 ≠0 (see Fig. 6.5), the IPSA segment located in region VIII is considered as an actuator, and the rest are considered as sensors. For regions I, II, III, VII, VIII and IX, they can be considered to be made up of two component segments (i.e., IPSA and host beam segment), and for regions IV, V, and VI, each segment consists of three component segments (i.e., IPSA, upper delaminated beam and lower delaminated beam). Each segment can be modeled as a beam using the E32

x

E38

E35

IPSA

z I

II

III

IV

tsa tad Zd

V

Id

Xd

Composite beam

VI

VII

VIII

tg

IX

Adhesive layer

tb

Delamination

Lb

6.5 A schematic for a beam system with delamination and bonded with IPSA layer [22].



147

Bernoulli-Euler theory. For simplicity, the contact and friction between the upper and lower delaminated beam segments are not considered, and thus it is assumed that there is no stress transferring between these two segments. The motion for each region is described by the IPSA transverse displacements wksa, the host beam transverse displacements wkb due to bending, and the pure longitudinal displacement uksa for the IPSA and ukb for the host beam, where k = 1 to 9 stand for regions I to IX, respectively. It is worth pointing out that for regions IV, V, and VI, the host beam transverse displacement wkb should be replaced by the upper and lower delaminated beam transverse displacements (i.e., wkub and wklb) whereas the pure longitudinal displacement ukb for the host beam should be replaced by those for the upper and lower delaminated beam segments (i.e., ukub and uklb). By using the classical beam theory and the corresponding free-body diagram, the dynamic equations of motion for the IPSA and host beam in region K (K = I to IX) can be obtained, such as for the case of region I, the corresponding dynamic equations for the IPSA can be written as follows:

∂T1 sa ˙˙1 sa + τ 1u b , ρ1 sa A1 sa w ∂x

ρ1 sa A1 sa u˙˙1 sa = =

∂Q1 sa ∂M1 sa τ bt + σ 1u b , – Q1 sa + 1u sa = 0 2 ∂x ∂x

6.1

where ρ1sa, A1sa, b, tsa, Q1sa, τ1u and σ1u stand for the IPSA density, crosssectional area of IPSA, width of beam system, the IPSA thickness, shear force acting on the IPSA, shear and peel stress between the IPSA and host beam, respectively. The longitudinal force T1sa and bending moment M1sa acting on the IPSA (i.e., T1sa and M1sa) are given by

T1 sa = bYsa t sa

∂u1 sa bY t 3 ∂ 2 w1 sa – e31 bt sa E31, M1 sa = – sa sa 12 ∂x ∂x 2

6.2

where Ysa is the complex Young’s modulus of IPSA which is required in the following frequency domain analysis, and E31 is the electric field applied to the IPSA segment located in region I. It is worth pointing out that if the IPSA segment located in region I is considered as a sensor, then the value of E31 in Equation 6.2 should be set equal to zero. Under the well-known assumptions of constant peel and shear strains through an adhesive thickness in adhesively bonded joint [17], the shear and peel stresses in the adhesive layer between the IPSA layer and host beam can be obtained by

τ1u = Gγ1u

6.3

Yad (1 – ν ad ) σ 1u = ( wb – wt ) (1 – 2 v ad )(1 + ν ad )t ad

6.4


148


∂wb   ∂w 1  ∂w t + t ∂ w b  + u b – ut 6.5 γ 1u = 1  t +  + 2t  tt  b t ad 2  ∂x ∂x  ∂x ∂x  ad 

where Yad and νad, tad stand for complex Young’s modulus, Poisson’s ratio and bondline thickness, subscripts t and b denote the top and bottom layer (i.e., IPSA and host beam), respectively. By substituting Equations (6.2–6.5) into the corresponding dynamic equations of all host beam and IPSA segments, followed by taking Fourier transform with respect to time for the equations of motion, we have the following differential equations:

dU k = Ak U k dx

6.6

where the over-bar represents the Fourier transform with respect to time, U k is the state vector and Ak is a matrix for the case of region k (see Section 6.9 for the case of region I). By solving Equation 6.6, we have

U k ( x k ) = Ck e Ax k

6.7

in which κ = 1 – 9 for regions I to IX, respectively; xk = 0 to lk (lk is the length of region κ), Ck = U k ( x k )| x k =0 . For the considered cantilever beam system, a total of 12 applicable boundary conditions can be obtained as listed below: At the fixed end, T1sa = 0, Q1sa = 0, M1sa = 0, u1b = 0, w1b = 0,

∂ w1 b =0 ∂x

6.8

At the free end, T9sa = 0, Q9sa = 0, M9sa = 0, T9b = 0, Q9b = 0, M9b = 0

6.9

A total of 114 continuity conditions exist at the interfaces between regions I and II to VIII and IX in order to ensure the displacements and forces to be identical at the interfaces, such as for the host beam, the continuity conditions exist at the interface between regions I and II are given as follows: u1b = u2b, w1b = w2b,

∂ w1 b ∂w2 b = , ∂x ∂x

T1b = T2b, Q1b = Q2b, M1b = M2b

6.10

By numerically solving the governing Equation 6.7 together with their boundary and continuity conditions, the required natural frequencies and the absolute values of the longitudinal or axial strain distribution |εs(x, ω)| along the sensor segment surfaces can be obtained. Hence, for a selected natural



149

frequency ω, the SCO for a sensor segment with zero electric flux and segment length lk can be obtained by

Qk ( ω ) = b

lk

∫

e31 s |ε s ( x , ω )| dx

6.11

0

where e31s is the piezoelectric constant for a PZT sensor with linear and orthotropic properties. It is worth mentioning that for the sake of obtaining SCO distribution along the sensor segments located at region k, it is necessary to further divide region k into Nk elements. Following the similar procedure described above, the required data for each element can be obtained. Thus, the solution for the beam system as a whole is obtained in terms of the solutions of all elements using the classical beam theory, appropriate boundary conditions and continuity conditions. In addition, it is known that the SCO distribution can only be measured through the electrodes on the piezoelectric sensor [16]. In order to obtain a continuous curve of the SCO along the beam length, a number of electrode strips are required to be evenly distributed along the beam length. Hence, for a selected natural frequency ω, the SCO measured from an electrode strip, which is bonded on a sensor element with the length of ∆x and located at x = xa, is given by

q ( x a ,ω ) = b

∫

xa +

xa –

∆x 2

∆x 2

e31 s |ε s ( ξ , ω )| dξ

where ∆x ≤ x ≤ L – ∆ x a b 2 2

6.12

Using Equation 6.12, the SCO distribution for delamination locations and lengths can be predicted. It is worth pointing out that in practice, accurate detection of a delamination using evenly distributed electrode strips, may be dependent of the number and size of electrode strips.

6.4

Parametric study

The baseline case considered in this study is a cantilever beam system with a beam length Lb = 0.3 m, width b = 0.02 m and host beam thickness tb = 1.9 mm. A delamination with a delamination length ld = 0.04 m and a delamination gap tg = 0 is located at the host beam’s mid-plane and mid-length Xd = 0.15 m (where Xd is the distance measured from the delamination center to the fixed end). The IPSA thickness tsa is selected to be 0.4 mm and that of adhesive layer tad is chosen to be 0.15 mm. The length of actuator segment © 2008, Woodhead Publishing Limited

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la is selected to be 0.02 m. The beam and IPSA are made of T300/GY260 plain weave composite and PZT layers, respectively. Their mechanical properties are assumed to be linear. The required complex Young’s modulus for the host beam, IPSA and the adhesive layers are 65.68(1+0.011i)GPa, 69.2(1+0.011i)GPa and 2.15(1+0.011i)GPa, respectively. The piezoelectric constant e31s for the PZT IPSA is chosen to be 44.37C/m2. The density for the host beam, IPSA and adhesive layers are selected to be 1527.38, 7600 and 1600 kg/m3, respectively [23–24]. The stiffness of the electrodes strip is neglected because their small size relative to those for the IPSA and laminated beam. The electric field applied through the IPSA thickness in region k, E3k, is chosen to be 1000 V/m.

6.4.1

Identification of the presence, size and axial location of single delamination using integrated piezoelectric sensor/actuator (IPSA)

Using the present model and required data for the baseline case, the corresponding SCO distributions for the host beams with and without delamination can be obtained and plotted in Fig. 6.6 [22] for the case of E32 = 1000 V/m (i.e., E3k = 0 except for E32 = 1000 V/m). In Fig. 6.6, the blank parts in the curves indicate the location and size of the actuator segment. The abrupt axial discontinuities in the SCO distributions measured from the IPSA sensor segments clearly indicate the tips of a delamination. For the case of vibration mode I, the difference of SCO for the beams with and without delamination is about 11.4% at the left delamination tip and 12% at the right delamination tip. For the case of vibration mode III, the difference is about 28% at the left delamination tip and 27% at the right delamination tip. For the case of mode II, the difference is minor. Thus, the presence, size and axial location of the delamination are easily identified for the cases of mode I and III. This proves that the present method can be used to effectively predict the presence, size and axial location of delamination occurred in a cantilever beam system bonded with IPSA. The predicted SCO distributions for vibration modes I and III are very sensitive to the presence of a delamination.

6.4.2

Effect of Xa on the sensor charge output (SCO)

In order to discuss the influence of the IPSA actuator segment location on the SCO for the first three natural frequencies, three cases, i.e., the IPSA segment located at region II, V and VIII as actuator segment respectively, are considered here (i.e., the actuator segments with length la = 0.02 m are located at Xa = 0.065, 0.15 and 0.235 m, respectively, where Xa is the distance of the IPSA actuator segment center from the fixed end). The applied electric field E3k is chosen to be 1000 V/m. The rest of the required geometric © 2008, Woodhead Publishing Limited


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3.50E–09 Perf Del

3.00E–09

SCO (C)

2.50E–09 2.00E–09 1.50E–09 1.00E–09 5.00E–10 0.00E+09 0 7.00E–11

0.05

0.1

0.15 0.2 0.25 X (m) (a) For the case of mode I

0.3

Perf Del

6.00E–11

SCO (C)

5.00E–11 4.00E–11 3.00E–11 2.00E–11 1.00E–11 0.00E+00 0

0.05

0.1

0.15 0.2 0.25 X (m) (b) For the case of mode II

2.50E–09

0.3

Perf Del

SCO (C)

2.00E–09

1.50E–09

1.00E–09

5.00E–10

0.00E+00 0

0.05

0.1

0.15 0.2 0.25 X (m) (c) For the case of mode III

0.3

6.6 The SCO distributions for the beams with and without a single delamination in the case of E32 = 1000 V/m [22]. © 2008, Woodhead Publishing Limited

152


parameters and electromechanical properties are the same as those for the baseline case. Using the present model, the corresponding SCO distribution are obtained and plotted in Fig. 6.7 [22], in which E32, E35 and E38 stand for the case of Xa = 0.065, 0.15 and 0.235 m, respectively. It is interesting to note from Fig. 6.7 that among these three cases, the largest peak SCO value is obtained at Xa = 0.065 m for the case of mode I, Xa = 0.15 m for the case of mode II, and Xa = 0.235 m for the case of mode III, and the smallest peak SCO value is obtained at Xa = 0.235 m for the case of mode I, Xa = 0.065 m for the case of mode II, and Xa = 0.15 m for the case of mode III. This finding reveals that the SCO value is closely related to the actuator location. In general, a larger value of SCO can be obtained if the IPSA actuator segment is located at the region of larger longitudinal strain. Following a similar procedure, it is noted that the SCO distributions are closely related to la, ld, tg and E. However, due to limitations of space, the corresponding SCO distribution diagrams and discussion are not shown here.

6.4.3

Comparison between beams bonded with integrated or discontinuous piezoelectric (PZT) or piezoelectric fiber reinforced composite (PFRC) sensor/actuators

For the sake of comparing the SCO distribution for a beam bonded with an IPSA and that bonded with DPSAs, a beam system with nine DPSAs is considered here and shown in Fig. 6.8. It is worth mentioning that for simplicity, the gap between two consecutive discontinuous DPSA patches is assumed to be zero. It is similar to that for a beam bonded with IPSA, the required dynamic equations of motion for the DPSA and host beam in the region K (K = I to IX) shown in Fig. 6.8 can be derived using the classical beam theory. It is worth mentioning that the required constitutive equations for the SCO obtained from an IPSA segment are the same as those from the DPSA segment, and the continuity conditions for the IPSA cases are different from those for the DPSA case, in which there is no stress transferring between two consecutive DPSA segments. The major geometrical parameters for the case of DPSAs are chosen to be the same as those for the baseline case of IPSA. Thus, by solving numerically the dynamic equations of motion together with their 60 boundary and 66 continuity conditions, the required SCO for a beam bonded with nine DPSAs and with or without single delamination can be obtained and plotted in Fig. 6.9 as well as Fig. 6.10 for the range of X = 0.12 to 0.18 m. From Figs 6.9 and 6.10, it is noted that for the beam segments without single delamination (e.g., beam segments located in regions I, II, III, VII, VIII, IX in Fig. 6.8), the differences between the values of SCO measured from the host beams with and without delamination are minor. For the beam



153

3.50E–09 E32 E35 E38

3.00E–09

SCO (C)

2.50E–09 2.00E–09 1.50E–09 1.00E–09 5.00E–10 0.00E+09 0 3.50E–09

0.05

0.1

0.15 0.2 0.25 X (m) (a) For the case of mode I

0.3

E32 E35 E38

3.00E–09

SCO (C)

2.50E–09 2.00E–09 1.50E–09 1.00E–09 5.00E–10 0.00E+09 0 3.00E–09

0.05

0.1

0.15 0.2 0.25 X (m) (b) For the case of mode II

0.3

E32 E35 E38

2.50E–09

SCO (C)

2.00E–09 1.50E–09 1.00E–09 5.00E–10 0.00E+00 0

0.05

0.1

0.15 0.2 0.25 X (m) (c) For the case of mode III

0.3

6.7 Comparison of the SCO distributions for the cases of E32 = 1000 V/m, E35 = 1000 V/m and E38 = 1000 V/m [22].


154

Delamination behaviour of composites x DPSAS

z I

II

III

IV

V

VI

VII

VIII

IX

tsa tad

Zd

Id tg

Xd

Composite beam

Lb

Delamination

tb

Adhesive layer

6.8 A schematic for a beam system divided into nine regions and bonded with nine distributed MFRC patches.

segments embedded with a delamination (e.g., beam segments located in regions IV, V, VI in Fig. 6.8), the differences of the SCO obtained from the host beams with and without delamination are noticeable. In addition, Figs 6.9 and 6.10 reveal that for the beam segments located in regions IV, V and VI, the SCO measured from the host beam without delamination is greater than that with a delamination for the case of vibration mode I and II, but is less than that with a delamination for the case of vibration mode III. This is different from the findings shown in Fig. 6.6 for the case of IPSA. In addition, a comparison of Fig. 6.6 and Fig. 6.9 shows that the values of SCO measured using DPSAs are larger than those obtained using IPSA. This is expected because the stiffness of the beam system for the case IPSA is greater than that for DPSAs.

6.4.4

Comparison between beams with single and triple delaminations

To compare the SCO distribution for a beam having a single delamination with that having triple delaminations and bonded with an IPSA, we consider a laminated composite beam system with triple through-the-width delaminations and bonded an IPSA on its upper surface as shown in Fig. 6.11. The beam system is divided into nine regions. The values of delamination length Ldel-1, Ldel-2 and Ldel-3 are 0.01 m, 0.02 m and 0.03 m, and those of delaminated beam component thickness tdel-1, tdel-2, tdel-3, and tdel-4, are selected to be 0.475 mm. The values of delamination gap tgap-1, tgap-2, tgap-3 (see tgap-3, in Fig. 6.11) are chosen to be zero. Because only the delamination opening mode is considered, the stress transferring between any two adjacent delaminated beam components is ignored. The rest parameters are the same as those of the baseline case defined previously in Section 6.4. Following similar procedure described in Section 6.3, the corresponding dynamic © 2008, Woodhead Publishing Limited


155

3.00E–08 Perf Del

2.50E–08

SCO (C)

2.00E–08 1.50E–08 1.00E–08 5.00E–09 0.00E+00 0

0.1

0.2 X(m) (a) Vibration mode I

0.3

1.60E–08 Perf Del

1.40E–08

SCO (C)

1.20E–08 1.00E–08 8.00E–09 6.00E–09 4.00E–09 2.00E–09 0.00E+00 0

0.1

0.2 X(m) (b) Vibration mode II

0.3

6.00E–09 Perf Del

5.00E–09

SCO (C)

4.00E–09 3.00E–09 2.00E–09 1.00E–09 0.00E+00 0

0.1

0.2 X(m) (c) Vibration mode III

0.3

6.9 The SCO distributions for three vibration modes of the beams with and without a single delamination for the case of DPSAs.


156

Delamination behaviour of composites 1.40E–08 1.20E–08

SCO (C)

1.00E–08 8.00E–09 6.00E–09 4.00E–09 2.00E–09 0.00E+00 0.12

Perf Del

0.14

0.16 X (m) (a) Vibration mode I

0.18

1.20E–08 1.00E–08

SCO (C)

8.00E–09 6.00E–09 4.00E–09 2.00E–09 0.00E+00 0.12

Perf Del

0.14

0.16 X (m) (b) Vibration mode II

0.18

3.00E–09 2.50E–09

SCO (C)

2.00E–09 1.50E–09 1.00E–09 5.00E–10 0.00E+00 0.12

Perf Del

0.14

0.16 X (m) (c) Vibration mode III

0.18

6.10 The SCO distributions for three vibration modes of the beams with and without a single delamination for the case of DPSAs and the range of X = 0.12 to 0.18m. © 2008, Woodhead Publishing Limited


157

Ldel–3 Adhesive layer

Ldel–2

E32

IPSA

Ldel–1 I

II

III

IV

V

VI

VIII

IX

tm tad

tdel–1 tdel–2 tdel–3 tdel–4

tgap–3

tb

Lb

6.11 A schematic of a beam system bonded with an integrated PZT sensor/actuator (for the case of original-pattern).

analytical model for the beam system with triple delaminations and bonded with an IPSA can be developed, and is then employed to evaluate the required SCO distribution for the beam with triple delaminations. Figures 6.12–6.14 show the SCO and its normalized distributions (i.e., NSCO) for the cases of vibration modes I, II and III, respectively. It is noted from Figs 6.12–6.14 that the abrupt axial discontinuities in the SCO or NSCO distributions for the cases of modes I and III clearly indicate the tips for multiple delaminations embedded in the beam system. For example, the distances between point a and a′, b and b′ as well as c and c′ in Fig. 6.14(c) are consistent with the delamination length Ldel-1, Ldel-2 and Ldel-3, respectively. It is similar to the beam with single delamination that the SCO for the case of vibration mode II is not sensitive to the existence of the triple delaminations. In addition, by comparing the SCO distribution between the beams with single delamination (see Fig. 6.6) and triple delaminations (see Figs 6.12(a), 6.13(a) and 6.14(a)), we note that the values of SCO for the beam with triple delaminations are larger than those with single delamination as expected. In addition, Figs 6.12(c)–6.14(c) reveal that the leap occurring in the SCO distribution curves is more obvious for the case with longer delamination than that with shorter delamination.

6.5

Experimental verification

In order to discuss the feasibility of the present analytical model for delamination identification, an experimental program was conducted for the case of vibration mode I. Figure 6.15(a) [25] shows the photo for a laminated composite beam specimen with delamination and bonded with PZT sensor and actuator. For comparison of the output voltage distribution for the beams with and without a delamination, six laminated composite specimen coupons (three with a delamination and three without delamination) were made by © 2008, Woodhead Publishing Limited

158

Delamination behaviour of composites 3.00E–08 Perf Del

2.50E–08

SCO (C)

2.00E–08 1.50E–08 1.00E–08 5.00E–09 0.00E+00 0

0.05

0.1

0.15 0.2 0.25 0.3 X (m) (a) SCO distribution

1.20E+00 Perf Del

1.00E+00

NSCO

8.00E–01 6.00E–01 4.00E–01 2.00E–01 0.00E+00 0

0.05

0.1

0.15 0.2 0.25 X (m) (b) NSCO distribution

0.3

5.00E–01 Perf Del

NSCO

4.00E–01

3.00E–01

2.00E–01 0.12

0.14

0.16 0.18 X (m) (c) NSCO distribution for the range of x = 0.12 to 0.18m

6.12 The SCO and NSCO distributions for the beams with and without triple delaminations for the case of vibration mode I.


Piezoelectric sensor/actuator for delamination detection 1.60E–08

Perf Del

1.40E–08 1.20E–08

SCO (C)

1.00E–08 8.00E–09 6.00E–09 4.00E–09 2.00E–09 0.00E+00 0

0.05

0.1

0.15 0.2 0.25 X (m) (a) SCO distribution

0.3

1.20E+00

Perf Del

1.00E+00

SCO (C)

8.00E–01 6.00E–01 4.00E–01 2.00E–01 0.00E–00 0

0.05

0.1


0.3

9.00E–01

Perf Del

SCO (C)

8.00E–01

7.00E–01

6.00E–01

5.00E–01 0.12

0.14

0.16 0.18 X (m) (c) NSCO distribution for the range of x = 0.12 to 0.18 m

6.13 The SCO and NSCO distributions for the beams with and without triple delaminations for the case of vibration mode II.


159

160

Delamination behaviour of composites 6.00E–09

Perf Del

5.00E–09

SCO (C)

4.00E–09 3.00E–09 2.00E–09 1.00E–09 0.00E+00 –1.00E–09 0

0.05

0.1

0.15 0.2 0.25 X (m) (a) SCO distribution

0.3

1.20E+00 Perf Del

1.00E+00

SCO (C)

8.00E–01 6.00E–01 4.00E–01 2.00E–01 0.00E–00 –2.00E–01 0

0.05

0.1


0.3

8.00E–01 Perf Del

SCO (C)

6.00E–01

c′ c b′

4.00E–01

b a

a″

2.00E–01

0.00E+00 0.12

0.13

0.14

0.15 0.16 0.17 0.18 X (m) (c) NSCO distribution for the range of x = 0.12 to 0.18 m

6.14 The SCO and NSCO distributions for the beams with and without triple delaminations for the case of vibration mode III. © 2008, Woodhead Publishing Limited


161

(a) Photo of a composite beam segment bonded with PZT sensor and actuator PZT actuator

PZT sensor Laminated composite beam

Signal conditioner SCXI-1120

PI E-507.00 power amplifier

A/D

DAQ board

Computer with LabView software

D/A Cable (b) A schematic of the experimental set-up

(c) A photo of the experimental set-up

6.15 Experimental set-up for delamination identification in a laminated composite beam [25].


162


laying-up eight layers of T300/F593 plain weave composite prepregs. The dimension of the laminated composite beam is 0.6 m in length, 25 mm in width and 1.44 mm in thickness. The delamination with gap of 0.1 mm was created by inserting three small pieces of Teflon film with a total thickness of 0.1 mm in the mid-plane of the laminated composite beam during the composite prepreg lay-up. The delamination with a length of 60 mm and a width of 25 mm is located at Xd = 0.3 m. A PZT actuator with sputtered Cr/ Au metallization on its top and bottom surfaces was bonded at Xa = 0.12 m. The dimension of the actuator is 30 mm long, 25 mm wide and 0.4 mm thick. A PZT sensor with 60 mm in length, 25 mm in width and 0.4 mm in thickness and with a gridded electrode pattern on its top surface was bonded on the specimen section embedded with a delamination. The PZT sensor is located at Xs = 0.27 m. The gridded electrode pattern is 3 × 25 mm electrode strips separated by 1 mm gaps. Another surface of the sensor is fully covered by the Cr/Au metalization. The PZT sensors and actuators were custommanufactured by TRS Ceramics, Inc, USA. Figures 6.15(b) [25] and (c) [25] show the schematic and photo for the experimental set-up, which includes a National Instruments Signal Conditioner, PI E-507.00 power amplifier (three channels, gain gp = 100) and a computer with a data acquisition system. The specimen was clamped at one end and free at the other to match the boundary condition used in the analytical model presented in Section 6.3. Both actuation and data acquisition were performed using a data acquisition (DAQ) board, and a computer running LabVIEW as a virtual controller. Two LabVIEW VI-files were created in this testing program. The first one was used to evaluate the first three order of frequencies for the specimen coupons with and without delamination, and the second was generated for loading a sinusoidal voltage V(t)=VaSinωo1t at the PZT actuator through its thickness and thus exciting the beam system. By using the second LabVIEW VI-files, the variations trend of the output voltage with time for a select frequency and an electrode strip can be obtained by recording 25 samples per second in the computer with the data acquisition system. For example, for the case of vibration mode I and a laminated composite beam with a through-the-width delamination, a typical variation trend of the output voltage vs time, which was measured from an electrode strip coated on the PZT sensor, is plotted in Fig. 6.16 [25] for the time period ranging from T = 10s to T = 15s. By taking discrete Fourier transform with respect to time for the discrete output voltage data measured from the electrode strips, we can obtain the required variation trend of output voltage vs electrode location for a selected frequency ωoi. Figure 6.17(a) [25] shows the output voltage distribution vs electrode strip location, which were obtained from experimental test and present analytical model for the case of vibration mode I and the beam without delamination, and Fig. 6.17(b) [25] shows those for the beam with a delamination. Owing © 2008, Woodhead Publishing Limited


163

0.8 0.6

Output voltage (V)

0.4 0.2

0 –0.2 –0.4 –0.6 –0.8 10

11

12

13

14

15

Time (s)

6.16 A typical variation trend of the output voltage measured from an electrode strip coated on the PZT sensor (for the case of vibration mode I and a beam with delamination [25].

to failure in recording the output voltage from several electrode strips, only those recorded results measured from the third to the fifteenth electrode strips are plotted in Fig. 6.17(a) and those from the second to the fifteenth electrode strips are plotted in Fig. 6.17(b). It is worth mentioning that the average values and standard deviations for those measured results shown in Figs 6.17(a) and 6.17(b) are obtained by averaging ten data recorded from their corresponding electrodes. It is noted from Fig. 6.17(a) that there is a good agreement between the predicted and measured results. However, it is interesting to note from Fig. 6.17(b) that the output voltage distribution shifts upwards slightly compared to that predicted using the present analytical model even though they have similar variation pattern. One contributing factor could be the delamination gap that is assumed to be constant in the analytical model but it may change during testing. In addition, some factors such as stress concentration at the delamination tip are not considered in the analytical model. All these could contribute to the difference between the predicted and experimental results. For the purpose of verifying the present analytical model, a total of 60 tests (30 for the case of the beam with delamination, and another 30 for the beam without delamination) were conducted using the LabVIEW program to evaluate the first three frequencies. The corresponding average values (i.e., Avg.) and their standard deviations (i.e., Std.) are tabulated in Table 6.1 [25]. © 2008, Woodhead Publishing Limited

164

Delamination behaviour of composites 0.045 0.04

Output voltage (V)

0.035 0.03 0.025 0.02 0.015 0.01 Exp.

0.005

Model

0 0.27

0.28

0.29

0.3 0.31 0.32 X(m) (a) For the beam without delamination

0.33

0.06

Output voltage (V)

0.05

0.04

0.03

0.02

0.01

Model Exp.

0 0.27

0.28

0.29

0.3 0.31 0.32 X(m) (b) For the beam with a single delamination

0.33

6.17 Output voltage distributions obtained from experimental test and present analytical model for the case of vibration mode I [25].

The first three frequencies predicted using the present analytical model are also listed in Table 6.1 for comparison. From Table 6.1, it is noted that there is a good agreement between the measured and predicted results. The difference between the predicted and measured results ranges from 2.3% to 8.2% except for the case of vibration mode III for the beam without delamination, whose difference is 13.1%. © 2008, Woodhead Publishing Limited


165

Table 6.1 Comparison of the first three frequencies between the experimental and predicted results [25] Perfect beam Experimental result

Mode I (Hz) Mode II (Hz) Mode III (Hz)

6.6

Avg.

Std.

3.9 21.88 76.7

0.011 0.01 0.36

Analytical model

3.99 23.3 66.67

Delaminated beam Experimental result Avg.

Std.

3.86 21.53 63.06

0.007 0.019 0.075

Analytical model

3.99 23.3 66.51

Conclusions

In this investigation, several dynamic analytical models are developed for the identification of single delamination or triple delaminations embedded within a laminated composite beam using surface-bonded with integrated or discontinuous piezoelectric sensor/actuator. The feasibility of using integrated and discontinuous PZT sensor and actuator to detect through-the-width delaminations beneath the surface of a laminated composite beam is explored through analytical and experimental investigation. Numerical study shows that the present analytical model with PZT sensor and actuator can be employed to identify the presence, size and axial location of single delamination or triple delaminations within a laminated beam. It is also revealed that the values of SCO measured from the discontinuous PZT sensors are larger than those from the integrated PZT sensors, whereas the SCO distributions obtained using the integrated PZT sensors can indicate more clearly the tips of a delamination than those using discontinuous PZT sensors. For the first three frequencies and the output voltage distribution along the PZT sensor, a comparison between the predicted and measured results shows that there exists a reasonable good agreement. This demonstrates that the existence of a through-the-width delamination can be identified experimentally using the PZT sensor and actuator, and the present analytical model can be employed to detect delaminations embedded in a laminated composite beam.

6.7

Acknowledgments

The authors are grateful to the support of University of Sydney via University Postdoctoral Research Fellowship and of ARC via a Discovery-Project Grant Scheme (Grant No. DP0666683).

6.8

References

1. Zou Y, Tong L and Steven G P (2000), ‘Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures – A review’, Journal of Sound and Vibration, 230, 357–378. © 2008, Woodhead Publishing Limited

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2. Saravanos D A and Hopkins D A (1996), ‘Effects of delaminations on the damped dynamic characteristics of composite laminates: Analysis and experiments’, Journal of Sound and Vibration, 192, 977–993. 3. Williams T O and Addessio F L (1998), ‘A dynamic model for laminated plates with delaminations’, International Journal of Solids and Structures, 35, 83–106. 4. Lee J (2000), ‘Free vibration analysis of delaminated composite beams’, Computers & Structures, 74, 121–129. 5. Todoroki A and Tanaka Y (2002), ‘Delamination identification of cross-ply graphite/ epoxy composite beams using electric resistance change method’, Composite Science and Technology, 62, 629–639. 6. Todoroki A, Tanaka Y and Shimamura Y (2002), ‘Delamination monitoring of graphite/ epoxy laminated composite plate of electric resistance change method’, Composites Science and Technology, 62, 1151–1160. 7. Takeda S, Okabe Y, Yamamoto T and Takeda N (2003), ‘Detection of edge delamination in CFRP laminates under cyclic loading using small-diameter FBG sensors’, Composites Science and Technology, 63, 1885–1894. 8. Takeda S, Okabe Y and Takeda N (2002), ‘Delamination detection in CFRP laminates with embedded small-diameter fiber Bragg grating sensors’, Composites Part A: Applied Science and Manufacturing, 33, 971–980. 9. Ling H Y, Lau K T and Cheng L (2004), ‘Determination of dynamic strain profile and delamination detection of composite structures using embedded multiplexed fibre-optic sensors’, Composite Structures, 66, 317–326. 10. Lee D C, Lee J J and Yun S J (1995), ‘The mechanical characteristics of smart composite structures with embedded optical fiber sensors’, Composite Structures, 32, 39–50. 11. Teboub Y and Hajela P (1992), ‘A neural network based damage analysis of smart composite beams’, Fourth AIAA/USAF/NASA/OAI symposium on multidisciplinary analysis and optimisation, AIAA paper 92–4685. 12. Keilers C H and Chang F K (1993), ‘Damage detection and diagnosis of composites using built-in piezoceramics’, Proceedings of the 1993, North American conference on smart structures and materials, 1917, 1009–1019. 13. Keilers C H and Chang F K (1995), ‘Identifying delamination in composite beam using built-in piezoelectrics: Part I – experiments and analysis’, Journal of Intelligent Material Systems and Structures, 6, 649–663. 14. Keilers C H and Chang F K (1995), ‘Identifying delamination in composite beam using built-in piezoelectrics: Part II – an identification method’, Journal of Intelligent Material Systems and Structures, 6, 664–672. 15. Saravanos D A, Birman V and Hopkins D A (1994), ‘Detection of delaminations in composite beams using piezoelectric sensors’, NASA Technical Memorandum 106611, AIAA–94–1754. 16. Yin L, Wang X M and Shen Y P (1996), ‘Damage-monitoring in composite laminates by piezoelectric films’, Computers & Structures, 59, (4), 623–630. 17. Tong L and Steven G P (1999), Analysis and Design of Structural Bonded Joints, Dordrecht, Kluwer. 18. Hagood N W, Kindel R, Ghandi K and Gaudenzi P (1993), ‘Improving transverse actuation of piezoceramics using interdigitated surface electrodes’, SPIE paper No.1917–25, Proceedings of 1993 North American Conference on Smart Structures and Materials, Albuquerque, NM.



167

19. Bent A A, Hagood N W and Rodgers J (1995), ‘Anisotropic actuation with piezoelectric fiber composites’, Journal of Intelligent Material Systems and Structures, 6, 338– 349. 20. Chee C, Tong L and Steven G P (1998), ‘A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures’, Journal of Intelligent Material Systems and Structures, 9, (1), 3–19. 21. Tan P and Tong L (2001), ‘Micro-electromechanics models for the piezoelectric fiber reinforced composite materials’, Composites Science and Technology, 61, 759– 769. 22. Tan P and Tong L (2004), ‘Identification of delamination in a composite beam using integrated piezoelectric sensor/actuator layer’, Composite Structures, 66, 391–398. 23. Tong L, Sun D C and Atluri S N (2001), ‘Sensing and actuating behaviours of piezoelectric layers with debonding in smart beams’, Smart Materials and Structures, 10, 713–723. 24. Tan P, Tong L and Steven G P (1997), ‘A flexible 3D FEA modelling approach for predicting the mechanical properties of plain weave unit cell’, Proceedings of the Eleventh International Conference on Composite Materials (ICCM11), Vol. V, 67–76. 25. Tan P and Tong L (2007), ‘Experimental and analytical identification of a delamination using isolated PZT sensor and actuator patches’, Journal of Composite Materials, 41(4), 477–492.

6.9

Appendix ∂ w1 sa ∂w   U1 =  u1 sa w1 sa T1 sa Q1 sa M1 sa u1 b w1 b 1 b T1 b Q1 b M1 b  ∂ x ∂ x  

T

A1 =  0   0   0   bGad – ρ1  t  ad  0   bGad t s   2 t ad  0   0   0   bGad  – t ad   0   bt G b ad   2 t ad

0

0

0

0

0

0

0

1

1 a1 0

0

0

0

0

0

0

0

0 1 D1

0

0

0

0

0

0

0

bGad t s 2 t ad

0

0

0

0

0

0

0

0

0

0

0

0 0 0

–

bYad – p1 t ad 0

–

0 –

bGad t s2 4 t ad

0

0

0

0

1

0

–

bGad t ad 0

–

bGad t s 2 t ad

0

–

bY – ad t ad 0

bGad t b 2 t ad 0

–

bGad t b t s 4 t ad

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1 A3 0

0

0

0

0

0

0

0

0

0

0

0

bGad t s 2 t ad

0

0

0

bGad – ρ3 t ad

0

bGad t b 2 t ad

0

0

bYad t ad

0

0

0

0

0

bYad – ρ3 t ad

0

0

0

0

bGad t b2 – 4 t ad

0

1

0

–

bGad t b t s 4 t ad


0

0

0

–

bGad t b 2 t ad

0 0

0   0   0    0   0    0   0   0   1  D3   0   0    0  

168


a1 = bYsa t sa , D1 = – A3 = bYb t b , D3 = –


3 bYsa t sa , ρ1 = ρsa Asa ω 2 , 12

bYb t b3 , ρ3 = ρb Ab ω 2 12

7 Lamb wave-based quantitative identification of delamination in composite laminates Z S U, The Hong Kong Polytechnic University, Hong Kong and L Y E, The University of Sydney, Australia

7.1

Introduction

With such features as comparatively high strength-to-weight and stiffnessto-weight ratios, excellent capability of resisting corrosion and tailorable mechanical properties, the use of fibre-reinforced composite structures is increasing exponentially among various communities, offering significant weight saving and immense performance enhancement. In the latest Airbus models, A-350 and A-380, employment of composite materials has reached the unprecedented levels of 40–50% and 22–25%, respectively (Pora, 2003). However, it must be borne in mind that composite structures are highly vulnerable to structural damage, in particular the delamination that can be introduced during manufacturing, tooling, processing or service. The occurrence of delamination is a great threat to the functional operation of composite structures, potentially leading to catastrophic failure of the whole structure if it accumulates above a critical extent. With safety regarded as the paramount factor for engineered structures, it is of vital importance to identify delamination in composite structures at an early stage so as to prevent any potential failure. This has motivated the rapid development of non-destructive evaluation (NDE) techniques, witnessed over the past three decades. At present, prevailing NDE techniques for identification of delamination in composite laminates include ultrasonic inspection, acoustic emission, eddy-current, radioscopy, infrared thermograph, modal data-based analysis, etc. However, these techniques have increasingly been challenged as to their capacities to meet both the desired detection precision and practical feasibility. With excellent capability of propagating a reasonably long distance even in materials with high attenuation ratio (e.g. CF/EP composites) and very high sensitivity to internal structural damage (e.g. delamination), Lamb waves, guided elastic waves in plate-like structures, have been examined as a means of establishing novel NDE tools, with laudable potential to cost-effectively identify delamination in composite structures. 169 © 2008, Woodhead Publishing Limited

170


With the benefit of various propagation modes of Lamb waves, the interlaminar position of the delamination can also be pinpointed. Given the above benefits, from the late 1990s there has been increasing awareness of the use of Lamb waves to evaluate delamination in composite laminates, leading to an impressive number of successful attempts. This chapter has the goal of describing Lamb wave-based quantitative evaluation of delamination in composite laminates, covering key knowledge in the sphere from the fundamental mechanism to signal processing and data fusion. Emphasis is particularly placed upon the propagation characteristics of Lamb waves in laminated composites, methods of wave generation and collection, mode selection for detectability, signal processing and data fusion, and sensor networks, followed by case studies. A novel data processing and fusion approach, ‘digital damage fingerprints’ (DDF), is discussed and its effectiveness in quantitatively identifying delamination is demonstrated.

7.2

Lamb waves in composite laminates

It was Horace Lamb in 1917 who first captured the waves in plate-like structures, named after him as Lamb waves, and established the theoretical rudiments in his historic publication, On Waves in an Elastic Plate (Lamb, 1917). A comprehensive solution to Lamb waves was presented by Mindlin in 1950, followed by great detail provided by Viktorov who also first evaluated the dispersive properties of Lamb waves (Lowe, 1995). Firestone and Ling inaugurated the use of Lamb waves for damage evaluation in the 1950s, and its effectiveness in seismology and NDE was soon confirmed by follow-up tests conducted by Worlton in 1961 and Frederick in 1962 (Chimenti, 1997). With progressive advances in computing devices, the period from the 1990s to the present day has seen a renaissance in Lamb wave-based applications.

7.2.1

Theory and fundamentals

The term ‘Lamb wave’ refers to the wave type existing in thin plate (with planar dimension far greater than thickness) that requires free upper and lower boundaries to maintain the propagation (Lamb, 1917). Mathematically, Lamb waves, like other elastic waves, can generally be described in a form of Cartesian tensor notation (Rose, 1999)

µ · ui,jj + (λ + µ) · uj,ji + ρ · fi = ρüi (i, j = 1, 2, 3)

7.1

where ui and fi are the deformation and body force in the xi direction, respectively, ρ and µ are the density and shear modulus of the plate, respectively, and λ = 2 µ v /(1 – 2 v ) (v: Poisson’s ratio). As a result of plane strain and applying boundary conditions at both free surfaces of the plate, a general solution to Equation (7.1), describing Lamb waves in a homogeneous plate, © 2008, Woodhead Publishing Limited

Lamb wave-based quantitative identification of delamination

171

can be obtained as below (since the field variables involve trigonometric functions of sines and cosines, it is a corollary to split the solution into two parts with symmetric and anti-symmetric properties, leading to two different Lamb wave modes, shown in Fig. 7.1)

where

tan( qh ) 4 k 2 qp = – 2 tan( ph ) (k – q 2 )2

for symmetric wave modes

7.2a

tan( qh ) (k 2 – q 2)2 = – tan( ph ) 4 k 2 qp

for anti-symmetric wave modes

7.2b

2 2 p 2 = ω2 – k 2 , q 2 = ω2 – k 2 , k = ω / c p cL cT

h, k and ω are the half thickness of the plate, wavenumber and wave circular frequency, respectively. The wave phase velocity, cp, is related to the wavelength, λwave, by cp = (ω/2π) · λwave: cL and cT are the velocities of

Symmetric mode

u3

u1

Anti-symmetric mode

7.1 Symmetric and anti-symmetric Lamb modes.


172


longitudinal and transverse modes, respectively (Lamb waves are the superposition of longitudinal and transverse modes), defined by cL =

E (1 – ν ) and c T = ρ (1 + ν )(1 – 2ν )

E 2 ρ (1 + ν )

These equations are known as ‘Rayleigh-Lamb frequency equations’. Though simple in appearance, they can be analytically solved for very simple cases only. At a given frequency, there are an infinite number of wavenumbers, either real or purely imaginary, to satisfy Equation (7.2), representing an infinite number of wave modes. Hereinafter we use the symbols Si and Ai to denote the symmetric and anti-symmetric modes, respectively, with the subscript representing the order. Expanding Equation (7.1) for a N-layered laminate, the displacement field, u, within each layer must satisfy Navier’s displacement equations (Rose, 1999), and for the nth layer it has 2 n µn∇2un + (λn + µn) ∇(∇ · un) = ρ n ∂ u2 ∂t

where

(n = 1, 2, …, N)

7.3

2 2 2 ∇ = ∂ + ∂ + ∂ , and ∇ 2 = ∂ 2 + ∂ 2 + ∂ 2 ∂ x1 ∂x2 ∂ x3 ∂ x1 ∂x2 ∂ x3

Variables in Equation (7.3) are distinguished by the superscript for each layer. The Helmholtz principle can be used to decompose the displacement fields so as to achieve a solution to describe Lamb waves in multi-layered laminate. For the sake of brevity, the detail of Helmholtz decomposition is not presented here, as it can be found elsewhere (Rose, 1999).

7.2.2

Dispersion and attenuation

When Lamb waves propagate in an elastic medium under the influence of material viscoelasticity, attenuation in wave magnitude, variations in velocity and changes in wavenumber are often observed, i.e. dispersion. The dispersion phenomenon is mainly referred to as the observation that the wave velocity is subject not only to the material and structural geometry but also to the excitation frequency, in particular in a relatively low frequency range where the wavelength is of the order of 100 µm to 1 mm. It has been shown that the velocity of Lamb waves, regardless of mode, is a function of the algebraic product of excitation frequency and plate thickness. As a paradigm, dispersion curves of different Lamb modes in a quasi-isotropic [0/±45/90]s carbon-fibre (T800)-resin (924) composite laminate (1.275 mm in thickness) are compared in Fig. 7.2. It is further noticed that Si and Ai propagate at very different velocities in composite laminates with various configurations. For example,


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9

S1

Group velocity [km/s]

S0 6

A1 S0

3

A0 FEM Experiment

0

0.0

0.5

1.0 1.5 2.0 Frequency thickness [MHz.mm]

2.5

3.0

7.2 Dispersion curves of Lamb waves in an 8-ply quasi-isotropic CF/ EP (T650/F584) composite laminate (S0 stands for the fundamental horizontal shear mode).

Table 7.1 Attenuation of Lamb waves in composite materials (Pierce et al., 2000) Materials

Lamb mode

Excitation frequency [KHz]

Attenuation coefficient [mm–1]

Distance of decaying to 10% of original amplitude [mm]

CFRP woven (8-ply)

S0

250

0.0014

1700

CFRP woven (10-ply)

A0

285

0.027

S0 S0

250 250

0.00078 0.0016

GFRP random

S0

220

0.0035

660

CFRP/GFRP hybrid (RTM) sandwich foam core

S0

250

0.013

182

CFRP/GFRP hybrid (RTM) sandwich honeycomb core

S0 S0

250 150

0.0036 0.0015

640 1600

GFRP filament wound pipe

S0 S0

250 150

0.015 0.011

150 210

CFRP woven 10-ply with T-stringers parallel to stringers perpendicular to stringers


85

3000 1500

174


S0 propagates at circa 10 300 m/s along the fibre in a laminate with configuration of [0]8, whereas it reduces to 6000 m/s in the quasi-isotropic [0/±45/90]s laminate (Percival and Birt, 1997). Both Si and Ai modes are synchronously available at any frequency, while higher order modes appear as an increase in frequency. A relatively low frequency region often exists, in which S0 and A0 modes propagate at constant velocities; it is therefore termed the nondispersion region. In Fig. 7.2, a non-dispersion region ( 60 dBAE A > 40 dBAE

–1.0 0

10

20 30 X-location (mm)

40

8.12 Approximated size of AE process zone of UD GF/PA-12 in stable mode I delamination propagation.

dispersion of the nanoscale filler in the matrix and its compatibility with matrix and reinforcing fibers. A study with a commercial type of organo-silicate added to an epoxy resin showed promising improvement of the fracture toughness of the nanocomposite epoxy resin, but an effective reduction in the delamination resistance, mainly of the propagation values when used as a matrix in a laminate with woven glass-fiber reinforcement (Barbezat et al. 2007a). Results from AE monitoring of mode I fracture tests on GFRP laminates with nanomodified and pristine epoxy matrix are shown in Fig. 8.13. The size of the AE process zone does seem to be slightly less and with slightly fewer signals of high amplitudes (filtered between 60 and 90 dBAE) located behind the process zone in the laminate with nanomodified epoxy matrix (Fig. 8.13 a,b). Mechanical bending tests indicated a slight increase of the laminate modulus, but a clear reduction in strength and strain at failure for the laminate with modified matrix. The appearance of the fracture surfaces of the laminates after the mode I DCB test is similar; both types of laminates yielded a fiber-rich side and a resin-rich counterpart with no indication of clear differences in fiber-matrix adhesion. The microscopic mechanisms responsible for the reduction in delamination resistance and the slightly reduced AE process zone size in the laminates with modified matrix are not yet fully clarified. Of course, some organosilicate nanomodified matrices were reported to yield improvements of both, the resin fracture toughness and the delamination resistance of the laminate, e.g. (Siddiqui et al. 2007). Such matrix materials, besides improving mechanical and fracture properties, may also contribute to fire resistance (Morgan 2006). © 2008, Woodhead Publishing Limited


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Another approach for improved delamination resistance is based on throughthickness reinforcement by, e.g., stitching, use of three-dimensional reinforcement (Mouritz et al. 1999), or inserting pins normal to the plies (Meo et al. 2005). Delamination resistance tests of a carbon-fiber epoxy composite with and without z-pins monitored with AE have been discussed in (Cartié et al. 2003). The results shown in Fig. 8.14 are from specimens

Energy [eu]

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5E7 1

0

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10E7 10

5E7 1

0

0 0

100

200

300

400 (b)

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600

700

8.13 Mode I DCB delamination tests on woven GF/EP composite specimens monitored with two AE sensors mounted on the load block and towards the end of the specimen, respectively, 20 cm apart, sensor positions are indicated by +; energy distribution as a function of linear location and time (a and b) with organo-silicate clay modified matrix, (c and d) without nanomodified matrix, only signals with amplitudes between 60 and 90 dBAE are plotted, the AE process zone appears somewhat narrower in the nanomodifed GF/EP.


Delamination behaviour of composites Time [s]

Energy [eu]

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lo [m cat m ion ]

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30E7 25E7 20E7 15E7

20

10E7 10

5E7 0

0 100

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400 (c)

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2

20

10E7 10

5E7 1

0

0 0

100

200

300

400 (d)

500

600

700

8.13 (Continued)

(250 mm × 20 mm × 4 mm) with two pinned areas (25 mm long), the first 15 mm from the tip of the starter film with the same (0.5%), the second 40 mm from the first with different areal density of pins (0.5% versus 4.0%). For both specimens, the first pinned area induces unstable delamination growth between the pinned areas as shown by the time period without any recorded and located AE events and the discontinuity in the AE process zone. This unstable delamination is stopped by the second pinned area. The pin-density of 4.0%, however, affects the failure mode, the delamination does not propagate through that area and further loading induces a bending failure in the beam; as indicated by stationary event locations and increasing energy per located event. This change in mechanism results in distinctly lower AE activity and higher AE intensity for the bending failure compared to delamination propagation (Fig. 8.14). © 2008, Woodhead Publishing Limited


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Energy [eu]

X

-lo [m cat m ion ]

Time [s]

10E6 9E6 8E6 7E6 6E6 5E6 4E6 3E6 2E6 1E6 0

30 20 10 0 0

100 200 300 400 500 600 700 800 (a)

24 1 22

X-location [cm]

20 18 16 14 12 10 8 6 4

2

2 100

200

300

400 500 Time [s] (b)

600

700

800

8.14 Mode I DCB delamination test of UD CF/EP with two z-pinned areas (25 mm long) separated by 40 mm, different AE signal source location plots from linear source location monitored with two AE sensors mounted on the load block and towards the end of the specimen, respectively, the sensor positions are indicated by +; (a and d) AE energy (bars) as a function of location and time, (b and e) AE source location (circles) as a function of time, (c and f) AE signal activity (bars) as a function of location and time; only signals with amplitudes between 75 and 95 dBAE were used for location, (a,b,c) both areas with 0.5% pin density, (d,e,f) first area with 0.5%, second with 4.0% pin density. Note the difference in AE activity and AE intensity for the different pin densities.



X

-lo [c cat m io ] n

248

Hits

6 5 4 3

30

1

20

2

10

1

2

0

0 100 200 300 400 500 600 700 800 (c)

Time [s]

X

-lo [m cat m ion ]

0

Energy [eu]

10E6 9E6 8E6 7E6 6E6 5E6 4E6 3E6 2E6 1E6 0

30 20 10 0 0

500

1000

1500 (d)

2000

2500

26 24 22

1

X-location [cm]

20 18 16 14 12 10 8 6 4 2 0

2 500

8.14 (Continued) © 2008, Woodhead Publishing Limited

1000

1500 Time [s] (e)

2000

2500


249

X

-lo [c cat m io ] n

Time [s]

Hits

6 5 4 3

30 20

2

10

1 0

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1000

1500 (f)

2000

2500

8.14 (Continued)

8.4.3

Extended analysis of acoustic emission signals from fiber-reinforced, polymer matrix composites

Often the question is asked whether the source mechanisms can be identified based on waveform and spectral analysis of AE? In the opinion of the authors this is at best feasible as a statistical approach applicable to AE data sets with a sufficient number of signals for one mechanism at a time, because of the complex nature of the sources and the relation between source mechanism and recorded AE signal, as discussed below. In a specific frequency range the amplitude of the voltage signal from the piezoelectric sensor element is proportional to the mechanical force of the acoustic wave hitting the sensor. The maximum displacement amplitude umax of, e.g., a fracture impulse source resulting from the rebound of elastically stored energy can be approximated by Equation 8.2 with shear modulus G and density ρ0 of material, involved fracture volume vf, wave velocity c and distance r from source to AE sensor.

u max ~

G ⋅ Vf 4 π ⋅ ρ0 ⋅ c 3 ⋅ r

8.2

The AE signal amplitudes of (fiber or matrix) crack sources show a broad scatter due to fracture stress distribution and varying crack size. Fiber breakage is mostly accompanied by interface debonding and/or matrix cracking in the same short-term process resulting in a complex source behavior. Generally, the fracture of a brittle material with high stiffness and high strength (e.g., fibers) causes AE sources with fast rebound of elastically stored energy and, hence, high displacement velocities. Visco-elastic materials with low stiffness and © 2008, Woodhead Publishing Limited

250


low strength (e.g., polymer matrix) fail under slower and much lower elastic energy release, except after formation of fiber-like morphologies, such as fibrils in polymer blends or fibrils of thermoplastic composite matrices (Bohse 2000). Spectral analysis of AE source mechanisms has shown that process times of AE sources from polymer-matrix composites are in the range of less than about one microsecond. During failure the sources stimulate different acoustic wave modes. A wave hitting an AE sensor generates ringing with all excited sensor resonance frequencies. Provided that the full source spectrum was transported by the acoustic wave from the event source to the sensor location the activation of a sensor resonance indicates how wide the frequency characteristic of the AE source was. For unstable crack growth the upper cut-off-frequency f0 can be approximated by Equation 8.3 with source duration τ. f0 ~ 1 τ

8.3

Process times for the creation of polymer matrix cracks and fiber breakage of equal size should be in the ratio between about two and three (assuming crack velocities to be proportional to velocities of transverse waves). That means matrix cracking creates lower wideband-frequencies than fiber breakage. In plate-like structures of composite materials the AE energy is transferred by plate (Lamb) wave modes and surface (Rayleigh) waves. In-plane microfailure (e.g., fiber breakage) stimulates a higher amount of extensional plate wave modes with a higher content of wideband-frequencies. Out-of-plane failure (e.g., matrix cracking in mode I delamination) preferably stimulates flexural plate wave modes of lower frequencies than extensional wave modes (proportional to the wave velocities). An example is shown in Fig. 8.15 and further discussed in (Krietsch and Bohse 1998; Bohse 2004). Considering the spectral sensitivity of the wideband AE sensor it is concluded that matrix cracking in delamination processes mainly produces flexural waves and dominant energy contributions up to frequencies of about 300 kHz. Fiber breakage gives rise to extensional waves with dominant energy contributions above 300 kHz if dispersion and frequency-dependent attenuation are negligible, i.e., in sufficiently small-sized specimens. Composites with a visco-elastic polymer matrix possess a comparatively high material attenuation. The attenuation is dependent on a number of factors such as the direction of wave propagation (parallel or perpendicular to fiber orientation), the particular wave mode involved, and the relaxation behavior of the matrix which is frequency and temperature dependent, see (Chen et al. 1993) for AE from GFRP with a soft thermoplastic matrix. Identification of AE source mechanisms using power spectra is hence restricted to short distances between source and sensor. © 2008, Woodhead Publishing Limited


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While the acoustic waves propagate from the source (origin) to the sensor position they are often greatly altered, as noted earlier. The AE signal waveform at a distance from the source is hence often highly distorted from what would be obtained by a sensor located very close to the source. In addition to these geometrical and materials influences, modulations of waveforms and spectra due to the sensitivity of the AE sensors which can vary strongly with frequency, as well as due to the electronic frequency filters used in preamplifiers and data acquisition systems must be accounted for. AE waveform analysis often uses pattern recognition (Dzenis and Saunders 2002; Kostopoulos et al. 2007) for classification of transient waveforms and for identification of different damage mechanisms. If prototype signals are needed for classification, the quality of the classification depends on the selection of these signals. Unambiguous identification of the source mechanism for a particular signal is difficult, except in simplified model experiments (Nordstrom et al. 1996). Further development of classification approaches and algorithms, as well as modeling tools may contribute to improved discrimination of mechanisms. The main obstacle is wave propagation effects which affect frequency content and amplitudes, or even wave modes. In concrete, another type of composite, the method of moment tensor inversion, originally developed for geophysical applications, has been successfully used for AE source and damage characterization (Shigeishi and Ohtsu 2001; Grosse et al. 2004). This method requires volumetric source

Amplitude (V)

6.0 × 10–5

4.0 × 10–5

2.0 × 10–5

0.0 100

200

300 400 Frequency (kHz) (a)

500

600

700

8.15 Spectral sensitivity of the wideband AE sensor (a) and averaged spectra from hundreds of signals (b) from a mode I fracture and (c) a tensile test on UD GF/PP with strong fiber-matrix adhesion.


252


58%

40%

Power spectrum

30

10

100

300 Frequency [cycles/msec] (b) 30%

500

69%

Power spectrum

100

60

20

100


300 500 Frequency [cycles/msec] (c)

700


253

location using at least six AE sensors recording the longitudinal and transverse bulk waves along three mutually orthogonal spatial dimensions. Moment tensor inversion is, in principle, able to discriminate between tensile and shear fracture mechanisms. For plate-like or thin-walled FRP composite specimens and structures, the method is not applicable, because bulk waves do not propagate.

8.5

Acoustic emission investigation of delaminations in structural elements and structures

The results presented below from laboratory tests shall demonstrate the advantages and limits of delamination investigation of structures or structural elements using AE monitoring. The first example is a small, cylindrical, allcomposite pressure vessel; a filament wound CF-epoxy shell with a thermoplastic liner inside (Fig. 8.16). Such pressure vessels are, e.g., used for storage of compressed air at 300 bar. The AE characteristics of model defects (Fig. 8.16) before and after different service periods simulated by applying pressure cycling were investigated during pressure testing. The circumferential rough surface region probably represents a different manufacturing quality. The notches were machinesawed into the surface layer of the pressure vessel wall. In the case shown, both circumferential and axial notch were of 50 mm length, 1 mm width and 3 mm depth, i.e. cut through the outer surface plies only.

Y

S4

X P

Notch

Rough surface region Impact

S2/(S1)

S6/(S5) Notch

S3

8.16 Cylindrical pressure vessel with positions of the six AE sensors and of the model defects, i.e., either rough surface region or impact or two surface notches (one circumferential and one axial), the impact was applied at the same position as the axial notch; a separate vessel was tested for each type of damage (rough surface, impact, surface notches); the coordinate system used for planar location of AE sources on the cylindrical wall of the pressure vessel has the x-axis in axial, and the y-axis in circumferential direction (compare Fig. 8.18).


254


Impacts and notches produce local delamination. ‘Micro’-delaminations are generated in different composite layers during impact and ‘macro’delamination in a specific area at the tip of the surface notches during pressurization. Results of AE testing with the pressure cycle shown in Fig. 8.17 shortly after the introduction of the defects and after 5000 pressure cycles in the service-pressure range 20–300 bar which simulate a service period of several years are shown in Figs 8.18 and 8.19, respectively. Planar AE source location with six AE sensors was used on the cylindrical shell of the pressure vessel. Of the pressure vessels tested, the one carrying the impact damage and the other with axial notch yielded AE of high energy. Note that the axial notch was machine-sawed and cut across fibers which run in the circumferential direction. After 5000 pressure cycles defect detection and AE signal source location are still possible. All defects produce significant AE during testing after 5000 pressure cycles. The AE sources in the area with a roughened surface probably arise from micro-defects and micro-delaminations as a result of fatigue. AE sources are located using planar and zone location procedures in parallel (Figs 8.18 and 8.19, d,e,f and g,h,i, respectively). The planar location algorithm yields specific coordinates for each located source, but their accuracy is limited because the wave velocities are direction dependent. The greater the distance between AE source and sensor, less reliable the source location will be. Zone location uses the AE channel that first detects an AE signal since it is nearest to the AE sources. Plots of cumulative hits or energy of

350

Pressure [bar]

300 250 200 150 100 50 0 0

500

1000

1500

2000 2500 Time [s]

3000

3500

8.17 Pressure cycles for AE testing of CFRP pressure vessel.


4000

Energy [eu]

Energy [eu]


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26E5 24E5 22E5 20E5 18E5 16E5 14E5 12E5 10E5 8E5 6E5 4E5 2E5 0 0

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1500

2000 2500 Time [s] (b)

3000 3500 4000

8.18 Detection of defects or damaged regions by location of AE sources with high AE activity and high AE signal energy during pressure testing directly after introduction of damage (a,d,g) rough surface region; (b,e,h) ‘micro’-delamination after impact damage; (c,f,i) ‘macro’-delamination caused by both surface notches; a,b,c: cumulative AE energy of located events versus time; d,e,f: AE energy of located events (planar location); g,h,i: AE energy of located events (zone location for each sensor).



Energy [eu]

256

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oc Y-l

o ati

m n[

500

1000

X-lo

m]

catio

1500

2000 2500 3000 Time [s] (c)

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[m

X-lo

m]

catio

]

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n [m m

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400 300 200 100

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200 300

8.18 (Continued)


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n tio

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catio

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n [m m]

Energy [eu]

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1

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2

3

4

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Energy [eu]

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2

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4 Channel (h)

5

6

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Delamination behaviour of composites 22E5 20E5 18E5

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Channel (i)

8.18 (Continued)

first-hit sensors usually yield a good indication of damaged areas. The zone location procedure is also preferred for larger composite structures, since planar location would require a much larger number of sensors. Due to the Felicity effect, composite structures with damage which is not critical only yield significant AE activity during a successive load cycle if the maximum load of the previous load cycles is exceeded. For the vessels this should be the case above the maximum cycling pressure of 300 bar. A Felicity-ratio of 1.0 is observed for the vessel with rough surface region (Fig 8.20). This feature is hence not relevant for the burst behavior. In the event of severe damage, however, a strong increase of cumulative AE energy is already recorded at lower pressures. From AE retesting with hydraulic pressurization after 5000 pressure cycles Felicity-ratios of 0.9 and 0.7 were obtained for the vessel with impact damage and with notches, respectively. These values indicate significant damage and a strong decrease of burst pressures. Rubbing or friction between delaminated surfaces causes a progressive increase of AE energy during periods of depressurization, in this case below pressures of about 150 bar (Fig. 8.21). It is hence recommended to record and analyze AE during the whole pressurization cycle until complete depressurization. In any case low pressures (or loads) correlating with high AE energy and progressive AE from rubbing or friction indicate severe damage. Adhesively bonded GFRP double-lap joints, e.g. used in civil engineering applications, are an example of AE monitoring of FRP composite structural elements. The joints all consisted of two types of pultruded GFRP flat profiles (500 mm × 100 mm) 5 mm and 10 mm thick for two outer and one inner © 2008, Woodhead Publishing Limited


259

profile, respectively, and had an overlap length of 100 mm (Fig. 8.22). Joints with three different bond-line thicknesses (0.3, 0.5 and 1.0 mm) were manufactured. The fiber architecture comprised mainly unidirectional rovings towards the center and, depending on the profile thickness, one (5 mm thickness) or two (10 mm thickness) combined mats towards the outside. Tensile tests (Keller and Vallée 2005) had shown that the joints fail by delamination inside the GFRP plate near the adhesive layer (Fig. 8.22).

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In order to gain further insight, three tensile tests per bond-line thickness were monitored by AE. The correlation between linear AE source location, AE signal energy distribution and time was investigated (Fig. 8.23). With the exception of one test, the AE sensors were removed after loading to 140 kN and unloading because the ‘explosive’ failure of the joints would destroy the AE sensors. Even though the failure loads range between about 155 and 239 kN, AE source location yields of damage clusters near the top of the overlap © 2008, Woodhead Publishing Limited


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region for all joints. When tested to failure, all primary failures by delamination were inside the GFRP (Fig. 8.22), no adhesive failure was observed. Fig. 8.24 shows the AE plots for the test monitored until failure (continuous loading up to 239 kN). Both, the width of the band of locations and the energy of the recorded signals are continuously increasing. This provides strong indications that the delamination in the GFRP observed after the test initiates near the top edge of the adhesive bond. AE monitoring of tensile or compressive tests on sandwich T-joints has been discussed in detail by Brunner and Paradies (2000) for a three-point load configuration. The sandwich elements consisted of thick balsa-wood cores with CFRP facings. Tensile or compressive loads (Fig. 8.25 a,b) were introduced into the top of the thin sandwich plate and 13 AE sensors were distributed among three zones, five sensors each on the top CFRP facing on the left and right side, and three sensors on the CFRP at the bottom center (Fig. 8.25 a,b). Bending then resulted in deformation of the thick sandwich plate, concave (i.e., compressive strains) at the bottom in case of tensile, and convex (i.e., tensile strains) at the bottom in case of compressive loads. The aim of AE monitoring was a prediction of the failure zone location during the test. This was achieved by analyzing the Felicity-ratio from a special stair-step load sequence (cycles of loading – unloading – reloading to a higher load). © 2008, Woodhead Publishing Limited

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8.22 Photographs of GFRP double lap joint during tensile testing in a servo-hydraulic machine, showing two of four AE-sensors, the remaining two are in comparable positions on the back side (a) and of delamination fracture of the center GFRP plate near the adhesive layer to the outer plate of the double lap joint; the second outer plate is partly delaminated; due to the explosive type of failure, the center plate also delaminated partly along its central plane (b).

The failure typically originated in the balsa wood core of the base (shear failure of adhesively bonded wood blocks) and resulted in partial delamination of the CFRP facings (Fig. 8.25 c,d). The Felicity-ratios as a function of peak load per cycle of one tensile and one compression test are shown in Fig. 8.26. Assuming a Felicity-ratio of 0.95 yields 56–63% of the failure load of 160 kN for the tensile and 36–50% of the failure load of 140 kN for the compressive test, respectively as indications of critical loading. During compression loading in the three-point bending configuration, the CFRP facing at the bottom of the joint is subject to the largest strain and tensile stress. This area, as a consequence of the convex bending deformation, is identified as critical zone by the Felicity-ratio analysis already at load levels below 50% of the failure load (Fig. 8.26). This is probably due to the fact that compressive loading of FRP composite specimens and elements, © 2008, Woodhead Publishing Limited


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contrary to tensile loading, often does not yield large amounts of AE, as noted in, e.g. (Chen et al. 1993; Roos et al. 2007). In this case, a Felicityratio of 0.95 probably overestimates the criticality of the loading level.

8.6

Advantages and limitations for acoustic emission delamination investigations

The examples presented in this chapter and available in the literature illustrate the following advantages and limitations of AE monitoring of delaminations in FRP composite specimens and structural parts. 1. AE can identify the presence of delaminations in specimens and elements, as long as they change under the applied load. Stationary delaminations can produce AE signals from friction. These are probably best detected during unloading but may not be identifiable in all cases. 2. If delaminations do initiate, propagate in a stable or unstable manner under the applied load, AE can identify the load level at which this happens. Extraneous AE, e.g. noise from hydraulics in fatigue tests, sets a limit on the sensitivity of detection but mostly can be eliminated by using so-called ‘guard’ sensors on the load-introduction or by AE signal source location procedures. During combined thermo-mechanical or purely thermal loading, generation of noise signals from pyroelectric effects in piezoelectric AE sensors is possible. 3. AE source location (requiring at least two sensors) can provide information on the position and size of the AE process zone, and when monitoring delamination propagation or growth over a certain time or load interval, © 2008, Woodhead Publishing Limited


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8.24 AE plots as in Fig. 8.23 but energy scale increased by a factor of six and time-scale extended to failure (at 239 kN) for the joint with 0.3 mm bond-line thickness (sensor positions indicated by +), this provides clear evidence that continuous damage accumulation near the top of the overlap region (around 40 cm location length) leads to the delamination failure.

on the size/extent of the delamination. The accuracy of the location depends on the FRP material, the geometry of the structure or component, sensor type and sensor arrangement, as well as on the location procedure and AE noise. In pressure vessels, signals may propagate at different speeds along the walls and through the medium inside the vessel. In plate-like or thin-walled composite specimens and elements, the resolution of AE location is not sufficient to yield information on the throughthickness position, i.e., the layers or plies which delaminate. This is best obtained from ultrasonic testing, but AE can limit the area to be inspected. © 2008, Woodhead Publishing Limited


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4. Unambiguous identification of the AE source mechanism is difficult even with AE waveform analysis and advanced pattern recognition techniques. Delamination propagation or growth in FRP composites is the result of a large number of microscopic damage events involving

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8.25 Schematic side-view of the load configuration in three-point bending and AE sensor positions (shaded circles), for tensile load (a) and compressive load (b); the forces on top (thick line) are moving, the forces on the bottom (thin line) are unmoving reactions in the supports; differently shaded circles (light gray up front, medium gray center, dark backside) indicate AE sensor positions across width; (c) schematic of the T-joint shape and overall dimensions (in mm), and (d) delamination failure in the balsa wood core and CFRP facing resulting from tensile loading in a three-point configuration. © 2008, Woodhead Publishing Limited

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several mechanisms. Identification based on empirical criteria (e.g., amplitude range) or pattern recognition (waveform or power spectrum features) is only possible with a certain probability. Different source mechanisms can yield similar amplitudes or waveform features at the AE sensor if waves propagate beyond a material-dependent distance in the FRP composite. 5. Detection and characterization of delaminations in structural elements or structures made from FRP composites reported in the literature mainly comprise laboratory case studies. Standard AE tests on FRP composite elements or structures typically assess the overall integrity without attempts at identifying the type of defect or the microscopic AE source mechanism. The preferred approach is by determination of the Felicity-ratio combined with AE activity and AE intensity analysis and with AE source location. © 2008, Woodhead Publishing Limited


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The data are usually evaluated based on empirical criteria, often proprietary databases. AE testing is very sensitive for identifying areas or volumes of activity, e.g., indicated by spatio-temporal clusters of AE source locations, which shall then be inspected and assessed with other nondestructive test methods.


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Related nondestructive acoustic methods for delamination investigations

Delaminations in FRP composites can also be detected with other nondestructive test (NDT) methods. Acoustic NDT methods are particularly suitable for detection of delaminations. Beside ultrasonic C-scan, there is the tapping test (Hsu et al. 2001), probably the simplest approach, but it requires experience and does not yield test documentation, unless results are recorded in writing or signals recorded with AE sensors (Raju et al. 1993). Electrical impedance testing with a piezoelectric transducer coupled to a structure (Peairs et al. 2004) yields information on changes in the mechanical impedance of the structure, e.g., due to delamination propagation. Model experiments with known location and size of the delamination yielded clear changes in the electrical impedance spectra with increasing delamination size (Barbezat et al. 2006). In thin-walled or plate-like structures, acousto-ultrasonics or guided waves excited with piezoelectric elements can indicate delamination growth in the area of wave propagation when recorded at another location (Barbezat et al. 2007b). As for electrical impedance testing, two signals recorded at different times are compared to detect changes. Identifying defect types and determining their size, however, is difficult with both methods and may require additional techniques. Rough zonal location of the defect similar to that from AE is possible with networks of electrical impedance transducers or acousto-ultrasonic emitter-sensor pairs.

8.8

Summary and outlook

In structures manufactured from fiber-reinforced polymer matrix (FRP) composite materials, impact is one of the main causes for delaminations. Impact damage inside may be much more severe than its appearance on the surface, and detecting delaminations is hence crucial for ensuring safe operation or use of FRP composite elements and structures. Acoustic emission (AE) has been shown to be a useful method for detection of damage accumulation in FRP composites. This is illustrated with examples comprising mainly fracture mechanics specimens and laboratory-scale structural elements. Specifically, AE identification of delaminations requires propagation or growth under the applied load and corresponding AE source location with sufficient resolution. A minimum of two AE sensors is required to yield linear source locations characterizing delamination propagation in specimens, but due to the specific properties of FRP composites, a considerable number of sensors may be needed for monitoring structural elements with sufficient sensitivity and location resolution. AE in-service or field testing for delamination detection in large-scale FRP composite structures also requires consideration of, e.g., noise interference and environmental effects. Often, for FRP composite © 2008, Woodhead Publishing Limited


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structures, AE is successfully used for an assessment of structural integrity indicating damaged areas or volumes instead of attempting detection and detailed characterization of defects, such as delaminations. AE laboratory testing is a mature technology for development of composite materials, comparative delamination resistance characterization, as well as validation and optimization of designs for FRP composite structures.

8.9

Acknowledgments

One of the authors (JB) acknowledges support for testing and data analysis by former doctoral students and colleagues T. Krietsch and L. Qiao (BAM), and J. Chen (NRC, CNRC, Canada). Specimen supply by and discussions with T. Keller and T. Vallée (Composite Construction Lab EPFL, GFRP double lap joints); and by the members of ESIS TC4 on polymers and composites, in particular D.D.R. Cartié (Cranfield University, z-pinned DCB-specimens), as well as materials characterization and data analysis by M. Barbezat and technical support of M. Heusser, B. Jähne, M. Rees, K. Ruf, and D. Völki at Empa for specimen preparation and testing is also gratefully acknowledged by the other author (AJB).

8.10

References

Arul S, Vijayaraghavan L, Malhotra S K, Krishnamurthy R ‘The effect of vibratory drilling on hole quality in polymeric composites’ Intl. J. Machine Tools Manuf. 2006 46(3-4) 252–9. ASNT ‘Acoustic Emission Testing’ in Miller R K, Hill E, Moore P O, ASNT Handbook of Nondestructive Testing, Vol. 6 (3rd edn.), American Society for Nondestructive Testing (2005). ASTM D3039/D3039-M ‘Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials’, American Society for Testing and Materials International 15.03 (2006). ASTM E1067 ‘Standard Practice for Acoustic Emission Examination of Fiberglass Reinforced Plastic Resin (FRP) Tanks/Vessles’, American Society for Testing and Materials International 03.03 (2007). ASTM E1888/E1888M ‘Standard Test Method for Acoustic Emission Examination of Pressurized Containers Made of Fiberglass Reinforced Plastic with Balsa Wood Cores’, American Society for Testing and Materials International 03.03 (2002). ASTM E2076 ‘Standard Test Method for Examination of Fiberglass Reinforced Plastic Fan Blades Using Acoustic Emission’, American Society for Testing and Materials International 03.03 (2005). ASTM E2191 ‘Standard Test Method for Examination of Gas-Filled Filament-Wound Composite Pressure Vessles Using Acoustic Emission’, American Society for Testing and Materials International 03.03 (2002). ASTM Subcommittee E07.04, Work item #12759 ‘Standard Guide for Acoustic Emission Examination of Plate-like and Flat Panel Composite Structures Used in Aerospace Applications’, American Society for Testing and Materials International (2007). © 2008, Woodhead Publishing Limited

274


Awerbuch J ‘On the identification of failure mechanisms in composite laminates through acoustic emission’ in Eisenblätter J, Acoustic Emission, DGM Informationsgesellschaft 1998, 47–58. Barbezat M, Birchmeier M, Santschi Y, Brunner A J ‘Damage detection in fiber-reinforced laminates using impedance testing with piezoelectric Active Fiber Composite elements’, in Bernadou M, Cagnol J, Ohayon R Proceedings 16th International Conference on Adaptive Structures and Technologies ICAST, DESTech Publications, Lancaster 2006 109–16. Barbezat M, Brunner A J, Necola A, Rees M, Gasser Ph, Terrasi G P ‘Fracture behaviour of GFRP laminates with nanocomposite epoxy resin matrix’, in: Saravanos D Proceedings 6th International Symposium on Advanced Composites, COMP-07 2007a, paper #030. Barbezat M, Brunner A J, Huber Ch, Flüeler P ‘Integrated active fiber composite elements: Characterization for acoustic emission and acousto-ultrasonics’, J. Intell. Mat. Syst. Struct. 2007b 15(5) 515–25. Barre S, Benzeggagh M L ‘On the use of acoustic emission to investigate damage mechanisms in glass-fiber-reinforced polypropylene’, Compos. Sci. Technol. 1994 52(3) 369–76. Bohse J ‘Acoustic Emission characteristics of micro-failure processes in polymer blends and composites’, Compos. Sci. Technol. 2000 60(8) 1213–26. Bohse J ‘Acoustic emission examination of polymer matrix composites’, J. Acoustic Emission 2004 22 208–23. Bohse J, Chen J ‘Acoustic emission examination of mode I, mode II and mixed-mode I/ II interlaminar fracture of unidirectional fiber-reinforced polymers’, J. Acoustic Emission 2001 19 1–10. Bohse J, Krietsch T ‘Damage analysis of composite materials by acoustic emission examination’, in Grellmann W and Seidler S, Deformation and Fracture Behaviour of Polymers, Springer Verlag New York, Berlin, Heidelberg 2000 385–402. Bohse J, Krietsch T, Chen J, Brunner A J ‘Acoustic emission analysis and micro-mechanical interpretation of mode I fracture toughness on composite materials’, in Williams J G and Pavan A, Fracture of Polymers, Composites and Adhesives, ESIS Publication 27, Elsevier 2000 15–26. Bourchak M, Farrow I R, Bond I P, Rowland C W, Menan F, ‘Acoustic emission energy as a fatigue damage parameter for CFRP composites’, Intl. J. Fatigue, 2007 29(3) 457–70. Brunner A J, Nordstrom R, Flüeler P ‘A study of acoustic emission-rate behavior in glass fiber-reinforced plastics’, J. Acoustic Emission 1995 13(3-4) 67–77. Brunner A J, Nordstrom R A, Flüeler P ‘Fracture phenomena characterization in FRPcomposites by acoustic emission’, in R. Pick Proceedings European Conference on Macromolecular Physics: Surfaces and Interfaces in Polymers and Composites, European Physical Society 1997 21B, 83–4. Brunner A J, Paradies R ‘Comparison of designs of CFRP-sandwich T-joints for surface effect ships based on acoustic emission analysis from load tests’, in Grant P and Rousseau C.Q. ‘Composite Structures: Theory and Practice’ Special Technical Publication STP 1383, American Society for Testing and Materials International 2000 366-81. Brunner A J, Hack E, Neuenschwander J ‘Nondestructive testing’, in Wiley Encyclopedia of Polymer Science & Technology, 3rd Edition, J. Wiley 2004 27 pp. Brunner A J, Barbezat M ‘Acoustic emission monitoring of delamination growth in fibrereinforced polymer-matrix composites’, in Gdoutos E E ‘Proceedings 16th European © 2008, Woodhead Publishing Limited


275

Conference of Fracture, ECF-16 Fracture of Nano and Engineering Materials and Structures’ 2006 1–8. Brunner A J, Blackman B R K, Davies P ‘A status report on delamination resistance testing of polymer-matrix composites’, Engng. Fract. Mech. 2008 75(9) 2779–94. CARP, Committee on Acoustic Emission from Reinforced Plastics, Aerospace/Advanced Composites Subcommittee ‘Guidance for development of AE applications on composites’, J. Acoustic Emission 1993 11(3) C1–C24. CARP, Committee on Acoustic Emission from Reinforced Plastics, a Division of the Technical Council of The American Society for Nondestructive Testing ‘Recommended practice for acoustic emission evaluation of fiber reinforced plastic (FRP) tanks and pressure vessels’, Draft I, October 1999. Cartié D D R, Brunner A J, Partridge I ‘Effects of mesostructure on crack growth control characteristics in Z-pinned laminates’, in Blackman B R K, Williams J G, Pavan A, Proceedings 3rd ESIS TC4 Conference on Fracture of Polymers, Composites and Adhesives, Elsevier 2003 503–514. CEN/TC138/WG7 WI00138141 ‘Acoustic Emission’, ‘Non-destructive testing – Acoustic Emission – General principles for testing of fibre-reinforced polymers – Methodology and general evaluation criteria’, European Committee for Standardization (2008). Chen F, Bazhenov S, Hiltner A, Baer E, ‘Flexural failure mechanisms in unidirectional glass fibre-reinforced thermoplastics’, Composites 1993 25(1) 11–20. Dzenis Y, Saunders I, ‘On the possibility of discrimination of mixed mode fatigue fracture mechanisms in adhesive composite joints by advanced acoustic emission analysis’, Intl. J. Fract. 2002 117(4) 23–8. Green R E ‘Non-contact ultrasonic techniques’, Ultrasonics 2004 42(1–9) 9–16. Grosse C U, Finck F, Kurz J H, Reinhardt H W ‘Improvements of AE technique using wavelet algorithms, coherence functions and automatic data analysis’, Constr. Build. Mats. 2004 18(3) 203–13. Hamstad M A ‘An examination of acoustic emission evaluation criteria for aerospace type fiber/polymer composites’, in Proceedings Fourth International Symposium on Acoustic Emission from Composite Materials, AECM-4, American Society for Nondestructive Testing, Columbia, Ohio 1992 436–49. Hsu D K, Barnard D J, Peters J J ‘Nondestructive evaluation of repairs on aircraft structures’, in Proceedings SPIE ‘NDE of Materials and Composites’ 2001 4336, 100–7. Hussain F, Hojjati M, Okamoto M, Gorga R E (2006) ‘Polymer-matrix nanocomposites, processing, manufacturing, and application: An overview‘, J. Comp. Mats. 2006 40(17) 1511–75. Karger-Kocsis J, Czignay T ‘Effects of interphase on the fracture and failure behaviour of knitted fabric reinforced composites produced from commingled GF/PP yarn’, Composites Part A – Appl. Sci. Manuf. 1998 29(9–10) 1319–30. Keller T, Vallée T ‘Adhesively bonded lap joints from pultruded GFRP profiles. Part I: Stress-strain analysis and failure modes’, Composites Part B – Eng. 2005 36(4) 331– 40. Kostopoulos V, Tsotra P, Karapappas P, Tsantzalis S, Vavouliotis A, Loutas T H, Paipetis A, Friedrich K, Tanimoto T ‘Mode I interlaminar fracture of CNF or/and PZT doped CFRPs via acoustic emission monitoring’, Compos. Sci. Technol. 2007 67(5) 822–8. Kotsikos G, Evans J t, Gibson A G, Hale J M‚ Environmentally enhanced fatigue damage in glass fibre reinforced composites characterised by acoustic emission’, Composites Part A – Appl. Sci. Manuf. 2000 31(9) 969–77. © 2008, Woodhead Publishing Limited

276


Krietsch T and Bohse J ‘Selection of acoustic emissions and classification of damage mechanisms in fiber composite materials’ in Progress in Acoustic Emission IX, Pace Publication Arts, Anaheim 1998 IV–30 – IV–39. Lachaud F, Lorrain B, Michel L, Barriol L ‘Experimental and numerical study of delamination caused by local buckling of thermoplastic and thermoset composites’, Compos. Sci. Technol. 1998 58(5) 727–33. Martin T, Jones A, Read I, Murray S, Haynes D, Lloyd P, Foote P, Noble R, Tunnicliffe D ‘Structural health monitoring of a carbon fibre structure using low profile piezoelectric, optical and MEMS sensors’, Key Engineering Materials 2001 204–2 371–80. Meo M, Achard F, Grassi M ‘Finite element modelling of bridging micro-mechanics in through-thickness reinforced composite laminates’, Compos. Struct. 2005 71(3–4) 383–7. Mizutani Y, Nagashima K, Takemoto M, Ono K ‘Fracture mechanism characterization of cross-ply carbon-fiber composites using acoustic emission analysis’, NDT&E Intl. 2000 33(2) 101–10. Mizutani Y, Hiratsuka T, Tanabe H, Takemoto M ‘Damage analysis of CFRP plates exposed to cryogenic shock by AE monitoring’, Advanced Composite Materials 2005 14(1) 99–111. Morgan A B ‘Flame retarded polymer layered silicate nanocomposites: a review of commercial and open literature systems’, Polymers for Advanced Technologies 2006 17(4) 206–17. Mouritz A P, Bannister M K, Falzon P J, Leong K H ‘Review of applications for advanced three-dimensional fibre textile composites’, Composites Part A – Appl. Sci. Manuf. 1999 30(12) 1445–61. Nam K W, Ahn S H, Moon C K ‘Fracture behavior of carbon fiber reinforced plastics determined by the time-frequency analysis method’, J. Appl. Polym. Sci. 2003 88(7) 1659–64. Ndiaye I, Maslouhi A, Denault J ‘Characterization of interfacial properties of composite materials by acoustic emission’, Polym. Compos. 2000 21(4) 595–604. Nordstrom R A, Sayir M B, Flüeler P ‘Comparison of AE from glass fiber breaks in bundles and in the single fiber fragmentation test’, in Rogers L M, Tscheliesnig P, Proceedings 22nd European Conference on Acoustic Emission Testing 1996 93–8. Olsson R, Thesken J C, Brandt F, Jonsson N, Nilsson S ‘Investigations of delamination criticality and the transferability of growth criteria’, Compos. Struct. 1996 36(3-4) 221–47. Peairs D M, Park G, Inman D J ‘Improving accessibility of the impedance-based structural health monitoring method’, J. Intell. Mat. Systs. Structs. 2004 15(2) 129–39. Perrot Y, Baley C, Grohens Y, Davies P ‘Damage resistance of composites based on glass fibre reinforced low styrene emission resins for marine applications’, Appl. Compos. Mats. 2007 14(1) 67–87. Raju P K, Patel J R, Vaidya U K ‘Characterization of defects in graphite fiber based composite structures using the acoustic impact technique (AIT)’, J. Test. Eval. 1993 21(5) 377–95. Roos R, Kress G, Barbezat M, Ermanni P ‘Enhanced model for interlaminar normal stress in singly curved laminates’, Compos. Struct. 2007 80(3) 327–33. Shigeishi M, Ohtsu M ‘Acoustic emission moment tensor analysis: development for crack identification in concrete materials’, Build. Constr. Mats. 2001 15(5-6) 311–19. Siddiqui N A, Woo R S C, Kim J K, Leung C C K, Munir A ‘Mode I interlaminar fracture © 2008, Woodhead Publishing Limited


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behavior and mechanical properties of CFRP’s with nano-clay filled epoxy matrix’, Composites Part A – Appl. Sci. Manuf. 2007 38(2) 449–60. Tsamtsakis D, Wevers M ‘Acoustic emission to model the fatigue behaviour of quasiisotropic carbon-epoxy laminate composites’, INSIGHT 1999 41(8) 513–16. Wevers M ‘Listening to the sound of materials: Acoustic emission for the analysis of material behaviour’, NDT&E Intl. 1997 30(2) 99–106. Xiao Y, Ishikawa T ‘Bearing strength and failure behavior of bolted composite joints (part I: Experimental investigation)’, Compos. Sci. Technol. 2005 65(7-8) 1022–31.


Part III Analysis of delamination behaviour from tests


9 Experimental study of delamination in cross-ply laminates A J B R U N N E R, Empa-Swiss Federal Laboratories for Materials Testing and Research, Switzerland

9.1

Introduction

Fiber-reinforced polymer-matrix (FRP) laminates made of a stack of plies or layers are labeled cross-ply, if fiber orientation from one ply or layer to the next alternates in the 0° and 90° direction, respectively. Cross-ply laminates thus constitute a case between unidirectional (UD) laminates with all fibers aligned in the same direction, and laminates with non-orthogonal angle-ply (e.g., ±θ °, e.g., ±30°), or multidirectional plies in the stacking sequence (e.g., quasi-isotropic laminates). In cross-ply, as for all FRP laminates, constituents (fiber and matrix material, additives) and their content, stacking sequence (symmetric or asymmetric with respect to the midplane), number of plies or layers, and layer thickness or number of plies per layer with the same fiber orientation can be varied. Depending on the application, various shapes and sizes (e.g., thickness reduction requiring ply-drop off), modifications (e.g., machining of holes) of the crossply laminate and combination with other components are possible. Together with the processing conditions, a specific choice from the above selections will yield a cross-ply laminate specimen or element whose delamination behavior can be assessed experimentally. With respect to formation of delaminations, their stable propagation or unstable growth in cross-ply laminates, the effects of load-type (e.g., tensile, compressive, biaxial), the rate of load-application (e.g., monotonic or quasistatic, high-rate or impact, fatigue, combined), and of environmental conditions (e.g., constant or varying, elevated or low temperature, exposure to humidity or other gaseous or fluid media) can be investigated. While delamination propagation represents an important damage mechanism in FRP laminates and frequently is responsible for failure, it usually does not occur in isolation. Damage in cross-ply laminates, as in FRP composites in general, involves mechanisms at all length scales, from the molecular level (chemical bonds and physical interaction), to microscopic crack formation and fiber breaks, damage accumulation and growth on the meso-scale and all 281 © 2008, Woodhead Publishing Limited

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the way up to macro-scale. Damage can be induced by mechanical or thermal loads, environmental influences, or result from residual stresses or from combinations such as, e.g., thermomechanical stresses. The range of different mechanisms and scales in damage accumulation in FRP composites poses certain problems for a detailed experimental characterization as well as for modeling of damage or delaminations in cross-ply laminates. From the point of view of applications, there is rather scant literature on cross-ply laminates manufactured from unidirectional plies with alternating fiber orientation. Facings of sandwich elements, see, e.g. (Vaidya et al. 1999; Shiau and Kuo 2004), asymmetric cross-ply laminates used in snapthrough devices or cross-ply laminates for aeroelastic elements (e.g., Qin and Himbrescu 2002; Schultz and Hyer 2003), and cylindrical shells or elements (Shen 2001) are examples. Composites with woven reinforcement in two orthogonal directions are more widely used, e.g. (Le Page et al. 2004; Welsh et al. 2004; Epaarachchi and Clegg 2006). Cross-ply laminates, on the other hand, represent a model laminate to assess the performance of models for mechanical behavior or damage, (e.g., Kyoung et al. 1998; Hallett et al. 1999; Berthelot and Le Corre 2000; Gamstedt and Sjögren 2002; Rebière et al. 2002; Zou et al. 2002; Banks-Sills et al. 2003; Andersons et al. 2008). First, a brief literature review summarizes the current state of the damage behavior of cross-ply laminates, and then experimental methods for delamination characterization are introduced. The results of fracture mechanics round robin tests on cross-ply laminates are then discussed in detail. The chapter ends with a brief note on experimental characterization of delaminations in cross-ply laminate structural elements and parts.

9.2

Summary of current state

There is ample literature on damage mechanisms and their effect on the properties and behavior of FRP cross-ply laminates. Recent reviews (e.g., Lafarie-Frenot et al. 2001; Kashtalyan and Soutis 2002; McCartney 2003) cover various load cases and damage effects in cross-ply laminates while simultaneously relating experiments with models. Damage formation and failure behavior of FRP cross-ply laminates has been investigated for various types of mechanical and thermomechanical loading. Uniaxial tensile loading first results in transverse matrix cracks (see, e.g., Tong et al. (1997) investigating the load rate dependence of transverse crack formation, or Andersons et al. (2008) modeling the experimentally observed behavior). Frequently, the direction of tensile loading coincides with one of the fiber directions (0° or 90°), but off-axis loads have been considered as well (e.g., Belingardi and Cavatorta 2006) observing different damage mechanisms for on-axis and off-axis loading and similarly (Tohgo et al. 2006) comparing static and fatigue on-axis and off-axis loading. © 2008, Woodhead Publishing Limited


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Interlaminar shear strength from Iosipescu testing (Khashaba 2004) or from four-point bending (Feraboli and Kedward 2003), the strain-rate dependence of the interlaminar shear strength (Hallett et al. 1999) and transverse loading with an indenter inducing bending and shear cracks leading to delaminations (Kuboki et al. 2005) are examples of other types of mechanical load applied to cross-ply laminates. The damage sequence from transverse to longitudinal cracks and to delamination initiation has also been investigated and modeled (Rebière et al. 2002). The behavior of cross-ply laminates under compressive load has been compared with other types of layup (e.g., Hsiao et al. 1998; Hosur et al. 2001), both reporting strain rate dependent effects. Buckling of cross-ply laminate plates with cylindrical holes with and without delaminations around the hole under uniaxial compression has been assessed experimentally and modeled (Arman et al. 2006). An extensive numerical post-buckling analysis of cross-ply laminated shells including geometrical imperfections under external pressure and uniaxial compression has been performed, however without validation by experiments (Shen 2001). Separate uniaxial tensile fatigue loading and thermal cycling yielded analogous damage development, i.e., formation of edge cracks (Henaff-Gardin and Lafarie-Frenot 2002). Thermomechanical loading at room-temperature and +100°C was compared for unidirectional and cross-ply laminates, both under static and fatigue loads along the fiber direction and a predictive model for fatigue failure of cross-ply laminates agreed well with experiments (Kawai and Maki 2006). Damage behavior under fatigue loads is important for many applications of FRP composites and has been extensively investigated for cross-ply laminates. Beside the investigations noted above (Kawai and Maki 2006; Tohgo et al. 2006), tension-tension fatigue of cross-ply laminates and the resulting transverse crack formation has been reported by (Kobayashi and Takeda 2002). Similar experiments have been complemented with modeling (Lafarie-Frenot et al. 2001). Interleafing of cross-ply laminates delayed crack propagation under fatigue loading (Meijer and Ellyin 2004) and stitching affected behavior under fatigue but not that under static loading (Aymerich 2004). Fracture mechanics experiments with mode II fatigue shear loading using the end-notch flexure specimen (a three-point bending configuration with a starter crack at midplane on one side of the beam) have also been reported (Beghini et al. 2006). Impact or high-rate compressive loading is a damage mechanism that can induce delaminations in FRP composites. Low-velocity impact (about 0.5 to 3.1 m/s) on symmetric cross-ply laminated plates yielded little differences for different stacking sequences (Mili and Necib 2001). These experiments were also correlated with a simple model yielding reasonable agreement. Varying strain rates in compressive loading of cross-ply specimens with different fibers (glass and carbon) showed a strain-rate dependence for both, © 2008, Woodhead Publishing Limited

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but differences in energy absorption attributed to different mechanisms (Ochola et al. 2004). Comparative impact testing of stitched and unstitched cross-ply laminated plates indicated that stitching reduced the damage area but induced fiber fracture and lowered the penetration resistance (Aymerich et al. 2007). Residual stresses/strains and their effects in cross-ply laminates have been modeled by, e.g., Bank-Sills et al. (2003) for residual stress from curing and transverse cracks, Hobbiebrunken et al. (2004) for initial matrix failure in 90° plies, and Zhang et al. (2004) again for residual stress from curing. Among the effects of exposure to environmental conditions, thermal, thermomechanical and hygrothermal effects have been investigated in crossply laminates. Fatigue loading at elevated temperatures has already been noted above (Kawai and Maki 2006). Aging at +150°C or +200°C reduced the onset stress for transverse cracking (Ogi 2003) while aging at +150°C in air and nitrogen showed less effect in tension after impact testing following storage in nitrogen (Parvatareddy et al. 1996). In tension-tension fatigue at room and various elevated temperatures (+75°C to +150°C) cross-ply laminates yielded higher values than a quasi-isotropic layup, but on the other hand proved more notch-sensitive (Jen et al. 2006). A fracture mechanics-based model predicting the microcracking onset temperature for cryogenic exposure showed reasonable agreement with experiments (Timmerman and Seferis 2004). Low velocity impact at room and low temperatures (–60°C, –120°C) compared unidirectional, cross-ply and quasi-isotropic laminates (Gómezdel Río et al. 2005). It was observed that cooling was equivalent to increasing impact energy. Also, residual stresses affected the behavior of cross-ply and quasi-isotropic, but not that of unidirectional laminates. There is also literature on exposure to humidity or fluid media, and to ultraviolet radiation. A shear lag model described experimental trends for delamination formation under fatigue loading of dry, saturated and immersed cross-ply laminates (Selvarathinam and Weitsman 1999). Hygrothermal aging of different woven cross-ply laminate specimens with machined or molded edges yielded a complex behavior in tensile tests with decreasing or increasing properties depending on the specific combination of moisture and temperature (Cândido et al. 2006). Trends of flexural strength after various combinations of UV and moisture exposure have been identified, however, with limited statistical significance (Nakamura et al. 2006). Depending on the kind of environmental exposure, matrix and fiber types are probably more relevant for damage resistance than the use of a specific layup (e.g., cross-ply), for example in space applications (Felbeck 1995). In cross-ply beam specimens quasistatic tensile loading along the 0° fiber orientation first results in the formation of transverse matrix cracks in the plies oriented perpendicular to the load-axis (90° plies). Once the number of transverse matrix cracks per unit length has reached saturation, e.g., indicated by a drop in the acoustic emission activity recorded during the test, other © 2008, Woodhead Publishing Limited


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damage mechanisms become active leading to formation of longitudinal cracks (parallel to the loading direction), progressive matrix splitting, and, finally, to failure (Brunner et al. 1997). This damage sequence can also involve a certain amount of delamination (e.g. Dharani et al. 2003). Acoustic emission analysis has proven useful for characterizing the delamination behavior and damage mechanisms in cross-ply laminates (Prosser et al. 1995; Mizutani et al. 2000; Romhany et al. 2006). The failure sequence observed in experimental testing of cross-ply laminates is quite complex, and may be influenced by edge (e.g., Prosser et al. 1995; Kobayashi and Takeda 2002; Cândido et al. 2006; Stevanovic et al. 2006; Zhang et al. 2006) and notch effects (e.g. Liu et al. 2002) or interaction between damage types (e.g., Ogihara and Takeda 1995). Even though there seems to be reasonable agreement between damage models and experiments for FRP cross-ply laminates (Lafarie-Frenot et al. 2001; Kashtalyan and Soutis 2002; McCartney 2003), certain details of delamination formation are still debated, see e.g., Lim and Li (2005).

9.3

Experimental methods for studying delaminations

Experimental methods applicable for studying delaminations in cross-ply laminates are no different from the methods used for other types of FRP laminates (see corresponding chapters in this book). Analysis of the behavior under mechanical or combined thermomechanical loading can yield indications of delamination initiation and propagation in the measured response, e.g. stiffness or compliance. The same holds for mechanical or thermomechanical load tests on structural elements made from FRP cross-ply laminates. With respect to detection and sizing of delaminations, the same methods that are applicable to FRP composites, i.e., the methods described in Part II Delamination: Detection and Characterization, (Brunner et al. 2004) are also applicable to FRP cross-ply laminates. One approach to experimentally determine the delamination behavior of composite materials with respect to delaminations is the use of fracture mechanics tests. In principle, this yields a quantitative measure of delamination resistance under various types of loads or load combinations, e.g., in the form critical energy release rate or fracture toughness (Davies et al. 1998). Such data can be used for qualitative comparison of laminates or for designing composite elements. While a number of test procedures have been standardized, and others are under development, most test scopes specify that the procedures are applicable to UD laminates only (Brunner et al. 2008). Fracture mechanics of cross-ply laminates has been studied, among others, by (Bazhenov 1995) determining inter- and intralaminar fracture values from tensile mode I tests, (La Saponara et al. 2002) correlating mode I tests with a finite element © 2008, Woodhead Publishing Limited

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model, however neglecting mixed-mode phenomena (crack branching) observed in the experiments, and (Beghini et al. 2006) performing mode II fatigue tests and noting that they resulted in a Paris type law with no large difference between different stacking sequences.

9.4

Fracture mechanics study of delamination in cross-ply laminates

9.4.1

Introduction

A comprehensive study has been undertaken by a technical committee of the European Structural Integrity Society to investigate the applicability of the standardized tensile opening (mode I) load test method for fracture toughness or delamination resistance of unidirectional FRP (ISO 2001) to cross-ply laminates. The applicability of this procedure to cross-ply laminates has to be investigated, since experiments had indicated the mode I test method, as well as tests in other modes, were not always applicable to multi-directional laminates because of crack branching, multiple cracking, or deviation from the midplane (Choi et al. 1999; Pereira and de Morais 2004).

9.4.2

Test specimens and test parameters

The test specimens and test parameters are defined in the respective standard or draft procedures (Brunner et al. 2008). The test results reported here are for the quasistatic, opening tensile mode I load test using carbon-fiber reinforced polymer (CFRP) double cantilever beam (DCB) specimens with nominal dimensions of 150–200 mm × 20 mm × 3–4 mm. A minimum length of 125 mm, a width of 20 mm and a total thickness of 3 and 5 mm, for CFRP and glass-fiber reinforced polymer (GFRP), respectively, are recommended (ISO 2001). Effective dimensions have to be determined with a defined precision, and load-rates, i.e., cross-head speeds are prescribed to be in the range between 1 and 5 mm/min. Tests reported in the literature may deviate from these prescriptions or recommendations, potentially affecting the validity of the data. This can be attributed to a number of effects, e.g. unsuitable dimensions, in particular of the starter delamination, typically a polymer film with less than 13 mm thickness (ISO 2001), quoting values without distinguishing between initiation and propagation (R-curve effects), or neglecting correction factors and environmental effects (mainly humidity). As shown in (Brunner 2006) the scatter in fracture mechanics data reported in the literature for CFRP was reduced considerably, once the standardized test procedures were followed. The data presented in this section are all from round robin testing of Technical Committee 4 on Polymers and Composites of ESIS. Data analysis © 2008, Woodhead Publishing Limited


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was performed with a spreadsheet based routine (Brunner et al. 1994) incorporating the data analysis and criteria of the ISO test procedure (ISO 2001). Partial results of the round robin tests on cross-ply laminates have been published by (Blackman et al. 1998; de Morais et al. 2002; Brunner and Blackman 2003).

9.4.3

Data analysis and results for T300/970 carbon fiber epoxy

The first round robin was on a CFRP laminate with epoxy matrix (T300/ 970). Four layup configurations were considered: 1. 2. 3. 4.

symmetric layup [0°/90°]6s asymmetric layup [0°/90°]12 woven fabric layup UD laminate for comparison.

All specimens contained an insert film as starter crack. The nominal thickness of this film was 25 µm. Note that this is greater than that recommended in ISO (2001). A second round robin with a limited number of laboratories used the same material and compared unidirectional specimens with thick (25 µm) and thin (12 µm) insert films with a symmetric cross-ply layup with thin insert. Figure 9.1 shows the results as obtained in the first round robin by six laboratories (sets 1 and 2 were tested in the same laboratory as separate sets of four specimens each, the others are sets of five specimens each). Most laboratories reported the initiation value of GIC corresponding to the maximum load value (MAX), one laboratory based it on the 5%-compliance increase for those specimens for which this value was lower than the MAX (as prescribed by ISO 2001). Yet another laboratory used that value of load at which a nonlinearity is seen in the load-displacement curve. Figure 9.1 clearly shows the differences among the GIC-values versus delamination lengths (so-called Rcurve) for the different layups. The symmetric cross-ply yields the lowest average initiation and propagation values (313 ± 65 and 258 ± 47 J/m2, respectively) among the cross-ply laminates, the asymmetric the highest (396 ± 85 and 443 ±93 J/m2, respectively with one laboratory yielding clearly lower propagation values than the others), and the woven fabric values inbetween (352 ± 68 and 322 ± 74 J/m2, respectively). The only set of data available for the corresponding unidirectional laminate with a 25 mm thick insert (Fig. 9.1) yields average initiation from four specimens tested just below 300 J/m2 with average propagation values at 388 ± 34 J/m2 and average maximum propagation at 473 ± 61 J/m2, i.e., a clearly rising R-curve behavior. Observations of unstable stick-slip delamination growth were reported in © 2008, Woodhead Publishing Limited

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several of the tested specimens (Fig. 9.2). These are reflected by the average propagation values that are typically lower than the average initiation values. For the asymmetric layup, several data sets show the typical ‘rising’ R-curve behavior (initiation lower than average propagation), but there, at least one laboratory reported observation of delamination propagation deviating from the midplane, i.e., a delamination oscillating between the neighboring 0°800 700

GIC [J/m2]

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9.1 Critical energy release rates GIC of carbon-fiber epoxy (T300/970) cross-ply and unidirectional laminates obtained from mode I opening tests of four to five specimens per laboratory; data from six laboratories testing seven sets of specimens (sets 1 and 2 tested at the same laboratory by different operators); shown are averages and standard deviations for (a) symmetric layup, (b) asymmetric layup, (c) woven fabric layup, (d) unidirectional layup; left column (light gray) initiation, center column (dotted) average propagation, right column (dark gray) maximum propagation values for each set.



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4 (c)

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GIC [J/m2]

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9.1 (Continued)

plies. Since the unidirectional specimens with thin insert from the second round robin also show unstable stick-slip (Figs 9.2, 9.3), it is unlikely that this behavior is due to the thick insert used in the first round robin, but seems typical for this combination of fiber (T300) and epoxy resin (970). Since stick-slip behavior was typical for this material, no attempt was made to reanalyze the data and to eliminate the arrest values which, as shown in Fig. 9.2 may slightly change the GIC-propagation values. In most specimens, elimination of arrest points would leave only about half of the recorded data points for the analysis, usually less than the recommended number of 10–15 (ISO 2001). However, as long as the data points are still distributed over the full range of delamination lengths which is the case here, the results do not vary much as indicated in Fig. 9.2. This is in agreement with observations by Brunner et al. (1994) made on unidirectional FRP laminates. © 2008, Woodhead Publishing Limited

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Comparing the results for cross-ply and unidirectional carbon-fiber epoxy (T300/970) laminates with thin insert to the results from the first round robin with thick inserts (Figs 9.1 and 9.3), the GIC-values for the symmetric layup seem to increase when testing from a mode I precrack (average initiation 396 ± 46 J/m2, average propagation 408 ± 79 J/m2 and maximum propagation 576 ± 118 J/m2). The same holds for initiation values of the unidirectional laminate (average initiation 402 ± 25 J/m2). Average propagation values of

500 450 400

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40 60 80 Delamination length [mm] (a)

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0

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40 60 80 Delamination length [mm] (b)

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GIC [J/m2]

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9.2 Critical energy release rates GIC of carbon-fiber epoxy (T300/970) cross-ply and unidirectional laminates obtained from mode I tensile tests; shown are two specimens each plotted as a function of delamination length (so-called R-curves) for (a) symmetric cross-ply layup with thick insert from laboratories 1 and 4, (b) the same data reanalyzed after elimination of arrest values, (c) unidirectional layup with thin insert from laboratory 1, and (d) the same data reanalyzed after elimination of arrest values, the distinct ‘zig-zag’-variation in GIC indicates unstable stick-slip behavior.



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GIC [J/m2]

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40 60 80 Delamination length [mm] (c)

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40 60 Delamination length [mm] (d)

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100

9.2 (Continued)

the unidirectional laminate are affected by unstable stick slip and lower than initiation values (277 ± 17 J/m2). Overall, even though the inter-laboratory comparison seems to indicate reasonable agreement between laboratories for the carbon-fiber epoxy (T300/ 970) cross-ply laminates tested, the observations of unstable delamination growth and of delaminations deviating from the mid-plane raise doubts about the validity of the data. The standard procedure (ISO 2001) notes that deviation of the crack from the midplane will invalidate results; and arrest values, i.e., data points taken after unstable delamination growth and before reloading to propagation, shall be eliminated from the analysis. This will be investigated in more detail in the next section; the results in Figs 9.1 and 9.3 include all data points as reported. To which extent a delamination path oscillating between adjacent 0° plies does violate assumptions essential for data analysis, is not clear at this point. Delaminations permanently deviating from the midplane or even branching into two ore more delaminations propagating in parallel can not be analyzed with the standard procedures (Choi et al. 1999). © 2008, Woodhead Publishing Limited

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9.4.4


Data analysis and results for IM7/977-2 carbon fiber epoxy

The results of a third round robin with laminates made from a different CFRP epoxy (IM7/977-2) are shown in Fig. 9.4. This laminate was expected to yield more stable delamination propagation than that tested previously. Nevertheless, occasional unstable delamination propagation was observed in 800 700

GIC [J/m2]

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9.3 Critical energy release rates GIC of carbon-fiber epoxy (T300/970) cross-ply and unidirectional laminates obtained from mode I tensile tests of four to five specimens per laboratory; shown are averages and standard deviations for (a) symmetric layup with thick insert identical to top left of Fig. 9.1, (b) unidirectional layup with thick insert identical to bottom right of Fig. 9.1, (c) symmetric layup with thin insert, (d) unidirectional layup with thin insert; left column (gray) initiation, center column (dotted) average propagation, right column (dark gray) maximum propagation values for each set.



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9.3 (Continued)

some specimens, even for unidirectional layup at several laboratories. For this laminate, a few laboratories determined initiation values both, from the insert and from the resulting mode I precrack. The data analysis (Fig. 9.4) clearly shows R-curve effects for the values determined from the mode I precrack. From the insert, average initiation yields 325 ± 124 J/m2 and 411 ± 161 J/m2 for the symmetric and asymmetric layup, respectively, while initiation from the mode I precrack yields 716 ± 360 J/m2 and 540 ± 152 J/m2. It has to be noted that all standard deviations are relatively large, indicating considerable scatter within each laboratory. For the tests from the precrack, average and maximum propagation values for the symmetric and asymmetric layup, respectively, do differ by much less than one standard deviation (959 ± 211 J/m2 and 1318 ± 233 J/m2 for the symmetric and 980 ± 198 J/m2 and 1249 ± 186 J/m2 for the asymmetric layup). The causes of the large scatter will be investigated in more detail below. For the unidirectional © 2008, Woodhead Publishing Limited

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laminate (Fig. 9.5), no significant difference between testing from the insert or from the mode I precrack is observed (average initiation 283 ± 37 J/m2 and 299 ± 34 J/m2, respectively, i.e., with standard deviations of 11 and 13%). Another observation was that the mode I delamination path in the 90°plies viewed on the edge of the specimens was ‘oscillating’ between the adjacent 0°-plies (Fig. 9.6). Several specimens showed fiber bridging in the 1800 1600

GIC [J/m2]

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9.4 Critical energy release rates GIC of carbon-fiber epoxy (IM7/977-2) cross-ply laminates obtained from mode I tensile tests of four to five specimens per laboratory; data from five laboratories each testing one set of specimens; shown are averages and standard deviations for (a) symmetric layup tested from the insert, (b) asymmetric layup tested from the insert, (c) symmetric layup tested from the mode I precrack, (d) asymmetric layup tested from the mode I precrack; left column (gray) initiation, center column (dotted) average propagation, right column (dark gray) maximum propagation values for each set.



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9.4 (Continued)

90°-plies transverse to the direction of delamination propagation. ‘Amplitude’ and ‘wavelength’ of the oscillatory delamination propagation vary with layup, the symmetric with a thicker 90°-layer yielding larger values than the asymmetric (Brunner and Flüeler 2005). Visual inspection of the fracture surfaces of the specimens after mode I testing showed distinct changes in some specimens (Fig. 9.7). The changes could be attributed to a deviation of the delamination into the neighboring 0°-plies instead of the oscillatory behavior creating regular ‘hillock’-type topography. The photographs in Fig. 9.7 show that, at least in some cases, the deviation only takes place on part of the fracture surface, and may not be noticeable when observing the delamination on the edge of the specimen. Further data analysis showed that the permanent deviation into the 0°-plies, even on part of the fracture surfaces yields R-curves with decreasing propagation values with increasing delamination lengths (Fig. 9.8). Such © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites 800 700

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9.5 Critical energy release rates GIC of carbon-fiber epoxy (IM7/977-2) unidirectional laminates obtained from mode I tensile tests of four to five specimens per laboratory for comparison with the cross-ply laminates shown in Fig. 9.4 (note the reduced scale); data from three to four laboratories each testing one set of specimens; shown are averages and standard deviations for (a) unidirectional layup tested from a thin insert, (b) unidirectional layup tested from a mode I precrack; left column (gray) initiation, center column (dotted) average propagation, right column (dark gray) maximum propagation values for each set.

deviations have been observed for the symmetric and asymmetric layup, and the propagation values may drop to a level comparable to that of the unidirectional laminate. Figure 9.9 shows results after identifying and eliminating specimens with delaminations deviating permanently from the midplane based on R-curve behavior as shown in Fig. 9.8. In a second step, arrest values were eliminated from the remaining specimens. Identification of arrest values was based on recorded load curves (if available) or schematic load-displacement plots © 2008, Woodhead Publishing Limited


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(a)

(b)

9.6 Photographs of the delamination path under mode I opening loading of double cantilever beam cross-ply specimens made from carbon-fiber epoxy (IM7/977-2) with (a) symmetric and (b) asymmetric layup; note the oscillating delamination path and the transverse fiber bridging (normal to the direction of delamination propagation) in the 90° plies at the center of the beam (photographs courtesy of Dr D. D. R. Cartié, Cranfield University).


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(a)

(b)

9.7 Photographs of the fracture surfaces after mode I opening loading of cross-ply specimens made from carbon-fiber epoxy (IM7/ 977-2) with (a) symmetric and (b) asymmetric layup; note the increasing portion of less corrugated, shiny surface appearing with increasing delamination length indicating deviation into the adjacent 0° plies (insert on left hand side of each specimen); for (a), deviation starts at the insert and soon covers the whole width, for (b) the deviation starts beyond the tip of the insert and does not reach the edge of the specimen (photographs courtesy of Mr C. J. Murphy, Imperial College).



299

derived from measured data. The analysis without specimens with delaminations deviating from the midplane yielded average initiation values of 667 ± 291 J/m2 and 504 ± 127 J/m2, average propagation values of 1087 ± 171 J/m2 and 1109 ± 94 J/m2, and maximum propagation values around 1489 ± 196 J/m2 and 1348 ± 99 J/m2 for the symmetric and asymmetric layup, respectively. Both cross-ply layups show a strongly rising R-curve with comparable propagation values. The initiation values differ even from 1600 1400 1200

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9.8 Critical energy release rates GIC of carbon-fiber epoxy (IM7/977-2) cross-ply and unidirectional laminates obtained from mode I tensile tests from two laboratories; shown are two specimens each plotted as a function of delamination length (so-called R-curves) for (a) symmetric cross-ply layup from laboratory 1, (b) symmetric crossply layup from laboratory 4, (c) asymmetric layup from laboratory 1, and (d) unidirectional layup from laboratory 4; note that of the two R-curves in each graph for cross-ply layup, one shows a trend toward lower propagation values with increasing delamination lengths, reaching values comparable to those of the unidirectional layup (d) at least in one case (c).


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GIC [J/m2]

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1400 1200 1000 800 600 400 200 0

9.8 (Continued)

one laboratory to the other; this is attributed to R-curve effects from precracking. It is expected that testing from the insert would yield similar values for initiation among the laboratories and for the different cross-ply layups. Eliminating the arrest values from the remaining specimens in which the delamination did not deviate into the 0°-plies did not change the GIC propagation values significantly, as expected from the arrest analysis for the T300/970 cross-ply laminate (Fig. 9.2). Average initiation values were 753 ± 327 J/m2 and 499 ± 124 J/m2, average propagation of 1121 ± 130 J/m2 and 1082 ± 80 J/m2, and maximum propagation values 1457 ± 209 J/m2 and 1345 ± 97 J/m2 for the symmetric and asymmetric layup, respectively. However, scatter in the propagation values is reduced.

9.5

Discussion and interpretation

A dependence of the strain energy release rate on the fiber orientation at the delaminating interface of carbon-epoxy double cantilever beam (DCB) © 2008, Woodhead Publishing Limited


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specimens under mode I opening tensile load was investigated experimentally and probably first noted by (Laksimi et al. 1991). Whether the standard fracture mechanics tests developed for unidirectional CFRP and GFRP laminates are really applicable to FRP cross-ply laminates made from the same FRP can be discussed at different levels. As shown above, the delamination can partly or fully deviate into the adjacent 0°-plies. At the interior of the DCB-specimen, this is not noticeable by visual observation along the edge. Noticing these deviations requires 1800 1600

GIC [J/m2]

1400 1200 1000 800 600 400 200 0 1

2

3 (a)

4

5

1

2

3 (b)

4

5

1800 1600

GIC [J/m2]

1400 1200 1000 800 600 400 200 0

9.9 Critical energy release rates GIC of carbon-fiber epoxy (T300/970) cross-ply laminates obtained from mode I tensile tests of two to three specimens per laboratory; shown are averages and standard deviations for (a) symmetric layup, specimens with delamination deviating into the 0° plies eliminated, (b) asymmetric layup, specimens with delamination deviating into the 0° plies eliminated, (c) symmetric layup, arrest values also eliminated, (d) asymmetric layup, arrest values also eliminated; left column (gray) initiation, center column (dotted) average propagation, right column (dark gray) maximum propagation values for each set.


302


GIC [J/m2]

1400 1200 1000 800 600 400 200 0 1

2

3 (c)

1

2

3 (d)

4

5

1800 1600 1400

GIC [J/m2]

1200 1000 800 600 400 200 0 4

5

9.9 (Continued)

visual inspection of the fracture surface after the test (Fig. 9.7). Interestingly, if the mode I delamination path increasingly deviates into the adjacent 0°plies, the data analysis yields a decreasing R-curve with increasing delamination length (Fig. 9.8). However, even if specimens with delamination deviating into 0°-plies and arrest values in the R-curves of the remaining specimens are excluded, the results for the cross-ply laminates could be formally considered invalid, since the delamination is oscillating between adjacent 0°-plies which constitute deviations from the midplane (ISO 2001). The oscillations in the delamination path inside the plies with 90° fiber orientation may formally invalidate the materials data (delamination resistance) but further raise the question whether they represent a pure interlaminar mode I delamination behavior. An extensive discussion based on experiments using the carbon-fiber epoxy (IM7/977-2) of the round robin presented above, complemented with finite element simulations concluded that in spite of the observed oscillation of the delamination path the corrected beam theory © 2008, Woodhead Publishing Limited


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analysis (ISO 2001) seemed applicable (de Morais et al. 2002). This resulted in higher GIC values for both, delamination initiation and propagation. Another conclusion was that the total intralaminar energy release rate G was lower than the corresponding interlaminar G (by about 30%). The authors also note that the measured interlaminar GIC-values might be affected by periodic occurrence of transverse cracking causing the stick-slip instabilities that were clearly observed. A different analysis argued that the observed oscillatory delamination path might, at least in part, be due to mixed-mode I/II effects which would result in an increase in apparent GIC compared with unidirectional fiber-reinforced laminates (Brunner and Flüeler 2005). One argument in support of this hypothesis is the amount of additional fracture surface created by the oscillatory path of the delamination in cross-ply laminates. The increase in fracture surface amounts to between 10 and 20%, essentially independent of the approach used for quantitatively estimating it (Brunner and Flüeler 2005). It is difficult to reconcile the observed increase in apparent GIC of around 100% with this modest increase in delamination surface while maintaining that the behavior is dominated by interlaminar mode I delamination propagation. Separating the assumed mixed-mode I/II interlaminar delamination into the respective components is difficult, and intra-laminar delamination provides another possible contribution (de Morais et al. 2002; Brunner and Flüeler 2005). If delamination in cross-ply laminates is regarded as a case of bi-material fracture (as suggested in a comment by a reviewer), separating the contributions from different modes would also be difficult. It is, however, questionable whether this point of view is applicable because of the oscillatory delamination path within the 90° plies. The modest increase in peak load observed in the mode I tests of cross-ply laminates (about +10%) compared with the unidirectional laminate would be compatible with a slight increase in initiation values and also point to propagation effects as a cause for the significant increase in GIC, i.e., the rather steep R-curve behavior of crossply laminates. Independent of the assessment of the validity of the corrected beam theory analysis for the mode I delamination of cross-ply laminates and of the resulting GIC, there is no reason to doubt that unidirectional laminates yield conservative mode I data, and this provides the motivation for the development of test procedures for this rather artificial type of laminate (Davies et al. 1998). In the opinion of the author, it will prove necessary to look into alternative approaches such as the J-integral (see, e.g., Schapery et al. 1986; de Morais et al. 2002; Romhany et al. 2006), maybe complemented with nondestructive test methods for the determination of delamination length or size (Romhany et al. 2006) for quantitative determination of delamination resistance for laminates with cross-ply and multi-directional fiber-orientation. This might apply to non-orthogonal angle-ply laminates as well, but has not been investigated in detail yet. © 2008, Woodhead Publishing Limited

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Depending on the intended use, determination of reliable delamination resistance values of fiber-reinforced cross-ply laminates, e.g. for modeling and design, may require the development of standardized test methods capable of dealing with the complex behavior, involving delamination propagation oscillating around the midplane. Delamination resistance for mode I opening load calculated according to (ISO 2001) seems to indicate relatively high GIC values for cross-ply compared with unidirectional composites. However, due to their questionable validity, these values should be treated as ‘apparent’ delamination resistance and used with due caution in designs.

9.6

Structural elements or parts with cross-ply laminates

At best, fracture mechanics data from cross-ply laminate specimens may yield information for a relative comparison of delamination resistance of different fiber-matrix systems. As detailed above, the apparent G values are of questionable validity and probably not suitable for design purposes. Experimental investigations and empirical formulations based thereon offer a viable alternative for studying the delamination resistance of structural elements or parts made from or using cross-ply laminates. In this case, the primary focus is not on the explicit detection of delaminations and the quantitative characterization of their behavior. Structural integrity and overall damage behavior involving a sequence of interacting damage mechanisms can be characterized by suitable load tests. Experimental investigations on structural elements can be monitored with nondestructive test methods (see Part II Delamination: Detection and Characterization, Brunner et al. 2004) and, of course, be complemented by modeling or simulation. Experimental investigations of structural elements with FRP cross-ply laminates are reported for different types of elements. Buckling behavior is one issue for cross-ply FRP thin shells, thin or thick plates, or sandwich elements (e.g. Kyoung et al. 1998; Jones 2005; Arman et al. 2006), behavior under impact another (e.g., Vaidya et al. 1999; Vaidya and Hosur 2003; Sanchu-Saez et al. 2005). Some applications of cross-ply laminates for structural elements had already been noted in the introduction, e.g. facings of sandwich elements and snapthrough or aeroelastic devices (Vaidya et al. 1999; Qin and Librescu 2002; Schultz and Hyer 2003). Other parts or structural elements with FRP crossply laminates that have been investigated include, e.g., an asymmetrically loaded cantilever structure (Shan and Pelegri 2003), cross-ply joints (Aymerich 2004), a cross-ply CFRP cylinder under compression (Nilsson et al. 2001), or cross-ply woven roving elements (Epaarachchi and Clegg 2006).



9.7

305

Summary and outlook

Delamination initiation and stable propagation or unstable growth is one mechanism in a sequence of damage mechanisms occurring in cross-ply laminates under mechanical or thermomechanical loading to sufficiently high levels. Delaminations can be initiated by impact as in other composite laminates, but also arise from transverse matrix cracking due to normal or off-axis loads on plies with fiber orientation differing from the load axis. A quantitative, experimental determination of the delamination resistance of cross-ply laminate specimens is considered difficult based on extensive round robin testing. Measured delamination resistances do seem to be significantly affected by R-curve effects whose origin is debatable. It is hence doubtful whether fracture mechanics testing can yield useful data for design purposes. It may even be questionable whether comparative delamination resistance testing of different cross-ply laminates based on quantitative determination of ‘apparent’ GIC–values is feasible. Published examples of engineering applications of cross-ply laminates do seem to be rather scant and the damage behavior of structural elements with cross-ply laminates, including formation and propagation of delaminations, is probably best assessed experimentally, or with experiments complemented by modeling.

9.8

Acknowledgments

Contributions by members of Technical Committee 4 of the European Structural Integrity Society to the fracture mechanics round robin results, notably by B. R. K. Blackman and C. J. Murphy (Imperial College), by D. D. R. Cartié (Cranfield University), A. B. de Morais (University of Aveiro), A. Pavan (Politecnico di Milano), M. v. Alberti (MPA Stuttgart), L. Warnet and E. Van de Ven (University of Twente), as well as material supply by ICI Fiberite (through D. R. Moore), and technical support for testing at Empa by M. Heusser, B. Jähne, K. Ruf and D. Völki, as well as partial support by the Federal Institute for Materials Research and Testing (BAM) in Berlin during a sabbatical is gratefully acknowledged.

9.9

References

Andersons J, Joffe R, Spārninš E ‘Statistical model of the transverse ply cracking in cross-ply laminates by strength and fracture toughness based failure criteria’ Engng. Fract. Mech. 2008, 75(g) 2651–65. Arman Y, Zor M, Aksoy S ‘Determination of critical delamination diameter of laminated composite plates under buckling loads’ Compos. Sci. Technol. 2006, 66(15) 2945–53. Aymerich F ‘Experimental investigation into the effect of edge stitching on the tensile strength and fatigue life of co-cured joints between cross-ply adherends’ Adv. Compos. Lett. 2004, 13(3) 151–61.


306


Aymerich F, Pani C, Priolo P ‘Damage response of stitched cross-ply laminates under impact loadings’ Engng. Fract. Mech. 2007, 74(4) 500–14. Banks-Sills L, Boniface V, Eliasi R ‘Effect of residual stresses on delamination cracks in fiber reinforced composites’ Interface Science 2003, 11(3) 339–48. Bazhenov S L ‘Interlaminar and intralaminar fracture modes in 0/90 cross-ply glass/ epoxy laminates’ Composites 1995, 26(2) 125–33. Beghini M, Bertini L, Forte P ‘Experimental investigation on the influence of crack front to fiber orientation on fatigue delamination growth rate under mode II’ Compos. Sci. Technol. 2006, 66(2) 240–7. Belingardi G, Cavatorta M P, ‘Bending fatigue stiffness and strength degradation in carbon-glass/epoxy hybrid laminates: Cross-ply vs. angle-ply specimens’ Intl. J. Fatigue 2006, 28(8) 815–25. Berthelot J-M, Le Corre J-F ‘A model for transverse cracking and delamination in crossply laminates’ Compos. Sci. Technol. 2000, 60(14) 1055–66. Blackman B R K, Brunner A J ‘Mode I Fracture Toughness Testing of Fibre-Reinforced Polymer Composites: Unidirectional versus Cross-ply Lay-up’ in: Brown M W, de los Rios E R, Miller K J ‘Proc. 12th European Conference on Fracture (ECF-12): Fracture from Defects, Volume III, EMAS Publ. 1998, 1471–6. Brunner A J ‘Fracture testing of polymer-matrix composites’ in: Proc. 10th Portuguese Conference on Fracture, University of Minho 2006, 11 pp. Brunner A J, Blackman B R K ‘Delamination fracture in cross-ply laminates: What can be learned from experiment?’ in: Blackman B R K, Williams J G, Pavan A ‘Proc. 3rd ESIS Conference, Fracture of Polymers, Composites and Adhesives II’ ESIS Publication No. 32 Elsevier 2003, 433–44. Brunner A J, Flüeler P ‘Prospects in fracture of “engineering” laminates’ Engng. Fract. Mech. 2005, 72(6) 899–908. Brunner A J, Tanner S, Davies P, Wittich H ‘Interlaminar fracture testing of unidirectional fibre-reinforced composites: results from ESIS round robins’ in: Hogg P J, Schulte K, Wittich H Proc. European Conference on Composite Materials: Composites testing and Standardization, Woodhead Publ. 1994, 523–32. Brunner A J, Nordstrom R A, Flüeler P ‘Fracture Phenomena characterization in FRPcomposites by acoustic emission’ in: Pick R Proc. European Conference on Macromolecular Physics: Surfaces and Interfaces, European Physical Society 1997, 21B 83–84. Brunner A J, Hack E, Neuenschwander J ‘Nondestructive Testing’, in: Wiley Encyclopedia of Polymer Science & Technology, 3rd Edition, J. Wiley 2004, 27 pp. Brunner A J, Blackman B R K, Davies P ‘A status report on delamination resistance testing of polymer-matrix composites’, Engng. Fract. Mech. 2008, 75(g) 2779–94. Cândido G M, Leali Costa M, Cerqueira Rezende M, de Almeida S F M, ‘Hygrothermal and stacking sequence effects on carbon epoxy composites with molded edges’ PolymerPlastics Technol. Engng. 2006, 45(10) 1109–15. Choi N S, Kinloch A J, Willams J G ‘Delamination fracture of multidirectional carbonfiber/epoxy composites under mode I, mode II and mixed mode I/II loading’ J. Compos. Mater. 1999, 33(1) 73–100. Davies P, Blackman B R K, Brunner A J ‘Standard test methods for delamination resistance of composite materials: current status’ Appl. Compos. Mats. 1998, 5(6) 345–64. de Morais A B, de Moura M F, Marquez A T, de Castro P T ‘Mode I interlaminar fracture of carbon/epoxy cross-ply laminates’ Compos. Sci. Technol. 2002, 62(5) 679–86. Dharani L R, Wei J, Ji F S, Zhao J H ‘Saturation of transverse cracking with delamination © 2008, Woodhead Publishing Limited


307

in polymer cross-ply composite laminates’ Intl. J. Damage Mechanics, 2003, 12(2) 89–114. Epaarachchi J A, Clegg R ‘An experimental investigation of the properties of cross-ply laminate used for manufacturing of small aircraft components’ Compos. Struct. 2006, 75(1-4) 93–9. Felbeck D K, ‘Toughened graphite-epoxy composites exposed in near-earth orbit for 5.8 years’ Journal of Spacecraft and Rockets 1995, 32(2) 317–23. Feraboli P, Kedward K T ‘Four-point bend interlaminar shear testing of uni- and multidirectional carbon/epoxy composite systems’ Compos. Part A – Appl. Sci. Manuf. 2003, A34(11) 1265–71. Gamstedt E K, Sjögren B A ‘An experimental investigation of the sequence effect in block amplitude loading of cross-ply composite laminates’ Intl. J. Fatigue 2002, 24(2–4) 437–46. Gómez-del Río T, Zaera R, Barbero E, Navarro C ‘Damage in CFRPs due to low velocity impact at low temperature’ Compos. Part B – Engng. 2005, 36(1) 41–50. Hallett S R, Ruiz C, Harding J, ‘The effect of strain rate on the interlaminar shear strength of a carbon/epoxy cross-ply laminate: comparison between experiment and numerical prediction’ Compos. Sci. Technol. 1999, 59(5) 749–58. Henaff-Gardin C, Lafarie-Frenot M C ‘Specificity of matrix cracking development in CFRP laminates under mechanical or thermal loadings’ Intl. J. Fatigue 2002, 24(2-4) 171–7. Hobbiebrunken T, Hojo M, Fiedler B, Tanaka M, Ochiai S, Schulte K ‘Thermomechanical analysis of micromechanical formation of residual stresses and initial failure in CFRP’ JSME Intl. J. Series A – Solid Mech. Mat. Engng. 2004, 47(3) 349–56. Hosur M V, Alexander J, Vaidya U K, Jeelani S ‘High strain rate compression response of carbon/epoxy laminate composites’ Compos. Struct. 2001, 52(3–4) 405–17. Hsiao H M, Daniel I M, Cordes R D ‘Dynamic compressive behavior of thick composite materials’ Exptl. Mech. 1998, 38(3) 172–80. ISO 15024 ‘Fibre-reinforced plastic composites – Determination of mode I interlaminar fracture toughness, GIC, for unidirectionally reinforced materials’ 2001, pp. 1–24. Jen M H R, Tseng Y C, Lin W H, ‘Thermo-mechanical fatigue of centrally notched and unnotched AS-4/PEEK APC-2 composite laminates’ Intl. J. Fatigue 2006, 28(8) 901–9. Jones R M ‘Thermal buckling of uniformly heated unidirectional and symmetric crossply laminated fiber-reinforced composite uniaxial in-plane restrained simply supported rectangular plates’ Compos. Part A – Appl. Sci. Manuf. 2005, 36(10) 1335–67. Kashtalyan M L, Soutis C ‘Mechanisms of internal damage and their effect on the behaviour and properties of cross-ply composite laminates’ 2002, Intl. Appl. Mech. 38(6) 641– 57. Kawai M, Maki N ‘Fatigue strengths of cross-ply CFRP laminates at room and high temperatures and its phenomenological modeling’ Intl. J. Fatigue 2006, 28(10) 1297– 306. Khashaba U A ‘In-plane shear properties of cross-ply composite laminates with different off-axis angles’ Compos. Struct. 2004, 65(2) 167–77. Kobayashi S, Takeda N ‘Experimental and analytical characterization of transverse cracking behavior in carbon/bismaleimide cross-ply laminates under mechanical fatigue loading’ 2002 Compos. Part B – Engng. 33(6) 471–8. Kuboki T, Jar P Y B, Cheng J J R ‘Damage development in glass-fiber reinforced polymers (GFRP) under transverse loading’ Adv. Compos. Mats. 2005, 14(2) 131–45. © 2008, Woodhead Publishing Limited

308


Kyoung W-M, Kim C-G, Hong C-S ‘Buckling and postbuckling behavior of composite cross-ply laminates with multiple delaminations’ Compos. Struct. 1998, 43(4) 257– 74. Lafarie-Frenot M C, Hénaff-Gardin C, Gamby D ‘Matrix cracking induced by cyclic ply stresses in composite laminates’ Compos. Sci. Technol. 2001, 61(15) 2327–36. Laksimi A, Benzzegagh M L, Jing G, Hecini M, Roelandt J M ‘Mode I interlaminar fracture of symmetrical cross-ply composites’ Compos. Sci. Technol. 1991, 41(2) 147–64. La Saponara V, Muliana H, Haj-Ali R, Kardomateas G A ‘Experimental and numerical analysis of delamination growth in double cantilever laminated beams’ Engng. Fract. Mech. 2002, 69(6) 687–99. Le Page B H, Guild F J, Ogin S L, Smith P A ‘Finite element simulation of woven fabric composites’ Compos. Part A – Appl. Sci. Manuf. 2004, 35(7-8) 861–72. Lim S H, Li S ‘Energy release rates for transverse cracking and delaminations induced by transverse cracks in laminated composites’ Compos. Part A – Appl. Sci. Manuf. 2005, 36(11) 1467–76. Liu C J, Sterk J C, Nijhof A H J, Marissen R ‘Matrix-dominated damage in notched crossply composite laminates: Experimental observations’ Appl. Compos. Mats. 2002, 9(3) 155–68. McCartney L N ‘Physically based damage models for laminated composites’ Proc. Inst. Mech. Eng. Part L – J. Mats. Design Appl. 2003, 217(3) 163–99. Meijer G, Ellyin F ‘Effect of interleave on transverse cracks in thick glass-fiber epoxy cross-ply laminates’ J. Compos. Mats. 2004, 38(24) 2199–211. Mili F, Necib B ‘Impact behavior of cross-ply laminates composite plates under low velocities’ Compos. Struct. 2001, 51(3) 237–44. Mizutani Y, Nagashima K, Takemoto M, Ono K ‘Fracture mechanism characterization of cross-ply carbon-fiber composites using acoustic emission analysis’ NDT&E Intl. 2000, 33(2) 101–10. Nakamura T, Singh R P, Vaddadi P, ‘Effects of environmental degradation on flexural failure strength of fiber reinforced composites’ Exptl. Mech. 2006, 46(2) 257–68. Nilsson K F, Asp L E, Alpman J E, Nystedt L ‘Delamination buckling and growth for delaminations at different depths in a slender composite panel’ Intl. J. Solids Structs. 2001, 38(17) 3039–71. Ochola R O, Marcus K, Nurick G N, Franz T ‘Mechanical behaviour of glass and carbon fibre reinforced composites at varying strain rates’ Compos. Struct. 2004, 63(3–4) 455–67. Ogi K ‘Influence of thermal history on transverse cracking in a carbon fiber reinforced epoxy composite’ Adv. Compos. Mats. 2003, 11(3) 265–75. Ogihara S, Takeda N ‘Interaction between transverse cracks and delamination during damage progress in CFRP cross-ply laminates’ Compos. Sci. Technol. 1995, 54(4) 395–404. Parvatareddy H, Tsang P H W, Dilard D A ‘Impact damage resistance and tolerance of high-performance polymeric composites subjected to environmental aging’ Compos. Sci. Technol. 1996, 56(10) 1129–40. Pereira A B, de Morais A B ‘Mode I interlaminar fracture of carbon/epoxy multidirectional laminates’ Compos. Sci. Technol. 2004, 64(13–14) 2261–70. Prosser W H, Jackson K E, Kellas S, Smith B T, McKeon J, Friedman A ‘Advanced waveform-based acoustic-emission detection of matrix cracking in composites’ Mats. Eval. 1995, 53(9) 1052–8. © 2008, Woodhead Publishing Limited


309

Qin Z M, Librescu L ‘On a shear deformable theory of anisotropic thin-walled beams: further contribution and validations’, Compos. Struct. 2002, 56(4) 345–58. Rebière J-L, Maâtallah M-N, Gamby D ‘Analysis of damage mode transition in a crossply laminate under uniaxial loading’, Compos. Struct. 2002, 55(1) 115–26. Romhany G, Czigany T, Karger-Kocsis J ‘Determination of J-R curves of thermoplastic starch composites containing crossed quasi-unidirectional flax fiber reinforcement’ Compos. Sci. Technol. 2006, 66(16) 3179–87. Sanchu-Saez S, Barbero E, Zaera R, Navarro C ‘Compression after impact of thin composite laminates’ Compos. Sci. Technol. 2005, 65(13) 1911–19. Schapery R A, Goetz D P, Jordan W M ‘Delamination analysis of composites with distributed damage using a J integral’ in: Loo T T, Sun C T Proc. Intl. Symp. on Comp. Mater. Structs. Technomic Publ. 1986, 543–9. Schultz M R, Hyer M W ‘Snap-through of unsymmetric cross-ply laminates using piezoceramic actuators’ J. Intell. Mat. Systs. Structs. 2003, 14(12) 795–814. Selvarathinam A S, Weitsman Y J ‘A shear-lag analysis of transverse cracking and delamination in cross-ply carbon-fibre/epoxy composites under dry, saturated and immersed fatigue conditions’ Compos. Sci. Technol. 1999, 59(14) 2115–23. Shan B X, Pelegri A A ‘Assessment of the fracture behavior of an asymmetrically loaded cantilever composite structure’ J. Engng. Mats. Technol. Transactions ASME 2003, 125(4) 353–60. Shen H S ‘Postbuckling of shear deformable cross-ply laminated cylindrical shells under combined external pressure and axial compression’ Intl. J. Mech. Sci. 2001, 43(11) 2493–523. Shiau L C, Kuo S Y, ‘Thermal postbuckling behavior of composite sandwich plates’, Journal of Engineering Mechanics 2004, 130(10) 1160–7. Stevanovic M. Gordic M. Sekulic D. Djordevic I ‘The effect of edge interlaminar stresses on the strength of carbon/epoxy laminates of different stacking geometry’ J. Serbian Chem. Soc. 2006, 71(4) 421–31. Timmerman J F, Seferis J C ‘Predictive modeling of microcracking in carbon-fiber/epoxy composites at cryogenic temperatures’ J. Appl. Polym. Sci. 2004, 91(2) 1104–10. Tohgo K., S. Nakagawa, K. Kageyama ‘Fatigue behavior of CFRP cross-ply laminates under on-axis and off-axis cyclic loading’, Intl. J. Fatigue 2006, 28(10) 1254–62. Tong J, Guild F J, Ogin S L, Smith P A ‘On matrix crack growth in quasi-isotropic laminates – I. Experimental investigation’ Compos. Sci. Technol. 1997, 57(11) 1527–35. Vaidya U K, Hosur M V ‘High strain rate impact response of graphite/epoxy composites with polycarbonate facing’ J. Thermopl. Compos. 2003, 16(1) 75–95. Vaidya U K, Kamath M V, Hosur M V, Mahfuz H, Jeelani S ‘Low-velocity impact response of cross-ply laminated sandwich composites with hollow and foam-filed Zpin reinforced core’ J. Compos. Sci. Technol. & Res. 1999, 21(2) 84–97. Welsh J S, Mayes S, Key C T, McLaughlin R N ‘Comparison of MCT failure prediction techniques and experimental verification for biaxially loaded glass fabric-reinforced composite laminates’ J. Compos. Mats. 2004, 28(24) 2165–81. Zhang D X, Ye J Q, Sheng H Y ‘Free-edge and ply-cracking effect in cross-ply laminated composites under uniform extension ad thermal loading’ Compos. Struct. 2006, 76(4) 314–25. Zhang Y Xia Z, Ellyin F ‘Evolution and influence of residual stresses/strains of fiber reinforced laminates’ Compos. Sci. Technol. 2004, 64(10–11) 1613–21. Zou Z, Reid S R, Li S, Soden P D ‘Application of a delamination model to laminated composite structures’ Compos. Struct. 2002, 56(4) 375–89. © 2008, Woodhead Publishing Limited

10 Interlaminar mode II fracture characterization M F S F d e M O U R A, Faculdade de Engenharia da Universidade do Porto, Portugal

10.1

Introduction

The application of composite materials in the aircraft and automobile industries has led to an increase of research into the fracture behaviour of composites. One of the most significant mechanical properties of fibre reinforced polymer composites is its resistance to delamination onset and propagation. It is known that delamination can induce significant stiffness reduction leading to premature failures. Delamination can be viewed as a crack propagation phenomenon, thus justifying a typical application of fracture mechanics concepts. In this context, the interlaminar fracture characterization of composites acquires remarkable relevancy. There are several tests proposed in the literature in order to measure the interlaminar strain energies release rates in mode I, mode II and mixed mode I/II. Whilst mode I has already been extensively studied and the double cantilever beam (DCB) test is universally accepted, mode II is not so well studied, which can be explained by some difficulties inherent to experimental tests. Moreover, in many real situations delaminations propagate predominantly in mode II, as is the case of composite plates under low velocity impact (Choi and Chang, 1992). This gives relevancy to the determination of toughness propagation values instead of the initiation ones commonly considered in design. Some non-negligible differences can be achieved considering the R-curve effects (de Morais and Pereira, 2007). These issues make the fracture characterization in mode II an actual and fundamental research topic. However, problems related to unstable crack growth and to crack monitoring during propagation preclude a rigorous measurement of GIIc. In fact, in the mode II fracture characterization tests the crack tends to close due to the applied load, which hinders a clear visualization of its tip. In addition, the classical data reduction schemes, based on beam theory analysis and compliance calibration, require crack monitoring during propagation. On the other hand, a quite extensive fracture process zone (FPZ) ahead of crack tip exists under mode II loading. This non-negligible FPZ affects the measured toughness as a non-negligible amount 310 © 2008, Woodhead Publishing Limited


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of energy is dissipated on it. Consequently, its influence should be taken into account, which does not occur when the real crack length is used in the selected data reduction scheme. To overcome these difficulties a new data reduction scheme based on crack equivalent concepts and depending only on the specimen compliance is presented in the next section. The main objective of the proposed methodology is to increase the accuracy of experimental mode II fracture tests on the GIIc measurements. In fact, a rigorous monitoring of the crack length during propagation is one of the complexities of these tests.

10.2

Static mode II fracture characterization

There are three fundamental experimental tests used to measure GIIc. The most popular one is the end notched flexure (ENF), which was developed for wood fracture characterization (Barrett and Foschi, 1977). The test consists of a pre-cracked specimen under three-point bending loading (see Fig. 10.1). δ, P

a

ENF

2h

L

L

δ, P

ELS

2h

a L δ, P

4ENF

2h

a

d L

L

10.1 Schematic representations of the mode II tests.


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Unstable crack propagation constitutes one of the disadvantages of the ENF test. Another possibility is the end loaded split (ELS) test which is based on cantilever beam geometry (see Fig. 10.1). Although the ELS test involves more complexities during experiments relative to the ENF test, it provides a larger range of crack length where the crack propagates stably. In fact, the ENF test requires a0/L>0.7 to obtain stable crack propagation (Carlsson et al., 1986), whereas in the ELS test a0/L>0.55 is sufficient (Wang and VuKhanh, 1996). However, both of these tests present a common difficulty related to the crack length measurement during the experimental test. Different methods have been proposed in literature to address these difficulties. Kageyama et al. (1991) proposed a stabilized end notched flexure (SENF) test for experimental characterization of mode II crack growth. A special displacement gage was developed for direct measurement of the relative shear slip between crack surfaces of the ENF specimen. The test was performed under constant crack shear displacement rate, which guarantees stable crack propagation. Yoshihara et al. (Yoshihara and Ohta, 2000) recommended the use of crack shear displacement method (CSD) to obtain the mode II R-curve since the crack length is implicitly included in the CSD. Tanaka et al. (1995) concluded that to extend the stabilized crack propagation range in the ENF test, the test should be done under a condition of controlled CSD. Although the CSD method provides the measurement of the mode II toughness without crack length monitoring, this method requires a servo valve-controlled testing machine and the testing procedure is more complicated than that under the loading point displacement condition. Alternatively the four-point end notched flexure test (4ENF) (Fig. 10.1) can be used to evaluate the mode II R-curve. This test does not require crack monitoring but involves a more sophisticated setup and larger friction effects were observed (Shuecker and Davidson, 2000). In the following, a summary of the classical reduction schemes used for these experimental tests is presented.

10.2.1 Classical methods Compliance calibration method (CCM) The CCM is the most used. During the test the values of load, applied displacement and crack length (P-δ-a) are registered in order to calculate the critical strain energy release rate using the Irwin-Kies equation (Kanninen and Popelar, 1985) 2 10.1 GIIc = P dC 2 B da where B is the specimen width and C = δ/P the compliance. In the ENF and ELS tests a cubic relationship between the compliance (C) and the measured crack length a is usually assumed (Davies et al., 2001)



C = D + ma3 where D and m are constants. GIIc is then obtained from GIIc =

3 P2m a2 2B

313

10.2

10.3

For the 4ENF test a linear relationship (Yoshihara, 2004) between the compliance (C) and the measured crack length a is used C = D + ma

10.4

D and m being the respective coefficients. It should be noted that relations C = f(a) given by Equations 10.2 and 10.4 are based on the beam theory approach, as it will be shown in the next sub-section. GIIc is given by 2 GIIc = P m 2B

10.5

The three tests require the calibration of the compliance as a function of the crack length. This can be done by measurement of crack length during propagation or, alternatively, considering several specimens with different initial cracks lengths to establish the compliance–crack length relation, which is regressed by cubic (Equation 10.2) and linear (Equation 10.4) functions. Beam theory Beam theory methods are also frequently used to obtain GIIc in mode II tests. In the case of ENF test Wang and Williams (1992) proposed the Corrected Beam Theory (CBT)

GIIc =

9 ( a + 0.42 ∆ I ) 2 P 2 16 B 2 h 3 E1

10.6

where E1 is the axial modulus and ∆I a crack length correction to account for shear deformation E1 11G13

∆I = h

(

 Γ 3 – 2 1 + Γ 

)  2

10.7

with

Γ = 1.18

E1 E 2 G13

10.8

where E2 and G13 are the transverse and shear moduli, respectively. In the ELS case a similar expression is proposed (Wang and Williams, 1992)

GIIc =

9 ( a + 0.49 ∆ I ) 2 P 2 4 B 2 h 3 E1


10.9

314


For the 4ENF test the beam theory leads to the following equation (Silva, 2006) C=

d (18 d a – 20 d 2 + 60 L2 – 6 d L ) 24 E1 I

10.10

where I is the second moment of area and d represents the distance between each support and its nearest loading actuator (Fig. 10.1). Using Equation [10.1] GIIc can be obtained from 2 2 GIIc = 9 P d2 3 16 E1 B h

10.11

In summary, the application of beam theory to ENF and ELS tests involves the crack length, which does not occur in the 4ENF test. However, it should be emphasized that 4ENF setup is more complex. Also, friction effects (Shuecker and Davidson, 2000) and system compliance (Davidson and Sun, 2005) can affect the results. Owing to these drawbacks of the 4ENF test, the ENF and ELS tests emerge as the most appropriate to fracture characterization of composites in mode II. In this context, a new data reduction scheme, not depending on the crack length measurements, is proposed in the following section for these experimental tests.

10.2.2 Compliance-based beam method (CBBM) In order to overcome the difficulties associated to classical data reduction schemes a new method is proposed. The method is based on crack equivalent concept and depends only on the specimen compliance. The application of the method to ENF and ELS tests is described in the following. ENF test Following strength of materials analysis, the strain energy of the specimen due to bending and including shear effects is U=

∫

2L

0

M 2f dx + 2Ef I

2L

∫ ∫ 0

h

–h

τ 2 B dy dx 2 G13

10.12

where Mf is the bending moment and y2  V  τ = 3 i 1 – 2  2 Ai  ci 

10.13

where Ai, ci and Vi represent, respectively, the cross-section area, half-thickness of the beam and the transverse load of the i segment (0 ≤ x ≤ a, a ≤ x ≤ L or © 2008, Woodhead Publishing Limited


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L ≤ x ≤ 2L). From the Castigliano theorem, the displacement at the loading point for a crack length a is 3 PL P (3 a 3 + 2 L3 ) δ = dU = + 10 G13 Bh dP 8 E f Bh 3

10.14

Since the flexural modulus of the specimen plays a fundamental role on the P-δ relationship, it can be calculated from Equation 10.14 using the initial compliance C0 and the initial crack length a0 Ef =

–1 3 a 03 + 2 L3  3L  C – 0 10 G13 Bh  8 Bh 3 

10.15

This procedure takes into account the variableness of the material properties between different specimens and several effects that are not included in beam theory, e.g. stress concentration near the crack tip and contact between the two arms. In fact, these phenomena affect the specimen behavior and consequently the P-δ curve, even in the elastic regime. Using this methodology their influence are accounted for through the calculated flexural modulus. On the other hand, it is known that, during propagation, there is a region ahead of crack tip (fracture process zone), where materials undergo properties degradation by different ways, e.g. micro-cracking, fibre bridging and inelastic processes. These phenomena affect the material compliance and should be accounted for in the mode II tests. Consequently, during crack propagation a correction of the real crack length is considered in the equation of compliance (10.14) to include the FPZ effect C=

3( a + ∆a FPZ ) 3 + 2 L3 3L + 3 10 G 8 E f Bh 13 Bh

10.16

and consequently, C  C   a eq = a + ∆ a FPZ =  corr a 03 + 2  corr – 1 L3  C C 3  0 corr    0 corr

1/3

10.17

where Ccorr is given by Ccorr = C –

3L 10 G13 Bh

10.18

GIIc can now be obtained from

GIIc =

2 9 P 2 a eq 16 B 2 E f h 3


10.19

316


This data reduction scheme presents several advantages. Using this methodology crack measurements are unnecessary. Experimentally, it is only necessary to register the values of applied load and displacement. Therefore, the method is designated as compliance-based beam method (CBBM). Using this procedure the FPZ effects, that are pronounced in mode II tests, are included on the toughness measurement. Moreover, the flexural modulus is calculated from the initial compliance and initial crack length, thus avoiding the influence of specimen variability on the results. The unique material property needed in this approach is G13. However, its effect on the measured GIIc was verified to be negligible (de Moura et al., 2006), which means that a typical value can be used rendering unnecessary to measure it. ELS test Following a procedure similar to the one described for the ENF test, the applied P-δ relationship is

3 PL P (3 a 3 + L3 ) δ = dU = + dP 5 BhG13 2 Bh 3 E1

10.20

In order to include the root rotation effects at clamping and the details of crack tip stresses or strains not included in the beam theory, an effective beam length (Lef) can be achieved. In fact, considering in Equation 10.20 the initial crack length (a0) and the initial compliance (C0) experimentally measured, it can be written C0 –

3a 03 L3ef 3 Lef = + 3 3 5 BhG 2 Bh E1 2 Bh E1 13

10.21

To take account for the FPZ influence a correction to the real crack length (∆aFPZ) should be considered. From Equation 10.20 the compliance (C) during crack propagation can be expressed as C–

L3ef 3( a + ∆a FPZ ) 3 3 Lef = + 5 BhG13 2 Bh 3 E1 2 Bh 3 E1

10.22

Combining Equations 10.22 and 10.21, the equivalent crack length can be given by 2 Bh 3 E1   a eq = a + ∆a FPZ =  ( C – C0 ) + a 03  3  

1/3

10.23

GIIc can now be obtained from

GIIc =

2 9 P 2 a eq 4 B 2 h 3 E1


10.24


317

Following this procedure GIIc can be obtained without crack measurement during propagation which can be considered an important advantage. Equation 10.24 only depends on applied load and displacement during crack growth. Additionally, the influence of root rotation at the clamping point and singularity effects at the crack tip are accounted for, through initial compliance C0. During propagation, the effect of FPZ on the compliance is also included using this methodology. In this case (ELS test) it is necessary to measure the longitudinal modulus.

10.2.3 Numerical simulations In order to verify the performance of the CBBM on the determination of GIIc of unidirectional composites, numerical simulations of the ENF and ELS tests were performed. A cohesive mixed-mode damage model based on interface finite elements was considered to simulate damage initiation and propagation. A constitutive relation between the vectors of stresses (σ) and relative displacements (δ) is postulated (Fig. 10.2). The method requires local strengths (σu,i, i = I, II, III) and the critical strain energy release rates (Gic) as inputted data parameters [8, 9]. Damage onset is predicted using a quadratic stress criterion 2

2

2

 σI   σ II   σ III  σ  + σ  + σ  =1  u ,I   u , II   u , III  2

 σ II   σ III  σ  + σ   u , II   u , III 

if σ I ≥ 0

10.25

2

=1

if σ I ≤ 0

where σi, (i = I, II, III) represent the stresses in each mode. Crack propagation was simulated by a linear energy criterion

GI G G + II + III = 1 GIc GIIc GIIIc

10.26

Basically, it is assumed that the area under the minor triangle of Fig. 10.2 is the energy released in each mode, which is compared to the respective critical fracture energy represented by the bigger triangle. The subscripts o and u refer to the onset and ultimate relative displacement and the subscript m applies to the mixed-mode case. More details about this model are presented in de Moura et al. (2006). Three-dimensional approaches (Figs 10.3 and 10.4) were carried out to include all the effects that can influence the measured GIIc. The interface elements were placed at the mid-plane of the specimens to simulate damage progression. Very refined meshes were considered in the region of interest corresponding to crack initiation and growth. The specimens’ geometry and


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Delamination behaviour of composites σi σ u ,i Pure mode model

i = I, II, III σ um,i

Gic

Mixed-mode model

Gi δi σ om,i

δo,i

δum,i

δu , i

10.2 Pure and mixed-mode damage model.

10.3 The mesh used for the ENF test: global view and detail of the refined mesh at the region of crack initiation and growth.

10.4 The mesh used for the ELS test.

material properties are listed in Tables 10.1, 10.2 and 10.3, respectively. An analysis of G’s distributions at the crack front showed a clear predominance of mode II along the specimens’ width, although some spurious mode III exists at the specimens edges (de Moura et al., 2006 and Silva et al., 2007). © 2008, Woodhead Publishing Limited


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Table 10.1 Specimens’ geometry

ENF ELS

L (mm)

b (mm)

h (mm)

a0 (mm)

100 100

10 10

1.5 1.5

75 60

Table 10.2 Material properties E1 (GPa)

E 2 = E3 (GPa)

ν12 = ν13

ν23

G12 = G13 (GPa)

G23 (GPa)

150

11

0.25

0.4

6

3.9

Table 10.3 Strength properties σu,i (i = I,II,III) (MPa)

GIc (N/mm)

GIIc (N/mm)

GIIIc (N/mm)

40

0.3

0.7

1.0

140 120

P (N)

100 80 60 40 20 0 0

2

4

6

8

10

δ (mm)

10.5 P-δ curve of the ENF specimen.

Appropriate values of critical strain energy release rates were considered for each of the three modes, respectively (see Table 10.3). Consequently, the efficacy of the proposed data reduction scheme can be evaluated by its capacity to reproduce the inputted GIIc from the P-δ results obtained numerically. The application of the CBBM is performed by three main steps. The first one is the measurement of the initial compliance C0 from the initial slope of the P-δ curves (Figs (10.5) or (10.7)). This parameter is then used to estimate the flexural modulus in the ENF test (Equation 10.15). The next step is the © 2008, Woodhead Publishing Limited

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evaluation of the equivalent crack length (Equations 10.17 or 10.23) as a function of the current (C) and initial compliance (C0). Finally, the R-curves, Figs (10.6) and (10.8), can be obtained from Equations 10.19 and 10.24, respectively. It should be noted that crack propagation occurs after peak load in both tests. During crack growth P decreases with the increase of equivalent crack length. This originates a plateau on the R-curves, which corresponds to the critical strain energy release rate in mode II (GIIc). These plateau values are compared with the reference value (Figs (10.6) and (10.8)), which represents the inputted GIIc. The excellent agreement obtained in both cases demonstrates

1.0 0.9 0.8

GII (N/mm)

0.7 0.6 0.5 0.4 0.3 0.2

GII (CBBM)

0.1

GIIc (Reference value)

0.0 75

80

85

90

95

100

aeq (mm)

10.6 R-curve of the ENF specimen. 70 60

P (N)

50 40 30 20 10 0 0

2

4

6

8 δ (mm)

10.7 P-δ curve of the ELS specimen.


10

12

14

16


321

1.0 0.9 0.8

GII (N/mm)

0.7 0.6 0.5 0.4 0.3 0.2

GII (CBBM)

0.1

GIIc (Reference value)

0.0 60

65

70

75

80 aeq (mm)

85

90

95

10.8 R-curve of the ELS specimen.

the effectiveness of the CBBM as a suitable data reduction scheme to determine GIIc, without crack length monitoring during propagation. As the ENF test is much simpler to execute than the ELS one, it can be concluded that using the CBBM, the ENF test is the most suitable for the determination of GIIc and it should be considered as the principal candidate for standardization.

10.3

Dynamic mode II fracture characterization

The research on dynamic crack propagation in composites has become the focus of several authors in the recent years. The dynamic fracture characterization of composites is not easy to perform. In fact, it is experimentally difficult to induce high speed delamination growth in a simple and controlled manner (Guo and Sun, 1998). However, the determination of dynamic fracture toughness of composites is of fundamental importance in the prediction of the dynamic delamination propagation in composite structures. In addition, it is known that the impact delamination is mainly governed by mode II fracture (Wang and Vu-Khanh, 1991). However, there are several unclear phenomena related to dynamic crack propagation. One of the most important issues is related to the influence of rate effects on the propagation of dynamic cracks. An example of this occurrence is the dynamic delamination propagation occurring in composites submitted to low velocity impact. In this case, rate effects in the FPZ can interact with the well known rate-dependency of polymers leading to a very complicated phenomenon. In addition, Kumar and Narayanan (1993) verified that when glass fibre reinforced epoxy laminates are impacted, the total delamination area between the various plies multiplied by the quasi-static energy release rate exceeds the energy of the impacting © 2008, Woodhead Publishing Limited

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mass. This suggests that under high crack speeds, delamination propagates at lower toughness which leads to larger damaged areas. In order to explain this behaviour, Maikuma et al. (1990) suggest that the calculation of critical strain energy release rate should account for the kinetic energy (Ekin) in Equation 10.1 2 dE kin GIIc = P dC – 2 B da B da

10.27

The kinetic energy expression can be obtained from E kin = 1 ρ B 2 h 2

∫

Lt

0

2

 dw ( x )  dx  dt 

10.28

where ρ and Lt are the mass density and the total length of the specimen, respectively, t represents the time and w(x) the displacement field. The quasistatic approach may provide an adequate approximation to the dynamic problem if the contribution of kinetic energy is small. Wang and Vu-Khanh (1995) have suggested that the dynamic fracture behaviour of materials depends on the balance between the energy released by the structure over a unit area of crack propagation (G) and the material resistance (R), which can be viewed as the energy dissipated in creating the fracture surface. When unstable crack growth occurs, the difference G-R is converted into kinetic energy. If G increases with crack growth the crack speed also increases because more energy is available. Crack arrest will occur when G becomes lower than R and, consequently, no kinetic energy is available for crack growth. Thus, it can be affirmed that fracture stability depends on the variations of the strain energy release rate and the materials resistance during crack growth. On the other hand, the fracture resistance of polymer composites is generally sensitive to loading rate. Under impact load or during rapid delamination growth, the strain rate at the crack tip can be very high and the material toughness significantly reduced. The fracture surface exhibited ductile tearing and large scale plastic deformation of the matrix. The dynamic fracture surface in the initiation exhibits less plastic deformation; during propagation even less deformation is observed. It was also verified that plastic zone size at the crack tip diminishes with increasing rate. Consequently, the decrease in mode II interlaminar fracture toughness is attributed to a transition from ductile to brittle matrix dominated failure with increasing rate. The decreasing trend of toughness with increase of crack speed was also observed by Kumar and Kishore (1998). The authors used a combination of numerical and experimental techniques on the DCB specimens to carry out dynamic interlaminar toughness measurements of unidirectional glass fibre epoxy laminate. They observed a sharp decrease of dynamic toughness values



323

relatively to the quasi-static ones. In fact, they measured dynamic toughness initiation values of 90–230 J/m2 against quasi-static values of 344–478 J/m2. Propagation values of 0–50 J/m2 were obtained for crack speed ranging between 622–1016 m/s. The majority of the experimental studies consider unidirectional laminates. Lambros and Rosakis (1997) performed an experimental investigation of dynamic crack initiation and growth in unidirectional fibre-reinforced polymeric-matrix thick composite plates. Edge-notched plates were impacted in a one-point bend configuration using a drop-weight tower. Using an optical method the authors carried out a real-time visualization of dynamic fracture initiation and growth for crack speeds up to 900 m/s. They verified that the elastic constants of the used material are rate sensitive and the measured fracture toughness values are close to those typical of epoxies. This was considered consistent, because in unidirectional lay-ups crack initiation and growth occurs in the matrix. Tsai et al. (2001) used a modified ENF specimen to determine the mode II dominated dynamic delamination fracture toughness of fiber composites at high crack propagation speeds. A strip of adhesive film with higher toughness was placed at the tip of interlaminar crack created during laminate lay-up. The objective was to delay the onset of crack extension and produce crack propagation at high speeds (700 m/s). Sixteen pure aluminium conductive lines were put on the specimen edge side using the vapour deposition technique, to carry out crack speed measurements. The authors concluded that the mode II dynamic energy release rate of unidirectional S2/8553 glass/epoxy composite seems to be insensitive to crack speed within the range of 350 and 700 m/s. The authors also simulated mixed mode crack propagation by moving the pre-crack from the mid-plane to 1/3 of the ENF specimen thickness of unidirectional AS4/3501-6 carbon/epoxy laminates. The maximum induced crack speed produced was 1100 m/s. They found that that the critical dynamic energy release rate is not affected by the crack speed and lies within the scatter range of the respective static values. For numerical simulations of the dynamic crack propagation the cohesive damage models emerge as the most promising tools. The major difficulty is the incorporation of the rate-dependent effects in the constitutive laws. Corigliano et al. (2003) developed a cohesive crack model with a ratedependent exponential interface law to simulate the nucleation and propagation of cracks subjected to mode I dynamic loading. The model is able to simulate the rate-dependent effects on the dynamic debonding process in composites. The authors concluded that the softening process occurs under larger relative displacements in comparison to rate-independent models. They verified that the type of rate-dependency can affect dynamic crack processes, namely the time to rupture and fracture energy. They also state that these effects diminish when inertial terms become dominant. © 2008, Woodhead Publishing Limited

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In summary, dynamic fracture toughness characterization of composite materials has been the centre of attention of several authors with no apparent consensus on the results. Although the majority of the studies point to a decrease of the fracture toughness with increasing load rate there is no unanimity about this topic. Some authors observed the opposite trend (Corigliano et al., 2003) and others detected no remarkable influence of crack speed on toughness (Tsai et al., 2001). Although some of these discrepancies can eventually be explained by different behaviour of the tested materials, and the attained crack speed values, it is obvious that more profound studies about the subject are necessary.

10.4

Conclusions

Interlaminar fracture characterization of composites in mode II acquires special relevancy under transverse loading such as low velocity impact. Up to now there is no standardized test in order to measure the critical strain energy release rate in mode II. Owing to their simplicity, the ENF and ELS tests become the principal candidates to standardization. However, they present a common difficulty associated with crack monitoring during propagation which is fundamental to obtaining the R-curves, following the classical data reduction schemes. To surmount these difficulties a new data reduction scheme based on specimen compliance is proposed. The method does not require crack length measurement during propagation, and accounts for the effects of the quite extensive FPZ on the measured critical strain energy release rate. Numerical simulations of the ENF and ELS tests demonstrated the adequacy and suitability of the proposed method to obtain the mode II Rcurves of composites. Due to its simplicity the ENF test is proposed for standardization. Little work has been done on dynamic fracture of composite materials, namely under mode II loading. This is due to experimental difficulties related to inducing high crack speeds in a monitored way. Although the majority of the published works point out to a decrease of the dynamic toughness with increase of crack speed, it appears that dynamic toughness can be similar to the respective quasi-static value up to a given crack speed (Tsai et al., 2001). Undoubtedly, more research about this topic is necessary. In fact, an unsafe structural design can occur if the quasi-static values of toughness are used in a dynamically loaded structure.

10.5

Acknowledgements

The author thanks Professors Alfredo B. de Morais (UA, Portugal), José Morais (UTAD, Portugal) and Manuel Silva for their valorous collaboration, advices and discussion about the matters included in this chapter. The author © 2008, Woodhead Publishing Limited


325

also thanks the Portuguese Foundation for Science and Technology for supporting part of the work here presented, through the research project POCI/EME/56567/2004.

10.6

References

Barrett JD, Foschi RO (1977), ‘Mode II stress-intensity factors for cracked wood beams’, Engng. Fract. Mech., 9: 371–378. Carlsson LA, Gillespie JW, Pipes RB (1986), ‘On the analysis and design of the end notched flexure (ENF) specimen for mode II testing’, J Compos Mater, 20: 594–604. Choi HY, Chang FK (1992), ‘A model for predicting damage in graphite/epoxy laminated composites resulting from low-velocity point impact’, J Compos Mater, 26: 2134– 2169. Corigliano A, Mariani S, Pandolfi A (2003), ‘Numerical modelling of rate-dependent debonding processes in composites’, Compos. Struct., 61: 39–50. Davidson BD, Sun X (2005), ‘Effects of friction, geometry and fixture compliance on the perceived compliance from three- and four-point bend end-notched flexure tests’, J. Reinf. Plastics Compos., 24: 1611–1628. Davies P, Blackman BRK, Brunner AJ (2001), Mode II delamination. In: Moore DR, Pavan A, Williams JG, editors. Fracture Mechanics Testing Methods for Polymers Adhesives and Composites, Amsterdam, London, New York: Elsevier; 307–334. de Morais AB, Pereira AB (2007), ‘Application of the effective crack method to mode I and mode II interlaminar fracture of carbon/epoxy unidirectional laminates’, Composites Part A: Applied Science and Manufacturing, 38: 785–794. de Moura MFSF, Silva MAL, de Morais AB, Morais JJL (2006), ‘Equivalent crack based mode II fracture characterization of wood’, Engng. Fract. Mech., 73: 978–993. Guo C, Sun CT (1998), ‘Dynamic mode-I crack propagation in a carbon/epoxy composite’, Composites Science and Technology, 58: 1405–1410. Kageyama K, Kikuchi M, Yanagisawa N (1991), ‘Stabilized end notched flexure test: characterization of mode II interlaminar crack growth’. In: O’Brien TK, editor. Composite Materials: Fatigue and Fracture, ASTM STP 1110, Vol. 3. Philadelphia PA: ASTM, p. 210–225. Kanninen MF, Popelar CH (1985), Advanced Fracture Mechanics, Oxford: Oxford University Press. Kumar P, Narayanan MD (1993), ‘Energy dissipation of projectile impacted panels of glass fabric reinforced composites’, Compos. Struct., 15: 75–90. Kumar P, Kishore NN (1998), ‘Initiation and propagation toughness of delamination crack under an impact load’, J. Mech. Phys. Solids, 46: 1773–1787. Lambros J, Rosakis AJ (1997), ‘Dynamic crack initiation and growth in thick unidirectional graphite/epoxy plates’, Composites Science & Technology, 57: 55–65. Maikuma H, Gillespie JW, Wilkins DJ (1990), ‘Mode II interlaminar fracture of the center notch flexural specimen under impact loading’, J. Compos. Mater., 24: 124– 149. Schuecker C, Davidson BD (2000), ‘Effect of friction on the perceived mode II delamination toughness from three and four point bend end notched flexure tests’, ASTM STP, 1383: 334–344. Silva MAL (2006), ‘Estudo das Propriedades de Fractura em Modo II e em Modo III da Madeira de Pinus pinaster Ait.’, Master Thesis, FEUP, Porto.


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Silva MAL, Morais JJL, de Moura MFSF, Lousada JL (2007), ‘Mode II wood fracture characterization using the ELS test’, Eng. Fract. Mech., 74: 2133–2147. Tanaka K, Kageyama K, Hojo M (1995), ‘Prestandardization study on mode II interlaminar fracture toughness test for CFRP in Japan’, Composites, 26: 243–255. Tsai JL, Guo C, Sun CT (2001), ‘Dynamic delamination fracture toughness in unidirectional polymeric composites’, Composites Science & Technology, 61: 87–94. Wang H, Vu-Khanh T (1991), ‘Impact-induced delamination in [05, 905, 05] carbon fiber/Polyetheretherketone composite laminates’, Polymer Engineering and Science, 31: 1301–1309. Wang H, Vu-Khanh T (1995), ‘Fracture mechanics and mechanisms of impact-induced delamination in laminated composites’, J. Compos. Mater., 29: 156–178. Wang H, Vu-Khanh T (1996), ‘Use of end-loaded-split (ELS) test to study stable fracture behaviour of composites under mode II loading’, Compos. Struct., 36: 71–79. Wang Y, Williams JG (1992), ‘Corrections for Mode II Fracture Toughness Specimens of Composite Materials’, Composites Science & Technology, 43: 251–256. Yoshihara H, Ohta M (2000), ‘Measurement of mode II fracture toughness of wood by the end-notched flexure test, J Wood Sci., 46: 273–278. Yoshihara H (2004), ‘Mode II R-curve of wood measured by 4-ENF test’, Engng. Fract. Mech., 71: 2065–2077.


11 Interaction of matrix cracking and delamination M F S F de M O U R A, Faculdade de Engenharia da Universidade do Porto, Portugal

11.1

Introduction

Laminated composite materials have high strength-to-weight and stiffnessto-weight ratios. They can be considered as laminar systems with weak interfaces. Consequently, they are very susceptible to interlaminar damage. In the presence of delamination, material stiffness, and consequently, the associated structure, can be drastically reduced, which can lead to its catastrophic failure. Moreover, delamination is a form of internal damage and is not easily detected, which increases the associated risks.

11.1.1 Damage mechanism In the majority of real applications delamination does not occur alone. It is known that matrix cracking inside layers and delamination are usually associated and constitute a typical damage mechanism of composites (Takeda et al., 1982; Joshi and Sun, 1985), especially when structures are subjected to bending loads. In fact, although the phenomenon can occur under tensile loading, it acquires a marked importance under bending loads, e.g. low velocity impact (Choi et al., 1991a). It is commonly accepted that there is a strong interaction between matrix cracking inside layers and delamination between layers. This coupling phenomenon is initiated by matrix cracking, i.e., shear and/or bending cracks in the early stages of the loading process. These cracks can cause delamination and significantly affect its propagation. Delamination occurs between layers with differing fibre orientations. In fact, when a bending or shear crack in a layer reaches an interface between two differently oriented layers it is unable to easily penetrate the other layer, thus propagating as a delamination. On the other hand, two adjacent laminae having different fibre angles induce extensional and bending stiffness mismatch which, combined with the low strength of the matrix, make composite materials very sensitive to delamination at those interfaces. In general, the delamination resistance of a given interface decreases with the increase of its stiffness 327 © 2008, Woodhead Publishing Limited

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mismatching degree (Liu and Malvern, 1987). During delamination propagation extensive micro-matrix cracks are generated in the adjacent layers. It is worth noting that this complex interaction damage mechanism can occur in different forms depending on the bending stiffness of the structure. Flexible components respond primarily in a flexural mode, inducing significant tensile stresses at farther layers from the loaded surface. Therefore, the cracks located in these outermost layers are vertical and caused by bending effects (Choi and Chang, 1992) and their initiation and growth occur in an almost mode I fracture process. These cracks will generate a delamination along the upper adjacent interface, which will interact with other matrix cracks of the neighbour upper ply, leading to delamination on the second interface and so on (see Fig. 11.1). Delamination patterns in flexible laminates present a frustum-conical shape, where delamination size increases towards the lower face of the laminate (Levin, 1991). A different failure mechanism occurs for stiffer laminates. In this case, only small deflections take place and damage initiates near to the loaded surface as a result of contact forces. Shear cracks develop near to the impact indentation and propagate through the upper ply up to the neighbour interface degenerating in a delamination. This delamination extends from the loaded region until it also deflects into a lower ply by shear and inclined cracks (see Fig. 11.2). Further delamination growth in stiff laminates occurs in a barrel-shaped region by growth of delaminations around the midplane (Olsson et al., 2000). Matrix bending cracks are a predominantly mode I fracture process governed by transverse normal tensile stresses, whilst matrix shear cracks are governed by interlaminar shear and transverse normal tensile stresses (Choi et al.,

11.1 Schematic representation of damage initiation mechanism in flexible laminates.



329

11.2 Schematic representation of damage initiation mechanism in stiff laminates.

1991b). Delamination initiation is controlled by mode I although its growth is typically a mixed shear mode (II and III) fracture process (Choi and Chang, 1992), which is explained by the bending stiffness mismatching between adjacent differently oriented layers.

11.1.2 Classical prediction methodologies Two main approaches have been used in order to simulate the damage mechanism described above. One is based on strength of materials concepts, where materials are assumed to be free of defects. However, in many situations the problem of stress concentrations near to a notch or a flaw leads to mesh dependency in numerical approaches. To overcome this problem, the stresses obtained analytically or numerically are used in a point stress or average stress criteria (Whitney and Nuismer, 1974) in order to evaluate the occurrence of failure. In the point stress criterion the stresses are evaluated at a characteristic distance whereas in the average stress criterion they are averaged over a distance. An example is given by Choi et al. (1991b) where quadratic average stress criteria were used to predict matrix cracking when 2

2

 n σ 23   n σ 22   n  +  n  =1  Y   Si 

11.1

and delamination when   n σ 23  2  Da   n  +     Si  © 2008, Woodhead Publishing Limited

2

σ 13   n σ 22  +  n   n+1  Y  Si 

n+1

2

  =1 

11.2

330


The subscripts 1, 2 and 3 are the orthotropic local coordinates of the nth or (n + 1)th layers, which correspond, respectively, to the upper and lower plies of the nth interface. Y and Si are the in situ ply transverse and shear interlaminar strengths, respectively, and Da is an empirical constant which was determined from experiments using a fitting procedure. The stress components are averaged within the ply thickness n

σ ij = 1 hn

∫

tn

σ ij dz

t n –1

11.3

where tn and tn–1 correspond to the position of the upper and lower interfaces of the nth ply, h is the ply thickness and z the axis normal to the plate. These criteria were applied in two steps: • •

the matrix cracking criterion is initially applied in each layer; if matrix cracking is predicted in a given layer, the delamination criterion is applied subsequently considering two different circumstances; this criterion is applied to the lower adjacent interface if a shear crack was predicted and to the upper one if a bending crack was found.

Unlike what happens in strength of materials based approaches, the fracture mechanics approach assumes the presence of an inherent defect in the material. The majority of the proposed works are based on the concepts of strain energy release rate. It is usually assumed that damage propagation occurs when the strain energy at the crack front is equal to the critical strain energy release rate, which is a material property. The strain energy release rates are commonly obtained by using the virtual crack closure technique (VCCT) (Krueger, 2002). Considering a two-dimensional problem (see Fig. 11.3), the strain energies (GI and GII) can be calculated by the product of the relative displacements at the ‘opened point’ (nodes l1 and l2) and the loads at the ‘closed point’ (node i) GI =

1 Y ∆v 2 B∆ a i i

GII =

1 X ∆u 2 B∆ a i l

11.4

B∆a being the area of the new surface created by an increment of crack propagation (see Fig. 11.3). It should be evident that similar propagation occurs, and an adequate refined mesh should be used. Liu et al. (1993) used the VCCT to obtain the strain energy release rates and a linear mixed-mode criterion

GI G + II = 1 GIc GIIc


11.5


331

y

Thickness = B

v l2

l2 ul 2

ul1

Yi v l1

Xi

x

i

l1

a

∆a

∆a

11.3 Scheme of the VCCT.

to simulate the propagation of a small initial delamination crack introduced after the occurrence of the initial matrix cracking failure. GIc and GIIc represent the interlaminar critical strain energy release rates in modes I and II, respectively. A similar approach was followed by Zou et al. (2002) where a mixed-mode delamination growth is considered when α β γ  GI  +  GII  +  GIII  = 1  GIc   GIIc   GIIIc 

11.6

and α, β and γ are mixed-mode fracture parameters determined from material tests. The stress and fracture mechanics based criteria present some disadvantages. The stress-based methods present mesh dependency during numerical analysis due to stress singularities. On the other hand, the point/average stress criteria require the definition of a critical dimension which depends on the material and stacking sequence (Tan, 1989), and do not have a physically powerful theoretical foundation. Fracture mechanics approach relies on the definition of an initial flaw or crack length. However, in many structural applications the locus of damage initiation is not obvious. On the other hand, the stressbased methods behave well at predicting delamination onset, and fracture mechanics has already demonstrated its accuracy in delamination propagation modelling. In order to overcome the referred drawbacks and exploit the usefulness of the described advantages, cohesive damage models and continuum damage mechanics emerge as suitable options. These methodologies combine aspects of stress-based analysis to model damage initiation and fracture © 2008, Woodhead Publishing Limited

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mechanics to deal with damage propagation. Thus, it is not necessary to take into consideration an initial defect, and mesh dependency problems are minimized.

11.2

Mixed-mode cohesive damage model

Cohesive damage models are frequently used to simulate damage onset and growth. They are usually based on a softening relationship between stresses and relative displacements between crack faces, thus simulating a gradual degradation of material properties. They do not depend on a predefined initial flaw, unlike conventional fracture mechanics approaches. Typically, stress-based and energetic fracture mechanics criteria are used to simulate damage initiation and growth, respectively. Usually cohesive damage models are based on spring (Cui and Wisnom, 1993 and Lammerant and Verpoest, 1996) or interface finite elements (Mi et al., 1998, Petrossian and Wisnom, 1998; de Moura et al., 2000) connecting plane or three-dimensional solid elements. Those elements are placed at the planes where damage is prone to occur which, in several structural applications, can be difficult to identify a priori. However, an important characteristic of delamination is that its propagation is restricted to a well-defined plane corresponding to the interface between two differently oriented layers, thus leading to a typical application of cohesive methods. Taking this into consideration, a cohesive mixed-mode damage model based on interface finite elements is presented. The formulation is based on the constitutive relationship between stresses on the crack plane and the corresponding relative displacements

σ = E δr

11.7

where δr is the vector of relative displacements between homologous points, and E a diagonal matrix containing the penalty parameter e introduced by the user. Its values must be quite high in order to hold together and prevent interpenetration of the element faces. Following a considerable number of numerical simulations (Gonçalves et al., 2000), it was found that e = 107 N/mm3 produced converged results and avoided numerical problems during the non-linear procedure. The interface finite element includes a damage model to simulate damage onset and growth. Equation 11.7 is only valid before damage initiation. The considered damage model combines aspects of strength-based analysis and fracture mechanics. It is based on a softening process between stresses and interfacial relative displacements and includes a mixed-mode formulation. After peak stress the material softens progressively or, in other words, undergoes damage (see Fig. 11.4). To avoid the singularity at the crack tip and its effects, a gradual rather than a sudden degradation, which would result in mesh-dependency, is considered. It is assumed that failure occurs gradually © 2008, Woodhead Publishing Limited


333

σi σ u, i Pure mode model

i = I, II, III

σ um,i

Gic Mixed-mode model

Gi δi σ om,i

δo , i

δum,i

δu , i

11.4 Pure and mixed-mode damage model.

as energy is dissipated in a cohesive zone behind the crack tip. This is equivalent to the consideration of a ‘fracture process zone’, defined as the region in which the material undergoes softening deterioration by different ways, e.g. micro-cracking, fibre bridging and inelastic processes. Numerically, this is implemented by a damage parameter whose values vary from zero (undamaged) to unity (complete loss of stiffness) as the material deteriorates. For pure mode (I, II or III) loading, a linear softening process starts when the interfacial stress reaches the respective strength σu,i (see Fig. 11.4). The softening relationship can be written as = (I – D)Er

11.8

where I is the identity matrix and D is a diagonal matrix containing, on the position corresponding to mode i (i = I, II, III), the damage parameter,

di =

δ u , i (δ i – δ o , i ) δ i (δ u , i – δ o , i )

11.9

where δo,i and δu,i are, respectively, the onset and ultimate relative displacements of the softening region (see Fig. 11.4), and δi is the current relative displacement. The maximum relative displacement, δu,i, at which complete failure occurs, is obtained by equating the area under the softening curve to the respective critical strain energy release rate

Gic = 1 σ u ,i δ u ,i 2

11.10

In general, structures are subjected to mixed-mode loadings. Therefore, a formulation for interface elements should include a mixed-mode damage model, which, in this case, is an extension of the pure mode model described


334


above (see Fig. 11.4). Damage initiation is predicted by using a quadratic stress criterion 2

2

2

 σI   σ II   σ III  σ  + σ  + σ  = 1 if σ 1 ≥ 0  u ,I   u ,II   u ,III  2

2

 σ II   σ III  σ  + σ  =1  u ,II   u ,III 

11.11

if σ 1 ≤ 0

assuming that normal compressive stresses do not promote damage. Considering Equation 11.7, the first equation 11.11 can be rewritten as a function of relative displacements 2

2

2

 δ om ,I   δ om ,II   δ om , III   δ  +  δ  +  δ  =1  o ,I   o ,II   o , III 

11.12

δom,i (i = I, II, III) being the relative displacements corresponding to damage initiation. Defining an equivalent mixed-mode displacement

δm =

2 δ I2 + δ II2 + δ III

11.13

and mixed-mode ratios

βi =

δi δI

11.14

the mixed-mode relative displacement at the onset of the softening process (δom) can be obtained combining Equations 11.12–11.14.

δ om = δ ο ,I δ o ,II δ o ,III

2 1 + β II2 + β III (δ o ,II δ o ,III ) 2 + ( β II δ o ,I δ o ,III ) 2 + ( β III δ o ,I δ o ,II ) 2

11.15 The corresponding relative displacement for each mode, δom,i, can be obtained from Equations 11.13–11.15.

δ om , i =

β i δ om 2 1 + β II2 + β III

11.16

Once a crack has initiated the above stress-based criterion cannot be used in the vicinity of the crack tip due to stress singularity. Consequently, the mixed-mode damage propagation is simulated using the linear fracture energetic criterion

GI G G + II + III = 1 GIc GIIc GIIIc © 2008, Woodhead Publishing Limited

11.17


335

The released energy in each mode at complete failure can be obtained from the area of the minor triangle of Fig. 11.4. Gi = 1 σ um ,i δ um ,i 2

11.18

Considering Equations 11.7, 11.13 and 11.14, the energies (Equations 11.10 and 11.18) can be written as a function of relative displacements. Substituting into Equation 11.17, it can be obtained

δ um =

2 2 2 (1 + β II2 + β III ) 1 β II2 β III  + +   G G G e δ om IIc IIIc   Ic

11.19

which corresponds to the mixed-mode displacement at failure. The ultimate relative displacements in each mode, δum,i, can be obtained from Equations 11.13, 11.14 and 11.19

δ um , i =

β i δ um 2 1 + β II2 + β III

11.20

The damage parameter for each mode can be obtained substituting δom,i and δum,i in Equation 11.19. The interaction between matrix cracking and delamination in (04, 904)s carbon-epoxy laminates under low velocity impact was simulated using a cohesive damage model (de Moura and Gonçalves, 2004). Circular clamped plates of 50 mm diameter were tested and damage, identified by X-ray method, was constituted by: • •

a longitudinal long crack parallel to fibre direction in the outermost group of equally oriented layers and caused by bending loading; an extensive delamination located at the distal interface between differently oriented layers relative to the loaded surface; the delamination has a characteristic two-lobed shape with its major axis oriented on the direction of the lower adjacent ply.

In the numerical model only half plate was considered due to geometrical and material symmetrical conditions. With the aim to numerically simulate this damage mechanism, interface finite elements were placed at the critical interface in order to simulate the observed delamination, and at the vertical symmetry plane of the used mesh. The objective of these vertical elements was to model the onset and growth of the longitudinal bending crack in the outermost group of equally oriented layers which was observed to be the initial damage (see Fig. 11.5). This crack induces delamination in the adjacent interface as it can be seen in Fig. 11.6. This damage mechanism occurs in a progressive way, i.e., the growth of the vertical crack is associated with an increasing delamination. Numerical results agreed with the experiments. © 2008, Woodhead Publishing Limited

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11.5 Detail of the initial bending crack.

11.6 Delamination induced by the vertical crack.

(a)

(b)

11.7 Experimental and numerical delamination and crack length.

The shape of the delamination was accurately predicted (see Fig. 11.7). The delamination and crack lengths as a function of the maximum load were also in agreement (see Figs 11.8 and 11.10), although some non-negligible differences were noted in the delamination width (see Fig. 11.9). The global trend was captured for all cases. © 2008, Woodhead Publishing Limited


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50

l (mm)

40 30 20 10 0 1600

Experiment Numerical 1800

2000

2200

2400

2600

Pmax (N)

11.8 Delamination length (l) as a function of maximum load.

50

W (mm)

40 30 20 10 0 1600


2000

2200

2400

2600

Pmax (N)

11.9 Delamination width (W) as a function of maximum load.

50

lc (mm)

40 30 20 10 0 1600


2000

2200

2400

2600

Pmax (N)

11.10 Longitudinal crack length (lc) as a function of maximum load.


338


Although promising results were obtained with this approach, it should be emphasized that it cannot be considered an adequate prediction model for more general applications. In fact, in laminates with several different oriented layers it is not suitable to include numerous vertical interface elements in all layers to predict matrix cracking in any layer. The alternative is to consider that solid elements can also include a damage model in order to simulate damage inside layers. This can be done using continuum damage mechanics which is discussed in the next section.

11.3

Continuum damage mechanics

The classical approaches based on strength of materials usually assume that once matrix cracking arises, a sudden loss of material properties occurs, which is generally denominated by ply discount models (Hwang and Sun, 1989). However, it is known that in presence of matrix cracking, the composite does not lose its load-carrying capacity immediately. In fact, damage can be considered as the progressive weakening mechanism which occurs in materials prior to failure. It can be constituted by micro-cracking, voids nucleation and growth, and several inelastic processes that deteriorate the material. The analysis of cumulative damage is fundamental in life prediction of components and structures under loading. Tan (1991) proposed a progressive damage model relating the material elastic properties with internal state variables DiT and DiC (i = 1, 2, 6), ranging between 0 and 1, that are function of the type of damage. When a given failure criterion is satisfied, the material properties are abruptly reduced according to the respective residual strength experimentally observed. Each damage mode is predicted by the subsequent expressions: Fibre tensile fracture d d d E11 = D1T E11 ; ν 12 = ν 13 =0

11.21

Fibre compressive fracture d d d E11 = D1C E11 ; ν 12 = ν 13 =0

11.22

Matrix tensile failure d d d d E 22 = D2T E 22 ; E33 = D2T E33 ; ν 12 = ν 23 =0 d T d T d T G12 = D6 G12 ; G13 = D6 G13 ; G23 = D6 G23

11.23

Matrix compressive failure or shear cracking d d d d E 22 = D2C E 22 ; E33 = D2C E33 ; ν 12 = ν 23 =0 d C d C d C G12 = D6 G12 ; G13 = D6 G13 ; G23 = D6 G23


11.24


339

The author (Tan, 1991 and Tan and Perez, 1993) obtained good agreement with experimental results considering D1T = 0.07, D2T = D6T = 0.2, D1C = 0.14 and D2C = D6C = 0.4 . Although these types of model consider residual strength in accordance with the physical reality, they are mesh dependent during numerical analysis. To avoid the sudden loss of material properties and mesh dependency, continuum damage models combining strength of materials and fracture mechanics concepts are an appealing alternative. Material damage is simulated by introducing damage variables into the constitutive equations (Lemaitre and Chaboche, 1985). After the matrix cracking initiation is predicted, a gradual softening post-failure analysis is also performed by appropriately reducing the material properties within the elements where matrix cracking onset occurred. Ladevèze and Le Dantec (1992) developed a continuum damage mechanics formulation for orthotropic materials to account for ply degradation. Considering a damaged layer in a state of plane stress the strain-stress relationship take the general form = S

11.25

where and are vectors of elastic strain and stress, respectively. In a local system associated with orthotropy axes it can be written = (σ11, σ22, σ12)T; = (ε11, ε22, 2ε12)T

11.26

and S the compliance matrix

1   E1 (1 – d1 )  υ S =  – 12 E1   0 

υ 12 E1 1 E 2 (1 – d 2 ) –

0

   0   1  G12 (1 – d12 )  0

11.27

The damage parameters (d1, d2 and d12) define the damage state for the three types of stress loading and vary between 0 (undamaged state) and 1 (complete loss of stiffness). No additional parameter is used to simulate degradation in Poisson’s ratios as they are intrinsically affected during damage progression; υ12 is reduced by the factor (1-d1), since for a uniaxial stress σ11 it can be shown from Equations 11.25, 11.26 and 11.27 that –ε22/ε11 = υ 12(1 – d1); similarly it can be easily demonstrated that υ21 is affected by (1 – d2). In order to establish the evolution of damage parameters as a function of the damage growth the concept of strain energy density ϕ is used

ϕ = 1 T S 2


11.28

340


The theory also includes the concept of ‘thermodynamic forces’, Y1, Y2, Y12, associated with the internal damage variables d1, d2, d12. Those parameters are also considered as driving forces for damage development and are defined by Y=

∂ϕ ∂d

11.29

where Y = (Y1, Y2, Y12)T and d = (d1, d2, d12)T. Combining Equations 11.28 and 11.29 it follows

Y1 =

2 2 2 σ 11 σ 22 σ 12 ; Y = ; Y = 2 12 2 E1 (1 – d1 ) 2 2 E 2 (1 – d 2 ) 2 2 G12 (1 – d12 ) 2

11.30 In the absence of fibre breakage d1 is zero throughout the load history and the longitudinal modulus does not degrade; therefore only Y2 and Y12 driving forces should be considered to model matrix cracking. A linear combination Yˆ = Y12 + bY2 11.31 where b is a material constant, can be used to account for coupling between transverse tension and shear effects. To avoid healing phenomena the maximum value of Yˆ up to the current time t is defined as Yˆ ( t ) = max ( Y12 + bY2 )

11.32

τ ≤t

Experimental results for carbon-epoxy laminates showed that damage parameters can be written as d12 =

Yˆ – Y0 ; d2 = YC

Yˆ – YC′

Y0′

11.33

where Y0, YC, Y0′ and YC′ are damage evolution parameters. They are determined experimentally performing tension tests on [±45]s and [±67.5]s laminates (Ladevèze and Le Dantec, 1992). The model was tested on [±45]2s, [67.5, 22.5]2s and [–12, 78]2s laminates under tensile loading and excellent agreement was obtained with the respective experimental σ–ε curves. An approach similar to the one used in the cohesive damage model described in Section 11.2 is proposed by other authors (Crisfield et al., 1997; Pinho et al., 2006; de Moura and Chousal, 2006). In this case there is a softening relationship between stresses and strains instead of between stresses and relative displacements considered in the cohesive model. Consequently, in this case a characteristic length lc must be introduced to transform the relative displacement into an equivalent strain (see Fig. 11.4) © 2008, Woodhead Publishing Limited


Gic = 1 σ u ,i ε u ,i lc 2

341

11.34

This parameter was considered to be equal to the length of influence of a Gauss point in the given direction and physically can be regarded as the dimension at which the material acts homogeneously. The stress–strain relation can be written considering an equation similar to Equation 11.8 = (I – D)C

11.35

C being the stiffness matrix of the undamaged material in the orthotropic directions. Assuming that matrix cracking occur in mixed-mode I+II the damage model described in Section 11.2 can be adopted. The damage parameter is calculated by an expression similar to Equation 11.9 but considering strains instead of relative displacements. A linear softening law is also used. The properties are smoothly reduced due to the energy released at the FPZ. The material properties at a given Gauss point are degraded according to the assumed criterion. This leads to load redistribution for the neighbouring points, thus simulating a gradual propagation process. In summary, it can be affirmed that these methods allow simulating damage inside solid finite elements used to model composite layers and can be used to simulate matrix cracking phenomenon. A gradual degradation of properties instead of a sudden one avoids the singularity effects and minimizes the consequent mesh sensitivity.

11.4

Conclusions

Matrix crack and delamination are intrinsically associated in composite materials, namely under bending loads. The interaction between these two modes of damage constitutes a complex damage mechanism that has not been addressed at a realistic level. Such interaction is fundamental to be considered in a failure model prediction because one mode may initiate the other and they may intensify each other. Two different models come out to deal with the referred damage modes. Mixed-mode cohesive damage models join the positive arguments of stress-based and fracture mechanics criteria, overcoming their inherent difficulties. These models have been used with success to simulate delamination initiation and growth. They are usually based on interface finite elements including a softening relationship between stresses and relative displacements. Continuum damage mechanics is being applied to the simulation of matrix cracking. These models are based on the introduction of damage parameters into the constitutive equations in order to simulate material damage. These damage parameters increase smoothly with growing damage, leading to a slow degradation of material properties, instead of an abrupt one, which is not realistic, and cause mesh dependencies.


342


In summary, cohesive and continuum damage models are actually the most prominent numerical tools in order to simulate matrix cracking inducing delamination damage mechanism of composites. However, it should be recognized that an accurate methodology addressing all the realistic issues of this complex damage mechanism is still lacking. The solution points out to the development of a numerical tool incorporating the two kinds of models. The two methods (cohesive elements and damage mechanics) can coexist. In fact, both models can be implemented via user subroutines in commercial software. When the selected damage criterion in a solid element is satisfied the element fails simulating matrix cracking. This induces important relative displacements at the adjacent interfaces leading to delamination initiation and propagation according to the damage criterion of the cohesive elements. Some aspects should require special attention, such as the influence of stress concentration at the intersection of a critical matrix crack with a given interface. It is not clear up to now if the coexistence of the two methods will be able to accurately model such particularity. Some efforts should also be dedicated to the development of stress and fracture criteria adequate for the specificities of this damage mechanism.

11.5

References and further reading

Choi H Y, Chang F K (1992), ‘A model for predicting damage in graphite/epoxy laminated composites resulting from low-velocity point impact’, Journal of Composite Materials, 26, 2134–2169. Choi H Y, Downs R J, Chang F K (1991a), ‘A new approach toward understanding damage mechanisms and mechanics and mechanics of laminated composites due to low-velocity impact: part I – experiments’, Journal of Composites Materials, 25, 992–1011. Choi H Y, Wu H-Y, Chang F K (1991b), ‘A new approach toward understanding damage mechanisms and mechanics and mechanics of laminated composites due to low-velocity impact: part II – analysis, Journal of Composites Materials, 25, 1012–1038. Crisfield M, Mi Y, Davies G A O, Hellweg H B (1997), ‘Finite element methods and the progressive failure modelling of composites structures’, in Owen D R J, Oñate E and Hinton E, Computational Plasticity – Fundamentals and Applications, Barcelona, CIMNE, 239–254. Cui W, Wisnom M R (1993), ‘A combined stress-based and fracture-mechanics-based model for predicting delamination in composites’, Composites, 24, 467–474. de Moura M F S F, Chousal J A G (2006), ‘Cohesive and continuum damage models applied to fracture characterization of bonded joints’, International Journal of Mechanical Sciences, 48, 493–503. de Moura M F S F, Gonçalves J P M (2004), ‘Modelling the interaction between matrix cracking and delamination in carbon-epoxy laminates under low velocity impact’, Composites Science and Technology, 64, 1021–1027. de Moura M F S F, Marques A T (2002), ‘Prediction of low velocity impact damage in carbon-epoxy laminates’, Composites: Part A, 33, 361–368. de Moura M F S F, Gonçalves J P M, Marques A T, de Castro P M S T (2000), ‘Prediction © 2008, Woodhead Publishing Limited


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of compressive strength of carbon-epoxy laminates containing delamination by using a mixed-mode damage model’, Composite Structures, 50, 151–157. Gonçalves J P M, de Moura M F S F, de Castro P M S T, Marques A T (2000), ‘Interface element including point-to-surface constraints for three-dimensional problems with damage propagation’, Engineering Computations, 17, 28–47. Hwang W C, Sun C T (1989), ‘Failure analysis of laminated composites by using iterative three-dimensional finite element method’, Computers and Structures, 33, 41–47. Joshi S P, Sun C T (1985), ‘Impact induced fracture in a laminated composite’, Journal of Composite Materials, 19, 51–66. Krueger R (2002), ‘The Virtual Crack Closure Technique: History, Approach and Applications’, Nasa/CR-2002-211628, Icase Report No. 2002-10. Ladevèze P, Le Dantec E (1992), ‘Damage modelling of the elementary ply for laminated composites’, Composites Science and Technology, 43, 257–267. Lammerant L, Verpoest I (1996), ‘Modelling of the interaction between matrix cracks and delaminations during impact of composite plates’, Composites Science and Technology, 56, 1171–1178. Lemaitre J, Chaboche J-L (1985), ‘Mécanique des matériaux solides’, Paris, Dunod. Levin, K (1991), ‘Characterization of delamination and fiber fractures in carbon reinforced plastics induced from impact’, in M Jono, T Inoue, N Z Gakkai, and K K S Zaidan Mechanical Behavior of Materials-VI, Pergamon Press, Oxford, 1, 615–622. Liu D, Malvern L E (1987), ‘Matrix cracking in impacted glass/epoxy plates’, Journal of Composite Materials, 21, 594–609. Liu S, Kutlu Z, Chang F-K (1993), ‘Matrix cracking and delamination in laminated composite beams subjected to a transverse concentrated line load’, Journal of Composite Materials, 27, 436–470. Mi Y, Crisfield M A, Davies G A O, Hellweg H-B (1998), ‘Progressive delamination using interface elements’, Journal of Composite Materials, 32, 1246–1272. Olsson R, Asp L E, Nilsson S, Sjögren A (2000), ‘A review of some key developments in the analysis of the effects of impact upon composite structures’, in P Grant and C Q Rousseau Composite Structures: Theory and Practice, ASTM STP 1383, 12–28. Petrossian Z, Wisnom M R (1998), ‘Prediction of delamination initiation and growth from discontinuous plies using interface elements’, Composites Part A, 29A, 503–515. Pinho S T, Iannuci L, Robinson P (2006), ‘Physically based failure models and criteria for laminated fibre-reinforced composites with emphasis on fibre kinking. Pat II: FE implementation’, Composites Part A, 37, 766–777. Takeda N, Sierakowski R L, Malvern L E (1982), ‘Microscopic observations of cross sections of impacted composite laminates’, Composites Technology Review, 4(2), 40– 44. Tan S C (1989), ‘Effective stress fracture models for unnotched and notched multidirectional laminates’, Journal of Composite Materials, 23, 1082–1104. Tan S C (1991), ‘A progressive failure model for composite laminates containing openings’, Journal of Composite Materials, 25, 556–577. Tan S C, Perez J (1993), ‘Progressive failure of laminated composites with a hole under compressive loading. Journal of Reinforced Plastics and Composites, 12, 1043–1057. Whitney J M, Nuismer R J (1974), ‘Stress fracture criteria for laminated composites containing stress concentrations’, Journal of Composite Materials, 8, 253–265. Zou Z, Reid S R, Li S, Soden P D (2002), ‘Modelling interlaminar and intralaminar damage in filament-wound piped under quasi-static indentation’, Journal of Composite Materials, 36, 477–499. © 2008, Woodhead Publishing Limited

12 Experimental studies of compression failure of sandwich specimens with face/core debond F A V I L É S, Centro de Investigación Científica de Yucatán, A C, México; and L A C A R L S S O N, Florida Atlantic University, USA

12.1

Introduction

A sandwich panel is a composite structure consisting of two thin and stiff face sheets bonded to a thick and low weight core. Sandwich structures offer great potential for the development of optimized structures with high stiffness and strength-to-weight ratios. Sandwich structures are found in ground transportation, aircraft, naval and aerospace structures where low weight is required. A major concern when using such materials is the bonding between the face sheets and core. Adequate bonding is necessary to warrant load transfer between the face sheets in order to utilize the full potential that a sandwich structure offers. Face-to-core debonds may arise as a consequence of improper manufacturing procedures, or in-service loads such as hard object impact loading or blast situations. A face/core debond degrades the structural integrity of the sandwich and may led to premature failure, especially when the sandwich is loaded in compression. Under in-plane compressive loading, the face sheet over the debond may buckle and grow triggering structural collapse. The compressive failure mechanism of sandwich specimens containing impact damage or artificially created face/core debonds has been the subject of several research papers published in recent years. In this chapter, we will review experimental studies of the failure mechanisms of debonded sandwich columns and panels loaded in compression. Both in-plane and out-of-plane compressive loading will be examined.

12.2

Compression failure mechanism of debonded structures

Research on local buckling of a face/core (F/C) debond, and subsequent growth of the debond in a sandwich structure subject to in-plane compression loading is a fairly recent topic that shares some common characteristics with a previously much investigated topic, viz. delamination buckling and growth 344 © 2008, Woodhead Publishing Limited

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in composite laminates. Such studies date back to the early works of Chai et al., (1981), and Whitcomb, (1981). Another predecessor of the debond buckling and growth problem in sandwich structures is related to coatings, see e.g. Evans and Hutchinson (1984), where, in addition to mechanically induced stresses, residual compression stresses in the coating may contribute to buckling and subsequent growth. These studies showed that a near-surface delamination will buckle at a certain critical load, called the local buckling load, and that the out-of-plane deflection of the buckled delamination will lead to intense stresses at the interface at the delamination front, Fig. 12.1. As shown in Fig. 12.1, prior to buckling of the debond the crack does not disturb the stress field since the stresses act parallel to the interface. After buckling of the debond, however, large tensile and shear stresses are induced at the debond front, which may result in propagation of the debond. Figure 12.2 shows that the difference in load-displacement paths for the unbuckled and buckled states constitutes a source of strain energy for propagation of the debond. Several of the early works on delamination buckling of composites consider the simplified geometry of a laminate coupon containing a through-width, near-surface delamination loaded in uniaxial compression. Whitcomb (1981), was probably the first to address such a case using finite element analysis to examine a unidirectional (0°) carbon/epoxy laminate, adhesively bonded to a 6 mm thick aluminum substrate loaded in uniaxial compression. Whitcomb’s

Before buckling

P < PCR

Face/core debond

a

P

δ

σ

τ

After local buckling

τ

P > PCR

a P

δ

12.1 Schematic illustration of a compression loaded debonded structure before and after buckling of the debond.


346


Applied load, P

Unbuckled Released energy, ∆U

PCR

Buckled

Displacement, δ

12.2 Schematic illustration of edge load-displacement response for a plate containing a debond. The shaded area (∆U) represents the energy released upon debond buckling.

numerical results provide valuable insights in the load transfer mechanism occurring after delamination buckling. With increasing compressive load, Whitcomb observed a tendency of the crack tip to first open and then close after a substantial buckling deflection is reached. This is the result of an eccentricity in the load path due to a decrease in the fraction of load carried by the buckled sublaminate.

12.3

Compression failure of debonded sandwich columns

The basic mechanics of delamination buckling and propagation along the interface for composite laminates loaded in compression discussed above apply also to F/C debonds in sandwich structures. An in-plane compression loaded sandwich panel containing a sufficiently large F/C debond may fail by local buckling of the face sheet over the debond, followed by debond propagation and final collapse of the structure. The simplest configuration allowing the study of debond failure in sandwich structures loaded in in-plane compression is a sandwich column containing a F/C debond that runs over the entire width of the column, known as a through-width F/C debond. Such a configuration is schematically shown in Fig. 12.3. The test specimen is relatively simple to manufacture and the test provides valuable insights into the failure mechanism and fracture behavior of debonded sandwich structures.

12.3.1 Specimen design and test rig Proper design of test specimens and test fixtures are necessary elements of the experimental study of debonded sandwich columns. Large sandwich panels with composite face sheets are normally manufactured by a resin © 2008, Woodhead Publishing Limited


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P

Deflectometer (LVDT)

Strain gage

Debond

12.3 Sandwich column in a compression end-loading test fixture.

injection process known as vacuum assisted resin transfer molding (VARTM), and cured at room temperature. It is also possible to adhesively bond face sheets to the core. The F/C debond is commonly defined using a non-stick thin film (typically Teflon or Kapton) placed at the F/C interface prior to assembly of the sandwich. A film of the desired size is carefully positioned on the surface of the core, before bonding of the face sheets. After the panel has been assembled, the column specimens are cut to the required dimensions. It is recommended to mill machine the loaded edges of the specimens to achieve parallelism. Non-parallel specimen edges cause non-uniform load distribution in the specimen during compressive loading between parallel platens and may trigger premature failure of the specimen. Certain specimen conditions should be met in order to trigger failure by debond buckling rather than other failure mechanisms, viz. global buckling, face sheet compression or face sheet wrinkling. To avoid failure by global buckling, the specimen should be sufficiently thick. A core thickness of at least 25 mm is recommended for such test specimens. Another undesired failure mode is face sheet compression failure, which may be prevented by using enough 0° plies in the face laminates. The specimen length is another critical parameter. Too long specimens may fail by global buckling, although a certain length is needed in order not to constrain out-of-plane deflections of the debonded face sheet. Thick face sheets are stiff in bending and may not buckle. On the other hand, overly thin face sheets will buckle but may not propagate the debond due to insufficient supply of strain energy. Hence, if propagation is to be examined, a certain range of face sheet thicknesses is required. Failure maps are thus useful tools to select the specimen dimensions according to the desired failure mode. For example, Vaddakke and Carlsson (2004) achieved debond buckling and propagation using sandwich columns © 2008, Woodhead Publishing Limited

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consisting of 2 mm thick glass/vinylester face sheets over 25 mm thick PVC foam cores with 25 and 50 mm long debonds. The specimens were 150 mm long (100 mm span length and 25 mm for each gripping region), and 37.5 mm wide. Once the specimens have been designed, manufactured, and cut to their final dimensions, the next step is mounting the specimen in a specially designed test rig, Fig. 12.3, and setting up the required instrumentation. The sandwich specimen is end-loaded, which requires that the fixture and ends of the column are properly aligned and parallel. Rectangular steel clamps help to position the specimen at the center of the fixture and clamp the sandwich column. The load is applied to the end surfaces of the sandwich through steel platens attached to the crosshead and base of the test machine. Strain gages can be attached to both face sheets (normally at the center of the specimen) to monitor the axial strain response during loading, and to track possible signs of global buckling or wrinkling. Additionally, the tip of a deflectometer may be put in contact with the debonded face sheet to monitor lateral (outof-plane) deformation of the face sheet upon debond buckling.

12.3.2 Failure mechanism As mentioned in the previous section, an in-plane compression loaded sandwich column containing a sufficiently large F/C debond may fail by local buckling of the face sheet over the debond, followed by debond propagation and final collapse of the sandwich structure. The mechanism for debond buckling and collapse of the sandwich column may be explained with the aid of Fig. 12.4. For loads below the critical load for local buckling (P < Pcr) the sandwich is compressed uniformly and displaces along the axial (loading) direction P1 < P2 < P 3 ≈ P4 P2 P3

P1

P4

δ F/C debond

Initial (a)

Buckling (b)

Growth (c)

Face compression collapse (d)

12.4 Mechanism for debond failure in a sandwich column. (a) Initially straight configuration; (b) debond buckling; (c) debond propagation; (d) collapse by face sheet compression.



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without noticeable lateral deflection, Fig. 12.4a. When the load reaches Pcr, local buckling is observed which is identified by a sudden increase in the lateral deflection over the debonded region, Fig. 12.4b. If the strain energy stored is sufficient, debond growth parallel to the F/C interface and towards the fixture grips typically occurs immediately after buckling, Fig. 12.4c. Once the debond has propagated to its final length, the load carried by the buckled face sheet becomes negligible, while the load carried by the intact face sheet increases drastically until it becomes unable to carry the compressive load and the column collapses by face sheet compression failure, indicated by the horizontal line across the intact face sheet in Fig. 12.4d. Experimental investigations, see Vaddakke and Carlsson (2004), show that for low density (compliant) cores, the column may also collapse by wrinkling of the intact face and/or core shear. Accurate identification of the onset of local buckling requires some dedicated instrumentation. The buckling load can be identified experimentally by monitoring the strains at the specimen surfaces and the lateral deflection of the debonded face sheet. Figure 12.5 shows examples of plots of load vs. inplane strain (Fig. 12.5a) and load vs. lateral deflection (Fig. 12.5b) of a debonded sandwich column. The magnitude of the compression strain in Fig. 12.5 is indicated on the x-axis. At small loads (P < Pcr) the strains in both face sheets are indistinguishable, indicating uniform compression of the column, Fig. 12.5a. At the critical load Pcr (about 13 kN in this case), the strain records bifurcate and a strain reversal is observed for the debonded face sheet indicating that the strain changes from compression to tension, caused by outward buckling in the fundamental mode. This point is identified as the bifurcation point, and the corresponding load is the buckling (critical) load. Pcr can also be extracted from the load vs. lateral deflection plot, Fig. 12.5b. The lateral deflection is negligible until the critical load is reached, after which a substantial increase in the deflection is noticed while the load decreases. The rapid decrease in load indicates that the structure does not 8

Debonded face

10

Deflection, mm

Load, kN

15

Intact face

5

6 4 2 0

0 0

2

4 6 Strain × 10–3 (a)

8

0

4

8 Load, kN (b)

12

12.5 Load-strain (a) and load-lateral deflection; (b) responses of a typical sandwich column loaded in in-plane compression.


16

350


possess post-buckling strength, i.e. the critical and collapse loads are essentially the same. Such behavior is due to the lack of redundancy (single local path) of the through-width debonded column specimen.

12.3.3 Influence of face and core thickness, debond size and core stiffness The influence of debond size and core stiffness on the failure mode and critical load will be discussed here based on experimental findings. Avery and Sankar (2000), considered specimens with plain woven fabric carbon/ epoxy face sheets over a Nomex (Aramid) honeycomb core (cell size = 3.2 mm). The face sheets consisted of 1, 3, 5 or 7 plies, each nominally 0.221 mm thick. The core thicknesses examined were 6.4, 9.5 and 12.7 mm. The face sheets were bonded to the core using an autoclave cure process. To define a F/C debond, a thin Teflon strip of 12.7, 25.4, 38.1 or 50.8 mm width was introduced between face sheet and core before cure of the panel. The specimens were 102 mm long and 50.8 mm wide. The density of each core material and mechanical properties of the face sheets and cores are listed in Table 12.1. The mechanical properties listed in Table 12.1 refer to the loading direction of the column specimens. Experimental testing revealed a variety of buckling failure modes depending on the specific combination of face sheet thickness, core thickness and density and debond size. The failure modes observed are schematically illustrated in Fig. 12.6. The local antisymmetric (LA) failure mode refers to local failure of the column in a mode shape that is antisymmetric with respect to a plane through the center of the specimen and perpendicular to the specimen axis. Local symmetric buckling (LS) means that local failure occurred in a symmetric local mode. Global buckling was also observed in some specimens and this could involve the antisymmetric (GA) and symmetric (GS) modes, shown in Fig. 12.6. The LA mode is a quite uncommon failure mode which involves localized buckling of both face sheets in the way shown in Fig. 12.6. This Table 12.1 Face sheet and core mechanical properties for sandwich specimens tested by Avery and Sankar (2000). E = Young’s modulus, Xc = Compressive strength Material

Density (kg/m3)

E (MPa)

Xc (kPa)

Face Core Core Core

N/A 28.8 48 96

53,000 2.30 3.86 7.72

531 × 103 18.9 42.0 157


Experimental studies of compression failure P

P

P

P

GS

GA

LS

LA

351

12.6 Schematic of buckling modes observed in compression testing of sandwich column specimens containing a one-sided debond.

failure mode is related to local instability of an overly compliant structure, rather than debond buckling. Thus, only the LS mode, where the fully bonded face sheet remains straight, may be related to debond buckling. It should be pointed out that the specimen configuration and test program were designed according to a ‘statistical design approach’ as opposed to more traditional programs where one parameter at the time is systematically varied. The test program was designed to maximize the amount of empirical data on failure mechanisms and strengths, and the data were analyzed using a statistical correlation approach. Unfortunately, the local buckling loads were not experimentally determined. Only the ultimate loads were reported by Avery and Sankar, see Table 12.2. In this table, hc is the core thickness, hf the face sheet thickness, ρc the core density, a the debond size, b the specimen width, and Pult the ultimate load. The results listed in Table 12.2 reveal that only one specimen (#1) failed in the local antisymmetric mode and such a failure occurred at the lowest load reported. This specimen has very thin face sheets and core, as well as small debond size, which may promote such a failure mode. It is noted that all other specimens with thin face sheets (# 2–8) failed in a local symmetric (LS) mode. The debond did not propagate after local buckling. Given the variety of specimens and failure modes experimented, a large range of failure loads was observed (0.9 – 40.2 kN). The minimum failure load corresponded to the LA mode, followed by the specimens that failed in the LS mode. The global buckling (and face compression) failure modes required the largest loads. More recently, Vadakke and Carlsson (2004) conducted testing of sandwich columns containing a through-width debond implanted at one F/C interface. The sandwich columns examined consisted of 2 mm thick plain weave S2glass/vinylester face sheets over 50 mm thick PVC foam cores of various © 2008, Woodhead Publishing Limited

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Table 12.2 Test results for debonded sandwich column specimens. After Avery and Sankar, (2000) Specimen

hc (mm)

hf (mm)

ρc (kg/m3)

a (cm)

Pult/b (kN/m)

Failure mode

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

6.35 9.53 12.7 9.53 9.53 6.35 9.53 12.7 9.53 12.7 9.53 6.35 12.7 9.53 6.35 9.53

0.221 0.221 0.221 0.221 0.663 0.663 0.663 0.663 1.11 1.11 1.11 1.11 1.55 1.55 1.55 1.55

29 48 48 96 48 96 29 48 48 29 96 29 96 48 48 29

1.27 2.54 3.81 5.1 1.27 2.54 3.81 5.1 1.27 2.54 3.81 5.1 1.27 2.54 3.81 5.1

17.2 28.4 28.7 34.0 212 87.0 63.2 76.8 442 213 242 156 792 406 295 277

LA LS LS LS LS LS LS LS GS GA GA GS FC* GA GS GA

* FC = Face sheet compression failure.

densities (H45, H80, and H100, where the number following ‘H’ represents the foam density in kg/m3). The specimen length was 150 mm (with a 100 mm span length) and its width was 37.5 mm. Debonds examined were 25 and 50 mm long, centered at the midspan between the test rig grips, see Fig. 12.3. A set of sandwich columns without debond were also manufactured and tested as a baseline for strength reduction. The baseline specimens with H45, H80 and H100 PVC foam cores (without debond) failed by face sheet compression at average loads of 21.5, 23.5 and 26.5 kN, respectively. The failure mechanism of the debonded columns was local buckling of the region over the F/C debond, following the sequential steps shown in Fig. 12.4. Critical (failure) loads of the debonded sandwich columns (Pult) normalized with the corresponding failure load of the specimen without debond (Po) are presented in Fig. 12.7. A clear tendency of the (local) buckling load to decrease with increasing debond size is observed. Since no post-buckling strength was observed, the critical (buckling) load corresponds to the ultimate load. To highlight the role of the foam density, Fig. 12.8 shows the buckling load of the debonded column specimens normalized by the theoretical face sheet compression failure load ( PFC = 2 X cf h f b , where X cf is the compression strength of the glass/vinylester face sheets), for two debond sizes, a = 25 and 50 mm. It is clear that the compression strength of debonded sandwich columns increases with increasing foam density, and hence, increased core © 2008, Woodhead Publishing Limited

Experimental studies of compression failure 1.0

1.0

Pult0.5

H45 core hc = 50 mm

P0 0

1.0

Pult 0.5 P0

H80 core hc = 50 mm

Pult 0.5 P0

0 0

25 50 75 Debond length, mm (a)

353

H100 core hc = 50 mm

0 0

25 50 75 Debond length, mm (b)

0

25 50 75 Debond length, mm (c)

12.7 Normalized failure loads of debonded sandwich columns. (a) H45 core; (b) H80 core; (c) H100 core. 1.0 0.8

a = 25 mm

0.6

a = 50 mm

Pult PFC 0.4

0.2 0 0

50 100 Foam density, kg/m3

150

12.8 Normalized failure loads of debonded sandwich columns as a function of the core density.

stiffness. The core contributes to the compression strength of the sandwich by providing lateral elastic support of the bonded ends of the debonded face sheet (Avilés and Carlsson, 2005). Post mortem analysis of the failed columns showed that the initial crack (debond) in sandwich columns containing a low density core, as H45, exhibit a tendency to propagate into the core, while for higher density cores, the crack tends to propagate at the interface between the core and the resin rich layer near the face/core interface, see Vadakke and Carlsson (2004).

12.4

Compression failure of debonded sandwich panels

Debonds in sandwich panels are typically located in the interior, away from the edges of the panel. Such a debond is less critical than the through-width debond in a column discussed earlier, since a debond embedded in a panel is surrounded by intact material that provides a load path around the debond. © 2008, Woodhead Publishing Limited

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As stated before, a F/C debond is most severe when the panel is loaded in compression, since the debond may buckle and grow and cause structural collapse. The compression failure of sandwich panels containing a F/C debond has been examined by several investigators. Berggreen (2004) conducted linear and nonlinear finite element analysis as well as testing to investigate the behavior of such panels subject to uniform and non-uniform in-plane compression. Avilés and Carlsson (2006a) conducted a systematic experimental test program on debonded sandwich panels subject to uniform in-plane compression, and performed analytical and numerical analyses to estimate the local buckling loads and investigate the fracture parameters in the postbuckling regime (2005, 2006b, 2007). Although the majority of the studies of debonded sandwich panels consider in-plane compression, Falk (1994), and later Jolma et al. (2007), investigated sandwich panels loaded by lateral pressure (through-thickness compression). The results of these investigations will be discussed herein.

12.4.1 Specimen design and test rig Experimental study of debond sandwich panels requires design of test specimens and fixture to ensure that the desired failure mode, local buckling, occurs. For sandwich panels, the general design and manufacturing procedures to promote local buckling failure follow those outlined for columns (see Section 12.2.1). The face sheet and core thickness and face sheet stiffness are critical to promote debond buckling. Successful studies on debond buckling and propagation have been reported using 25 mm thick (or thicker) cores and 2 mm thick plain weave glass fibers/vinylester for the face sheets, see Avilés and Carlsson (2006a). Figure 12.9 shows a test rig used for in-plane compression testing of debonded (or intact) sandwich panels by Avilés and Carlsson (2006a). The fixture possesses flexibility to accommodate variations in the panel width and thickness. Rounded metal edge supports are used to constrain out-of-plane deflections of the vertical unloaded edges. The lower horizontal edge rests on a steel platen that constrains rotations. Load is introduced at the upper horizontal edge using a steel platen with a central groove (adjustable to the panel thickness). The steel platen is hinged at the center to promote uniform loading across the panel width. Controlled compressive displacement of the upper steel platen is applied using the movable crosshead of the testing machine. The high loads required to achieve local buckling in sandwich panels and the stiff contact surfaces of the test often lead to crushing of the loaded sandwich panel edges. In order to prevent this undesirable failure mode, aluminum or wooden inserts can be fit to the loaded (top and bottom) edges of the panel, which promote a more gradual load introduction into the panel suppressing crushing failure, see Fig. 12.9. © 2008, Woodhead Publishing Limited


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Strain gage

12.9 In-plane compression test fixture for sandwich panels.

Local buckling

Debond propagation

Face compression (fiber microbuckling)

12.10 Failure mechanism of a debonded sandwich panel subject to in-plane compression.

As for the columns, the onset of buckling may be detected by bonding ‘back-to-back’ strain gages on the debonded and intact face sheets. Deflectometers for monitoring the out-of-plane deflection of the debonded face sheet is also an effective method to detect buckling. In addition, advanced optical techniques may be used to obtain a full-field depiction of the in-plane and out-of-plane displacements, see e.g. Berggreen (2004).

12.4.2 Failure mechanism The debond failure mechanism for sandwich panels subjected to in-plane compression, Fig. 12.10, can be divided in three steps: local buckling, debond propagation and collapse. First, local buckling instability of the debonded © 2008, Woodhead Publishing Limited

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face sheet occurs, with the face sheet adopting a deflection shape with zero transverse displacement near the panel edges and maximum displacement at the center. Local buckling of the face sheet is evidenced by a significant increase in the out-of-plane deflection of the panel and a strain reversal, see Fig. 12.11. Once the debonded face sheet buckles, rapid propagation of the debond occurs towards the side panel edges, predominantly perpendicular to the direction of applied load. The debond propagates near the F/C interface for high density cores but may propagate in the core for low density cores. The final collapse occurs by face sheet compression and/or core cracking, depending on the core density and debond size. Sandwich panels with relative small debonds (e.g. 50 or 75 4

In-plane strain, × 10–3

Intact face 2

0 H45, 25 mm thick core Debond diameter = 75 mm

–2

Debonded face

–4 0

10

20 30 Load, kN (a)

40

50

Out-of-plane deflection, mm

4 Debonded face 3

0

–2 H45, 25 mm thick core Debond diameter = 75 mm –4 0

10

20 30 Load, kN (b)

40

50

12.11 In-plane and out-of-plane responses of a sandwich panel with a 75 mm debond loaded in in-plane compression. (a) in-plane strain vs. load, (b) out-of-plane deflection vs. load.



357

mm diameter) do not present post-buckling strength, i.e. the load bearing capacity of the panel is diminished (or lost) as soon as the face sheet buckles locally. On the other hand, panels with large debonds tend to present some post-buckling strength. Those panels typically present early signs of local buckling without noticeable loss of load-bearing capacity, and the load continues to increase up to the point where final collapse occurs. For such panels, the exact buckling load may be more difficult to asses given the continuous and smooth increase of the out-of-plane displacement, as opposed to the sudden increase presented for panels with smaller debonds, Fig. 12.11b.

12.4.3 Influence of debond size and core stiffness The influences of debond size and core stiffness on the critical local buckling load of debonded sandwich panels loaded in in-plane compression were examined by Avilés and Carlsson [2006a]. In their study, sandwich panels of 20 cm × 15 cm (length × width) containing a central implanted circular F/C debond were investigated. The panels used 2 mm thick glass/vinylester face sheets over PVC foam cores (H45, H100, and H200) and a 150 kg/m3 balsa wood core (SL910). The core thickness was mostly 25 mm, although two sets of panels utilized thinner (12.7 mm) and thicker (50 mm) H45 foam. The debond diameters were 50, 62.5, 75 and 100 mm. Panels were tested in the fixture described earlier, Fig. 12.9. The local buckling loads are shown versus debond diameter in Fig. 12.12. Recall that, in this case, local buckling resulted in collapse of the panels.

120 Balsa H200 80

Pc , kN

H100

P

H45

40

0 4

6

8 10 Debond diameter, cm

12

12.12 Local buckling loads for sandwich panels with different cores and debond sizes.


358


It is observed that, regardless of the core material, the compression strength decreases with increasing debond diameter. Since all the sandwich panels in Fig. 12.12 have identical face sheets and core thickness, the effect of the core stiffness can be isolated. The compression strength at a given debond size is observed to increase as the core stiffness increases, showing a strong influence of the core material which acts as a foundation support against local buckling. Although the balsa core is of lower density than the H200 PVC foam core, it has greater out-of-plane elastic modulus and compression strength than H200. It is also seen that the influence of core stiffness is more pronounced at smaller debond sizes.

12.4.4 Influence of debond geometry In order to investigate how the debond shape influences the compression strength of sandwich panels, the buckling (collapse) loads for panels with circular and square debonds were measured and displayed versus the debonded area in Fig. 12.13. The core employed was H100 PVC foam. For any specific debond area, sandwich panels with square debonds are weaker than those with circular debonds. The uniform compressive stress applied at the top edge of the sandwich panel is disturbed by the presence of the debond. A circular debond promotes a smoother transition of the stress field around the debond. The more drastic transition in the stress field associated with the sharp corners of the square debonds may be a cause for their earlier local instability. The earlier buckling of square debonds is consistent with finite element analysis results, see Avilés and Carlsson (2006b).

90 Square debond Circular debond

Pc , kN

60

30

H100, 25 mm thick core 0 10

30

50 70 Debond area, cm2

90

12.13 Critical (buckling) loads for panels with square and circular debonds centered in the panel. Core is H100 PVC foam.



359

12.4.5 Non-uniform in-plane compression Long ships tend to experience ‘sagging’ and ‘hogging’ loading conditions referring to longitudinal bending due to rough sea conditions. Boat hulls consist of panels attached to the remainder of the structure, and panels placed near the neutral axis will be subject to non-uniform horizontal tension and compression loads as illustrated in Fig. 12.14. Notice that the loading is sketched vertical for consistency with the nomenclature presented previously in this Section. Failure investigation of debonded sandwich panels subjected to non-uniform compression requires a specially designed test rig fitted into a robust testing machine. Such a fixture arrangement has been designed by Berggreen and Simonsen (2005). The test rig utilized a rigid steel frame with the loading beams being able to rotate. The loaded horizontal edges were reinforced with plywood to minimize end crushing. Sandwich panels of dimensions 58 cm × 80 cm were examined in such a study. The panels consisted of 3.2 mm thick glass/polyester face sheets bonded to 45 mm thick PVC foam cores (H80 and H200). Of the eight panels with H80 core, six had circular debonds of 10, 20, or 30 cm diameter (two replicates) while two panels contained no debonds. For the two panels with H200 core, one contained no debond and one a 20 cm circular debond. The panels were fitted into the non-uniform compression test rig and instrumented with strain gages. After testing, the panels were examined non-destructively using an air-coupled ultrasonic scanning equipment. Table 12.3 lists the failure mode, buckling load (when it was possible to detect it) and failure load of the sandwich panels examined. P

F/C debond

P

12.14 Debonded sandwich panel subjected to non-uniform compression.


360


Table 12.3 Failure modes, buckling loads and failure loads for sandwich panels loaded in non-uniform compression Core type

Debond dia. (cm)

Failure mode*

Buckling load (kN)**

Failure load (kN)

H80 H80 H80 H80 H80 H80 H80 H80 H200 H200

0 0 10 10 20 20 30 30 0 20

FC FC FC BDP BDP BDP BDP BDP FC BDP

N/A N/A N/A 260 168 113 – 99 N/A –

248 268 307 318 180 171 167 173 333 195

*

FC = Face compression, BDP = Buckling-induced debond propagation N/A = Not applicable, “–” not possible to determine.

**

All panels without debond failed by face compression. One panel with H80 core and a 10 cm diameter debond also failed by face sheet compression. Panels with larger debonds displayed local buckling of the debond followed by face/core debond growth. The difference between the failure load and the buckling load indicates that the panels showed post-buckling strength. It is noted that the test replicates are few and reported data show quite a bit of scatter, but some trends are clear. From the data for the H80 core panels with 10, 20 and 30 cm debonds, it is observed that buckling and failure loads decrease with increasing debond diameter. For the panels with large (20 and 30 cm) debonds, the compression strength was reduced by more than 30%. For the panels with H200 foam core, the compression strength of the panel with a 20 cm debond was 59% of the baseline. Post-mortem investigations of the tested panels revealed that the debond propagated perpendicular to the loading direction towards the highest loaded edge. Similar to the panels loaded uniformly, the crack propagated slightly underneath the interface for sandwich panels with low density foam cores (e.g. H80) and very close to the F/C interface for the denser core (H200).

12.4.6 Through-thickness compression In this section we will examine panels containing debonds loaded by a uniform pressure perpendicular to the plane of the panel, i.e. in throughthickness compression. Early work on through-thickness compression of debonded sandwich panels was conducted by Falk (1994). His research involved finite element analysis and testing of square 80 cm × 80 cm sandwich panels with a F/C debond located at the panel center, Fig. 12.15. © 2008, Woodhead Publishing Limited


361

y F/C debond

x

Top view

Side view

12.15 Sandwich panel with a F/C debond at the panel center loaded in through-thickness compression. Table 12.4 Failure mode and failure loads for sandwich panels loaded in throughthickness compression Core

Debond location

Debond diameter (cm)

Failure mode

Failure load (kN)

H80

Intact Center Edge Corner

0 20 10 20

Core shear Core shear Debond propagation Debond propagation

258 279 64 90

H200

Intact Center Edge Corner

0 20 10 20

Premature core shear No failure Debond propagation Debond propagation

263 >350 158 218

The panel rested horizontally over two simple supports near the panel edges. Pressure loading was achieved using a water-filled rubber bladder which distributed the pressure load uniformly. The F/C debond was placed between the pressurized face sheet and core. The set-up is then put it in a press and loaded through prescribed displacement of the platens. Seven panels were manufactured using glass/polyester face sheets bonded to H60 and H100 PVC foam cores, and a PMI foam core (WF51 or WF71). The face sheets were 2, 3 and 4 mm thick and the cores 15 and 30 mm thick. The debond diameters were 40 and 50 cm. Based on finite element analysis and limited experimental testing, Falk concluded that the fracture process for © 2008, Woodhead Publishing Limited

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this type of loading is mode II dominated and that the center location of the debond has a relative small influence on the compression strength of the sandwich panel. Post mortem examination of the tested panels revealed that the initial crack kinked deeply into the core. More recently, Jolma et al. (2007) examined 1.1 m × 0.85 m sandwich panels in a similar through-thickness compression test set-up. The panels consisted of 4.5 mm thick glass/polyester face sheets bonded to 30 mm thick H80 and H200 PVC foam cores. Debonds of 10 and 20 cm diameter as well as panels without debonds were investigated. The panels investigated can be divided into four groups depending on the debond location: (a) no debond (intact panel); (b) center debond; (c) edge debond with the debond located at mid-span near the long edge of the panel; (d) corner debond. Only one panel for each configuration was investigated, making a total of eight sandwich panels. The panels with no debond and centered debond failed by core shear failure or did not fail, while the panels with edge and corner debonds failed by debond propagation which triggered core shear failure. The failure loads and failure mechanism are listed in Table 12.4. The data in Table 12.4 indicate that a F/C debond located at the panel center does not influence the failure mechanism and compression strength of the panel, regardless of the core. On the other hand, when the debond is located at a panel edge or corner, failure is governed by debond propagation, and the strength of the sandwich panel is substantially reduced. The strength reduction is severe and can amount to 50% or more of that of the intact panels. The core shear stress is maximum near the test rig supports and minimum at the panel center. Thus, the larger shear stresses near the panel edges (close to the supports) promote mode II (shear) dominated debond propagation in the panels.

12.5

Acknowledgments

The research conducted by the authors reported herein was sponsored by the Office of Naval Research with Dr Yapa D. S. Rajapakse as the program manager. Thanks are also due to Shawn Pennell for producing the artwork and Alejandra Quesada for typing assistance.

12.6

References

Avery, J.L., Sankar, B.V. (2000). ‘Compressive Failure of Sandwich Beam with Debonded Face-Sheets’, J. Compos. Mater., Vol. 34 (14), 1176–1199. Avilés, F., Carlsson, L.A. (2005). ‘Face Sheet Buckling of Debonded Sandwich Panels using a Two-Dimensional Elastic Foundation Approach’, Mech. Advanced Mat. Struct., Vol. 12 (5), 349–361. Avilés, F., Carlsson, L.A. (2006a). ‘Experimental Study of Debonded Sandwich Panels Loaded in Compression’, J. Sandwich Struct. Mater., Vol. 8 (1), 7–31. © 2008, Woodhead Publishing Limited


363

Avilés, F., Carlsson, L.A. (2006b). ‘3D Finite Element Analysis of Debonded Sandwich Panels’, J. Compos. Mater., Vol. 40 (11), 993–1008. Avilés, F., Carlsson, L.A. (2007). ‘Post-buckling and Debond Propagation in Sandwich Panels Subject to In-plane Loading’, Eng. Fract. Mech., Vol. 74 (5), 794–806. Berggreen, C. (2004). ‘Damage Tolerance of Debonded Sandwich Structures’, PhD Thesis, Technical University of Denmark, Copenhagen. Berggreen, C., Simonsen, B.C. (2005). ‘Non-uniform Compressive Strength of Debonded Sandwich Panels – II. Fracture Mechanics Investigation’, J. Sandwich Struct. Mater, Vol. 7 (6), 483–517. Chai, H., Babcock, C.D. and Knauss, W.G. (1981). ‘One Dimensional Modeling of Failure in Laminated Plates by Delamination Buckling’, Int. J. Solids Struct., Vol. 17 (11), 1069–1083. Evans, A.G., Hutchinson, J.W. (1984). ‘On the Mechanics of Delamination and Spalling in Compressed Films’, Int. J. Solids Structures, Vol. 20 (5), 455–466. Falk, L. (1994). ‘Foam Core Sandwich Panels with Interface Disbonds”, Compos. Struct., Vol. 28 (4), 481–490. Jolma, P., Sergercrantz, S., Berggreen, C. (2007). ‘Ultimate Failure of Debond Damaged Sandwich Panels Loaded with Lateral Pressure – An Experimental and Fracture Mechanics Study’, J. Sandwich Struct. Mater, Vol. 9 (2), 167–196. Vadakke, V., Carlsson, L.A. (2004). ‘Experimental Investigation of Compression Failure of Sandwich Specimens with Face/Core Debond’, Compos, Part B, Vol. 35 (6–8), 583–590. Whitcomb, J.D. (1981). ‘Finite Element Analysis of Instability Related Delamination Growth”, J. Compos. Mater., Vol. 15 (9), 403–426.


Part IV Modelling delamination


13 Predicting progressive delamination via interface elements S H A L L E T T, University of Bristol, UK

13.1

Introduction

The significance of delamination has amply been expounded in other sections of this book and will not therefore be further described here. Suffice to say that it is of great concern for the design and structural integrity of composite components. Thus it is highly beneficial to have predictive techniques which are capable of determining the onset and progression of delamination growth. One such technique, which has been gaining prominence and credibility in the analysis community in recent years, is that of interface or cohesive zone elements. The advancement of this technique is such that it has now moved out of the academic and research environment and is being implemented in many of the major commercial finite element software packages for use in real engineering applications. This chapter reviews some of the background behind the development of interface elements with application to composites delamination. It then considers some of the formulations which have been proposed in the literature and looks at some applications of these to composite materials and structures.

13.2

Background to the development of interface elements

Interface elements are elements which model a thin or zero thickness layer between continua in a finite element analysis. The location of the interface elements are not themselves a prediction of the crack path but a plane of potential delamination which may or may not fail depending on the loading and the relevant failure criteria. For laminated composites this approach works well as the planes of possible delamination are generally well defined between the plies or at adhesive bond lines. Figure 13.1 shows how interface elements can be inserted between coincident or adjacent nodes in a finite element mesh. In 3D this can either be done discretely (effectively as a nonlinear spring representing the nodal area) or as an eight (or more)-noded 367 © 2008, Woodhead Publishing Limited

368


Ply elements Interface elements

Ply elements

(a)

Nodal area

(b)

13.1 Interface elements: (a) discrete or 1D form; (b) continuum or 3D form.

element with either finite or zero thickness. 2D formulations are of course also possible. The relative nodal displacements under an applied load can be separated out into normal and shearing components. The normal component represents a mode I type crack opening and the shearing a combination of mode II and III since these cannot be distinguished without knowledge of the direction of the crack front. For simplicity this shearing mode is referred to as mode II throughout the rest of this chapter. The constitutive relation of the interface element is based on a traction-displacement law which is generally elastic up to a stress-based failure criterion (initiation) and then undergoes a softening behaviour which describes an area under the curve equal to the critical fracture energy (GC) at complete failure (propagation).

13.3

Numerical formulation of interface elements

It has been argued that for compatibility with the cohesive zone model the initial stiffness should be infinitely high to model a perfect connection across the interface. The introduction of a finite stiffness to the interface element is, however, necessary for the finite element formulation. This may affect the overall compliance of a laminate if it is not sufficiently high, however excessively high values may cause numerical problems and so a balance needs to be struck.1 In practice the delamination will occur in the resin-rich region that occurs between the plies which does have a physical stiffness. This can then be used to set the initial stiffness of the elastic region of the element constitutive model. Fracture mechanics has long been used to predict crack propagation in a variety of materials including composites and this has been directly applied to the finite element method through the virtual crack closure technique (VCCT).2 This method allows a crack to propagate when the local crack tip strain energy release rate, as calculated from nodal displacements and forces, exceeds a critical value. A limitation of this approach is that it requires the © 2008, Woodhead Publishing Limited


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introduction of an initial flaw. By contrast the interface element approach combines a stress-based initiation criterion with a fracture mechanics-based propagation criterion such that no initial flaw is required and both initiation and propagation are captured in a single coherent analysis. This requires the introduction of a cohesive or ‘process’ zone ahead of the crack tip, a concept introduced by Dugdale3 and Barrenblatt4 in the 1960s. Material or elements within the cohesive zone that have exceeded the initiation criterion are governed by a traction-displacement relation. In order to satisfy the propagation criterion, the area under the curve should be equal to the critical fracture energy when traction is reduced to zero. The element or material is then deemed to have failed completely. There is experimental evidence of a cohesive or process zone ahead of a crack tip in polymer and composite materials5,6 which takes the form of a zone of plasticity or micro-cracking (Fig. 13.2). The size of this region has been shown to be somewhat smaller than that predicted by closed form solutions or observed in numerical analyses but it does show that the technique is indeed based on physical reality. Crack bridging effects, not evident in Fig. 13.2, are known to occur in some tests and may effectively increase the cohesive zone length, thus accounting for some of the difference between analytical predictions and physical measurements. To date a large number of different formulations for interface elements have been proposed in the literature. In concept they all follow the same principle of stress-based initiation followed by energy based propagation. One of the fundamental differences between formulations is the shape of the softening part of the curve used after initiation. The most commonly used and numerically simplest is the bi-linear law which is used here to describe the various features of a typical interface element formulation. Figure 13.3a shows the traction-displacement relationship for a single mode and Fig. 13.3b for a mixed mode case, which will be discussed later. The element is assumed to be elastic (linear in this case) up to the point at which a maximum stress is reached. This is the initiation criterion. After this is exceeded the element enters the propagation regime on further loading. The slope of the curve is such that at the point at which the tractions are returned to zero (the point of maximum displacement) the area under the curve is equal to the critical fracture energy. A definition of single mode behaviour for each of the normal and shearing modes is required, as well as an algorithm for the interaction between them. For the mixed-mode initiation criterion many authors use the quadratic stress interaction proposed by Brewer and Lagace:7 2

2

 max (σ I , 0)   σ II    +  max  = 1 max σ    σ II  I

13.1

where σI is the normal interlaminar tensile stress, σII the shear stress resultant © 2008, Woodhead Publishing Limited

370


10 µm

Crack growth (a)

10 µm

Crack growth (b)

13.2 Experimental evidence of ‘cohesive zone’ ahead of crack tip: (a) neat resin plasticity; and (b) fibre reinforced composite microcracking.5

of the interface, σ Imax the interlaminar tensile strength, and σ IImax the interlaminar shear strength. This assumes that the normal interlaminar compressive stress does not affect the onset of delamination. The Von Mises equivalent stress has also been used to define the maximum stress occurring in the interface element.8 Even though it is being applied to an anisotropic material the assumption is that failure occurs in the pure resin zone in between the oriented composite plies. There is more variation amongst the different interaction criteria proposed for mixed mode propagation. One generally used equation is the power law in the form


Predicting progressive delamination via interface elements Mode I maximum force

Traction

371

Traction

Mode II maximum force

Fmax

Gc X element area

Mode II r displa elative cemen t

M o di de sp I la rel ce at m iv en e t

Relative displacement

df

GI G + II = 1 GIc GIIc

(a)

(b)

13.3 Bi-linear traction displacement relationship for: (a) single mode; and (b) mixed mode. α

α

 G I  +  G II  = 1  G IC   G IIC 

13.2

with α being an empirical parameter derived from a best fit to data from mixed-mode tests. α is generally used in the range 1 to 2 with values of 1 proposed by Hallett and Wisnom,9 1.21 by Pinho et al.10 and 1.23 by Borg et al.11 Another widely used criterion is that of Benzeggagh and Kenane:12 η

G II  G IC + ( G IIC – G IC )  = GC  G I + G II 

13.3

where η is an empirically derived parameter. As well as the input parameters which govern the initiation and propagation criteria a further feature of the numerical formulation which can have an influence on results is the shape of the traction displacement curve (Fig. 13.4). A variety of different models have been proposed, the simplest in numerical terms being the bi-linear behaviour already described and used by a large number of authors.10,11,13–16 Tvergaard and Hutchinson17 used a trapezoidal rule and Xu and Needleman18 an exponential rule, though not with application specifically to composites whilst Cui and Wisnom19 proposed a perfectly plastic rule and Davies et al.20 a monotonically increasing one. Sudden changes in stiffness or force can result in numerical instabilities so a number of ‘smoother’ curves have been proposed such as the exponential form of Goyal et al.21 or the polynomial and linear/polynomial forms of Pinho et al.10 Shet and Chandra22 provide a detailed study on the implementation of three different shapes (exponential, trapezoidal and bi-linear) in finite © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites σ

σ

Exponential18

δ

σ

Trapezoidal17

δ

Linear/polynomial10

δ

σ

Perfectly plastic19

δ

13.4 Selected traction-displacement curves from interface element formulations.

element analyses which shows that shape of the traction displacement curve is an important factor albeit only shown in metals. Owing to the highly non-linear nature of the failure process, numerical analysis faces significant challenges to achieve stable and converged solutions. For implicit finite element analyses a number of different procedures have been proposed such as complex path following algorithms23 or viscous regularisation, with a parameter introduced in the commercial finite element code ABAQUS to aid convergence.24 Viscous regularisation induces stability in the analysis but care should be taken to ensure that excessively high values do not adversely affect global response. Other authors have adopted the use of explicit solvers to overcome difficulties with convergence in implicit analyses9,10 even though their applications have been predominantly quasi-static. As well as considering the stability of the solution sequence other features of the numerical models, such as the shape of the traction displacement curve or the number of elements in the process zone ahead of the crack tip, can also affect stability. Minimising runtime is always of great importance for the use of finite element techniques in real engineering applications. As a result there have been a number of attempts to address this issue specifically for interface elements. The formulations described up to this point have predominantly been for use between solid or continuum elements since coincident nodes are required. There are also formulations for use between layers of shell elements that assume that the material is initially connected at a position offset from © 2008, Woodhead Publishing Limited


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the surface by the shell thickness. The traction-displacement relation for separation acts at this point, controlled by the deformation of the adjacent shells.25 A series of stacked shells can then be built up to create a laminate which has delamination capability between the plies. This can either be done using a contact formulation for the cohesive model15,26 or continuum elements between the layered shells.27 An alternative strategy for model size reduction is to increase the size of elements that one can use. Mi et al.16 reported that at least two elements are required in the process zone at the crack tip to obtain a smooth solution and that the length of the process zone can be artificially increased by reducing the maximum stress to obtain greater model stability. Turon et al.1 have further formalised this as a strategy to allow users to increase the minimum element size. This is based on using existing closed form solutions to estimate the cohesive zone length. For a given mesh size, the maximum stress of the interface element is then appropriately reduced to ensure that there are sufficient elements in the cohesive zone. The predictions of cohesive zone length from closed form solutions are dependent on material properties (E, GC, σmax) and in slender specimens, the depth of material surrounding the crack.1,28,41 It has been shown that the analytical cohesive zone size is not always in agreement with that predicted by the interface element analyses20,28 thus indicating the need for caution when using this strategy. In the case of fracture-dominated problems the results from analyses with interface elements show little dependence on the stress based initiation criterion27 and strategies such as that proposed above are likely to be successful. There are, however, a range of strength-based or initiation-dominated problems for which these input parameters will have a more significant effect, as has been demonstrated by Mi et al.16 for overlap joints.

13.4

Applications

Interface elements have found application in a large variety of areas of composite failure prediction involving delamination. This technique is particularly successful in situations where high stress areas occur which can lead to initiation and subsequent propagation of delamination. Such high stress areas can occur in composite structures due to geometric stress raisers, discontinuities such as ply drops and property mismatches leading to free edge stresses.

13.4.1 Double cantilever beam (DCB), end loaded split (ELS) or end notched flexure (ENF) and mixed mode bending (MMB) In the simplest form, interface elements have been shown to be applicable for single or mixed mode tests which are characteristically used for generating © 2008, Woodhead Publishing Limited

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fracture toughness data. These can take the form of double cantilever beam (DCB), end notched flexure (ENF), end loaded split (ELS) and mixed mode bending (MMB) tests amongst others.29 It is tests such as these that have been used to develop and validate the numerical formulations described above and investigate such effects as variation in maximum stress, traction displacement shape and mesh size. Figure 13.5 shows some typical models and results, compared to analytical solutions and experimental results30 for mode I (DCB) and analytical solution only for mode II (4 point ENF). Such analyses are a useful verification of the implementation of cohesive zone models. They are, however, not in themselves generally useful predictive tools since they are largely a reproduction of the tests which themselves are used to generate the input data for the models. For true validation, applications and predictive capability we need to look at more complex models which represent real engineering examples.

13.4.2 Free edge delamination and other geometrically simple tests Probably the simplest application in terms of geometry, though not in terms of stress state, is the case of free edge delamination. Very high local stresses exist at the free edge of a laminate between plies of different orientations due to the different directional properties and the compatibility of strain requirement. This was one of the first applications of interface elements to composites by Schellekens and De Borst31 to predict failure strains in a number of different carbon/epoxy laminates. This study was used to demonstrate the independence of results on mesh refinement as well as the lack of influence of maximum traction. Fracture toughness was deemed to be the controlling parameter in this situation. Several ply thicknesses were analysed and results showed very good agreement with the inverse dependence of ultimate strain on the square root of ply thickness, as expected from fracture mechanics. Another study on scaling of ply thickness was carried out by Hallett et al.32 A range of failure modes were obtained experimentally which included both delamination and fibre fracture. In this case the fracture mechanics scaling could therefore not be applied. Full 3D analyses using interface elements for delamination between the plies were conducted as in the above study but additionally matrix cracks were selectively included within each ply using coincident nodes. This significantly changed the delamination pattern from extending along the entire length of the specimen to being localised at the point at which the matrix crack intersected the free edge (see Fig. 13.6). The inclusion of the matrix cracks further allowed the delaminations to progress through the thickness of the laminate to different ply interfaces as is commonly observed experimentally. For the delamination-dominated failure cases correlation was extremely good. For the fibre-dominated cases the © 2008, Woodhead Publishing Limited

Mode II-ENF

Mode I-DCB

700

50

600 500 Load (N)

Load (N)

40 30 20

300 200

0.5 mm mesh Experimental results Corrected beam theory

10

400

0.5 mm mesh Analytical propagation load

100 0

0 0

1

2

3 4 5 Displacement (mm)

6

7

8

0

1

2 3 Displacement (mm)

4


375

13.5 Mode I and mode II delamination models and results compared to experimental and analytical solutions.

5


60

376


Matrix crack

Delamination y zx

13.6 Localised delamination at the intersection of a matrix crack with the specimen free edge.

inclusion of the delamination was shown to significantly improve stress level predictions since the local delamination damage resulted in a stress concentration which was the cause of ultimate failure. Wisnom33 has used interface elements in models of short beam shear tests. This has shown that small cracks can have a significant effect on the apparent interlaminar shear strength and that their influence is variable, depending on their length and location relative to the loading. These results can be used to explain the scatter observed in experimental results in terms of small defects which typically occur in composite specimens. This work has been extended to look at delaminations starting from discontinuous (cut) plies, first in straight specimens in three point bending and then in curved specimens in four point bending.34 This provides a simple configuration which represents the typical high stress concentrations observed at ply drops in realistic structures. The straight specimens give a configuration in which mode II dominates whilst in the curved specimens mode I dominates. It was further seen that the mode ratio changed considerably as the analysis progressed and the interface element formulation must necessarily be able to take account of this.

13.4.3 Stiffened composite panels In aerospace applications composites are frequently used as large panels, for example as wing skins. These require the application of stiffeners (either cocured or post-bonded) to prevent out of plane deformations and increase © 2008, Woodhead Publishing Limited


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buckling strength (Fig. 13.7a). A common mode of failure is debonding of the stiffener from the panel, particularly in the runout region where local stresses can be relatively high. Tests which approximate the stringer flange bonded to the skin have been developed35 (Fig. 13.7b) and interface elements applied to predict failure of this simplified case.36 Good agreement was achieved for both the tensile and bending cases examined, with delamination initiating and propagating from the flange edge. It was found necessary to include thermal residual stresses resulting from the cool down from processing temperature to achieve good correlation, particularly for the bending case. The interface element analysis was further used to investigate the effect of including small defects in the region of damage initiation. A similar bending case that explicitly modelled the path of the crack from the flange/skin joint into the skin laminate and along the interface between the first two plies was investigated in 2D by Chen.37 It was found that the interface element maximum stress had a strong effect on results. A range of values was used for maximum stress and it was found that the predicted failure loads from these models encompassed the wide range of experimental data obtained.

13.4.4 Notched and open hole failure Another application which has made good use of interface elements is that of notched tension, though it is perhaps not immediately obvious why a delamination model is important to predict an in-plane tensile failure. It has been shown that the sub-critical damage which occurs local to the notch has a profound effect on redistribution of the in-plane stresses. This sub-critical damage takes the form of splitting within the plies and delamination between plies,38 and occurs at load levels well below ultimate failure. There have been many attempts to model this sub-critical damage growth and subsequent ultimate failure, some of which have included the intra-ply splits and Flange

Panel Stiffener blade

Run-out (a)

(b)

13.7 Typical blade stiffened panel and simplified experimental specimen.


378


delamination explicitly modelled with interface elements.9,39–41 Hallett and Wisnom9 used shell element models in which potential splits were inserted into the plies in the fibre direction, starting tangentially to a v-shaped edge notch. Interface elements were inserted between duplicate nodes to predict the growth of these splits. Further interface elements were inserted between plies to predict delamination. This model has been able to capture the damage patterns which have been observed experimentally as can be seen in Fig. 13.8. This technique has been extended to a full 3D analysis using solid elements and has been applied to the case of scaled open hole tensile tests.14 Exceptionally good correlation was obtained for all results where the failure was dominated by delamination (Fig. 13.9). In the case of failures dominated by fibre fracture, the delamination and splitting is still important as it significantly modifies the stress state used to evaluate failure. Again good correlation was obtained for these cases when combined with a statistically based fibre failure criterion.42 In all cases predictions were generated from independently measured material properties without the need for any fitting parameters. Existing predictive techniques for open hole tensile strength and size effects are predominantly based on elastic stress distributions and fitting to empirical data.43 This

Split

Delamination

[90/0]s 94% mean test failure load

Delamination

[+45/90/–45/0]s 98% mean test failure load

Split

13.8 Splitting and delamination damage at notch tip for a tensile specimen and predictions at the same scale and load level using interface elements.8



379

800 1-D scaling

2-D scaling

3-D scaling

700

Failure stress (MPa)

600 × ×

× × × ×FEM (Fibre failure) FEM (delamination) Test

500 400 300 200 100 0 P11 P21 P41 P81

P41 P42 P44 P48

P11 P22 P44 P88

Speciman size

13.9 Finite element analysis comparison to experimental results for scaled open hole tension tests.12

results in predictions which are limited to specific layups for which the data was generated. When interface elements are used, as above, then it is further possible to predict the effect of layup on open hole tensile strength.44 An important feature of this modelling technique is the inclusion of the intra-ply splits, which allow the delaminations at the different interfaces to join up through the thickness and thus propagate and cause ultimate failure. A similar approach has been adopted by Satyanarayana et al.45 but with a continuum damage model used for the intra-laminar splits. As a result differences in layup are not well captured by the analysis.

13.4.5 Impact Delamination as a result of impact to composites is one of the most commonly considered failure modes and is of great concern in practical design. This is due to the extensive internal damage which can occur as a result of an impact which leaves very little surface damage and is therefore hard to detect, so called Barely Visible Impact Damage (BVID). There is therefore a strong requirement to be able to predict the delamination damage resulting from impact and the associated reduction in Compression After Impact (CAI) strength. Interface elements have been applied to impact loading scenarios by a number of authors.15,26,46 Good predictions have generally been obtained for the damage modes occurring but quantitative peak force predictions remain only approximate. To reduce CPU time Iannucci46 has modelled an © 2008, Woodhead Publishing Limited

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impacted plate with a [45/-45/90/0]s layup in the explicit finite element code LS-Dyna using only one solid-shell element through the thickness either side of the midplane, representing sub-laminates, with interface elements in between. This was shown to correctly model the reduction in peak force which accompanies delamination propagation and associated loss in bending stiffness as well as reduced overall peak force values when compared with models without interface elements. Johnson et al.15 have used their stacked shell model to simulate a 16 ply laminate with four sub-laminates giving three possible planes of delamination to produce similar results. Borg et al.26 have modelled up to eight plies, each separated by interface elements. They have shown that by including variation in fracture toughness for different fibre orientations adjacent to the interface a more realistic delamination pattern can be obtained. In all of these studies in-plane damage was also considered using failure algorithms embedded in the laminate elements. Aymerich et al.47 and Nishikawa et al.48 have further explicitly modelled a central in-plane matrix crack in a cross-ply laminate using interface elements as has been done for other applications discussed earlier.14 Aoki et al.49 have embedded a more complex pattern of cracks and systematically compared results to those without matrix cracks and concluded that their inclusion has a significant effect in the model’s ability to accurately reproduce test results (Fig. 13.10).

13.5

Enhanced formulations

The areas of application for interface elements are expanding as interface elements become part of mainstream analysis techniques and incorporated in

2.34 kN

0°

Delamination 3

No matrix cracks

Delamination 4

Delamination 6

Matrix cracks included

2.25 kN

13.10 Delamination patterns with and without matrix cracks.49


0°


381

commercial codes. Work is also continuing on developing formulations for use in wider and a more diverse range of applications. One such enhancement is to take account of the effect of compressive stress on the shear failure behaviour. It is widely reported that as out of plane compressive stress on a laminate increases so does the resistance to delamination. This is often accounted for by an increase in the shear failure stress in failure prediction criteria. Goyal et al.50 have embedded an equation proposed by Sun et al.51 in their interface element formulation to enhance the maximum stress before softening. This has proved successful in modelling compressive tests on thick specimens with the fibre direction inclined at various angles to the direction of loading, thus giving different ratios of compressive to shear stress. It has further been shown by Cui et al.52 that out of plane compressive stress also has an enhancing effect on mode II critical fracture energy (GIIC). This enhancement has been incorporated into the interface element formulation of Li et al.53 together with a similar criterion for enhancement of interfacial strength to that of Goyal et al.50. It was shown that both the enhancement of initiation stress and fracture energy were necessary to obtain good correlation with results for cut ply and dropped ply experiments described by Cui et al.52. With the modified formulation four different test configurations could be modelled with a single set of input parameters which, without the enhancements, required four independent sets of input values. A second area of investigation to take the interface element into new applications is that of fatigue modelling. A number of investigators have included formulations which account for accumulation of fatigue damage as a function of a number of cycles.54–57 It is generally not feasible to model the fatigue loading on a cycle by cycle basis. These investigators have therefore adopted various forms of an ‘envelope’ strategy, illustrated in Fig. 13.11. Only the maximum load level is modelled and damage is accumulated over

Load applied in model remains constant

Load (P)

Pmax Load ramped from zero to Pmax

Additional model inputs required • Frequency • R-ratio

R = Pmin/Pmax Time (T) Fatigue law inactive

Fatigue law active

13.11 Schematic showing load ‘envelope’ for fatigue modelling.


382


a number of load cycles using a previously established relationship, taking account of the R-ratio as necessary. These models have been able to successfully reproduce Paris law plots for crack growth rates with good agreement to experimental data. Analysis to date has predominantly been limited to simple specimens such as DCB and ENF but as this technology is developed so it is being extended to structural applications where it is likely to prove a valuable tool. Further detailed information on one particular interface element formulation for fatigue can be found in Chapter 17.

13.6

Conclusions

In this chapter an introduction has been given to the concept of prediction of delamination initiation and propagation through finite element analysis and the use of interface elements. Many different formulations have been presented in the literature and here only a general overview has been given. Readers with an interest in numerical analysis are encouraged to follow up the details in the references provided. More important perhaps than the details of each and every formulation is the predictive capabili that this technique opens up to designers and engineers. Here several different applications have been presented which show scenarios in which interface elements have proved a useful tool. In general this is in regions of high local stress concentration where traditional stress-based failure criteria yield mesh-dependent results and fracture mechanics requires knowledge of a pre-existing starter crack. The ability to predict both initiation and subsequent propagation is highly compatible with a ‘no-growth’ design philosophy adopted for delamination in many structural applications. What has been less common in the literature to date, are examples of larger scale structural applications. This remains a computational challenge whilst mesh size is limited by the requirement to preserve several elements in the process zone. Computer power is, however, increasing at an exponential rate and it will perhaps not be long before fullscale structures are within the grasp of this powerful and effective technique.

13.7

Acknowledgements

The author would like to acknowledge the contribution of colleagues and students at the University of Bristol whose work has made writing this chapter possible.

13.8

References

1. Turon A, Dávila CG, Camanho PP and Costa J, An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models, Engineering Fracture Mechanics, 74(10), 2007, 1665–1682.



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2. Rybricki EF and Kanninen MF, A finite element calculation of stress intensity factors by a modified crack closure integral, Engineering Fracture Mechanics, 9, 1977, 931–938. 3. Dugdale DS, Yielding of steels containing slits, J. Mech. Phys. Solids, 8, 1960, 100– 104 4. Barenblatt GI, The mathematical theory of equilibrium cracks in brittle fracture, Adv. Appl. Mech., 7, 1962, 55–129. 5. Datta S, Investigation of the micromechanics of delamination in fibre reinforced composites, PhD Thesis, Imperial College, 2005. 6. Bradley WL, Relationship of matrix toughness to interlaminar fracture toughness, in Application of fracture mechanics to composite materials, ed. Friedrich K, Elsevier, 1989, 159–187. 7. Brewer JC and Lagace PA, Quadratic stress criterion for initiation of delamination, Journal of Composite Materials, 22, 1988, 1141–1155. 8. Petrossian Z and Wisnom MR, Prediction of delamination initiation and growth from discontinuous plies using interface elements, Composites Part A, 29, 1998, 503–515. 9. Hallett SR and Wisnom MR, Numerical Investigation of Progressive Damage and the Effect of Layup in Notched Tensile Tests, Journal of Composite Materials, 40(14), 2006, 1229–1245. 10. Pinho ST, Iannucci L and Robinson P, Formulation and implementation of decohesion elements in an explicit finite element code, Composites Part A, 37, 2006, 778–789. 11. Borg R, Nilsson L and Simonsson K, Simulation of delamination in fiber composites with discrete cohesive failure model, Composites Science and Technology, 61, 2001, 667–677. 12. Benzeggagh ML and Kenane M, ‘Measurement of Mixed-Mode Delamination Fracture Toughness of Unidirectional Glass/Epoxy Composites with Mixed-Mode Bending Apparatus,’ Composites Science and Technology, 56, 1996, 439–449. 13. Camanho PP, Dávila CG and De Moura MF, Numerical simulation of mixed-mode progressive delamination in the composite materials, Journal of Composite Materials, 37(16), 2003, 1415–1438. 14. Jiang WG, Hallett SR, Green BG and Wisnom MR, A concise interface constitutive law for analysis of delamination and splitting in composite materials and its application to scaled notched tensile specimens, Int. J. Numer. Meth. Engng, 69, 2007, 1982– 1995. 15. Johnson AF, Pickett AK and Rozycki P, Computational methods for predicting impact damage in composite structures, Composites Science and Technology, 61(15), 2001, 2183–2192. 16. Mi Y, Crisfield MA, Davies GAO and Hellweg HB, Progressive delamination using interface elements, Journal of Composite Materials, 32(14), 1998, 1246–1272. 17. Tvergaard V and Hutchinson JW, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, Journal of the Mechanics and Physics of Solids, 40(6), 1992, 1377–1397. 18. Xu XP and Needleman A, Numerical simulation of fast crack growth in brittle solids, Journal of the Mechanics and Physics of Solids, 42(9), 1994, 1397–1434. 19. Cui W and Wisnom MR, A combined stress-based and fracture-mechanics-based model for predicting delamination in composites, Composites, 24, 1993, 467– 474. 20. Davies GAO, Hitchings D and Ankerson J, Predicting delamination and debonding © 2008, Woodhead Publishing Limited

384

21.

22.

23.

24. 25.

26.

27.

28.

29. 30.

31.

32.

33. 34.

35.

36.

37.

Delamination behaviour of composites in modern aerospace composite structures, Composites Science and Technology, 66, 2006, 846–854. Goyal VK, Johnson ER and Davila CG, Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities, Composite Structures, 65(3-4), 2004, 289–305. Shet C and Chandra N, Effect of the shape of traction-displacement cohesive zone curves on the fracture response, Mechanics of Advanced Materials and Structures, 11, 2004, 249–275. Alfano G and Crisfield MA, Finite element interface models for the delamination analysis of laminated composites: mechanics and computational issues, International Journal for Numerical Methods in Engineering, 50, 2001, 1701–1736. Abaqus V6.5 Analysis User’s Manual, 2005, HKS Inc. Borg R, Nilsson L and Simonsson K, Simulating DCB, ENF and MMB experiments using shell elements and a cohesive zone model, Composites Science and Technology, 64(2), 2004, 269–278. Borg R, Nilsson L and Simonsson K, Simulation of low velocity impact on fiber laminates using a cohesive zone based delamination model, Composites Science and Technology, 64(2), 2004, 279–288. Reedy ED, Mello FJ and Guess TR, Modeling the Initiation and Growth of Delaminations in Composite Structures, Journal of Composite Materials, 31, 1997, 812–831. Harper P and Hallett SR, A detailed analysis of the cohesive zone in numerical simulations of composite delamination using interface elements, Engineering Fracture Mechanics in Press. Allix O, Ladevéze P and Corigliano A, Damage analysis of interlaminar fracture Specimens, Composite Structures, 31, 1995, 61–74. Asp LE, The effects of moisture and temperature on the interlaminar delamination toughness of a carbon/epoxy composite Composites Science and Technology, 58 (6), 1998, 967–977. Schellekens JCJ and De Borst R, A non-linear finite element approach for the analysis of mode I free edge delamination in composites, Int. J. Solids and Structures, 30(9), 1993, 1239–1253. Hallett SR, Jiang WG, Khan B and Wisnom MR, Modelling the interaction between matrix cracks and delamination damage in scaled quasi-isotropic specimens, Composites Science and Technology, 68, 2008, 80–89. Wisnom MR, Modelling the effect of cracks on interlaminar shear strength, Composites Part A, 27, 1996, 17–24. Petrossian Z and Wisnom MR, Predicting delamination initiation and growth from discontinuous plies using interface elements, Composites Part A, 29, 1998, 503– 515. Krueger R, Cvitkovich MK, O’Brien TK and Minguet P, Testing and analysis of composite skin/stringer debonding under multi-axial loading, J. Composite Materials, 34, 2000, 1263–1300. Camanho PP, Dávila CG and Pinho ST, Fracture analysis of composite co-cured structural joints using decohesion elements, Fatigue and Fracture of Engineering Materials and Structures, 27, 2004, 745–757. Chen J, Predicting progressive delamination of stiffened fibre-composite panel and repaired sandwich panel by decohesion models, Journal of Thermoplastice Composite Materials, 15, 2002, 429–442.



385

38. Hallett SR and Wisnom MR, Experimental Investigation of Progressive Damage and the Effect of Layup in Notched Tensile Tests, Journal of Composite Materials, 40(2), 2006, 119–141. 39. Wisnom MR and Chang FK, Modelling of splitting and delamination in notched cross-ply laminates, Composites Science and Technology, 60, 2000, 2849–2856. 40. Yashiro S, Murai K, Okabe T and Takeda N, Numerical study for identifying damage in open-hole composites embedded FBG sensors and its application to experimental results, Advanced Composite Materials, 16(2), 2007, 115–134. 41. Cox B and Yang Q, Cohesive models for damage evolution in laminated composites, International Journal of Fracture, 133, 2005, 107–137. 42. Wisnom MR, Green B, Jiang WG and Hallett SR, Specimen size effects on the notched strength of composite laminates loaded in tension, International Conference for Composite Materials 16, Kyoto 2007. 43. Awerbuch J and Madhukar MS, Notched strength of composite laminates: predictions and experiments – a review, Journal of Reinforced Plastics and Composites, 4, 1985, 153–159. 44. Hallett SR, Jiang WG and Wisnom MR, The effect of stacking sequence on thickness scaling of tests on open hole tensile composite specimens, 48th AIAA/ASME/ASCE/ AHS/ASC Structures, Structural Dynamics, and Materials Conference, Hawaii, 2007. 45. Satyanarayana A, Bogert PB and Chunchu PB, The effect of delamination on damage path and failure load prediction for notched composite laminates, 48th AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Hawaii, 2007. 46. Iannucci L, Dynamic delamination modelling using interface elements, Computers and Structures, 84, 2006, 1029–1048. 47. Aymerich F, Dore F and Priolo P, Prediction of impact-induced delamination in cross-ply composite laminates using cohesive interface elements, Composites Science and Technology, 2007, doi:10.1016/j.compscitech.2007.06.015 48. Nishikawa M, Okabe T and Takeda N, Numerical simulation of interlaminar damage propagation in CFRP cross-ply laminates under transverse loading, International Journal of Solids and Structures, 44, 2007, 3101–3113. 49. Aoki Y, Suemasu H and Ishikawa T, Damage propagation in CFRP laminates subjected to low velocity impact and static indentation, Adv. Composite Mater, 16 (1), 2007, 45–61. 50. Goyal VK, Schubel PM, Rome JI and Keough MR, Enhancement to the interfacial element formulations for the prediction of delamination, 48th AIAA/ASME/AHS/ ASC Structures, Structural Dynamics and Materials Conference, Honolulu, 2007. 51. Sun C, Quinn B and Oplinger D, Comparative evaluation of failure analysis methods for composite laminates, DOT/FAA/AR-95/109, 1996. 52. Cui W, Wisnom MR and and Jones MI, Effect of through thickness tensile and compressive stresses on delamination propagation fracture energy, Journal of Composites Technology & Research, 16(4), 1994, 329–335. 53. Li X, Hallett SR and Wisnom MR, Predicting the effect of through-thickness compressive stress on delamination using interface elements, Composites Part A: Applied Science and Manufacturing, 2008, 39(2), 218–230. 54. Robinson P, Galvanetto U, Tumino S, Bellucci G and Violeau D, Numerical simulation of fatigue-driven delamination using interface elements, International Journal for Numerical Methods in Engineering, 63(13), 2005, 1824–1848. 55. Muñoz JJ, Galvanetto U and Robinson P, On the numerical simulation of fatigue © 2008, Woodhead Publishing Limited

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driven delamination with interface elements International Journal of Fatigue, 28(10), 2006, 1136–1146. 56. Turon A, Costa PP, Camanho PP and Dávila CG, Simulation of delamination in composites under high-cycle fatigue, Composites Part A: Applied Science and Manufacturing, 38(11) 2007, 2270–2282. 57. Harper P and Hallett SR, A robust interface element law for delamination propagation under fatigue loading, ECCOMAS Thematic Conference on Mechanical Response of Composites, Porto, 2007.


14 Competing cohesive layer models for prediction of delamination growth S S R I D H A R A N, Washington University in St. Louis, USA and Y L I, Intel Corporation, USA

14.1

Introduction

The cohesive layer methodology has proven to be a powerful tool for the study of delamination growth in laminated composites. It is based on the postulation of a layer of cohesive material between the surfaces liable to delaminate and an energy release criterion for its decohesion. The literature abounds with successful applications of this model with variations in the cohesive law employed, the type of cohesive elements employed and the criterion employed to predict delamination initiation and growth. No attempt will be made here to provide a comprehensive review, but it behooves us to recognize salient landmarks. The basic concept, albeit in a simplified form, was proposed by Barenblatt (1962) and Dugdale (1960). Further developments are due to Williams (1963), and Schapery (1975) who modeled crack growth in visco-elastic media. Unguwarungasri and Knauss (1987) proposed a cohesive layer composed of a series of nonlinear springs. Extensive analyses of void nucleation and void coalescence in metals have been performed by Needleman (1987, 1990) using cohesive layer models. Further significant contributions came, among others, from Shahwan and Wass (1997), Ortiz and Pandolfi (1999), Yu (2001) and El-Sayed and Sridharan (2001). A majority of the investigators have used cohesive layers of zero thickness (see, for example, Hillerborg et al., 1976; Needleman, 1987, 1990; Alfano and Crisfield, 2001), although there is an implicit recognition of a cohesive zone upstream of the crack tip which must necessarily have some thickness, however small. The cohesive law takes the form of a relationship between the relative displacements between the delaminating layers and the interlaminar stresses. On the other hand Sprenger et al. (2000) and El-Sayed and Sridharan (2001, 2002) used cohesive layers of finite thickness. Here the cohesive law takes the form of a stress-strain relationship. Notwithstanding the extreme thinness of the cohesive layer used in the latter model, there appear to be significant differences in the predictions of these two types of models. In short the latter model predicts an early termination of crack growth and 387 © 2008, Woodhead Publishing Limited

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indicates a dependence of the extent of crack length attained on the initial crack length. Given the current interest in establishing a reliable methodology of tracing delamination growth, it is of considerable interest, to examine the individual capabilities and relative performances of these two classes of models. This chapter summarizes the results of an investigation on the relative merits of these types of models, using test cases of delamination in a double cantilever beam and a pre-cracked laminate under low velocity impact. The evidence offered by the test results indicates that the model with zero initial thickness is more reliable.

14.2

User material model

In the context of finite element analysis, the model with finite thickness takes the form of a set of standard elements available as a part of any finite element package (such as four noded elements in 2-D analysis), but with user supplied material properties. This will be referred to as the UMAT model (UMAT being the abbreviation for user material – a terminology borrowed from ABAQUS finite element package). Typically it receives (GreenLagrange) strain increments from the main program and returns the stresses and tangential stiffnesses of the material, these being co-rotational with the element. The strains are approximately logarithmic as their increments are based on the current geometry. Apart from the modal components of the critical values of strain energy release rates (SERR), the model is characterized by several parameters some of which are given, while others must be determined or assumed. Considering first the opening mode (mode I), El-Sayed and Sridharan (2001, 2002) noted that there are six parameters or characteristics of the cohesive layer. These are respectively: (1) (2) (3) (4) (5) (6)

GIc (SERR in mode I), E ( the initial stiffness), σmax, (the tensile strength), εf, the failure strain and initial thickness h0 of the layer The cohesive stress (σ) – strain (ε) relationship: σ = f (ε)

Of these item (1), GIc is generally available. Item (6), i.e. the cohesive law has to be assumed and is taken, typically, in the form of a cubic parabola which includes a strain softening phase. Items (1) and (6), taken together, provide two equations involving the other four unknowns:

σ = E εf (ξ – 2ξ2 + ξ3) where ξ = ε εf © 2008, Woodhead Publishing Limited

14.1

Competing cohesive layer models

G Ic = ε f h0

389

1

∫ σ dξ

14.2

0

The initial stiffness, E, and the strength σmax, are taken as those of the matrix or the matrix dominated transverse properties; the remaining items, viz. εf and h0 can be found from Equations 14.1 and 14.2. In the interests of realism, h0 may be taken as close to the process zone thickness as possible, making if need be some adjustments to the strength and stiffness values. In a mixed mode problem, one may select the maximum shear strength τmax to reflect that of the matrix or matrix dominated shear strength of the composite. Again using the SERR in mode II, GIIc and the assumed cohesive relationship, i.e. from equations similar to 14.1 and 14.2, one can determine shear stiffness and the failure strain in shear γf. While the cohesive material response in shear and tension are not coupled, an interactive failure criterion is used to reflect the mixed mode nature of delamination failure. Cohesive layer response in the present study: In the present study a highly simplified of cohesive law is employed. Both the stress-strain relationships in the two modes (σ – ε and τ – γ relationships) are linear up to the proportional limit where after the stress remains constant at a value equal to σmax or τmax (Fig. 14.1) as the case may be. This bilinear stress-strain relationship, denoted as ‘epp’ in a later section, is of the same form as that of an elastic-perfectly plastic material, but the material response is assumed to be elastic in the present study for simplicity. Though alternative forms of cohesive law including τ

τmax σ

σmax

ε (a)

(b)

14.1 (a) Normal stress-strain (σ – ε) relationship of the cohesive layer; (b) shear stress-strain (τ – γ) relationship of cohesive layer.


390


a strain softening phase are used for making comparisons, the ‘epp’ version of the model is the principal one adopted in the present study. There are some advantages to the form of the cohesive law chosen. The value of failure strain εf or γf need not be known a priori. The material stiffness never becomes negative and the stress values remain steady as the material is failing, thus promoting numerical stability in the computations. Yet another merit of this approach is that an estimate of the critical value of SERR that must be used in conjunction with the cohesive layer model can be obtained by simply computing strain energy release rate at an experimentally determined load corresponding to crack initiation ( Sridharan and Li, 2004).

14.2.1 Selection of parameters In contrast to the approach proposed by El-Sayed and Sridharan, there is freedom in the present approach to select h0 and σmax independently. ho is taken as either the actual thickness of a thin adhesive film between the layers in question or as mentioned earlier, the observed characteristic dimension of the process zone of composite materials which is around 0.02 mm. Again, the strength and initial stiffness are chosen as those of the more compliant of the delaminating layers in the transverse direction. To compensate for the absence of an unloading phase in the present model (‘epp’ version), the strength must be stepped down appropriately. It would seem, for example in pure mode I situation, σmax must be taken as one half of the actual strength of the material. We shall return to this issue in a later section in this chapter. Typically the transverse modulus E2 of the layer characterizes the initial stiffness. The other properties, viz. E1 and G12 (longitudinal and shear moduli), α1, α2 (coefficients of thermal expansion in the longitudinal and transverse directions) are all taken as the weighted averages of the respective values of the layers above and below the delamination plane.

14.2.2 Strain energy stored and failure criterion Modal components of strain energy stored in the elements are computed incrementally using the current thickness as the basis. Thus the values of the strain energy stored per unit area in the two modes at the end of the nth increment are given respectively by: n

GI = Σ σ ( i ) ∆ε ( i ) h ( i ) i =1

14.3a

n

G II = Σ τ ( i ) ∆γ ( i ) h ( i ) i =1

14.3b

There is no need to stipulate the ultimate strain, and failure is deemed to © 2008, Woodhead Publishing Limited


391

occur as soon as a certain mixed mode fracture energy criterion is satisfied, taken in the form:

GI G + II = 1 G Ic G IIc

14.4

As soon as this criterion is satisfied at the crack tip element, the stresses in the element are relieved and the crack tip moves to the next element.

14.3

User supplied element model

The second type of model considered in the present study is one of zero thickness initially, but the thickness is deemed to evolve from zero value and is equal to the current opening between the layers (Li and Sridharan, 2005b). A special element is developed that encapsulates all the characteristics of the debonding material. This is designated as the UEL (user supplied element) model. The UEL element module (see ABAQUS, Version 6.3) receives the total nodal displacements at the end of the previous increment of loading and iterative estimates of the incremental displacements for the current increment of loading. The module returns, at the end of an iteration, the set of nodal forces and tangential stiffness matrix to the main program. In a two-dimensional (plane stress or plane strain) analysis, elements employed are four-noded with two displacement degrees of freedom per node. The incremental deformation of the model is characterized in terms of two parameters, viz. effective normal relative displacement δ1 and effective shear displacement δ2, computed from the incremental nodal displacements and the current thickness of the cohesive layer. The normal and shear stresses in the element are related to δ1 and δ2 respectively via bilinear elastic stressrelative-displacement relationships, as in the UMAT model. Nodal forces and the stiffness matrix are found in terms of the stress state in the element. These are transformed to the global axes and returned to the main program.

14.3.1 Degrees of freedom, transformation matrix Figure 14.2 shows the four-node element. The degrees of freedom associated with the ith node are (ui, vi). The degrees of freedom are arranged in the form: {d} = {u1 v1 u2 v2 u3 v3 u4 v4}T

14.5

The rotation of the element is given with sufficient accuracy by:

θ = tan –1  

[ v 2 + v 3 ] – [ v1 + v 4 ]  2 le 

14.6

The displacement components at the ith node referred to the local axes can be obtained from: © 2008, Woodhead Publishing Limited

392

Delamination behaviour of composites v 3 ; F6

u 3;

F5

C′ v 4; F 8

y, v

v 2;

u 4;

D′

F7

B′

F4 F3 u 2;

θ v 1, F2

A′

u 1;

F1

x, u

14.2 Tangential and normal displacements and nodal forces of the cohesive layer element.

ui   c  =   vi   – s

s  ui    where c = cos (θ) and s = sin (θ) c   v i 

14.7

An 8 × 8 transformation matrix [α] of block diagonal form can be constructed using the 2 × 2 transformation block in Equation (14.7).

14.3.2 Incremental deformation parameters Incremental normal strain in the element may be approximated as: ∆ε n = 1 [ ∆v 3 + ∆v 4 – ∆v 2 – ∆v1 ] 2 hc

14.8

where hc is the current thickness of the cohesive layer element and approximated at the end of previous increment as: hc = 1 [ v 3 + v 4 – v1 – v 2 ] 2

14.9

Incremental normal relative displacement (crack opening in mode I) takes the form ∆δ 1 = 1 [ ∆v 3 + ∆v 4 – ∆v 2 – ∆v1 ] 2

14.10

Incremental shear strain in the element may be approximated as: ∆γ = 1 [ ∆u3 + ∆u 4 – ∆u1 – ∆u 2 ] + 1 [ ∆v 2 + ∆v 3 – ∆v1 – ∆v 4 ] 2 hc 2 le 14.11 © 2008, Woodhead Publishing Limited


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Incremental shear slip across the element takes the form:

h ∆δ 2 = 1 [ ∆u3 + ∆u 4 – ∆u1 – ∆u 2 ] + c [ ∆v 2 + ∆v 3 – ∆v1 – ∆v 4 ] 2 2 le 14.12 Li and Sridharan (2005b) recognized the second term on the right hand side which depends on the current thickness of the element, to be of some importance, but it appears to have been ignored by previous investigators. It is not generally negligible and is responsible for the agreement in the crack initiation predictions of the UMAT and UEL models, as will be seen in a later section.

14.3.3 Nodal forces and stiffness matrix The normal stress – the crack opening displacement σ – δ1 relationship and the shear stress – shear slip relationship τ – δ2 are taken in the same forms as the σ – ε and τ – γ relationships of the UMAT model (Fig. 14.1.a,b). The nodal force vector {F} is obtained from stresses assumed uniform over the element and is given by (subscripts 1–8 correspond to 8 degrees of freedom, refer to Fig. 14.2): F1 = F3 = – F5 = – F7 = –

τ le 2

σ l e τ hc σl τh – ; F4 = e + c ; 2 2 2 2 σ l e τ hc σ l e τ hc + ; F8 = – – F6 = – 2 2 2 2

F2 =

14.13a-h

where

τ=

δ2 τ (δ 2 < δ 20 ), δ 20 max

τ = τmax (δ2 > δ20)

σ=

δ1 σ (δ 1 < δ 10 ), δ 10 max

σn = σmax (δ1 > δ10) In tangential stiffness, matrix can be obtained from the expression of nodal forces as:

K ij =

∂Fi ∂d j

14.14

In order to maintain the positive definiteness of the matrix, suitably assumed small positive values replace zero values along the leading diagonal. © 2008, Woodhead Publishing Limited

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14.3.4 Strain energy stored and failure criterion At the nth increment of loading/time the total strain energy stored per unit area in each mode in the element is given as a sum of incremental contributions as follows: n

G I = Σ σ ( i ) ∆δ 1( i ) i =1 n

G II = Σ τ ( i ) ∆δ 2( i )

14.15

i =1

The mixed-mode failure criterion in Equation 14.4 is once gain employed to signal failure of the crack tip element.

14.3.5 Selection of parameters The proportional limit stresses σmax, τmax must reflect the actual strength of the material and the selection of these values is discussed in a later section. δ10, δ20 are the proportional limits of δ1, δ2 and are prescribed taking into consideration of the stiffness of the material and the thickness of the process σ σ zone. They can be estimated as δ 10 ≈ max t ; δ 20 ≈ max t where t represents E2 E12 process zone thickness. Only an order of magnitude estimate of t is necessary (≈ 10–2 mm). In the following sections, the analytical predictions of the two models are compared with test results. The selection of model parameters in each case is also briefly discussed. The examples considered are a double cantilever specimen and delamination of a pre-cracked layered plate under lateral impact.

14.4

Double cantilever problem

Robinson and Song (1991) give details of an experimental program on the study of crack growth in a double cantilever beam (DCB) problem. The specimen is a (00)24 unidirectional laminate made of carbon fiber epoxy XAS-913C. Figure 14.3 shows the specimen geometry. The length of the specimen is taken as 130 mm. The significant material properties of the composite listed by Robertson and Song are as follows: E1 = 126 GPa ; E2 = 9.5 GPa; ν12 = 0.263, Tensile strength = 57 MPa and GIc = 0.281 N/mm.

14.5

UMAT model: details of the study and discussion of results

It was found earlier on in the course of the investigation that the UMAT model consistently predicted smaller crack growth than observed in the test. © 2008, Woodhead Publishing Limited


395

P 2h = 3.0 mm w = 30 mm a = 29 mm

δ 2h

a

P

14.3 Robinson-Song (1991) double cantilever beam test specimen.

With a view to identify the root cause of this discrepancy, a detailed parametric study was undertaken. Finite element analysis The finite element mesh used in the present study consisted of four noded plane strain elements with reduced integration (CPE4R, ABAQUS, version 6.3, 2002) throughout. Two alternative finite element configurations were used, one with element size of 0.05 mm × 0.05 mm and the other with size 0.02 mm × 0.02 mm (with the aspect ratio of unity). Loads are applied by displacement control by prescribing the crack opening, δ at the free end. The increment was such that not more than a single element in the cohesive layer failed in an increment. The stress-strain relations of the cohesive layer Three alternative forms of elastic (perfectly reversible) stress-strain relationships were tried to judge not only their accuracy but also the facility with which numerical convergence is achieved during crack growth. In all the cases considered, a linear stress-strain relationship was assumed till the maximum stress is reached. Beyond the proportional limit, the following alternative stress-strain relationships were considered: (a) A softening relationship designated as ‘ecu’- standing for ‘elastic cubic unloading’ in the form of a cubic parabola with zero slopes at the beginning and ending of this phase. The ultimate strain εf can be obtained in terms of GIc which is given. (b) An stress-strain relationship designated ‘epu’ – standing for ‘elastic plastic unloading’ in the form of a yield plateau up to 0.7εf followed by a linear unloading phase to a value of stress equal to σmax/10 and strain of 0.95 ε f is attained. Thereafter the stress remains constant till failure. •


396

Delamination behaviour of composites σ

ε

σ

ε σ

ε

14.4(a) σ– ε ‘ecu’ relationship; (b) σ – ε ‘epu‘ relationship; (c) σ – ε ‘epp’ relationship.

(c) A bilinear relationship in which stress remains constant beyond the linear range at σmax till failure occurs designated as ‘epp’ as it is reminiscent of elastic perfectly plastic material response. Parameters of the model and the cases studied The most significant parameters that influence the final predictions are: σmax, ho and the assumed stress-strain relationship of the cohesive layer. To this we add another parameter, ᐉo the initial crack length as it was suspected that the relationship between the load and crack length may depend on this parameter. Table 14.1 lists the various combinations of these giving us in all 13 cases to be studied. The following scheme was adopted in the designation of the cases: σmax_ho_ cohesive law_ᐉo. E was taken as E2 of the composite material in all cases. © 2008, Woodhead Publishing Limited


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Table 14.1 UMAT model family: member designation and properties (GIc = 0.281N/ mm and E2 = 9500 MPa for all cases) Member # (designation)

σmax

h0

Constitutive relationship

Initial crack length

1 (57_05_ecu_29) 2 (57_02_ecu_29) 3 (57_05_epu_29) 4 (57_05_epp_29) 5 (57_05_epp_35) 6 (57_05_epp_40) 7 (57_05_epp_45) 8 (57_05_epp_50) 9 (28.5_05_epp_29) 10 (28.5_05_epp_35) 11 (28.5_05_epp_40) 12 (28.5_05_epp_45) 13 (28.5_05_epp_50)

57 57 57 57 57 57 57 57 28.5 28.5 28.5 28.5 28.5

0.05 0.02 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

1 1 2 3 3 3 3 3 3 3 3 3 3

29 29 29 29 35 40 45 50 29 35 40 45 50

120

100 #4 80

#3

Load P, N

#1 60

40 # 1 (57_05_ecu_29) # 3 (57_05_epu_29) # 4 (57_05_epp_29) Experimental results

20

0 0

0.5

1

1.5 2 2.5 Displacement δ, mm

3

3.5

4

14.5 A comparison of cohesive layer models with differing stressstrain relationships.

Discussion of results Figure 14.5 shows the P vs. δ (crack opening displacement) for the members 1, 3 and 4 of the UMAT family listed in Table 14.2. The cohesive layer models for the three cases have the same σmax and ho, but differ in the form of the assumed stress-strain relationship beyond the proportional limit. The © 2008, Woodhead Publishing Limited

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Table 14.2 Loads at crack initiation for different initial crack lengths (UMAT results) Station point

Crack length (mm)

P/2w (N/mm)

P/h1.5 (N/mm1.5)

GI (N/m)

GII (N/m)

GI G + II G Ic G IIc

Test configuration – 1 A 1.92 B 4.8 C 7.16 D 11.12

2.25 2.58 2.93 3.80

58.84 67.47 76.63 99.38

204.2 204.2 203.9 203.1

36.5 36.6 36.9 37.2

1.00 1.00 1.00 1.00

Test configuration – 2 A 12.72 B 16.24 C 20.40 D 23.40

7.90 8.74 10.06 11.25

71.79 79.43 91.42 102.24

197.1 197.1 196.4 196.3

55.1 55.8 56.3 57.0

1.00 1.00 1.00 1.00

initial loading phase of the models is computed using the nominal initial crack length of 29 mm. Also shown in the figure is the unloading branch of the P-δ relationship obtained in the experiment since the emphasis here is to compare the rate of unloading as the crack grows. (The initial linear P vs. δ relationship reported by Robinson and Song is not consistent with the initial crack length of 29 mm and is therefore not shown here.) Of the three members with differing constitutive relationships, the member #4 with ‘epp’ constitutive relationship gives an unloading curve which captures the trend of the experimental result at least in the initial phase of crack growth. The results from the other two members (#1 and #3 with elastic-cubic-unloading (ecu) and elastic-plastic-unloading (epu) relationships respectively) are almost the same, both indicating earlier crack initiation with a lower value of P than that given by member #4 and the experimental results. However the corresponding unloading characteristics run somewhat parallel to that given by member #4. From the point of view of numerical convergence, member #4 (epp model) traces the unloading phase with the greatest facility with a couple of iterations for each prescribed increment of δ.. (crack opening). The convergence characteristics of ‘ecu’ model is better than those of ‘epu’ model. The most significant feature of all the results is that all the UMAT models indicate that after the initiation and some initial growth, the crack growth comes to a standstill and the load begins to increase with crack opening. Such a prediction is clearly at variance from experimental results which indicate steady crack growth and a continued unloading after crack initiation. Influence of ho Figure 14.6 shows a comparison between the P-δ relationships given by members #1 and #2 which differ only in the value of h0 (0.05 mm vs. 0.02 mm). It is clearly seen the thickness of the cohesive layer does not affect the © 2008, Woodhead Publishing Limited


399

120

100

#2

Load P, N

80 #1 60

40

20

#2 (57_02_ecu_29) #1 (57_05_ecu_29)

0 0

0.5

1 1.5 Displacement δ, mm

2

2.5

14.6 A comparison of results from models with differing cohesive layer thickness.

performance of the model provided its value is small enough. Once again it is seen that crack growth is shut off and the P begins to increase with δ after a certain amount of crack growth. Influences of σmax and ᐉ0 Figure 14.7 compares the P-δ relationships given by two sets of cases of the UMAT model studied. Of these, the first set (set I : cases #4, 5, 6, 7, 8) has σmax equal to the reported transverse strength of the material (57 MPa), while the second set (set II : cases #9, 10, 11, 12, 13) has a value equal to one-half (i.e., 28.5 MPa) of this value. Within each set, the initial crack length is varied from 29 mm to 50 mm, through 35, 40 and 45 mm respectively. Note that h0 (= 0.05 mm) and the cohesive law (‘epp’) are the same for all the cases (thus in Fig. 14.7 each case is identified by σmax and ᐉ0 only). In Fig. 14.8, P-δ characteristics given by the first group are plotted alongside the experimental unloading curve. It is interesting to see that as far as the crack initiation and the crack growth immediately thereafter are concerned, set I is seen to closely follow the experimental result. Members of set II consistently predict earlier crack initiation with the unloading branch running roughly parallel to that of the corresponding member of group I. The discrepancy is small, but cannot be ignored. The results depicted in Fig. 14.8 indicate that it is possible to trace the entire history of crack growth using the UMAT model by a program of loading consisting of the following phases: © 2008, Woodhead Publishing Limited

400

Delamination behaviour of composites #4

100 #5 #9

80

#6

Load P, N

#10

#7

#8

#11

60

#12 #13

Experimental

40 # 9 (28.5_29) # 11 (28.5_40) # 13 (28.5_50) # 5 (57_35) # 7 (57_45) Experimental results

20

# 10 (38.5_35) # 12 (28.5_45) # 4 (57_29) # 6 (57_40 # 8 (57_50)

0 0

1

2 3 Displacement δ, mm

4

5

14.7 A comparison of results from two sets of cohesive layer models with differing values of strength (28.5 MPa vs 57 MPa) for various initial crack lengths (29 – 50 mm). 120

100

#4 #5

Load P, N

80

#6 #7 #8

60 Experimental

40

Experimental results # 5 (57_05_epp_5) # 7 (57_05_epp_45)

20

# 4 (57_05_epp_29) # 6 (57_05_epp_40) # 8 (57_05_epp_50)

0 0

1


4

5

14.8 A comparison of simulation results with experimental results for the DCB problem.

(a) Loading phase with an initial crack length ᐉo consisting of a largely linear elastic response (not withstanding some nonlinear material response in the crack tip elements), crack growth initiation, a small crack extension (ᐉo to R1). © 2008, Woodhead Publishing Limited


401

(b) Unloading (assuming the cohesive layer material is nonlinearly elastic, the strains in the cohesive layer element that have not yet failed and elsewhere in the structure are completely recovered). (c) Reloading with the new crack length R1, in the sanse manner as described in (a), and attaining a new crack length of R2. and so on. Used this way, the UMAT model could perform the role of identifying directions of crack growth in a 3-D problem but not of tracing significant crack extensions. It is observed that the ‘epp’ version of the model using full strength predicts crack initiation and the initial unloading phase associated with crack growth correctly. But in every case considered, the crack growth shuts off soon after initiation and the load begins to increase. This happens irrespective of the initial crack length, the values of other model parameters or the form of the cohesive material response postulated. Thus the UMAT model is unable to trace large crack extensions. Diagnosis of deficiency of the UMAT model It is natural ask why the model predicts (a) a smaller load for the crack initiation in some cases (cases #1, 3, 9, 10, 11, 12 and 13), and (b) a termination of crack growth after some growth has occurred. These questions are considered in the sequel. The issue of earlier crack initiation The model (the ‘epp’ version) postulates for the cohesive material constant stress once the proportional limit is attained, i.e. it does not consider the phenomenon of strain softening. If a linear or a cubic unloading law (such as the ‘ecu’ relationship) is indeed realistic, then, it appears logical that in order to maintain the same crack opening at failure, the strength used in the ‘epp’ model must be stepped down to half its actual value. On the other hand, it is observed from simulations using ‘epp’ version, that when a reduced strength is used, 28.5 MPa vs 57 MPa (cases #9 – #13), an earlier crack initiation is predicted; whereas, using the full strength, results in good agreement with experimental results are obtained (cases #4 – #8). It is also interesting to observe that when the full strength is used in conjunction with strain softening, such as in ‘ecu’ and ‘epu’ versions of the model the crack initiation load is again under predicted (cases #1 and #3). With a view to gain an insight into such model behavior, states of stress and deformation of the cohesive layer are examined in the vicinity of crack tip. The stress contours of the cohesive layer were obtained from a few elements downstream of the crack tip element to about 100 elements (5 mm) upstream of the crack. Figure 14.9(a,b) shows a plot of typical the transverse


402


(a) σ max = 28.5; Current crack length: 46.65 mm

Stress free

Tension

Transition

Compression

(b) σ max = 57; Current crack length: 46.65 mm

Stress free

Tension

Transition

Compression

14.9 (a-b) Distribution of transverse stresses near the crack tip downstream in two typical cases (with differing strengths of cohesive layer).

stress variation in the vicinity of the crack for two typical members from group I and II respectively, namely #4 (57_05_epp_29) and #9 (28.5_05_epp_29). The two contour plots have several common features: They both exhibit tensile stress equal to the corresponding σmax at the crack tip and for a short distance upstream of the crack tip. We may call this region the ‘cohesive zone’. Beyond the cohesive zone the stress gradually diminishes, goes through zero and becomes compressive and remains compressive over the length examined, though gradually diminishing as we move upstream of the crack. In the present model the compression zone downstream of the crack tip is in active contact with the material in the vicinity of the crack tip, unlike in the models that do not consider a physical cohesive layer with finite thickness. It appears that the slight compressive deformation over an extended length of the layer produces a ‘wedge effect’ which facilitates, however slightly, the opening of the crack. This somewhat artificial effect may be countered in the ‘epp’ version of the model by providing additional resistance at the crack tip by postulating the full rather than half the strength for the cohesive material and thereby reducing the length of the cohesive zone. (Note that the length © 2008, Woodhead Publishing Limited


403

of the cohesive zone with σmax = 57 MPa is one half of that with σmax = 28.5 MPa.) Thus an ‘epp’ version of the model with σmax = 57 MPa gives a more accurate prediction. The ‘ecu’ version of the model would call for a σmax greater than the actual strength of the material in order to predict crack initiation load with accuracy. A far more serious lacuna of the model is its inability to predict continued crack growth. It is apparent that this has to do with some significant deformation suffered by the cohesive layer as the crack tip elements continue to fail successively and the crack advances. In order to gain an insight into this phenomenon, transverse deflections of the nodes at the top and bottom surfaces of the cohesive layer are examined with respect to the distance along the delamination plane (x-coordinate) for the two cases considered previously #4 (57_05_epp_29) and #9 (28.5_05_epp_29) as the crack grows. The plots of deflections shown in Fig. 14.10(a) correspond to a situation when the crack length has reached a value of 46.5 mm starting from the initial value of 29 mm. At this point the crack growth has ceased and the load is beginning to increase. It is seen that just past the crack tip there has developed a slight ‘neck’ in the region of compression. There is an abrupt change of slope and curvature of the deflection variation at this region. This feature is illustrated more clearly in the locally blown-up view of displacements in Fig. 14.10(b). Once a ‘neck’ is formed in the vicinity of crack tip, any further rotation of the beam stabilizes and accentuates the compressive stress block locking the crack in its current position. The necking phenomenon appears to be due to the manner in which the cohesive material in the vicinity of the crack tip deforms as the crack tip elements fail as the crack grows.

14.6

UEL model: details of the study and discussion of results

Parameters of UEL model For all the UEL models investigated in this study, the constitutive relationship is taken to be the same as the ‘epp’ version discussed earlier. For a given GIc, the only parameters that need to be chosen are the strength of the cohesive layer, σmax and δ1o, the proportional limit crack opening. Alfano and Crisfield (2001) have shown that there is considerable latitude in their selection. Selection of σmax As discussed earlier, insofar as the strain softening is a real phenomenon, the total strain energy released per unit area at failure as the element fails may be approximated as ησultδult, where σult is the tensile fracture strength of the cohesive material and η is a fraction ≈ 1 (Fig. 14.11). Since in the present •


2

404

Delamination behaviour of composites 0.2 No. 4 (57_05_epp_29)

0.15

Distance from center, mm

Top surface of the cohesive layer 0.1 This region is blown up in Fig. 14.10(b) 0.05 0 –0.05 Bottom surface of the cohesive layer –0.1 –0.15 –0.2 40

42

44

46 48 50 Distance along the axis, mm (a)

52

54

0.0251 No. 4 (57_05_epp_29)

Distance from center, mm

0.02508 0.02506 0.02504 Top surface of the cohesive layer prior to deformation

0.02502 0.025 0.02498 0.02496 47.1

47.6

48.1 Distance along the axis, mm (b)

48.6

49.1

14.10(a) Transverse deflections of the cohesive layer surfaces near the crack tip with the current crack length = 46.65 mm for the case #4 (57_05_epp_29); (b) The deformation of the cohesive layer upstream of crack tip in the vicinity of the ‘neck’.

model no softening phase is included, the strain energy released may be approximated as σmaxδult, while the crack opening displacement at fracture remains unchanged being a material parameter. This means that the σmax must be taken as roughly half the strength of the material.



405

σ

σf

ησ f δ ult

δ δult (a) Typical response of cohesive material

σ

σ max

σmaxδult

δ δult (b) The assumed response of the material in the model

14.11(a-b) The selection of cohesive material strength of the model.

Selection of δ1o Theoretically speaking, δ1o, the elastic limit of normal relative displacement must be very small in view of the initial zero thickness of the cohesive layer. But extremely small values, such as 1 × 10–9 are known to cause convergence problems. Values such as 1 × 10–7 and higher have been successfully used by Alfano and Crisfield (2001) and the present authors. One approach to determine δ1o is to assume a process zone thickness h of (≈ 0.02 mm) so that δ1o · σmaxh/E2.


406


Finite element configuration The finite element mesh consists of four-noded plain strain elements of dimensions 0.1 mm × 0.1 mm throughout. Thus we have 30 layers of 1300 elements each. As discussed in the description of UEL model, four node user formulated elements are used for the cohesive layer and these are of width 0.1 mm each and zero initial thickness covering the middle plane of the DCB specimen. Simulation results With σmax taken equal to 28 MPa and 15 MPa respectively and the corresponding δ1o values to be 5.9 × 10–5 and 3.2 × 10–5, nonlinear finite element analyses were conducted of the Robertson-Song DCB specimen using displacement control. Once again, the increment size is so adjusted as to ensure no more than one element failed in a loading increment. The results of simulation and the experiment are shown in Fig. 14.12. It is seen that the model is able to capture with a fair degree of accuracy the unloading branch of the experimental result of the DCB specimen. We notice once again, albeit in a diminished form, the dependence of the load corresponding to crack initiation on the value of σmax selected but that the crack propagation rate is virtually unaffected by this value. σmax controls the length of the cohesive zone – the zone that carries a tension equal to 120

Analysis with σmax = 15 Analysis with σmax = 28

100

Experimental results

Load P, N

80

60

40

20

0 0

2

4


10

12

14.12 A comparison of UEL model results with Robinson-Song test results.



407

σmax – and the crack opening displacement, i.e. the geometry of the cohesive zone. Differing levels of σmax would correspond to cohesive zones of differing lengths at crack initiation and hence the dependence of crack initiation load on σmax. But once the crack has begun to propagate, it does so translating the cohesive zone in a more or less self-similar manner releasing equal amounts of energy for equal advances of crack tip whatever the value of σmax. This makes the load reductions for a given crack growth independent of σmax.

14.7

Delamination of composite laminates under impact

14.7.1 Jih_Sun experimental study Jih and Sun (1993) and Jih (1991) give detailed description of a test program of quasi-static and impact loading on a layered carbon-epoxy plate. Two typical test configurations used in the study are illustrated in Fig. 14.13(a) and 14.13(b) respectively. These are designated as TC-1 and TC-2 respectively. It is seen that the chosen configurations incorporate discontinuities which mimic transverse cracks. An initial delamination of unspecified length is introduced during fabrication in each case. The specimens have widths (w = 38.1 mm) that are of the same order of magnitude as the span and are subjected to a line load of magnitude P, i.e. of intensity P/w. The material of the specimens is AS4/3501-6 graphite/epoxy. The material properties are as follows: E1 = 139 GPa, E2 = 9.86 GPa, G12 = 5.24 GPa,

ν12 = 0.3, α1 = –0.54 µε/°C, α2 = 27.2 µε/°C Temperature drop from curing condition was reported as ∆T = –156°C. The effects of curing stresses are important (Jih and Sun, 1993) and are duly considered in the present study. TC-1 (Fig. 14.13.a) consists of three sublaminates, the outer ones being composed of five plies with fibers running in the transverse direction (90° plies), while the interior one has fibers running in the longitudinal direction of the beam (0° plies). The bottom ply is cut right at the middle and a pre-implanted delamination of certain length ‘2a’ is installed between the bottom 90° and the next 0° sublaminates during fabrication. Because of the symmetry of the configuration, it is sufficient to deal with one half of the beam carrying a load of P/2w per unit width. TC-2 (Fig. 14.13.b) is designed to be significantly asymmetric, ostensibly with the objective of making it distinctly different from TC-1. It is built of an upper sublaminate with a ply sequence of [(0/90)4/0] of length 101.6 mm and a lower sublaminate with a ply sequence of [(90/05] of length 63.5 mm. A pre-implanted delamination of certain length ‘a’ is installed between these sublaminates. The loading arrangement and other details of the set up are given in Fig. 14.13(b). © 2008, Woodhead Publishing Limited

408

Delamination behaviour of composites P/w d

L

L

d

[90]5 [0]5 [90]5 2a

P/2w 25.4 mm Plane of symmetry

12.7 mm

[90]5 [0]5

Structural analysis model [90]5

Free

a Cohesive layer Dimensions: d = 12.7 mm; L = 25.4 mm, h = 2.04 mm; w = 38.1 mm (a)

P/ w d

L

2L

[(0/90)4/0]

d

h1 [(0/90)5]

h2

Cohesive layer

a Dimensions: d = 12.7 mm; L = 25.4 mm, h1 = 1.2 mm; h2 = 1.4 mm; h = h1 + h2 (b)

14.13 (a) Configuration of test specimen 1 (TC-1); (b) configuration of test specimen 1 (TC-2).

Experimental results for crack growth under static loading During the quasi-static tests specimens were loaded in steps and unloaded to measure crack extensions. Thus Jih and Sun were able to associate a given crack length with a certain load and an increase in crack length was possible © 2008, Woodhead Publishing Limited


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only under the increase of load. Thus the crack growth occurred under stable conditions. The transverse load P and the corresponding crack length (lcr) relationships are given in Fig. 14.14(a) and 14.14(b) for TC-1 and TC-2 respectively. The results show significant scatter. 200

150

P/h1.5 (N/mm1.5)

Exp. upper band

L14 L17 L7 L15 L2 L12

Exp. lower band

100

50

0 0

5

10 Crack length (mm) (a)

15

200

150

P/h1.5 (N/mm1.5)

Exp. upper band

T1 T3 T5 T4 T7 T8

Exp. lower band

100

50

0 10

15

20 Crack length (mm) (b)

25

14.14(a) Experimental transverse load versus crack length relationship for TC-1 (Jih, 1991); (b) experimental transverse load versus crack length relationship for TC-2 (Jih, 1991). © 2008, Woodhead Publishing Limited

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14.7.2 Approach to finding GIc and GIIc using the UMAT and UEL models For the material on hand, measured values of the critical values of strain energy release rates are not available. For the purposes of the present study, these values are determined to fit the experimental data of loads (P) at incipient crack growth using the initial crack length and are subsequently used to explain the entire range of behavior. Since the cohesive layer model is selected as the tool for tracing the crack growth, it is at once consistent and expedient to determine the values of GIc and GIIc using the same tool. Thus the cohesive layer model performs in the present work the dual function of identifying the critical values of SERR and tracing the crack growth as well (Sridharan and Li, 2004; Li, 2005; Li and Sridharan, 2005a). The values of the (smallest) initial crack lengths in TC-1 and TC-2 respectively that were used in the calculations are 1.92 mm and 12.72 mm. A short crack length is preferred, as at the beginning of loading, the crack surfaces are expected to be smooth and free of crack bridging effects. A nonlinear finite element analysis with the cohesive layer model in place is undertaken. Prior to the application of lateral load, P, a thermal stress analysis is performed to compute the curing stresses as they do influence the results, especially for TC-1. Several discrete values of P (corresponding to crack initiation) within the core of the scatter band for the chosen crack length were selected for each test configuration. For each value of P, a pair of GI and GII values is found from the analysis using the cohesive layer model. Such a pair of values of GI and GII from TC-1 is paired with a pair of values from TC-2. Using the linear interactive failure criterion:

GI G + II = 1 G Ic G IIc

14.4

a set of GIc and GIIc values is obtained from each of the two pairs of GI and GIT. After eschewing unrealistic and inadmissible sets of GIc and GIIc thus obtained and averaging the rest, we finally arrive at an estimate of GIc and GIIc. The values thus found were associated only with the model (UMAT or UEL) used in their determination.

14.7.3 Static analysis and predictions using UMAT model Model parameters First the parameters of the model must be selected. The thickness of this layer h0, as mentioned earlier, is based on the observed dimensions of the process zone of composite materials (Charalambides et al., 1992) viz, h0 = © 2008, Woodhead Publishing Limited


411

0.02 mm. This dimension then plays the role of the length scale in the study. The cohesive layer consists of four node elements h0 × h0 (aspect ratio = 1). The transverse modulus E2 is the same as that of the composite material. The other properties, viz. E1 and G12 (longitudinal and shear moduli), α1, α2 (coefficients of thermal expansion in the longitudinal and transverse directions) are all taken as the weighted averages of the respective values of the layers above and below the delamination plane. The transverse and shear strengths of the cohesive layer material are those of the graphite-epoxy and are taken as 41 MPa (σmax) and 97 MPa (τmax) respectively (Vinson and Sierakowsky, 1986). The cohesive law used is bilinear of ‘epp’ type already discussed in the earlier DCB problem. Critical values of SERR (strain energy release rates) The general approach to identification of GIc and GIIc using a cohesive layer model was outlined earlier. Full details of the calculations using the UMAT model were given in earlier publications of authors (Sridharan and Li, 2004; Li and Sridharan, 2005a, Li, 2005) and will not be repeated here. The calculations reveal that the phenomenon is mode-I dominated, so much so GIc is determined more reliably than GIIc, as the GII values are relatively small. The values of GIc and GIIc respectively are found to be: 216 N/m and 620 N/m respectively. UMAT model prediction of crack growth initiation Static nonlinear analyses are run for both TC-1 and TC-2 starting with differing values of crack lengths, with crack tips at the four station points A, B, C and D respectively. Table 14.2 gives the corresponding loads at incipient crack growth which were determined using the criterion in Equation (14.4) and the already determined values of GIc and GIIc. It is interesting to note that modemixity remains approximately the same, i.e. more or less independent of the initial crack length for each test configuration. These numerical results would correspond to a ‘discontinuous’ test program of loading which consists of a loading phase which culminates in crack growth initiation and some tiny crack growth followed by an unloading phase. The P vs. Pcr (crack length) relationships are plotted in Fig. 14.15(a,b) for the two cases. It is seen that the analytical predictions of loads corresponding to crack initiation for the four crack lengths considered lie within the scatter band. However, the line joining the four data points has a noticeably smaller curvature than the bounding curves of the scatter bands. It is believed that this is due to the increased resistance to crack growth due to some bridging mechanisms which come into play as crack growth continues in the experiments. © 2008, Woodhead Publishing Limited

412


140

P/h1.5 (N/mm1.5)

120 100 80 60

Exp. lower band Exp. upper band Numerical results

40 20 0

2

4

6 8 Crack length (mm) (a)

10

12

14

140

P/h1.5 (N/mm1.5)

120

100

80

60 Exp. lower band Exp. upper band Numerical results

40

20 10

12

14

16 18 20 Crack length (mm) (b)

22

24

26

14.15(a) A comparison of load versus crack length relationship for TC-1 given by UMAT model with experimental results (Jih, 1991); (b) a comparison of load versus crack length relationship for TC-2 given by UMAT model with experimental results (Jih, 1991).

UMAT model: crack growth analysis under continuous loading Next, a crack growth analysis is undertaken. With the given initial crack lengths, static nonlinear analyses are conducted till significant crack growth occurs. The crack growth occurs by virtue of Equation (14.4) being satisfied at the integration point of the crack tip element at which instant, the stresses are dropped to zero. The element is in effect eliminated and the crack tip moves to the next element. No convergence difficulties were experienced as © 2008, Woodhead Publishing Limited


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the structure loses its equilibrium. Apparently the element size is sufficiently small for the Newton procedure to recapture the equilibrium with a few iterations. Results of continuous crack growth analysis Nonlinear analyses were conducted with the initial crack tip at station points A, B, and C respectively (Table 14.2) for the two test configurations. This time the analysis was continued till significant crack growth occurred. Typical P- ᐉcr relationships thus obtained are shown alongside the results for crack initiation for the two test configurations in Fig. 14.16 (a,b). It is seen that P- ᐉcr relationship depends on the initial crack length. For small crack growths, these tend to follow the crack initiation line, but soon turn upward indicating a retardation of crack growth. Similar deviations, if less spectacular, of the P- ᐉcr relationship under continuous loading from that due to discontinuous loading are seen for TC-2. The deviation of the P- ᐉcr relationship of a continuous loading program from that for crack initiation was reported by the authors in earlier papers (Sridharan and Li, 2004, Li and Sridharan, 2005), where the authors conjectured that this is due to the continuous nonlinear variation of the stiffness of the structure as delamination progresses which tends to steer the structure in a different path associated with an increasing resistance to crack growth. This indeed is true, but as we have seen in the example of the DCB specimen the phenomenon is caused by a subtle deformation of the cohesive layer of finite thickness in the neighborhood of the crack tip. The increasingly greater resistance to crack growth seen in the continuous program of loading is attributable, once again, to the locking mechanism of the UMAT model which retards and shuts off the crack growth. Such increased resistance to crack growth under continuous loading is not observed when the UEL model (which has zero initial thickness) is employed, as already seen in the case of the DCB specimen.

14.7.4 Static analysis and predictions using UEL model Next the same problem is investigated using the UEL model. Selection of model parameters: σmax, τmax, δo and γo The maximum normal stress σmax and maximum shear stress τmax are selected as follows: Trial values of GIc and GIIc (216 N/mm and 620 N/mm) are assumed. A trial analysis with the following cohesive layer parameters is run: σf = 41 MPa and τf = 97 MPa; a linear softening phase is assumed. The values of © 2008, Woodhead Publishing Limited

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normal and shearing stress , viz. σ* and τ* at the instant when the failure criterion is satisfied are determined. Assuming the crack opening displacements in normal and shear modes to be equal to their respective values in the model with no strain-softening, σmax and τmax may be estimated as:

σ max = 1 (σ f + σ *) 2 τ max = 1 (τ f + τ *) 2

14.16

In the present problem they are found to be 26 Mpa and 95 Mpa respectively. Since the opening mode dominates, the reduction for σmax from σf (≈36%) is much more appreciable than that for τmax from τf. The other parameters that need to be estimated for the model are δ10 and δ20, the proportional limit displacements for the opening and shear mode respectively. These are made equal to the respective values determined on the basis of an assumed process zone thickness of 0.02 mm. Thus the values of δ10 and δ20 are found respectively to be 5.3 × 10–5 and 3.8 × 10–4 respectively. Finite element configuration The finite element mesh configuration is the same as the one used for our investigation of the UMAT model – four noded plane strain elements were used; the size of the element is 0.02 mm × 0.02 mm. Identification of GIc and GIIc The same procedure detailed for the UMAT model is followed for the identification of GIc and GIIc. The details are given in Li (2005). These values are found to be 222 N/m and 597 N/m respectively. These are not far different from the values obtained using UMAT (216 N/m and 620 N/m). Prediction of crack initiation With the now available values of GIc and GIIc, UEL model is used in conjunction with nonlinear finite element analysis, to predict crack initiation under various initial crack lengths for both the test configurations. The results of loads corresponding to the four station points (A–D) for each test configuration are shown in Table 14.3 and these are very close – within 1.5% in all casesalong with those already obtained using the UMAT model (Table 14.2 and Fig. 14.16(a,b)). Thus the earlier observations made in the context of UMAT hold good again: the mode-mixity does not change as the crack lengths are varied; the results of loads obtained fall well within the scatter band; and for higher crack lengths, the loads predicted by the model are smaller than the © 2008, Woodhead Publishing Limited


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Table 14.3 Loads at incipient crack growth for different crack lengths (UEL predictions) Station point

Crack length (mm)

P/2w (N/mm)

Test configuration – 1 A 1.92 B 4.8 C 7.16 D 11.12 Test configuration – 2 A 12.72 B 16.24 C 20.40 D 23.40

2.254 2.588 2.943 3.80 7.96 8.83 10.18 11.40

P/h1.5 (N/mm1.5)

GI (N/m)

GII (N/m)

GI G + II G Ic G IIc

58.95 67.68 76.97 99.38

211.52 211.41 210.98 210.62

28.5 29.1 29.8 30.6

1.00 1.00 1.00 1.00

72.34 80.25 92.52 103.60

198.86 198.51 198.33 197.35

62.3 63.3 63.9 66.4

1.00 1.00 1.00 1.00

averaged value of the experimental scatter band, due presumably to the increasing resistance to crack growth offered by bridging effects in the experiments. Continuous crack growth UEL model is now employed to trace crack growth with various initial crack lengths. Typical results of crack growth from station points A and C (Table 14.3) for the TC-1 and TC-2 are shown Fig. 14.17(a,b) and Fig. 14.18(a,b) respectively. Also shown in these figures, are the respective P-. Pcr relationships for crack initiation, i.e., those obtained from ‘discontinuous’ program of loading involving loading-minute crack growth-unloading-reloading. It is seen that P-. Pcr relationships for continuous crack growth do coincide with those for crack initiation. This is in contrast to the UMAT predictions of increasingly greater resistance to crack growth in ‘continuous’ program of loading. The UEL results indicate that for quasi-static delamination it is possible to associate a certain load P with a certain crack length. Pcr whatever the initial crack length Po, as was implicitly assumed by Sun and Jih (1993) . The results are consistent with the simplicity of model, in that it does not account for continuous enhancement of resistance crack growth due to such phenomena as fiber bridging. The conclusions at this point are: (a) Both UMAT model and UEL model are capable of predicting crack initiation. (b) UMAT model can be used only to predict small crack extension, whereas UEL model can be used for predicting large crack growth. (c) For the structural configurations studied, P-. Pcr relationships under static loading, as given by the UEL model for continuous crack growth and crack initiation respectively, do coincide. (d) In view of the DCB results discussed earlier, UEL results must be viewed as more credible. © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites Continuous crack growth from A Continuous crack growth from B

110

Continuous crack growth from C

P/h1.5 (N/mm1.5)

Discontinuous crack growth D 90

C 70 B A 50 0

2

4


8

10

12

120 Continuous crack growth from A Continuous crack growth from B Continuous crack growth from C

110

P/h1.5 (N/mm1.5)

Discontinuous crack growth

D

100

C

90

80 B A 70 12

14

16

18 20 Crack length (mm) (b)

22

24

26

14.16(a) A comparison of load versus crack length relationship as obtained from continuous loading program starting at station points A, B and C respectively with that given by the discontinuous loading program for TC-1; (b) A comparison of load versus crack length relationship as obtained from continuous loading program starting at station points A, B and C respectively with that given by the discontinuous loading program for TC-2.



417

110 Continuous crack growth from A Discontinuous crack growth

P/h1.5 (N/mm1.5)

100

90 D 80 C 70 B 60

A

50 0

2

4


8

10

12

110 Continuous crack growth from C Discontinuous crack growth

P/h1.5 (N/mm1.5)

100

90 D 80 C 70 B 60 A 50 0

2

4

6 Crack length (mm) (b)

8

10

12

14.17(a) A comparison of load versus crack length relationship as obtained from continuous loading program starting at station point A with that given by the discontinuous loading program for TC-1 (UEL model); (b) A comparison of load versus crack length relationship as obtained from continuous loading program starting at station point C with that given by the discontinuous loading program for TC-1 (UEL model).


418


120

Continuous crack growth from A Discontinuous crack growth

P/h1.5 (N/mm1.5)

110 D 100

C

90

80 B A 70 12

14

16

18 20 Crack length (mm) (a)

22

24

26

130 Continuous crack growth from C Discontinuous crack growth

P/h1.5 (N/mm1.5)

120

110 D

100

C

90

80

B A

70 12

14

16


24

26

28

14.18(a) A comparison of load versus crack length relationship as obtained from continuous loading program starting at station point A with that given by the discontinuous loading program for TC-2 (UEL model); (b) A comparison of load versus crack length relationship as obtained from continuous loading program starting at station point C with that given by the discontinuous loading program for TC-2 (UEL model).



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14.7.5 Impact loading Jih (1991) gives a detailed report of an investigation of delamination growth under impact loading for the two test configurations. Since the mass of the impactor (1 kg) was greater than the mass of the laminate (typically 0.1 kg) by an order of magnitude and the contact time (typically 10 m sec) was found to be many times greater than half-period of the laminate (typically 0.5 m sec), it was argued that the impacts came under the category of lowvelocity impact; and an analysis in which the maximum impact load was treated as quasi-statically applied was considered adequate to deal with the problem. While the assumption of ‘low-velocity impact’ is certainly valid, the comparisons of quasi-static analysis results with test results do reveal significant discrepancies with numerical predictions giving appreciably higher values in some cases and lower values in others. It is therefore of interest to revisit the problem and examine what conclusions, a detailed dynamic analysis employing the cohesive layer model may yield. Load versus time and load versus impact velocity relationships Given the focus on the delamination problem under a dynamic load, the present analysis does not attempt to predict the impact load history by considering the interactive contact between the impactor and laminate. The delamination behavior of the laminate is examined under an experimentally determined form of impact load versus time relationships, in which the amplitude, Pmax performs the role of a loading parameter and may be suitably prescribed. The non-dimensional forms of impact load history (Pmax/P vs. time) for the two test configurations are given in Fig. 14.19(a,b). These shapes are experimentally determined and are approximately sinusoidal. For a given test configuration, there is a certain duration of contact and neither the shape of the P vs time relationship nor the duration of contact is found to vary noticeably with the impact velocity or the maximum load imposed or the deformation of the specimen. The only variable that determines the loading history is Pmax, the maximum load sustained by the specimen. Jih (1991) has given plots of load amplitude vs. impact velocity relationships for the two test configurations and in the range of interest (0.6 m/sec< v < 1.8 m/sec) the plots can be represented by the following linear relationships: (a) For TC-1 Pmax = 5 + 94 v h 1.5 (b) For TC-2 Pmax = 25 + 43 v h 1.5 © 2008, Woodhead Publishing Limited

14.17.a

14.17.b

420


1

0.8

P/Pmax, λ

λ = GI /GIc + GII /GIIc 0.6

Pmax = 2.4 N/mm

0.4

0.2

0 0

1

2

3

4

5 6 Time, ms (a)

7

8

9

10

11

1

P/Pmax, λ

0.8

λ = GI /GIc + GII /GIIc 0.6

Pmax = 7.07 N/mm 0.4

0.2

0 0

2

4

6 Time, ms (b)

8

10

12

14.19(a) P/Pmax and λ versus time relationship for TC-1; (b) P/Pmax and λ versus time relationship for TC-2.

where h is the overall thickness of the laminate (2.04 mm and 2.6 mm for TC-1 and TC-2 respectively) and v is the impact velocity in m/sec. Thus it is possible to represent the results in terms of P or v.



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14.7.6 Dynamic analysis for initiation of crack growth In the first phase of the analysis the objective is limited to picking up the load ( Pl0 ) at which the crack growth just initiates, for given initial crack length ᐉo. The analysis proceeds along the following lines: First, a static analysis is performed to account for thermal stresses and strains associated with curing – as in the static delamination problem. In the next step, dynamic analysis with implicit integration available in ABAQUS/standard is invoked. For a given crack length, a certain Pmax is selected. The dynamic load follows the load time history already discussed (Fig. 14.19a,b). The crack growth criterion index, λ is computed at the crack tip at the end of each increment of loading throughout the analysis. If it remains below 1.0 throughout, no crack growth occurs, in which case Pmax is increased; if it exceeds 1.0 at any time during the loading, Pmax is decreased. The analysis is rerun thus adjusting Pmax till a value is obtained for which λ just reaches 1.0 with just the crack tip element failing. Typical results obtained using UMAT model (with the same finite element mesh and model parameters used for static analysis) are shown in Fig. 14.19(a, b). Fig. 14.19(a) gives the variation of λ with time for a crack length of 4.8 mm (station point B) and Pmax = 2.4 N/mm of TC-1. Fig. 14.19(b) likewise plots of the variation of λ for a crack length of 12.72 mm (station point A of TC-2) and gives the variation of λ with time for a crack length of 4.8 mm (station point B) and Pmax = 7.07 N/mm. The loads in each case are those which just fail the crack tip element at a certain point in the loading history. The variation of λ with time in the two cases is oscillatory in nature, though increasing with load overall. This oscillation is due, apparently to the triggering of a mode of a higher frequency localized near the crack tip region, leading to oscillations in the crack opening displacements. For TC1, the value of λ has a head start, starting with a value of 0.2 at the beginning of the loading due to curing stresses and strains. This effect is not significant for TC-2. In general, loads at which crack growth initiation occurs under dynamic loading are smaller than the corresponding loads for static loading. This is due to the oscillatory nature of λ which shoots to a value of unity at the crest of an oscillation causing the crack tip element to fail. The differences in the static and dynamic values of P corresponding to crack initiation (Po) are not large but significant from the point of view of engineering predictions. For example, for TC-1, with ᐉo = 4.8 mm, the static and dynamic values of Po are 2.58 N/mm and 2.40 N/mm respectively; for TC-2, with ᐉ = 12.72 mm, they are respectively 7.90 N/mm and 7.07 N/mm. The results of the impact analysis are best expressed in the form of plots of v versus ᐉo and these are shown in Fig. 14.20(a,b) In order to compare the static and dynamic results (loads/velocities) •

•


422


16 14

Crack length, mm

12 10 8 6 4 Static analysis Dynamic analysis

2 0 0.5

0.6

0.7

0.8 0.9 1 Impact velocity, m/s (a)

1.1

1.2

1.3

23

Crack length, mm

20

17

14 Static analysis Dynamic analysis

11 0.8

0.9

1

1.1

1.2 1.3 1.4 Impact velocity, m/s (b)

1.5

1.6

1.7

1.8

14.20(a) v -– lo relationship (impact velocity versus crack length corresponding to incipient crack growth) for TC-1 as given by UMAT model. (b) v – lo relationship (impact velocity versus crack length corresponding to incipient crack growth) for TC-2 as given by UMAT model.



423

corresponding to crack initiation, one may compute a fictitious velocity corresponding to quasi-statically applied P corresponding to crack growth initiation using Equation 14.17a,b. These results of v are plotted against the corresponding ᐉo in Fig. 14.20(a,b) along with the results of dynamic analyses. It is seen that for any given crack length the ‘fictitious’ (static) velocity is larger than those of real (dynamic) velocity indicating that static analysis underestimates the load required to cause crack growth initiation.

14.7.7 Comparison of the UEL and UMAT results The calculations are repeated using the UEL model and the impact velocities corresponding to crack initiation for various initial crack lengths are obtained for TC-1 and TC-2. The finite element mesh and model parameters used are the same as for the static analysis. These results are compared with those obtained from the UMAT model in Fig. 14.21(a,b). It is seen that for TC-1 the UMAT results are in close agreement with the UEL results. As far as TC2 is concerned, when ᐉo < 17 mm, the ᐉo vs. v relationships coincide with each other; for ᐉo > 17 mm, the UMAT model predicts a relatively higher resistance to crack growth. For example with ᐉo = 19 mm, V is 1.26 m/s and 1.3 m/s for UEL model and UMAT model respectively. The predictions from both models tend to get closer again for ᐉo > 21 mm. Such discrepancies are minor, especially when we consider the oscillatory nature of dynamic behavior and the local effects that influence the crack growth. Continuous crack growth for TC-1 and TC-2 under impact load In order to examine the continued crack growth under dynamic loading, a certain crack length l0 is selected and the impact velocity (i.e. Pmax) is increased in steps to obtain the maximum crack length attained in each case (ᐉmax). Thus a relationship between ᐉmax and impact velocity v is developed. Such relationships obtained using UMAT model are plotted in Fig. 14.22(a,b) for various values of ᐉo for TC-1 and TC-2 respectively. Calculations were repeated using the UEL model and the corresponding results are shown in Fig. 14.23(a,b). Plotted also in these figures, are (a) the relationship between the ᐉo and v corresponding to crack initiation; (b) the upper limit as well as the mean values of ᐉmax for various impact velocities as determined experimentally (Jih and Sun, 1993). It is seen that under dynamic loading, the crack growth does depend on the initial crack length and a single blow with a given impact velocity causes less delamination than a sequence of impacts with gradually increasing impact velocity leading up to the given value. This dependency on ᐉo is stronger for TC-2 than for TC-1. Besides, the v -ᐉcr characteristics for smaller ᐉo tend to lie below those for higher ᐉo indicating the role played by the initial crack © 2008, Woodhead Publishing Limited

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14

12

Crack length, mm

10

8

6

4

2

0 0.500

Crack initiation from UEL Crack initiation from UMAT 0.600

0.700

0.800 0.900 Impact velocity, m/s (a)

0.1000

1.100

1.200

25

23

Crack length, mm

21

19

17

15

13 Crack initiation from UEL Crack initiation from UMAT

11 0.800

0.900

1.000


1.400

1.500

1.600

14.21(a) v – lo relationship (impact velocity versus crack length corresponding to incipient crack growth) for TC-1 as given by UMAT and UEL models; (b) v – lo relationship (impact velocity versus crack length corresponding to incipient crack growth) for TC-2 as given by UMAT and UEL models.



425

18 16

ᐉ° = 12 mm

Crack length, mm

14 12 ᐉ° = 8 mm

10 8

Crack initiation Initial crack length = 0+ mm Initial crack length = 4 mm Initial crack length = 8 mm Initial crack length = 12 mm Experimental: mean Experimental: upper bound

ᐉ° = 4 mm 6 4 2 0 0.5

0.7

0.9 1.1 Impact velocity, m/s (a)

1.3

1.5

25 23

Crack length, mm

21 ᐉ° = 19 mm 19

ᐉ° = 17 mm

17 ᐉ° = 15 mm 15

ᐉ° = 12 mm

13 11 0.800

1.000

Crack initiation Initial crack length = 12 mm Initial crack length = 15 mm Initial crack length = 17 mm Initial crack length = 19 mm Experimental: mean ... – ... Experimental: upper bound

1.200 1.400 Impact velocity, m/s (b)

1.600

1.800

14.22(a) Impact velocity versus crack length relationships for TC-1 (results from UMAT model); (b) impact velocity versus crack length relationships for TC-2 (results from UMAT model).

length: the maximum crack length attained for any given impact velocity or Pmax, increases with the initial crack length ᐉo. The UMAT model predicts smaller crack growth for a given impact velocity than the UEL model. and exhibits greater dependence on initial crack length. This translates itself into a wider spectrum of crack lengths attained for a given velocity. The UEL © 2008, Woodhead Publishing Limited

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16 14 ᐉ° = 12 mm

Crack length, mm

12 10

ᐉ° = 8 mm

8 6

Crack initiation Initial crack length = 1.92 mm Initial crack length = 4 mm Initial crack length = 8 mm Initial crack length = 12 mm Experimental: mean Experimental: upper bound

ᐉ° = 4 mm

4 2 ᐉ° = 1.92 mm 0 0.500

0.600

0.700

0.800 0.900 1.000 Impact velocity, m/s (a)

1.100

1.200

1.300

25 23

Crack length, mm

21

ᐉ° = 19 mm

19

ᐉ° = 17 mm

17

Crack initiation Initial crack length = 12 mm Initial crack length = 15 mm Initial crack length = 17 mm Initial crack length = 19 mm Experimental: mean Experimental: upper bound

ᐉ° = 15 mm 15 ᐉ° = 12 mm 13

11 0.800

0.900

1.000

1.100


1.500

1.600

1.700

1.800

14.23(a) Impact velocity versus crack length relationships for TC-1 (results from UEL model); (b) impact velocity versus crack length relationships for TC-1 (results from UEL model).

result is clearly more reliable in view of the inherent deficiency of the UMAT model discussed earlier. Figures offer a comparison of the numerical results of dynamic crack growth with experimental results. Insofar as the UEL model considers neither © 2008, Woodhead Publishing Limited


427

the increasing resistance to crack growth due to fiber bridging and other mechanisms, nor the geometrical changes suffered by the cohesive layer unlike the UMAT model, we anticipate that this model must give an upper bound estimate of crack growth. This indeed seems to be the case, as it gives results that tend to be close to upper bound to the experimental crack lengths for a given impact.

14.8

Conclusion

The two distinctly different cohesive layer models, one having finite thickness and governed by a stress-strain relationship (UMAT model) and the other having zero initial thickness and governed by stress-relative displacement relationship (UEL model) were studied in comparison. It turns out that either model can predict delamination growth initiation accurately. The UMAT model always predicts a cessation of delamination growth after some growth. This is not borne out by test results which are well predicted by the UEL model. It was shown that the there develops a neck in the cohesive layer of the UMAT model as the crack grows and this is the reason the cessation of crack growth occurs. This indeed must be viewed as an intrinsic deficiency of the model. The extent of delamination growth under dynamic conditions does appear to be dependent on initial crack length, but the UMAT model predicts a greater dependence than the UEL model. If, however, a layer of cohesive matrix material, however thin, does actually exist between the delaminating surfaces, the predictions of UMAT model may very well be mirroring the reality. Further experimental work is called for to confirm such a hypothesis.

14.9

References

ABAQUS/Standard User’s Manual, UMAT: Define a Material’s Mechanical Behavior, Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, RI, Vol. 6.3, 2002, pp. 24.2.30.124.2.30.14. ABAQUS/Standard User’s Manual, UEL: Define an Element, Hibbitt, Karlsson and Sorensen, Inc., Pawtucket, RI, Vol. 6.3, 2002, pp. 24.2.19.1-24.2.19.17. Alfano, G and Crisfield, M A (2001), ‘Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues’, International Journal for Numerical Methods in Engineering, 50, 1701–1736. Barrenblatt, G I (1962), ‘The mathematical theory of equilibrium cracks in brittle fracture’, Advances in Applied Mechanics, 7, 55–129. Charalambides, M, Kinloch, A J, Wang, Y and Williams, J G (1992), ‘On the analysis of mixed-mode failure’, International Journal of Fracture, 54, 269–291. Dugdale, D S (1960), ‘Yielding of sheets containing slits’, Journal of Mechanics and Physics of Solids, 8, 100–104. El-Sayed, S and Sridharan, S (2001), ‘Predicting and tracking interlaminar crack growth in composites using a cohesive layer model’, Composites: Part B, 32, 545–553. © 2008, Woodhead Publishing Limited

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El-Sayed, S and Sridharan, S (2002), ‘Cohesive layer models for predicting delamination growth and crack kinking in sandwich structures’, International Journal of Fracture, 117 Sept (1), 63–84. Hillerborg, A, Modeer, M and Peterson, P-E (1976), ‘Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements’, Cement and Concrete Research, 6, 773–782. Jih, C J (1991), ‘Analysis of delamination in composite laminates under low velocity impact’, PhD Dissertation, School of Aeronautics and Astronautics, Purdue University, Lafayette, IN. Jih, C J and Sun, C T (1993), ‘Prediction of delamination in composite laminates subjected to low velocity impact’, Journal of Composite Materials, 27(7), 684–701. Li, Y (2005), ‘Delamination of Composite Structures under lateral impact and inplane compression’, DSc Dissertation, The Henry Edwin Sever Graduate School, Washington University in St. Louis, St. Louis, MO. Li, Y and Sridharan, S (2005a), ‘Investigation of delamination due to low velocity impact using a cohesive layer model’, AIAA Journal, 43(10), 2243–2251. Li, Y and Sridharan, S (2005b), ‘Performance of two distinct cohesive layer models for tracking composite delaminations’, International Journal of Fracture, 36, 99–131. Needleman, A (1987), ‘A continuum model for void nucleation by inclusion debonding’, J. Appl. Mech., 54, 525–531. Needleman, A (1990), ‘An analysis of decohesion along an imperfect interface’, International Journal of Fracture, 42, 21–40. Ortiz M and Pandolfi, A (1999), ‘Finite-deformation irreversible cohesive elements for three dimensional crack propagation analysis’, International Journal of Numerical Methods in Engineering, 44, 1267–1282. Robinson, P and Song, D Q (1991), ‘A modified DCB specimen for mode I testing of multi-directional laminates’, Journal of Composite Materials, 26, 1554–1577. Schapery, R A (1975), ‘A theory of crack initiation and growth in visco-elastic media’, International Journal of Fracture, 11(1), 141–159. Sprenger, W, Gruttman, F and Wagner, W (2000), ‘Delamination growth analysis in laminated structures with continuum-based 3D-shell elements and a visco-plastic softening model’, Computer Methods in Applied Mechanics and Engineering, 185, 123–139. Shahwan, K L and Waas, A M (1997), ‘Non-self-similar decohesion along a finite interface of unilaterally constrained delaminations,’ Proceedings of the Royal Society of London, 453, 515–550. Sridharan, S and Li, Y (2004), ‘The dual role of cohesive layer model in delamination investigation’, 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamic & Materials Confer, Palm Springs, California. Ungsuwarungsri, T and Knauss, W G (1987), ‘The role of damage-softened material behavior in the fracture of composites and adhesive’ Int. J. Frac., 35, 221–241. Vinson, J R and Sierakowski, R L (1986), The Behavior of Structures Composed of Composite Materials, Martinus Nijhoff, Dordrecht, The Netherlands, p. 60. Williams, M L (1963), ‘The fracture of visco-elastic material’, In: Drucker, Gilman (ed.), Fracture of Solids, Wiley Interscience Publishers, 157–188. Yu, C (2001), ‘Three dimensional cohesive modeling of impact damage of composites’, PhD thesis, Caltech, Pasadena, CA.


15 Modeling of delamination fracture in composites: a review R C Y U, Universidad de Castilla-La Mancha, Spain and A P A N D O L F I, Politecnico di Milano, Italy

15.1

Introduction

For their attractive mechanical properties and significant weight savings, composites have been employed increasingly in different structural applications, such as: thin films and layered structures in micro-electro-mechanical systems (MEMS); ship hulls in naval engineering; thermal coatings in aeronautical and aerospace engineering; strengthening of concrete and steel members in civil structures; and many others. Mechanical loads, impacts, drilling during manufacturing processes (Arul et al., 2006, Abrao et al., 2007), moisture or temperature variations (Garg, 1988) may induce separation of the composite layers, leading to delamination, i.e. the formation of a mica-like structure. Unnoticeable from the external surface, delamination may be particularly insidious. In particular, it may reduce the stiffness and the strength – and consequently the life – of a structure without warning. Delamination of composites is of special concern in both industry and the scientific community (Carroll et al., 2003). An overview of interlaminar properties characterization of composites used for marine structures, as well as a detailed description of the tests available to measure the resistance to delamination, can be found in the paper by Baley et al. (2004). According to industrial applications, composites can be grouped in fiber composites and layered composites. As delamination may cause local buckling and the occurrence of high interlaminar shear and normal stresses at the edges of the buckled regions, it may play a critical role in the overall compressive behavior (Hutchinson et al., 2000). Such behavior is of particular interest in the area of thermal barriers and wear coatings (Balint and Hutchinson, 2001; 2003; 2005; Balint et al., 2006). Additionally, it has been observed that fiber composites may fail under compression-compression fatigue loading because of micro-buckling (Slaughter et al., 1993; Slaughter and Fleck, 1993a) and may be affected by multi-axial loading and creep (Talreja, 2006). Local delamination mechanisms are not unique. Different loading conditions may induce mode I delamination, when two layers are pulled apart; mode II 429 © 2008, Woodhead Publishing Limited

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delamination, when sliding is observed; or mode III delamination, when the two layers are torn apart. If the loading is time dependent, the local failure mode may change in time, and typically it shows a mixed-mode feature. Furthermore, a strong coupling between roughness-wedging at the interface and frictional sliding, which affects the apparent mode II fracture toughness, has been observed (Balint and Hutchinson, 2001). Significant interactions may arise between multiple through-width delaminations (Andrews et al., 2006), and delamination often combines with the cracking of the matrix or of the layers, and renders even more complex the detection and the prediction of the overall failure. Clearly, the mechanical strength of the matrix-layer interface is the basis of the strength of the composite material. Nevertheless, the size of such interface is of the order of microns or nanometers, and the description of the mechanical interaction is of some complexity, thus any numerical model definitely will bear a multiscale character (Mariani et al., 2005). Under various aspects, many efforts in modeling delamination failure are documented in the literature. With reference to low speed impacts, an interesting review of models for delamination in composites is reported by Elder et al. (2004). Roughly, there are two main approaches for the numerical modeling of delamination processes, based on damage mechanics and fracture mechanics respectively. The damage approach has been pursued, for example, by Allix and Ladeveze (1992). Allix and collaborators introduced a meso-model of the laminate, where the laminae are considered as a stacking sequence of homogeneous layers and inter-laminar interfaces throughout the thickness. The material models for both ply and interface are based on damage theories, and embed all the degenerative mechanisms, such as fiber breaking, matrix microcracking, adjacent layer debonding, in the constitutive laws. Internal variables (i.e., inelastic strains, damage and hardening) are used to describe the state of the material. Fatigue induced damage has been investigated by Spearing and Beaumont (Spearing and Beaumont, 1992a, 1992b, Spearing et al., 1992a, 1992b). They proposed to quantify the tip damage by the extent of individual failure processes, and presented an approach able to assess the post-fatigue strength and stiffness of the composite. Recently, Suiker and Fleck (2006) proposed a relationship between fatigue crack growth rate and remote cyclic stress to predict fatigue crack tunneling and plane-strain delamination. Models based on fracture mechanics can be traced back to the early 1980s. Chai et al. (1981) introduced a one-dimensional ‘thin-film model’ to analyze the failure in composite laminated plates induced by low speed impacts. They adopted an energy release rate criterion to explain the local buckling and the growth of delamination in a beam-column connection. The model was able to describe several interesting behaviors, depending on the extension of the delamination, the load level, and the fracture energy. Mi et al. (1998) © 2008, Woodhead Publishing Limited


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proposed a model of progressive mixed-mode delamination in fiber composites, based on the use of interface elements and softening relationships between stresses and displacement jumps. The area enclosed by the stress-displacement jump curve represents the critical fracture energy, and provides an indirect link with fracture mechanics. The same approach has been followed recently by Oguni and Ravichandran (2001b) in proposing a failure model, based on micromechanical considerations, for unidirectional fiber reinforced composites. In their review paper, Yang and Cox (2005) pointed out the last decade trend towards models that describe explicitly nonlinear processes at the crack tip. In particular, cohesive formulations have been successful to some practical problems in composite engineering, where conventional tools such as linear elastic fracture mechanics and the virtual crack closure technique have been proved of scarce efficiency (Cox and Yang, 2006; Bennati and Valvo, 2002; Yam et al., 2004). An important feature of cohesive formulations is that the shape of the evolving delamination surface can be explicitly modeled within the simulation, according to loading conditions and the failure modes. Cohesive models are able to describe various damage mechanisms, including: ply delamination; large shear-induced splitting cracks; multiple cracking of the matrix; fiber rupture or micro-buckling (such as kink band formation); friction dissipation between delaminated plies; crazing or other nonlinearities in the process zones ahead of the crack tips; large scale bridging by through-thickness reinforcement; and oblique crack-bridging fibers. In the next section we give more details on the cohesive approach. The rest of the chapter is structured as follows. In Section 15.3, we focus on the delamination failure and modeling in fiber composites. In Section 15.4, we review the delamination behavior in layered structures, including thin films, thermal barrier coatings, and sandwich structures. In Section 15.5, we conclude by presenting the cohesive composite shell formulation and its potentials.

15.2

The cohesive approach

On the wake of the pioneering works by Dugdale (1960) and Barenblatt (1962), cohesive models regard fracture as a gradual process, where the separation of the incipient crack flanks is resisted by cohesive traction. By focusing specifically on the separation process, cohesive theories make a sharp distinction between fracture, described by recourse to cohesive laws, and bulk material behavior, described through an independent set of constitutive relations. As a noteworthy consequence, the use of cohesive models is not limited by any consideration of material behavior, finite kinematics, nonproportional loading, dynamics, or geometry of the specimen. In addition, they fit naturally within the conventional framework of finite-element analysis, © 2008, Woodhead Publishing Limited

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and have been proved effective in the simulation of complex fracture processes (Camacho and Ortiz, 1996; Ortiz and Pandolfi, 1999; Pandolfi et al., 2001; Ruiz et al., 2000, 2001). It bears emphasis that upon closure the cohesive surfaces are subjected to the contact unilateral constraint, including friction. Contact and friction are regarded as independent phenomena, to be modeled outside the cohesive law. Friction may significantly affect the sliding behavior in closed cohesive surfaces. In particular, the presence of friction may result in a steady – or even increasing – sliding resistance while the normal cohesive strength simultaneously weakens. One important numerical requirement pertaining to the use of cohesive theories is that the selected discretization must resolve adequately the cohesive zone in order to obtain mesh-size independent results (Camacho and Ortiz, 1996). This requirement is satisfied when the mesh is at least five times smaller than the intrinsic length scale of the material lch, defined as: lch = EGc / σ c2

15.1

where E is the elastic modulus, Gc the specific fracture energy, σc the tensile strength of the material, respectively. In materials such as composites, the length lch can be comparatively large, therefore the mesh-size requirements are less severe.

15.3

Delamination failure in fiber reinforced composites

15.3.1 Experimental background Failure in fiber composites may be characterized by several mechanisms, such as: compression-compression fatigue loading (Slaughter and Fleck, 1993a); remote compressive loading (Fleck, 1997); tensile axial loading (Daniel and Anastassopoulos, 1995); multi-axial loading and creep (Slaughter et al., 1993); thermal expansion misfit (Lu et al., 1991); and micro-buckling or plastic kinking (Budiansky and Fleck, 1993). Failure mechanisms may be related to the geometry of the composite, in particular to waviness, diameter, length of the fibers and the global architecture (Fleck et al., 1995b); to the fiber stiffness or the fiber-matrix interface strength; to loading conditions, such as temperature (Ningyun and Evans, 1995); or to existence of holes (Soutis et al., 1993, Soutis and Fleck, 1990, Soutis et al., 1991a, 1991b, 1991c). Budiansky and Fleck (1993) reviewed experimental data and elementary theoretical formulas for compressive failure of polymer matrix fiber composites. In their study, they observed that the dominant failure mode is the plastic kinking, where the initial local fiber misalignment plays the central role. Fractographic features of fracture in multidirectional and unidirectional © 2008, Woodhead Publishing Limited


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laminates can be identified by using scanning electron microscopy (SEM). Using SEM, Gilchrist, Svensson and coworkers (Gilchrist and Svensson, 1995) (Svensson et al., 1998, Svensson and Gilchrist, 1998, Gilchrist et al., 1998) observed that a large amount of fiber pull-out is the dominant feature of a mode I fracture, whereas many cusps with large cusp angles are the main characteristics of mode II fracture. They argued that the energy associated with cusp formation provides a large proportion of the mode II fracture toughness, while the amount of fiber pull-out and fracture has a considerable influence on the mode I fracture toughness. Also Oguni and Ravichandran (2001a) adopted SEM to study the dynamic compressive behavior of unidirectional E-glass/vinylester composites. Figure 15.1 shows two SEM micrographs of the composite with 30% fiber volume fraction, failed in uniaxial compression, under quasi-static loading (Fig. 15.1a) and at 500 /s strain rate (Fig. 15.1b). The former micrograph shows axial splitting, whereas the latter shows fiber-matrix debonding and matrix rupture. When the loading condition changes from uniaxial to proportional multiaxial compression, the failure mode may change remarkably, as observed by Oguni et al. (2000). In a specimen loaded uniaxially, they observed simple axial splitting; contrariwise, under mutiaxial conditions, they noticed the formation of conjugate kink bands, possibly induced by splitting, without any axial splitting. From experimental investigations, Moran et al. (1995) discovered that, under compressive loading, aligned-fiber composites may present two kinds of kink-band propagation: (a) band broadening, i.e., growth of an uniform kink band in the direction of loading at constant stress; (b) transverse kink propagation, i.e., formation of a transverse kink band under constant contraction. Although the compressive failure of fiber-reinforced composite laminates is a cumulative effect of a series of events consisting of a complex pattern of delaminations, matrix cracks, fiber breaks, and others, nevertheless delamination buckling and growth are considered to be the key factors that upperbound the compressive load of a panel (Murty and Reddy, 1993). Ningyun and Evans (1995) assessed the thermal behavior of continuous graphite fiber/thermoplastic matrix composites in the range 20–300 °C, by performing failure tests on short beam specimens. They observed that, at low temperatures, collapse and fracture of the beam is initiated by inter-laminar shear and delamination; whereas failure at high temperatures is characterized by large-scale inelastic deformations.

15.3.2 Analysis and modeling of delamination in fiber composites In order to describe the failure behavior in fiber composites, the bulk material must be modeled by proper constitutive laws. In particular, for unidirectional fiber composites, the material anisotropy has to be taken into consideration. © 2008, Woodhead Publishing Limited

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100µm

100µm

15.1 SEM micrographs of 30% fiber volume fraction E-glass/ vinylester composite, failed in uniaxial compression: (a) quasi-static loading, producing axial splitting; (b) 500 m/s strain rate loading, producing fiber-matrix debonding and matrix rupture (after Oguni and Ravichandran, 2001a).



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By developing the concept of spatial rescaling, that reduces orthotropic problems into equivalent cubic symmetry problems, Suo et al. (1991) derived approximate solutions for orthotropic materials. Such approach provides insights in the interplay between anisotropy and finite geometry. In particular, several technical problems such as delamination induced by a surface flaw, cracking related to stress concentrations, effective contraction of orthotropic material specimens, and crack deflection onto preferential fracture planes, can be addressed efficiently. Nishiwaki et al. (1995a, 1995b) modeled laminated composites accounting for their heterogeneous nature. Slaughter and Fleck (1993a, 1993b, 1994; Slaughter et al., 1993) conducted a theoretical study on visco-elastic micro-buckling of fiber composites. Their analysis is formulated in terms of general linear visco-elastic behavior within the kink band. Assuming that the surrounding material behaves elastically, two specific forms of linear visco-elastic behavior are considered to describe the kink band: a standard linear visco-elastic model and a logarithmically creeping model. They showed that failure is due to either the attainment of a critical failure strain in the kink band or to the intervention of plastic micro-buckling. The initiation of failure in fiber composites can be marked by the first acoustic signal, or the non-linearity point on the load/displacement curve (Ducept et al., 1997; 1999; 2000). Ducept et al. (1997) performed delamination experiments using the DCB specimen on unidirectional glass/epoxy composites to determine a mixed-mode initiation failure criterion, while Sutcliffe et al. (Sutcliffe et al., 1997, Sutcliffe and Fleck, 1997) treated the micro-buckling at the macro scale as a sliding mode II crack. Fleck et al. (1995a) used a coupled stress theory to predict the width of the kink band that occurs in the compressive failure by micro-buckling of a fiber composite. The composite is assumed to be inextensible in the fiber direction, and to deform as a Ramberg-Osgood solid under shear and transverse tension. Predictions for the kink width are given as functions of diameter, elastic modulus and strength of the fibers. Additionally, the material nonlinearities of the composite and amplitude and wavelength of fiber waviness were proved to be significant. Fleck and Shu (Fleck and Shu, 1995; Shu and Fleck, 1997) employed the Cosserat theory to include the role of the resistance due to fiber bending, assuming the fiber diameter as internal length scale of the constitutive law. They analyzed the micro-buckling strength of long fiber-polymer matrix composites under multi-axial in-plane loading using FEM. The analyses showed that the dominant geometrical feature is the length of the initial imperfection in the transverse direction. Cohesive modeling in fiber composites Within the cohesive framework, Siegmund et al. (1997) analyzed the growth of a crack: (a) across an elastic solid; (b) through an interface; and (c) across © 2008, Woodhead Publishing Limited

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an elastic-viscoplastic solid. The numerical analyses show that, without any ad hoc assumptions concerning crack growth or crack path selection criteria, crack initiation, crack growth and crack arrest emerge as natural outcomes of the imposed load. Walter et al. (1997) developed a composite unit cell model based on cohesive theories to analyze the deformation of fiber reinforced composites. They found that strong interfaces led to high stress concentrations in the fiber. In such case, catastrophic brittle failure may occur since the crack nucleated in the matrix would propagate preferentially through the fibers rather than along the interface. Additionally, they observed that the toughness of the matrix has great influence on the onset of debonding and on the trapping of the propagating crack. Finally, they pointed out the importance of providing a debonding criterion able to account for the strength and toughness of both interface and matrix. Numerical modeling of the dynamic delamination of a single lamina from a rigid substrate, based on rate-dependent cohesive theories, has been presented by Corigliano et al. (2003, 2006) and Mariani et al. (2005). Such studies pointed out the space-time multi-scale nature of the delamination process and proposed numerical strategies for the prediction of crack nucleation, growth and arrest.

15.3.3 A numerical application for fiber reinforced composites A detailed example of a combined experimental and numerical analysis of dynamic mode II failure in plates made of unidirectional fiber reinforced composite is documented in Yu et al. (2002). Figure 15.2 depicts the orthonormal coordinate system linked to the plate geometry and to the fiber structure. The x1 axis lies along the fibers in the plane of the plate, the x3 axis is normal to the middle plane of the plate, while the x2 axis is normal to the x1–x3 plane. x3

x2

x1

15.2 Definition of the coordinate axes with respect to the material symmetry axes (after Yu et al., 2002).



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If one assumes that the fibers are randomly distributed in the x2–x3 plane, the material can be considered as transversally isotropic in such plane. The finite element model of the composite plate, discretized into ten-node tetrahedral elements, was analyzed with a 3D finite deformation code. A self-adaptive procedure is used to reproduce explicitly the formation and propagation of cracks. The original mesh is originally fully coherent, and cohesive elements are inserted in a self-adaptive fashion during the analysis at the locations where a suitable failure criterion is satisfied. The experimental planar fracture surface is explicitly described in the discretization, in order to allow the numerical formation of a smooth main crack. A transversally isotropic material model for the bulk Within a nonlinear kinematics framework, the continuum is assumed to be hyper-elastic, characterized by the following free energy density A: A = W ( F , T ) = 1 ( ε i j – αδ ij T ) Cijhk ( ε hk – αδ hk T ) 2

15.2

where Cijhk is the transversely isotropic elastic tensor, F the deformation gradient, T the temperature, δij the Kronecker symbol, and α the thermal expansion coefficient. The logarithmic strain εij is given by:

ε i j = log

C = 1 log ( F T F ) 2

15.3

where C is the right Cauchy-Green strain tensor. The material properties, provided from the experiments and adopted in the calculations, are listed in Table 15.1. A transversally isotropic cohesive model for interfaces Unidirectional composites are designed so as to increase the strength of the matrix material in the direction of the fibers, while the strength parallel to the fibers remains ostensibly identical to that of the matrix. As a consequence, Table 15.1 Elastic material properties of the graphite-epoxy unidirectional composite Property

Value

E1 E 2 = E3 ν12 = ν13 ν23 µ12 = µ13 µ23

80 MPa 8.9 MPa 0.25 0.43 3.6 3.1


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the toughness of the material attains a maximum in the direction of the fibers and a minimum in the direction normal to the fibers. The dependence of the toughness of fiber reinforced composites on the orientation has been analyzed by other authors in previous studies. Allix et al. (Allix and Ladeveze, 1992; Allix et al., 1995; Corigliano, 1993) introduced and analyzed a meso-model for composite, characterized by a cohesive law whose parameters are dependent on the direction of fibers in adjacent layers. The transversely isotropic behavior manifests itself clearly during fracture events. Recent works by Ravichandran (Oguni and Ravichandran, 2000, Oguni and Ravichandran, 2001a, Oguni et al., 2000, Oguni and Ravichandran, 2001b) enhanced the strong dependence of the fracture energy and fracture toughness on the relative orientation between fracture surface and fibers. A very simple way to render such dependence is to assume that the tensile strength varies sinusoidally with the inclination of the fracture surface (see Fig. 15.3). Denoting with α the angle between the fibers and the normal to the fracture surface, one can write:

σcα = σmin + (Qmax – σmin) cos2 α

15.4

where σmax denotes the maximum tensile strength, in the direction parallel to the fibers, and σmin the minimum tensile resistance, in the direction normal to the fibers. On the interelement surfaces, the characteristic cohesive strength varies locally according to the orientation α of the normal to the surface. Following Ortiz and coworkers (Camacho and Ortiz, 1996), in the calculation an effective cohesive law able to account for mixed mode fracture is adopted. Specifically, the displacement jump across a fracture surface is decomposed into normal component δn and tangential component δS, and a scalar opening displacement δ is derived as:

FD

N

CS

N

σ cα

α

σ min α σ max

x1

(a)

CS

(b)

15.3 (a) Definition of the angle α. FD denotes the fiber direction. (b) Visualization of the orthotropic cohesive strength described in Eq. (4). CS denotes the orientation of the cohesive surface (after Yu et al., 2002).



δ=

δ n2 + β 2 δ S2

439

15.5

where the weighting coefficient β defines the ratio shear to normal strength of the cohesive material. The existence of a free energy density φ(δ, σcα) per unit undeformed cohesive surface, dependent on the scalar δ, is assumed. In a thermodynamically consistent way, the cohesive law for the cohesive surface is obtained by differentiating the free energy density with respect to the effective opening displacement: tα =

∂ Φ(δ , σ cα ) ∂δ

15.6

The previous considerations apply to the initiation criterion for fracture as well. In agreement with Eq. (15.5), the effective scalar traction t is computed as:

t n2 + β –2 δ S2

t=

15.7

where tn is the normal component and tS the tangential component of the traction vector with respect to the cohesive surface. The insertion criterion reads: t ≥ σcα

15.8

The scalar cohesive law adopted in the numerical calculation is reported in Fig. 15.4. For each orientation, the cohesive law is characterized by two parameters only, i.e. the cohesive strength and the energy cohesive rate, represented by the peak of the tractions and by the area enclosed by the cohesive law respectively. The fracture parameters for the graphite-epoxy material are collected in Table 15.2. t/σc

1

Gc = σ cδ c /2

0

δmax/δ c

1

δ /δ c

15.4 Cohesive law adopted in the simulation of failure in transversally isotropic plates: linearly decreasing cohesive envelope and irreversible behavior under unloading and reloading (after Yu et al., 2002).


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Delamination behaviour of composites Table 15.2 Fracture parameters of the graphite-epoxy unidirectional composite Property

Value

Fracture toughness parallel to the fibers KIc Fracture energy parallel to the fibers GIc Cohesive strength parallel to the fibers σc Weighting coefficient β Ratio σc,max/σc,min

2.2 MPaVm 474 N/m 35.8 MPa 0.726 100

Validation and verification The dynamic experiments performed by Coker and Rosakis (Coker and Rosakis, 2001; Yu et al., 2002) on composite plates were conducted in an asymmetric configuration, leading to intrasonic mode-II crack growth. An impacting projectile launched at 30 m/s speed was fired from a gas gun. The coherent gradient sensing (CGS) optical technique recorded the gradient of the out-of-plane displacements on one side of the specimen and was able to capture the formation of a double shock wave structure typical of intersonic crack propagation. The specimen size is described in Fig. 15.5a. The mesh adopted in the calculations is reported in Fig. 15.5b. The experiments showed that, at the early stage of the crack propagation, a broad band of compressive stresses clearly forms at the top side of the crack defining a wave structure inclined about 30 degrees. The strong stress discontinuity testifies that the crack tip speed is larger than the shear wave speed. Figure 15.6 compares the double shock wave structure recorded in the experiments with the stress contour levels obtained in the calculations. A noteworthy correspondence was also observed in the curve crack tip position versus crack tip velocity. In particular, the simulation was able to capture the intersonic speed of the mode II crack, see Fig. 15.7.

15.4

Delamination failure in layered structures

According to their applications, layered structures can be classified in three main categories: films, thermal barrier coatings and multi-layered structures (Hutchinson and Suo, 1992). In particular, thin films and multi-layered structures made of different classes of material are often used for various functional requirements. As these become relatively large in section and geometrically more complex, the thermo-mechanical integrity is a major concern. Delamination failure is one of the most important issues for such applications. In the following, after a brief introduction able to describe the physical origin of delamination in films and thermal barrier coatings, we present the application of cohesive models to the numerical analysis of sandwich structures, a special case of layered stuctures. © 2008, Woodhead Publishing Limited


Fiber direction

a

441

H = 20.4 cm W = 12.7 cm a = 2.5 cm

50 mm

H V

Mode II

Field of view

W

Projectile Steel (a)

(b)

15.5 Experiments and numerical simulations of the mode II loaded composite plates. (a) Specimen size and loading conditions; (b) initial mesh adopted in the calculations. Minimum mesh size 1 mm. The mesh consists of 44 593 nodes and 24 685 10-node tetrahedral elements (after Yu et al., 2002).

15.4.1 Delamination in thin films and coatings Debonding of films attached to substrates typically involves three phases: (a) initiation, (b) steady-state propagation, and (c) final transient, as the debond converges towards an edge or another debond (He et al., 1997). Debonding may be originated by several causes, such as the elastic mismatch (i.e., difference in the elastic properties) or the thermal mismatch (i.e., difference in the thermal expansion properties) between films and substrate. The role of elastic mismatch The elastic mismatch between film and substrate can be described by Dundurs´ constants: αD and βD, which represents the mismatch of Young’s moduli and the dissimilarity of shear moduli and Poisson’s ratio respectively (Suo and Hutchinson, 1990). In most cases αD appears to be more important, thus © 2008, Woodhead Publishing Limited

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(a)

(b)

15.6 Comparison between the experimental and numerical double shock wave structure in transversally isotropic composite plates loaded in mode II (after Yu et al., 2002).

generalized delamination charts as a function of the sole constant αD have been constructed. Such charts are used to characterize composite systems and to maximize both the transverse strength and the longitudinal toughness. Yu and Hutchinson (2002) analyzed a straight-sided delamination buckling, focusing on the influence of the substrate compliance. In particular, they calculated the critical buckling condition, the energy release rate and the mode mix of the delamination crack of the interface as a function of the elastic mismatch between film and substrate. Their analysis showed that the more compliant the substrate, the higher the values of the energy release © 2008, Woodhead Publishing Limited


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8

C LII 7

Crack-speed [km/s]

6 5 4 Experiment 1 Experiment 2 Experiment 3 Experiment 4 Coarse mesh

3 2

CR

1 0 20

30

40 50 60 Crack-extension [mm]

70

80

15.7 Comparison between the experimental and numerical crack-tip velocity versus crack-tip position curves (after Yu et al., 2002). The crack velocity reaches the intersonic values.

rates and the easier to observe film buckling. Such effects become even more significant when the elastic modulus of the substrate is appreciably smaller than the one of the film. He et al. (1994) analyzed the role of residual stress on the behavior of a crack intersecting an interface between two different materials. The main result of their work was that expansion of elastic modulus misfit may have severe consequences in systems with planar interfaces, but is of little significance in fiber composites. The role of thermal mismatch As a result of the thermal expansion mismatch, ceramic coatings deposited on metal substrates generally develop significant compressive stresses when cooled from the temperature at which they are processed. One of the main failure modes for such coatings is edge delamination (Balint and Hutchinson, 2001). For ideally brittle interfaces, the edge delamination of a compressed thin film typically implies mode II interface cracking. The crack surfaces are in contact, under compressive normal stresses acting behind the advancing crack tip. It has been observed that the frictional shielding of the crack tip increases the overall apparent fracture toughness. In their work, Lu et al. addressed the cracking of the matrix originated by thermal expansion misfit © 2008, Woodhead Publishing Limited

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in brittle matrix composites (Lu et al., 1991). They observed the existence of a critical reinforcement size, able to limit or impede the cracking of the matrix. This concept was summarized in terms of a non-dimensional group R, which accounts for reinforcement size, misfit strain, elastic modulus and matrix toughness. Additionally, Lu et al. (1991) proved that, by relaxing the constraints though debonding, the interface may exerts a major influence on the cracking of the matrix. In the definition of material combinations, they propose to use a failsafe value of R in order to suppress matrix cracking. Balint and Hutchinson (2005) developed a mechanistic model as analytical approximation of undulation growth during cyclic thermal histories in multilayer thermal barrier coatings, deposited on super-alloy turbine blades and exposed to combustion temperatures greater than 1500 °C. In their model, one of the dominant failure modes accounts for cracking originated from undulation growth (rumpling) of the highly compressed oxide layer that grows between the ceramic top coat and the inter-metallic bond coat. The energetic perspective By recourse to von Karman theory of moderate deflections of a plate, Ortiz and Gioia provide an energetic interpretation of the shapes and patterns of blisters in thin films and coatings under a state of residual compression (Ortiz and Gioia, 1994). Their energy functional contains two terms, i.e., the membrane energy and bending energy. The latter term is seen as a singular perturbation of the former. The membrane energy functional is non-convex and, consequently, its infimum is in general not attained. The solutions were constructed through a matched asymptotic expansion. The outer solution, obtained by minimization of the membrane energy, determines the essential folding pattern of the film. The inner solution is obtained by fitting boundary layers at the sharp edges within the membrane solution. The constructed film deflections matched, in surprising detail, the observed complex folding patterns of delaminated films. In addition, in the membrane solution, the boundary layer analysis permits linking a well-defined tension line to sharp edges and to the boundary of the blister. This provides a simple device for assessing the configurational stability of some blister morphologies. In particular, the analysis predicts the transition from straight-sided to telephone-cord morphologies at a critical mismatch strain. Subsequently, the same authors proposed alternative methods for the determination of the fracture energy and kinetic coefficient of thin film/substrate interfaces from measurements performed on telephone cord blisters (Gioia and Ortiz, 1997a, 1997b). The methods proposed by Gioia and Ortiz stem from the following facts: (a) telephone cord blisters are spontaneous debonding features, and therefore more amenable than artificially contrived features to yield realistic debonding parameters; (b) they are very commonly observed in compressed thin films; (c) their boundaries are © 2008, Woodhead Publishing Limited


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characterized by a constant fracture mode mixty; and (d) the driving force for debonding equals the fracture energy everywhere on their boundaries. In order to elucidate how the folding patterns of thin-film diaphragms depend on shape, thickness and size of the diaphragms, Gioia et al. (2002) performed experiments by applying in-plane isotropic and anisotropic compressive strains to thin-film diaphragms. The tests showed that the folding differences between isotropic and anisotropic cases can be traced back to the structure of the membrane energy density function. In the isotropic case, it is possible to identify folding patterns that minimize the membrane energy and satisfy the boundary conditions. Such foldings do not exist in the anisotropic case, but it is possible to construct sequences of increasingly fine foldings able to satisfy the boundary conditions, whose membrane energies converge to the infimum. Resistance to delamination and stability of the delamination growth Since delamination resistance in composites can be strongly dependent on the specimen size and geometry, it cannot be considered as a material property. Suo et al. (1992a) pointed out that the resistance to delamination can be enhanced by a variety of bridging mechanisms. In general, the size of the bridging zone is usually several times larger than the thickness of the lamina. In slender beams, Suo et al. (1992b) found that the resistance plateau is independent of the beam thickness, whereas the size of the steady-state bridging zone increases with the beam thickness. Furthermore, the failure mode mechanism may also show size dependence. By exploiting methods based on complex variables, Suo et al. (1992b) addressed the concept of stability. Considering two semi-infinite solids bonded along a planar cohesive interface, they investigated the conditions of uniqueness of the solutions for quasi-static boundary value problems. Since the interface is characterized by a traction-displacement jump relation, from dimensional analysis it is possible to identify the presence of a characteristic length. As a peculiar result, Suo et al. (1992b) showed that the minimum wavelength for the instability mode may be one to two orders of magnitude larger than the characteristic length of the cohesive interface. In brittle adhesive layers joining two substrates, cracks may propagate in a variety of ways, i.e., they may follow straight or wavy paths within the adhesive layer or along one of the interfaces, and they may switch from interface to interface crossing the layer. In such materials the effective toughness of the joint depends strongly on the feature of the crack path. Fleck et al. (1991) investigated the problem of the crack stability, by analyzing the conditions that drive a straight crack path within a brittle adhesive layer. They consider an elastic problem, where an adhesive layer between two semi-infinite blocks contains a semi-infinite straight crack. The asymptotic © 2008, Woodhead Publishing Limited

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analysis provides the location of the crack, as a function of the combination of applied intensity factors and of the elastic modulus mismatch between the layer and the adjoining material. A phenomenological continuum model of film growth, based on a series expansion of the deposition flux in powers of the profile gradient, on consideration of the energetics of the film-substrate interface and on the enforcement of Onsager’s reciprocity relations, was presented by Ortiz et al. (1999). The interfacial term, which operates at very small thicknesses, is nonconservative and breaks the +/–h symmetry proper of the remaining terms of the kinetic equation. By virtue of this term, very thin films are predicted to be stable within an appropriate range of geometrical parameters, while they lose stability and become rough at a well-defined critical thickness. Such instability effectively provides an island nucleation mechanism. Contrariwise, for thick films, the rate processes envisioned in the model suggest a characteristic slope for the film profile, feature in keeping with observations for a number of systems, including YBCO (YBa2Cu3O7) films.

15.4.2 Sandwich structures Sandwich structures are more complex in comparison to laminated structures. Owing to their special structure, often the damage does not develop uniformly across the thickness. Additionally, typical failure modes of sandwich structures include crushing and facet sheets debonding, besides penetration and delamination. Frequently, the shear load transfer through the core is characterized by the presence of unsymmetrical damage, which requires deep investigations in order to understand the evolution mechanisms and to evaluate the residual strength. Composite sandwich beams, made of glass-vinylester face sheets and a PVC foam core, have been manufactured and tested in quasi-static configuration by Tagarielli et al. (2004). In their work, Tagarielli et al. studied the initial collapse modes, the processes that govern the post-yield deformation, and the parameters that affect the ultimate strength. In particular, they observed that the initial collapse is due to the competition of three distinct mechanisms: micro-buckling of the faces, shear in the core, and indentation. In a subsequent work, Tagarielli and Fleck (2005) showed that the presence of clamped boundaries may drive the deformation mechanism towards plastic stretching of the face sheets. As a consequence, the ultimate strength and the level of energy absorption in sandwich beams are defined by the ductility of the face sheets. Tilbrook et al. (2006) developed an analytical model to classify the response of sandwich beams to impulsive loads. The model is based on the relative time scales of the core compression and of the bending/stretching response of the beams. The analysis suggested that an overlap in time scales © 2008, Woodhead Publishing Limited


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may lead to a coupled response and to the possibility to enhance the shock resistance. Yam et al. (2005) gave an interpretation to the mechanism of mode dependent energy dissipation due to delamination in composite plates, providing experimentally the evidence of the dependence of model parameters on size and location of damage. Espinosa et al. (Espinosa et al., 2000; Dwivedi and Espinosa, 2003) analyzed numerically the dynamic delamination in woven glass fiber reinforced plastic composite (GRP), by using a 3D finite deformation anisotropic viscoplastic model in conjunction with interfacial cohesive/contact laws. The heterogeneity of composite materials, modeled by considering a layered structure, leads to dispersion and scattering of pressure and shear waves. Delamination and matrix cracking in sandwich structured composites Xu and Rosakis (2002) conducted an experimental program to investigate the generation and subsequent evolution of dynamic failure modes in layered materials subjected to impact. To simplify the complex analysis of impact processes in real sandwich structures, they performed model experiments in plane stress configuration. The nature and the sequence of failure modes were analyzed by means of high-speed photography and dynamic photoelasticity. The experiments revealed a series of complex failure modes, and pointed out that the dominant dynamic failure mode is the interlayer failure, i.e. the delamination between panel and core. Interlayer failure is a sheardriven process, and cracks propagate at intersonic speed even under moderate impact speeds. Shear induced interlayer cracks tend to kink into the core layer; propagate in the form of opening-dominated intralayer cracks; and, when they attain sufficiently high propagation speeds, eventually branch, causing diffused fragmentation. The sandwich specimen used in Xu and Rosakis’s experiments is a thin plate obtained by bonding two 4340-steel external plates to a Homalite-100 plate. The geometry for a typical specimen used in the experiments is shown in Fig. 15.8a. The elastic material constants for steel and Homalite are listed in Table 15.3. The fracture parameters for the two bulk materials, together with the properties for the interface bond (Weldon-10) are listed in Table 15.4 Yu et al. (2003) performed finite element simulations of the sandwich structure experiments. In the numerical analysis, the effect of the impacting bullet is approximated by prescribing a velocity profile to the nodes on the impacted area. The maximum impact speed used in the simulation is 32.4 m/s. Figure 15.8 shows a comparison between experimental and numerical photo-elastic fringes. The images clearly indicate a shock wave structure, which testifies a crack speed greater than the Rayleigh wave speed. The © 2008, Woodhead Publishing Limited

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W = 114.3 mm

L = 254 mm

Steel Homalite

T = 6.35 mm

Steel

V Area = 57.15 (a)

mm2

(b)

15.8 Sandwich structure experiments and numerical calculations: (a) specimen geometry; (b) comparison between experimental and numerical photo-elastic fringes (after Yu et al., 2003).

crack pattern obtained in the simulation of a larger sandwich beam is shown in Fig. 15.9 (Yu, 2001). Interestingly, the delamination buckling and the crack kinking into the central core are explicit outcomes of the simulations.



449

Table 15.3 Elastic material properties for Homalite-100 and 4340 Steel Material Young’s modulus Poisson’s ratio Mass density Longitudinal speed Shear wave speed Rayleigh wave speed

E (GPa) ν ρ (kg/m3) cd (m/s) cs (m/s) cR (m/s)

Homalite-100

4340 Steel

5.3 0.35 1230 2200 1255 1185

208 0.3 7830 5500 3320 2950

Table 15.4 Fracture and cohesive parameters for Homalite-100, 4340 Steel and Weldon-10 Material Tensile strength Shear strength Fracture energy Opening displacement Characteristic length

σc (MPa) τc (MPa) Gc (N/m) δc (µm) R (mm)

Homalite-100

4340 Steel

Weldon-10

35 40 88.1 6.6 2.8

1490 1054.0 10620 14.3 14

14 22 45.0 176.6 18

15.9 Numerical simulation of branching and delamination buckling in a sandwich structure (after Yu, 2001).

15.4.3 The use of shell elements in delamination modeling Composite cylindrical shells and panels are widely used in aerospace and submarine structures. The use of three-dimensional finite elements for predicting the delamination buckling of these structures is computationally expensive. One alternative is the use of shell elements. In comparison with plate elements, shell elements may provide more reliable results when adopted in the simulation of delamination processes (Engblom et al., 1989). Tafreshi (2004a, b) combined double-layer and single-layer of shell elements to study the effect of delamination. He showed that through-the-thickness delamination can be modeled and analysed effectively without requiring a great deal of computing time and memory (Tafreshi, 2006). In the analysis of post-buckled delaminations, Tafreshi applied the virtual crack closure technique to find the distribution of the local strain energy release rate along the delamination front. More applications of shell elements in the analysis of failure process and delamination buckling can be found in Tanimoto et al., (2002), Chattopadhyay and Gu (1995) and Gu and Chattopadhyay (1996). © 2008, Woodhead Publishing Limited

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By using the subdivision cohesive shell model developed by Cirak et al. (2005), Elsayed (2007) proposed an innovative composite shell model, as shown in Fig. 15.10. The bulk material for the shell is described by Ogdentype deviatoric elasticity and rate-dependent porous plasticity, as presented in Weinberg et al. (2006). The energy of the shell accounts for both the deformation energy associated with membrane and shell behavior and the cohesive energy of the fracture surfaces. The shell elements are based on the subdivision algorithm and use non-local shape functions. Fracture surfaces are allowed to develop only along the element boundaries, and their insertion requires the pre-fracturing of the shell and the inclusion of cinematic constraints to guarantee the continuity of the domain. At the onset of fracture, the kinematic constraints are removed and the progressive opening of the fracture is controlled by cohesive surfaces. An example of application of the composite shell element is the simulation of impact in a ship hull structure, see in Fig. 15.11(a) (front) and (b) (rear).

15.5

Summary and conclusions

We presented a synthetic review of delamination fracture modeling in fiber composites and in layered structures. Approaches based on damage theories and on fracture mechanics have been analyzed. Cohesive approaches have been discussed in more detail, since in numerical application of debonding and delamination processes they have been proved to be the most appropriate to capture several failure mechanisms such as: fiber breaking, fiber-matrix debonding; matrix cracking, etc. For the analysis of thin layered structures, shell elements are computationally more efficient compared to solid finite elements. In this respect, we briefly discussed an innovative composite shell

Cohesive surface

a 3–

+

a3

a3

Quadrature points PU

h Steel

15.10 Formulation of the composite shell element (after Elsayed, 2007).



451

0

y

–1

–2

–3

0 –1 z

–2

–4

–2

0 x

2

4

(a)

y

0

–1 –2

0 –1 z –2 –4

–3

–2

x

0

2

4

(b)

15.11 Modeling of the impact failure with composite shell elements (after Elsayed, 2007).

element with fracture features based on cohesive theories. This type of cohesive composite shell elements incorporate the efficiency of shell kinematics with the versatility of cohesive theory, are therefore appropriate for modeling the delamination failure in layered composites.

15.6

Acknowledgements

The authors wish to thank Prof. Michael Ortiz for the valuable suggestions he gave in carrying out this work. Yu acknowledges the financial support from the Universidad de Castilla-La Mancha for her two-month stay in Caltech where this manuscript took its original form. © 2008, Woodhead Publishing Limited

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15.7


References

Abrao, A. M., Faria, P. E., Rubio, J. C. C., Reis, P. & Davim, J. P. (2007) Drilling of fiber reinforced plastics: A review. Journal of Materials Processing Technology, 186, 1–7. Allix, O. & Ladeveze, P. (1992) Interlaminar interface modeling for the prediction of delamination. Composite Structures, 22, 235–242. Allix, O., Ladeveze, P. & Corigliano, A. (1995) Damage analysis of interlaminar fracture specimens. Composite Structures, 31, 61–74. Andrews, M. G., Massabo, R. & Cox, B. N. (2006) Elastic interaction of multiple delaminations in plates subject to cylindrical bending. International Journal of Solids and Structures, 43, 855–886. Arul, S., Raj, D. S., Vijayaraghavan, L., Malhotra, S. K. & Krishnamurthy, R. (2006) Modeling and optimization of process parameters for defect toleranced drilling of GFRP composites. Materials and Manufacturing Processes, 21, 357–365. Baley, C., Davies, P., Grohens, Y. & Dolto, G. (2004) Application of interlaminar tests to marine composites. A literature review. Applied Composite Materials, 11, 99–126. Balint, D. S. & Hutchinson, J. W. (2001) Mode II edge delamination of compressed thin films. Journal of Applied Mechanics – Transactions of the ASME, 68, 725–730. Balint, D. S. & Hutchinson, J. W. (2003) Undulation instability of a compressed elastic film on a nonlinear creeping substrate. Acta Materialia, 51, 3965–3983. Balint, D. S. & Hutchinson, J. W. (2005) An analytical model of rumpling in thermal barrier coatings. Journal of the Mechanics and Physics of Solids, 53, 949–973. Balint, D. S., Xu, T., Hutchinson, J. W. & Evans, A. G. (2006) Influence of bond coat thickness on the cyclic rumpling of thermally grown oxides. Acta Materialia, 54, 1815–1820. Barenblatt, G. I. (1962) The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, 7, 55–129. Bennati, S. & Valvo, P. S. (2002) An elastic-interface model for delamination buckling in laminated plates. Experimental Techniques and Design in Composite Materials 5, 221–2, 293–306. Budiansky, B. & Fleck, N. A. (1993) Compressive Failure of Fiber Composites. Journal of the Mechanics and Physics of Solids, 41, 183–211. Camacho, G. T. & Ortiz, M. (1996) Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures, 33, 2899–2938. Carroll, H. E., Matthams, T. J. & Davies, A. J. (2003) Delamination growth during fatigue of advanced polymer matrix composites. Plastics Rubber and Composites, 32, 139– 144. Chai, H., Babcock, C. D. & Knauss, W. G. (1981) One dimensional modeling of failure in laminated plates by delamination buckling. International Journal of Solids and Structures, 17, 1069–1083. Chattopadhyay, A. & Gu, H. Z. (1995) Modeling of delamination buckling in composite cylindrical-shells with a new higher-order theory. Composites Science and Technology, 54, 223–232. Cirak, F., Ortiz, M. & Pandolfi, A. (2005) A cohesive approach to thin-shell fracture and fragmentation. Computer Methods in Applied Mechanics and Engineering, 194, 2604– 2618. Coker, D. & Rosakis, A. (2001) Experimental observations of intersonic crack growth in asymmetrically loaded unidirectional composite plates. Philosophical Magazine A, 81, 571–595.



453

Corigliano, A. (1993) Formulation, identification and use of interface models in the numerical-analysis of composite delamination. International Journal of Solids and Structures, 30, 2779–2811. Corigliano, A., Mariani, S. & Pandolfi, A. (2003) Numerical modeling of rate-dependent debonding processes in composites. Composite Structures, 61, 39–50. Corigliano, A., Mariani, S. & Pandolfi, A. (2006) Numerical analysis of rate-dependent dynamic composite delamination. Composites Science and Technology, 66, 766–775. Cox, B. & Yang, Q. D. (2006) In quest of virtual tests for structural composites. Science, 314, 1102–1107. Daniel, I. M. & Anastassopoulos, G. (1995) Failure mechanisms and damage evolution in crossply ceramic-matrix composites. International Journal of Solids and Structures, 32, 341–355. Ducept, F., Davies, P. & Gamby, D. (1997) An experimental study to validate tests used to determine mixed mode failure criteria of glass/epoxy composites. Composites Part A – Applied Science and Manufacturing, 28, 719–729. Ducept, F., Gamby, D. & Davies, P. (1999) A mixed-mode failure criterion derived from tests on symmetric and asymmetric specimens. Composites Science and Technology, 59, 609–619. Ducept, F., Davies, P. & Gamby, D. (2000) Mixed mode failure criteria for a glass/epoxy composite and an adhesively bonded composite/composite joint. International Journal of Adhesion and Adhesives, 20, 233–244. Dugdale, D. S. (1960) Yielding of steel sheets containing slits. Journal of the Mechanics and Physics of Solids, 8, 100–104. Dwivedi, S. K. & Espinosa, H. D. (2003) Modeling dynamic crack propagation in fiber reinforced composites including frictional effects. Mechanics of Materials, 35, 481– 509. Elder, D. J., Thomson, R. S., Nguyen, M. Q. & Scott, M. L. (2004) Review of delamination predictive methods for low speed impact of composite laminates. Composite Structures, 66, 677–683. Elsayed, T. (2007) Constitutive models for polymers and soft biological tissues PhD Thesis. Pasadena, California Institute of Technology. Engblom, J. J., Fuehne, J. P. & Hamdallah, J. M. (1989) Transverse stress calculations for laminated composite shell structures using plate/shell finite element formulations. Jounral of Reinforced Plastics and Composites, 8, 446–457. Espinosa, H. D., Dwivedi, S. & Lu, H. C. (2000) Modeling impact induced delamination of woven fiber reinforced composites with contact/cohesive laws. Computer Methods in Applied Mechanics and Engineering, 183, 259–290. Fleck, N. A. (1997) Compressive failure of fiber composites. Advances in Applied Mechanics, Vol 33. San Diego, Academic Press Inc. Fleck, N. A. & Shu, J. Y. (1995) Microbuckle Initiation In Fiber Composites – A FiniteElement Study. Journal of the Mechanics and Physics of Solids, 43, 1887–1918. Fleck, N. A., Hutchinson, J. W. & Suo, Z. G. (1991) Crack path selection in a brittle adhesive layer. International Journal of Solids and Structures, 27, 1683–1703. Fleck, N. A., Deng, L. & Budiansky, B. (1995a) Prediction of kink width in compressed fiber composites. Journal of Applied Mechanics – Transactions of the ASME, 62, 329– 337. Fleck, N. A., Jelf, P. M. & Curtis, P. T. (1995b) Compressive failure of laminated and woven composites. Journal of Composites Technology & Research, 17, 212–220.


454


Garg, A. C. (1988) Delamination – a damage mode in composite structures. Engineering Fracture Mechanics, 29, 557–584. Gilchrist, M. D. & Svensson, N. (1995) A fractographic analysis of delamination within multidirectional carbon/epoxy laminates. Composites Science and Technology, 55, 195–207. Gilchrist, M. D., Svensson, N. & Shishoo, R. (1998) Fracture and fatigue performance of textile commingled yarn composites. Journal of Materials Science, 33, 4049–4058. Gioia, G. & Ortiz, M. (1997a) Delamination of compressed thin films. Advances in Applied Mechanics, Vol 33. San Diego, Academic Press Inc. Gioia, G. & Ortiz, M. (1997b) Determination of thin-film debonding parameters from telephone-cord measurements. Acta Materialia, 46, 169–175. Gioia, G., Desimone, A., Ortiz, M. & Cuitino, A. M. (2002) Folding energetics in thinfilm diaphragms. Proceedings of the Royal Society of London Series A – Mathematical Physical and Engineering Sciences, 458, 1223–1229. Gu, H. Z. & Chattopadhyay, A. (1996) Delamination suckling and postbuckling of composite cylindrical shells. AIAA Journal, 34, 1279–1286. He, M. Y., Evans, A. G. & Hutchinson, J. W. (1994) Crack deflection at an interface between dissimilar elastic-materials – role of residual-stresses. International Journal of Solids and Structures, 31, 3443–3455. He, M. Y., Evans, A. G. & Hutchinson, J. W. (1997) Convergent debonding of films and fibers. Acta Materialia, 45, 3481–3489. Hutchinson, J. W. & Suo, Z. (1992) Mixed-mode cracking in layered materials. Advances in Applied Mechanics, Vol 29. San Diego, CA, Academic Press Inc. Hutchinson, J. W., He, M. Y. & Evans, A. G. (2000) The influence of imperfections on the nucleation and propagation of buckling driven delaminations. Journal of the Mechanics and Physics of Solids, 48, 709–734. Lu, T. C., Yang, J., Suo, Z., Evans, A. G., Hecht, R. & Mehrabian, R. (1991) Matrix cracking in intermetallic composites caused by thermal-expansion mismatch. Acta Metallurgica Et Materialia, 39, 1883–1890. Mariani, S., Pandolfi, A. & Pavani, R. (2005) Coupled space-time multiscale simulations of dynamic delamination tests. Materials Science – Poland, 23, 509–519. Mi, Y., Crisfield, M. A., Davies, G. A. O. & Hellweg, H. B. (1998) Progressive delamination using interface elements. Journal of Composite Materials, 32, 1246–1272. Moran, P. M., Liu, X. H. & Shih, C. F. (1995) Kink band formation and band broadening in fiber composites under compressive loading. Acta Metallurgica Et Materialia, 43, 2943–2958. Murty, A. V. K. & Reddy, J. N. (1993) Literature review: compressive failure of laminates and delamination buckling: a review. The Shock and Vibration Digest, 25, 3–12. Ningyun, W. & Evans, J. T. (1995) Collapse of continuous fiber-composite beams at elevated-temperatures. Composites, 26, 56–61. Nishiwaki, T., Yokoyama, A., Maekawa, Z. & Hamada, H. (1995a) A new numerical modeling for laminated composites. Composite Structures, 32, 641–647. Nishiwaki, T., Yokoyama, A., Maekawa, Z. & Hamada, H. (1995b) A quasi-3-dimensional elastic-wave propagation analysis for laminated composites. Composite Structures, 32, 635–640. Oguni, K. & Ravichandran, G. (2000) An energy-based model of longitudinal splitting in unidirectional fiber-reinforced composites. Journal of Applied Mechanics – Transactions of the ASME, 67, 437–443.



455

Oguni, K. & Ravichandran, G. (2001a) Dynamic compressive behavior of unidirectional E-glass/vinylester composites. Journal of Materials Science, 36, 831–838. Oguni, K. & Ravichandran, G. (2001b) A micromechanical failure model for unidirectional fiber reinforced composites. International Journal of Solids and Structures, 38, 7215– 7233. Oguni, K., Tan, C. Y. & Ravichandran, G. (2000) Failure mode transition in unidirectional E-glass/vinylester composites under multiaxial compression. Journal of Composite Materials, 34, 2081–2097. Ortiz, M. & Gioia, G. (1994) The morphology and folding patterns of buckling-driven thin-film blisters. Journal of the Mechanics and Physics of Solids, 42, 531–559. Ortiz, M. & Pandolfi, A. (1999) Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. International Journal for Numerical Methods in Engineering, 44(9), 1267–1282. Ortiz, M., Repetto, E. A. & Si, H. (1999) A continuum model of kinetic roughening and coarsening in thin films. Journal of the Mechanics and Physics of Solids, 47, 697– 730. Pandolfi, A., Guduru, P. R., Ortiz, M. & Rosakis, A. J. (2000) Three dimensional cohesiveelement analysis and experiments of dynamic fracture in C300 steel. International Journal of Solids and Structures, 37(27), 3733–3760. Ruiz, G., M. Ortiz, & Pandolfi, A. (2000) Three-dimensional finite-element simulation of the dynamic Brazilian tests on concrete cylinders. International Journal for Numerical Methods in Engineering, 48(7), 963–994. Ruiz, G., Pandolfi, A. & Ortiz, M. (2001) Three-dimensional cohesive modeling of dynamic mixed-mode fracture. International Journal for Numerical Methods in Engineering, 52(1–2), 97–120. Shu, J. Y. & Fleck, N. A. (1997) Microbuckle initiation in fibre composites under multiaxial loading. Proceedings of the Royal Society of London Series A – Mathematical Physical and Engineering Sciences, 453, 2063–2083. Siegmund, T., Fleck, N. A. & Needleman, A. (1997) Dynamic crack growth across an interface. International Journal of Fracture, 85, 381–402. Slaughter, W. S. & Fleck, N. A. (1993a) Compressive fatigue of fiber composites. Journal of the Mechanics and Physics of Solids, 41, 1265–1284. Slaughter, W. S. & Fleck, N. A. (1993b) Viscoelastic microbuckling of fiber composites. Journal of Applied Mechanics – Transactions of the ASME, 60, 802–806. Slaughter, W. S. & Fleck, N. A. (1994) Microbuckling of fiber composites with random initial fiber waviness. Journal of the Mechanics and Physics of Solids, 42, 1743–1766. Slaughter, W. S., Fleck, N. A. & Budiansky, B. (1993) Compressive failure of fiber composites – the roles of multiaxial loading and creep. Journal of Engineering Materials and Technology – Transactions of the ASME, 115, 308–313. Soutis, C. & Fleck, N. A. (1990) Static compression failure of carbon-fiber T800/924C composite plate with a single hole. Journal of Composite Materials, 24, 536–558. Soutis, C., Fleck, N. A. & Curtis, P. T. (1991a) Hole hole interaction in carbon-fiber epoxy laminates under uniaxial compression. Composites, 22, 31–38. Soutis, C., Fleck, N. A. & Smith, P. A. (1991b) Compression fatigue behavior of notched carbon-fiber epoxy laminates. International Journal of Fatigue, 13, 303–312. Soutis, C., Fleck, N. A. & Smith, P. A. (1991c) Failure prediction technique for compression loaded carbon fiber-epoxy laminate with open holes. Journal of Composite Materials, 25, 1476–1498. Soutis, C., Curtis, P. T. & Fleck, N. A. (1993) Compressive failure of notched carbon© 2008, Woodhead Publishing Limited

456


fiber composites. Proceedings of the Royal Society of London Series A – Mathematical Physical and Engineering Sciences, 440, 241–256. Spearing, S. M. & Beaumont, P. W. R. (1992a) Fatigue damage mechanics of compositematerials. 1. Experimental-measurement of damage and post-fatigue properties. Composites Science and Technology, 44, 159–168. Spearing, S. M. & Beaumont, P. W. R. (1992b) Fatigue damage mechanics of compositematerials. 3. Prediction of post-fatigue strength. Composites Science and Technology, 44, 299–307. Spearing, S. M., Beaumont, P. W. R. & Ashby, M. F. (1992a) Fatigue damage mechanics of composite-materials. 2. A damage growth-model. Composites Science and Technology, 44, 169–177. Spearing, S. M., Beaumont, P. W. R. & Smith, P. A. (1992b) Fatigue damage mechanics of composite-materials. 4. Prediction of post-fatigue stiffness. Composites Science and Technology, 44, 309–317. Suiker, A. S. J. & Fleck, N. A. (2006) Modelling of fatigue crack tunneling and delamination in layered composites. Composites Part A – Applied Science and Manufacturing, 37, 1722–1733. Suo, Z. G. & Hutchinson, J. W. (1990) Interface crack between two elastic layers. International Journal of Fracture, 43, 1–18. Suo, Z., Bao, G., Fan, B. & Wang, T. C. (1991) Orthotropy rescaling and implications for fracture in composites. International Journal of Solids and Structures, 28, 235–248. Suo, Z., Bao, G. & Fan, B. (1992a) Delamination R-curve phenomena due to damage. Journal of the Mechanics and Physics of Solids, 40, 1–16. Suo, Z., Ortiz, M. & Needleman, A. (1992b) Stability of solids with interfaces. Journal of the Mechanics and Physics of Solids, 40, 613–640. Sutcliffe, M. P. F. & Fleck, N. A. (1997) Microbuckle propagation in fibre composites. Acta Materialia, 45, 921–932. Sutcliffe, M. P. F., Fleck, N. A. & Xin, X. J. (1997) Prediction of compressive toughness for fibre composites (vol 452, pg 2443, 1996). Proceedings of the Royal Society of London Series A – Mathematical Physical and Engineering Sciences, 453, 1119– 1120. Svensson, N. & Gilchrist, M. D. (1998) Mixed-mode delamination of multidirectional carbon fiber/epoxy laminates. Mechanics of Composite Materials and Structures, 5, 291–307. Svensson, N., Shishoo, R. & Gilchrist, M. (1998) Interlaminar fracture of commingled GF/PET composite laminates. Journal of Composite Materials, 32, 1808–1835. Tafreshi, A. (2004a) Delamination buckling and postbuckling in composite cylindrical shells under external pressure. Thin-Walled Structures, 42, 1379–1404. Tafreshi, A. (2004b) Efficient modelling of delamination buckling in composite cylindrical shells under axial compression. Composite Structures, 64, 511–520. Tafreshi, A. (2006) Delamination buckling and postbuckling in composite cylindrical shells under combined axial compression and external pressure. Composite Structures, 72, 401–418. Tagarielli, V. L. & Fleck, N. A. (2005) A comparison of the structural response of clamped and simply supported sandwich beams with aluminium faces and a metal foam core. Journal of Applied Mechanics – Transactions of the ASME, 72, 408–417. Tagarielli, V. L., Fleck, N. A. & Deshpande, V. S. (2004) Collapse of clamped and simply supported composite sandwich beams in three-point bending. Composites Part B – Engineering, 35, 523–534. © 2008, Woodhead Publishing Limited


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Talreja, R. (2006) Damage analysis for structural integrity and durability of composite materials. Fatigue & Fracture of Engineering Materials & Structures, 29, 481–506. Tanimoto, Y., Nishiwaki, T., Nishiyama, N., Nemoto, K. & Maekawa, Z. (2002) A simplified numerical simulation method of bending properties for glass fiber cloth reinforced denture base resin. Dental Materials Journal, 21, 105–117. Tilbrook, M. T., Deshpande, V. S. & Fleck, N. A. (2006) The impulsive response of sandwich beams: Analytical and numerical investigation of regimes of behaviour. Journal of the Mechanics and Physics of Solids, 54, 2242–2280. Walter, M. E., Ravichandran, G. & Ortiz, M. (1997) Computational modeling of damage evolution in unidirectional fiber reinforced ceramic matrix composites. Computational Mechanics, 20, 192–198. Weinberg, K., Mota, A. & Ortiz, M. (2006) A variational constitutive model for porous metal plasticity. Computational Mechanics, 37, 142–152. Xu, L. R. & Rosakis, A. J. (2002) Impact failure characteristics in sandwich structures. Part I: Basic failure mode selection. International Journal of Solids and Structures, 39, 4215–4235. Yam, L. H., Wei, Z., Cheng, L. & Wong, W. O. (2004) Numerical analysis of multi-layer composite plates with internal delamination. Computers & Structures, 82, 627–637. Yam, L. H., Cheng, L., Wei, Z. & Yan, Y. J. (2005) Damage detection of composite structures using dynamic analysis. Measurement Technology and Intelligent Instruments, 295–296, 33–38. Yang, Q. D. & Cox, B. (2005) Cohesive models for damage evolution in laminated composites. International Journal of Fracture, 133, 107–137. Yu, C. (2001) Three-dimensional cohesive modeling of impact damage of composites. Aeronautics. Pasadena, California Institute of Technology. Yu, H. H. & Hutchinson, J. W. (2002) Influence of substrate compliance on buckling delamination of thin films. International Journal of Fracture, 113, 39–55. Yu, C., Pandolfi, A., Ortiz, M., Coker, D. & Rosakis, A. J. (2002) Three-dimensional modeling of intersonic shear-crack growth in asymmetrically loaded unidirectional composite plates. International Journal of Solids and Structures, 39, 6135–6157. Yu, C., Ortiz, M. & Rosakis, A. J. (2003) 3D Modeling of impact failure in sandwich structures Fracture of Polymers, Composites and Adhesives, II, 527–538.


16 Delamination in adhesively bonded joints B R K B L A C K M A N, Imperial College London, UK

16.1

Introduction

In this chapter, the fracture of adhesively bonded composite joints is reviewed and discussed. The review is concerned with fracture when the failure path is either through the adhesive layer or along the adhesive-fibre composite substrate interface. It should be noted at the outset that an adhesive joint in a composite structure is frequently not the weakest link and failure does not necessarily occur at the joint. However, in such a situation, delamination in the composite may occur and this is described in detail elsewhere in the present book. The failure of joints between metallic substrates is referred to but is not the central theme in the present chapter. The effects of loading mode (mode I, mixed mode I/II and mode II) are discussed, as are the effects of test rate, service environment (including cyclic fatigue loading and exposure to hostile environments). Throughout, a fracture mechanics approach is taken and it is shown how very powerful such an approach is when applied in a systematic manner.

16.2

Adhesive bonding of composites

16.2.1 Why adhesively bond? When one material has to be joined to another in a component or structure then clearly the designer has a selection of joining methods to choose from including welding, brazing, bolting, riveting, adhesive bonding or a combination of these. For polymer matrix fibre-composite materials the choice is usually between bolting or adhesive bonding. Several factors have led to adhesive bonding becoming a popular joining technology for composite parts. Adhesives will readily bond the thermosetting polymer matrix composite materials commonly employed by the aerospace industry and will also successfully bond composites to many of the dissimilar substrate materials employed in the automotive industry. With more advanced surface treatments adhesives 458 © 2008, Woodhead Publishing Limited


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will also bond thermoplastic matrix composites. The requirement to drill or punch a hole in the substrate is avoided with adhesives leading to a more even stress distribution and also the avoidance of damage to the composite (Kinloch 1987). In addition, significant weight savings can be achieved by the use of adhesive bonding. Adhesives are also commonly used to repair composite parts and the design and integrity of repair joints has been the subject of considerable research. Inevitably joint failures do occur and these can be caused by poor joint design or adhesive selection, the use of no (or inappropriate) surface pre-treatments or by exposure to certain types of loading and/or hostile service environments (e.g., impact or cyclic fatigue loading and hot-wet conditions). Most adhesive manufacturers now formulate a range of adhesives specifically marketed towards the composites bonding sector including a wide range of epoxy, acrylic and urethane adhesives.

16.2.2 Surface treatments for composite bonding The adhesive bonding of carbon fibre reinforced plastics (CFRP) or glass fibre reinforced plastics (GFRP) requires some attention to the surface preparation of the composite part if strong and durable bonds are to be attained. It is now widely accepted that for composites consisting of an epoxy matrix, such treatment should consist of a degreasing stage with a solvent followed by a light abrasion. For composites consisting of a thermoplastic matrix such as polyether-etherketone (PEEK), a more complex surface pre-treatment is required such as corona discharge or plasma treatment (Kodokian and Kinloch 1989). Where significant controversy has existed however, is in the use of peel plies on cured composite parts and the ability to adhesively bond to these composites with simply the peel ply layer removed. In the detailed study by Hart-Smith, the limitations of the use of peel plies for composite bonding were well highlighted (Hart-Smith, Redmond et al. 1996). The authors concluded that strong and durable adhesive bonds could never be achieved by adhesively bonding to a composite surface which had simply been exposed by removing a peel ply layer. It was highlighted that in the case of peel plies which had been treated with releasing agents to facilitate ply removal, then many of these releasing agents are simply transferred to the composite surface and will compromise any subsequent bonding process. Also, the use of what are known as non-released peel plies was investigated and it was concluded that regardless of the peel ply used, the surface of the composite should always be lightly abraded following the removal of the peel ply and prior to bonding.


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16.3


Fracture of adhesively bonded composite joints

16.3.1 Introduction An adhesively bonded composite joint may fail cohesively through the adhesive layer, by a crack propagating along the interface between the adhesive and the composite substrate or by interlaminar fracture within the composite substrate. In this latter mode of failure, the crack by-passes the adhesive layer and the resistance to crack propagation is provided solely by the composite. Combinations of the above failure modes are of course also possible. The fracture path will have a significant bearing on the resistance to crack propagation. Interfacial failure is frequently attributed to an inadequate surface pre-treatment but it may also be caused by a weakening of the interface due to environmental conditions, as is discussed later in the present chapter.

16.3.2 Fracture mechanics Fracture mechanics – introduction Fracture mechanics concepts have proved invaluable to the study of delamination in composites and also to the study of fracture in adhesive joints. Indeed, the use of the strain energy release rate parameter, G, has proved to be a remarkably powerful tool for understanding and predicting joint failure. Many of the complexities that are encountered when a stress intensity factor (K field) approach is followed, e.g. the proximity to the adhesive/substrate interface, are avoided when an energy balance approach is used. The pioneering work of Griffith (1920) and then of Irwin and Kies (1954) laid the foundations for many of the fracture mechanics developments which have since followed. It was demonstrated that for a material exhibiting linear elastic behaviour the strain energy release rate could be related to the load applied in a cracked plate of width b via: 2 G = P ⋅ dC 2 b da

16.1

where P is the applied load, C is the compliance and a the crack length, where: C= δ P

16.2

and δ is the displacement. The fracture criterion via the Griffith energy balance was: GC =

Pc2 dC ⋅ 2 b da


16.3


461

where Gc is the value of the critical strain energy release rate at fracture and Pc is the associated load at fracture. Within a linear elastic fracture mechanics (LEFM) framework, the loading applied to a crack is typically described in terms of the components of tensile opening (mode I), in-plane shear (mode II) and anti-plane shear (mode III) which may exist. These are shown schematically in Fig. 16.1. The critical strain energy release rate in modes I, II and III are written as GIc, GIIc and GIIIc respectively and mode I loading is usually considered the most damaging, followed by modes II and III as will be described later. Whilst it is uncommon for a crack to propagate under mode II loading in a bulk material, such behaviour is not uncommon in adhesive joints as often the crack will be directionally constrained by the presence of the nearby substrates. Hence mode II and mixed-mode I/II fractures do frequently arise in adhesively bonded joints. Mixed-mode combinations are also important with I/II loading having received the most attention. Mode I loading Ripling, Mostovoy, and co-workers were the first to apply fracture mechanics principles to the study of adhesive joint failure in the 1960s (Ripling, Mostovoy et al. 1964; Mostovoy, Crosley et al. 1967). Their work established two popular test specimens known as the adhesively bonded double cantilever beam (DCB) and tapered double cantilever beam (TDCB). These are shown in Fig 16.2. Their work led to an ASTM standard (ASTM 1990) which first appeared in 1973 for the determination of GIc, the critical value of the strain energy release rate in mode I, also described in the present chapter as the adhesive fracture energy. The analysis employed a simple shear corrected beam theory in which it was shown that for thin adhesive layers, the compliance of the beam could be expressed as: C=

8( a 3 + h 2 a ) E s bh 3

16.4

where h was the height, b the width and Es the axial modulus of the substrate respectively. Differentiating Equation 16.4 and substituting into Equation 16.3 led to:

Mode I

Mode II

Mode III

16.1 Schematic figure depicting modes I, II and III loading on a crack.


462


(a)

(b)

16.2 Mode I adhesive joint test specimens: (a) the double cantilever beam (DCB) and (b) the tapered double cantilever beam (TDCB).

G Ic =

4 P 2  3a 2 + 1 2  3 h Es b h

16.5

This result was used to determine the shape of the TDCB specimen as the term in brackets in Equation 16.5 was held constant to simplify the analysis and was defined as m, the shape factor of the joint. The increased interest in the failure of fibre composite laminates during the 1970s and 1980s led to improvements in the experimental techniques and to more accurate beam theory analyses being derived, e.g. Kanninen (1973) and Williams and co-workers (Williams 1988; Hashemi, Kinloch et al. 1990). These workers modelled the specimen as a beam on an elastic foundation and in addition to correcting for shear effects, the analysis corrected for the effects of crack root rotation. Williams considered specifically the case of orthotropic fibre-composite arms in the DCB specimens and considered also the effects of large displacements and stiffening effects when metallic endblocks were used to apply load to the composite arms. A corrected analysis for the DCB specimen took into account all of these effects, correcting for beam root rotation via an additional length term, ∆, which is added to the crack length, and also the effects of end-block stiffening and large deflections via two corrections factors N and F respectively (Hashemi, Kinloch et al. 1990). The expression for GIc was thus written for the composites DCB as:

G Ic =

3 Pδ ⋅ F 2 b( a + ∆ ) N

16.6

This analysis was shown to be also applicable to adhesively bonded fibre reinforced composites laminates with relatively thin bondlines (Blackman, Dear et al. 1991) and this was subsequently adopted for the analysis in a © 2008, Woodhead Publishing Limited


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revised standard for determining GIc in structural adhesive joints (BSI 2001). This standard was developed as a project co-ordinated by the European Structural Integrity Society (ESIS) Technical Committee 4 (TC4) which develops fracture mechanics test methods for polymers, composites and adhesive joints. The adhesively bonded DCB with fibre composite substrates is shown schematically in Fig. 16.3, with loading blocks applied to the joint to facilitate load introduction. The analysis for the TDCB test specimen was similarly corrected for the effects of root rotation (Blackman, Hadavinia et al. 2003) and the expression for GIc was thus written for the TDCB as: G Ic =

( )

4P2 m  1 + 0.43 3 ma E s b 2 

1/3

  

16.7

The results of an inter-laboratory evaluation of the British Standard (Blackman, Kinloch et al. 2003a) showed good repeatability and therefore it was agreed internationally that the test should be adopted as an ISO standard. At the time of writing, the document is a draft international standard (ISO 2007). The test determines the resistance to crack initiation from an inserted defect and then additionally determines the resistance to the re-initiation of the crack from the pre-crack formed above. The resistance curve (R-curve) is also determined as the crack propagates through the joints, i.e. values of GIc are determined as a function of crack length. The re-initiation from the precrack represents more natural initiation conditions and the R-curve demonstrates whether or not there is an increasing resistance to fracture as the crack propagates. The procedure is analogous to that applied to mode I composite delamination. In the above analyses the restriction was imposed that the adhesive layer should be relatively thin such that a thin layer is sandwiched between stiff substrates. Under these conditions the adhesive makes negligible contribution to the specimen compliance. A practical limit of 1 mm was stated in the British Standard (BSI 2001) but this will depend on the toughness of the adhesive and the stiffness of the substrates. Analyses have been performed which also consider the effects of the adhesive layer, (e.g. Penado 1993; Williams 1995) and their adoption may be justified when thick or very

16.3 The adhesively bonded double cantilever beam (DCB) specimen with end blocks (as usually used with fibre-composite substrates).


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ductile adhesive layers are used. Such analyses require additional information, i.e. the elastic properties of the adhesive but for thick adhesive layers, as pointed out by Steinbrecher, Buchman et al. (2006) such analyses may give improved accuracy. For fibre-composite substrates the DCB specimen is far easier to manufacture than the TDCB specimen and is the specimen of choice in most published research (e.g. Ashcroft, Hughes et al. 2001). It is possible, however, to manufacture a TDCB joint with constant height substrates bonded to tapered backing beams (e.g. Qiao, Wang et al. 2003b) and also to use an adhesively bonded width tapered composite DCB specimen (e.g. Jyoti, Gibson et al. 2005). A frequent complication in the fracture behaviour in mode I is the occurrence of stick-slip crack growth as described by Gledhill and Kinloch (1979) in which the crack grows in bursts. During this type of growth, it is possible to define values of GIc associated with crack initiation and values associated with crack arrest. Such behaviour has been variously attributed to crack-tip blunting and also to a rate effect, where the resistance to crack initiation (when the crack speed is zero) is greater than the resistance to a propagating crack (where the crack speed >0). Additionally, it has been proposed that it is caused by the superposition of the visco-elastic losses and dynamic effects (Maugis 1985). A final word on mode I fracture in adhesive layers is that frequently during stable crack propagation GIc remains approximately constant with increasing crack length. Thus under mode I loading conditions rising R-curves are not frequently observed. Mixed-mode (I/II) loading A number of different LEFM test methods have been employed to measure the mixed-mode fracture toughness of adhesive joints and are shown schematically in Fig. 16.4. A wide variety of adhesives and substrate materials have been investigated. One of the most popular mixed-mode tests for composite delamination testing has been the mixed-mode bending (MMB) test developed at NASA Langley by Reeder and Crews (1992) and now an ASTM standard (ASTM 2004). The popularity stems from the ability to vary the mixed-mode ratio (or mixity) over a wide range with a single test apparatus. The mixed-mode ratio can be varied from almost pure mode I to almost pure mode II by simply adjusting the length of the lever arm. This test has also been adopted for use in testing adhesively bonded joints as shown schematically in Fig. 16.4a. Ducept and co-workers (Ducept et al., 2000) used the MMB test to investigate mixed-mode failure criteria for composite-composite joints bonded with an epoxy adhesive. They used glass/epoxy composite substrates with a two part epoxy adhesive (Redux 420) and fitted empirical failure criteria to their data. Liu, Gibson et al. (2002a,b) used the MMB test to investigate mixed-mode fracture of adhesively bonded aluminium alloy © 2008, Woodhead Publishing Limited


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(a)

(b)

Clamp

(c)

16.4 Mixed-mode (I/II) adhesive joint test specimens: (a) the mixedmode bend (MMB) specimen, (b) the mixed-mode flexure specimen and (c) the asymmetric double cantilever beam (ADCB), also know as the fixed-ratio mixed-mode (FRMM) specimen.

substrates. These authors made some modifications to the test specimen to avoid plastic deformation of the substrate arms and additionally refined the analytical model to incorporate the effects of the adhesive layer, elastic foundation and shear deformation ahead of the crack tip. However, the MMB test is not without problems and various difficulties remain with the method, perhaps the most serious of which is the degree of scatter in the data when the mode-mix is substantially mode II, as is discussed in the following section. Another popular mixed-mode test for adhesive joints has been the mixedmode flexure (MMF) test proposed by Fernlund and Spelt (1994) and shown © 2008, Woodhead Publishing Limited

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schematically in Fig. 16.4b. Results have been reported for joints consisting of aluminium bonded with both a brittle epoxy adhesive and a tough epoxy adhesive (Papini, Furnlund et al. 1994). Parvatareddy and Dillard (1999) investigated the effect of mode mix on joints consisting of titanium substrates bonded with an epoxy adhesive, FM-5. The MMF test was used, together with mode I DCB and mode II ENF tests. Their results indicated that GIC > GIIC or GI/IIC – an unusual result which the authors suggested may have been due to the crack interacting with the woven glass scrim carrying the adhesive. Finally, the asymmetric double cantilever beam (ADCB) specimen, also known as the fixed-ratio mixed mode (FRMM) test specimen has also been employed by various workers, as shown in Fig. 16.4c. This test provides a constant mixed-mode ratio of GI/GII = 4/3. Mode II loading Standard test methods for mode II fracture in composite laminates and adhesively bonded joints have been slower to develop than mode I test methods for various reasons. Firstly, as mode II fracture represents the growth of a crack via in-plane shear loading, there is no crack opening component to the applied G and not only will the crack tip be more difficult to define, but friction will also oppose the shear displacements. Secondly, various workers have reported on the complexity of the damage zone ahead of the crack tip in adhesive layers when subjected to mode II loading. This damage zone may incorporate an extended region of microcracking and also a region of plastically sheared adhesive. The test methods proposed to measure the mode II fracture resistance in adhesive joints are mainly adapted from the composite delamination test methods. Of these, the end-notch flexure (ENF) test, (e.g. Chai 1988; Parvatareddy and Dillard 1999) and the end-loaded split (ELS) test (Blackman, Kinloch et al. 2005a) have been the most popular. Various other geometries have been used including the four point loaded end-notch flexure (4ENF) (Martin and Davidson 1997) and the contoured end-notch flexure (CENF) test (Edde and Verreman 1995; Qiao, Wang et al. 2003a). These mode II test specimens are shown schematically in Fig. 16.5. Chai was one of the first researchers to report a detailed investigation into shear fracture in adhesive joints (Chai 1988). Three different adhesives were used to bond metallic substrates and the effect of the adhesive layer thickness on the measured values of Gc was studied for both modes II and III loading. Chai described the fracture process as starting with the development of tensile microcracks ahead of the crack tip, and as continuing via the advance of damage through interfacial microcrack linkage and adhesive yielding and then finally, when the damage zone was fully developed, via stable crack advance. An interesting observation that is made in this work is that GIIc ≈ © 2008, Woodhead Publishing Limited


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(a)

(b)

Clamp

(c)

(d)

16.5 Mode II adhesive joint test specimens: (a) the ENF, (b) the 4-ENF, (c) the ELS and (d) the tapered ENF test specimens.

GIIIc and that as the adhesive layer thickness, t → 0, then the values of GIc, GIIc and GIIIc all converged to a common intrinsic value. This would suggest that the fundamental fracture mechanism is common for the three loading modes and that the different energy absorption values often measured are due to additional deformation mechanisms occurring in the adhesive layer. © 2008, Woodhead Publishing Limited

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Chai subsequently investigated a number of these additional mechanisms: the shear deformation in the adhesive layer (Chai 1992); crack kinking (crack growth at an angle governed by the direction of the principal stresses) and voiding (controlled by the mean stresses) (Chai and Chiang 1998). These were all shown to contribute to the energy absorption in mode II. Although the ENF test is straightforward to perform, it tends to be unstable and provides only initiation values of GIIc. Thus, it is usually not possible to measure an R-curve with an ENF test and hence it is not possible to fully observe how GIIc may develop during crack growth. Blackman and co-workers (Blackman, Kinloch et al. 2005a) used the ELS test to measure GIIc for adhesive joints consisting CFRP substrates bonded with either an epoxypaste or epoxy-film adhesive. It was argued that due to the difficulty of identifying the crack tip in the presence of the characteristic mode II microcracked zone, it was more reliable to use corrected beam theory coupled with an effective crack length approach to evaluate GIIc. The effective crack length approach uses a calculated crack length, based on the experimental compliance and an independently measured value of the substrate modulus. It was argued that the elevation of the R-curve was at least partly due to the development of microcracking in the adhesive layer and thus the creation of additional fracture surface area (Blackman, Kinloch et al. 2005a). Most studies have concluded that whilst friction does increase the values of GIIc, the effects are relatively minor and that the extra energy dissipated during mode II fracture (as compared to mode I fracture) is due to formation of tensile microcracks and the subsequent shear failure of the ligaments as suggested by Chai. Similar conclusions have been drawn for the interlaminar fracture of composites under mode II loading (O’Brien 1998) who argued that GIIc was not a material property but was strongly system dependent. The notion that GIIc = GIc + X, where X is the additional energy dissipated via the various additional mechanisms induced by mode II loading appears to be correct. However, despite these arguments there is still an urgent requirement to be able to characterise mode II fracture (and the associated deformations) in adhesive joints, and as such reliable test and analysis methodologies are required. When comparing values of GIIc quoted for adhesively bonded joints in the literature, various complications arise. Firstly, it is important to note whether the values of GIIc quoted correspond to crack initiation or to crack propagation values. Secondly, it is important to know how these are defined, as different common definitions of crack initiation can led to very different values of GIIc being determined (Blackman, Kinloch et al. 2005a). Thirdly, as the values of GIIc are very sensitive to the bondline thickness (Chai 1988), then this values needs to be taken into account when comparing values. Finally, the analytical model employed to deduce GIIc can lead to widely differing results. A major challenge in mode II testing is always to ensure that the substrates do not © 2008, Woodhead Publishing Limited


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plastically deform when tough adhesives are used. Such plastic deformation would invalidate the LEFM analyses frequently used. Of course, many of these complications would vanish if a common test standard were agreed internationally for the determination of this fracture parameter. The ESIS TC4 working group on polymers, composites and adhesives is developing test protocols based on the ELS method for both composite laminates and adhesive joints. At the time of writing, the methods are currently under evaluation in a series of round-robin tests using CFRP composite substrates and two structural epoxy adhesives. Table 16.1 summarises some results taken from the literature where both values of GIc and GIIc have been reported for a common adhesive joint. Indeed, some workers have indeed shown that values of GIIc are very much greater than GIc for a given joint (e.g. Papini, Fernlund et al. 1994), but also Table 16.1 Values of GIc and GIIc reported in the literature for various adhesive joints Adhesive and Manufacturer (reference)

Substrate material and adhesive thickness where known

J/m2 ——————————————— GIc GIIc

FM300, American Cyanamid (Liechti and Freda, 1989)

AA

1120

Epoxy Cybond 4523GB American Cyanamid (Fernlund and Spelt, 1994)

AA (7075-T6) ha = 0.4 mm

213 ± 19

Epoxy ESP310 Permabond (Papini, Fernlund et al., 1994)

AA (7075-T6)

794

5605

Epoxy, Goodrich (Swadener, Liechti et al., 1999)

GFRP

4000

4000

Dow Betamate 4601 Essex Speciality Products (Liu, Gibson et al., 2002)

AA (5754-0)- no surface treatment ha = 0.254 mm

2657

3229

AF126, 3M Inc. (Blackman, Kinloch et al., 2005 a,b)

CFRP ha = 0.08 mm

1449 ± 113

1624 ± 520 †

ESP110, Bondmaster Plc, UK. (Blackman, Kinloch et al., 2005 a,b)

CFRP

945 ± 28

2672 ± 534 †

ha = 0.4 mm

1400 577 ± 68

3381 ± 372 §

4280 ± 128 §

Notes: AA: Aluminium alloy, ha: adhesive layer thickness, where known. † GIIc at crack initiation (defined as a 5% change in initial compliance of the joint) § Mean plateau value of GIIc measured on the R-curve. © 2008, Woodhead Publishing Limited

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some workers have shown that the values are similar (e.g. Liechti and Freda 1989; Swadener, Liechti et al. 1999; Liu, Gibson et al. 2002a,b). Recent work has also shown that values of GIIc at crack initiation (defined here as the point corresponding to a 5% change in initial joint compliance) can be very similar to the values of GIc for a joint bonded with a thin layer of epoxy film adhesive, but that subsequently a strong rising R-curve was measured leading to mean propagation values of GIIc being between 2-3 times the value of GIc (Blackman, Kinloch et al. 2005a,b) as shown in Table 16.1. This was partly explained in terms of the increased fracture surface area generated when inclined tensile micro-cracks form across a bondline. Whether or not GIc ≈ GIIc at crack initiation appears to depend on the bondline thickness and the deformation mechanisms present. It is clear that whilst LEFM test methods have been successfully applied to the study of fracture in adhesive joints, challenges still exist, especially when the loading contains a significant proportion of mode II.

16.3.3 High rate fracture Early work investigating the effect of test rate on the fracture resistance of adhesively bonded joints (e.g., Gledhill, Kinloch et al. 1978) focused on joints between metallic substrates (i.e. using the TDCB test geometry) and bulk adhesive specimens (using a double torsion test specimen). These studies revealed that the fracture behaviour in the bulk adhesive specimens and in the joints were similar and that transitions from stable, continuous to unstable, stick-slip crack growth could be induced by altering the loading speed. The test rates used in these studies were of the order of 10–6 to 10–4 m/s. Khalil and Bayoumi investigated the effects of somewhat higher loading rates (10–4 to 10–3 m/s) on the fracture toughness of bonded joints using a cleavage test (Khalil and Bayoumi 1991) and these results indicated much more significant reductions in fracture energy for the joints. In the work of Khalil, a three-fold decease in values GIc over the narrow range of test rates was reported, an unusually strong rate dependence. Blackman and co-workers performed high speed loading tests on adhesively bonded composite substrates using the DCB test geometry at displacement rates from 10–5 to 101 m/s (Blackman, Dear et al. 1995). Two different structural epoxy adhesives were used to bond the joints. High speed photography was used to monitor the load-line opening displacement of the beam together with the crack length and the loads were measured using a high frequency response load cell. The authors commented that a main challenge faced when performing a high rate DCB test was to measure accurate experimental parameters when severe flexural oscillation of the substrate arms occurred during the test. These oscillations led to highly transient load values that could not be used in the calculation of GIc. This problem was © 2008, Woodhead Publishing Limited


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overcome by using a displacement analysis – the need to use measured load values was avoided by deriving expressions for GIc which were independent of the measured load. For the load-independent beam theory expressions, accurate values of the beam opening displacement and crack length were required and these were obtained via high speed photography of the test. Additionally, the variation of substrate modulus with test rate was required by this approach and this was achieved via an ultrasonic method. A second challenge was to accurately account for the effects of kinetic energy in the test, which have to be included in the energy balance at high rates. Kinetic energies were derived for the DCB test specimen using Berry’s method (Berry 1960) and the kinetic corrections became significant at test rates of about 10 m/s. A toughened epoxy-film adhesive was shown (see Fig. 16.6a) to be relatively insensitive to test rate over this speed range. However, a toughened epoxy-paste adhesive was shown to exhibit almost a 50% reduction in the value of GIc over the same speed range of test rates, as shown in Fig. 16.6b. It is common for adhesive joints to undergo a transition from stable, continuous crack growth at slow speeds to unstable stick-slip growth at higher rates. During stick-slip crack propagation, a series of initiation and arrest points may be determined, where the arrest points occur at a lower value of load than the initiation points. This behaviour is due at least in part to the rate sensitivity of the adhesive as discussed previously. The open data points represent crack initiation values and the closed data points represent crack arrest values of GIc. This transitional behaviour is depicted in Fig. 16.6 where a transition from stable, continuous to unstable, stick-slip crack growth occurred. The transitions are marked by the vertical dashed lines in Figs 16.6a and 16.6b. The epoxy-paste adhesive (Fig. 16.6b) shows a second transition back to stable dynamic crack growth at displacement rates faster than about 2 m/s. At rates faster, stick-slip growth did not occur and the joint failed by a continuous rapid fracture following initiation. The adhesive with the lower glass transition temperature in this study was shown to exhibit the greatest loss of toughness. In this case, a ductile-brittle transition was observed but the failure always remained cohesive in the adhesive layer. The dynamic effects in the tests were analysed in detail in a subsequent paper (Blackman, Dear et al. 1996). The use of the TDCB test specimen has the advantage that higher crack velocities can be obtained for a given loading rate (c.f. the DCB test) and values of GIc versus test rate have been published for a range of automotive grade adhesives (Blackman, Kinloch et al. 2000). Additionally, the dynamic crack growth in this specimen has been modelled both analytically and numerically (Wang and Williams 1996). Various researchers have used rate dependent cohesive zones to model the behaviour of the joints under high speed loading (e.g. Xu, Siegmund et al. 2003; Cooper, Ivankovic 2005). Both the DCB and to a lesser extent the TDCB test suffer from a loss of © 2008, Woodhead Publishing Limited

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GIC init GIC arrest

3.5 3

GIc (kJ/m2)

2.5 2

1.5

Stable

1

Stick-slip

0.5 0 10–5

0.0001

0.001 0.01 0.1 1 Displacement rate (m/s) (a)

10

100

5

GIC init GIC arrest

GIc (kJ/m2)

4

3 Stable (quasi-static)

Stick-slip

2

Stable (dynamic)

1

0 10–5

0.0001

0.001 0.01 0.1 1 Displacement rate (m/s) (b)

10

100

16.6 Values of GIc versus test rate for two adhesively bonded composite joints (a) an epoxy-film adhesive, FM73M and (b) an epoxy-paste adhesive, EA9309. The substrates were unidirectional carbon-fibre reinforced epoxy. (Open data points: crack initiation values, closed data points: crack arrest values.) Adapted from Blackman, Dear et al. 1995.



473

symmetry as the test rate is increased above about 1 m/s if loading is only applied to one half of the specimen. For this reason, there is an advantage in using a symmetric loading system or the use of wedge loading. Blackman and co-workers reported on the use of the impact wedge test for structural adhesive joints (Blackman, Kinloch et al. 2000) and more recently Simon, Johnson et al. (2005) and Dillard, Jacob et al. (2007) have used a driven wedge test to acquire rate dependent fracture energies in adhesive joints. Indeed, Simon, Johnson et al. (2005) have used high speed DCB, ELS and a single leg bend (SLB) test to measure the fracture resistance of adhesively bonded joints over a range of mixed-mode ratios at elevated test rates. They used a modified drop tower apparatus and for the DCB test they used a falling wedge to open the joint. Both aluminium alloy and composite substrates bonded with a two part automotive epoxy adhesive were studied. A high speed video camera was employed to measure beam displacements and an edge detection algorithm was used to deduce crack lengths. Values of GIc, GIIc and GI/IIc were measured under static and impact loading rates. Modest reductions in the values of Gc were reported, but the results measured at high speeds showed significant scatter. Much of the current high rate research effort in the area of adhesively bonded composite joints is directed towards automotive applications, stemming from the drive towards the development of light weight vehicle structures and improved vehicle occupant safety (Wall, Sullivan et al. 2004). Here, a major challenge is to employ the fracture mechanics parameters GIc, GIIc and GI/IIc etc. to predict structural failure loads and energy dissipation in bonded components during high speed impacts. Such fracture mechanics criteria are often coupled to 2D and 3D finite element analysis models using commercially developed codes.

16.3.4 Fatigue loading The fatigue resistance of adhesively bonded joints is usually superior to other methods of joining such as welding, bolting or riveting due to the elimination of the stress concentrations. However, it is clear that fatigue crack growth (FCG) in adhesive joints does occur at applied stress levels far below those required to cause failure under monotonic loading conditions (Kinloch 1987). As such, there has been a great deal of research interest directed towards characterising FCG in adhesively bonded joints and in the use of such data to predict the fatigue life of a joint. Two approaches have typically been followed to study the fatigue performance of bonded joints: (a) a classical stress-life characterisation in which an S-N curve is determined for a particular component (i.e. applied stress amplitude is plotted against the number of cycles to failure) and (b) the fatigue crack growth (FCG) characterisation in which a cracked joint is © 2008, Woodhead Publishing Limited

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subjected to fatigue loading and the rate of crack growth per cycle (da/dN) is plotted against the cyclic range of the applied strain energy release rate, ∆G, i.e. ∆G = Gmax–Gmin. This latter approach typically results in a sigmoidal relationship when plotted on a log-log axis curve. Region I of this curve is characterised by the applied ∆G falling below the so-called threshold value, ∆Gth, i.e. to an applied strain energy release rate range insufficient to cause crack growth. The existence of a fatigue threshold is clearly valuable from a design perspective as joints can be designed never to fail by fatigue using this approach. Region II is the part of the curve well described by a Paris Law of the form:

da = C ⋅ ( ∆G ) m dN

16.8

where C and m are material constants. Region III is when rapid crack growth occurs as the applied ∆G approaches the critical strain energy release rate in monotonic loading, i.e. GC. The FCG approach was followed by Osiyemi and Kinloch for the study of the fatigue behaviour of adhesively bonded fibre-reinforced polymer composite joints (Osiyemi and Kinloch 1993). In this work, DCB test specimens were cyclically loaded at constant displacement amplitude at a frequency of 5 Hz. The rate of crack growth per cycle, da/dN, was recorded as a function of Gmax rather than ∆G. It was noted that Gmax was a more consistent parameter to use than ∆G as debris propping open the crack could lead to facial interference on the unloading part of the cycle, leading to erroneous values of ∆G. Figure 16.7 shows the form of data reported by Osiyemi and Kinloch for a CFRP joint bonded with the rubber toughened epoxy adhesive (Osiyemi and Kinloch 1993). Regions I-III are depicted in the figure. These data were then used to predict the service life of a lap joint. Such a prediction is essentially a three stage process. Firstly, fracture mechanics tests are performed under cyclic loading. The crack growth rate per cycle, da/dN, is measured as a function of Gmax during the fatigue crack growth test. A modified Paris law is employed (Martin and Murri 1990) of the type:

da = DG n max dN

  Gth  n1  1–  G  max ⋅ n2  1–  Gmax    Gc 

    

16.9

where the values of the fatigue coefficients D, n, n1 and n2 are calculated from the experimental data. Secondly, this relationship is then combined with a numerical description of the variation of Gmax with crack length and applied load in the joint. This is obtained either analytically or via a finite element analysis of the joint. Finally, an integration of the combined expression © 2008, Woodhead Publishing Limited


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GIc

–2

Region III

log da/dN (m/cycle)

–4

–6 Region II

–8

–10 Region I

–12 2.4 Gth

2.6

2.8

3.0 3.2 log Gmax(J/m2)

3.4

3.6

16.7 FCG data for a bonded composite joint conforming to a modified Paris law. Adapted from Osiyemi and Kinloch 1993.

allows a prediction for the fatigue lifetime of the bonded component between certain limits of integration. One of the main issues in the published research has been the question of how much of the fatigue life of a joint is taken up by the initiation of a fatigue crack and how much is taken up by its subsequent propagation to a critical crack length, leading to failure. The use of region II of the da/dN versus ∆G (or Gmax) curve alone ignores the contribution to fatigue life of the crack initiation stage although it has been argued that lifetime predictions based purely on region II will be conservative. Hadavinia and co-workers (Hadavinia, Kinloch et al. 2003a,b) investigated the cyclic fatigue loading of aluminium alloy joints bonded with an epoxy adhesive and FCG tests were performed using cyclically loaded TDCB joints and additionally single-lap joints were tested using a back-face strain gauge technique to detect crack initiation. These authors concluded that it was crack propagation, rather than crack initiation that occupied the dominant part of the lifetime of the single lap-joints. Reasonable service lifetime predictions for the lap-joints were achieved by neglecting the initiation phase and assuming that the lifetime was dominated by the crack propagation phase. © 2008, Woodhead Publishing Limited

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Several other workers have studied the fatigue failure of adhesively bonded composite joints. Ashcroft and Shaw studied the effect of temperature on the FCG in CFRP substrates bonded with an epoxy adhesive and used the data to predict failure in uncracked lap joints (Ashcroft and Shaw 2002). Reasonable predictions were achieved for the service lifetimes, except at the highest exposure temperatures of 90°C, when a creep mechanism also became significant. Quaresimin and Ricotta have investigated the influence of various design parameters on fatigue behaviour of single lap joints, i.e. joint overlap length, spew fillet corner geometry and surface preparation and found these all significantly influenced the fatigue performance of the joints (Quaresimin and Ricotta 2006a). They concluded however, that between 20–70% of joint life was taken up by crack initiation and in a subsequent paper these authors have used a two stage model to predict fatigue lifetime (Quaresimin and Ricotta 2006b). The first stage of the model, relating to crack initiation, used a stress intensity factor approach and the second stage, relating to crack propagation, used the FCG approach and an integration of the Paris-law as described previously. The authors claimed that such an approach gave more accurate and less conservative lifetime predictions. Another issue raised in the literature is that FCG tests are usually performed under cyclic mode I opening conditions and if these data are then used to predict fatigue life then the mixed-mode ratio of the loading situation in the component might not be adequately accounted for if only mode I data are used in the prediction. Pirondi and Nicoletto investigated FCG in adhesively bonded aluminium joints under cyclic mixed-mode (I/II) loading using a compact tension shear specimen (Pirondi and Nicoletto 2006). These authors reported that fatigue crack growth rates were higher under mixed-mode loading than under pure more I loading and this surprising result was attributed to greater energy dissipation in the crack tip region under mode I loading. Clearly this is an area in which further research is required.

16.3.5 Service environment effects It is well documented that the durability of adhesive joints is very sensitive to the presence of moisture in the environment (Comyn 1983; Kinloch 1987). However, it is also apparent that it is the joints between metallic substrates that present the main concern and that adhesive joints between plastics, glass and carbon fibre reinforced plastics are far less susceptible to environmental attack (Kinloch 1979). Kinloch goes on to explain that this does not mean that joints between composites do not suffer from environmental attack but that when they do, it is usually the substrate rather than the adhesive-substrate interface that is more readily attacked. This conclusion was reinforced by the work of Parker, who investigated the strength of single lap shear joints consisting of carbon fibre epoxy composite substrates bonded with epoxy © 2008, Woodhead Publishing Limited


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adhesives exposed to high humidity (Parker 1990). The joints were exposed to 50°C/96% RH for up to three years. No evidence of interfacial failure between the adhesive and the substrate was found and the crack propagated in the adhesive, or in the composite, or a combination of these depending on the adhesive. It was further found that adhesives cured at 175°C gave better strength retention than those cured at 120°C. John and co-workers also investigated the durability of carbon fibre/epoxy composite substrates bonded with one of two room temperature curing epoxy adhesives and exposed to 40°C/90% RH for up to one year (John, Kinloch et al. 1991). A steady decline in joint failure load was reported and the main mechanism of environmental attack was identified as water ingress into, and the plasticisation, of the adhesive and the matrix of the fibre composite. Pethrick used a dielectric method to assess the durability of adhesively bonded CFRP structures when subjected to various solvents including ethylene glycol, aviation fuel, water and dichloromethane (Pethrick, Armstrong et al. 2004). The authors showed that the ingressing solvents led invariably to swelling of the joints and plasticisation of the adhesive layer. In some cases this was reversible on drying but in other cases the recovery was limited. An example of an irreversible change was when a significant swelling of the adhesive created voids that were not eliminated on subsequent drying. A good correlation was found between the dielectric constant associated with the joint and the change in mechanical strength observed after exposure to water and ethylene glycol, confirming the value of the dielectric technique. Literature describing the durability of joints between thermoplastic matrix fibre composites is less abundant. A study investigating the durability of carbon fibre reinforced PEEK composite substrates following a surface pretreatment using a corona discharge technique and bonding with one of two structural epoxy adhesives was reviewed in 2004 (Blackman and Kinloch 2004). PEEK is the thermoplastic polymer, poly-ether-etherketone. The retained mode I fracture energies of the adhesively bonded DCB joints after water immersion at 50°C for durations of up to one year were measured. A weakening of the adhesive-substrate interfacial adhesion was observed when an epoxyfilm adhesive was used to bond the joints. The interface remained fully intact when the same adhesive was used to bond carbon fibre reinforced epoxy composite substrates. These results imply that the environmental durability of joints between thermoplastic matrix composites may indeed be inferior to joints between thermosetting matrix composites when various structural epoxy adhesives are employed. Finally, a very effective method to assess the environmental durability of adhesively bonded joints is to perform FCG tests in the required environment. This combination of fatigue loading and exposure to the hostile environment very readily accelerates the ageing process. The cyclic loading drives water to the crack tip and potentially to the interface, thus long term exposure is © 2008, Woodhead Publishing Limited

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effectively accelerated. In particular, ageing in wet environments whilst performing FCG tests, has been shown to lower the Gth value significantly for metallic substrates (Hadavinia, Kinloch et al. 2003a.b). Such data are very valuable for the purposes of joint design.

16.3.6 Pre-bond moisture effects An important additional environmental consideration when adhesively bonding composites is the potential for significant amounts of absorbed moisture to be already present in the composite substrates prior to bonding. If a high temperature curing adhesive is then employed to bond the substrates then this so-called pre-bond moisture can act to weaken the joint during curing. A related problem may also be experienced when using cold stored film adhesives which, if not adequately sealed from the environment, can absorb significant quantities of moisture when exposed to ambient conditions prior to joint cure. This latter problem is, of course, not restricted to the bonding of composites substrates. The problem relating to pre-bond moisture in composite substrates has been investigated by a number of researchers. Pre-bond moisture has been reported to cause a ‘bubble problem’ in adhesive layers which reduced the static strength of the joint (Sage and Tiu 1982). These authors found that the severe voiding problem was avoided by drying out the composite or curing the joint under isostatic pressure. Parker identified three possible deleterious effects of pre-bond moisture in the composite substrates: (a) voiding, (b) plasticisation of the adhesive and (c) a reduction in interfacial adhesion (Parker 1983). Which of these occurred depended upon the adhesive used. Drying out the composite substrates prior to joint curing was found to be successful, although aggressive drying cycles (e.g., drying substrates with high levels of absorbed moisture at 180°C) was found to induce blistering of the composite. Robson investigated the effect of pre-bond moisture on the strength of adhesively bonded repair patches and reported that low levels of pre-bond moisture (below 0.5% w/w) had little effect on the repair strength or microstructure, but as moisture content increased, so the repair strength fell and the volume fraction of voids increased with increasing levels of prebond moisture (Robson, Matthews et al. 1994). More recently, the effects of prebond moisture have been discussed by Hart-Smith (Hart-Smith 2004), and investigated further by Blackman (Blackman, Kinloch et al. 2003b; Blackman, Kinloch et al. 2005b) and Van Voast (Van Voast 2002; Van Voast and Barnes 2005). In these studies, composite substrates were moisture conditioned and were then adhesively bonded to form joints. Van Voast studied three different 177°C curing adhesives and reported reductions in measured values of GIc and glass transition temperature, Tg, as the degree of exposure increased. Blackman and co-workers investigated © 2008, Woodhead Publishing Limited


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Table 16.2 Values of GIc and glass transition temperature of the adhesive for the CFRP joints Substrate

GIc (J/m2)

Tg (°C)

CRRP (‘wet’: 0.6% w/w) CFRP (‘as-received’) CFRP (‘fully dried’: 0% w/w)

102 ± 12 202 ± 23 963 ± 20

82.5 ± 1.0 87.6 ± 1.3 103.6 ± 0.5

For comparative purposes: Aluminium alloy Mild steel

685 ± 86 892 ± 108

100.6 ± 0.6 103.0 ± 0.9

Notes: a. Locus of joint failure was cohesive, approximately in the centre of the adhesive layer for all specimens.

in detail a 150°C curing adhesive and found an extreme moisture sensitivity. When the CFRP substrates were fully dried prior to making the joint, then the adhesive joint possessed a value of GIc = 963 ± 20 J/m2 via (BSI 2001) and the values of Tg measured via differential scanning calorimetry (DSC) of samples of adhesive removed from the adhesive layer were 103.6 + 0.5°C, as shown in Table 16.2. However, when water was present in the composite substrates as pre-bond moisture, then severe reductions in both the values of GIc and Tg were measured, as shown in Table 16.2, leading to about a 90% reduction in GIc values and a 20°C reduction in the glass transition temperature of the adhesive following the introduction of pre-bond moisture into the composite substrates The substrates marked as ‘as-received’ were uncontrolled, being allowed to pick up moisture in a standard laboratory atmosphere of 23°C, 55% RH over a period of six months and those marked as ‘wet:0.6% w/w’ had been immersed in water until they substrates possessed 0.6% by weight of water. The dangers posed by pre-bond moisture in composite parts is now well known to the aerospace industry and drying procedures are usually specified prior to composite bonding. It is also clear that the adhesive manufacturers are aware of the problems posed by pre-bond moisture in composite substrates and modern structural adhesives are often formulated to impart a particular resistance to pre-bond moisture during curing.

16.4

Future trends

It is clear that the development of fracture mechanics test methods and analyses have provided a sound framework for the characterisation of adhesively bonded joints over a wide range of loading conditions and service environments. Challenges still remain, however, in various areas. Firstly, the mixed-mode and mode II characterisation of adhesive joint fracture remains a challenge and as the understanding of the associated failure mechanisms in these modes improves, then this should lead to the establishment of validated © 2008, Woodhead Publishing Limited

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and widely accepted test methods and improved analyses for these loading modes. This will lead to more accurate and reproducible data being measured and more reliable values of Gc being determined. Secondly, extending these test methods and analyses to higher rates is challenging and is a very active current research area and one in which great progress is being made. Thirdly, the ability to determine the fracture parameters accurately when dissimilar substrates materials are employed is becoming increasingly important, as is the determination of adhesive joint performance under impact and high speed loading conditions. Fourthly, the development of additional test methods and analysis procedures when there is substantial plasticity in the test is another area in which future development is required. Such methods are needed when the conditions of LEFM cannot be met, such as when one substrate deforms plastically during the test. Finally, the development of improved analytical and numerical models to describe fracture in adhesive joints under a wide range of loading and service conditions will continue.

16.5

Sources of further information and advice

A number of detailed texts have been published which provide a wide coverage of the different aspects of the analysis of fracture in adhesive joints and the development of test protocols and standards. These multi-authored works provide an excellent foundation in the subject and additionally present detailed works by individual specialist authors in the field. The following are particularly recommended for further reading: 1. Adhesive Bonding: Science, Technology and Applications, editor: R. D. Adams, published by CRC Press, Woodhead Publishing, 543 pages, 2005. 2. The Mechanics of Adhesion, volume 1, editors: D. A. Dillard and A. V. Pocius, published by Elsevier Science, 790 pages, 2002. 3. Fracture Mechanics Testing Methods for Polymers, Adhesives and Composites, editors: D. R. Moore, A. Pavan and J. G. Williams, ESIS Publication 28, published by Elsevier Science, 375 pages, 2001. 4. The Application of Fracture Mechanics to Polymers, Adhesives and Composites, editor: D. R. Moore, ESIS Publication 33, Published by Elsevier Science, 288 pages, 2004. The following organisations have websites which provide information on future conferences, seminars, technical meetings and publications relevant to the study of fracture in adhesive joints: • • • •

The Adhesion Society (http://www.adhesionsociety.org) The Society for Adhesion and Adhesives UK (http://www.uksaa.org) British Adhesives and Sealants Association (http://www.basa.org) Adhesion, Adhesives and Composites Group at Imperial College (http:/ /www.me.ic.ac.uk/AACgroup)



•

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ESIS TC4 on Polymers, Composites and Adhesives (http:// www.tc4pca.elsevier.com)

16.6

References

Ashcroft, I. A. and Hughes, D. J. et al. (2001). ‘Mode I fracture of epoxy bonded composite joints 1. Quasi-static loading,’ International Journal of Adhesion & Adhesives, 21: 87–99. Ashcroft, I. A. and S. J. Shaw (2002). ‘Mode I fracture of epoxy bonded composite joints 2. Fatigue loading,’ International Journal of Adhesion & Adhesives, 22: 151–167. ASTM (1990). ASTM D3433, Annual book of ASTM standards. Adhesives section 15. Philadelphia, 15.06. ASTM (2004). ‘Standard test method for mixed mode I-mode II interlaminar fracture toughness of unidirectional fibre reinforced polymer matrix composites,’ ASTM Standard D6671M, 1–14. Berry, J. P. (1960). ‘Some kinetic considerations of the Griffith Criterion of fracture-I; eqns of motion at constant deformation,’ J. Mech. Phys. Solids, 8: 207–216. Blackman, B. R. K. and Kinloch A. J. (2004). ‘The durability of adhesive joints in hostile environments,’ The application of fracture mechanics to polymers, adhesives and composites, D. R. Moore. Amsterdam, Elsevier. ESIS Publication, 33: 143–148. Blackman, B. R. K., Dear, J. P. et al. (1995). ‘The failure of fibre-composites and adhesivelybonded fibre-composites under high rates of test. Part I: Mode I Loading-Experimental studies,’ Journal of Materials Science, 30: 5885–5900. Blackman, B. R. K., Dear, J. P. et al. (1996). ‘The failure of fibre-composites and adhesively bonded fibre-composites under high rates of test. Part II: Mode I Loading-dynamic effects,’ Journal of Materials Science 31: 4451–4466. Blackman, B. R. K., Dear, J. P. et al. (1991). ‘The calculation of adhesive fracture energies from DCB test specimens,’ Journal of Materials Science Letters, 10: 253–256. Blackman, B. R. K., Hadavinia, H. et al. (2003). ‘The calculation of adhesive fracture energies in mode I: Revisiting the tapered double cantilever beam (TDCB) test,’ Engineering Fracture Mechanics, 70(2): 233–248. Blackman, B. R. K., Kinloch, A. J. et al. (2000). ‘The impact wedge-peel performance of structural adhesives,’ Journal of Materials Science, 35, 1867–1884. Blackman, B. R. K., Kinloch, A. J. et al. (2003a). ‘Measuring the mode I adhesive fracture energy, GIC, of structural adhesive joints: – The results of an International round-robin,’ International Journal of Adhesion and Adhesives, 23: 293–305. Blackman, B. R. K., Kinloch, A. J. et al. (2003b). The influence of pre-bond moisture on the adhesive bonding of CFRP, 26th Annual Meeting of the Adhesion Society, Myrtle Beach, SC, USA, The Adhesion Socity. Blackman, B. R. K., Kinloch, A. J. et al. (2005a). ‘The determination of the mode II adhesive fracture energy, GIIC, of structural adhesive joints: An effective crack length approach,’ Engineering Fracture Mechanics, 72: 877–897. Blackman, B. R. K., Kinloch, A. J. et al. (2005b). The effects of pre-bond moisture on the fracture behaviour of adhesively bonded CFRP joints: further considerations, 28th Annual Meeting of the Adhesion Society, Mobile, AL, USA, The Adhesion Society. BSI (2001). ‘Determination of the mode I adhesive fracture energy, GIC, of structural adhesives using the double cantilever beam (DCB) and tapered double cantilever beam (TDCB) specimens,’ BS 7991. Chai, H. (1988). ‘Shear fracture,’ International Journal of Fracture, 37: 137–159. © 2008, Woodhead Publishing Limited

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Chai, H. (1992). ‘Micromechanics of shear deformations in cracked bonded joints,’ International Journal of Fracture, 58: 223–239. Chai, H. and Chiang, M. Y. M. (1998). ‘Finite element analysis of interfacial crack propagation based on local shear, Part II – Fracture,’ International Journal of Solids & Structures, 35(9–10): 815–829. Comyn, J. (1983). Ch. 3 ‘Kinetics and Mechanisms of Environmental Attack’, Durability of Structural Adhesives, A. J. Kinloch, London and New York, Applied Science Publishers, 1: 85–129. Cooper, V., Ivankovic, A., Murphy, N. and Regan, S. (2005). ‘Measurement of tractionseparation curves for structural adhesives’, Ninth International Conference on the Science and Technology of Adhesion and Adhesives (Adhesion OS), September, Oxford, UK. Dillard, D. A., Jacob, G. C. et al. (2007). ‘The use of a driven wedge test to acquire ratedependent fracture energies of bonded beam specimens,’ Journal of Adhesion (in review). Ducept, F., Davies, P. et al. (2000). ‘Mixed mode failure criteria for a glass/epoxy composite and an adhesively bonded composite/composite joint,’ International Journal of Adhesion and Adhesives, 20(3): 233–244. Edde, F. C. and Verreman, Y. (1995). ‘Nominally constant strain energy release rate specimen for the study of mode II fracture and fatigue in adhesively bonded joints,’ International Journal Adhesion and Adhesives, 15: 29–32. Fernlund, G. and Spelt, J. K. (1994). ‘Mixed-mode fracture characterization of adhesive joints,’ Composites Science and Technology, 50: 441–449. Gledhill, R. A. and Kinloch, A. J. (1979). ‘Mechanics of crack growth in epoxide resins,’ Polymer Engineering And Science, 19(2): 82–88. Gledhill, R. A., Kinloch, A. J. et al. (1978). ‘Relationship between mechanical properties and crack propagation in epoxy resin adhesives,’ Polymer, 19: 574–582. Griffith, A. A. (1920). ‘The phenomena of rupture and flow in solids,’ Phil. Trans. R. Soc., A221: 163–189. Hadavinia, H., Kinloch, A. J. et al. (2003a). ‘The prediction of crack growth in bonded joints under cyclic-fatigue loading I: Experimental studies,’ International Journal of Adhesion & Adhesives, 23: 449–461. Hadavinia, H., Kinloch, A. J. et al. (2003b). ‘The prediction of crack growth in bonded joints under cyclic-fatigue loading II: Analytical and finite element studies,’ International Journal of Adhesion & Adhesives, 23, 463–471. Hart-Smith, L. J. (2004). The critical factors contolling the durability of bonded composite joints- surface preparation and the presence pr absence of pre-bond moisture, MILHDBK-17 & FAA Meeting, Seattle, Washington, DC. USA. Hart-Smith, L. J., Redmond, G. et al. (1996). The curse of the nylon peel ply. 41st SAMPE International Symposium. Anaheim, CA. Hashemi, S., Kinloch, A. J. et al. (1990). ‘The analysis of interlaminar fracture in uniaxial fibre-polymer composites,’ Proceeding of the Royal Society London, A427: 173–199. Irwin, G. R. and Kies, J. A. (1954). ‘Critical energy release rate analysis of fracture strength of large welded structures,’ Welding Journal, 33: 193–198. ISO (2007). ‘Adhesives – Detemination of the mode I adhesive fracture energy GIC of structural adhesive joints using double cantilever beam and tapered double beam specimens,’ ISO Standard DIS 25215. John, S. J., Kinloch, A. J. et al. (1991). ‘Measuring and predicting the durability of bonded carbon fibre/epoxy composite joints,’ Composites, 22(2): 121–127. Jyoti, A., Gibson, R. F. et al. (2005). ‘Experimental studies of mode I energy release rate © 2008, Woodhead Publishing Limited


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in adhesively bonded width tapered composite DCB specimens,’ Composites Science and Technology, 65, 9–18. Kanninen, M. F. (1973). ‘An augmented double cantilever beam model for studying crack propagation and arrest,’ International Journal of Fracture 9(1): 83–92. Khalil, A. A. and Bayoumi, M. R. (1991). ‘Effect of Loading Rate on Fracture Toughness of Bonded Joints,’ International Journal of Adhesion and Adhesives, 11(1): 25–29. Kinloch, A. J. (1979). ‘Interfacial fracture: Mechanical aspects of adhesive bonded joints – A review,’ Journal of Adhesion, 10: 193–219. Kinloch, A. J. (1987). Adhesion & Adhesives: Science & Technology, London and New York, Chapman Hall. Kodokian, G. K. A. and Kinloch, A. J. (1989). ‘The adhesive fracture energy of bonded thermoplastic fibre-composites,’ Journal of Adhesion, 29, 193–218. Liechti, K. M. and Freda, T. (1989). ‘On the use of laminated beams for the determination of pure and mixed-mode fracture properties of structural adhesives,’ Journal of Adhesion, 29: 145–169. Liu, Z., Gibson, R. F. et al. (2002a). ‘Improved analytical models for the mixed-mode bending tests of adhesively bonded joints,’ Journal of Adhesion, 78: 224–268. Liu, Z., Gibson, R. F. et al. (2002b). ‘The use of a modified mixed-mode bending test for characterisation of mixed-mode fracture behaviour of adhesively bonded metal joints,’ Journal of Adhesion, 78, 223–244. Martin, R. H. and Davidson, B. D. (1997). Mode II fracture toughness evaluation using a four point bend end notched flexure test, 4th International Conference on deformation and fracture of composites, Manchester. Martin, R. H. and Murri, G. B. (1990). ‘Characterization of mode I and mode II delamination growth and thresholds in AS4/PEEK composites,’ Composite Materials Testing and Design, ASTM STP 1059, Philadelphia, PA, ASTM, 9: 251–270. Maugis, D. (1985). ‘Suncritical crack growth, surface energy, fracture toughness, stickslip and embrittlement – a review,’ Journal of Materials Science, 20: 3041–3073. Mostovoy, S., Crosley, P. B. et al. (1967). ‘Use of crack-line loaded specimens for measuring plane-strain fracture toughness,’ Journal of Materials, 2(3): 661–681. O’Brien, T. K. (1998). ‘Composite interlaminar shear fracture toughness, GIIC: Shear measurement or sheer myth?’ Composite Materials: Fatigue and Fracture, Vol. 7: ASTM STP 1330: 3–18. Osiyemi, S. O. and Kinloch, A. J. (1993). ‘Predicting the fatigue life of adhesively bonded joints,’ Journal of Adhesion, 43: 79–90. Papini, M., Fernlund, G. et al. (1994). ‘Effect of crack growth mechanism on the prediction of fracture load of adhesive joints,’ Composites Science and Technology, 52: 561–570. Parker, B. M. (1983). ‘The effect of composite prebond moisture on adhesive-bonded CFRP-CFRP joints,’ Composites, 14(3): 226–232. Parker, B. M. (1990). ‘The strength of bonded carbon fibre composite joints exposed to high humidity,’ International Journal of Adhesion and Adhesives, 10(3): 187–191. Parvatareddy, H. and Dillard, D. A. (1999). ‘Effect of mode-mixity on the fracture toughness of Ti-6Al-4V/FM-5 adhesive joints,’ International Journal Adhesion and Adhesives, 96: 215–228. Penado, E. J. (1993). ‘A closed form solution for the energy release rate of the double cantilever beam specimen with an adhesive layer,’ Journal of Composite Materials, 27(4): 383–407. Pethrick, W. R., Armstrong, W. G. et al. (2004). ‘Dielectric and mechanical studies of the durability of adhesively bonded CFRP structures subjected to aging in various solvents,’ © 2008, Woodhead Publishing Limited

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Proceedings of the Institution of Mechanical Engineers. Proceedings part L, Journal of Materials, Design and Applications 218(L4): 273–281. Pirondi, A. and Nicoletto, G. (2006). ‘Mixed-mode I/II fatigue crack growth in adhesive joints,’ Engineering Fracture Mechanics, 73: 2557–2568. Qiao, P., Wang, J. et al. (2003a). ‘Analysis of tapered ENF specimen and characterization of bonded interface fracture under Mode II loading,’ International Journal of Solids & Structures, 40: 1865–1884. Qiao, P., Wang, J. et al. (2003b). ‘Tapered beam on elastic foundation model for compliance rate of change of TDCB specimen,’ Engineering Fracture Mechanics, 70: 339–353. Quaresimin, M. and Ricotta, M. (2006a). ‘Fatigue behaviour and damage evolution of single lap bonded joints in composite material,’ Composites Science and Technology, 66: 176–187. Quaresimin, M. and Ricotta, M. (2006b). ‘Life prediction of bonded joints in composite materials,’ International Journal of Fatigue, 28: 1166–1176. Reeder, J. R. and Crews, J. H. (1992). ‘Redesign of the Mixed-mode bending delamination test to reduce nonlinear effects,’ Journal of Composites Technology and Research, 14: 12–19. Ripling, E. J., Mostovoy, S. et al. (1964). ‘Measuring fracture toughness of adhesive joints,’ Materials Research & Standards (ASTM Bulletin), 4(3, March): 129–134. Robson, J. E., Matthews, F. L. et al. (1994). ‘The bonded repair of fibre composites: Effect of composite moisture content,’ Composites Science and Technology, 52: 235–246. Sage, G. N. and Tiu, W. P. (1982). ‘The effect of glue-line voids and inclusions on the fatigue strength of bonded joints in composites,’ Composites, 13: 228–232. Simon, J. C., Johnson, E. et al. (2005). ‘Characterising dynamic fracture behaviour of adhesive joints under quasi-static and impact loading,’ Journal of ASTM International, 2(7 (July/August)): 53–71. Steinbrecher, G., Buchman, A. et al. (2006). ‘Characterisation of the mode I fracture energy of adhesive joints,’ International Journal of Adhesion & Adhesives, 26: 644–650. Swadener, J. G., Liechti, K. M. et al. (1999). Mixed-mode fracture of automotive bonded joints, SAMPE-ACCE-DOE Advanced Composites Conference, Detroit, MI. Van Voast, P. J. (2002). Effect of prebond humidity exposure of cured composite on fracture toughness, morphology and cure kinetics, 34th International SAMP Technical Conference, Baltimore, MD, 4–7 November, 1098–1110. Van Voast, P. J. and Barnes, S. R. (2005). Development of a thick adherend composite specimen to evaluate shear strength and the effect of prebond humidity and bondline thickness, 28th Annual Meeting of the Adhesion Society, Mobile, AL, USA., The Adhesion Society. Wall, E., Sullivan, R. et al. (2004). ‘Progress report for automotive lightweighting materials,’ O. o. F. C. V. t. US Department of Energy. FY2003. Wang, Y. and Williams, J. G. (1996). ‘Dynamic crack growth in TDCB specimens,’ International Journal of Mechanical Sciences, 38(10): 1073–1088. Williams, J. G. (1988). ‘On the calculation of energy release rates for cracked laminates,’ International Journal of Fracture, 36: 101–119. Williams, J. G. (1995). Fracture in adhesive joints: The beam on elastic foundation model, ASME symposium on mechanics of plastics and plastics composites, San Francisco, USA. Xu, C., Siegmund, T. et al. (2003). ‘Rate dependent crack growth in adhesives I. Modeling approach,’ International Journal of Adhesion & Adhesives, 23: 9–13.


17 Delamination propagation under cyclic loading P P C A M A N H O, Universidade do Porto, Portugal and A T U R O N and J C O S T A, University of Girona, Spain

17.1

Introduction and motivation

The increased use of advanced composite laminates in primary structures of commercial aircraft requires a thorough understanding of the inelastic response of composites under general loading conditions. One of the most relevant mechanisms that contribute to the loss of stiffness and to the structural collapse of composite structures is delamination. Besides degrading the structural integrity of composites, delamination is also difficult to detect using traditional non-destructive inspection methods. The majority of the analytical and experimental investigations of delamination have been focused on the study of delamination growth under quasi-static loads (Pipes and Pagano 1970; Brunner et al., 2008). However, under cyclic loading, delamination may grow up to a critical size for loads well below the critical load for quasi-static loading conditions. Currently, the design methodologies barely consider the possibility of interlaminar crack growth under fatigue loading, being more oriented to prevent fatigue damage by assuring that a stress or strain threshold for delamination onset is not exceeded. This approach is evolving towards more powerful physically based analyses implemented in advanced computational models. This is a result of the industrial interest to partially replace the experimental tests required to certify new composite components (qualification of materials, design allowables, sub-components and full components) by virtual (numerical) tools able to simulate the inelastic response of the composite materials up to structural failure. In the framework of the development of virtual tools, enhanced computational models for the prediction of delamination onset and growth under fatigue loading emerges as one of the most important topics to assess the lifetime of the structure in service, and to evaluate the structural integrity of composite laminates after damage events (e.g. impact). This chapter describes the main experimental observations of delamination propagation under cyclic loading, an enhanced cohesive zone model (CZM) 485 © 2008, Woodhead Publishing Limited

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to predict delamination growth under cyclic loading, and the computational implementation of the CZM in a finite element code. An overview of the fatigue tests that are required to characterize delamination growth rates under different load conditions is presented. These experimental tests provide the basic information required for the analytical and numerical models of delamination growth under cyclic loading. The numerical simulation of delamination growth under cyclic loading is normally performed using models based on damage mechanics. Under general cyclic loading, total damage is taken as the sum of the damage caused by quasi-static loads and the damage that result from the cyclic loads. A damage model previously developed by the authors for quasi-static loads (Turon et al., 2006) is enhanced to incorporate a damage evolution law for highcycle fatigue (Turon et al., 2007b). The evolution of the damage variable resulting from cyclic loading is derived from a fracture mechanics description of the fatigue crack growth rate. The fatigue model relates damage accumulation to the number of load cycles while taking into account the loading conditions (load ratio, energy release rate, and mode mixity). The fatigue model is validated comparing its predictions with experimental data obtained in test specimens cyclically loaded under pure mode I, mode II and mixed-mode I and II. In addition, the predicted fatigue response of a skin-stiffener composite structure loaded in tension is presented.

17.2

Experimental data

The computational models should be based on the information obtained in experimental tests. The experimental tests and analysis methods developed for the study of delamination growth under quasi-static loads are addressed in other chapters of this book. A brief review of the experimental and modelling efforts related to delamination onset and propagation under fatigue loading is presented here. Even though the microstructure of composite materials is heterogeneous and the damage zone close to a macrocrack may be large, most of the experimental and modelling studies of delamination propagation under fatigue loading use linear elastic fracture mechanics (LEFM). In fact, most of the traditional approaches attempt to extend methods that are applicable for metals to composites.

17.2.1 Delamination onset The only existing standard for the analysis of delamination under fatigue loading is focused on delamination onset. Using the ASTM D6115 standard (2004) it is possible to determine the maximum number of cycles, N, which a double cantilever beam (DCB) specimen will withstand at a certain load © 2008, Woodhead Publishing Limited


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level, characterized by the peak cyclic value of the energy release late, Gmax, before the onset of delamination. The test is performed under mode I tensiontension constant amplitude and the damage onset criterion is established in terms of a relative increase in the specimen compliance (1% or 5%), or by the visual observation of crack growth. As a result, a Gmax-N curve is generated which may serve for design purposes, in an equivalent way as the S-N curves are used in metals. The ASTM D6115 standard (2004) does not recommend measuring crack growth rates in mode I because fibre bridging develops during the test. Fibre bridging may slow down or even interrupt crack growth. Notwithstanding this difficulty, most of the studies published in the literature focus on crack propagation.

17.2.2 Delamination propagation The objective of the tests promoting delamination propagation under cyclic loading is to determine the dependence of the crack growth rate, defined as da/dN where ‘a’ is the crack length, on the cyclic loading conditions. There are different variables that define the fatigue loading state but only three of them are mutually independent. Firstly, it is necessary to define the mode loading (I, II, III or mixed mode) which is usually expressed as GII/GT because most of the studies are limited to mode I, mode II or mixed-mode I and II. For a particular mode ratio, the amplitude and mean value of the load may be defined using two additional variables. Some authors (Pagano and Schoeppner, 2000; Gustafson and Hojo, 1987; Ramkumar and Whitcomb, 1985) use the energy release rate, G, and define the loading conditions using Gmax and ∆G = Gmax – Gmin, or using Gmax and the load ratio, R. The load ratio corresponds to the ratio between the minimum and maximum values of the applied loads, Pmin/Pmax or, equivalently to the ratio between the stress intensity factors, Kmin/Kmax. Under small displacements, R may be approximated as the ratio between the minimum and the maximum displacements, ∆min/ ∆max. Other authors (Hojo et al., 1987; 1997; Matsuda et al., 1997, Tanaka and Tanaka, 1995) establish the dependence of da/dN on the stress intensity factor (∆K = Kmax – Kmin and R). Quite often the effect of the load ratio is disregarded and the load state is simply expressed with the peak cyclic parameter (Gmax or Kmax) (Krueger et al., 2001). The objective of the experimental investigations is to define the complete dependence of da/dN on the three selected parameters that determine the loading state (for example, da/dN = f(Gmax, R, GII/GT)). With this information, and for a particular fatigue loading case, the determination of the lifetime of the component before the crack growth reaches a critical length is straightforward. The definition of the dependence of the crack growth rate da/dN on Gmax, R and GII/GT implies a large amount of experimental tests. © 2008, Woodhead Publishing Limited

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Therefore, most of the studies concern only a limited part of this function by keeping some of the variables constant. Usually, a modified Paris law (Blanco et al., 2004, and references therein; Schön et al., 2000; Andersons et al., 2001, 2004) is used to describe the delamination growth rate as a power law function of the applied energy release rate range, ∆G: da = C  ∆G  dN  Gc 

m

17.1

where Gc is the critical energy release rate for the particular mode ratio, and C and m are parameters that depend on the mode ratio and on the load ratio (Hojo et al., 1997; Andersons et al., 2004). Some authors (Ramkumar and Whitcomb, 1985; Russell and Street, 1989; Dahlen and Springer, 1994) propose similar expressions in which the crack growth rate for mixed mode results from a combination of the crack growth rates for pure mode I and for pure mode II loading. The power-law function accounts only for the linear part (Region II) of the typically observed crack growth curves, as shown in Fig. 17.1. Another relevant observation provided by the experimental data is the existence of a threshold value below which crack growth is not observed (Benzeggagh and Kenane, 1996).

17.3

Damage mechanics models

The majority of damage models that have been developed to simulate fatigue use a damage variable that evolves with the number of cycles. In cohesive

Crack growth rate, log (da/dN)

Region I

Region II

Region III

Gc

m 1

Gth

Normalized energy release rate, log (Gmax/Gc)

17.1 Typical crack growth rate regions.



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zone models the material separation under cyclic loading is described by a constitutive equation formulated in the context of the thermodynamics of irreversible processes. Among other classifications, the models developed in the damage mechanics context can be classified in two different groups (Lemaitre et al., 1999): high-cycle fatigue models and low-cycle fatigue models.

17.3.1 Low-cycle fatigue According to Lemaitre et al. (1999), low-cycle fatigue occurs when the damage is localized in domains of stress concentrations but it can be measured and evaluated at the mesoscale. The number of cycles to failure is smaller than 104. Low-cycle fatigue models account for fatigue damage evolution on a cycle by cycle analysis defining, in the majority of the models, an evolution of the damage variables during the unloading paths. Within the context of the cohesive zone model, there are several models that extend cohesive laws that were derived for monotonic loading into forms suitable for cyclic loading. Yang et al. (2004) developed a cohesive law that describes separately the unloading and reloading processes and creates a hysteresis loop between unloading and reloading paths. Roe and Siegmund (2003) describe fatigue crack growth by incorporating a damage evolution equation for cyclic loading. Nguyen et al. (2001) developed a cohesive zone model for cyclic loads in which the irreversible material degradation is represented as a loss of stiffness in the cohesive zone during the unloading portion of the load cycle. Similarly, Goyal-Singhal et al. extended the capability of their cohesive constitutive model (Goyal-Singhal et al., 2004) to account for fatigue damage accumulation during unloading (Goyal-Singhal and Johnson, 2003). In all of these references, the fatigue damage accumulation is accounted for in a cycle-by-cycle analysis. For high-cycle fatigue, where the number of cycles is larger than 106, a cycle-by-cycle analysis would be computationally intractable.

17.3.2 High-cycle fatigue According to Lemaitre et al. (1999), high-cycle fatigue occurs when the damage is localized at the microscale as a few micro-cracks. The number of cycles to failure is greater than 104. For high-cycle fatigue, a cycle-by-cycle analysis is computationally impractical. Therefore, a two-scale computation is usually adopted (Lemaitre et al., 1999): damage evolution is extrapolated over a given number of cycles. Using this strategy, called cycle-jump strategy, the computation of the whole load history is reduced to a selected number of cycles. High-cycle fatigue models require the definition of the relation between the damage variable and the number of cycles as an input for the cycle-jump © 2008, Woodhead Publishing Limited

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strategy. The cycle-jump strategy controls the accuracy of the solution by means of the number of cycles to be skipped in the analysis. There are different approaches in the literature to control the accuracy of the solution. Van Paepegem and Degrieck (2001), computed the cycle-jump length by limiting the maximum increment of the damage variable at each cycle-jump. Mayugo (2003) developed a similar cycle jump strategy where the cycle jump starts after checking, in a cycle-by-cycle calculation, that the damage variable evolves monotonically. Cocojaru and Karlsson (2006) presented a similar model with an improved control of the cycle jump length. The majority of the models relating the damage variable (or mechanical properties) to the number of cycles use a phenomenological law established a priori and formulated as a function of the number of cycles. The damage evolution law is a function of several parameters that have to be adjusted to calibrate the numerical model with experimental results, usually by trial and error. An example of these models is the Peerling’s law (Peerlings et al., 2000):

∂d = Ce Λd ε β a ∂N

17.2

where C, Λ and β are parameters to be determined by means of experimental data and εa is a strain norm. Peerling’s law has been adapted successfully to simulate high-cycle fatigue by means of an irreversible cohesive zone model (Robinson et al., 2005). However, Peerling’s law is only valid to simulate region II of the crack growth regime, and it is not sensitive to stress ratio variations. Muñoz et al. (2006), improved Peerling’s law to account for variations in the stress ratio. However, in these references, a damage evolution law expressed in terms of the number of cycles is established a priori by adjusting several parameters through a trial-and-error calibration process.

17.4

Simulation of delamination growth under fatigue loading using cohesive elements: cohesive zone model approach

The cohesive zone model (CZM) approach (Dugdale, 1960; Barenblatt, 1962; Hillerborg et al., 1976) is a common tool for the simulation of quasi-brittle fracture. Cohesive zone models represent a damage zone that develops near the tip of a crack assuming that all inelastic material response can be lumped to a surface ahead of the crack tip. Cohesive damage zone models relate tractions, τ, to displacement jumps, ∆ , at the interfaces where crack propagation occurs. Damage initiation is related to the interfacial strength of the material, τ o. When the energy dissipated © 2008, Woodhead Publishing Limited


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is equal to the fracture toughness of the material, Gc, the traction is reduced to zero and new crack surfaces are formed.

17.5

Numerical representation of the cohesive zone model

The constitutive law used to simulate delamination onset and growth under fatigue loading is based on a bilinear relation between the tractions and the displacement jumps (Camanho et al., 2003; Turon et al., 2006; Mi et al., 1999; Reedy et al., 1997). The bilinear cohesive law uses an initial linear elastic response before damage initiation, as shown in Fig. 17.2. This linear elastic part is defined using a penalty stiffness parameter, K, that ensures a stiff connection between the surfaces before crack propagation. The interfacial strength and the penalty stiffness define an onset displacement jump, ∆o, related to the initiation of damage. The area under the cohesive law is equal to the fracture toughness of the material, Gc. Therefore, the final displacement jump, ∆f, is obtained from the fracture toughness of the material and from the interfacial strength. Further details on the determination of ∆o and ∆f under mixed-mode loading can be found in Turon et al. (2006).

17.5.1 Kinematics and constitutive model for quasi-static loading The displacement jump across the interface [[ui]], is obtained from the displacements of the points located on the top and bottom sides of the interface, ui+ and ui– , respectively:

[[ ui ]] = ui+ – ui–

17.3

Traction, τ

τo

Gc

∆o

Displacement jump, λ

∆f

17.2 Bilinear constitutive law used for quasi-static loads.


492


where ui± are the displacements with respect to a fixed Cartesian coordinate system. A co-rotational formulation is used to express the components of the displacement jumps with respect to the deformed interface. The coordinates x i of the deformed interface are (Ortiz and Pandolfi, 1999):

x i = X i + 1 ( ui+ – ui– ) 2

17.4

where Xi are the coordinates of the undeformed interface. The components of the displacement jump vector in the local coordinate system on the deformed interface, ∆m, are expressed in terms of the displacement field in global coordinates: ∆m = Θmi[[ui]]

17.5

where Θmi is the rotation tensor, defined in Camanho et al. (2003) and Turon et al. (2006). The constitutive operator of the interface, Dji, relates the element tractions, τj, to the displacement jumps, ∆i:

τj = Dji∆i

17.6

The constitutive model must compute accurately the energy dissipated by fracture. Under mixed-mode loading, a criterion established in terms of an interaction between components of the energy release rates associated with each loading mode is used to predict crack propagation. The formulation of the damage model previously proposed by the authors (Turon et al., 2006) is summarized in Table 17.1, where ψ and ψ0 are the free energy per unit surface of the damaged and undamaged interface, respectively. δij is the Kronecker delta, and the variable d is a scalar damage variable. The parameter Table 17.1 Definition of the constitutive model Free Energy

ψ(∆, d) = (1 – d) ψ0(∆i) – dψ0(δ3i〈– ∆3〉)

Constitutive equation

τi =

Displacement jump norm

λ=

Damage criterion

F(λ , r ): = G(λt) – G(r t) ≤ 0 t

∂ψ = (1 – d) δijK∆j – dδijKδ3j〈–∆3〉 ∂∆ i 〈 ∆ 3 〉 2 + ( ∆ shear ) 2 t

G (λ ) =

Evolution law Load/unload conditions

λ (∆ f – ∆ o )

∂F (λ , r ) ∂G (λ ) d˙ = µ˙ = µ˙ ∂λ ∂λ

µ˙ ≥ 0; F (λ t , r t ) ≤ 0; µ˙ F (λ t , r t ) = 0 r t = max{∆o, maxs λs} 0 ≤ s ≤ t

*t is the pseudo-time variable used in the analysis


∀t ≥ 0

∆ f (λ – ∆ o )


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λ is the equivalent displacement jump norm. The equivalent displacement jump is a non-negative, continuous scalar function defined as:

λ=

〈 ∆ 3 〉 2 + ( ∆ shear ) 2

17.7

where 〈 〉 is the MacAuley bracket defined as 〈 x 〉 = 1 ( x + | x |) . The 2 displacement jump in mode I, i.e., normal to midplane is ∆3. The displacement jump tangent to the midplane, ∆shear, is the Euclidean norm of the displacement jump in mode II and mode III:

∆ shear =

( ∆ 1) 2 + ( ∆ 2 ) 2

17.8

The damage variable increases when the current equivalent displacement jump, λt, is higher than the current threshold, r t. The evolution of damage is defined by a suitable monotonic scalar function, G(·), ranging from 0 to 1. A damage consistency parameter, µ˙ , is used to define loading-unloading conditions according to the Kuhn-Tucker relations (Simo and Ju, 1987). Under crack closure during load reversal, the constitutive model prevents interpenetration of the faces of the crack by restoring the normal penalty stiffness of the element even in the presence of damage. Further details regarding the damage model can be found in Turon et al. (2006). Under loading conditions, the damage variable is calculated as:

d=

∆ f ( λ – ∆o ) λ ( ∆ f – ∆o )

17.9

The relation between the damage variable d, representing the loss of stiffness, and the damage variable d , representing the ratio between the energy dissipated and the fracture toughness is given as (Turon et al., 2007b): d =

Ade = Ξ = 1 – λo (1 – d ) e Gc ∆ A

17.10

where Ade and Ae are respectively the damaged and undamaged areas associated with the local discretization (Kachanov, 1958), and Ξ is the energy dissipated. Using Equations 17.9 and 17.10:

Ade d∆o = f e A ∆ (1 – d ) + d∆o

17.6

17.11

Constitutive model for high-cycle fatigue

The total damage evolution is considered to be the sum of the damage created by the quasi-static loads and the damage created by the cyclic loads (Turon et al., 2007b): © 2008, Woodhead Publishing Limited

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dd = d˙ = d˙ ˙ static + d cyclic dt

17.12

The first term in the right hand side of Equation [17.12] is obtained from the integration of the constitutive model presented in Table 17.3, while the second term needs to be defined as a function of the cyclic loads. The evolution of the damage variable, d, is related to the surface crack growth rate, dA as (Turon et al., 2007b): dN e dd = ∂d ∂Ad dN ∂Ade ∂N

17.13

∂Ade is the growth rate of the damaged area. The term ∂de can be ∂N ∂Ad obtained from Equation [17.11]:

where

f o 2 ∂d = 1 [ ∆ (1 – d ) + d∆ ] e e f o ∂Ad A ∆ ∆

17.14

17.6.1 Determination of the growth rate of the damaged area Under cyclic loading, the damaged area grows as the number of cycles increase: after ∆N cycles, the damaged area ahead of the crack tip increases by ∆Ad as schematically represented in Fig. 17.3. It is assumed that the increase in the crack area ∆A is equal to the increase of the damaged area. Therefore, the surface crack growth rate can be assumed to be equal to the sum of the damaged area growth rates of all damaged elements ahead of the crack tip (Turon et al., 2007b): e ∂A = Σ ∂Ad ∂N e∈ACZ ∂N

Crack tip position

17.15

Load

After ∆N cycles

Ae

A

∆A d

A

17.3 Schematic representation of the increase in the damaged area (after Turon et al., 2007b).



495

where Ade is the damaged area of the element e, and the term ACZ is the area of the cohesive zone. Taking ∂Ad/∂N as the mean value of the damaged area growth rate ∂Ade / ∂N of the elements over the cohesive zone, and assuming that the mean area of the elements in the cohesive zone is Ae, Equation (17.16) can be written as (Turon et al., 2007b): e ∂A = Σ ∂Ad = ACZ ∂Ad ∂N e∈ACZ ∂N A e ∂N

17.16

where the ratio ACZ/Ae represents the number of elements spanning the cohesive zone. Rearranging terms in Equation [17.16]: e ∂Ad = A ∂A A ∂N CZ ∂N

17.17

17.6.2 Evolution of the damage variable under cyclic loading Using Equations 17.14 and 17.17 in Equation 17.13 the evolution of the damage variable as a function of the number of cycles is given as: f o 2 ∂d = 1 ( ∆ (1 – d ) + d ∆ ) ∂A f o ACZ ∂N ∂N ∆ ∆

17.18

The area of the cohesive zone under general loading conditions is given as:  (1 – B )  max ACZ = b n + 1 Q  + B  Go 2 π a11  (τ )  a11 a 22

17.19

where Gmax is the maximum energy release rate in the loading cycle, τo is the interfacial strength, n is a material parameter that defines the distribution of tractions ahead of the crack tip (Bazant and Planas 1998), and B is the mixed-mode ratio:

B=

G II G I + G II

17.20

Q is an elastic constant that reads: 1  a 2 2 a12 + a 66   22 Q=  + 4 a11   4 a11  

   

–1 2

17.21

The parameters aij are the material constants of the ply compliance matrix. For plane stress problems, © 2008, Woodhead Publishing Limited

496


ν a11 = 1 , a 22 = 1 , a12 = – 12 , and a 66 = 1 E11 E 22 E 22 G12 For plane strain problems the parameters aij read: (ν ) 2 (ν ) 2 ν ν ν a11 = 1 – 31 , a 22 = 1 – 32 , a12 = 12 – 31 32 , E11 E33 E 22 E33 E11 E33 and

a 66 = 1 G12

Parameter b is the width of the delamination front. In the implementation in a finite element code, b is taken as the characteristic element length.

17.6.3 Crack growth rate The crack growth rate under fatigue loading, ∂A/∂N, is a function of the loading conditions, geometry and material, and can be related to the Paris Law using the crack front width: ∂A = b ∂a ∂N ∂N

17.22

The growth rate defined by the Paris Law given in Equation 17.1 represents crack propagation in region II of the typical pattern of the crack growth rate (see Fig. 17.1). In region I, crack growth is not observed if the maximum energy release rate is smaller than the fatigue threshold of the energy release rate, Gth. In region III, the crack growth rate increases because the maximum energy release rate approaches the fracture toughness. Tearing fracture controls the crack growth rate in region III instead of fatigue propagation. The crack growth rate ∂a/∂N used in the fatigue damage model, Equation 17.22, is defined as a piecewise function defined as (Turon et al., 2007b):

  ∆G  m , ∂a =  C  Gc  ∂N  0, 

Gth < G max < Gc

17.23

otherwise

where C, m and Gc are parameters that depend on the mode-ratio. Defining the load ratio, R as R2 = Gmin/Gmax, the variation of the energy release rate is given as: o  ( ∆ f – λ max ) 2 ∆G = τ  ∆ f – 2  ∆ f – ∆o


 2  (1 – R ) 

17.24


497

17.6.4 Mixed-mode loading The material parameters, c, m, Gth used in Equation 17.23 depend on the mode ratio. Under mode I loading, the crack growth rate parameters are CI, mI, and GIth. Under mode II loading, the crack growth rate parameters are CII, mII, and GIIth. Under mixed-mode loading, the crack growth rate parameters C, m, and Gth are given as (Blanco et al., 2004): G G 2 CII log C = log CI +  II  log C m +  II  log  GT   GT  Cm CI

17.25

G G 2 m = m I + m m  II  + ( m II – m I – m m )  II   GT   GT 

17.26

where Cm and mm are mode-ratio material parameters that must be determined by curve-fitting experimental data. The dependence of the energy release rate threshold is given as (Turon et al., 2007b): η2 G Gth = GIth + ( GIIth – GIth )  shear   GT 

17.27

where η2 is a material parameter obtained from a curve-fit of experimental results.

17.6.5 Cycle jump strategy The cycle jump strategy proposed computes the damage variable d iJ at an integration point J and the current position of the crack tip after Ni cycles using the quasi-static constitutive equations. The predicted evolution of the damage variable with the number of cycles, ∂d/∂N, is calculated using Equation 17.18. The damage variable at an integration point J after ∆Ni cycles is: d iJ+ ∆N i = d iJ +

∂d ij ∆N i ∂N

17.28

To determine the number of cycles ∆Ni that can be skipped with a controlled level of accuracy, the following equation is used: ∆N i =

∆d max  ∂d J  max  i  J  ∂N 

where ∆dmax is a small pre-established value.


17.29

498

17.7


Examples

The model proposed was implemented as a user-written finite element in ABAQUS® (Hibbit et al., 1996) by adding the fatigue damage model to the constitutive equations implemented in cohesive element previously developed (Camanho et al., 2003; Turon et al., 2006). Several analyses of specimens under mode I, mode II and mixed-mode loading were performed to verify the response of the fatigue damage model, and demonstrate that when the constitutive damage model is used in a structural analysis, the analysis can reproduce the response of the test specimens without the use of any model specific adjustment parameters. Finally, the analysis of a substructure component of an aircraft, a composite skin-stiffener panel, is performed to simulate fatigue delamination onset and growth in a specimen without an initial crack and in a scenario where variable mode ratio may occur.

17.8

Mode I loading

Two different specimens were simulated to analyze mode I fatigue delamination. The specimens have the same geometry corresponding to a double cantilever beam, 20 mm wide, 150 mm long, with two 1.55 mm thick arms with an initial crack of 35 mm. Different loading conditions were used in the models. The first model, shown in Fig. 17.4, corresponds to a specimen loaded by constant moments to ensure a constant value of the energy release rate, regardless of the crack length (Robinson et al., 2005). This model is required to obtain the crack growth rates for different values of the applied energy release rate. The second model represents a specimen loaded by a constant displacement instead of a constant moment. In this case, the energy release rate decreases with increasing crack length. This model is used to predict delamination onset, following the procedure explained in the ASTM D6115 standard (ASTM, 2004). The finite element models are composed of four-node plane strain elements for the arms, connected by four-node cohesive elements representing the interface. Two elements through the thickness, h, of each arm are used. The length of the element is chosen as 0.05 mm. This small length is chosen because a minimum number of elements are required in the cohesive zone to accurately capture the softening behavior ahead of the crack tip M

M

17.4 DCB specimen loaded by constant moments.



499

(Turon et al., 2007a). Under fatigue loading the crack growth occurs for energy release rates smaller than the fracture toughness of the material. Therefore, the length of the cohesive zone under fatigue loading is smaller than the one under static loading and smaller elements are required for the accurate simulation of delamination growth under fatigue loading. The experimental data on fatigue-driven delamination growth reported by Asp et al. (2001) was selected for the validation of the numerical model. The specimen was fabricated with HTA/6376C carbon/epoxy tape produced by Hexcel. The layup consisted in [012//(±5/04)s], where the symbol // refers to the plane of the artificial delamination. The specimen was 150 mm long, 20.0 mm wide, with two 1.55 mm thick arms, and an initial crack length of 35 mm. A description of the experimental procedure is reported by Asp et al. (2001). The material properties are shown in Table 17.2 (Juntti et al., 1999; Asp et al., 2001; Robinson et al., 2005). These fatigue properties of the material used in the analysis are obtained by curve fitting the results from tests performed under mode I, mode II, and mixed-mode loading with a mode-mixity ratio of 50% given by Asp et al. (2001) and Juntti et al. (1999). Using the experimental data reported by Asp et al. (2001) data for a mode-mixity ratio of 50% it is possible to use Equations 17.25, 17.26 and 17.27 to calculate the constants Cm, mm, and η2. The material properties used in the analysis are summarized in Table 17.3. Table 17.2 Material properties for HTA/6376C carbon/epoxy laminate E11 (GPa)

E22 = E33 (GPa)

G12 = G13 (GPa)

G23 (GPa)

ν12 = ν13

ν23

120

10.5

5.25

3.48

0.3

0.51

GIc (kJ/m2)

GIIc (kJ/m2)

o τ shear (MPa)

τ 3o (MPa)

0.260

1.002

30

30

Table 17.3 Fatigue material properties for HTA/6376C carbon/epoxy laminate CI (mm/cycle)

CII (mm/cycle)

Cm (mm/cycle)

0.0308

0.149

22904

mI

mII

mm

5.4

4.5

4.94

GIth (kJ/m2)

GIIth (kJ/m2)

η2

0.060

0.100

2.73


500


The load is applied in two steps, as shown in Fig. 17.5. The first loading step corresponds to the application of the maximum load under quasi-static conditions. No fatigue damage accumulation occurs during this first step. A second loading step is applied where the maximum load is held constant. The pseudo-time increment is assumed to be proportional to the number of loading cycles so that the fatigue damage model accounts for the accumulation of the damage caused by the cyclic loads. The maximum variation in the damage variable ∆dmax (Equation 17.29) allowed in a cycle jump is set to 0.001.

17.8.1 Delamination growth under mode I loading Applying constant moments to the specimen, the energy release rate is related to the moment as: 2 GI = M bEI

17.30

where b is the specimen width, E is the longitudinal flexural Young’s modulus, and I is the second moment of area of the specimen’s arm. The application of constant moments to the arms of the DCB specimen results in a linear relation between the crack length and the number of cycles. For example, the relation between the crack growth and the number of cycles is shown in Fig. 17.6 for a ratio of 40% between the applied energy release rate and the fracture toughness. Several simulations corresponding to different levels of the applied energy release rate for four different load ratios were conducted to simulate the crack growth under mode I loading. The results obtained and the experimental data obtained by Asp et al. (2001) are compared in Fig. 17.7. It is observed that the constitutive model accounts for all three regions of fatigue crack 1st step

2nd step

Load ∆N1

∆N2

t

17.5 Cycle jump strategy.



501

1.50

Crack growth, a -a0 (mm)

1.25

1.00

0.75

0.50

0.25

0.00 2.0 × 104

4.0 × 104

6.0 × 104 Cycles

8.0 × 104

1.0 × 105

17.6 Predicted relation between the crack length and the number of cycles.

0.005

Crack growth rate, da/dN (mm/Cycle)

5E-4 5E-5

Experimental (R = 0) Numerical (R = 0.0) Numerical (R2 = 0.2 Numerical (R2 = 0.5) Numerical (R2 = 0.7)

5E-6 5E-7 5E-8 5E-9 5E-10 5E-11 5E-12 0.2

0.3 0.4 0.5 0.6 Normalized energy release rate, GImax/GIc

0.7

0.8 0.9 1

17.7 Comparison of the experimental data with the predicted crack growth rate; sensitivity to the load ratio under mode I loading.


502


growth. In region II, where crack growth rates follow the Paris Law, a good agreement between the predictions and the experimental data is obtained. In region I there is negligible crack growth rate for small values of the normalized energy release rate and the numerical data follows the trend of the experimental data. A difference between the numerical and the experimental data is observed in region III. One of the reasons for this difference is that the crack growth rates present in region III are very high. Therefore, a low-cycle instead of a high-cycle fatigue model is likely to be more appropriate for this region. However, in spite of this difference, the model can also predict region III crack growth rate, where the Paris law equation is not valid. Moreover, the sensitivity of the constitutive model to the load ratio observed in Fig. 17.7 is an asset of the model. The sensitivity of the propagation rate to the load ratio derives directly from the quasi-static model rather than from a fatigue model defined as a function of the load ratio, as has been done in previous investigations (Robinson et al., 2005; Muñoz et al., 2006).

17.8.2 Delamination onset under mode I loading According to the ASTM D6115 standard (2004), delamination onset under fatigue loading occurs when an increase of 1% or 5% of the global compliance of a DCB specimen is measured. The load cycles necessary to increase 1% or 5% the global compliance of the DCB specimen are the cycles corresponding to the onset of delamination. Since the crack growth after this number of cycles is negligible, the maximum applied energy release rate is approximately constant over the load history. The predicted relation between the applied energy release rate and the number of cycles for delamination onset is shown in Fig. 17.8. Therefore, the model proposed can be used to construct the Gmax vs. N curves for the prediction of delamination onset for any type of structure.

17.9

Mode II loading

Several tests were conducted to simulate the crack growth rate under mode II loading for different ranges of the energy release rate. The experimental data on fatigue driven delamination growth reported in Asp et al. (2001) was selected for comparison. The dimensions and the material of the specimen are the same used for the DCB specimen described in the previous section. To impose pure mode II loading conditions, the four-point end notched flexure (4ENF) test shown in Fig. 17.9 was used. The energy release rate is related to the applied moment, cP , as (Robinson 2 et al., 2005):



503

1.0

Normalized energy release rate

0.9 0.8 0.7 0.6 0.5 0.4 0.3 100

103 104 105 Cycles for an increase of 1% in the global compliance

17.8 Load cycles for an increase of 1% the compliance of the standard DCB test. P

P

C

P

P

17.9 Loading pattern for mode II 4ENF specimen.

 cP   2  G II = 3 4 bEI

2

17.31

The finite element model used is similar to that used in the simulation of the mode I test, the only differences are in the boundary conditions and loads. The load is applied in two steps, as previously described, and the material properties used in the simulations are given in Tables 17.2 and 17.3. The crack growth rates obtained from the different simulations and the experimental data selected for comparison (Asp et al., 2001) are shown in Fig. 17.10. A good correlation between the experimental data and the numerical predictions is observed. © 2008, Woodhead Publishing Limited

504

Delamination behaviour of composites Experimental data Numerical prediction

Crack g rowth rate da/dN (mm)

0.005

5E-4

5E-5

5E-6

5E-7 0.06

0.08

0.1 0.2 0.4 0.6 Normalized energy release rate, (GIImax /GIIc )

0.8

1

17.10 Comparison of the experimental data with the crack growth rate obtained from the numerical simulation for a mode II 4ENF test. M ρM

17.11 Loading pattern for mixed-mode specimen.

17.10 Mixed-mode I and II loading Several tests were conducted to simulate the crack growth rate under mixedmode loading with GI = GII for different energy release rates. Experimental data on fatigue driven delamination growth reported in Asp et al. (2001) was selected for comparison. The dimensions and the material of the specimen are the same used for the DCB specimen previously described. In the standard mixed-mode bending test, the applied energy release rate changes with the crack length. To obtain an energy release rate independent of the crack length, the boundary conditions of the FEM were introduced using different moments applied to the arms of the specimen, as shown in Fig. 17.11. In Fig. 17.11, where M is the applied moment and ρ is the ratio between the two applied moments. For a mode-ratio of 50% ρ is: 3 2 ρ= 3 1+ 2 1–


17.32


505

The energy release rate is related to the applied moment, M, as (Robinson et al., 2005): G I = G II =

3  3 4 1 + 2  

2

M2 bEI

17.33

The finite element model used was similar to that used in the simulation of the DCB test, where the moments applied to the arms of the specimen were modified according to Fig. 17.11. The load is applied in two steps and the material properties used in the simulations are given in Tables 17.2 and 17.3. The results obtained from the simulations and the experimental data (Asp et al., 2001) are shown in Fig. 17.12. Like in the examples of specimens loaded under pure mode I and mode II, a good correlation predictions and experiments is observed.

17.11 Fatigue delamination on a skin-stiffener structure Several aerospace composite structures are made of panels with co-cured or adhesively bonded frames and stiffeners. Testing of stiffened panels has 0.01

Crack growth rate, da/dN (mm/cycle)

Numerical Experimental 1E-3

1E-4

1E-5

1E-6

1E-7 0.1

0.2 0.3 0.4 0.5 Normalized energy release rate, Gmax/Gc

0.6

0.7

17.12 Comparison of the experimental data with the crack growth rate obtained from the numerical simulation for a mixed-mode test with GI = GII.


506


shown that bond failure at the tip of the stiffener flange is a common failure mode (Krueger et al., 2001). Stiffened panels used in aerospace applications are subjected to fatigue loading and the prediction of skin-stiffener debonding under cyclic loading is a relevant step in the design of such structures. The model presented in the previous sections is used in the prediction of skin-stiffener debonding under fatigue loading. This is a challenging problem because in the structure under investigation the mode ratio may change throughout the analysis and the structure does not have a pre-crack. The test specimens used to further illustrate the capabilities of the fatigue model consists of a stiffener flange bonded onto a skin that has been tested by Krueger et al. (2001). The skin is 101.6 mm long, 25.4 mm wide and the stiffener is 50.8 mm long and 25.4 mm wide. The taper angle of the stiffener edges is 25°. Both skin and flange were made from IM7/8552 graphite/ epoxy. Prepreg tape with a nominal ply thickness of 0.142 mm was used for the skin and the lay-up used was [45/-45/0/-45/45/90/90/-45/45/0/45/-45]T. The flange consisted of a plain-weave fabric with a 0.208 mm nominal ply thickness and a lay-up [45/0/45/0/45/0/45/0/45]T, where 0 represents a 0-90 fabric ply. The properties of the graphite/epoxy and the properties of the interface are shown in Tables 17.4, 17.5 and 17.6, respectively. A complete analysis of the delamination growth in skin-stiffener structures requires three-dimensional models with a high degree of complexity. Several alternatives to fully three-dimensional models have been proposed. Krueger et al. (2002) compared two-dimensional finite element modeling assumptions Table 17.4 Material properties for IM7/8552 unidirectional graphite/epoxy prepreg (Krueger et al. 2001) E11 (GPa)

E22 = E33 (GPa)

G12 = G13 (GPa)

G23 (GPa)

ν12 = ν13

ν23

161.0

11.38

5.17

3.92

0.32

0.45

Table 17.5 Material properties for IM7/8552 graphite/epoxy plain weave fabric (Krueger et al. 2001) E11 = E22 (GPa)

E33 (GPa)

G12 (GPa)

G23 = G13 (GPa)

ν12

ν23 = ν13

71.7

10.3

4.48

4.14

0.04

0.35

Table 17.6 Properties of the interface for IM7/8552 (Hansen and Martin 1999) GIc (kJ/m2)

GIIc (kJ/m2)

o (MPa) τ shear

τ 3o (MPa)

η

0.200

1.400

61

68

1.45



507

with fully three-dimensional analysis in the simulation of composite skinstiffener debonding. The two-dimensional models are based on plane-stress, plane-strain, generalized plane strain elements. In addition, the authors analyzed the results obtained using a mesh of three-dimensional finite elements where the generalized plane-strain conditions were imposed by means of kinematic constrains. The results obtained by Krueger et al. (2002) indicate that the threedimensional-generalized plane-strain model provides the closest agreement with fully the three-dimensional model in terms of the distribution of the energy release rate. Based on the results reported by Krueger et al. (2002), three-dimensional generalized plane-strain models are used in the analysis of the skin-stiffener specimen. As exemplified in Fig. 17.13, the model consists in a line of eightnode three-dimensional elements, using two different constraints to impose the generalized plane strain state: the movement of the nodes located at the midplane 1–3 is constrained in the two-direction, and the nodes located in the surface parallel to the 1–3 midplane are linked by a kinematic constraint that ensures that this surface is always parallel to the 1–3 midplane. Eightnode cohesive elements are used along the skin-stiffener interface to simulate debonding under fatigue loading. The specimen is loaded applying a displacement at one end of the specimen while keeping the opposite end of the specimen clamped. A quasi-static tension test was conducted to obtain the flange debond load. The results obtained are shown in Fig. 17.14. The load-displacement

U2 = 0 U1 and U3 free

Cohesive elements 3 2

U2 = f(kinematic constraint) U1 and U3 free

1

17.13 Mesh and plane strain boundary conditions used in the skinstiffener simulation.


508


Debond load

16 14

Load, kN

12 10 8 6 4 2 0 0.0

0.1

0.2

0.3

0.4 0.5 0.6 Displacement, mm

0.7

0.8

0.9

1.0

17.14 Load-displacement curve of the skin-stiffener under quasi-static tension.

curve is almost linear until the flange debond load, approximately 16 kN, corresponding to an end displacement of 0.7 mm. Once the debond load has been determined, several tension fatigue tests were conducted for load levels corresponding to 60%, 70%, 83% and 90% of the quasi-static debond load. The load R-ratio was set to zero. The fatigue tests were conducted using the same procedure explained in previous sections. The simulation is divided in two steps: firstly, a quasi-static step until the desired displacement is reached, and then, a second step keeping the displacement constant and introducing fatigue degradation. Since the fatigue properties used for the IM7/8552 were not measured the properties given in Table 17.3 were used in the analyses. The crack growth length with the number of cycles for an applied load of 83% the quasi-static debond load is shown in Fig. 17.15. It is observed that the model predicts crack growth for a specimen without initial crack. The number of cycles required to increase in 1% the global compliance of the specimen are plotted in Fig. 17.16 for load levels corresponding to 60%, 70%, 83% and 90% of the quasi-static debond load. It is observed that the number of cycles increases exponentially as the load decreases. The results of the simulations on an aircraft substructure component show that the presented model is able to predict delamination onset and growth on complex structures without initial pre-crack. © 2008, Woodhead Publishing Limited


509

9 8 7

Crack length, mm

6 5 4 3 2 1 0 1000

10 000 Cycles

17.15 Crack growth length of the skin-stiffener under fatigue with a load of 83% the debond load.

90

Normalized load

85

80

75

70

65

60 100

1000 10 000 Cycles to increase the compliance 1%

17.16 Load cycles to increase the global compliance 1%.


100 000

510


17.12 Conclusions This chapter presented a methodology to simulate the propagation of delamination under cyclic loading. The overview of the experimental observations of delamination onset and propagation under cyclic loading, and the critical assessment of the corresponding analytical models previously developed provided the basis for the development of a cohesive model able to represent delamination growth under high-cycle fatigue. The constitutive equations developed for quasi-static loading were enriched with additional terms that account for the material degradation that results from cyclic loading. A new closed-form solution to predict the length of the damage zone ahead of the crack tip under general loading conditions was presented. This solution is used in the enhanced constitutive model, and plays a fundamental role in the accuracy of the model. The model proposed uses the Paris law as an input. Therefore, the model can be used to predict the mechanical response of composite structures using standard data measured in simple test specimens. In addition, the formulation can be used with virtually any form of the Paris law because the required terms are computed internally by the model. The model was implemented in ABAQUS non-linear finite element code as a user-defined cohesive element. The accuracy of the model was assessed by comparing its predictions with experimental data obtained in test specimens loaded in pure mode I, pure mode II, and mixed-mode I and II. A good agreement between predictions and experiments was obtained for all loading situations. The model is able to predict the threshold value of the energy release rate below which no delamination propagation is observed, as well as the increase on the crack growth rate for energy release rates close to the material fracture toughness. In addition, the model accounts for the effect of the load ratio on the crack growth rate without any additional parameters. The model has also been used to simulate fatigue degradation over a specimen without initial crack. The simulations over a skin-stiffener substructure component of an aircraft show that the model is able to predict delamination onset and growth in a specimen without an initial crack under complex loading scenarios. Future work should address the effect of friction between the faces of the delamination and the effects of high-frequency loading on crack growth rates.

17.13 Acknowledgments The financial support of the Portuguese Foundation for Science and Technology (FCT) under the project PDCTE/50354/EME/2003 and of the Spanish Government through DGCYT under the contract MAT2006-14159-C02-01 is acknowledged.



511

The authors would like to acknowledge the contribution of Dr Carlos G. Dávila, NASA Langley Research Center, USA. The authors would also like to thank Soraia Pimenta, University of Porto, for the collaboration in the preparation of the finite element models.

17.14 References and further reading Andersons J, Hojo M, Ochiai S (2001), Model of delamination propagation in brittlematrix composites under cyclic loading, Journal of Reinforced Plastics and Composites, 20(5), 431–450. Andersons J, Hojo M, Ochiai S (2004), Empirical model for stress ratio effect on fatigue delamination growth rate in composite laminates, International Journal of Fatigue, 26(6), 597–604. Asp L, Sjögren A, Greenhalgh E (2001), Delamination growth and thresholds in a carbon/ epoxy composite under fatigue loading, Journal of Composites Technology and Research, 23, 55–68. ASTM Standard D6115 (2004), Standard test method for mode I fatigue delamination growth onset of unidirectional fiber-reifnorced polymer matrix composites. Barenblatt G (1962), The mathematical theory of equilibrium cracks in brittle fracture, Advances in Applied Mechanics, 7, 5–129. Bazant ZP, Planas J (1998), Fracture and Size Effect in Concrete and Other Quasibrittle Materials, CRC Press, Boca Raton. Benzeggagh ML, Kenane M (1996), Measurement of mixed-mode delamination fracture toughness of unidirectional glass/epoxy composites with mixed-mode bending apparatus, Composites Science and Technology, 49, 439–449. Blanco N, Gamstedt EK, Asp LE, Costa J (2004), Mixed-mode delamination growth in carbon-fibre composite laminates under cyclic loading, International Journal of Solids and Structures, 41, 4219–4235. Brunner AJ, Blackman BRK, Davies P (2008), A status report on delamination resistance testing of polymer-matrix composites, Engineering Fracture Mechanics, 75(a), 2779– 2794. Camanho PP, Dávila CG, de Moura M (2003), Numerical simulation of mixed-mode progressive delamination in composite materials, Journal of Composite Materials, 37, 1415–1438. Cojocaru D, Karlsson AM (2006), A simple numerical method of cycle jumps for cyclically loaded structures, International Journal of Fatigue, 8(12), 1677–1689. Dahlen C, Springer GS (1994), Delamination growth in composites under cyclic loads, Journal of Composite Materials, 28(8), 732–781. Dugdale D (1960), Yielding of steel sheets containing slits, Journal of Mechanics and Physics of Solids, 8, 100–104. Goyal-Singhal V, Johnson E (2003), Cohesive-decohesive interfacial constitutive law for the analyses of fatigue crack initiation and growth, 44th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference AIAA-2003-1678, 1–11. Goyal-Singhal V, Johnson E, Dávila CG (2004), Irreversible constitutive law for modeling the delamination process using interfacial surface discontinuities, Composite Structures, 64, 91–105. Gustafson CG, Hojo M (1987), Delamination fatigue crack-growth in unidirectional graphite epoxy laminates, Journal of Reinforced Plastics and Composites, 6(1), 36–52. © 2008, Woodhead Publishing Limited

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Hansen P, Martin R (1999), DCB, 4ENF and MMB Delaminaton characterisation of S2/ 8552 and IM7/8552, United States Army (European Research Office of The US Army), N68171-98-M-5177. Hibbitt, Karlsson, Sorensen (1996), ABAQUS 6.2 Users’ Manual, Pawtucket, RI, USA. Hillerborg A, Modéer M, Petersson P (1976), Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and Concrete Research, 6, 773–782. Hojo M, Tanaka K, Gustafson CG, Hayashi R (1987), Effect of stress ratio on nearthreshold propagation of delamination fatigue cracks in unidirectional CFRP, Composites Science and Technology, 29(4), 273–292. Hojo M, Matsuda S, Ochiai S (1997), Delamination fatigue crack growth in CFRP laminates under mode I and II loadings – effect of mesoscopic structure on fracture mechanism, Proceedings of the International Conference on Fatigue of Composites, Paris (France), pp. 15–26. Juntti M, Asp L, Olsson R (1999), Assessment of evaluation methods for the mixed-mode bending test, Journal of Composites Technology and Research, 21, 37–48. Kachanov LM (1958), Time of the rupture process under creep conditions, ZV Akad Nauk – SSR Od Tech Nauk 8, 21–31. Krueger R, Paris IL, O’Brien TK, Minguet PJ (2001), Fatigue life methodology for bonded composite skin/stringer configurations, NASA/TM-2001-210842. Krueger R, Paris IL, O’Brien TK, Minguet PJ (2002), Comparison of 2D finite element modeling assumptions with results from 3D analysis for composite skin-stiffener debonding, Composite Structures, 57, 161–168. Lemaitre J, Sermage J, Desmorat R (1999), A two scale damage concept applied to fatigue, International Journal of Fracture, 97, 67–81. Matsuda S, Hojo M and Ochiais (1997), Mesuscopic fracture 572 mechanism of mode II delamination fatigue crach propagation in interlayer-toughened CFRP, JSME International Journal Series A – Solid Mechanics and Material Engineering, 40(4), 423–429. Mayugo JA (2003), Estudio constitutivo de materiales compuestos laminados sometidos a cargas cíclicas, Universitat Politècnica de Catalunya, PhD thesis. Mi U, Crisfield M, Davies G (1999), Progressive delamination using interface elements, Journal of Composite Materials, 32, 1246–1272. Muñoz JJ, Galvanetto U, Robinson P (2006), On the numerical simulation of fatiguedriven delamination using interface elements, International Journal of Fatigue, 28, 1136–1146. Nguyen O, Repetto E, Ortiz M, Radovitzky R (2001), A cohesive model for fatigue crack growth, International Journal of Fracture, 110, 351–369. Ortiz M, Pandolfi A (1999), Finite-deformation irreversible cohesive elements for threedimensional crack propagation analysis, International Journal for Numerical Methods in Engineering, 44, 1267–1282. Pagano N.J, Schoeppner GA (2000), Delamination of polymer matrix composites: problems and assessment, Comprensive Composite Materials, 2. (ed.) Kelly A, Zweben C. Elsevier Science Ltd., Oxford (UK). Peerlings R, Bredelmans W, de Borst R, Geers M (2000), Gradient-enhaced damage modelling of high-cyclic fatigue, International Journal of Numerical Methods in Engineering, 9, 1547–1569. Pipes RB, Pagano NJ (1970), Interlaminar stresses in composite laminates under uniform axial extension, Journal of Composite Materials, 4, 538–548. © 2008, Woodhead Publishing Limited


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Ramkumar RL, Whitcomb JD (1985), Characterization of mode I and mixed-mode delamination growth in T300/5208 graphite/epoxy, Delamination and Debonding of Materials, ASTM STP 876. (ed.) Johnson, W.S. American Society for Testing and Materials, Philadelphia, PA (USA), pp. 315–175. Reedy E, Mello F, Guess T (1997), Modeling the initiation and growth of delaminations in composite structures, Journal of Composite Materials, 31, 812–831. Robinson P, Galvanetto U, Tumino D, Bellucci G (2005), Numerical simulation of fatiguedriven delamination using interface elements, International Journal of Numerical Methods in Engineering, 63, 1824–1848. Roe K, Siegmund T (2003), An irreversible cohesive zone model for interface fatigue crack growth simulation, Engineering Fracture Mechanics, 70, 209–232. Russell AJ, Street KN (1989), Predicting interlaminar fatigue crack growth rates in compressively loaded laminates, Composite Materials: Fatigue and Fracture II, ASTM STP 1012. (ed.) Lagace, P.A. American Society for Testing and Materials, Philadelphia, PA (USA), pp. 162–178. Schön J, Nyman T, Blom A, Ansell H (2000), A numerical and experimental investigation of delamination behaviour in the DCB specimen, Composites Science and Technology, 60(2), 173–184. Simo JC, Ju JW (1987), Strain and Stress-based continuum damage models-I. Formulation, International Journal of Solids and Structures, 23(23), 821–840. Tanaka H, Tanaka K (1995), Mixed-mode growth of interlaminar cracks in carbon/epoxy laminates under cyclic loading, Proceedings of the 10th International Conference on Composite Materials, Whistler, BC (Canada), pp. 181–189. Turon A, Camanho PP, Costa J, Dávila CG (2006), A damage model for the simulation of delamination in advanced composites under variable-mode loading, Mechanics of Materials, 38, 1079–1089. Turon A, Dávila CG, Camanho PP, Costa J (2007a), An engineering solution for solving mesh size effects in the simulation of delamination with cohesive zone models, Engineering Fracture Mechanics, 74(10), 1665–1682. Turon A, Costa J, Camanho PP, Dávila CG (2007b), Simulation of delamination in composites under high-cycle fatigue, Composites Part A, 38(11), 2270–2282. Tvergaard V, Hutchinson J (1992), The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, Journal of Mechanics and Physics of Solids, 40, 1377–1397. Van Paepegem W, Degrieck J (2001), Fatigue degradation modelling of plain woven glass/epoxy composites, Composites Part A, 32(10), 1433–1441. Yang Q, Shim D, Spearing S (2004), A cohesive zone model for low cycle fatigue life prediction of solder joints, Microelectronic Engineering, 75, 85–95.


18 Single and multiple delamination in the presence of nonlinear crack face mechanisms R M A S S A B Ò, University of Genova, Italy

18.1

Introduction

Delamination and interfacial fracture are dominant failure mechanisms of laminated composites and multilayered systems. They may occur as localized or diffuse events and are often in the presence of nonlinear crack face mechanisms, which include bridging by fibers and through-thickness reinforcements, contact and friction. The nonlinear mechanisms typically develop along extended regions of the crack surfaces so that delamination becomes a large scale bridging problem that cannot be described by Linear Elastic Fracture Mechanics or characterized by a single fracture parameter. In addition, they introduce characteristic length and time scales that control the fracture characteristics and govern the transitions in the mechanical response and modes of failure. The first part of the chapter (Sections 18.2–18.4) presents the modelling approaches currently used to study fracture in the presence of extensive nonlinear crack face mechanisms, namely the cohesive- and bridged-crack models, and some of the techniques used to model delamination that lead to analytic or semi-analytic solutions for special configurations (Section 18.2.1). In Section 18.3, characteristic length scales are presented for mode I and mode II delamination; these length scales differ from those in non-slender bodies being material/structure properties that scale with the thickness of the laminate. The methods used to infer the bridged-crack model parameters from experimental tests are recalled in Section 18.4, focusing on macromechanical inverse approaches and using through-thickness reinforced laminates as a prime example. The second part of the chapter (Section 18.5) shows how nonlinear mechanisms acting along extended regions of the crack surfaces may profoundly affect fracture characteristics and the mechanical response. Different problems of static and dynamic, single and multiple delamination are discussed and the consequences of large scale bridging and delamination interaction on life prediction and design are highlighted. Section 18.5.1 describes some unusual 514 © 2008, Woodhead Publishing Limited

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515

characteristics of mixed mode delamination in the presence of bridging created by z-pins and stitching, which lead to crack arrest at large crack lengths. Section 18.5.2 presents quasi-static and dynamic interaction effects of multiple delaminations on fracture parameters, macrostructural response and key properties, such as damage tolerance and energy absorption. Finally, Section 18.5.3 shows two examples of the important effects that large-scale bridging may have on the dynamic response of systems of stationary and propagating delaminations.

18.2

The cohesive- and bridged-crack models

Fracture processes in brittle-matrix composites and quasi-brittle materials are always associated with the formation of a very narrow band, or process zone, where nonlinear deformation occurs. The nonlinear mechanisms, which include the formation, coalescing and branching of microcracks, crazing, debonding, yielding, sliding and pulling-out of the reinforcing phases and frictional contact, can dissipate a considerable amount of energy so that additional external work is required for sustained growth of the macrocrack. As a consequence, the intrinsic fracture toughness of the material is increased. In the delamination fracture of conventional 2D laminated composites, the nonlinear processes consist mainly of microcracking and crazing in the polymer matrix between the plies. A nice example of S-shaped microcracks at a mixed mode crack tip is reproduced in Fig. 18.1a (Rugg et al., 2002). These processes typically occur over a short length, which might be up to ≈ 1–3 mm, and they can be represented as an increment in the intrinsic fracture energy of the material that in a tough laminate is on the order of 1 kJ/m2. Greater enhancement of delamination fracture resistance can be due to extended friction along the crack wake and to the action of in-plane fibers crossing the fracture plane at shallow angles if the crack meanders from the interplay layer into the plies themselves. In these cases the length of the process zone varies as the crack grows. Much greater enhancement of delamination fracture resistance is achieved in through-thickness reinforced laminates, or 3D laminates. In these materials, a trans-laminar reinforcement is incorporated that will bridge any delamination cracks. A through-thickness reinforcement is applied by various techniques, including stitching or weaving continuous fiber tows or implanting discontinuous fibrous or nonfibrous rods (z-pins) (Evans and Boyce, 1989; Freitas et al., 1994, 1996; Dransfield et al., 1994). A picture of titanium z-pins bridging a mixed mode delamination crack in a carbon-epoxy laminate is presented in Fig. 18.1b (Rugg et al., 2002). The bridging ligaments shield the crack tip from the applied loading, so reducing the driving force for crack propagation. The process zone length can grow in these material up to ≈ 50–100 mm and the fracture toughness can become even two orders of © 2008, Woodhead Publishing Limited

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0.5 mm (a)

2 mm

(b)

18.1 (a) S-shaped microcracks at the tip of a delamination crack loaded in mixed mode in a carbon-epoxy laminate. (b) Titanium z-pins bridging a mixed mode crack in a carbon-epoxy laminate (adapted from Rugg et al., 2002).

magnitude greater than that of conventional 2D laminates. This has important effects on crack growth characteristics, failure modes and macrostructural behaviour (Jain and Mai, 1994; Massabò and Cox, 1999, 2001; Massabò et al., 1998; Rugg et al., 2002; Mouritz and Cox, 2000). Figure 18.2 compares experimental load versus mid-span deflection curves taken in End Notched Flexural tests on conventional 2D (a) and stitched carbon-epoxy quasi-isotropic laminates (b). In the conventional laminate, the crack grew catastrophically at a critical value of the applied load and the macrostructural response is characterized by a snap-back instability (a). In the stitched laminate, instead, the crack grew stably up to the midspan of the specimen and the macrostructural response is described by a hardening curve (b). The enhancement in fracture toughness due to the bridging ligaments was measured to be up to 100 times the fracture toughness of the base unstitched laminate (see Section 18.4) (Massabò et al., 1998). © 2008, Woodhead Publishing Limited

Single and multiple delamination Critical load

P (KN)

Unstitched Snap-back

1.5

P 1.0 2L = 240 mm 2h = 6.64 mm d = 23.72 mm a0 = 27 mm

0.5

0.0 0.0

4.0

8.0

12.0 16.0 20.0 24.0 28.0 32.0 36.0 Total deflection (mm) (a)

Stable growth

Glass stitches

3.0

P

Initial propagation

P (KN)

517

2.0

2L = 120 mm 2h = 7.2 mm d = 24.07 mm a0 = 20 mm

1.0

0.0 0.0

1.0

2.0

3.0 4.0 5.0 6.0 7.0 Total deflection (mm) (b)

8.0

9.0

18.2 (a) Load versus mid-span deflection curves for ENF specimens of length 2L, height 2h, width d and notch length a0: (a) no stitches; (b) glass stitches with area fraction 0.062. Base material: carbonepoxy quasi isotropic laminate with 48 plies (adapted from Massabò et al., 1998).

Nonlinear crack processes acting over such long lengths must be represented explicitly in the models rather than being assigned to a point process at the crack tip (as in Linear Elastic Fracture Mechanics, LEFM). Two alternative nonlinear crack models have been used in the literature: the bridged-crack model and the cohesive-crack model (Hillerborg et al., 1976; Carpinteri, 1989; Bao and Suo, 1992; Cox and Marshall, 1994; Massabò, 1998). The models are based on LEFM concepts and derive from the original works of Barenblatt (1959) and Dugdale (1960). Both models replace the process zone by a fictitious crack ahead of the pre-existing crack, and represent the © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites Continuous bridging tractions

Rea

σ

l cra ck Crack tip Bridging mechanisms mechanisms (a) σ

Brittle material (b)

σ σ0

σ0

σ0

σ1 σ1

wc (c)

wc (d)

w

w

wc

w1

w

(e)

18.3 (a) Schematic of crack wake and crack tip mechanisms in a stitched laminated composite. (b) Bridged-crack model. (c,d,e) Exemplary traction laws: (c) cohesive law for a strain softening material (e.g., conventional 2D laminate); (d) bridging law for a fiber reinforced composite (e.g., through-thickness reinforced laminate); (e) two-part cohesive law for a fiber reinforced composite (e.g., through-thickness reinforced laminate).

nonlinear mechanisms which control the fracture process as a distribution of traction vectors, T, acting along the fictitious crack (Fig. 18.3a,b). The tractions are related to the relative crack displacement vector, w, through a bridging traction law, T(w), which is generally nonlinear. In this chapter only plane problems will be considered and the bridging tractions will be represented by closing σ (wN) and shear τ (wS) tractions, acting normally and tangentially to the crack plane and depending on their respective opening wN and sliding wS relative displacements. The areas beneath the laws: G bI =

∫

w Nc

σ ( w N ) dw N

18.1a

τ ( w S ) dw S ,

18.1b

0

G bII =

∫

wSc

0

where wNc and wSc are the critical crack displacements beyond which the closing tractions vanish, are material properties that describe the toughening mechanisms developed in the process zone under mode I and mode II conditions (Fig. 18.3c,d,e). In general mixed mode problems, the bridging traction law may not necessarily be separable, so that σ (wN, wS) and τ (wS, wN) (opening induces shear tractions and vice versa), and in dynamic/fatigue processes it may be rate/cycle dependent. © 2008, Woodhead Publishing Limited


519

From a mathematical point of view, the only difference between the above mentioned models is in the form of the assumed crack tip stress field. In the bridged crack model this is singular and when the crack is at the onset of growth the singular field is measured by an intrinsic fracture energy, Gc = Gc(GIc, GIIc, GI/GII), which depends on the mode I and mode II intrinsic fracture energies, GIc and GIIc, and energy release rates, GI and GII. In the cohesive crack model, the crack tip stress field is finite (non-singular) and when the crack is at the onset of propagation the stress at its tip equals the value of the cohesive tractions at zero displacement, e.g. σ (wN = 0). In this sense the cohesive-crack model can be considered as a particularization of the bridged-crack model under the assumption of a vanishing crack tip stressintensity factor. However, the two models are conceptually different as they presuppose different descriptions of the material leading to a different significance of the bridging tractions (Cox and Marshall, 1994; Massabò, 1999). In the bridged-crack model, the composite is theoretically simulated as a biphase material and two distinct factors contribute to its global toughness. The first is the toughness due to the near crack tip mechanisms, described by Gc. The second is due to the crack wake mechanisms and is represented by the shielding the bridging tractions develop on the crack tip stress field. In the bridged-crack model, crack growth is governed by the intrinsic toughness, and the bridging tractions, which control crack opening and sliding, are governed by the properties of the reinforcing phase and by its interaction with the matrix. The bridged-crack model is then suitable for the description of separation processes that involve distinct physical phenomena (crack tip toughness + shielding), such as those of through-thickness reinforced laminates. In the cohesive-crack model, the composite material is theoretically simulated as being homogeneous. Only the global toughening mechanism of the whole composite is defined, and it is represented by the shielding due to the cohesive tractions. The crack tip mechanisms, explicitly represented in the previous model by Gc, are now merged with the toughening mechanisms developed in the process zone through the cohesive law. Therefore the damage process producing the growth of the crack is the same as that governing opening and sliding along the process zone. The model is suitable to describe the separation process of materials characterized by a wide zone of microcracking, plastic deformation or crazing, when the matrix toughness is negligible compared to the energy dissipated in the nonlinear processes. However, the definition of an appropriate cohesive law (e.g., a two-part law, Fig. 18.3e) allows utilization of the cohesive-crack model in the description of materials characterized by distinct mechanisms of crack control. A composite laminate reinforced through the thickness, for instance, is well described by a two-part cohesive law where the first branch, for small relative displacements, describes cohesive mechanisms due to microcracking and crazing in the © 2008, Woodhead Publishing Limited

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matrix and the second branch describes the bridging mechanisms due to the through-thickness reinforcement. The choice between the two models will ultimately depend on the material being studied, the scale of work and the modelling techniques used to solve the fracture problem. As already mentioned above, the bridged crack model can facilitate description of materials characterized by distinct toughening mechanisms that would require the computationally more expensive use of two-part cohesive laws. However, this description removes the possibility of studying crack initiation, since a crack must already exist, and has a well defined range of applicability when the matrix is not perfectly brittle (Massabò, 1999). A cohesive crack model, instead, allows the description of the entire fracture process from crack initiation to final separation. A bridged crack model can be the preferred choice when a weight function approach is used to analyze fracture, while the cohesive crack model is apt to descriptions based on the use of interface zone elements (see next section).

18.2.1 Modeling large scale bridging delamination fracture Different modeling techniques have been used in the literature to study quasi-static and dynamic delamination fracture in the presence of nonlinear crack face mechanisms (see Tay, 2003; Yang and Cox, 2005 for reviews). Among the fully numerical approaches, the Finite Element Method has been applied extensively with special crack tip elements able to reproduce the singular crack tip fields of LEFM and cohesive zone (interface) elements able to describe process regions; adaptive or initially elastic interfaces have been used in combination with 2D, 3D and beam and plate elements. This chapter focuses on plane problems and refers to modelling techniques based on particularizations of the theories of bending of beams and plates and on the weight function method. Within the models based on structural theories, two different approaches will be used, here referred to as classical and interface models. The classical model decomposes the body into sub-laminates defined by the delamination planes and the crack tip locations and represents the intact part of the member ahead of each delamination tip as a single plate/beam (e.g., sublaminates i, j and k in Fig. 18.4a). The interface model represents the delaminated body as an assembly of layers joined by cohesive interfaces that define the actual and potential fracture planes (Fig. 18.4b). The approximate models yield analytic or semi-analytic solutions in the quasi-static cases (Andrews et al., 2006) and in special dynamic problems (Sridhar et al., 2002), and considerably simplified computations in the dynamic cases (Andrews, 2005; Andrews et al., 2008). They allow the boundaries of different behavior domains to be mapped exhaustively and accurately. This is not easily done with fully numerical © 2008, Woodhead Publishing Limited


521

q

q i j

k Cohesive interfaces

(a)

(b)

18.4 Schematics for a multiply delaminated beam subject to out of plane loads: (a) classical model; (b) interface model. ∆ϕ0,1

h1

x

1

h2

2 z

a

M 1 V1 N1

h1 0

c

h0 h2

h2

M0 h0

N0 V0

∆ x →0

∆ϕ0,2 (a)

N2 M 2 V2

h1

(b)

(c)

18.5 (a) Edge-cracked beam subject to generalized end forces. (b) Relative rotations of the beam segments at the crack tip cross section (root rotations). (c) Crack tip stress resultants.

solutions, due to numerical noise, which can obscure which behavioral domain a particular case belongs to. In addition, the approximate models may be very accurate also for problems with short cracks. Classical model Consider, for sake of simplicity, the singly delaminated beam subject only to arbitrary end forces shown in Fig. 18.5a. The classical model assumes the beam composed of three sub-beams (0, 1 and 2) joined by continuity conditions at the crack tip cross-section. The elementary solution then imposes the bending rotations of the three sub-beams to coincide at the crack tip. This assumption neglects the near tip elastic deformations and corresponds to the well known built-in assumption in the case of a Double Cantilever Beam loaded in mode I. Models based on this assumption lead to accurate predictions of the energy release rate for short and moderately long cracks only if the problem is statically determinate (namely if the crack tip stress resultants are defined by equilibrium only) and the shear stress resultants at the crack tip cross-sections are zero or negligible. Accurate predictions of the mode ratio and the stress intensity factors in the absence of shear forces can be obtained if, in addition, a mode decomposition based on the local two-dimensional crack tip fields is performed, following, for instance, the methods developed © 2008, Woodhead Publishing Limited

522


by Hutchinson and Suo, (1992): the kinematic constraints imposed on the crack tip fields by the structural theories does not accurately describe the local crack tip conditions. In the presence of crack tip shear forces, assuming the sub-beam rotations to coincide at the crack tip cross-section leads to an underestimation of the energy release rate and erroneous predictions of the mode ratio also in beams with long cracks. The problem can be solved and accurate predictions of energy release rate and stress intensity factors can be obtained following the methods proposed by Li et al. (2004), for isotropic bilayer beams, and by Andrews and Massabò (2007), for homogeneous orthotropic beams. The method by Andrews and Massabò (2007) accounts for the near tip deformations through localized rotations of the sub-beams at the crack tip or root rotations, ∆ϕ0,1 and ∆ϕ0,2 in Fig. 18.5b. The root rotations depend on the crack tip stress resultants (Fig. 18.5c) through compliance coefficients that have been derived with accurate finite element analyses for a wide range of orthotropic materials and thickness-wise positions of the delamination (Table 18.1). Semi-analytic expressions have then been obtained for the energy release rate and the stress intensity factors, which depends on the crack tip stress resultants, the elastic constants and the root rotation compliance coefficients (Table 18.2). The semi-analytic expressions in Tables 18.1 and 18.2 are very accurate also for beams with short cracks (provided a ≥ cmin given in Table 18.1), when the crack tip stress resultants are calculated accounting for the root rotations in all statically indeterminate problems. Fig. 18.13 compares predictions of energy release rate and stress intensity factors by the classical model formulated by Andrews (2005) and Andrews et al. (2008) which includes root rotations, and accurate two-dimensional finite element results in a multiply delaminated beam subject to out of plane loading. The agreement is excellent also when the crack tips are closed. For a degenerate orthotropic DCB specimen with thickness 2h, longitudinal and transverse Young’s Moduli Ex and Ez subject to transverse end forces P, the expressions in Table 18.2 yield: 2 G DCB E x h  a  1 + 0.673 λ –1/4  h   = 12  h    a   P2

2

18.2a

that in the case of isotropy (λ = Ez/Ex = 1) matches very well, with only a difference in the third significant digit, with existing 2D results for the DCB specimen. The expression of the energy release rate for an ENF specimen turns out to be:

G ENF E x h a 2 h  = 9   1 + 0.208 λ –1/4    2 16  h    a P again in excellent agreement with existing results. © 2008, Woodhead Publishing Limited

2

18.2b

Table 18.1 Root rotation compliance coefficients of an orthotropic edge cracked beam subject to generalized end forces acting at a distance ≥ cmin = hiλ–1/4 (i = 0, 1, 2) from the crack tip  a1M  VS VD N  h M + a1 N + a1 V S + a1 V D   1 

∆ϕ 0,1 =

1 E x h1

∆ϕ 0,2 =

M 1  a 2 M + a N N + a VS V + a VD V  S D 2 2 2 E x h1  h1 

a1M , a 2M , a1N , a 2N = bλ –1/4 , = bλ

–1/2

a

VS 1

= bλ

–1/2

a V2S = bλ –1/2

with λ =

– ν xz /κ S , a

VD 2

Modified crack tip stress resultants = bλ

–1/2

+ ην xz /κ S z

η ν xz + 1 + η κS

E xE z Ez ,ρ= – 2G xz Ex

M

x

ν xz – 1 , 1 + η κS

VS + VD

N

h1

N

h2

h0

VS

M* VD

M* = M + N(h1 + h2)/2

ν zx ν xz and κS = 5/6

(see Table 18.2 for the modified stress resultants) Constant b

for:

a 2M

7.439 7.529 7.672 7.817 7.952 8.077

0.000 –0.372 –1.399 –3.034 –5.262 –8.077

η = h 1/ h 2 → 0.0 0.2 0.4 0.6 0.8 1.0

a 1N

a 2N

a 1V D

a V2 D

a 1V S

a V2 S

2.606 2.188 1.945 1.769 1.631 1.518

0.000 –0.137 –0.410 –0.746 –1.120 –1.518

2.606 2.208 1.758 1.382 1.087 0.858

0.000 –0.146 –0.350 –0.506 –0.604 –0.656

ρ = 1 and 0.025 ≤ λ ≤ 1


1.732 2.066 2.304 2.492 2.649 2.782

0.000 –0.331 –1.077 –2.163 –3.567 –5.280

523

a1M


a

VD 1

524

η → 0.0 0.2 0.4 0.6 0.8 1.0

ρ = 3 and 0.025 ≤ λ ≤ 1 10.046 10.234 10.487 10.705 10.884 11.029

0.000 –0.480 –1.860 –4.094 –7.156 –11.029

η → 0.0 0.2 0.4 0.6 0.8 1.0

2.107 2.604 2.935 3.183 3.379 3.541

0.000 –0.475 –1.544 –3.087 –5.069 –7.472

2.825 2.237 1.943 1.743 1.591 1.469

0.000 –0.126 –0.384 –0.709 –1.074 –1.469

2.825 2.684 2.200 1.716 1.306 0.977

0.000 –0.184 –0.408 –0.526 –0.544 –0.488

2.931 2.226 1.904 1.692 1.533 1.406

0.000 –0.115 –0.357 –0.668 –1.022 –1.406

2.931 3.105 2.613 2.034 1.518 1.093

0.000 –0.221 –0.463 –0.542 –0.475 –0.308

ρ = 5 and 0.025 ≤ λ ≤ 1 12.029 12.313 12.663 12.946 13.163 13.331

0.000 –0.566 –2.225 –4.927 –8.637 –13.331

2.408 3.043 3.449 3.745 3.975 4.161

0.000 –0.586 –1.902 –3.796 –6.221 –9.153

Uncertainties estimated on root rotations from FE solution and interpolation affect the fourth decimal digit. © 2008, Woodhead Publishing Limited


Table 18.1 (Continued)

Table 18.2 Energy release rate and stress intensity factors in an edge-cracked orthotropic beam subject to generalized end forces acting at a distance ≥ cmin = hiλ–1/4 (i = 0, 1, 2) from the crack tip Energy release rate:

Crack tip stress resultants

 2  M2  Vi 2 N2 M 02 V 02 N 02 i G=J = 1 Σ  + + i + 2V i ∆ϕ 0,i  – – – 3 3 i =1 2   E x h i /12 κ SG xz h i E x h i  E x h 0 /12 κ SG xz h 0 E x h 0 

  

M1

h1

N1 N2 M2

where: Ni, Mi, Vi (i = 1, 2, 3) = crack tip stress resultants (see schematic (a)), ∆ϕ 0,i =

V1

 1  M + a iN N + a Vi S V S + a Vi D V D  = root rotation sub-layer i (i = 1, 2) E x h 1  h 1 

M0 h0

N0

h2 V0

V2 ∆x → 0

a iM

(a)

a iM , a iN , a iV S , a Vi D = root rotation compliances (i = 1, 2) (Table 18.1), Ex, Ez, Gxz, νxz, = Young’s and shear moduli, Poisson’s ratio. κS = 5/6

M0 = M1 + M2 + 1 (h1N2 – h2N1), 2 N 0 = N 1 + N2 , V 0 = V 1 + V 2 .

Stress Intensity factors:

KI =

K II =

λ 3/ 8 1 + ρ    2 

1/ 4

λ 1/ 8 1 + ρ    2 

1/ 4

 f M sin (γ M + ω ) f NN cos (ω ) f V V D sin (γ  M + + D  h 13/ 2 h1 h1 

VD

 f M cos (γ M + ω ) f NN sin (ω ) f V V D cos (γ – M + – D 3/ 2  h h h1 1 1 

VD

+ω )

–

VS

+ω )    

f V S V S cos (γ h1

VS

+ω )    

525


+ ω ) f V S V S sin (γ + h1


N, M, VS, VD = modified crack tip stress resultants (see schematic (b)),

where: h 0 = h 1 + h 2, η = h 1/ h

526

with

Modified crack tip stress resultants

f M (η ) = 1 2

12 (1 + η 3 ) , f N (η ) = 1 2

f V S (η , λ , ρ ) =

f VD (η , λ , ρ ) =

1 + 4 η + 6 η 2 + 3η 3 , M

x

 V 1 1  ρ + ν  , S  a1 + xz   κ S 1 + η  λ  

z

 V  1+η ρ VD D + ν xz   ,  a1 – a 2 +  κS  λ  

VS + VD

N

h1

N

h2

h0

VS

M* VD

M* = M + N(h1 + h2)/2

(b)

where:

 3 (1 + η ) η 2 γ M (η ) = sin –1  f Mf N 

γ

VS

 , 

 a 1N  (η , ρ ) = sin –1  , γ  2 f Nf V S 

VD

 a N – a 2N (η , ρ ) = sin –1  1  2 f N f VD

ω = 52.1° – 3°η

λ=

E zE x Ez and ρ = – Ex 2G xz


ν xz ν zx

 , 

M = M1 –

1 M0 , (1 + 1/η ) 3

N = – N1 +

M0 1 1 , N0 – 6 1 + 1/η η (1 + 1/η ) 3 h 1

V S = V 0, V D = – V 2




527

Models based on the classical approach described here are suitable to study delamination fracture in beams that can be modeled as homogeneous, which include laminates with isotropic or orthotropic layers of the same material, unidirectionally reinforced laminates and quasi-isotropic laminates with a large number of layers. Interface model The interface model assumes the delaminated beam (Fig. 18.4b) composed of sub-layers joined by cohesive interfaces apt at describing perfect adhesion (in the intact portion of the member), the nonlinear process of material rupture and all crack wake mechanisms, including contact and friction. Interface models have been used extensively in the literature to model single delamination fracture (Tay, 2003 for a review) and recently have been applied to study problems of multiple delamination fracture under static and dynamic loading conditions (Andrews et al., 2008). The interface model allows the study of laminated and multilayered materials and it naturally accounts for the root rotations at the crack tip. However, the root rotations, because of the kinematic constraints imposed by the structural theories that are typically used to describe the sub-layers, only approximate the actual near tip deformations. Consequently, solutions obtained using an interface model do not always reach the level of accuracy that can be obtained using a classical approach that accounts for root rotations (Li et al., 2004; Andrews and Massabò, 2007). Weight function models Mixed mode delamination problems in beams and plates deforming in cylindrical bending can be formulated as a system of integral equations using the mode I and mode II weight functions of the problem. A mode I/II weight function defines the stress intensity factor at the crack tip produced by a pair of unit point forces acting normally/tangentially to the crack plane (Fig. 18.6a,b). In Massabò et al. (2003) and Brandinelli and Massabò (2006) approximate expressions for the mode I and mode II weight functions of orthotropic double cantilever beams have been derived using asymptotic matching through finite elements and an orthotropy rescaling technique. From these weight functions the net crack tip stress intensity factors due to the combined action of the applied loads and all crack wake mechanisms can be easily computed (Fig. 18.6c). The weight functions have the correct asymptotic form and therefore offer greatest control in numerical methods over singularities either at the crack tips or at discontinuities in the bridging mechanisms in the crack wake. The integral equation formulation reduces plane problems to one-dimensional © 2008, Woodhead Publishing Limited

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x3

x3 1 1

d

c

a

1 1 d

2h

x1

(a)

2h

x1 c

a (b)

x1 a0

2h

a (c)

18.6 (a,b) Schematics for the mode I and mode II weight functions of double cantilever beams; (c) delaminated beam in the presence of large scale bridging mechanisms along the crack faces.

problems, with distance from the crack tip the only dimensional variable. This allows very rapid scanning of large quantities of parametric problems. Using the approximate weight functions derived by Massabò et al. (2003) and Brandinelli and Massabò (2006) a large number of delamination problem can be simulated, including many standard fracture tests, e.g. DCB, ENF, MMB, ELS, …. The main limitation of the weight function approach to study delamination fracture is that solutions are restricted to cases for which weight functions have been derived (essentially only beams with centered delaminations).

18.3

Characteristic length scales in delamination fracture

Delamination growth in the presence of extensive bridging or cohesive mechanisms, such as those produced by a through thickness reinforcement, is a large scale bridging problem that cannot be described by LEFM or characterized by a single fracture parameter. A bridged-crack model is required, in which the essential material property is the bridging traction law. However, if the bridging ligaments fail in the wake of the crack in a sufficiently long specimen, the process zone may eventually evolve after some crack growth to a zone of constant length, the fracture toughness become constant and the solution of the problem approach that expected for a brittle crack when the material toughness includes the work required to fail the ligaments. This limiting configuration is known as the small-scale bridging limit and it has been defined analytically by Suo et al. (1992) and Massabò and Cox (1999) for bridged delamination cracks loaded in mode I and mode II. If the bridging ligaments do not fail, the crack may approach another limit configuration, the ACK limit, that is characterized by a constant critical load for crack propagation and has been defined analytically by Massabò and Cox (1999) for a mode II delamination problem. Two characteristic length scales are associated with these limits: lSSB is © 2008, Woodhead Publishing Limited


529

the length of the process zone in the small-scale bridging limit and lACK is related to the interval of crack growth over which the crack evolves into the ACK limit. These length scales are material/structure properties, which scale with the thickness of the laminate, and differ from those in non-slender bodies, which are material properties (Bao and Suo, 1992; Cox and Marshall, 1994). This is an important size-scale effect that characterizes the response of delamination beams. The characteristic lengths govern size-effects and scaling transitions in the mechanical behavior and failure mode of structural components, notch sensitivity of ultimate strength and the stability of crack growth. They are also essential for a priori estimations of the element sizes in numerical work involving cohesive zones: mesh-independent results require that the length of any cohesive zones, including the small-scale bridging zone, be large compared to the size of the elements on either side of the crack. In addition, knowing the characteristic length scales of the problem is necessary to validate the assumption of replacing the action developed by discrete bridging entities, e.g. stitches, with distributed tractions: the characteristic lengths must be larger than their spacing.

18.3.1 Small-scale bridging limit and characteristic length scales In the small-scale bridging configuration the crack propagates in a sufficiently long body with a process zone of constant size, lSSB, which is much smaller than the crack length. The composite fracture energy becomes a material constant, GSSB = Gc + Gb, given by the sum of two contributions: the intrinsic interlaminar fracture energy, Gc (= 0 in a cohesive crack where the intrinsic energy is merged with Gb, Section 18.2), and the energy supplied by the bridging/cohesive mechanisms, Gb, which is related to the area under the bridging/cohesive traction law. In this limit, the solution of the problem can be defined through LEFM by assuming that the bridged/cohesive zone has negligible size (Barenblatt, 1959; Willis, 1967; Smith, 1989). The mode of failure is usually catastrophic. Infinite bodies For a prescribed mode I bridging/cohesive law σ (wN), with σ0 the peak value and Gb the area under the curve, and intrinsic fracture energy, GIc, in an infinite, linear elastic, homogeneous and isotropic body under uniform remote loading, the limit length of the process zone is a material property given, to order of magnitude, by: I lSSB = a SI


18.3a

530


where a SI ≈

Gb E σ 02

a SI ≈

Gb E  G 1 + Ic – Gb σ 02 

for a cohesive crack with GIc = 0 (if σ(0) ≠ 0) 18.3b

or 2

G Ic  for a bridged crack with Gb ≠ 0 G b  18.3c

Gb is the energy supplied by the bridging mechanisms (Equation 18.1a) and E is the effective Young’s modulus. Equations 18.3a-c apply also for the II , for purely mode II cracks in infinite bodies, with characteristic length lSSB I II GIc → GIIc, a S → a S and σ0 → τ0, τ0 being the peak stress of the mode II bridging/cohesive law (Cox and Marshall, 1994). The dimensional group, GbE/ σ 02 , that defines the characteristic length scale when GIc = 0 has been noted by many in the literature as the parameter that characterizes the intrinsic brittleness of the material, with low values indicating a brittle material and high values a tough material (Bilby et al., 1963; Cottrell, 1963; Rice, 1968; Hillerborg et al., 1976). In materials with GIc ≠ 0, the dimensional group of Equation 18.3b is not sufficient to define the length of the bridged zone or to characterize the brittleness of the material, which are also controlled by the ratio Gb/GIc between the energy supplied by the bridging mechanisms and the intrinsic fracture energy: the characteristic length decreases for decreasing values of Gb/GIc indicating a progressive transition toward material brittleness. Equation 18.3b defines an upper bound to Equation 18.3c for GIc → 0. If σ(0) = 0 but GIc >> Gb, the length of the bridged zone is still approximated by Equation 18.3c, while if σ (0) = 0 and GIc > a, the solution of an edge crack in a semi-infinite sheet must hold and lSSB becomes a material constant lSSB = aS. Order of magnitude estimates of lSSB are 100 – 101 mm for conventional 2D laminates and 101 – 102 mm for through-thickness reinforced laminates. For a typical quasi isotropic carbon-epoxy laminate with GIb = 0.35 kJ/m2, GIIb = 0.7 kJ/m2, σ 0 = 50 MPa, τ0 = 35 MPa, GIc = GIIc = 0, E = 50 GPa and h = 3 mm, the mode I and mode II characteristic lengths turn out to be: I II a SI ≈ 7 mm and lSSB ≈ 4 mm and a SII ≈ 30 mm and lSSB ≈ 10 mm. In composite laminates reinforced through the thickness they are much larger. For instance, in the stitched composite laminate studied by Massabò et al. (1998), the mode II characteristic lengths turn out to be: a SII ≈ 500 mm and II lSSB ≈ 40 mm; the composite is a carbon/epoxy quasi-isotropic laminate with 48-ply, 2h = 7.2 mm, E = 49 GPa, stitched with glass fiber tows in a square array of side 3.2 mm with area fraction 0.062, GIIc = 0.37 kJ/m2, and τ (wS) = 12.7 + 102 wS MPa with τ0 = 55 MPa and GIIb = 30 kJ/m2. When multiple delamination cracks are anticipated, the best choice of h in Equations 18.4 must be considered. For a priori estimations of the mesh size in numerical work, the thickness of the thinnest sub-laminate would be a conservative choice. However, the actual size of lSSB probably lies in between the thickness of the thinnest sub-laminate and the thickness of the whole laminate. In cases where the cohesive law describes more than one nonlinear crack mechanism, such as the law in Fig. 18.3e, distinct characteristic lengths can be estimated for each of the mechanisms taken separately and if the cohesive law is used in numerical models, the elements should be properly sized to the smaller length.

18.3.2 ACK limit and characteristic length scales The ACK limit, named after the seminal work of Aveston, Cooper and Kelly (1971) on mode I cracks in fibrous composites, is characterized by a long © 2008, Woodhead Publishing Limited

532


delamination crack entirely bridged by intact ligaments. The limit can be approached in a long enough body if the applied load is uniform, the bridging ligaments do not fail in the wake of the crack and the bridging tractions are an increasing function of the relative crack displacement at least over a certain interval. In this limit the relative crack displacement takes that uniform value in the far crack wake required for the bridging tractions to balance the applied load. While the fracture toughness is not a material property in this limit, the critical load for crack propagation becomes constant and a material property, unaffected by the crack length or the presence of a notch, provided its length is much smaller than the crack length. The ACK limit configuration can be approached when the crack has propagated beyond any notch a distance which is typically several times a characteristic length, lACK. The limit defines a noncatastrophic mode of failure. Infinite bodies For a prescribed mode I power bridging law, σ ( w N ) = β w Nα , and intrinsic fracture energy, GIc, in an infinite linear elastic, homogeneous and isotropic body, the characteristic length is given by (Cox and Marshall, 1994): I l ACK = a mI

18.5a

where 1– α

a mI

–2

1+ α πE 1 + α = G Ic  β 1+α 4  2α 

18.5b

For linear bridging tractions, α (wN) = βwN, the characteristic length becomes I l ACK = a mI = π E /(4 β ). Length scales for the mode II problem in infinite bodies have not been derived but they will have similar forms. Slender bodies As for the small scale bridging limit, in a relatively slender body lACK becomes a material/structure property that scales with a function of the thickness (Massabò and Cox, 1999). For a centered mode II crack in a linear elastic, homogeneous and orthotropic slender beam of half thickness h, with a power bridging law, τ ( w S ) = β w Sα and intrinsic fracture energy, GIIc, it has been estimated as: II l ACK = ( a mII h )1/2


18.6a


533

where 1– α

a mII

–2

1+ α = E  1 + α G IIc  β 1+α . 4  2α 

18.6b

For the stitched laminate tested in (Massabò et al., 1998) and described above, the mode II characteristic lengths turn out to be: a mII ≈ 120 mm and II l ACK ≈ 20 mm. The solutions above apply to the End Notched Flexural specimen and other mode II standard specimens.

18.3.3 Large-scale bridging solutions and transition from non-catastrophic to catastrophic failure Whether crack propagation across an unnotched and long slender body (of length L >> lSSB, lACK) will be noncatastrophic (no ligament failure – ACK limit attainable) or catastrophic (extensive ligament failure – small scale bridging limit attainable) can be estimated by comparing the orders of magnitude of lSSB and lACK (Cox and Marshall, 1994; Massabò and Cox, 1999). If l ACK >> l SSB, failure will be catastrophic; if lSSB >> l ACK, noncatastrophic. The presence of a notch of length a0 will favour catastrophic cracking. If a0 is large enough (a0 >> lSSB), the transition can be estimated by comparing the ACK limit stress and the small-scale bridging stress for the given notch size. Failure will be catastrophic if the ACK limit stress is much larger than the small-scale bridging stress and noncatastrophic if the opposite is true. If neither limit can be approached (e.g., lSSB ≈ lACK and L ≈ lACK or L ≈ lSSB) then large-scale bridging conditions prevail and detailed calculations are required. The dimensionless diagram in Fig. 18.7 shows the critical load for crack propagation as a function of crack length in the mode II problem of a plate with an edge delamination subject to uniformily distributed shear tractions acting along the crack surfaces. Solutions for particular specimens can be constructed from this problem by superposing a trivial solution for a plate containing no crack or notch and subject to external loads. The diagram has been obtained in (Massabò et al., 1998) using a classical model approach. The critical load, τcr, has been normalized to the critical load in the ACK limit, τACK; the crack length, a, to the noncatastrophic characteristic length II (Equation 18.6a). The figure compares accurate solutions of the scale, l ACK large-scale bridging problem with the ACK and small-scale bridging limits for a laminate characterized by a linear bridging law, τ (wS) = τ (0) + βwS, with τ (0) and the critical sliding displacement beyond which the tractions vanish, wSc, consistent with the specific values for the stitched reinforced laminate mentioned above and studied by Massabò et al. (1998) (τ (0) = 2 G IIc β and Gb/GIIc = 80). The diagram also demonstrates the transition © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites a0 =0 II h )1/2 (a m z

Bridging tractions

x 2h

1.50

a0

τ0

τ (0)

Small-scale bridging limit

0.2

τ cr τ ACK

τ

a – a0

wSc wS

1.00 0.5 ACK limit 1.0 0.50

2.0

No bridging 3.0

5.0 10.0

0.00 0.0

4.0

8.0

12.0

16.0

a II h )1/2 (a m

18.7 Transition from noncatastrophic (ACK limit approached) to catastrophic (small-scale bridging limit approached) failure on varying the notch length in a stitched plate subject to pure mode II conditions (crack growth criterion: GII = GIIc; τ (0) = 2 GIIc β and Gb /GIIc = 80) (adapted from Massabò et al., 1998).

from noncatastrophic to catastrophic failure on varying the length of the notch. The different solid curves refer to different notch lengths. Three regimes of fracture are recognized: (a) unstable delamination crack growth with intact bridging ligaments in the crack wake for notch lengths lower than 0.33 ( a mII h )1/2 (or 7 mm for the case study defined above); (b) stable delamination crack growth in laminates whose notch lengths are in the range 0.33 < a 0 /( a mII h )1/2 < 2.3 (or 7 < a0 < 46 mm), with the ACK limit approached in long enough specimens; (c) stable delamination crack growth followed by ligament failure beginning at the notch root and further unstable crack growth to ultimate failure in laminates whose notch lengths satisfy a 0 > 2.3 ( a mII h )1/2 (or 46 mm); small scale bridging limit approached in long enough specimens. The response of typical laboratory specimens, e.g. the ENF test, would be roughly represented by the curve of stable crack growth with a 0 /( a mII h )1/2 = 1 (or 20 mm for the case study). This curve approaches within a few percent of the ACK limit only if a > 5 ( a mII h )1/2 (or a > 100 mm). For typical laminate depths, a specimen long enough to accommodate such a crack would be too long to avoid flexural failure of the laminate. Delamination cracks in ENF specimens of typical size, e.g., L ≈ 60 mm, will propagate stably in large-scale bridging conditions at critical stresses still 20% below © 2008, Woodhead Publishing Limited


535

the ACK limit even when they reach the center loading pin. Nevertheless, at this stage the critical stress is approximately eight times the critical stress for an unbridged crack. Thus, bridging effects are very strong and accurate analysis of the fracture process requires full solution of the bridged crack problem: asymptotic solutions will be inadequate. The small-scale bridging solution of the problem is represented in Fig. 18.7 by the dotted curve. This limit will be reached when stitches fail ( a 0 > 2.3 ( a mII h )1/2 ) to within 10% in the critical stress only after the crack has propagated over a length ≈ 10 ( a mII h )1/2 (or 200 mm). For the subject II case, the length of the bridged zone in this limit is lSSB ≈ 40 mm, and it represents a lower bound to the length of the bridged zone during crack propagation. Again, in specimens of common dimensions, the asymptotic solution (i.e., in this regime, LEFM) will be inadequate.

18.4

Derivation of bridging traction laws

The bridging traction law, T(w), and the interlaminar intrinsic fracture energy, Gc = Gc(GIc, GIIc, GI/GII), are the material characteristics needed to predict delamination cracking using a bridged crack model. In the case of throughthickness reinforced polymer matrix laminates, the bridging traction law will depend on the mechanisms of load transfer from the reinforcements to the surrounding material and will therefore be influenced by the reinforcement type (stitches, fibrous or metallic rods), area fraction, architecture and all damage mechanisms taking place in the reinforcement, the surrounding laminate and the interface; the interlaminar fracture energy is usually similar in magnitude to the matrix fracture energy, as the flaws originate and propagate in the matrix rich regions between the plies. Both macro- and micromechanical tests and models have been used in the literature to quantify bridging and cohesive mechanisms in different materials and systems (for macromechanical approaches see: Du et al., 1989; Rödel et al., 1990; Cox and Marshall, 1991; Guo et al., 1993; Fett et al., 1994; Massabò et al., 1998, Wittmann et al., 1987, 1988; Guinea et al., 1994; Uchida et al., 1995; Nanakorn and Horii, 1996; Fett 1995; for micromechanical studies: Li et al., 1991; Turrettini, 1996; Mouritz and Jain, 1999; Rugg et al., 1998; Cox et al., 1997; Cox 1999a,b; Cox and Sridhar, 2002; Cartie et al., 2004; Chang et al., 2006, 2007). The two approaches give insight into different aspects of the problem and the essential questions are therefore: in what detail must the bridging law be known; and what experiments/models suffice to determine it to the required level of detail? The answer depends on what needs to be predicted. Micromechanical tests and models are essential to the materials design problem, which needs to specify the optimal reinforcement, since they relate the bridging law to the properties and geometry of the constituents, the details of the fiber architecture © 2008, Woodhead Publishing Limited

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and the damage mechanisms associated with the bridging entities. Micromechanical models have been developed for general mixed mode loading conditions, based on observed mechanisms of deformations in single rods and stitches (Cox, 1999a,b; Cox and Sridhar, 2002). Essential mechanisms in fibrous rods bridging mixed mode cracks are: debonding of the rod from the laminate and pullout resisted by friction; development of axial tension in the rod during pullout; shear deformation of the rod through large strains, with matrix damage in the interior of the rod culminating in internal splitting; ploughing of the rod laterally through the laminate. The analytical results of the models (Cox, 1999a,b; Cox and Sridhar, 2002) agree well with empirical data for single stitch and single rod specimens (Turrettini, 1996; Cartiè et al., 2004) (Fig. 18.8). 1000 Micromech. model

900 800

Sliding rope

T1 (MPa)

700 600 500 From fracture data

400 300

Perfectly plastic solid

200 100 0 0

0.5 Sliding displacement, ws (mm)

1

18.8 Comparison of the predictions of different models with measurements from Turrettini (1996) on small cuboidal specimens of carbon-epoxy laminate reinforced by a single stitch tested in mode II. The traction in the stitch T1 (ws) is related to the bridging tractions τ (ws) by the area fraction of the through-thickness reinforcement. The sliding rope model, which assumes the stitch as an elastic string that can deform in the rectangular path shown in the inset, yields a law that is too stiff by a factor of approximately 7. The perfectly plastic model ignores the axial tension developed in the tow so that its deflection is resisted by plastic shear only; it leads to estimates that are too soft. The micromechanics model (Cox, 1999a,b) accounts for essential mechanisms of deformation. The line indicated with ‘From fracture data’ corresponds to the bridging law derived in Massabò et al. (1998) from crack profile and load vs. deflection data for the stitched laminate loaded in Notch End Flexure (adapted from Cox and Sridhar, 2002).



537

On the other hand, if the bridging law is to be embedded in a structural reliability model to predict large scale bridging delamination and define the performance of a structure, it should be as simple as possible, with the only requirement of accurately predicting the macrostructural behavior. The law should then be obtained by simple, macromechanical engineering tests whose geometry reflects that of the actual structural component and information on mechanisms and micromechanics should be used just to complement the fracture analyses. Indeed, micromechanical models can be misleading if they are used to predict service behavior, because they inevitably involve simplifications and idealizations which may not be universally applicable. The ideas above have been explored in Massabò et al. (1998), where a stitched carbon-epoxy laminate under mode II loading was taken as prime example. The bridging traction law and the intrinsic interlaminar fracture energy were deduced from crack profile measurements and load vs. loadpoint displacement data (shown in Fig. 18.2b) in an End Notched Flexure specimen, emphasizing the question of how well the experiments can determine the traction law and conversely the extent to which variations in the law bear on crack propagation and structural behavior. Inferring the bridging traction law from fracture data is an indirect procedure that requires the solution of an inverse problem. The solution is based on the following steps: experimental data are collected, e.g. crack profile measurements for assigned values of the applied load or load vs. load point displacement data; a bridged crack model is used to obtain numerical predictions of the experimental measurements for assigned bridging traction laws and intrinsic fracture energies; a general least square method (linear or nonlinear depending on the experimental data available) is applied and the best fitting fracture parameters are obtained as those that minimize a functional of the residuals between the experimental data and the numerical predictions. In solving inverse problems, special measures must be taken to treat experimental noise, since small oscillations in the data can lead to large oscillations in the deduced function, here the bridging law. Different techniques were examined in Massabò et al. (1998), e.g., Tikhonov linear regularization method, utilization of more than one profile in the minimization problem, use of parametric forms for the bridging law. The last technique maximizes the use of a priori knowledge when data contain only enough information to reveal the general shape for the function being sought and not its details. Figure 18.9a shows typically noisy crack sliding profiles (for a stable crack that grew past unfailed stitching in the crack wake), together with the profiles computed with the bridging traction law that has been deduced. The traction laws deduced from linear, quadratic and cubic trial forms are shown in Fig. 18.9b along with the inferred intrinsic interlaminar fracture energy. Their concurrence confirms that the law is essentially linear with an offset from the origin. In Massabò et al. (1998) the best fitting law and interlaminar © 2008, Woodhead Publishing Limited

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Notch 150.0

P = 3.2 (KN)

Crack sliding displacement Experimental Theoretical

ws(µ m)

100.0

P = 2.4 (KN)

50.0

P = 1.7 (KN) 0.00 0.00

10.0

20.0 30.0 40.0 Position along the crack (mm) (a)

50.0

III order 30.0

τ (MPa)

II order

20.0

10.0

0.00

I order

I order: τ (ws) = 12.7 + 102ws MPa GIIc = 0.37 kJ/m2 25.0

50.0

75.0 100.0 125.0 150.0 ws(µ m) (b)

18.9 (a) Comparison of experimental and fitted crack sliding profiles in an ENF test of a carbon-epoxy laminate with S-2 glass stitches (material and geometry are described in Fig. 18.2); (b) Best fitting bridging laws corresponding to different parametric forms (polynomials of different orders) (adapted from Massabò et al., 1998).

fracture energy were used to reproduce load versus deflection data obtaining good agreement. The use of load versus deflection data to deduce the traction law was also explored and, using a linear parametric traction law, fitted parameters values proved to be in excellent agreement with the traction law deduced from crack profile data, which is an entirely independent measurement. The bridging law deduced in Massabò et al. (1998) is compared with the experimental data on a single stitch in Fig. 18.8. There is some discrepancy at the lower end of the displacement range. However the deduced law very well describes the average behavior of the experimental data at larger displacements and it is in very good agreement with the predictions of the micromechanical model in (Cox, 1999a,b, Cox and Sridhar, 2002). Again independent measurements concur. © 2008, Woodhead Publishing Limited


539

By the analyses in Massabò et al. (1998), strong conclusions could be inferred about how many characteristics of the bridging law are relevant to fracture and what constitutes a sufficient test or set of tests for quantifying the delamination resistance of a given stitched laminate in general applications. •

•

•

•

Mode II delamination propagation, when the through-thickness reinforcement does not fail, as in the case examined, is sensitive to at most three parameters, which can be chosen conveniently to be: the magnitude of the tractions at incipient relative crack displacement, the slope of the traction law, and the intrinsic interlaminar fracture toughness. This indicates that a linear, non proportional bridging law, such as that in Fig. 18.9b, can adequately describe the bridging mechanisms in mode II and there appear to be no justification for choosing anything more complicated. If the reinforcement fails during delamination, good predictions of crack growth can probably be made if just one further parameter, the total work of fracture of the reinforcement or the critical crack displacement is included. This conjecture has not been tested in macromechanical experiments but seems to be confirmed by micromechanical tests and models (Fig. 18.8). It is recommended that the traction law be deduced in engineering practice from load vs. load-point displacement data from standard fracture tests, since they directly represent the performance of the laminate as an engineering structure and prediction from these data very well concur with predictions from local crack profile data; due regard must be taken to selecting the notch size and other specimen dimensions to ensure that crack growth is stable over a good part of the test and that crack sliding displacements comparable to those expected in service are achieved. In addition, the interlaminar fracture energy should be deduced from the same fracture tests because it is impossible to isolate the intrinsic fracture toughness from the bridging traction law for small values of relative crack displacement. The bridging law inferred for a typical through-thickness reinforced laminate imply that delamination cracks will commonly grow in conditions that are neither accurately nor properly described by linear elastic fracture mechanics (as already noted in Section 18.3).

18.5

Single and multiple delamination fracture

Nonlinear mechanisms acting along extended regions of the crack surfaces may have profound effects on fracture characteristics and the mechanical response of laminated systems. This section examines different cases of static and dynamic, single and multiple delamination fracture and highlights consequences on life prediction and design. © 2008, Woodhead Publishing Limited

540


18.5.1 Unusual characteristics of mixed mode delamination in the presence of large-scale bridging Mode I and mixed mode delamination in the presence of large-scale bridging mechanisms exhibit some unusual characteristics that have been observed in tests on laminated T-joints and Mixed Mode Bending (MMB) specimens reinforced by fibrous rods and mode I Double Cantilever Beam (DCB) specimens reinforced by stitching (Rugg et al., 2002; Cartiè and Partridge, 1999). In some of the mode I specimens tested, but not in all of them, a period of stable propagation of the crack was followed by crack arrest at large crack lengths (several tens of millimeters). The crack arrest was sometimes permanent and sometimes not. In the case of the permanent crack arrest the specimens failed at higher loads by other mechanisms, such as the failure of one of the loading arms. In the mixed mode specimens, detailed measurements of the crack profiles taken at large crack lengths during the stable propagation of the crack showed that the crack faces had shut down at the tip (Fig. 18.10). The curious phenomenon of crack arrest at large crack lengths that is maintained through large increases in the applied load probably has no precedent in delamination cracking of conventional 2D laminates. Its explanation lies in the profound effect on the characteristics of fracture exerted by large bridging zones of through-thickness reinforcement. The explanation of its presence in some but not all tests has been found in consideration of the length scales involved in the problem in Massabò and Cox (2001). In Massabò and Cox (2001) idealizations of the DCB and the MMB specimens were considered with simple but realistic mode I and mode II constitutive laws assumed for the bridging mechanisms. The problem was solved following a classical model approach (Section 18.2.1). When the bridging traction laws are linear functions of the respective crack displacements, σ (wN) = βNwN and τ (wS) = βSwS, the equation governing the crack opening in both the DCB and the MMB specimens is the wellknown equation that describes the deflection of a beam on a Winkler linear elastic foundation with modulus equal to the slope βN of the bridging law (assumed for simplicity to be the same in tension and compression). The solution is an oscillatory function of the coordinate that defines the position along the crack with wavelength λ = 2 π 4 4κ χ / β N , where κχ is the flexural stiffness of the beam. In beams of finite lengths subject to concentrated generalized forces at the free ends, the crack length for which first contact occurs between the crack faces is on the order of λ that therefore represents the length scale of the problem. For a typical laminate reinforced with 2% area fraction of Kevlar stitches λ is around 40–50 mm (Turrettini, 1996), but its value can vary substantially from material to material. In laminated © 2008, Woodhead Publishing Limited

Single and multiple delamination Mixed-mode bending

1800

c Grip failure

1600

Load (N)

541

1400 1200

a

1000 800

Crack initiation

P

2h

5% Z-fiber

2L

1.5% Z-fiber

600 400

Control

200 0 0

2

4 6 8 10 Cross-head displacement (mm) (a)

12

Displacement (µm)

MMB 5% Step 12, 1000 N 200 180 160 140 120 100 80 60 40 20 0

c 2h a

0

5

10

P

2L

15 20 Position (mm) (b)

Opening Sliding

25

30

35

18.10 (a) Experimental load versus load point displacement diagrams taken in MMB tests on a conventional carbon-epoxy laminate (control) and laminates with different area fractions of titanium z-pins (material described in Rugg et al., 2002). (b) Experimental crack profiles, taken during the stable crack growth in the laminate with 5% area fraction of z-pins, showing opening and sliding displacements as functions of the position along the crack (position 0 indicates the notch tip).

specimens, cracks are typically grown over quite long lengths and this explains why the phenomenon of crack arrest may be seen in some but not all tests. When the condition for crack face closure is approached in a mode I specimen, the critical load for crack propagation diverges. The crack then arrests and the specimen will fail at higher loads by mechanisms other than delamination. This behavior is described in the diagram of Fig. 18.11a that also highlights the influence of linear bridging mechanisms during the different phases of crack growth. A different response characterizes delamination growth when mixed mode conditions are present. In this case, when the limit crack length corresponding to crack face closure has been approached the crack may continue to propagate © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites P

Pcr β G Icr β Ν

P Bridging law

7.5

σ βN wN

5.0

2.5 No bridging 0.0 0.0

π /4

π /2 β (a – a0) (a)

3π /4

α1 = 0.2 α3 = 0.01

Pcr G Icr Eh

0.025 0.05 0.18 0.1 0.3

0.0075 c 2h

0.12

α3 = 0

a

P

2L GI

0.06

GII

= 1/4

No bridging 0.0 10.0

14.0

18.0 22.0 a /h (b)

26.0

18.11 (a) Dimensionless diagram of the critical load for crack propagation versus crack length in a DCB specimen with linear proportional bridging (crack growth criterion: G I = G Ic; β = 2π/λ). The response shows an initial unstable phase, when the bridging mechanisms have not yet been effective, followed by a phase of stable crack growth. The critical load then diverges when the crack length approaches the limit value corresponding to crack face closure. (b) Dimensionless diagram of the critical load for crack propagation versus crack length in a MMB specimen with Dugdaletype bridging laws. Crack propagation is rendered stable by the bridging mechanism in all cases examined; during crack growth the mode ratio varies from the initial value (G I /G II = 1/4) and progressively decreases and then vanishes when the solid curves reach the dotted curve, where G I = 0; at that point the crack faces come into contact at the crack tip; propagation then continues driven by the mode II component of the applied load and follows the dotted curve (obtained neglecting the presence of friction).



543

driven by the mode II component of the load and in the presence of regions of contact (as in the specimens tested in Rugg et al. (2002), Fig. 18.10). This behavior is described in the diagram of Fig. 18.11b that also highlights the influence of Dugdale-type bridging mechanisms during the various phases of growth. The diagram refer to a Mixed Mode Bending specimen with the applied load field chosen so as to give rise to a fairly constant mode ratio in an unreinforced specimen GI/GII = 1/4 (dashed curve). The bridging tractions are constants, σ = α 3 EG Icr / h and τ = α 1 EG Icr / h with values for α3 and α1 representative of typical stitched laminates, and the crack growth criterion is GI/GIcr + GII/GIIcr = 1 with GIIcr = 2GIcr. The specimen has an initial notch of length 10h. The possibility of crack tip closure has profound consequences for life prediction and the design of certifying tests for delamination resistance. If proper account is not taken of the possibility of crack tip closure, an unfortunately designed test could lead to severely non-conservative predictions of delamination resistance: crack tip closure may occur in a particular test configuration, but not in a structure of different geometry or loading configuration. Proper design rules will require the complete mapping of the regimes where crack tip closure occurs and does not occur. While crack tip closure is presumably very advantageous for delamination resistance, it should not be taken as the end of crack propagation, even if friction prevents subsequent immediate propagation under mode II loads. Fatigue effects may change the picture. Load cycling will generally degrade the bridging ligaments, softening the traction law even if the ligaments remain intact. This will result in the characteristic wavelength, λ, increasing and the possible resumption of delamination crack growth.

18.5.2 Static and dynamic interaction effects of multiple delaminations Damage in composite laminates and multilayered systems often occurs in the form of multiple delaminations, which are the typical outcome of severe dynamic loading, such as blast, impact and shock. The presence and interaction of multiple delaminations may have important effects on fracture parameters, crack growth characteristics and macrostructural response, as well as on key properties such as energy absorption, damage and impact tolerance, strength and stiffness. In spite of its technological importance, this problem has been neglected in the literature and fundamental characteristics of static and dynamic multiple delamination fracture have been highlighted only recently (Suemasu, 1993; Suemasu and Majima, 1996; Zheng and Sun, 1998; Larsson, 1991; Andrews et al., 2006, 2008; Andrews, 2005; Andrews and Massabò, 2008). An important interaction effect of multiple delaminations is extensive contact between the delamination surfaces. Contact has been observed for © 2008, Woodhead Publishing Limited

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both in-plane and out-of plane loading. Contact significantly affects the fracture parameters and introduces regions in which friction may be important. These nonlinear crack face mechanisms must be included in the solution of the problem, similar to what is done in the presence of large-scale bridging mechanisms, also when dealing with the perfectly brittle fracture of conventional 2D laminates. Strong interaction effects between delaminations include phenomena of shielding and amplification of the energy release rate and modifications of the mode ratio as compared to a structure with only a single delamination. These phenomena are also present when the crack tips are away from each other and in homogeneous systems they are controlled by the spacing of the delaminations only. The response of systems of multiple delaminations in cantilever and clamped-clamped beams subject to static and dynamic loading conditions has been studied in (Andrews, 2005; Andrews et al., 2006; Andrews and Massabò, 2008; Andrews et al., 2008). The study case of a plate with two built-in ends represents a skin structure supported by stiffening ribs, such as an airframe, ship hull, or armored vehicle shell. Dynamic loads imposed at mid-span might represent a ballistic impact, underwater blast, or blast in air. Figures 18.13a,b refers to an amplification case in the homogeneous and isotropic cantilever beam with two delaminations (Fig. 18.12); the beam is subject to a transverse force applied quasi-statically at the free end. The diagrams, as well as those in Figs 18.14-17, have been obtained using the semi-analytic model formulated in Andrews (2005) and based on the classical model approach described in Section 18.2.1. The energy release rate of the upper crack is strongly amplified by the presence of a lower crack. In addition, a sharp transition in energy release rate occurs when the two delaminations reach the same length. The diagram (b) show the phase angle Ψ = tan–1(KIIU/ KIU), where KIU and KIIU are the mode I and mode II stress intensity factors for the upper crack in the presence of a lower crack of fixed length. The single upper crack solution (in the absence of the lower crack) would give P hU h hL

aL aU L

18.12 Homogeneous cantilever beam with two arbitrarily spaced cracks. © 2008, Woodhead Publishing Limited

1000

100 FEM Proposed model Single crack limit

aU > aL

tan–1 (KIIU/KIU)

90

600 P2

GUEh

800

aU > aL

aU = varied a L = 5h hU = h/3 hL = h/3 isotropic

400 200

80

aU = varied a L = 5h hU = h/3 hL = h/3 isotropic

70 60 50

0 0

1

2

3

4

6

7

8

0

9

1

2

3

4

5 6 a U/ h (b)

7

8

9


aU > aL 90 tan–1 (KIIU/KIU)

800

aU > aL

P2

600

aU = varied a L = 5h hU = h/3 hL = h/3 λ = 0.05, ρ = 5

400 200

80

aU = varied a L = 5h hU = h/3 hL = h/3 λ = 0.05, ρ = 5


60 50

0 0

1

2

3

5 a U/ h (c)

6

7

8

9

0

1

2

3

4

5

a U/ h (d)

6

7

8

545


4

18.13 (a,c) Dimensionless energy release rate of the upper crack in the system of Fig. 18.12 as a function of its length with a lower crack of fixed length. (b,d) Relative 10 amount of mode II to mode I for the upper crack of the system in Fig. 18.12 as a function of its length. (a,b) refer to an isotropic material with Young’s modulus E; (c,d) to a highly orthotropic material with orthotropy ratios λ = Ez /Ex = 0.05 and ρ = E x E z 2G xz – ν zx ν xz = 5, with Ez, Ex, Gxz, νzx and νxz the through-thickness and longitudinal Young’s moduli, the shear modulus and Poisson coefficients (frictionless contact; perfectly 9 brittle material). The single crack limit is shown by the bottom line in (a)-(d).


5 a U/ h (a)

1000

GU E z h

FEM Proposed model

(A)

a U/ h aL/h 0 0.2 0.4 0.6 0.8 t

0.04

10 8 a6 h4 2

0.06

0

1

(C)

0.04

(A)

a U/ h aL/h 0

0.2 0.4 0.6 0.8 t

(B) 1

(C)

(B) 0.02

0.02

hU = 0.25h hL = 0.4h aL = 2.5h L = 10h aU = 5h

hU = 0.2h hL = 0.6h aL = 4h L = 10h aU = 5h 0

0 0

100

200 300 w (x = L )

400

500

0

100

200 w (x = L )

G cr h / E (a)

0.05 (C)

P

G cr Eh

0.04

(A) (B)

0.03

10 8 a6 h4 2 0

0.05

a U/ h aL/h 0 0.2 0.4 0.6 0.8 t

hU = 0.3h hL = 0.38h aL = 4h L = 10h aU = 6h

(A)

0.04

(C) 1

(D)

0.03 (B) 10 8 a 6 h 4 2 0

0.02 0.01

hU = 0.25h hL = 0.25h aL = 6h L = 10h aU = 5h

0 200

400 w (x = L ) G cr h / E (c)


600

800

a U/ h aL/h 0

0 0

400

G cr h / E (b)

0.02 0.01

300

0

200

400 w (x = L ) G cr h / E (d)

0.2 0.4 0.6 0.8 t 1

600

800

18.14 Dimensionless diagrams of the critical load for crack propagation versus load-point displacement in the two crack system of Fig. 18.12. Curves highlight different macrostructural phenomena resulting from local amplification and shielding effects (adapted from Andrews et al., 2006). Crack spacing and initial crack lengths are given in the diagrams; crack growth criterion: GI + GII = Gcr.


P

0.06

G cr Eh

0.08

10 8 a6 4 h 2 0

546

0.08


547

ΨUo = 66,7°, independent of the crack length; in the presence of the lower crack, the mode ratio of the upper crack varies as a function of its length, from ΨUo = 90° (pure mode II) when the upper crack is shorter than the lower crack, down to ΨUo = 62° when the upper crack is longer, and finally approaching the single crack solution when the upper crack is much longer than the lower. Figure 18.13c,d highlights similar phenomena in a highly orthotropic material such as a unidirectional boron-epoxy laminate. The phenomena of shielding and amplification of the fracture parameters induce interesting macrostructural effects in the beam of Fig. 18.12. Cases in which one crack shields the other and cases in which it leads to accelerated growth in its partner can be found, depending on where the two cracks reside in the laminate. The possibility of acceleration implies that design and life predictions based on solutions for a single crack cannot be safe. Figure 18.14 shows exemplary curves of the critical load for crack propagation versus load-point deflection in isotropic and perfectly brittle beams with propagating systems of two delaminations (Fig. 18.12): (a) the curve presents a snapback instability, due to the unstable propagation of the lower delamination, and a local increase in the critical load, due to a negative discontinuity (shielding) in the energy release rate when the two crack tips have the same length; (b) a snap-back instability, due to the unstable propagation of the lower crack, is followed by a sudden drop in the critical load, due to a positive discontinuity in the energy release rate (amplification); (c) a snapthrough instability due to a negative discontinuity in the energy release rate (shielding) leading to a critical load higher than that of initial propagation (strengthening effect); (d) crack pull along where the finite propagation of one of the cracks causes the propagation of the other crack of the system (Andrews et al., 2006). Similar phenomena can be observed in the experimental curve shown in Fig. 18.15 and described in the caption. In homogeneous beams with systems of equal length delaminations, the spacing of the delaminations controls the fracture response and the stability of the equality of length with respect to length perturbations (Andrews et al., 2006, 2008). In cantilever beams with edge delaminations loaded at the free end and clamped-clamped beams with central delaminations subject to concentrated forces at the mid-span, equal length and equally spaced delaminations grow with equal lengths under static and arbitrary dynamic loading conditions; in addition, the equality of length is stable with respect to length perturbations. Systems of unequally spaced delaminations can be characterized by the propagation of a single dominant crack. This behaviour has been synthesized in behavioural maps. The map in Fig. 18.16 has been obtained in closed form for an isotropic, perfectly brittle clamped-clamped beam with two delaminations subject to a quasi-static concentrated force and its validity for arbitrary dynamic loading conditions has been verified numerically in (Andrews et al., 2008). If the position of the delaminations, © 2008, Woodhead Publishing Limited

548

50

Experimental (Robinson et al., 1999)

40

Model (Andrews and Massabò, 2008) (shown as the lower of the two lines.)

20 mm

3.18 mm

20 mm

A

B

0 D

30

20

1.86 mm

100 mm

40 mm

Displacement increment

Load (N)

A

1.59 mm

C

Hexcel HTA913 carbon epoxy Ex = 115 GPa νxy = .29 Ey = 8.5 GPa GIc = 330 J/m2 Gxy = 4.5 GPa GIIc = 800 J/m2

10

B-C 100

D

200

300 400

0 0

5

10

15 20 25 End displacement (mm)

30

35

40

0

20

40

60 80 100 120 140 160 180 Crack tip position (mm)

18.15 Experimental (Robinson et al., 1999) and theoretical curves of the critical reaction load for crack propagation versus load-point displacement in the multiply delaminated DCB specimen shown on the right. At A the load reaches its peak value and the edge delamination begins to propagate; the crack propagates until it reaches the left tip of the inner delamination at point B; at B there is a discontinuity (amplification) in energy release rate leading to a sudden decrease in critical load to point C; the edge crack continues to propagate until it reaches approximately the center of the inner delamination at point D when also the right tip of the inner delamination begins to propagate; simultaneous propagation continues for the remainder of the test (Andrews and Massabò, 2008). Crack growth criterion used in the simulation: GI /GIcr + GII /GIIcr = 1. © 2008, Woodhead Publishing Limited


Width = 20 mm

60

Single and multiple delamination 1

P/2 hU

h

0.8

hL

0.6 hU h

549

a

G U > GL (I) G B > G U, G L

0.4

(II) 0.2

G L > GU (III)

0 0

0.2

0.4

0.6

0.8

1

h L/ h

18.16 Map of regions of different energy release rate for the clamped-clamped beam with two cracks shown in the inset. 9

8

L = 10h a o = 5h P2 GIIc = 2GIc = 100 Eh

P Load

a h

Time

7 P(t)/2

h

6

a L 5 150

200

250

300

tc L /h

18.17 Crack propagation history in the system of three equally spaced delaminations in a clamped-clamped beam subject to a step loading ( P / G IIcr Eh = 0.1; GIcr = 0.5 GIIcr; longitudinal wave speed cL, homogeneous, isotropic, perfectly brittle material, crack growth criterion GI/GIcr + GII/GIIcr ≥ 1).

defined by hU and hL, falls into the gray region, the delaminations will grow with equal lengths and the equality of length will be stable. Figure 18.17 shows the time history diagram of crack propagation for a system of three equally spaced, equal length delaminations subject to a step load. If the position falls into the white regions, delamination growth under quasi-static © 2008, Woodhead Publishing Limited

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loading conditions will be characterized by the propagation of only one of the delaminations of the system. If the load is applied dynamically, the localized growth of one of the cracks of the system may be followed by the growth of the other crack, but always at later times and reduced speed and only if the energy input into the system is large enough (Andrews et al., 2008). Figure 18.18 shows the time history diagrams of crack propagation for a system of two equal length, unequally spaced delaminations subject to triangular pulse forces of different magnitude. Whether a system of multiple equal length delaminations propagate selfsimilarly, while maintaining the equality of length, or with a single delamination growing dominantly, has important consequences on the mechanical behavior and energy absorption characteristics. Under static loading conditions, systems with single propagating delaminations have more brittle post-peak responses (Andrews et al., 2006). Under dynamic loading, preliminary results in (Massabò, 2007) lead to anticipation that homogeneous systems designed so that cracks will form at equal spacing and therefore propagate with equal P(t)/2 hU

hL

0.8

Load

h

1

P t

tp/2

tp

hU h

Plane of symmetry

L

G U > GL (I) G B > G U, GL 0.4 0.6

a L = 10h, a = 5h; hU = 0.1h, hL = 0.3h GIcr = GIIcr = Gcr tPcL/h = 425.7 = first period of vibration

(II) GL > GU (III)

0.2 0 0 (a)

10 8 a 6 h 4 2 0

10 8 6 4 2 0 0

200

400

10 8 6 4 2 0 0

200

400

0.2 0.4 0.6 0.8 h L/ h

1

10 8 6 4 2 0 0

200

400

0

200

400

tcL/h P / G crEh = 0.183 or

P / G crEh = 0.223 or

GcrEh/P 2 = 30 (b)

G crEh/P 2 = 20 (c)

P / G crEh = 0.447 or P / G crEh = 1.0 or G crEh/P 2 = 1.0 G crEh/P 2 = 5.0 (e) (d)

18.18 (a) Geometrical data and position of the crack configuration in the stability map of Fig. 18.16. (b-e) Crack propagation histories of a two delamination system in a clamped-clamped beam subject to a triangular pulse force for different magnitudes of the dimensionless applied load (lower crack, top lines; upper crack, lower lines. Shape and duration of the load are plotted in dotted lines with an arbitrary scale in the ordinate; cL = longitudinal wave speed; homogeneous, isotropic, perfectly brittle material, crack growth criterion GI/GIcr + GII/GIIcr ≥ 1).



551

lengths will have enhanced capabilities in terms of energy absorption through multiple delamination fracture: self-similar growth typically occurs at lower speed, the final size of the damaged areas is smaller and the energy expended into the creation of new surfaces is higher in cases where boundaries exist that prevent the extension of the singly propagating cracks beyond certain lengths.

18.5.3 Dynamic delamination with large-scale bridging Large-scale bridging mechanisms acting along the crack surfaces of delaminated beams loaded quasi-statically strongly affect fracture characteristics, failure modes and macrostructural behavior. These effects are well documented in the literature and a basic understanding of quasi-static large-scale bridging delamination fracture has been developed that includes the derivation of the characteristic length scales of the problem. On the other hand, the influence of such mechanisms on the characteristics of dynamically propagating cracks and the mechanical response of structures loaded dynamically is not completely understood. Preliminary results (Andrews et al., 2008) on the response of beams with single and multiple stationary delaminations subject to dynamic loads show that imposing a large-scale bridging mechanism along the wake of the delaminations reduces the driving force for crack propagation during the loading phase and strongly modifies the free motion that arises after the load has been removed. In the absence of bridging mechanisms, large amplification of the mode I component of the energy release rate and hammering of the crack faces (localized impacts at different times and locations) can occur due to out of phase vibrations of the delaminated sub-laminates. The hammering induces high frequency, high order oscillations in the fracture parameters. Bridging mechanisms similar to those produced by a typical through thickness reinforcement reduce amplification of the mode I component and prevent hammering so that the energy release rate tends to show smooth oscillations associated with waves propagating on the scale of the whole specimen. The diagram in Fig. 18.19 highlights this behaviour in a beam with a single offcenter delamination. In systems with dynamically propagating delaminations, shielding effects similar to those observed for quasi-static fracture can be observed and the behavior is strongly controlled by the crack velocity. A study performed by Sridhar et al. (2002) on the influence of large-scale bridging mechanisms on steady state delamination fracture in wedge loaded mode I DCB specimens, shows that the delamination resistance is considerably enhanced by a typical through thickness reinforcement for small to moderate crack speeds (up to 0.1–0.2 the longitudinal wave speed). On the other hand, for higher velocities the kinetic energy term dominates the overall energetics and the relative © 2008, Woodhead Publishing Limited

552

L 10h a = 5h hU = h/3 tPcL/h = 311

50

P(t)/2

hU

h

40

P2

GEh

a L 30

Load

20 G II-static 10 G I-static

Plane of symmetry

P

G II

tp/2

GI

tp

0 0

200

400

600 tcL/h

800

1000

1200

18.19 Time history diagram of the dimensionless energy release rate of the single off-center delamination in the clamped-clamped beam on the right. The applied load is a triangular pulse force with duration equal to 3/4 of the first natural period of vibration. Solid lines: response of the through-thickness reinforced system; dashed lines: response in the absence of a through thickness reinforcement. Homogeneous, isotropic, perfectly brittle material, cL = longitudinal wave speed. Frictionless contact. © 2008, Woodhead Publishing Limited


60


553

effect of the reinforcement on the delamination resistance becomes insignificant. This behavior is highlighted in the diagram of Fig. 18.20.

18.6

Final remarks

Delamination fracture in the presence of extensive nonlinear crack wake mechanisms cannot be described by Linear Elastic Fracture Mechanics or characterized by a single fracture parameter. Modeling approaches based on the cohesive- and bridged-crack models are required in which the essential material properties are the cohesive/bridging traction laws and the intrinsic fracture toughness of the base material. A cohesive/bridged crack model can be easily implemented in a computational model of the structure, for instance using an interface approach, and the parameters of the model derived by simple and effective engineering fracture tests. Linear Elastic Fracture Mechanics is able to describe only a limiting configuration of the delamination, the small-scale bridging limit, that in typical through-thickness reinforced laminates is not easily approached neither in standard fracture specimens or in typical structural components. The attempt, quite common in the literature on through-thickness reinforced laminates, of 7.0 Delamination crack tip

10 Gb = G Icr

5

F

1

2α

2h

x Bridging zone

5.0

0

a0 l

F Fu

a

µG Icr α 2Eh

3.0

µG Icr α 2Eh

=0 = 0.167

1.0

0.0

1.0

2.0

3.0

α v cL

4.0

5.0

Eh G Icr

18.20 Dimensionless diagram of the force applied at the wedge of a wedge-loaded DCB specimen as a function of the crack speed. Fu is the force required to drive the crack in the absence of large-scale bridging mechanisms, cL is the longitudinal wave speed, µ is the friction coefficient between the wedge and the beam v is the crack speed (adapted from Sridhar et al., 2002). Homogeneous, isotropic, perfectly brittle material. © 2008, Woodhead Publishing Limited

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describing their fracture toughness using a single fracture parameter is incorrect and can lead to misleading conclusions on life prediction and structural performance. Similarly, the fracture response of a through-thickness reinforced laminate cannot be described by an R-curve: R-curves are not material properties and they depend strongly on the geometry of the specimen used to define them. Progress toward standardized design methods for through-thickness reinforced systems, including the design of standard experiments, the formulation of design principles and advances in modeling, will be possible only if correct fracture mechanics is used. In addition, since delamination is often the result of dynamically applied loads, inertia effects should be accounted for and considerations of the regimes of crack speed where a through-thickness reinforcement may or not be effective should be included. The formulation of experimental/computational methodologies to derive the bridging traction law of a through-thickness reinforcement in the dynamic regime and to identify its rate dependency is one of the topics of current research. A cohesive crack model may be required also to study the mechanical response of conventional 2D laminates in all situations where the usual assumption of a perfectly brittle material is not applicable or satisfied, e.g. the problem of crack initiation or crack growth in regimes where the length of the cohesive zone is comparable to the length of the crack. The definition of the cohesive traction laws that describe a conventional 2D laminate is another open problem. Large-scale bridging conditions may also be due to extensive friction along the wake of the cracks. Friction effects dominate problems of dynamic fracture, but friction remains one of the less understood aspects of material behaviour (Cox et al., 2005). The problem of dynamic delamination fracture in the presence of extensive frictions is still unsolved. Delamination damage often occurs in the form of multiple delaminations, which are the typical outcome of severe dynamic loading, such as blast, impact and shock. The presence and interaction of multiple delaminations have important effects on structural response and performance. The presence of other cracks can lead to accelerated growth of a crack. The possibility of acceleration implies that design and life predictions based on solutions for a single crack cannot be safe. Preliminary results recalled in this chapter show that characteristic features of multiple delamination fracture could be exploited to maximize the energy absorption capabilities of laminates and multilayered systems. These ideas have not been explored before and the formulation of design principles for optimal energy absorbing structures and for through-thickness reinforced structures with optimal performance will be the topics of future researches.



18.7

555

Acknowledgement

The author was supported in writing this chapter, and for some of the work described here, by the US Office of Naval Research through contract no. N00014-05-1-0098, administered by Dr Y. D. S. Rajapakse.

18.8

References

Andrews, M.G. (2005), The Static and Dynamic Interaction of Multiple Delaminations in Plates Subject to Cylindrical Bending, Dissertation, PhD Degree, Northwestern University, Evanston, IL, USA. Andrews, M.G. and Massabò, R. (2007), The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers, Eng. Fracture Mechanics, 74, 2700–2720. Andrews, M.G. and Massabò, R. (2008), Delamination in flat sheet geometries in the presence of material imperfections and thickness variations, Composites Part B, 39, 139–150, special issue on Marine Composites. Andrews, M.G., Massabò, R., and Cox, B.N. (2006), Elastic interaction of multiple delaminations in plates subject to cylindrical bending, International Journal of Solids and Structures, 43(5), 855–886. Andrews, M.G., Massabò, R., Cavicchi, A. and Cox, B.N. (2008), The dynamic interaction of multiple delaminations in plates subject to cylindrical bending, International Journal of Solids and Structures (in press). Aveston, J., Cooper, G.A. and Kelly, A. (1971), Single and multiple fracture. In The properties of fiber composites, Conf. Proc., National Physical Laboratory, IPC Science and Technology Press Ltd., 15–24. Bao, G. and Suo, Z. (1992), Remarks of crack-bridging concepts, Applied Mechanics Review, 24, 355–366. Barenblatt, G.I. (1959), The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks, J. Applied Mathematics and Mechanics, 23, 622–636. Bilby, B.A., Cottrell, A.H. and Swinden, K.H. (1963), The spread of plastic yield from a notch, Proceedings Royal Society London, A272, 304–314. Brandinelli, L. and Massabò, R. (2006), Mode II Weight Functions for isotropic and orthotropic Double Cantilever Beams, Int. Journal of Fracture, 139, 1–25. Carpinteri, A. (1989), Cusp catastrophe interpretation of fracture instability, J. Mechanics Physics Solids, 37, 567–582. Cartié, D.D.R. and Partridge, I.K. (1999), Delamination Behaviour of z-Pinned Laminates, Proc. ICCM12, Paris, July, 1999, ed., T. Massard, Woodhead Publishing Limited, Melbourne. Cartie, D.D.R., Cox, B.N. and Fleck, N.A. (2004), Mechanisms of crack bridging by composite and metallic rods Composites Part A, 35(11), 1325–1336. Chang, P., Mouritz, A.P. and Cox, B.N. (2006), Properties and failure mechanisms of zpinned laminates in monotonic and cyclic tension, Composites Part A, 37(10), 1501– 1513. Chang, P., Mouritz, A.P. and Cox, B.N. (2007), Flexural properties of z-pinned laminates, Composites Part A, 38(2), 244–251. Cottrell, A.H. (1963), Mechanics of fracture, in Tewksbury Symposium of Fracture, University of Melbourne, Australia, 1–27. © 2008, Woodhead Publishing Limited

556


Cox, B.N. (1999a), Mechanisms and models for delamination in the presence of throughthickness reinforcement, Advanced Composites Letters, 8, 249–256. Cox, B.N. (1999b), Constitutive model for a fiber tow bridging a delamination crack, Mechanics of Composite Materials and Structures, 6, 117–138. Cox, B.N. and Marshall, D.B. (1991), The determination of crack bridging forces, Int. Journal of Fracture, 49, 159–176. Cox, B.N. and Marshall, D.B. (1994), Concepts for bridged cracks in fracture and fatigue, Acta Metall. Mater., 42, 341–363. Cox, B.N. and Sridhar, N. (2002), A traction law for inclined fiber tows bridging mixedmode cracks, Mechanics of Advanced Materials and Structures, 9(4), 299–331. Cox, B.N., Massabò, R., Mumm, D.R., Turrettini, A. and Kedward, K. (1997), Delamination fracture in the presence of through-thickness reinforcement, in Proc. 11th Int. Conf. Composite Materials, Gold Coast, Australia, 1997, ed. M. L. Scott, Woodhead Publishing, Melbourne, 159–177. Cox, B.N., Gao, H.J., Gross, D. and Rittel, D. (2005), Modern topics and challenges in dynamic fracture, Journal of the Mechanics and Physics of Solids, 53(3), 565–596. Dransfield, K., Baillie, C. and Mai, Y.-W. (1994), Improving the delamination resistance of CFRP by stitching – a review, Composite Science and Technology, 50, 305–317. Du, J., Hawkins, N.M. and Kobayashi, A.S. (1989), A hybrid analysis of fracture process zone in concrete, in Fracture of Concrete and Rock. Recent Developments, S.P. Shah, S.E. Swartz and B. Barr, eds., Elsevier Applied Science, Cambridge, UK. Dugdale, D.S. (1960), Yielding of steel sheets containing slits, J. Mechanics Physics Solids, 8, 100–104. Evans, D.A. and Boyce, J.S. (1989), Transverse reinforcement methods for improved delamination resistance, Intl. SAMPE Symp. and Expos., 34, 271–282. Fett, T. (1995), Determination of bridging stresses and R-curves from load-displacement curves, Engineering Fracture Mechanics, 52(5), 803–810. Fett, T., Munz, D., Yu, C-T. and Kobayashi, A.S. (1994), Determination of bridging stresses in reinforced Al2O3, J. of Am. Ceram. Soc., 77(12), 3267–3269. Freitas, G., Magee, C., Dardzinski, P. and Fusco, T. (1994), Fiber Insertion Process for Improved Damage Tolerance in Aircraft Laminates, J. Adv. Matls. 25(4), 36–43. Freitas, G., Fusco, T., Campbell, T., Harris, J. and Rosenberg, S. (1996), Z-fiber technology and products for enhancing composite design, AGARD Conference Proceedings, 17, 1–8. Guinea, G.V., Planas, J. and Elices, M. (1994), A general bilinear fit for the softening curve of concrete, Materials and Structures, 27, 99–105. Guo, Z.K., Kobayashi, A.S. and Hawkins, N.M. (1993), Further studies on fracture process zone for mode I concrete fracture, Engineering Fracture Mechanics, 46(6), 1041– 1049. Hillerborg, A., Modeer, M. and Petersson, P.E. (1976), Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and Concrete Research, 6, 773–782. Hutchinson, J.W. and Suo, Z.G. (1992), Mixed-Mode Cracking in Layered Materials, Adv. Appl. Mech., 29, 63–191. Jain, L.K. and Mai, Y.-W. (1994), Analysis of stitched laminated ENF specimens for interlaminar mode-II fracture toughness, Int. J. Fracture, 68, 219–244. Larsson, P.L. (1991), On multiple delamination buckling and growth in composite plates, Int. J. Solids Structures, 27(13), 1623–1637.



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Li, S., Wang, J. and Thouless, M.D. (2004), The effects of shear on delamination in layered materials, J. Mech. Phys. Solids, 52(1), 193–214. Li, V.C., Wang, Y. and Backers, S. (1991), A micromechanical model of tension-softening and bridging toughening of short random fiber reinforced brittle matrix composites, J. Mechanics Physics Solids, 39, 607–625. Massabò, R. (1999), The bridged-crack model, in Nonlinear Crack Models for Nonmetallic Materials, (ed. A. Carpinteri), Solid Mechanics and its Applications Series (ed. G. Gladwell), Kluwer Academic Publisher (ISBN 0-7023-5750-7), Dordrecht, The Netherlands, pp. 141–208. Massabò, R. (2007), Dynamic interaction of multiple damage mechanisms in multilayered systems, Proceedings of the Italian Conference of Theoretical and Applied Mechanics, Brescia, September 2007, CDRom. Massabò, R. and Cox, B.N. (1999), Concepts for bridged mode II delamination cracks, Journal of the Mechanics and Physics of Solids, 47(6), 1265–1300. Massabò, R. and Cox, B.N. (2001), Unusual characteristics of mixed mode delamination fracture in the presence of large scale bridging, Mechanics of Composite Materials and Structures, 8(1), 61–80. Massabò, R., Mumm, D. and Cox, B.N. (1998), Characterizing mode II delamination cracks in stitched composites, International Journal of Fracture, 92(1), 1–38. Massabò, R., Brandinelli, L. and Cox, B.N. (2003), Mode I weight functions for an orthotropic double cantilever beam, International Journal of Engineering Science, 41, 1497–1518. Mouritz, A. and Cox, B.N. (2000), A mechanistic approach to the properties of stitched laminates, Composites, A31, 1–27. Mouritz, A.P. and Jain, L.K. (1999), Further validation of the Jain and Mai models for interlaminae fracture of stitched composites, Composites Science and Technology, 59(11), 1653–1662. Nanakorn, P. and Horii, H. (1996), Back analysis of tension-softening relationship of concrete, J. Materials, Conc. Struct., Pavements, JSCE, 265–275. Rice, J.R. (1968), A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Applied Mechanics, 35, 379–386. Robinson, P., Besant, T. and Hitchings, D. (1999), Delamination growth prediction using a finite element approach. 2nd ESIS TC4 Conference on Polymers and Composites, Les Diablerets, Switzerland. Rödel, J., Kelly, J.F. and Lawn, B.R. (1990), In situ measurements of bridged crack interfaces in the scanning electron microscope, J. of Am. Ceram. Soc., 73(11), 3313– 3318. Rugg, K., Cox, B.N., Ward, K. and Sherrick, G.O. (1998), Damage mechanisms for angled through-thickness rod reinforcement in carbon-epoxy laminates, Composites Part A, 29A, 1603–1613. Rugg, K.L., Cox, B.N. and Massabò, R. (2002), Mixed mode delamination of polymer composite laminates reinforced through the thickness by z-fibers, Composites, part A, 33(2), 177–190. Smith, E. (1989), The size of the fully developed softening zone associated with a crack in a strain-softening material – I. A semi-infinite crack in a remotely loaded infinite solid, Int. J. Engineering Science, 27, 301–307. Sridhar, N., Massabò, R., Cox, B.N. and Beyerlein, I. (2002), Delamination dynamics in through-thickness reinforced laminates with application to DCB specimen, International Journal of Fracture, 118, 119–144. © 2008, Woodhead Publishing Limited

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Suemasu, H. (1993), Postbuckling behaviors of composite panels with multiple delaminations, J. of Compos. Mater., 27(11), 1077–1096. Suemasu, H. and Majima, O. (1996), Multiple delaminations and their severity in circular axisymmetric plates subjected to transverse loading, J. of Compos. Mater., 30(4), 441–463. Suo, Z., Bao, G. and Fan, B. (1992), Delamination R-Curve phenomena due to damage, J. Mech. Phys. Solids, 40, 1–16. Tay, T.E. (2003), Characterization and analysis of delamination fracture in composites: An overview of developments from 1990 to 2001, Appl. Mech. Rev., 56, 1–31. Turrettini, A. (1996), An investigation of the Mode I and Mode II stitch bridging laws in stitched polymer composites, Masters Thesis, Department of Mechanical and Environmental Engineering, University of California, Santa Barbara. Uchida, Y., Kurihara, N., Rokugo, K. and Koyanagi, W. (1995), Determination of tension softening diagrams of various kinds of concrete by means of numerical analysis, in proc. II Int. Conference on Fracture Mechanics of Concrete Structures, FRAMCOS 2, Zurich, Switzerland; F.H. Wittman, ed., Aedificatio Publisher, vol. I, 17–30. Willis, J.R. (1967), A comparison of the fracture criteria of Griffith and Barenblatt, J. Mechanics Physics Solids, 15, 151–162. Wittmann, F.H., Roelfstra, P.E., Mihashi, H., Huang, Y.Y., Zhang, X.H. and Nomura, N. (1987), Influence of age of concrete as determined by means of compact tension specimens, Materials and Structures, 20, 103–110. Wittmann, F.H., Rokugo, K., Bruhwiler, E., Mihashi, H. and Simoni, P. (1988), Fracture energy and strain-softening of concrete as determined by means of compact tension specimens, Materials and Structures, 21, 21–32. Yang, Q. and Cox, B.N. (2005), Cohesive zone models for damage evolution in laminated composites, International Journal of Fracture, 133(2), 107–137. Zheng, S. and Sun, C.T. (1998), Delamination Interaction in Laminated Structures, Eng. Fracture Mech., 59(2), 225–240.


Part V Analysis of structural performance in the presence of delamination, and prevention/ mitigation of delamination


19 Determination of delamination damage in composites under impact loads A F J O H N S O N and N T O S O - P E N T E C Ô T E, German Aerospace Centre (DLR), Germany

19.1

Introduction

This chapter describes the influence of delamination on the development of impact damage in composite plates and stiffened shells, from both experimental and theoretical considerations. Recent progress on materials modelling and numerical simulation of composite shell structures subjected to impact loads is presented and used to study the role of delamination under impact. A Continuum Damage Mechanics (CDM) model for unidirectional (UD) fibre reinforced composites is applied to model both in-ply damage and delamination failure during impact loading. The CDM model has been implemented in a commercial explicit finite element (FE) code in which a laminate is modelled by stacked shell elements with cohesive interfaces. Delamination failure is modelled by allowing the interfaces to damage and fracture when a delamination failure energy criterion is reached. The application of the code is described here to predict damage in composite panels subjected to low velocity drop tower and high velocity gas gun impact tests by steel impactors under a range of test conditions. A comparison of structural response and failure modes from numerical simulations and impact tests shows good agreement for the prediction of delamination damage and penetration at higher impact energies. To reduce development and certification costs for composite aircraft structures, efficient computational methods are required by the industry to predict structural integrity and failure under dynamic loads, such as crash and impact. Failure in polymer composites is initiated at the microscopic level, with length scales governed by fibre diameters, whilst the length scale of aircraft structures is in metres, which poses a severe challenge for FE analyses of composite structures. By using meso-scale models based on continuum damage mechanics (CDM), proposed by Ladevèze and co-workers [1] – [3], it is possible to define materials models for FE codes at the structural macro level which embody the salient micromechanics failure behaviour. CDM provides a framework within which in-ply and delamination failures 561 © 2008, Woodhead Publishing Limited

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may be modelled. In previous work by the authors [4] a ply failure model was developed for fabric reinforced plies with three scalar damage parameters representing ply microdamage, and damage evolution equations were introduced relating the damage parameters to strain energy release rates in the ply. A cohesive interface delamination model was developed by applying the CDM framework to the ply interface, as described in [2]. Failure at the interface is modelled by degrading stresses using two interface damage parameters corresponding to interfacial tension and shear failures, whilst fracture mechanics concepts are introduced as in [5] by relating the total energy absorbed in the damaging process to the interfacial fracture energy. The UD and fabric ply CDM models were implemented into shell elements and with delamination models have been introduced into a commercial explicit FE crash and impact code [6], which uses a numerical approach for delamination modelling based on stacked shell elements with cohesive interfaces. In this way ply and delamination failures in large composite structures may be modelled efficiently with stacked shells, avoiding both fine mesh solid models and interface elements. The reason for using shell elements to model impact damage is that composite aircraft structures are large thin-walled structures and detailed solid modelling in the damage regions with solid element dimensions typically equal to the ply thickness of 0.1 mm are not tractable. This raises the question of the suitability of a shell formulation to model impact damage and penetration. The authors show this is possible for the low and intermediate velocity impacts relevant to civil aircraft applications. Impact in a thin-walled structure causes through-thickness compression and shear waves, together with shell bending waves propagating outward from the impact point. If the impact velocity is high enough for the through-thickness waves to cause local fracture at the front or back face, the projectile passes through and there is no time for the bending waves to develop. This is the case for high velocity or ballistic impacts, where there is local fracture and penetration at the impact point. Lower velocity impacts may be defined as those with no local fracture, since the through-thickness failure strains are not reached. Then shell bending waves have time to develop and propagate, which may cause structural damage away from the impact point. In this case the composites shell models used here are appropriate, provided they take account of local damage in the plies and delamination damage. Shell models would be less appropriate for modelling ballistic penetration from hard projectiles, where through-thickness shear failures are dominant and solid models are needed. What is the critical impact velocity for composite structures? For rigid body impact with velocity V the initial compressive strain is approximately given as ε = V/c, where c is the through-thickness sound speed. On the basis of a through-thickness failure strain of 2% in a typical composite laminate, this gives a critical impact velocity V of about 40 m/s. For soft body impacts from birds, rubber or ice © 2008, Woodhead Publishing Limited

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the critical velocity would be much higher, since contact forces are reduced. Thus it is considered that the shell formulation discussed here is appropriate for low to intermediate impact velocities in civil aircraft structures where shell bending waves are observed and damage develops in the region surrounding the impact point. This contribution describes recent work on code validation in which impact simulations are compared with test data from impact tests of composite panel structures impacted with hard impactors. Emphasis is placed on the role of delamination damage in modifying the impact response and failure mode of the composite structure. The meso-scale ply damage models for UD composites are summarised in Section 19.2 along with the cohesive interface delamination model. Impact tests on composite plates in drop tower tests with large impacting masses at low velocities are discussed in Section 19.3 and gas gun tests at higher velocities ca. 100 m/s on composite panels in Section 19.4 which are in the intermediate velocity region discussed above These represent impact conditions relevant to civil aircraft structures ranging from tool drop up to foreign object damage (FOD). Test conditions and impact energies were chosen to give both delamination failures and penetration with fibre damage. Simulation results are presented which successfully predict composites failure modes and failure progression during impact from hard bodies in composite panels. Section 19.5 completes the chapter with the current status and future outlook for FE code developments for predicting impact damage in composite structures.

19.2

Composites failure modelling

19.2.1 Delamination model The two traditional methods of modelling interface failures in composites are to apply a material strength condition to the interface, which could be a fracture or yield stress at which the interface fails, and the fracture mechanics method which predicts how an existing flaw or crack at the interface will propagate. Neither method can model both the damage development causing interface cracks and then crack propagation leading to delamination failures. This has been studied more recently within the general damage mechanics formulation for composites proposed by Ladevèze and reviewed in [1]. By applying CDM principles to an elastic interface assumed to be infinitesimally thin as shown in Fig. 19.1, damage evolution equations are derived in [2] for through-thickness tensile damage and interlaminar shear damage. The interface damage model has been applied to analyse fracture mechanics tests in [3], and it is shown how to determine model parameters from fracture mechanics test data. The starting point is the general constitutive law for an orthotropic elastic material with damage. Consider the case when only the through-


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thickness stress components are non-zero. Referred to the materials’ orthotropic symmetry axes (x1, x2, x3), in which fibre reinforcements are in the form of thin plies in the (x1, x2)-plane, with x3 the through-thickness direction, throughthickness stress and strain components may be written as the vectors:

D = (σ13, σ23, σ33)T, D = (2ε13, 2ε23, ε33)T

19.1

and the orthotropic elastic constitutive equation becomes:

D = SD D

19.2

where the through-thickness elastic compliance matrix SD is assumed to have the form: 0 0   1/ G13 (1 – d13 ) SD =  0 1/ G23 (1 – d 23 ) 0   0 0 1/ E3 (1 – d 3 ) 

19.3

Here E3 is the through-thickness Young’s modulus and G13 and G23 the through-thickness shear moduli in the 1 and 2 directions. The damage parameters d3, d13 and d23 represent reductions in the corresponding throughthickness moduli due to degradation in the interface ply. Delamination failures take place in very thin layers between composite plies. In the delamination model it is assumed that the through-thickness stresses are constant through the layer thickness, and that instead of strains the layer kinematics is based on the jump in displacements across the thin layer. Thus the thin layer is modelled as a 2-D surface or interface, as shown schematically in Fig. 19.1. Such a formulation is then appropriate for numerical implementation as a tied interface or interface element between solid or laminated shell elements in FE codes. The constitutive law relating interface stresses and displacements is conveniently obtained from Equations 19.1– 19.3 above, applied to a layer of thickness h in the limit as h → 0. Let u + (x1, x2) be the displacement vector on the lower face of the upper ply in Fig. 19.1, and u – (x1, x2) the displacement on the upper face of the lower Upper ply + Interface

X3 X2 X1 Lower ply –

19.1 Schematic view of interface ply.



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ply, then the jump in displacement across the interface u = (u1, u2, u3) is defined as: u (x1, x2) = u + – u –

19.4

From the standard strain-displacement relations, the through-thickness strain components may be approximated as:

ε13 = 1/2 u1/h, ε23 = 1/2 u2/h, ε33 = u3/h

19.5

and by substitution in the layer stress-strain Equations 19.2 and 19.3 the interface stress-displacement relations are obtained in the limit as h → 0:

σ13 = k1 (1 – d13) u1, σ23 = k2 (1 – d23) u2, σ33 = k3 (1 – d3) u3 19.6 where k1, k2, k3 are the interface shear and normal stiffnesses defined by: k1 = G13/h, k2 = G23/h, k3 = E3/h

19.7

Equations 19.6 are the constitutive equations for an elastic damaging interface, relating the jumps in shear and normal displacement across the interface to the applied surface tractions and the interface stiffnesses. The interface damage parameters d13, d23 and d3 are assumed to have values in the range 0 ≤ di ≤ 1, which allows degradation at the interface during loading to be described. The model is completed by evolution equations for the interface damage parameters, which may be derived using the CDM framework described in [1]. This is based on an expression for the interface energy, which after partial differentiation by the damage parameters in [2] leads to the introduction of three thermodynamic ‘driving forces’ or damage energy release rate parameters. The damage evolution equations are obtained by assuming that the damage parameters are functions of these damage energy release rates. In [3] the latter parameters are shown to be connected to the delamination fracture energies GIC, GIIC, GIIIC under mode I, mode II and mode III interface failures. The interface model now consists of constitutive Equations 19.6 with damage evolution equations in which it is assumed that d13, d23 and d3 are functions of GIC, GIIC, GIIIC. The choice of this functional dependence may be based on interface test data or idealised physical models of the interface. The exact form chosen for the damage evolution equations here follows the approach of Crisfield et al. [5]. The equations of the model are given first for the case of mode I tensile failure at an interface, where it is assumed that tensile stress – displacement relation σ33 – u3 has the idealised triangular form shown in Fig. 19.2, where u30, u3m correspond to the displacement at the peak stress σ33m and at ultimate failure of the interface. On assuming the damage evolution equation for d3:


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d3 = 0, for 0 ≤ u3 ≤ u30, d3 = c1 (1 – u30/u3), for u30 ≤ u3 ≤ u3m 19.8 with the constants c1 = u3m/(u3m – u30), u30 = σ33m/k3, u3m = 2GIC/σ33m

19.9

it can be verified that the stress-displacement function (19.6)3 has the triangular form shown in Fig. 19.2. This damage evolution equation contains two independent parameters σ33m and GIC, the critical fracture energy under mode I interface fracture. From Equations 19.8 and 19.9 it can be shown that the area under the curve in Fig. 19.1 is equal to the fracture energy GIC. This interface model therefore represents an initially elastic interface, which is progressively degraded after reaching a maximum tensile failure stress σ33m so that the mode I fracture energy is fully absorbed at separation. For mode II interface shear fracture a similar damage interface law to Equation 19.8, Equation 19.9 is assumed for d13, with equivalent set of damage constants, u130, u13m and critical longitudinal shear fracture energy GIIC. An equivalent expression could be used for d23 to model transverse shear fracture at the interface. For mode I and mode II interply failures the interface energies GI , GII are defined as:

GI =

∫

u3

σ 33 du3

0

G II =

∫

u1

σ 13 du1

19.10

0

In general there will be some form of mixed mode delamination failure involving both shear and tensile failure. This is incorporated in the model by assuming a mixed mode failure condition [5], which for mode I/mode II coupling could be represented by the interface failure envelope:

GIC

Stress

σ 33m

Unload/reload

u3O

Displacement

u3m

19.2 Idealised mode I interface stress-displacement function.



 GI   G IC 

n

567

n

G +  II  = e D ≤ 1  G IIC 

19.11

where GI and GII are the monitored interfacial energies in modes 1 and 2 respectively, GIC and GIIC are the corresponding critical fracture energies and the constant n is chosen to fit the mixed mode fracture test data. Typically n is found to be between 1 and 2. Failure at the interface is imposed by degrading stresses when eD < 1 using Equation 19.8 and the corresponding shear relation. When eD ≥ 1 in Equation 19.11 there is delamination and the interface separates. Equation 19.11 could be generalised further to include coupling with mode III failure. However, the code implementation of the delamination referred to below is based only on modeI/modeII coupling.

19.2.2 Composite ply failure model The composite laminate is modelled by layered shell elements or stacked shells joined by cohesive interfaces which may fail by delamination, as depicted schematically in Fig. 19.1. The shells are composed of composite plies which are modelled as a homogeneous orthotropic elastic or elasticplastic damaging material whose properties are degraded on loading by microcracking prior to ultimate failure. Following Ladevèze [1] a CDM formulation is used in which ply degradation parameters are internal state variables which are governed by damage evolution equations. Constitutive laws for orthotropic elastic materials with internal damage parameters take the general form:

e = S

19.12

where and are vectors of stress and elastic strain, and S is the elastic compliance matrix. For shell elements a plane stress formulation with orthotropic symmetry axes (x1, x2) is required. The in-plane stress and strain components are: e

= (σ 11 , σ 22 , σ 12 ) T

e e e T e = ( ε 11 , ε 22 , 2 ε 12 )

19.13

Using a strain equivalent damage mechanics formulation, the plane stress elastic compliance matrix S may then be written: – ν 12 / E1 0   1/ E1 (1 – d1 ) S =  – ν 12 / E1 1/ E 2 (1 – d 2 ) 0   0 0 1/ G12 (1 – d12 ) 

19.14

where E1, E2, are the initial (undamaged) Young’s moduli in the fibre and transverse fibre directions, and G12 is the (undamaged) in-plane shear modulus. The principal Poisson’s ratio ν12 is assumed here not to be independently © 2008, Woodhead Publishing Limited

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degraded. The ply model introduces three scalar damage parameters d1, d2, d12 which have values 0 ≤ di < 1 and represent modulus reductions under different loading conditions due to microdamage in the material. For UD plies d1 and d2 are associated with damage or failure in fibre longitudinal and transverse directions, for fabric plies in the principal fibre directions, and d12 controls in-plane shear damage. In the general damage mechanics formulation [1] ‘conjugate forces’ or damage energy release rates Y1, Y2, Y12 are introduced corresponding to ‘driving’ mechanisms for materials damage, and it is shown that with the compliance matrix Equation 19.14 they take the form: 2 2 Y1 = σ 11 /(2 E1 (1 – d1 ) 2 ), Y2 = σ 22 /(2 E 2 (1 – d 2 ) 2 ), 2 Y12 = σ 12 /(2 G12 (1 – d12 ) 2 )

19.15

The ply model is completed by assuming damage evolution equations in which the three ply damage parameters d1, d2, d12 are functions of Y1, Y2, Y12. Specific forms for these functions are postulated based on study of ply specimen test data. The formulation of the damage evolution equations in the Ladevèze CDM models is physically based and allows generalisations to include features such as shear plasticity and rate dependence. Test data on unidirectional (UD) carbon fibre reinforced epoxy presented in [7] show that damage evolution equations for the transverse and shear damage d2, d12 are coupled through a linear dependence on √(Y2) and √(Y12). Test data on carbon and glass fabric/epoxy materials [4] lead to damage evolution equations in which fibre tension/ compression damage parameters d1, d2 are elastic damaging and linear in √(Y1) and √(Y2) respectively, but decoupled from elastic-plastic ply shear damage in which d12 is a nonlinear function of √(Y12). For in-plane shear, ply deformations are controlled by matrix behaviour which may be inelastic, or irreversible, due to the presence of extensive matrix cracking or plasticity. On unloading this can lead to permanent deformations in the ply. The extension of the meso-scale model to include these irreversible damage effects is based on the assumption that the total strain is written as the sum of elastic e and plastic strains p ( = e + p). The elastic strain components are given by the ply elastic-damage model defined in Equations 19.12–19.14 with appropriate damage evolution functions. The plastic strains are associated with the matrix dominated in-plane shear and transverse behaviour in UD plies. A classical plasticity model is used with an elastic domain function and hardening law applied to the ‘effective’ stresses in the damaged material. Cyclic loading tests are performed in which both the elastic and irreversible plastic strains are measured where the accumulated effective plastic strain p is determined over the complete loading cycle. The model is completed by specifying the plastic hardening function R(p). A typical form which models test data fairly well is an index function, which leads here to the general equation: © 2008, Woodhead Publishing Limited


R(p) = β pm

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19.16

so that the ply plasticity model for fabric plies depends on the parameters β, the power index m and the yield stress Ro. For UD plies there is an additional shear/transverse damage coupling parameter to determine. Ultimate ply failure is controlled by setting the damage parameters di = 1, at threshold energy values Y1f, Y2f, Y12f. From the definitions for the Yi in Equation 19.15 these ultimate failure conditions may be simply related to ply failure stresses or strains. Additionally the model allows well-known multiaxial ply failure criteria such as maximum strain, modified puck, etc., to be applied. In this case the CDM model is used to describe ply degradation such as matrix cracking, whilst ply ultimate failure may be controlled by fibre fracture. This is achieved by setting all the di equal to 1 when the multiaxial failure envelope is reached. Further details of the ply damage and plasticity model, the tests required to determine the required parameters, and the failure models are given in [7] and [8].

19.2.3 Code implementation and validation In order to apply the composites failure models developed above in the analysis of composite structures it is necessary to implement and validate the models in a suitable FE code. In recent years explicit FE methods have proved successful for the analysis of dynamic, highly non-linear problems, particularly where contact plays an important role. In collaboration with the software company Engineering Systems International the CDM ply and delamination models were implemented in the commercial explicit crash and impact code PAM-CRASHTM [6]. The code uses a bilinear four-node quadrilateral isoparametric shell element due to Belytschko with uniform reduced integration in bending and shear. Hourglass control is applied to compensate for the under-integration. A central difference explicit integration scheme is used in time with geometrical nonlinearities accounted for in an updated Lagrangian scheme with co-rotational description. A Mindlin-Reissner shell formulation is used with a layered shell description to model a composite ply, a sublaminate or the complete laminate, depending on the detail required. The layered shells contain one integration point per ply, so that at least four plies are required in a layered shell for the correct bending stiffness. The Mindlin-Reissner shell formulation contains an interpolated transverse shear strain, hence it is suitable for modelling moderately thick shell element stiffness behaviour. However, this shell formulation is not appropriate for modelling through-thickness penetration during impact, as in ballistic impact with hard projectiles which is dominated by through-thickness shear failure. A novel approach has been developed to implement the cohesive interface model of Section 19.2.1 into the PAM-CRASHTM code, in which the laminate


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is treated as a stack of shell elements. Each ply or sublaminate ply group is represented by a set of layered shell elements and the individual sublaminate shells are connected together by a cohesive interface with an interface tractiondisplacement law. The interface is modelled numerically as a contact constraint not an interface element in which a penalty force procedure is used to compute contact forces between adjacent shells. Contact is broken when the interface energy dissipated reaches the mixed mode delamination energy criteria. This ‘stacked shell’ approach is an efficient way of modelling delamination, with the advantage that the critical integration timestep is relatively large since it depends on the area size of the shell elements not on the interply thickness. Full details of the implementation and validation of the delamination model as a cohesive interface with failure between stacked shell elements is given in the paper by Greve and Pickett [9]. In order to model impacts the code has to simulate the contact and penetration between projectile and composite structure. In explicit FE codes this is achieved by use of contact interfaces between the projectile and each of the stacked shells which model the laminate. Between the upper composite shell and the projectile, which is usually modelled by solid elements, an interface thickness is defined. The contact interface uses penalty function methods to apply forces to prevent penetration of the projectile through the shell surface when it is within this thickness. The sliding friction between projectile and composite may also be included as a friction coefficient in the interface model. If the outer shell were fractured and damaged elements eliminated, the projectile will then come into contact with the second stacked shell with a similar contact interface. Impactor penetration is modelled in the stacked shells by elimination of damaged shell elements. The ply damage model discussed in Section 19.2.2 degrades the ply in-plane properties as the damage energy increases. Then at a critical point the damage parameters are set to a value 1 when the ply is eliminated from the layered shell element. When all plies in the element reach their failure point, this element is usually eliminated from the model. In this way it is possible for a projectile to ‘penetrate’ an outer shell, then make contact with the second shell in the stacked shell model of the composite laminate, which may also be eliminated if the plies reach their failure condition.

19.3

Delamination damage in low velocity impact

19.3.1 Drop tower impact tests on composite plates In order to provide test data for validation of the impact models with UD laminates where there is interaction between UD ply and delamination failures, a set of low velocity impact tests were carried out on cross-ply UD carbon/ epoxy plates within the EU HICAS project [10]. Figure 19.3 depicts the



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(a)

(b)

19.3 DLR drop tower impact test rig, showing damage to cross-ply plates (a) Plate CF0901: Fracture at impact; (b) Plate CF0902: Rear face damage; (c) Plate CF0903: Rear face damage.


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(c)

19.3 (Continued)

DLR drop tower test facility showing the support frame for the flat plates and the spherical indenter attached to the falling mass. During the test the force-time pulse and the vertical displacement-time at the impact head are measured and stored in a transient recorder. The plates tested had dimensions 300 mm × 300 mm. The composite was UD carbon/epoxy with cross-ply lay-up [0/90]20 giving a nominal plate thickness of 5 mm. A square steel frame with an inner diameter of 250 mm × 250 mm was used as support frame. A steel sphere with a diameter of 50 mm was used as impactor head, which was attached to the mass and guides giving a total impactor mass of 21 kg. Three tests were carried out with different normal impact velocities: 2.52 m/s, 3.11 m/s and 4.44 m/s. The impact velocities were chosen to give impact energy levels in the range 67–207 J so that extensive plate damage was observed with failure modes ranging from delamination to complete penetration. Test conditions and results are summarised in Table 19.1. Plate CF0901 was impacted at 4.44 m/s which led to complete fracture of the plate and impactor penetration (Fig. 19.3a). At 3.11 m/s, the impactor rebounded in Plate CF0902 which caused rear face cracking (Fig. 19.3b) with extensive delamination. The impactor also rebounded in Plate CF0903 at 2.52 m/s causing minor rear face damage with less delamination (Fig. 19.3c). Ultrasonic C-scans were carried out to determine the extent of delamination damage and Fig. 19.4 compares the size of the delaminated regions for the two rebound tests, showing clearly the greater delamination area after impact at a higher velocity. Test results indicate that for this 5 mm thick cross-ply UD carbon/epoxy plate with a steel ball impactor the threshold energy for significant damage © 2008, Woodhead Publishing Limited


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Table 19.1 Summary results of drop tower plate impact tests Test specimen

Test No.

Mass kg

V0 m/s

Initial Absorbed energy J energy J

Peak load

Max displ. mm failure mode

UD cross-ply Carbon/epoxy plate 50 mm diam. steel ball impact

CF0901 CF0902

21 21

4.44 3.11

207 102

160 70

7.5 9.0

45 - fracture 23 - rebound

CF0903

21

2.52

67

46

9.0

17 - rebound

is about 50 J in which nearly all the dissipated energy goes into delamination. By contrast in the high energy test failure was very brittle, with little delamination, and probably most of the absorbed energy of 160 J went into fibre fracture. This shows that the higher energy impacts with ‘massive damage’ are dominated by ply in-plane strength properties, whilst in the low energy impacts with extensive delamination it is through-thickness properties which are dominant. It is apparent in the impact test series that delamination damage is influencing the plate failure mode and failure load levels.

19.3.2 FE simulation of composite plate impact damage For the simulation a stacked shell laminate model is used with 40 plies. However, in order to reduce CPU time the laminate was modelled as eight sub-laminates of shell elements connected by cohesive interfaces. This model allows delamination at seven interfaces in the plate. A half-plate is modelled with 14 400 layered shell elements. The composite plate is simply supported over the frame and impacted at its centre with the sphere, modelled as a rigid impactor. Numerical results were presented in [11] for three impact test cases in Table 19.1. The ply failure model used was a ‘bi-phase’ model available in PAM-CRASHTM, which is a simplified form of CDM model as discussed in [11] in which the damage parameters depend on the shear strain invariant. Ply properties and delamination interface parameters required by the models were taken from HICAS test data [10] and DLR in-house sources. Simulation of test CF0901 at 4.44 m/s impact predicted the penetration of the plate by the impactor as seen in Fig. 19.3a. Nevertheless, the fracture of the plate into four pieces could not be assessed numerically. Simulation of test CF0903 at the lowest impact velocity 2.52 m/s successfully predicted the impactor rebound with delamination and some ply damage without penetration. The size of delamination predicted at the plate middle plane was also found to be in good agreement with the results of the C-scan test shown in Fig. 19.4b. However, the characteristic shape of the delamination contour shown in the C-scan could not be predicted in the simulation. This is probably © 2008, Woodhead Publishing Limited

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250

200

150

100

50

0 0

50

100

150

200

250

200

250

(a)

250

200

150

100

50

0 0

50

100

150 (b)

19.4 Plate C-scan test data after impact showing extent of delamination. (a) Plate CF0902: V0 = 3.11 m/s; (b) Plate CF0903: V0 = 2.52 ms.



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due to the fact that the current model does not couple delamination damage with in-ply damage in the adjacent plies. Both types of damage are in the model with their own damage criteria as discussed in Section 19.2, but the model allows no interaction between ply damage and delamination damage. They are independent failure conditions. It is known that ply microcracking can initiate delamination at the ply interface, which implies coupling between these modes. This is a refinement to the model which could be considered in the future. It was thus demonstrated that PAM-CRASHTM with the delamination model can predict reasonably well the failure modes in the impacted plates, by penetration or rebound, with convincing contour plots of ply damage and delamination damage. However, to validate the code it is necessary to compare quantitative predictions of impact loads and plate deflections with measured test data. The load cell on the impactor head records the contact load-time pulse during impact. The load test data at V0 = 3.11 m/s is plotted in Fig. 19.5. It is compared with simulations using the bi-phase model with a single laminated shell (without delamination), and with eight sublaminated stacked shells (with cohesive interfaces). The figure shows that the single laminated shell model over-predicts the measured failure loads by a factor of more than three after first delamination occurs. In contrast, the simulation with eightsublaminates including delamination shows a significant decrease of the

kN 20 17.5 Test 1-plate model 8-plate model

15 12.5 10 7.5 5 2.5 0 0

2

4

6

8

10 Time

12

14

16

18

ms 20

19.5 Comparison of the contact forces for CF0902 between test and simulations, including results from a 1-plate and 8-plate stacked shell models.


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loads after the peak, which is much closer to the test data. The simulation demonstrates that delamination in the plate reduces the plate bending stiffness near the impactor which leads to lower loads. It follows that the simulation with only laminate ply failure and no delamination gives poor agreement with test data, with a predicted peak load too high and delayed in the time. The results for the delamination model are encouraging and show that delamination effects are significant in low velocity plate impact and that these can be successfully modelled. Modifications to the delamination failure initiation stresses and interface fracture energies in the model leads to further changes in the peak loads, showing that the impact simulations are sensitive to the interface parameters.

19.4

Delamination damage in high velocity impact

19.4.1 Gas gun impact tests on stiffened composite panels In order to study the impact resistance of stringer stiffened panels, a gas gun impact test programme has been conducted at DLR with various impactors including metallic beams and cubes, rubber beams and ice projectiles. Part of this test programme has been used as validation tests for the modelling of the composite panels. Three cases taken out of the test programme are summarised in Table 19.2. The projectile considered here is a steel cube with 12 mm edge length and a weight of 13.6 g, which represents an aircraft impact scenario of hard debris runway impact on a lower fuselage panel at take-off or landing. Two impact locations are considered: impact on stringers and between stringers, which show different failure behaviour. In addition, impact energies were varied. All tests were carried out on a single 800 mm × 800 mm test panel with thickness of about 1.7 mm. It was manufactured from 14 composite plies mainly UD carbon/epoxy prepreg, with quasi-isotropic fibre orientations. It Table 19.2 High velocity impact test results with 13.6 g steel cube at 90° impact Shot number

Impact velocity [m/s]

Impact energy [J]

Impact location

Damage description

4

99.9

67.9

Between stringers

Perforation /delamination/cube flies through

5

97.2

64.2

On stringer

No perforation/delamination/cube rebounds

7

55.2

20.7

Between stringers

No perforation/delamination/cube rebounds



577

is stiffened on one face with seven parallel T-stringers with height 35 mm and spacing 100 mm, which are bonded to the panel. For test purposes, the panel was fully clamped on both opposite edges perpendicular to the stringers whereas both edges parallel to the stringers are free. This could reflect the constraints applied by the frames in an aircraft structure, with the clamped edges idealising the clamping at the frame. In addition, the feet of the stringers at the edge of the panels were supported in order to avoid an unrealistic peeling of the stringer feet from the panel skin. In the test cases considered here the steel cube projectile impacts the flat plate surface without stringers at an angle of 90° to the surface. Because the damage is localised near to the impact point, it was possible to carry out several tests with the small metal projectiles on a single panel. Figures 19.6–19.8 show the characteristic damage in the panel at the impact face (top) and on the rear stringer face (bottom) after the impact event at the three velocities 55.2 m/s, 99.9 m/s and 97.2 m/s. When considering the impact between two stringers at 99.9 m/s, the cube projectile penetrates through the plate, whereas it rebounds at the two lower velocities. Figure 19.9 shows the impact damage caused on the panel as detected by ultrasonic C-scan after testing. It should be noted that more than four impact tests were performed on the complete panel. The black regions on Fig. 19.9 represent delamination failures. As expected, it is seen that the damage extent due to delamination and fibre rupture increases with the impact energy. The shots 7 and 4 between stringers correspond respectively to the impact energies 20.7 J and 67.9 J and show this trend. Shot 5 (64.2 J) was on the stringer and rebounded, with the C-scan showing delamination damage near the stringer foot. The remaining damage seen in Fig 19.9 refers to that from additional tests on the complete panel not discussed in detail here.

19.4.2 Simulation of composite plate damage An extended test programme was carried out at the DLR on UD carbon epoxy and carbon fabric/epoxy specimens used in the plate impact programme in order to generate the necessary input data for the ply damage model described in Section 19.2. For UD specimens this comprised tension tests on specimens with fibres oriented at 0°, 90°, ±45° and ±62.5° to the load axis, with both monotonic loading and cyclic loading. For the delamination model inter-laminar failure properties were studied through standard Double Cantilever Beam (DCB) and End Notched Flexure (ENF) specimens with starter cracks. In order to study mesh dependency associated with fracture of the cohesive interfaces in the delamination model, individual delamination test specimens were modelled with different mesh densities and model parameters, which include fracture energies, interface thickness, interface stiffnesses and initiation


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(a)

(b)

19.6 Shot 7: Visible damage after an impact at V = 55.2 m/s – 20.7 J between two stringers (cube rebounds).

and propagation stresses. When a parameter set was determined which gave good agreement with the test curves, this was used in the panel models with the same mesh density. Such a procedure is used often in explicit code models due to their sensitivity to element and interface elimination. For example the interface failure energies used to propagate delamination between large elements in a coarse mesh may be different to those required for small elements in a fine mesh. © 2008, Woodhead Publishing Limited


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(a)

(b)

19.7 Shot 4: Visible damage after an impact at V = 99.9 m/s – 67.9 J between two stringers (cube passes through panel).

For the simulation of the stiffened composite panels a stacked shell laminate model is used with 14 plies. After some preliminary studies with different configurations the laminate was idealised by three sub-laminates of layered shells connected by two cohesive interfaces. This approximation limits the role of delamination damage to two interfaces, but keeps the model size within bounds. The T-stringers were attached to the panel laminate by a tied interface with failure to represent adhesive bonding. The stringer-stiffened panel FE model was parametrised such that fine scale meshes could be used © 2008, Woodhead Publishing Limited

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(a)

(b)

19.8 Shot 5: Visible damage after an impact at V = 97.2 m/s – 64.2 J on a stringer (cube rebounds but damages stringer/plate bond).

in the contact regions to better model the damage, with coarser meshes away from the impact region. Since from observing the tests the boundary conditions have an influence on the damage picture, it was decided to model the full panel requiring an FE model composed of approximately 107 000 layered shell elements. As in the tests the panel is fully clamped on opposite sides at the extremities of the stringers. Through experience, as the projectile (metallic cube) during the test does not undergo any plastic deformation, it is justified to define it as a rigid body in the simulations. © 2008, Woodhead Publishing Limited


(a)

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(b)

(c)

19.9 Ultrasonic C-scans of the panel (a) Shot 7; (b) Shot 5; (c) Shot 4.

Numerical results are presented for three selected impact test cases in Figs 19.10–19.12. In the simulation, when impacted at V = 55.2 m/s between two stringers, the cube (projectile) rebounds leaving delamination damage, whereas the panel is perforated and delaminated when impacted at V = 99.9 m/s. Concerning the simulation of the impact on a stringer at V = 97.2 m/s, the cube causes delamination on impact and rebounds with fibre fracture. The simulation results are in good agreement with the tests as seen by the photographs of post-test damage in Figs 19.6–19.8. The predicted delamination damage is seen in Figs 19.10–19.12 by the almost black region surrounding © 2008, Woodhead Publishing Limited

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19.10 Shot 7: Simulation sequence of the impact on a stringer at V = 55.2 m/s.

19.11 Shot 5: Simulation sequence of the impact on a stringer at V = 97.2 m/s.

19.12 Shot 4: Simulation sequence of the impact between two stringers at V = 99.9 m/s.

the grey damage contours. These zones should be compared with the black regions in the C-scans of Fig. 19.9 which show the measured delaminations. It is seen that the predicted delamination zones in the three tests Shots 4, 5 and 7 agree fairly well with the C-scan test data, in all cases. The cruciform shape of the calculated delamination corresponds with the detected delamination in the case of the impact between two stringers at 99.9 m/s in Shot 4. For the impact on a stringer at 97.2 m/s in Shot 5, the simulation is conservative and predicts a larger delaminated area than shown in the NDT results. To summarise the results for impact between stringers, there was a critical velocity at which penetration took place, which was well predicted by the © 2008, Woodhead Publishing Limited


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model. For impact at the stringer position the projectiles rebounded but caused considerable local damage with local stringer debonding possible, which was also predicted fairly well. A study was made of the role of the delamination interfaces in the total damage energy absorbed during impact. When cohesive interfaces were not included in the laminate model, the prediction of the penetration velocity was very poor. Only when the model included both delamination failure and ply damage failure was it possible to get good agreement with the observed failure behaviour. Delamination arising from local contact with the projectile reduces the plate flexural stiffness and hence influences subsequent deformation and failure of the plate and provides an additional mechanism for damage energy absorption.

19.5

Conclusions and future outlook

This chapter has studied the influence of delamination damage on the impact behaviour of composite plates and stiffened panels. Delamination damage was measured in low velocity drop tower impact tests and also in high velocity gas gun tests on composite plates with steel projectiles carried out at the DLR. In both types of test it was seen that at lower impact energies delamination damage was often the main damage mechanism, whereas at higher impact energies there was fibre fracture and plate penetration by the hard projectiles. In this case the presence of some delamination provides an additional energy absorption mechanism in the composite plate which can reduce complete penetration. It follows that modelling and analysis of impact damage require composites failure models which include both in-plane ply damage and delamination damage. These advanced mesoscale damage models were presented and some validation studies carried out with the DLR impact test data. The failure model for UD composites laminates includes both interply delamination and intraply damage and has been implemented in the dynamic FE code PAM-CRASHTM [8]. Detailed code validations were successfully carried out on the application of the model to predict impact damage to flat plates under low velocity impacts with large masses, typically 21 kg mass at 2.5–4.5 m/s. This was followed by a study of gas gun impacts with small masses, typically 13.6 gram steel cubes at 55–100 m/s on a stringer-stiffened composite panel. Two cases were considered: impacts on a stringer and between two stringers. In both test programmes projectile kinetic energies were chosen to give both delamination failures and penetration failures. The FE simulations gave good predictions of failure modes and observed damage. In particular it demonstrated the importance of delamination damage during impact. Along with earlier studies on fabric reinforced composite plates [7], the results indicate the general validity of the mesoscale damage mechanics and delamination models, and the associated code developments. © 2008, Woodhead Publishing Limited

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They also highlight the importance of delamination being included in the FE models. However, there are a number of issues to be solved before the methods may be applied with confidence to predict impact damage from both rigid and deformable impactors in composite aircraft structures such as fuselage panels, wings and tailplane leading edges. These include the effect of the number of delamination interfaces in the model, since the interface failures introduce a mesh dependency. Reliable delamination test data is difficult to acquire, particularly under mixed mode failures. Further work is also required to develop improved methods for including composites rate dependent properties in the ply and delamination failure models, with robust methods for corresponding FE code implementations involving significant localised damage. Developments in FE codes which are expected to be of future importance for safety studies in aircraft structures include multiscale FE modelling for studying damaged regions in large structures, and the application of stochastic methods for defining failure envelopes in structures under a range of crash and impact conditions. Although considerable progress has been made in the last decade, it should nevertheless be pointed out that it will take many years for composites modelling in dynamic FE codes to reach the same level as that currently achieved in crash and impact analysis of metallic structures. The reasons are historical. Metal plasticity was understood and established mathematical models date from the middle of the last century. Furthermore the automotive industry was the ‘driver’ for the improvement of FE models particularly in crash codes, so that considerable resources have been invested in materials models, development of test methods, code implementations, validations, etc. for metallic structures. Similar resources have not yet been made available for composites materials. In addition the diversity of composite materials, the complex anisotropic failure behaviour, the lack of standardised materials and dynamic test procedures, and the smaller more specialist market for composites all contribute to the delay in establishing good, reliable composites models in FE crash and impact codes.

19.6

References

1. Ladevèze P, ‘Inelastic strains and damage’, Chapt. 4 Damage Mechanics of Composite Materials, Talreja R (ed.), Composite Materials Series, Vol 9, Elsevier, 1994. 2. Allix O and Ladevèze P, ‘Interlaminar interface modelling for the prediction of delamination’, Composites Structures, 1992, 22, 235–242. 3. Allix O, Ladevèze P and Corigliano A, ‘Damage analysis of interlaminar fracture specimens’, Composite Structures, 1995, 31, 61–74. 4. Johnson A F, Pickett A K and Rozycki P, ‘Computational methods for predicting impact damage in composite structures’, Comp Sci and Tech, 2001, 61, 2183–2192. 5. Crisfeld M A, Mi Y, Davies G A O and Hellweg H B, ‘Finite Element Methods and the Progressive Failure Modelling of Composite Structures’, Computational Plasticity



6. 7. 8.

9. 10. 11.

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– Fundamentals and Applications, Owen D R J, Oñate E and Suárez B (eds), CIMNE Barcelona, 239–254, 1997. PAM-CRASH™ FE Code, Engineering Systems International, F-94578 Rungis Cedex, France. Ladevèze P and Le Dantec E, ‘Damage modelling of the elementary ply for laminated composites’, Comp Sci and Tech, 1992, 43, 257–267. Johnson A F, ‘Modelling impact damage in composite structural elements’ Chapt. 14 Multiscale Modelling of Composite Material Systems, Soutis C and Beaumont P W R (eds), 401–429, Woodhead Publishing, Cambridge, 2005. Greve L and Pickett A K, ‘Delamination testing and modelling for composite crash simulation’, Comp Sci Tech, 2006, 66, 816–826. HICAS: High Velocity Impact of Composite Aircraft Structures, CEC DG XII BRITEEURAM Project BE 96-4238, 1998–2000. Johnson A F, Pentecôte N and Körber H, ‘Influence of delamination on the prediction of impact damage in composites’, Proc. IUTAM Symposium Multiscale Modelling of Damage and Fracture Processes in Composite Materials, Sadowski T (ed.), Kazimerz Dolny, Poland, May 2005.


20 Delamination buckling of composite cylindrical shells A T A F R E S H I, The University of Manchester, UK

20.1

Introduction

Laminated composites are gaining importance in aircraft structural applications as a result of their very high strength-to-weight and high stiffness-to-weight ratios. However, owing to the lack of through-the-thickness reinforcement, structures made from these materials are highly liable to failures caused by delamination. Therefore, within a design process, a structure’s resistance to delamination should be addressed to maximize its durability and damage tolerance. Delaminations in composite cylinders may be due to manufacturing defects, transportation impacts and environmental effects during their service life. The presence of delaminations leads to a reduction in the overall buckling strength of the structure[1–6]. For the past two decades analytical and numerical analyses have been carried out by many researchers to analyse delaminated composite structures, considering their buckling and post-buckling behaviour [7–8]. The early work belongs to Chai et al. [9] who characterized the delamination buckling models by the delamination thickness and the number of delaminations through the laminate thickness. Almost all of the papers on delamination buckling deal with beams and flat plates [9–17]. Owing to its mathematical complexity and modelling, very limited information on the subject of delamination buckling of cylindrical shells and panels is currently available [18–25]. The most comprehensive analytical study is by Simitses et al. [20–22], where they analytically predicted delamination buckling of cylindrical shells and panels. The load cases considered in their study were uniform axial compression and uniform external pressure, applied individually. For both cases they did not account for the contact between delaminated layers during buckling. Thin-walled circular cylindrical shells are very often loaded in such a way that the three buckling membrane forces: axial compression, circumferential compression and shear, are not applied individually but in combination. Therefore, a designer not only has to consider the buckling characteristics of 586 © 2008, Woodhead Publishing Limited


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a cylindrical shell under fundamental loading conditions, but also the buckling interactions. There is a reasonable amount of work on the instability response of intact laminated cylindrical shells under axial compression, external pressure, bending and torsion, applied individually [26–29]. Thus, very limited information on the instability response of composite shells under combined loading is available and most of it does not consider the post-buckling response which, if considered, will result in significant cost savings. Most of the published data deals with the buckling response of compression-loaded composite shells. Reference [30] consists of a comprehensive study on the stability of metallic cylindrical shells under combined loading. Despite the relatively widespread attention given to the problem of delamination in laminated composites, to the author’s knowledge, there is hardly any information available on the effect of delamination in a composite cylindrical shell structure under combined loading. In recent studies by the author the delamination buckling and post-buckling analysis of laminated cylindrical shells under axial compression [1], external pressure [2], pure bending [3], combined axial compression and lateral pressure [4] and combined axial compression and bending [3] were investigated. In another study [6] the instability response of intact composite cylindrical shells, with or without geometric imperfections, subject to combined loading were presented. This chapter discusses and compares the computational modelling of delamination buckling and post-buckling of laminated composite cylindrical shells subjected to axial compression, bending and lateral pressure. The loads are either applied individually, or in combination. The numerical analysis, using the finite element method, has been carried out in conjunction with the CAE package ABAQUS [31]. The three-dimensional finite elements can model the delamination behaviour of laminates very well but they are computationally expensive. Here, it is shown that by employing a combined double-layer and single-layer of shell elements, the computational time and capacity, for the same level of accuracy, can be reduced. Section 20.2 describes the geometry, loading and applied boundary conditions. This is followed by the choice of mesh, the FE modelling, buckling and post-buckling procedures and also the method employed for calculation of the strain energy release rate (SERR) along the crack front, respectively. Section 20.3 presents the verification study of the proposed FE modelling approach where the results are compared with the available analytical results produced elsewhere. Analysis of delaminated cylinders under different types of loadings are described in Section 20.4. In this section the effects of loadings, material properties, delamination size and thickness on the instability response of delaminated composite cylinders are investigated. For the combined loading cases the interaction buckling curves and post-buckling response of delaminated © 2008, Woodhead Publishing Limited

588


cylinders have been obtained. In the analysis of post-buckled delaminations, a study using the virtual crack closure technique has been performed to find the distribution of the strain energy release rate along the delamination front. The final section provides a brief discussion of the results, presents the concluding remarks and describes the proposed future studies.

20.2

Finite element analysis

20.2.1 Geometry, loading and boundary conditions Figure 20.1a shows the geometry of a typical delaminated cylindrical shell with radius r and thickness t subjected to axial compressive load (R), lateral pressure (P) and bending moment (M), applied individually. t1 and t2 are the thicknesses of the upper and lower sublaminates, respectively. Figure 20.1b shows a differential element of an intact cylindrical shell segment with the coordinate axes. The axial coordinate is x, the thickness coordinate normal to the shell surface is z and the circumferential coordinate is y = rβ. Three types of delamination are considered, see Figs 20.1c–20.1e. Figure 20.1c shows a rectangular delamination (Ld × b) which is located symmetrically with respect to both ends of the shell, where Ld is in the axial direction and b = rα is in the circumferential direction. Angle α denotes the region of the delamination. When the pure bending is applied, the delamination surface is always symmetric with respect to the plane of bending moment. Figure 20.1d shows the geometry of a cylindrical shell with a delamination length of Ld extended along its entire circumference. Figure 20.1e shows a cylindrical shell with a longitudinal delamination of width b=rα extended over the entire length. Throughout this study the non-dimensional parameters h = t1/t and a = Ld/L are used to describe the delamination thickness and length, respectively. Here, the external pressure is assumed to be positive and the internal pressure is assumed to be negative. The cylinder has clamped ends unless otherwise stated. Using the constraint equations all the nodes on each end cross-section are kept in the same respective plane. This ensures that the cross-sections remain on the same planes after deformation, which properly models the actual experimental conditions. Also, the loads applied on the planes, to simulate bending or axial compression, would not create a very high local stress concentration. Figures 20.2a–20.2c show the first buckling mode shapes of an intact graphite-epoxy cylindrical shell subject to axial compression (Rc), external pressure (Pc) and bending (Mc), respectively, where the loads are applied individually. Subscript (c) for each loading case denotes the critical value of the applied load. Figure 20.2d shows the buckling mode shape of the same cylinder subject to combined axial compression and bending. The buckling mode shape of the cylinder subject to combined axial compression and external © 2008, Woodhead Publishing Limited

Ld dy R

R

z

dx

2r

2 3

M

M

L x

1

Mid-surface

2r β

y r

P (a)

(b)

t1 t2

t

α

r

L

L L

b α

Ld

r

r

α

Ld (c)

(d)

(e)

589

20.1 (a) Geometry of a cylindrical shell under axial compression, lateral pressure and bending; (b) A cylindrical shell differential element; (c) Rectangular delamination; (a × b); (d) Delamination of length (Ld), extended along its entire circumference; (e) Delamination of width (b) extended along the length of the cylinder. © 2008, Woodhead Publishing Limited


x

t

590

1 1

3

(a)

3

3 2

1 3

(d)

2

2

2 3

1

1

3 (b)

1

2

2 3

1

3

1

(e) L/r = 5, r/t = 30), [0/90/0]10T

1

1 3

3 2

(c)


2

20.2 First buckling mode of an intact graphite-epoxy cylindrical shell under (a) Axial compression (Rc); (b) External pressure (Pc); (c) Bending (Mc); (d) Combined axial compression (0.72Rc) and bending (0.33Mc); (e) Combined axial compression (0.8Rc) and external pressure (0.62Pc).


2

2


591

pressure is shown in Figure 20.2e. For the properties of the selected materials refer to Table 20.1.

20.2.2 Modelling and choice of mesh For the finite element analysis of a typical cylindrical shell, a single layer of shell elements can be employed and the corresponding buckling and postbuckling analysis can be performed subject to different loading conditions. Figures 20.3–20.4 show different FE models which may be employed for the analysis of a delaminated cylindrical shell. The following explains the advantages and disadvantages of each model [1–5]. The displacement field for an intact cylindrical shell, according to a first order shear deformation theory, is given by: u x = u 0x + zψ x

u y = u 0y + zψ y u z = u z0

20.1

ux, uy, uz are the displacements (in the x, y and z directions, respectively) and ψx, ψy are the rotations (about y and x axes, respectively), at arbitrary locations with the distance z to the shell’s mid-surface. u 0x , u 0y , u 0z are the mid-surface displacements of the shell in the respective directions. Therefore, it is also possible to model the entire laminate using two layers of shell elements forming an upper and lower sublaminates, where the nodes are located on the mid-surfaces of the laminates. In order to enforce compatibility along the interface, constraint equations based on Equations 20.1 can be imposed. The following constraint equations will tie the upper and lower sub-laminates in both displacements and rotations along the interface so that both sub-laminates together deform like an intact single laminate.  t 1  ψ 1 = u 2,0 +  t 2  ψ 2 u 1,0 x – x  2 x  2 x  t 1  ψ 1 = u 2,0 +  t 2  ψ 2 u 1,0 y – y  2 y  2 y 2,0 u 1,0 z = uz

20.2

Table 20.1 Lamina engineering constants for the selected materials Material

EL/ET

νLT

GLT/ET

GTT/GLT

ET(GPa)

Graphite-epoxy Kevlar-epoxy Boron-epoxy Aluminium

40 15.6 11.11 1.0

0.25 0.35 0.21 0.3

0.5 0.56 0.24 0.385

1.0 1.0 1.0 1.0

5.17 5.5 18.61 70.0


592

Delamination behaviour of composites Mid-surface Delaminated region (double layer)

on egi ct r layer) a t n I e ubl (do

Upper sublaminate

Inta (dou ct regio n ble laye r)

Gap elements

Lower sublaminate

Interface region (a)

Mid-surface

Delaminated region (double layer)

on egi r) e ct r Inta le lay g n i s (

Upper sublaminate

Inta c (sin t regio gle n laye r)

Gap elements

Lower sublaminate

Interface region (b) Delaminated region (double layer) Upper sublaminate

Transition zone intact (double layer) Inta (sin ct reg gle io lay n er)

Gap elements

Lower sublaminate Interface region (c)

20.3 Close-up view of a typical FE model to be used for a delaminated cylindrical shell. (a) with double layer of shell elements; (b) with single and double layer of shell elements; (c) with single, double layer and the transition zone © 2008, Woodhead Publishing Limited


Delaminated region

I

ct nta

reg

593

Delaminated surface

Inta

ion

ct r eg

ion

Upper sublaminate

Lower sublaminate

Delaminated zone

Interface region Gap element Mid-surface Reference surface (a)

a5 a4 b3

Crack front

a3 A5 a2

A4

c4

a1 A3 b2

c5

B3

b2

A2

d3

∆L

d2 c3

B2

∆L

c2

A1 B1

c1

∆c

d1 ∆c (b)

20.4 Close-up view of a typical FE model to be used for a delaminated cylindrical shell: (a) where nodes are located on the reference surface offset from the mid-surfaces of the laminate and sublaminates; (b) Schematic of delamination front region.

The first superscripts 1 and 2 of the displacements refer to the upper and lower sublaminates, respectively, and the second superscript 0 refers to the sublaminates’s mid-surface. For a delaminated cylinder, the intact regions can be represented by a © 2008, Woodhead Publishing Limited

594


double-layer of shell elements, and the delaminated regions can be modelled by upper and lower sub-laminates that are connected by contact elements. See Fig. 20.3a. To reduce the computer time and capacity and for the same level of accuracy, the intact regions can be represented by a single layer of shell elements. See Fig. 20.3b. For the interface region, a modified version of the sublaminate connection method, based on Equation 20.2, must be employed. Therefore, also the rotations (ψx, ψy, ψz) of all the nodes of the stacked layers at the transition border are coupled in addition to the displacements (ux, uy, uz). Thus, the mid-surfaces of the sub-laminates of the delaminated and the mid-surface of the laminate of the intact area are coupled by the following set of equations:  t 1  ψ 1 = u 2,0 +  t 2  ψ 2 = u 0 u 1,0 x – x x  2 x  2 x  t 1  ψ 1 = u 2,0 +  t 2  ψ 2 = u 0 u 1,0 y – y y  2 y  2 y 2,0 0 u 1,0 z = uz = uz

ψ 1x = ψ 2x = ψ 0x ψ 1y = ψ 2y = ψ 0y ψ 1z = ψ 2z = ψ 0z

20.3

Although this model can be easily used for single and multiple delaminations, it leads to the complexity of using constraint equations (20.3) to maintain compatibility of displacements on the delaminated front. The other disadvantage is for use with the virtual crack closure technique (VCCT) for calculation of the strain energy release rate (SERR) [32]. To overcome this problem a small transition zone made of two layers of shell elements, can be added in the intact region and around the crack front. See Fig. 20.3c. Because without having two stacked layers of elements the calculation of SERR will not be possible. The intact region is modelled by a single layer of laminated shell elements which is connected to the double layer by use of the Mindlin/ Kirchhoff equations and an additional coupling of the rotational degrees of freedom. The selection of the mid-surface as a reference surface in a shell element is due to tradition and the ease of defining bending and shear rigidities. An alternative method, which overcomes these difficulties and also simplifies the mesh generation, is to choose the reference surface at an arbitrary position within the shell’s thickness. According to the first order, shear deformation



595

theory (Equation 20.1), the deformation at any position within the thickness direction can be expressed by the deformation at any chosen surface, not necessarily the shell’s mid-surface. Figure 20.4a shows the close-up view of an FE model which is a combination of double-layer and single-layer of shell elements; however, the delaminated surface is chosen as the reference surface for both the intact and delaminated regions. In the delaminated region, nodes are separately defined on the upper and lower sublaminates although they may have the same coordinates. Gap elements are placed between the corresponding nodes of the upper and lower sublaminates to avoid the interpenetration of the delaminated layers. Interpenetration would not represent the real behaviour of laminates and result in incorrect estimations of the critical load and post-buckling response of the shell [2]. Within the intact and delaminated regions, the nodes located on the reference surface are offset from the mid-surfaces of the corresponding shells, either in the positive or negative direction. To maintain compatibility of displacements on the delaminated front, the nodes of the intact and delaminated regions are tied together to have equal deformations. This model has been used throughout this study which greatly simplifies the calculation of the SERR for the laminated plates and shells with single delaminations. It should be noted that the latest version of ABAQUS offers continuum shell elements with three displacement degrees of freedom, with nodes at the top and bottom surfaces of each element. This type of elements may be employed for the delamination buckling of cylindrical shells. However, continuum shell elements discretize the entire three-dimensional body, as do three-dimensional solid elements. The virtual crack closure technique (VCCT) is used to calculate the strain energy release rate (SERR). The nodal forces at the crack front and the displacements behind the crack front are required to calculate SERR. Figure 20.4b shows a model of a pair of four elements in the upper and lower sublaminates at the delamination front. The crack front is located beneath nodes cj (j = 1, 5). The components of the strain energy release rate for the three modes for a typical element are

GI =

1  F (w – w ) + F ( w – w ) B2 A3 Zd 2 b2 2 ∆A  Zc 3 a y +

G II =

1 [ F (w – w A 2 ) + FZc 4 ( w a 4 – w A 4 )] 2 Zc 2 a 2 

1  F (u – u ) + F (u – u ) B2 A3 Xd 2 b2 2 ∆A  Xc 3 a 3 +

1 [ F (u – u A2 ) + FXc 4 ( u a 4 – u A4 )] 2 Xc 2 a 2 


596


G III =

1 F (v – v ) + F (v – v ) B2 A3 Yd 2 b2 2 ∆A  Yc 3 a 3

+

1 [ F (v – v A 2 ) + FYc 4 ( v a 4 – v A 4 )] 2 Yc 2 a 2 

20.4

Here, eight noded shear deformable shell elements with reduced integration, which allows large rotations and small strains, are employed for modelling the laminated cylindrical shells.

20.2.3 Buckling and post-buckling analysis The linear eigenvalue buckling analysis can estimate the critical load of a structure subject to axial compression, bending, lateral pressure, etc. In simple cases, linear eigenvalue analysis may be sufficient for design evaluation; but geometrically nonlinear static problems sometimes involve buckling or collapse behaviour, where the load-displacement response shows a negative stiffness and the structure must release strain energy to remain in equilibrium. Therefore, a nonlinear load-displacement analysis is required to predict the post-buckling behaviour of the structure. Several methods are available for modelling such behaviour [33–35]. It is possible to treat the buckling response dynamically, therefore, modelling the response, with inertia effects included, as the structure snaps [31]. This approach can be carried out by restarting the terminated static procedure and switching to a dynamic procedure when the static solution becomes unstable. Another approach for finding static equilibrium states, during the unstable phase of the response, is the Riks method. The Riks method is based on moving with fixed increments along the static equilibrium path in a space defined by the displacements and a proportional loading parameter. This method is used for cases where the loading is proportional; therefore, the load magnitudes are governed by a single scalar parameter. Arc length methods such as the Riks method are global load-control methods that are suitable for global buckling and post-buckling analyses; they do not function well when buckling is localized. Another method is to use dashpots to stabilize the structure during a static analysis. In the current study, the automatic stabilization which applies volume proportional damping to the structure is used. In the dynamic case, the strain energy released locally from buckling is transformed into kinetic energy; in the damping case, the strain energy is dissipated. It should be noted that the initial small deflection that is necessary to make the structure buckle was established by imposing an imperfection to the original mesh. The applied imperfection rested on an eigenmode buckling analysis of the structure. The maximum initial perturbation was 5% of the thickness of the shell. © 2008, Woodhead Publishing Limited


597

The combined loading cases such as axial compression and bending or axial compression and lateral pressure are carried out in two steps. In the first step a live load such as external pressure or bending is applied to the shell, and in the second step the critical axial buckling load is determined. For the post-buckling analysis after application of the live load (bending or pressure) in the first step, the axial compressive load is set to increase incrementally up to 1.2 times the critical axial compressive load of a perfect cylinder subjected to axial compressive load alone. The lowest size of the delamination area considered in this study had the delamination length of a = 0.01 and delamination width of α = π/24. The analysis of cylinders with lower delamination areas was at the expense of the increase in the computational time and space. Because for very small delamination areas, regardless of the delamination thickness, the presence of delamination does not appreciably alter the critical load of a perfect geometry.

20.3

Validation study

The numerical results of variation of critical axial load with delamination length and delamination thickness of an isotropic cylindrical shell with clamped ends, obtained using the present method [1], are compared with those obtained by Simitses et al. [22]. See Figures 20.5a–20.5b. The delamination is located symmetrically with respect to both ends of the shell and it extends along its entire circumference. The dimensions of the shell are such that L/r = 5 and r/ t = 30. The critical loads are normalized with respect to the critical load of an intact cylinder. It is shown that although Simitses et al. [22] did not account for the contact, between delaminated layers during buckling, their results look reasonable. For the delamination thicknesses of h = 0.3 and 0.5, with delamination length of a > 0.04, the buckling load will increase when considering the effect of contact in the buckling mode. For the delamination thickness of h = 0.1, with delamination length of a > 0.04, the effect of contact in the buckling mode is almost negligible. This figure also shows that, for very small values of the delamination length (a < 0.01), regardless of the delamination thickness and also the effect of contact, the delamination has no significant effect on the critical loads. For the delamination length of (0.01 < a < 0.04), the buckling load will decrease when considering the effect of contact in the buckling mode.

20.4

Results and discussion: analysis of delaminated composite cylindrical shells under different types of loadings

20.4.1 Axial compression Two composite cylindrical shells made of graphite-epoxy and kevlar-epoxy laminates with clamped ends under axial compression are studied. The cylinders © 2008, Woodhead Publishing Limited

Graphite-epoxy, h = 0.3 Kevlar-epoxy, h = 0.3

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

Isotropic, h = 0.3 Isotropic, h = 0.3 [Ref. 22]

0.53–0.54

0.51

R/Rc

0.6

0.5 0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.64

0.5 0.38 0.31

0.1

L/r = 5, r/t = 30, [0/90/0]10T

Delamination length (a) 0

0 0

0.05

0.1

0.15

0.2 (a)

a = 0.125, h = 0.5, 0.72Rc

0.25

0.3

0.35

0.4

0

a = 0.125, h = 0.3, R = 0.85Rc (c)

0.05

0.1

0.15 0.2 0.25 0.3 Delamination length (a) (b)

0.35

0.4

a = 0.125, h = 0.1, R = 0.22Rc

20.5(a,b) Effect of delamination length (a), extended along the entire circumference on the critical buckling load of a cylindrical shell under axial compression with different delamination thicknesses and material properties, (c) Buckling mode shapes. © 2008, Woodhead Publishing Limited


R/Rc

Isotropic, h = 0.5 [Ref. 22] Isotropic, h = 0.1 [Ref. 22] Isotropic, h = 0.1

598

Graphite-epoxy, h = 0.5 Kevlar-epoxy, h = 0.5 Isotropic, h = 0.5


599

have the same geometry and boundary conditions as the isotropic cylinder studied earlier. The cross-ply laminates have the stacking sequence of [0°/ 90°/0°]10T. The buckling curves of the delaminated composite cylinders with the delamination thicknesses of h = 0.3 and h = 0.5 are also included in Figs 20.5a–20.5b. The results for the delamination thickness of h = 0.1 were obtained but are not presented. However, for h = 0.1, the trends of the variation of buckling loads, with respect to the delamination length in the laminated and isotropic cylindrical shells, are similar. The analysis of the selected materials showed that for very small values of the delamination length (a < 0.01), the delamination has no significant effect on the critical loads, regardless of the delamination thickness and material properties. For the delamination length of a > 0.01, when the delamination is located near the free surface of the shell (h = 0.1), regardless of the material properties, a sharp drop in the critical load is noticed. For h = 0.1 and a > 0.01, the normalized critical loads for the selected materials varied between 0.14 and 0.23. For the isotropic cylindrical shell, with delamination length of a > 0.04, the critical load decreases as the delamination thickness decreases. For the composite cylinders, except when delamination is located near the free surface, the critical load decreases as the delamination thickness increases. It can be said that in the composite cylinders, for 0.3 < h < 0.5, the lowest critical load occurs when delamination is located at midsurface. It was also observed that for the same delamination thickness, graphiteepoxy with a higher longitudinal-to-transverse modulus ratio (EL/ET = 40) and a higher transverse-to-shear modulus ratio (ET/GLT = 2.0) has a higher critical load. For a certain delamination thickness, when a > 0.4, the critical load is almost independent of the delamination length and material properties. Figure 20.5c shows the buckling mode shapes of the delaminated graphiteepoxy cylinder, with the delamination length of a = 0.125, extended along its entire circumference, for h = 0.5, 0.3 and 0.1, respectively. Results are generated for the aforementioned graphite-epoxy cylindrical shell with a rectangular delamination. Different delamination sizes with delamination thicknesses of h = 0.5, 0.3 are considered. The delamination length (a) and delamination width (α) vary as 0.04 < α < 0.5 and π/12 < α < π, respectively. See Figs 20.6a–b. The following observations are made from this study. For the delamination thickness of h = 0.5, the critical load gradually decreases with the increase of the delamination area. The delamination length and delamination width have almost the same effect in reducing the critical load. For the delamination thickness of h = 0.3, for very small values of the delamination area, when a < 0.1 and α < π/6, the delamination has no significant effect on the critical loads. When a > 0.1 and , α> π/6, there is a gradual drop in the critical load with respect to the increase of the delamination area. Except when the delamination is located near the free surface, the higher the © 2008, Woodhead Publishing Limited

0.9 0.8

0.7

0.70 0.7

0.70

0.6

0.58

0.6

0.58

0.5

0.5

L/r = 5, r/t = 30, [0, 90,0]10T

0.4

0.4

Delamination length, thickness

Delamination width (radians) and thickness

0.3

0.261, h = 0.5 1.57, h = 0.5 0.261, h = 0.3 1.57, h = 0.3

0.2 0.1

0.5233, h = 0.5 2.355, h = 0.5 0.5233, h = 0.3 2.355, h = 0.3

0.3

1.0466, h = 0.5 3.14, h = 0.5 1.0466, h = 0.3 3.14, h = 0.3

0.041, h = 0.5 0.25, h = 0.5 0.041, h = 0.3 0.25, h = 0.3

0.2

0.083, h = 0.5 0.375, h = 0.5 0.083, h = 0.3 0.375, h = 0.3

0.125, h = 0.5 0.5, h = 0.5 0.125, h = 0.3 0.5, h = 0.3

0.1 Delamination width (radians)

Delamination length (a) 0

0 0

0.05

0.1

0.15

0.2

0.25 0.3 (a)

0.35

0.95Rc, a = 0.125, h = 0.5

0.4

0.45

0.5

0

0.5

0.98Rc, a = 0.125, h = 0.3 (c)

1

1.5 (b)

2

2.5

3

0.98Rc, a = 0.125, h = 0.1

20.6 Effect of delamination size and thickness on the critical buckling load (R/Rc) of a graphite-expoxy cylindrical shell with a rectangular delamination subject to axial compression: (a) R/Rc vs. delamination length; (b) R/Rc vs. delamination width; (c) Buckling mode shapes. © 2008, Woodhead Publishing Limited


0.8

R/Rc

0.9

R/Rc

600

1

1


601

delamination thickness, the lower would be the critical load. It can be said that the lowest critical load occurs when the delamination is located at the laminate’s mid-surface. However, for very large delamination areas, when a > 0.4 and α> 5π/6, the critical load is almost independent of the delamination size and only depends on the delamination thickness. Figure 20.6c shows the buckling mode shapes of the cylinder with a rectangular delamination of (a = 0.125, α = π/4) and delamination thicknesses of h = 0.5, 0.3 and 0.1, respectively. Depending on the size and through the thickness position of delaminations of the cylindrical shell, three different modes of buckling behaviour occur. These three modes are referred to as local, global and mixed. The local mode occurs when the delamination is near the free surface of the laminate and the area of the delamination is large. The global mode occurs when the delamination has a small area and is deeper within the laminate. The mixed mode is a combination of local and global buckling.

20.4.2 Lateral pressure Next, a laminated graphite-epoxy cylindrical shell with longitudinal delamination over the entire length under external pressure, similar to the model employed by Simitses et al. [22] was studied [2]. The cylinder was simply supported at both ends. The dimensions of the shell were such that L/ r = 6 and r/t = 100. The cross-ply laminates had the stacking sequence of [90°/0°/90°]10T. The numerical results of variation of critical pressure with delamination width (α) and delamination thickness (h), calculated using the present method were compared with those obtained by Simitses et al. [22]. The critical loads are normalized with respect to the critical load of an intact cylinder (Pc). Figures 20.7a–b show that, for the delamination thicknesses of h = 0.5 and 0.1, although Simitses et al. [22] did not account for the contact (between delaminated layers during buckling) their results agree reasonably well with the present results. However, Fig. 20.7b shows that for the delamination thickness of h = 0.3 there is a large discrepancy of up to 50%, when considering the effect of contact in the buckling mode. The significant difference between the present results and those of Simitses et al. [22], for the case of h = 0.3, is obviously due to the material interpenetration at the delaminated layers, when the effect of contact is not considered. The buckling curve for h = 0.3 and 0 < α < 360°, for the case without gap elements is also shown in Fig. 20.7b. It can be seen that this curve is reasonably close to the buckling curve of reference 22. Similar results are obtained for a laminated cylinder made of Kevlar-epoxy with the same geometry, boundary conditions and stacking sequence as the cylinder studied earlier. See Figs 20.7a–20.7b. It can be observed that for a given delamination size and thickness, there is a slight discrepancy between the © 2008, Woodhead Publishing Limited

1

0.8 0.7

0.8 0.7

0.6

0.6

0.5

0.40 0.36

0.4

P/Pc

P/Pc

h = 0.3 (Ref. 22), G h = 0.3 (with contact elements), G h = 0.3 (without contact elements), G h = 0.3, K

0.9

K: Kevlar-epoxy G: Graphite epoxy

0.3

0.54 0.49

0.5 0.4

P 0.28 0.24

0.3

0.2

0.2

0.1

0.1

0

L/r = 6, r/t = 100, [90/0/90]10T

0 0

0.5

1

1.5 2 2.5 3 3.5 4 4.5 5 Delamination width (α) (radians) (a)

2 3

1

5.5

0

6

2

1

h = 0.5, α = 45° 2

3

(c) 3

0.5

1

1.5 2 2.5 3 3.5 4 4.5 5 Delamination width (α) (radians) (b)

6

1

h = 0.3, α = 45° 1

5.5

2

3

20.7 (a,b) Effect of delamination width (α), extended along the entire length on the critical buckling load of a laminated cylindrical shell under external pressure with different delamination thicknesses and material properties; (c) Buckling mode shapes. © 2008, Woodhead Publishing Limited


h = 0.5 (Ref. 22), G. h = 0.5, G h = 0.1, G h = 0.1 (Ref. 22), G. h = 0.5, K

0.9

602

1


603

critical loads of the two cylinders. Therefore, in this case the material properties do not have a great influence on the normalized critical loads. Fig. 20.7c shows the buckling mode shapes of the delaminated graphiteepoxy cylinder, with the delamination width of α = 45 degrees, extended along the entire length, for h = 0.5 and 0.3, respectively. It was observed that for h = 0.5, the delaminated cylinder mainly buckles in a global mode, with or without having contact elements between the delaminated layers. Therefore, the effect of contact in this case is almost negligible. For h = 0.3, when the contact elements are removed, the cylinder mainly buckles locally and there is serious penetration between the delaminated layers. As a result in this case, the influence of contact turns out to be high. For h = 0.1, the very thin upper delaminated layer tends to buckle independently, and the critical load is very low, whenever considering or ignoring the effect of contact.

20.4.3 Pure bending An example of a laminated cylindrical shell subjected to pure bending is presented here [3]. The delamination region has a rectangular shape and is placed on the compressive side of the cylinder, when subjected to bending. It is located symmetrically with respect to both ends of the shell and it is also symmetric with respect to the plane of the bending moment. The dimensions of the shell employed are L/r = 5, r/t = 30 with the stacking sequence of [0/90/0]10T. The delamination length (a) and delamination width (α) vary as 0.04 < a < 0.5 and π/12 < α < π, respectively. The variation of the critical bending moment (M/Mc) with respect to the delamination width (α) and delamination length (a) for the delamination thickness of h = 0.3, 0.5 are presented in Figs 20.8a–b. The results show that for a very small values of the delamination area, the presence of the delamination has no significant effect on the critical loads. For a large delamination area, especially when the delaminated layer is closer to the free surface of the laminate (h < 0.1), the critical load is very small. It was also observed that for h > 0.1, the higher the delamination thickness is, the lower the critical load is. It can be said that the lowest critical load occurs when the delamination is located at the laminate’s mid-surface. For a very large delamination area where a > 0.375 and α > π/2 , the critical load is almost independent of the delamination size and mainly depends on the delamination thickness. It was also observed that for a < 0.375 and α < π/2 , for two models with equal areas of delamination, the delamination with the larger width creates a slightly lower buckling load. Figure 20.8c shows the first buckling mode of two laminated cylinders with equal areas of delamination, but having different delamination lengths and widths. It can be observed that the cylinder with the larger delamination width, not only has a slightly lower buckling load but, also has a different © 2008, Woodhead Publishing Limited

1

0.9

0.9 0.8

0.7

0.7 0.61

0.5

0.50 L/r = 5,r/t = 30, [0, 90, 0]10T

0.61

0.6 M/Mc

0.6

0.4

0.5

Delamination length and thickness

0.50

0.4

0.3

0.3

Delamination width (radians) and thickness 0.2616, h = 0.5 1.57, h = 0.5 0.2616, h = 0.3 1.57, h = 0.3

0.2 0.1

0.5233, h = 0.5 2.355, h = 0.5 0.5233, h = 0.3 2.355, h = 0.3

1.0466, h = 0.5 3.14, h = 0.5 1.0466, h = 0.3 3.14, h = 0.3

0.0416, h = 0.5 0.25, h = 0.5 0.0416, h = 0.3 0.1666, h = 0.3 0.375, h = 0.3

0.2 0.1

0

0.0833, h = 0.5 0.5, h = 0.5 0.0833, h = 0.3 0.2083, h = 0.3 0.5, h = 0.3

0.125, h = 0.5 0.375, h = 0.5 0.125, h = 0.3 0.25, h = 0.3

0 0

0.05

0.1

0.15 0.2 0.25 0.3 0.35 Delamination length(a) (a)

0.4

0.45

0

0.5

0.5

1 1.5 2 Delamination width (radians) (b)

2

3

3 M = 0.84 Mc a = 0.22, α = 30°

3

1

1 2

1

2.5

1 (c)

2

3

2 3 M = 0.81 Mc a = 0.11, α = 60°

20.8 Effect of delamination size and thickness on the critical buckling load(M/Mc) of a graphite-expoxy cylindrical shell with a rectangular delamination subject to pure bending; (a) R/Rc vs. delamination length; (b) R/Rc vs. delamination width; (c) Buckling mode shapes. © 2008, Woodhead Publishing Limited


0.8

M/Mc

604

1


605

buckling shape. The cylinder with the larger delamination width buckles mainly locally, but the buckling mode of the cylinder with the larger delamination length is a combination of local and global buckling. Figure 20.9 shows the buckling curves of the cylinder with a square delamination subject to axial compression, bending and external pressure, applied individually. The parameter A* = Ad/A shows the fraction of the total area of the cylinder, where Ad is the delamination area and A is the total surface area of the cylinder. The variation of the normalized critical loads (M/Mc), (R/Rc), and (P/Pc), respectively, with respect to the delamination area A* shows that for very small delamination areas the presence of the delamination has small effect on the critical loads, regardless of the type of loading. However, with the increase of the delamination area, the critical load of the cylinder subject to bending alone drops dramatically, especially when the delamination is placed at the plate’s mid-surface. For larger delamination areas when A*>0.2, the critical load is almost independent of the delamination size and mainly depends on the delamination thickness.

20.4.4 Combined axial compression and bending Next, the analysis has been performed on the graphite-epoxy cylindrical shell (L/r = 5, r/t = 30) with the stacking sequence of [0/90/0]10T, subject to 1 0.9 0.8 0.69 0.64

R/Rc M/Mc, P/Pc

0.7 0.6

0.61 0.57

0.5

0.50

r/t = 30, L/r = 5, [0/90/0]10T

0.4

h = 0.5 (Axial compression) h = 0.3 (Axial compression) h = 0.5 (Bending) h = 0.3 (Bending) h = 0.5 (External pressure) h = 0.3 (External pressure)

0.3 0.2 0.1

Ad A

A* = Ad/A

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

A*

20.9 Normalized critical buckling load vs. delamination area of a graphite-epoxy cylindrical shell with a square delamination under axial compression, bending and external pressure, applied individually.


606


combined axial compression and bending [3]. Figure 20.10a shows the interaction buckling curve, relating (R/Rc) and (M/Mc), of the cylinder with square delamination, when h = 0.5. The delamination area varies as 0.0023 0.125, there is a sharp drop in the buckling load. The results show that for a < 0.08, the critical buckling curves for h = 0.3 and 0.5 are reasonably close. However, when a > 0.08, for the same delamination size, the delamination which is located on the mid-surface of the cylindrical shell, creates the lower buckling load. Figure 20.12b shows the interaction buckling curves for the delaminated cylinder, relating (P/Pc) and (R/Rc), through a range of values of P from buckling under external pressure (P = Pc) to internal pressure (P = –Pc). The delamination length varies as 0.02 a 0.42 when h = 0.5 and 0.3. The interaction curve of the intact cylinder is also included in Fig. 20.12b The results show that, for a very small delamination length (a 0.01), the presence of delamination does not appreciably alter the interaction buckling curve of an intact cylinder. For large delamination areas, a 0.375, the shape of the interaction curve is almost independent of the increase of the delamination length. For a delamination length of a 0.25 when P/Pc < 0.47, there is © 2008, Woodhead Publishing Limited

1.2 P R

1

R/Rc

1.2 1.1

R

1

0.8

Rcr/Rc

0.8

0.6

0.73 0.70 0.62 0.60 0.52 0.46 0.44

L/r = 5, r/t = 30, [0/90/0]10T, Internal/external pressure, delamination thickness

0.4

0.47 Pc, h = 0.5 0.16 Pc, h = 0.5 0.31 Pc, h = 0.3 (–0.16 Pc), h = 0.3 (–0.16 Pc), h = 0.5

0.2

0.31 Pc, h = 0.5 0.47 Pc, h = 0.5 0.16 Pc, h = 0.3 (–0.32 Pc), h = 0.3 (–0.31 Pc), h = 0.5

0.7 0.6

Intact a = 0.02, h = 0.5 a = 0.125, h = 0.5 a = 0.416, h = 0.5 a = 0.02, h = 0.3 a = 0.125, h = 0.3 a = 0.416, h = 0.3

0.19

0.3 0.2

–1

0.03 0.05 0.08 0.1 0.13 0.15 0.18 0.2 0.23 0.25 0.28 0.3 0.33 0.35 0.38 0.4

–0.8

–0.6

–0.4

0 –0.2

Delamination length (a) (a)

2 3

2 1

3 3 1 P= 0.78Pc, R = 0.18Rc, H = 0.5

2

2

2 1

3

1

3

P = 0.16Pc, R = 0.75Rc, h = 0.5

3

0 P/Pc

2

2 1

External pressure

0.1

Internal pressure

0 0

0.5 0.4

1

P = 0.78Pc, R = 0.30Rc, h = 0.3

3

0.2

0.4

0.6

0.8

2 1

3

1

P = 0.16Pc, R = 0.85Rc, h = 0.3

(c)


611

20.12 Effect of delamination length and thickness on: (a) the critical axial compressive load (Rcr); (b) the interaction buckling curves of a delaminated graphite-epoxy cylindrical shell subject to combined axial compression (R) and lateral pressure (P); (c) the buckling mode shapes.

1


0.9

612


almost a linear variation of R/Rc with respect to P/Pc, with a relatively small rate of change. However, for P/Pc > 0.47, the rate of change of R/Rc with respect to P/Pc is very high. Therefore, in this region for a slight increase of the external pressure the critical axial compressive load decreases dramatically. The same behaviour can be observed for a 0.25 when P/Pc > 0.31. For the delamination thickness of h = 0.3, the same trends of variation of R/Rc, with respect to P/Pc, can be observed. It can be seen that, for the same delamination length, the critical load for h = 0.5 is lower than the critical load for h = 0.3, as explained before. However, the trends of the variation of the curves are similar, regardless of the delamination thickness. It can also be observed that for a ≤ 0.02, the critical buckling curves for h = 0.3 and 0.5 are almost identical. Figure 20.12c shows the buckling mode shapes of a delaminated graphiteepoxy cylinder with a delmination length of a = 0.125, under different combinations of axial compression and external pressure. Obviously, the effect of the delamination on the buckling mode is clearly apparent. It can be seen that for higher values of the external pressure and lower values of the axial compressive load, the buckling is mostly in the global mode. On the other hand, for lower levels of external pressure and higher values of axial compressive load the buckling mode tends to be mainly in the local mode. The reduction of the external pressure will obviously result in the increase of the axial buckling load and therefore, the buckling mode tends to be mostly in the local mode. Next, the post-buckling analysis was performed on the cylinder with three different delamination lengths of a = 0.042, 0.08, 0.125, respectively. The external pressure was assumed to be 0.47Pc. For the postbuckling analysis, the external pressure was applied as a live load in the first step ,and in the next step, the axial compressive load was set to increase up to 1.2 times the critical axial compressive load of a perfect cylinder subjected to axial compressive load alone. Fig. 20.13a shows the load-shortening response (R/Rc vs. Ux/t) of the cylinder subject to the combined loading. Ux is the axial displacement of the cylinder. The effect of delamination length on the post-buckling response is clearly evident. It can be seen that as the delamination length increases, the load capacity drops sharply. This response also shows a dramatic loss of strength after the peak load. To investigate the effect of delamination length and external pressure on the SERR distribution along the crack front, the SERR results of six delaminated cylinders under the combined loading are presented. See Fig. 20.13b. All six delaminated cylinders are subject to the same compressive axial load. Three of the delaminated cylinders have the delamination length of a = 0.125 and are subjected to the external pressure levels of P = 0., 0.16Pc, 0.31Pc, respectively. The other three delaminated cylinders have the delamination lengths of a = 0.042, 0.08, 0.125, respectively, and are under the external pressure of 0.47Pc. The results show that with the increase of the delamination © 2008, Woodhead Publishing Limited


613

1 0.9

R

R

P

0.8 0.7

R/Rc

0.6

a = 0.042, P = 0.47 Pc a = 0.08, P = 0.47 Pc a = 0.125, P = 0.31 Pc a = 0.125, P = 0.16 Pc a = 0.125, P = 0.47 Pc a = 0.125, P = 0

0.5 0.4 0.3 0.2 0.1

L/r = 5, r/t = 30, h = 0.5, [0, 90, 0]10T

0 0.00

0.05

0.10

0.15

0.20

0.25 Ux/t (a)

0.30

0.35

0.40

0.45

0.50

60

Strain energy release rate (J/m2)

50 40 30 20 Angle (degrees)

10 0 –10

50

100

150

200

250

300

350

–20 –30 –40 –50 –60

a = 0.083, P = 0.47 Pc a = 0.125, P = 0.31 Pc

a = 0.125, P = 0.47 Pc a = 0.042, P = 0.47 Pc

a = 0.125, P = 0.16 Pc a = 0.125, P = 0

(b)

20.13 (a) Load-shortening response of a graphite-epoxy cylindrical shell under combined axial compression and external pressure; (b) Strain energy release rate distribution along the crack front of a delaminated graphite-epoxy cylindrical shell under the combined axial compression and external pressure.

length, the SERR changes dramatically. It can also be seen that the points on the crack front with zero values of SERR, displace with the increase of the delamination length. On the other hand, for the same delamination length, the increase of pressure level will also result to the increase of the SERR distribution along the crack front, as expected. © 2008, Woodhead Publishing Limited

614

20.5


Conclusion

Despite their advantages, composites suffer from layer separation or delamination which leads to loss of integrity and most likely to premature failure. The use of three-dimensional finite elements for predicting the delamination buckling of these structures is computationally expensive. Here, the combined single-layer and double-layer of shell elements are employed to study the delamination buckling and post-buckling of composite laminated shells subjected to axial compression, bending and lateral pressure, either applied individually or in combination. The automatic stabilization which applies volume proportional damping to the structure is used to predict the non-linear instability response of the structure. The total strain energy release rate along the delamination front was evaluated using the virtual crack closure technique. The present method provides an excellent framework for the buckling and post-buckling of delaminated plate and shell structures. It gives accurate results with simplicity of computer modelling, which requires relatively lower computer time and space. The results show that for very small delamination areas, the presence of delamination does not appreciably alter the critical load of a perfect geometry under the same loading condition. Irrespective of the type of load applied, the delamination located on the shell’s mid-surface creates a lower critical load. However, for larger delamination areas, as the position of delamination moves near the free surface of the laminate, the critical load is very small. In this case, the critical load is not related to the load-carrying capacity of the structure and failure will be due to the delamination growth, which depends on the fracture toughness of the material. Here, the post-buckling analysis was carried for the delamination thicknesses of 0.3 < h < 0.5. It would be desired to study the post-buckling response of the cylinder when h = 0.1. It is also observed that ignoring the effect of contact between the delaminated layers can lead to incorrect estimations of the critical loads. The results show that under pure bending, laminated cylindrical shells are more sensitive to the presence of delamination than they are under pure axial compression or external pressure. The interaction buckling curves of intact and delaminated composite cylindrical shells, relating the external pressure and axial compression, were obtained. The results show that the internal pressure increases the critical axial compressive load and delays the start of the buckling mode. On the other hand, external pressure advances the onset of the buckling mode and also delamination growth. For large delamination areas of a laminated cylinder under combined axial compression and external pressure, the SERR changes drastically with a slight increase of the external pressure. It was observed that the effects of delamination are more apparent when the composite cylindrical shells are subject to combined axial compression and bending. In this case with a slight increase of the applied bending moment, © 2008, Woodhead Publishing Limited


615

the SERR, on the compressive side of the cylinder, will change dramatically. Therefore, in some practical situations when the cylinder is subject to pure bending, once a delamination is identified, it is possible to simply rotate the cylinder to produce a useable cylinder. The interaction buckling curves of intact and delaminated composite cylindrical shells, relating the axial compressive load and bending, were obtained. The effect of the internal pressure on the interaction buckling curves was also studied. The results show that, for a very small delamination area, the presence of delamination does not appreciably alter the interaction buckling curve of an intact cylinder. For small delamination areas the interaction curve has a linear variation except those of the laminated cylinder under a very high compression load. When delamination is placed on the laminate’s mid-surface, for larger delamination areas, the shape of the interaction curve has almost a linear variation, but for very large delamination areas, the interaction curve is almost independent of the increase of the delamination area. When delamination is located closer to the free surface of the cylinder, the internal pressure can greatly increase the critical load of the cylinder and, therefore, can significantly change the interactive buckling curve. The results show that the increase of the delamination area greatly reduces the collapse load of the structure and the structure shows a dramatic loss of strength after the peak load. The FE modelling presented in this paper can be used to investigate multiple through-the-thickness delaminations in composite laminated cylinders. Therefore, multi-layer of shell elements must be employed in the delaminated region and contact elements must be placed between the delaminated layers to prevent overlapping of the layers. It is shown that computationally generated design curves can summarize the initial buckling loads of delaminated composite shell structures subject to different types of loadings. The curves should be useful for future design of shell structures subject to complex loading conditions. However, more research is needed to establish generally applicable, safe and correct guidelines for a purely numerical buckling design. The modelling approach established in this work offers high potential for further development. So far, the material properties are assumed to be linear. However, the structure of the model offers convenient extension to nonlinear behaviour, such as friction between the crack surfaces. In this study, only the geometric imperfections are considered. Therefore, it would be desirable to study the non-uniform distribution of the applied loads and various boundary conditions. It is also shown that the critical load was highly influenced by the laminate stacking sequence. Hence, the study to determine the influence of stacking sequence on SERR distribution and delamination growth in laminated composite shells needs further investigation. © 2008, Woodhead Publishing Limited

616

20.6


References

1. Tafreshi, A., ‘Efficient modelling of delamination buckling in composite cylindrical shells under axial compression’, Composite Structures, 2004; 64, 511–520. 2. Tafreshi, A., ‘Delamination buckling and post buckling in composite cylindrical shells under external pressure’, Thin-Walled Structures, 2004; 42/10, 1379–1404. 3. Tafreshi, A., ‘Instability of delaminated composite cylindrical shells under combined axial compression and bending’, Composite Structures, 2008, 82(3), 422–433. 4. Tafreshi, A., ‘Delamination buckling and post buckling in composite cylindrical shells under combined axial compression and external pressure’, Composite Structures, 2006; 72(4), 401–418. 5. Tafreshi, A. and Oswald T., ‘Global buckling behaviour and local damage propagation in composite plates with embedded delaminations’, International Journal of Pressure Vessels and Piping, 2003, 80(1), 9–20. 6. Tafreshi, A. and Bailey C.G., ‘Instability of imperfect composite cylindrical shells under combined loading’, Composite Structures, 2007, 80(1), 49–64. 7. Bolotin, V.V., ‘Delamination in composite structures: its origin, buckling, growth and stability’, Composites: Part B, 1996; 27B: 129–145. 8. Tay, T.E., ‘Characterization and analysis of delamination fracture in composites: An overview of developments from 1990–2001’, Applied Mechanics Review., 2003; 56(1): 1–31. 9. Chai, H., Babcock, C.A. and Knauss, W.G., ‘One dimensional modelling of failure in laminated plates by delamination buckling’, International Journal of Solids and Structures, 1981; 17(11); 1069–1083. 10. Klug, J., Wu, X.X. and Sun, C.T., ‘Efficient modelling of postbuckling delamination growth in composite laminates using plate elements’, AIAA J., 1996; 34(1)1, 178– 184. 11. Pavier M.J. and Clarke, M.P., ‘A specialised composite plate element for problems of delamination buckling and growth’, Composite Structures, 1996; 34, 43–53. 12. Chattopadhyay, A. and Gu, H., ‘New higher order plate theory in modelling delamination buckling of composite laminates’, AIAA J, 1994; 32(8), 1709– 1716. 13. Chai, G.B., Banks, W.M. and Rhodes J., ‘Experimental study on the buckling and post buckling of carbon fibre composite panels with and without interply disbonds’, In: Proceedings of the Institution of Mechanical Engineers: Design in Composite Materials. Mechanical Engineering Publications; 1989, 69–85. 14. Whitcomb, J.D., ‘Three-dimensional analysis of a post buckled embedded delamination’, Journal of Composite Materials, 1989; 23(9), 862–889. 15. Hu, N., ‘Buckling analysis of delaminated laminates with consideration of contact in buckling mode’, International Journal For Numerical Methods in Engineering, 1999; 44, 1457–1479. 16. Krueger, R. and O’Brien T.K., ‘A shell/3D modelling technique for the analysis of delaminated composite laminates’, Composites: Part A, 2001; 32, 25–44. 17. Kim, H.S., Chattopadhyay, A. and Ghoshal, A., ‘Characterization of delamination effect on composite laminates using a new generalized layerwise approach’, Computers and Structures, 2003; 81, 1555–1566. 18. Troshin, V.P., ‘Effect of longitudinal delamination in a laminar cylindrical shell on the critical external pressure’, Journal of Composite Materials, 1983, 17(5): 563– 567.



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19. Short, G.J. Guild, F.J. and Pavier,M.J., ‘Delaminations in flat and curved composite laminates subjected to compressive load’, Composite Structures, 2002; 58, 249–258. 20. Simitses, G.J. and Chen, Z., ‘Buckling of delaminated, long, cylindrical panels under pressure’, Computers and Structures, 1988; 28(2), 173–184. 21. Sallam, S. and Simitses, G.J., ‘Delamination buckling of cylindrical shells under axial compression’, Composite Structures, 1987; 7, 83–101. 22. Simitses, G.J., Chen, Z.Q. and Sallam S., ‘Delamination buckling of cylindrical laminates’, Thin-walled Structures, 1991; 11, 25–41. 23. Gu, H. and Chattopadhyay A., ‘Delamination buckling and postbuckling of composite cylindrical shells’, AIAA Journal, 1996, 34(6), 1279–1286. 24. Rasheed H.A. and Tassoulas J.L., ‘Delamination growth in long composite tubes under external pressure’, International Journal of Fracture, 2001; 108: 1–23. 25. Ju-fen Z., Gang, Z., Howson, W.P. and Williams, F.W., ‘Reference surface element modelling of composite plate/shell delamination buckling and postbuckling’, Composite Structures, 2003; 61, 255–264. 26. Arbocz, J., ‘The imperfection data bank, a means to obtain realistic buckling loads’. In Ramm E., Buckling of Shells. Stuttgart, Germany, 1982. 27. Simitses, G.J., ‘Buckling and postbuckling of imperfect cylindrical shells: a review’, Applied Mech. Rev., 1986; 39(10), 1517–1524. 28. Hilburger, M.W. and Starnes J.H. Jr., ‘Effects of imperfections of the buckling response of composite shells’, Thin-walled Structures, 2004; 42(3), 369–397. 29. Arbocz, J., ‘The effect of initial imperfections on shell instability’, in: Y.C. Fung, E.E. Scheler (Eds), Thin-shell Structures, Prentice-Hall, Inc., Englewood Chiffs, NJ, 1974, 205–245. 30. Winterstetter, Th. A. and Schmidt, H., ‘Stability of circular cylindrical shells under combined loading’, Thin-walled Structures, 2002; 40, 893–909. 31. ABAQUS User’s Manual, Version 6.4, Hibbit, Karlson and Sorenson, Inc. 1999– 2005. 32. Shivakumar, K.N., Tan, P.W. and Newman Jr J.C., ‘A virtual crack closure technique for calculating stress intensity factors for cracked three dimensional bodies’, International Journal of Fracture, 1988; 36, 43–50. 33. Riks, E., Rankin C.C., Brogan F.A. ‘On the solution of mode jumping phenomena in thin-walled shell structures’, Comput/Meth Appl Mech Engng, 1996, 136(1–2), 59– 62. 34. Powell G. and Simons J., ‘Improved iterative strategy for nonlinear structures’, Int. J. Numer Meth Engng, 1981; 17, 1455–1467. 35. Ramm E., ‘Strategies for tracing the nonlinear response near limit points’, in: Wunderlich E, Stein E, Bathe K.J. (eds), Nonlinear Finite Element Analysis in Structural Mechanics. Berlin: Springer, 1981.


21 Delamination failure under compression of composite laminates and sandwich structures S S R I D H A R A N, Washington University in St. Louis, USA, Y L I, Intel Corporation, USA and S E L - S A Y E D, Caterpillar Inc., USA

21.1

Introduction

The principal cause of strength degradation in laminated composites under inplane compressive loading is the activation of delamination sites by localized buckling and consequent growth of delamination. Experimental investigations on laminated composites in the 1980s at Caltech (Knauss et al., 1980, Chai et al., 1983) first highlighted the high degree of vulnerability of the composites to such a failure mechanism. Among the experimental studies on this problem that followed, the detailed work of Kutlu and Chang (Kutlu, 1991; Kutlu and Chang, 1992) in Stanford in the early 1990s is prominent. More recently, Carlsson and his associates at Florida Atlantic, see, for example, Vadakke and Carlsson (2004), Avilés and Carlsson (2006), have studied the problem extensively in the context of sandwich structures. Analytical studies of the problem have been undertaken by numerous investigators. Chai et al. (1981) in a first attempt suggested a simple onedimensional model which accounted for delamination buckling and the growth of delamination. They used the total strain energy release rate as a determinant of delamination growth. Detailed finite element analysis by Whitcomb (1981, 1986) and numerous others followed suit (Shahwan and Waas, 1995; ElSayed and Sridharan, 2001, 2002). In order to be realistic, it is vitally important to consider contact between delaminated surfaces and avoid unrealistic models of their behaviour. This is a serious stumbling block for mathematical solutions, but not for finite element codes that are prevalent today. A powerful methodology for tracing the growth of delamination is the cohesive layer modeling. This chapter describes investigations of the compressive delamination problem using cohesive layer model in differing versions. Though described briefly to provide the context, the objective here is not to expand on the methodology itself which is discussed elsewhere in the book (Chapters 13–15) in sufficient detail. The main motivation of this chapter is to examine the viability of this tool for studying the compressive delamination problem and assess the criticality of this type of failure in typical cases. 618 © 2008, Woodhead Publishing Limited

Delamination failure under compression

619

Three different case studies are presented. First is that of a composite laminate tested by Kutlu (1991) and studied in detail by El-Sayed and Sridharan (2001). One of the issues in the failure analysis is the interaction of material failure such as matrix cracking with delamination. A highly simplified analysis which accounts for the interaction is developed and employed in predicting the test results. Next, the chapter illustrates a catastrophic form of failure that occurs when a compressed sandwich column suddenly experiences delamination in its working condition, due to, say, an impact of a projectile. A nonlinear dynamic analysis incorporating a cohesive layer model is employed to investigate such a scenario. It is seen that for certain combinations of the axial compression and delamination size, the growth of delamination is abrupt and total. As a third and final example, the chapter presents an example of two-dimensional growth of delamination in a compressed laminate and highlights some features of the phenomenon. Differing versions of ABAQUS Standard/Explicit – a popular finite element analysis program – supplemented with user supplied subroutines are the analysis tools of the study and this is acknowledged by citing typical versions of the program in our reference list (Version 6.3, ABAQUS Standard, 2002; ABAQUS Explicit, 2002).

21.2

Case study (1): composite laminate under longitudinal compression

One of the several test specimens of composite laminates tested by Kutlu (1991) is selected for detailed study. The specimen has a pre-implanted delamination and is subjected to compression. A cohesive layer model in conjunction with a material damage model is employed in the study. In a first attempt only delamination is considered. This fell short of predicting accurately the behavior as given by the test. In the second stage the material damage model was added and the results improved significantly. We shall first briefly describe the cohesive layer model used.

21.2.1 Cohesive layer model for delamination (CLD model) The delaminated surfaces are connected by a thin cohesive layer having nonlinear normal and shear stress-strain responses. The strains are assumed to be small in the formulation of the model; stresses are co-rotational with the elements on which they are acting. Cohesive law The normal stress-strain relationship (Fig. 21.1) is taken in a simplified form of a cubic polynomial (El-Sayed and Sridharan, 2001): © 2008, Woodhead Publishing Limited

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σ / σ max

0.5

0 –0.2

0

0.2

0.4

0.6

0.8

1

–0.5

–1

ε / εf

21.1 Cohesive law for the CLD model.

σ = Eε f {ξ – 2 ξ 2 + ξ 3} σ=0

(ε > ε f )

21.1

where σ is the normal stress carried by the cohesive layer, E is the nominal Young’s modulus of the model material in tension, εf is the nominal tensile failure strain, and ξ = ε /ε f. A similar treatment is given for shear response of the cohesive layer, but with the important difference that the shear response is independent of the sign of the shear strain: the response for negative shear strain is a mirror image of that for positive strain. The significant parameters characterizing the stress-strain relationship are: G, the shear modulus, γf, the failure shear strain, and τmax, the maximum shear stress. The strain energy in Mode I, stored at failure per unit surface area is the strain energy release rate and is given by: G Ic =

Eε 2f h 12

21.2

where h is the thickness of the layer. The maximum stress, σmax is given by:

σ max = 4 Eε f 27

21.3

Similar equations can be written for shear (Mode II) response. The model is designated as the CLD (cohesive layer delamination) model to distinguish it from a more general model introduced later which accounts for matrix cracking. © 2008, Woodhead Publishing Limited


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Mixed-mode delamination The crack tip element is deemed to fail, as soon as the following interactive failure criterion is satisfied:

GI G + II > 1.0 G Ic G IIc

21.4

Selection of parameters Equations 21.2 and 21.3 are used to select the parameters of the model. In general, the stiffness (E) and the strength of the cohesive layer (σmax) are taken be the same as those of the composite in the transverse (out of plane) direction. Equation 21.3 gives the failure strain and Equation 21.2 gives the thickness. If the thickness turns out to be impractical, too small or too large, we use the closest practicable value for h and compute σmax. In a mixedmode problem, the thickness coming from the dominant mode, say Mode I is retained. In this case, maximum shear strain is obtained as:

γf =

12 G IIC Gh

21.5

and the maximum shear stress obtained from :

τ max = 4 Gγ 27

f

21.6

In general there was found to be considerable latitude in the selection of these parameters as along as Equations 21.2 and 21.3 for Mode I and similar ones for Mode II are satisfied.

21.2.2 Analysis details and results Specimen details The specimen under consideration is Fiberite T300/976 graphite/epoxy laminate, 2 in. (50.4 mm) long and 1 in (25.4 mm) wide, 0.1 in (2.54 mm) thick consisting of 20, 0° plies, i.e. [0]20, tested and reported on by Kutlu (1991). The material constants are given in Table 21.1. The specimen was clamped at both ends and tested in compression. In the experiment delamination was created by a thin Teflon strip 0.001 in. (0.0254 mm) thick which was pre-implanted in the specimen prior to curing. The specimen considered here had a single through-width delamination of length, d, of 0.75 in (19.05 mm) centered with respect to the middle of the plate (Fig. 21.2). The location of the delamination across the thickness is between the fourth and the fifth layers from the top.


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Table 21.1 Material properties of Fiberite T300/976 Elastic constants (msi/GPa)

Strengths (ksi/MPa)

Critical strain energy release rates (lbf/in/N/mm)

Longitudinal modulus, Ex Transverse modulus, Ey Out-of-plane modulus, Ez Poisson ratios: νxy, νxz, νyz Shear moduli: Gxy, Gxz, Gyz

0.81/ 5.585; 0.81/5.585; 0.50/ 3.45

Longitudinal tensile strength, Xt Longitudinal compressive strength, Xc Transverse tensile strength, Yt Transverse compressive strength, Yc Shear strength in XY plane S

220/1517 231/1593 6.46/44.54 36.7/253.0 15.5/106.9

Mode I, GIc

0.5/0.0876

Mode II, GIIc

1.8/0.3152

20.2/139.3 1.41/9.72 1.41/9.92 0.29, 0.29,0.40

d /2 0.2t

t

Line of symmetry

L/2

21.2 Kutlu-Chang test specimen.

Cohesive layer parameters and finite element configuration With a view to assessing the robustness of the model and the sensitivity of the simulation results to model parameters, models with differing sets of parameters were developed. A more detailed discussion is given elsewhere (El-Sayed, 2001) but here we cite in Table 21.2, two sets of model parameters and the details of the associated finite element configuration which give substantially the same results. Of these the first one (Model I) has h = 0.001 in (0.0254 mm) which is close to observed process zone thickness. This then becomes the length scale parameter of the model and acts as a delimiter for finite element refinement. From the values of GIc and GIIc, the maximum stress and the failure strain in mode I and II respectively are computed. For the second model (Model II) σmax is selected to be the same as the transverse strength of the material and results in a value of h that is 4.5 times as large as that of Model I and with about 50% reductions in failure strains. © 2008, Woodhead Publishing Limited


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Table 21.2 Details of models I and II Model Details

h, in (mm) σmax, (ksi MPa) E (msi/GPa) εf τmax G (msi/Gpa) γf

Model I Cohesive layer (CL) parameters 0.001 (0.0254) 13.63 (93.98) 1.41 (9.72) 0.065 19.6 (135.1) 0.81 (4.524) 0.1633

Finite element model CL Elements aspect ratio 1 Number of elements in CL 625 Number of elements in the model 40,000

Model II

0.00445 (0.1130) 6.46 (44.54) 1.41 (9.72) 0.031 9.3 (64.12) 0.81 (4.524) 0.0774 1 140 10,000

Calculations were performed with ABAQUS Standard version 5.9, using four-noded plane-strain reduced integration (CPE4R) elements. Meshes had more or less uniform refinement with aspect ratio maintained to be unity or close to unity. The cohesive layer model was incorporated as a user supplied subroutine defining the material properties (UMAT option). It was found that the results produced by the two models were very close (EL-Sayed and Sridharan, 2001). The second model was used for all computations in the study with the first model acting as a check. A static nonlinear analysis is used in this phase of the analysis using the arc length method. Initial imperfections in the form of the delamination buckling mode of magnitude 0.001 in. at the center of delamination are incorporated in the model. Results Figures 21.3(a–b) plot compression load versus strain on the top surface, (referred to as front strain) and the bottom surface (referred to as back strain) respectively at the line of symmetry as obtained from tests conducted by Kutlu (1991) and the cohesive model (CLD) analysis. Note the back strain predictions given by the CLD model are not plotted in the Fig. 21.3(b), as this model is unable to capture the load at which outward buckling of the lower sublaminate takes place. Considering Fig. 21.3(a) the history of strain variation can be divided into four stages. The first stage is the initial loading stage and corresponds to the portion of the graph in which the strain varies linearly with load and remains compressive. In the second stage, the front strain begins to decrease, signaling the buckling of the thin sublaminate. The buckling effect becomes pronounced at a load, P = 3500 lbf (15 569 N). The delaminated sublaminate continues © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites –0.004 0

–0.002

0

0.002

0.006

MCCL model CLD model Experimental data

–1000 Intial loading stage

Compression force, lbf

0.004

–2000

–3000 Unstable crack growth –4000 Post-buckling stage

A

–5000

–6000

Stable crack growth

–7000 Front strain (a)

–4.00E–03 0

–2.00E–03

0.00E+00

2.00E+03

6.00E+03

8.00E+03

MCCL model Experimental data

–1000

Compression force, lbf

4.00E+03

–2000

–3000

–4000

–5000

–6000

–7000 Back strain (b)

21.3 (a) Kutlu-Chang test specimen: front strain versus compression; (b) Kutlu-Chang test specimen: back strain versus compression.



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to bend out and the strain becomes tensile in the post-buckling range around 4000 lbf (17 793 N). The next important landmark is the initiation of delamination growth which occurs at 4550 lbf (20 239 N). This growth is sudden and abrupt with the load dropping slightly and must be deemed unstable. Up to this point the analysis and test results are close to each other. The discrepancy, however, begins as the extent of delamination in the unstable phase is underpredicted. The analysis predicts an unstable growth of delamination from 0.75 in (19.05 mm) to 1.06 in. (26.92 mm) with the corresponding front strain attaining a value of 0.003, but this is smaller than the value of 0.0038 recorded in the experiment. This earlier termination of delamination growth could at least be caused in part by an intrinsic deficiency of the cohesive layer models with finite thickness, as has been discussed elsewhere in the book (Chapter 14) However the analysis indicated that when the delamination reached this length of 1.06 in (26.92 mm) the material in a small but finite region in the vicinity of the crack tip exceeded its tensile strength. It is likely that this material damage contributed to the additional crack growth reported in the experiments. This point is further discussed in the next section. Continuing the analysis using the CLD model further, an increase in the load is necessary for delamination growth to continue – a stable growth. It is expected that more matrix cracks would develop around the crack tip as the crack advances. This would continue until the delamination spans the entire length of the specimens. Considering back strains and deformations of the lower sublaminate, we note that in the experiments, global buckling occurred at about 6000 lbf (26 689 N). This is indicated by the back strains increasing suddenly while the load remains more or less constant. The present CLD analysis predicts a much higher load of 7800 lbf (34 696 N) for this event. The global buckling occurred early in the experiment due to the failing of matrix in the region surrounding the crack tip as well as possible fiber fracture at the support which reduces the effective thickness of the lower sublaminate.

21.2.3 Matrix cracking model With a view to improve the predictions of the CLD model, a simplified 2-D micro-mechanical matrix cracking model was incorporated for all the elements. The following relationships between the matrix and fiber stress-strain quantities are consistent with equilibrium and compatibility conditions of 2-D elementary mechanics:

ε1c = εf = ε1m

21.7a

σ2c = σf = σ2m

21.7b

τ12c = τf = τm

21.7c


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where f and m refer the fiber and matrix respectively, subscript c refers to the composite values, 1 and 2 refer to the fiber and transverse directions respectively. From the global finite element analysis, the composite values of strain at any element, can be found. This is transformed to the principal material axes to find {ε1c, ε2c, γ12c}. Neglecting material nonlinearity, the corresponding stresses are found from material stiffness matrix:

 ε 1c   σ 1c       σ 2 c  = [ Qc ]  ε 2 c  γ  τ   12 c   12 c 

21.8

Thus we have all the stresses and strains of the composite, from which we can glean the values of {ε1m, σ2m, τm}. The corresponding conjugate quantities are found by partially inverting the following stiffness relationship for the matrix  ε1m  σ 1m      σ 2 m  = [ Qm ] ε 2 m  γ  τ   m   m 

21.9

σ 1 m   ε1m       ε 2 m  = [ Pm ] ε 2 m  τ  γ   m   m 

21.10

to read:

Note that similar relationship can be written for the fiber as well replacing Pm by Pf. The principal stresses of the matrix are then computed. If the major principal stress exceeds the tensile strength of the matrix, then the stiffness matrix written with reference to the principal axes is modified so as to relieve the normal and shear stresses. The matrix retains its stiffness and the stress parallel to the crack. The stiffness matrix and the stresses of the matrix are transformed to the principal axes of the composite and the former is designated as [ Qmcr ] . This matrix is once again partially inverted to obtain [ Pmcr ] . The composite stiffness matrix is then obtained in two steps: First find the ‘P’ matrix of the composite from: [Pc] = vf[Pf] + (1 – vf) [ Pmcr ]

21.11

where vf is the fiber volume fraction. The composite stiffness matrix is obtained by partially reinverting [Pc]. This is used in the next loading increment The composite stresses are found by volume averages of the stresses in fiber and matrix. A major difficulty arises because of the abruptness of the failure © 2008, Woodhead Publishing Limited


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processes and changes in stress locally which frequently result in loss of numerical convergence. The difficulty was overcome by resorting to dynamic analysis in its explicit form incorporating damping and applying the loading in extremely small steps. ABAQUS Explicit (version 5.8) supplemented with user supplied material subroutine (VUMAT option) was employed for the analysis. The VUMAT requires that the user supplies only the stresses at the end of each increment and all the intermediate element stiffness calculations are performed within the VUMAT module. This model which accounts for delamination via cohesive layer and matrix cracking as described above is designated as the MCCL (matrix cracking-cohesive layer) model (El-Sayed, 2001).

21.2.4 Comparison with experimental results We now revert back to considering the analytical predictions of the KutluChang test specimen, now using the MCCL model. The results given by the model are shown in Figs 21.3 and 21.4 along with those given by the CLD model. Two distinct improvements are noticeable: 1. The crack running through the cohesive layer opens up with relative ease in the absence of resistance from cracked elements adjoining the crack and a greater crack growth is recorded that accords with experimental observation.

S11

Region of highest tensile stress Failed cohesive layer elements

Rigid surface

VALUE –1.26E+06 –1.09E+06 –9.08E+05 –7.31E+05 –5.53E+05 –3.76E+05 –1.98E+05 –2.10E+04 +1.57E+05 +3.34E+05 +5.12E+05 +6.89E+05 +8.66E+05 +1.04E+06

21.4 Longitudinal stress distribution plotted on the deformed state of the laminate at P = 5900 lbf (26,244 N).


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2. The prediction of back strains as given by the CLD which was seriously inaccurate is replaced by a more reasonable one. The matrix cracking that occurs due to bending erodes both the bending and shear stiffness of the bottom sublaminate which now buckles out at a lower load of 6347 lbf (28 233 N) in contrast to 7800 lbf (34 696 N) predicted by the CLD model. In the experiment the overall buckling of the lower sublaminate occurs at 6000 lb (26 689 N) and the MCCL prediction still appears to have errors that are noticeable. Fig. 21.4 shows the longitudinal stress distribution (in psi) in the laminates which have come apart at a load of 5900 lb (26 244 N). It is easily seen that there are patches of severe stress (mostly carried by the fibers) far exceeding the material strength of 280 ksi (1931 MPa). It is obvious that fiber fracture near the support, not considered here, does occur and precipitates earlier buckling seen in the test.

21.3

Case study (2): dynamic delamination of an axially compressed sandwich column

In this section we study the response of a compressed 2-D sandwich column when subjected to a sudden through-width delamination of certain length. The compression carried by the column is small relative to the overall buckling load of the column. Because of sudden loss of equilibrium, the column exhibits a dynamic response as it adjusts itself to the new situation and significant additional growth of delamination occurs. A slightly different cohesive layer model (Sridharan and Li, 2006) which was found to have greater fidelity in sandwich problems is employed to study the delamination growth.

21.3.1 Geometry, materials and boundary conditions The details of the geometry of the column are shown in Fig. 21.5. The total length of the column investigated is 420 mm and it is considered clamped at either end. The thickness of the core and facings are 50 mm and 3.6 mm respectively. Because of symmetry involved – barring the unlikely scenario of sudden interference of antisymmetric modes of bifurcation – only half the column length of 210 mm included between the line of symmetry on the left and the clamped end on the right is considered. A rigid surface incapable of rotation and lateral translation is created to model the clamped end. Facings are made of E-glass/vinyl ester whereas the core material is H200 Divinycell PVC foam. The properties of this material system have been reported by Li (2001) and are summarized in Table 21.3. A symmetrically located initial delamination of total length 102 mm is



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Line of symmetry

u=v=0

Foam core

d = 50 mm P L = 210 mm Width = 38 mm

51mm Delamination

Composite facing, t = 3.6 mm

y,v x ,u

21.5 Sandwich column under compression. Table 21.3 Material properties of sandwich column Material property

Face sheet

Core

Young’s moduli (GPa) Ex, Ey, Ez

24.6, 11.8, 13.5

0.165, 0.263,0.165

Poisson’s ratios, υxy, υxz, υyz

0.31, 0.533,0.446

0.32, 0.32, 0.32

Shear moduli (GPa) Gxy, Gxz, Gyz

3.78, 9.19, 3.74

0.0644, 0.0644,0.0644

Critical strain energy release rate Mode I: GIc (J/m2)

1270

deemed to exist between the bottom facing sheet and the core. Mass density of the material of the composite facings and the core were assumed respectively to be 2.56 g/mm3 and 0.2 g/mm3. This constitutes significant mass scaling (by a factor of the order of 103) adopted here with the objective of slowing down the crack growth for capture by an implicit dynamic analysis. Stiffness proportional damping factor α was taken as 0.1.

21.3.2 Details of the analysis Finite element mesh configuration Plane strain elements (ABAQUS Standard, version 6.3, CPE4R) with reduced integration were employed in the analysis. The size of the elements is 0.25 mm in the longitudinal direction. The column is treated as having unit width (1 mm) and the load is reported as per unit width. The load is inducted by prescribing the end-shortening (relative to the plane of symmetry). © 2008, Woodhead Publishing Limited

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Cohesive layer: model description and parameters A version of the cohesive layer model which differs from that used in the last section, is employed to trace the delamination growth. In this model the cohesive layer has zero initial thickness and the cohesive law relates the relative displacement of the surfaces undergoing separation with the stress developed in the layer. Displacement-based mode separation technique (Sridharan, 2001) indicated that Mode I is the dominant one and therefore assumed to be the only mechanism of fracture. The normal stress varies linearly with the relative normal displacement up to the proportional limit and thereafter it remains constant (Fig. 21.6 ). For simplicity the behavior is deemed elastic (perfectly reversible stress-displacement relationship). Such a model is described in greater detail in Chapter 14. Once again ABAQUS/ Standard is employed in the analysis, but incorporating the cohesive layer model in a user supplied element (UEL option) module. The strain energy stored in the elements is found by summing up the incremental contributions in the form: m

G I = Σ σ i ∆δ i

21.12

i =1

where σi is the normal stress over the ith increment and ∆δi is the incremental relative displacement suffered by the element during the increment, m is the current increment number. The following parameters characterize the model: GIc, the proportional limit value of the relative displacement δp and the maximum stress, σmax

σ max

σ

δ, Relative normal displacement δp

21.6 Cohesive law of the model material in the sandwich problem.



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(which remains constant for δ > δp). The value of GIc for the sandwich is really controlled by the core properties as the delamination occurs not through the adhesive but runs through the core material a small distance away from the interface and this value was reported by Li (2000) as 1.21 N/mm. The value of maximum stress, σmax is taken as 10 MPa and δp is taken equal to σ 0.01 mm, which is estimated as ≈ max l , i.e., the relative displacement over Ey the process zone of thickness ᐉ (≈ 0.25 mm).

21.3.3 Static response Before the dynamic problem is studied, we shall briefly examine the static behavior of the column with a pre-existing delamination subjected to monotonic end-shortening. This will be of value in assessing the adverse effects of dynamic triggering of delamination. We note that this problem has been studied by El-Sayed and Sridharan (2002) using the UMAT type cohesive layer model discussed in Case study (1) which incorporates an unloading phase of the cohesive material response. The present study uses the UEL model for the problem and gives slightly different results. For both the static and dynamic analyses the model was perturbed slightly by incorporating imperfections in order to keep the delamination open and prompt the structure towards delamination buckling and growth. To this end, a linear stability analysis without the cohesive layer was conducted and the buckling loads and modes were obtained. The first three buckling loads were found to be 562 N, 2079 N and 3466 N respectively. Of these the first and the second pertain to delamination buckling – the former keeps the delamination fully open and the latter partially closed. The last one is an overall buckling mode. The first mode is shown in Fig. 21.7. It is clear that the first alone is critical and therefore an imperfection in the form of this mode is incorporated in the model with a maximum deflection of 0.5 mm at the center. Given the stoutness of the member (relative to the length, L/d = 7.3) and the magnitude of the overall buckling load, it is clear that overall bending of the column plays a relatively minor role in the problem. Significant features Figure 21.8 shows the variation of the load with end-shortening. The column is compressed in a quasi-static manner, so that end-shortening increases from zero to a value at which the facings undergo an average strain of 1.5%. Nonlinearity sets in as soon as the delamination tends to buckle outward and at a load of 650 N/mm (a longitudinal stress of 116 MPa in the facing sheets) delamination begins to grow. At first this is rather rapid and occurs with little increase in load (points A, B and C in the figure), but soon settles down to © 2008, Woodhead Publishing Limited

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21.7 First buckling mode of the sandwich column carrying delamination. 1800 1600 El_Sayed and Sridharan

Axial compression, P, N/mm

1400 Stable delamination growth

1200 1000

“Unstable” delemination growth A

800

B

D C

600 400

Delamination growth initiation ∆ = 0.8 mm, a = 2.50 mm ∆ = 1.0 mm, a = 21.75 mm ∆ = 1.2 mm, a = 37.50 mm

: : : :

A B C D

Incipient delamination buckling

200 a = Delamination growth 0 0

0.5

1

1.5 2 2.5 End shortening, mm

3

3.5

4

21.8 Load versus end-shortening response of sandwich column: static case.

a relatively modest rate with delamination growing under increasing load thereafter. Finally at load of 1380 N/mm (average stress of 246 MPa), a limit point is reached and thereafter delamination growth is accompanied by shedding of the load by the member. At a load of 1294 N/mm the delamination length recorded is 152.25 (the end-shortening = 3.15 mm). Compressed to an endshortening of say 3.5 mm, almost complete delamination of the bottom facing occurs, but this would necessarily be accompanied by other forms of failure not considered here. © 2008, Woodhead Publishing Limited


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Figure 21.8 also shows the results obtained earlier by El-Sayed and Sridharan (2002). It is seen that this result which is based on cohesive layer model of small but finite thickness overestimates the column stiffness at the more advanced delamination range. Also indicated in the figure are the delamination growths recorded at certain significant points in the history – in particular the values corresponding to end-shortening magnitudes of 0.8 mm, 1.0 mm and 1.2 mm respectively which are needed for comparison with those computed under dynamic delamination growth. Figure 21.12(a) shows the deformed shape when end shortening is 2 mm.

21.3.4 Dynamic response As mentioned earlier, the response of the sandwich column is studied in two steps. In the first step the nodes located on the bottom facing sheet carrying the delamination are restrained from lateral (vertical) translation, thus deactivating the delamination and a certain end-shortening, ∆ is applied. A static analysis is performed to model this situation. In the second step the restraints are released suddenly while the end shortening is kept unchanged. The sudden change in the boundary conditions results in a dynamic response resulting in a delamination that occurs in a dynamic mode. Within the duration of 0.1 sec, almost complete delamination of the bottom facing sheet occurs and this is accompanied by a significant reduction in the load carried by the column. Three cases of initially imposed end-shortening, ∆ are considered. These together with the corresponding axial loads, P obtained from the first step of the analysis are as follows: 1. ∆ = 0.8 mm, P = 804 N/mm 2. ∆ = 1.0 mm, P = 1005 N/mm 3. ∆ = 1.2 mm, P = 1206 N/mm The implicit dynamic analysis available in ABAQUS Standard is run releasing constraints of the bottom delamination. The total duration considered was 0.1 sec. Delamination process proceeds in an unstable manner under dynamic conditions. After some initial hesitation apparently due to inertial forces, delamination begins to occur at a fairly rapid rate with the load dropping rather abruptly. Thereafter the rate of crack growth somewhat moderates itself. Because this is an unstable process involving rapid crack growth, the final results obtained are dictated by the size of the time increment chosen: the larger the time increment, the smaller the crack growth. Figure 21.9 shows the variation of current delamination length with time obtained taking respectively 500, 1000, 2000 and 4000 increments over the duration of 0.1 sec for the case with ∆ = 1 mm. For a crack growth from 51 mm to 100 mm, the results do exhibit a semblance of convergence with there being only © 2008, Woodhead Publishing Limited

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200 180

n = 4000 Delamination length, mm

160

n = 2000 n = 1000

140

120

n = 500 100

n = Number of increments in 0.1 sec 80 60

40 0

0.01

0.02

0.03

0.04 0.05 Time, sec

0.06

0.07

0.08

21.9 Dynamic delamination: delamination versus time for differing time increments.

minor differences given by time increments of 50 and 25 microseconds (2000 and 4000 increments over 0.1 sec) where after the results gradually diverge. From the latter result it is seen that the delamination attains a length of 180 mm (Fig. 21.12b) and thus a good portion of the facing sheet (compare 180 mm and 210 mm) is torn away from the core. At this point, the axial load drops to 50% of the value at the end of the static phase. It is instructive to compare the extents of delamination under quasi-statically applied end compression of ∆, with the dynamic delamination under a fixed value of end shortening of ∆. Consider the case of ∆ = 1 mm; the delamination growth for the static case is 21.75 mm (Fig. 21.8) while that in the dynamic case is about 130 mm with the total delamination length recorded ≈180 mm. An end compression of 2.8 mm is needed for the same amount of delaminaton growth to occur under static conditions. It is believed that any unstable dynamic delamination growth must necessarily be very rapid and very minute time increments are needed to establish the speed of delamination growth. This has not been attempted in this study where the focus is merely on the extent of the total delamination and whether indeed a facing sheet can fully delaminate fairly abruptly. In order to overcome the rather extreme sensitivity to the time increment size, a small load is applied right at the center in lieu of the initial imperfection considered previously. This lateral load may be viewed as simulating a suction © 2008, Woodhead Publishing Limited


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presumably due to a sudden loss of pressure in the underside of the member. The small load considered is 1% of the axial load carried at the end of the static phase applied over duration of 0.1 second at the bottom of the facing sheet at the center of the column. For the case of ∆ = 1 mm, the load works out to be 5N for the half model studied. Figures 21.10 and 21.11 show respectively the variation of the load supported by the column with the delamination opening measured at the center of the column and time respectively for the three cases of end-shortening considered. Within a fraction of 0.1 second, the load in each case sustained drops to a value in the range 400N–550N and the delamination of the bottom facing is almost complete. The delamination length for the case of ∆ = 1 mm was found to be 200 mm (cf. the total length of the column = 210 mm) at 0.08 sec. The column in the deformed configuration at this stage is shown in Fig. 21.12b. This confirms the rapidity and completeness of the dynamic delamination under compression, though the actual speed has not been established.

21.4

Case study (3): two-dimensional delamination of laminated plates

The delamination studies on 2-D laminates in plane stress or plane strain do not offer a comprehensive insight into the actual delamination process in a 1400 End-shortening = 0.8 mm End-shortening = 1.0 mm End-shortening = 1.2 mm

Axial compression (N)

1200

1000

800

600

400

200

0 0

5

10 15 Crack opening (mm)

21.10 Axial compression versus crack opening.


20

25

636


1400

∆ = 1.2 mm

1200


∆ = 1.0 mm 1000

∆ = 1.0 mm ∆ = 1.0 mm Static load ∆ = 1.0 mm

∆ = 0.8 mm

800

600

400

∆ = End shortening

200

0 0

0.01

0.02

0.03

0.04 0.05 Time, seconds

0.06

0.07

0.08

21.11 Axial compression versus time.

(a)

(b)

21.12(a) Static response: deformed configuration at ∆ = 2 mm; (b) Dynamic response: deformed configuration at ∆ = 1 mm, t = 0.08 sec.



637

laminated plate in which delamination occurs in a certain lamination plane and the rate of growth of delamination is different in different directions. The cohesive layer model offers itself as a powerful tool for such an analysis. However this calls for a 3-D analysis which is demanding at once from standpoints of computation and visualization of the results. An early attempt at solving such a problem is by Nilsson and Storakers (1992). This work uses linear fracture mechanics concepts, von Karman plate theory and an evaluation of strain energy release rates along the delamination front. A cohesive layer approach to the problem was developed by Wagner et al. (2001) to study the crack growth in a plate carrying a circular delamination. In this study, the delamination is clearly overshadowed by overall buckling and the influence of delamination itself appears to be minimal. The work of Yu (2001) and Ortiz and Pandolfi (1999) appear to be the most comprehensive in this area, but their main focus is in the Mode II delamination of the composite plate under high speed impact. In this chapter relatively simple scenarios of twodimensional delamination propagation in a composite laminate under static compression are investigated using a cohesive layer model.

21.4.1 Motivation Consider a delamination in the form of, say, a circular region near the surface of laminate. The following issues are thought to be of interest: (a) Interaction of overall bending of the laminate with delamination buckling: This interaction can shut off delamination growth by virtue of contact of delaminated surfaces – so much so delamination ceases to be of serious concern at least in the static response. (b) The directional growth of delamination: The work of Nilsson et al. (1993) has indicated that delamination grows in a direction perpendicular to the direction of compression but their analysis is applicable only to a thin film bonded to a rigid substrate. This issue needs to be examined in a more general setting. (c) The capability of the cohesive layer model which is an extension of the one dimensional model discussed in the previous section and in Chapter 14 to trace the delamination growth efficiently and without numerical glitches.

21.4.2 Development of the cohesive layer model An element in the form of a triangular prism is selected to represent the cohesive layer. The element has six nodes with three displacement-degrees of freedom at each node, with linear interpolation functions both in the plane of the triangular cross-section and as well as across the thickness. As in the


638


earlier UEL model used in the context of one-dimensional delamination, initial thickness is taken as zero and relations between stress and the corresponding relative displacement (δ) are linear up to a certain δo where after the stress remains constant as the δ increases. Normal and tangential displacement components Consider an element DEFGHJ in the form of a triangular prism where the triangles DEF and GHJ are situated in the lower and upper surfaces of the delamination respectively in the undeformed configuration. These surfaces are separated by a small distance h → 0, (h = thickness of the cohesive layer). Let ∆abc be the middle surface of the prism where a, b and c are respectively the middle points of DG, EH and FJ respectively as shown in Fig. 21.13. The element has 18 degrees of freedom, three displacement components of the six nodes. Let them be denoted by: {q}T = {u1D , u 2D , u3D , u1E … … u1F , … … u1G … … u1H … … u1J … u3J}T

21.13 Consider the deformed configuration of the element. Let D′, E′… J′ be the displaced positions of D, E … J respectively and likewise let a′, b′ and c′ be the positions of a, b and c respectively. Their coordinates ( x ia , x ib , x ic ) are readily determined as the arithmetic mean of those of D′ and G′, E′ and H′ and F′ and J′ respectively. Let the sides of the triangle a′b′c′ be denoted by vectors: r r r r r r 21.14 {ab} = b – a ; {bc} = c – b; ( ca ) = a – c J c

F G a

H

D b E

21.13 3-D element for cohesive layer model.



639

r The unit normal vector, z to the plane a′b′c′ is obtained as a cross product vectors {ab} and {bc} as follows: r {ab} × {bc} z= |{ab} × {bc}|

21.15

r r Let the vector {ab} be denoted by x . A unit vector y orthogonal to both x and r zr is rso determined that x, y and z constitute a right-handed system. y=z×x rrr [ x y z ] constitute the local coordinate system for Now the three vectors rrr T the element and [ x y z ] constitutes the transformation matrix [α]. The displacements {u1(N) u2 (N) u3 (N)} at any node designated as N can be transformed to the local system by the transformation: {u′} = [α] {u}.

Relative displacements, nodal forces and stiffness matrix The relative displacements of the top and bottom surfaces of the element are:

∆i =

( u i′ ( G ) + u i′ ( H ) + u i′ ( J ) – u i′ ( D ) – u i′ ( E ) – u i′ ( F )) 3

(i = 1, 2, 3)

21.16

Here ∆3 is the normal relative displacement, positive when tensile and ∆1 and ∆2 are relative shear displacements and they give rise to normal and shear forces at the bottom and top surfaces of the element in opposite senses. The relationship between the normal stress (σn) and ∆3 and the shear stresses (τ1 and τ2) to ∆1 and ∆2 respectively are taken in a bilinear form:

∆3 σ δ 1 o max = σ max

σn =

∆ 3 ≤ δ 1o

21.17a

∆ 3 > δ 1o

Similarly, ∆i τ δ 2 o max = τ max

τi =

∆i ≤ δ 2o

21.17b

∆i > δ 2o

(i = 1, 2) where δ1o and δ2o are proportional limits of relative normal and shear displacements. The nodal forces at the top nodes G, H, J are taken to be equal respectively r r r in each of the x , y , z directions and are given by:

F1 = 1 τ 1 A; F2 = 1 τ 2 A; Fn = 1 σ n A 3 3 3 © 2008, Woodhead Publishing Limited

21.18

640


The forces in the bottom nodes D, E, F are equal and directed in the opposite directions respectively. Thus the load vector takes the form:

 – F1 – F2 – Fn | – F1 – F2 – Fn | – F1 – F2 – Fn |  {P}T =   + F1 + F2 + Fn | + F1 + F2 + Fn | + F1 + F2 + Fn  

T

21.19 The element tangential stiffness matrix coefficients are obtained by appropriate differentiation. A general expression for a typical coefficient may be written in the form:

K ij =

∂Pi ∂∆ 3 ∂Pi ∂∆ 1 ∂Pi ∂∆ 2 + + ∂∆ 3 ∂q j ∂∆ 1 ∂q j ∂∆ 2 ∂q j

(i, j = 1, …18)

21.20

When a certain ∆ exceeds the proportional limit, the corresponding diagonal terms in the tangential stiffness become zero. In order to avert possible numerical difficulties associated with a singular stiffness matrix, an arbitrarily small value is used for the stiffnesses in that case. The nodal load vector and the stiffness matrix are transformed to the global coordinate system as follows: {PG} = {α]T {P}

21.21a

[KG] = [α]T[K][α]

21.21b

These are returned to the main program at the end of each iteration. Only the load vector is modified from iteration to iteration whereas the stiffness matrix is kept unchanged during an increment of loading. Failure criterion The strain energies in the three modes stored in the element up to the mth increment are found by summing up the work of current stress acting over incremental relative displacements d∆’s. Thus: m

m

m

j =1

j =1

j =1

G I = Σ σ n j d∆ 3 j ; G II = Σ τ 1 j d∆ 1 j ; G III = Σ τ 2 j d∆ 2 j

21.22a–c

The interactive criterion for failure of the crack tip element and delamination extension is taken in the form:

GI G G + II + III = 1 G Ic G IIc G IIIc


21.23


641

21.4.3 Details of the examples studied The problem considered is one of delamination in a laminate (Fig. 21.14(a)) in the form of a rectangular plate (16 mm × 12 mm). A circular pre-implanted delamination of 1 mm radius is deemed to exist between the top sublaminate and the comparatively thick substrate at the bottom. The laminate is made of carbon-epoxy composite with the following properties: E1 = 139 GPa, E2 = E3 = 9.86 GPa, ν12 = ν13 = 0.3;

ν23 = 0.0; G12 = G13 = 5.24 GPa; G23 = 4.93 GPa. The fibers in all the laminae are oriented in the longitudinal (X-direction) of the plate. The critical values of the strain energy release rates are: GIc = 0.2 N/mm2, GIIc = 0.6 N/mm2.

12

a

16 (a) Note: the corner D is hidden. H

G C

Y

6

Z a F

E

X A

B

8 (b)

21.14 (a) The rectangular laminate with delamination: full model (Dimensions in mm). (b) The rectangular laminate with delamination: 1 /4 model (dimensions in mm).


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Scenarios studied The following scenarios of delamination with differing thickness of the bottom substrate and the support conditions are considered. Geometry The top layer in all the cases has a thickness of 0.1 mm and the bottom substrate has a thickness either of 0.8 mm (Cases I, II and III) or 0.3 mm (Cases IV and V). Cohesive layer model parameters The 3-D cohesive layer model (described in the previous section) has the following parameters: σmax = 26.0 N/mm2; τmax = 95.0 N/mm2; δ10 = 1 × 10–3 mm, δ20 = 1 × 10–5 mm (Case I); δ10 = 1 × 10–2 mm, δ20 = 1 × 10–4 mm (Cases II–V). Boundary conditions The out of plane displacement (w) is restrained over the edge surfaces in all the cases. Out of plane rotations of the surfaces are not restrained. In case V in addition, the bottom substrate is completely restrained from moving the z-direction and is thus prevented from buckling. Advantage was taken of the double symmetry of the plate and thus only a quarter of the plate was analyzed (Fig. 21.14b). The compression is inducted by requiring end sections perpendicular to x-axis to be rigid and letting it move in the x-direction relative to the plane of symmetry, but allowed to rotate about its center line (y-axis). Thus the face BFGC is moved relative to the plane AEHD for longitudinal compression. Contact condition The model is generally capable of generating compressive stresses at the interface to keep the interpenetration of delaminated surfaces to within acceptably low limits, but there is no internal control of interpenetration. In order to effectively eliminate interpenetration, the delaminated surfaces may be declared as ‘contact pair’ and thus invoking contact constraints option available in ABAQUS. This was done in case III and this is the only difference between Case II and Case III. Table 21.4 gives the foregoing details for ready reference.



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Table 21.4 Details for cases I–V Case I

Case II

Case III

Case IV

Case V

Thickness of substrate

0.8

0.8

0.8

0.3

0.3

Thickness of top layer

0.1

0.1

0.1

0.1

0.1

δ10

1.0E-3

1.0E-2

1.0E-2

1.0E-2

1.0E-2

δ20

1.0E-5

1.0E-4

1.0E-4

1.0E-4

1.0E-4

Bottom substrate restrained from deflecting in z direction

NO

NO

NO

NO

YES

Contact pair condition imposed

NO

NO

YES

NO

NO

21.15 The discretization of the laminated plate structure.

21.4.4 Finite element analysis Three-noded triangular facet thin shell elements – STRI3 element type in ABAQUS – are employed. The plate was subdivided into two layers with identical configuration of elements in the horizontal plane. The discretization was made fine in the vicinity of delamination front. The configuration is depicted in Fig. 21.15 The nodes of the top plate (sublaminate) are located at the bottom of the plate and those of the bottom plate (sublaminate) are located at the top of the plate by using the offset command available in ABAQUS. The cohesive layer element connects these nodes everywhere except in the region of preimplanted delamination. A geometrically nonlinear analysis is performed in several steps increasing the end-shortening. The total end-shortening in the first step is so selected that no delamination growth occurred, but the cohesive layer elements along the delamination front were in the verge of failure. The second step consisted of a large number of small increments with the objective of limiting failure © 2008, Woodhead Publishing Limited

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to first set of elements along the delamination front within an increment. Loads of 2N each were applied at the center in the z-direction in equal and opposite senses to keep the delamination open.

21.5

Results and discussion

The key results of all the cases under longitudinal compression are plotted in Fig. 21.16a, b. Figure 21.16a plots the axial compression versus end-shortening; Fig. 21.16b plots the axial compression versus crack opening, i.e., relative displacement of the delaminated surfaces in the z-direction at the center of the plate. Cases I, II and III The results of the Cases I, II and III are quite close except that the convergence difficulties were experienced in Case III: more iterations are needed (as the nodes come in and go out of contact) for each increment and greater tolerance in convergence criteria had to be accepted to move the solution forward. Numerical results, however, were not noticeably different. The first element failed initiating the delamination growth, at an axial compression of 1333N (indicated as Fi in the Fig. 21.16a). This is close to the critical load corresponding to delamination buckling which is 1316N as given by the linear stability analysis. The delamination growth continues till a load of 2517 N is reached (this load is indicated as Ff in the figure). The delamination buckling and growth are accompanied by an overall bending downwards of the laminate which tends keep the delaminated surfaces in contact. So much so the delamination growth shuts down, but bending of the laminate continues with increasing end-shortening. Figure 21.17 shows the deformed configuration of the plate at an axial compression of 2542 N. Case IV This is case of relatively thin laminate (thickness = 0.3 mm) in which the overall buckling plays a dominant role. The overall buckling load as given by the linear stability analysis is 144 N whereas the delamination buckling load is 546 N. At about a load of 468 N the delaminated surfaces come into contact and crack opening displacement ceases to grow there after (Fig. 21.16b) However, a small number of elements (four in fact) do fail at the delamination front over a relatively narrow range of loading (Fi = 652 N and Ff = 684 N) and a larger number fail in shear in a number of locations near the supports. Figure 21.18 shows the deformed configuration with clearly noticeable overall buckling.



645

3000


2500

Case V

2000 Case I, II & III 1500

1000 Case IV 500

0 0

0.01

0.02

0.03

0.04 0.05 0.06 End shortening (mm) (a)

3000

0.07

0.08

0.09

0.1

Ff = 2644

Case V


2500

Ff = 2517

Fi = 2043 2000

Case I, II & III

1500

Fi = 1333 1000

Ff = 684

500

Fi = 652

Case IV

0 0

0.1

0.2

0.3 0.4 Crack opening (mm) (b)

0.5

0.6

0.7

21.16 (a) Axial compression versus end-shortening at the center of the plate; (b) Axial compression versus crack opening at the center of the plate.

Case V This is once again a case of a laminate with the smaller thickness of 0.3 mm; the important difference, however, is that the plate is restrained at the bottom © 2008, Woodhead Publishing Limited

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21.17 Deformation configuration for case III at P = 2542 N (drawn with 3-D elements for clarity).

21.18 Deformed configuration for Case IV at P = 686 (drawn with 3-D elements for clarity).

from deflecting in the z-direction. Thus overall buckling is prevented and only delamination buckling is liable to occur. The crack initiation occurs at axial compression of 2043 N and continues without interruption, though not at a uniform rate (Fig. 21.16a,b) The maximum load attained is 2860 N. Figure 21.19a–c show the deformed configuration at several critical stages. It is observed that delamination grows rapidly in the transverse direction under axial (longitudinal) compression. Within a small increase 0.0027 mm of end-shortening (from 0.0724 mm to 0.0751 mm), the delamination hits the longitudinal boundaries. This is illustrated in Fig. 21.19a–b. It is only after the delamination in the transverse direction is complete, does it begin to grow in the longitudinal direction. An almost complete delamination of the top sheet occurs in this case as seen in Fig. 21.19c. Note that the preferential growth of delamination in the transverse direction was presaged in the work of Nilsson et al. (1993). The fact the present model is able to capture this © 2008, Woodhead Publishing Limited


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(a)

(b)

(c)

21.19 (a) Deformed configuration at end shortening = 0.070 mm; P = 2700N (drawn with 3-D elements for clarity); (b) Deformed configuration at end shortening = 0.075mm; P = 2666N (drawn with 3-D elements for clarity); (c) Deformed configuration at end shortening = 0.082 mm; P = 2820N (drawn with 3-D elements for clarity).

phenomenon does appear to confirm the validity and efficiency of the cohesive layer model.

21.6

Conclusion

1. The cohesive layer model used in conjunction with a material failure model was found to be a powerful tool for tracing with ease delamination growth and eventual failure of a composite laminate. © 2008, Woodhead Publishing Limited

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2. Sandwich columns carrying compression can fail by delamination buckling rather suddenly if a face sheet-core disbond with greater than a threshold length is suddenly triggered in them. 3. The triangular prismatic cohesive layer element is able to trace 2-D delamination growth with facility. The overall bending of the laminate can result in shutting down of delamination growth in a laminate under compression. When the overall buckling is prevented by appropriate constraints, the delamination grows rapidly. For a laminate under longitudinal compression, delamination grows first in the transverse direction. Only after hitting the longitudinal boundaries, does the delamination grow in the longitudinal direction.

21.7

References and further reading

ABAQUS/Standard/User’s manual, Hibbit, Karlsson and Sorensen, inc., Pawtucket, RI, (for example) version 6.3, 2002, also ‘UMAT: User subroutine to define a material’s mechanical behavior’, pp. 24.2.30.1-24.2.30.14; ‘UEL: Define an Element’, pp. 24.2.19.1–24.2.19.17. ABAQUS/Explicit/User’s manual, Hibbit, Karlsson and Sorensen, inc., Pawtucket, RI, (for example) version 6.3, 2002, also ‘VUMAT: User subroutine to define material behavior’, pp. 21.2.4.-1–21.2.4.-13. Avilés, F. and Carlsson, L.A. (2006), ‘Experimental study of debonded sandwich panels loaded in compression’, J. Sandwich Struct. Mater., 8(1), 7–31. Chai, H., Babcock, C.D. and Knauss, W.G. (1981), ‘One dimensional modeling of failure in laminated plates by delamination buckling’, International Journal of Solids and Structures, 1981, 17(11), 1069–1083. Chai, H., Knauss, W.G. and Babcock C.D. (1983), ‘Observations of damage growth in compressively loaded laminates’, Experimental Mechanics, 23, 329–337. El-Sayed, S. (2001), ‘Delamination-driven failure processes in two-dimensional Composite Structures’, D. Sc thesis, Washington University in St Louis. El-Sayed, S. and Sridharan, S. (2001), ‘Predicting and tracking interlaminar crack growth in composites using a cohesive layer model’, Composites: Part B, 32, 545–553. El-Sayed, S. and Sridharan, S. (2002), ‘Performance of a cohesive layer model in the prediction of interfacial crack growth in sandwich beams’, Journal of Sandwich Structures and Materials, 4, 31–48. Knauss, W.G., Babcock, C.D. and Chai, H. (1980), ‘Visualization of impact damage of composite plates by means of Moire technique’, NASA CR-159261, April. Kutlu, Z. (1991), ‘Compression response of laminated composite panels containing multiple delamination’, PhD Thesis, Stanford University. Kutlu, Z. and Fu-Kuo Chang (1992), ‘Modeling compression failure of laminated composite containing delamination’, Journal of Composite Materials, 26(3), 350–387. Li, X. (2000), ‘Debonding fracture of foam core the tilted sandwich structure’, PhD Thesis, Florida Atlantic University, Boca Raton, FL. Li, X. and Carlsson, L. (1999), ‘The tilted sandwich debond (TSD) specimen for face/ core interface fracture characterization’ Journal of Sandwich Structures and Materials, 1, 60–75.



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Nilsson, K.F. and Storakers, B. (1992), ‘On interface crack growth in composite plates’, Transactions of the ASME, 59, 530–538. Nilsson, K.F., Thesken, J.C., Sindelar, P., Giannakopoulos, A.E. and Storakers, B. (1993), ‘A theoretical and experimental investigation of buckling induced delamination growth’, J. Mech. Phys. Solids, 41(4), 749–782. Ortiz, M. and Pandolfi, A. (1999), ‘Finite-deformation irreversible cohesive elements for three dimensional crack propagation analysis’, International Journal of Numerical Methods in Engineering, 44, 1267–1282. Shahwan, K.L. and Waas, A.M. (1997), ‘Non-self-similar decohesion along a finite interface of unilaterally constrained delaminations’, Proceedings of the Royal Society of London, 453, 515–550. Sridharan, S. (2001), ‘Displacement-based mode separation of strain energy release rates for interfacial cracks in bi-material media’, International Journal of Solids and Structures, 38, 6787–6803. Sridharan, S. and Li, Y. (2006), ‘Static and dynamic delamination of foam core sandwich members’, AIAA Journal, 44(12), 2937–2948. Vadakke, V. and Carlsson, L.A. (2004), ‘Experimental investigation of compression failure of sandwich specimens with face/core debond’, Compos. Part B, 35(6–8), 583–590. Wagner, W., Gruttman, F. and Sprenger, W. (2001), ‘A finite element formulation for the simulation of propagating delaminations in layered composite structures’, International Journal of Numerical Methods in Engineering, 51, 1337–1359. Whitcomb, J.D. (1981), ‘Finite element analysis of instability related delamination growth’, J. Composite Materials, 15(9), 403–426. Whitcomb, J.D. (1986), ‘Parametric analytical study of instability-related delamination growth,’ Composites Science and Technology, 25(1), 19–48. Yu, C. (2001), ‘Three dimensional cohesive modeling of impact damage of composites’, PhD, Thesis, Caltech, Pasadena, CA.


22 Self-healing composites M R K E S S L E R, Iowa State University, USA

22.1

Introduction

Fiber-reinforced polymer (FRP) products offer multiple advantages over the metal components they are replacing in many market segments including aerospace, marine, automotive, civil infrastructure, construction, oil and gas, sports and recreation, electronics, and telecommunications. In the aerospace field, aircraft designers are continuing to incorporate a greater percentage of high performance composites into their designs. In the sports and recreation market, advanced composites are found in skis and snowboards, surfboards, golf club shafts, tennis and racquetball rackets, baseball bats, fishing poles, archery equipment, pole-vaulting poles and bicycle frames. In the marine market, composites, with their exceptional corrosion resistance, are used for boat hulls and sail masts. In the automotive industry, composites are meeting a need in enabling energy-efficient vehicles by reducing overall vehicle weight. FRPs for exterior structural panels, hoods, fuel storage tanks, composite driveshafts, and floor panels are being increasingly introduced into new vehicle models. Some of the advantages of using FRPs include vastly improved specific strength and stiffness, part count reduction, corrosion resistance, electromagnetic transparency and design flexibility. FRPs for structural applications typically consist of long fiber reinforcement (such as carbon, glass, or Kevlar) surrounded by a continuous polymer matrix. The polymer matrices are either thermosets, such as polyesters, vinyl esters, epoxies, phenolics, cross-linked cyclic olefins (such as poly-dicyclopentadiene), cyanate esters, bismaleimide and polyimides, or thermoplastics, such as polyetheretherketone (PEEK), polyetherketone (PEK), polyamide-imide (PAI), polyarylsuffone (PAS), polyetherimide (PEI), polyethersulfone (PES), and polyphenylene sulfide (PPS). Thermoset resins, which are more common in structural FRPs, are cured by an irreversible chemical reaction into a crosslinked polymer. While, thermosets typically have strong interfacial bonds and good chemical resistance, they often are relatively brittle. Matrix microcracking, 650 © 2008, Woodhead Publishing Limited


651

which is a problem in both classes of polymer matrix composites, becomes a precursor to other larger damage modes through crack coalescence. Because of the non-random orientation of the fibers, composite materials are intrinsically anisotropic, meaning their material properties vary with direction. For unidirectional composites, the difference between the material properties in the longitudinal direction (in the direction of the fibers) and the transverse direction (perpendicular to the fiber direction) can vary significantly. For this reason individual plies are stacked at varying orientations to form laminates with reinforcement in multiple directions. Alternately, bidirectional reinforcement can be achieved in one ply by using woven fabric. Often a lightweight structural core, such as synthetic honeycomb or foam, is sandwiched between outer plies for added weight savings and increased bending stiffness. FRPs are especially susceptible to matrix microcracking when subjected to repeated thermomechanical loading. When these matrix microcracks coalesce, other damage modes, including fiber/matrix debonding and ply delamination, develop within the composite. The result is a long term degradation of material properties. This damage is difficult to detect and even more difficult to repair because it often forms deep within the structure. Once this damage has developed, the integrity of the structure is greatly compromised. Currently composite parts that have been damaged in service are first inspected manually to determine the extent of damage. For critical parts this inspection may include such nondestructive testing (NDT) techniques as ultrasonics, infrared thermography, X-ray tomography, and computerized vibro thermography. If the damage is too severe the structural component is replaced entirely. For less extensive damage, repairs are attempted. If localized delamination has occurred it may be repaired by injecting resin via an access hole into the failed area. Another common repair method is the use of a reinforcing patch bonded or bolted to the composite structure. Numerous studies regarding these and other composite repair methods have been published, yet all require time-consuming and costly manual intervention by a trained technician. One area of composites that opens up new application possibilities is multifunctionality. Multifunctional composites are composite systems that provide a variety of functions beyond the role of acting exclusively as a structure. Examples include composites with embedded structural health monitoring or active fiber composites for suppression of vibration and acoustic control (Bent and Hagood, 1997). Another class of multifunctional composites is self-healing systems with the ability to heal cracks autonomically, without the need for external intervention. The first successful realization of selfhealing functionality to heal delamination damage in fiber-reinforced composites was recently reported by Kessler et al. (2003) using an embedded © 2008, Woodhead Publishing Limited

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microcapsule approach. In the following sections, this and alternative strategies to achieve self-healing multifunctionality are discussed.

22.2

Self-healing concept

When living systems are damaged, they often exude liquid to the site of injury to supply vital nutrients, filling, clotting and healing. Examples include the secretion of sap in damaged wood and blood in bone fractures. Over time, these living systems heal, often with strength and functionality approaching those of the undamaged system. Self-healing synthetic materials are inspired by such living systems, where damage elicits a healing response at the site of injury and the damage is repaired autonomously over time. While the traditional approach to material degradation in synthetic structural materials is to optimize the material for strength and toughness to slow down the process of micro- and macro-crack formation, these biomimetic materials offer a radically different approach. Rather than avoiding the formation of damage, the material is engineered to detect and repair the damage after the micro- and macro-cracks have formed. Three different approaches to achieving self-healing functionality have been successfully demonstrated to date in polymers and/or polymer matrix composites: (1) microcapsule approaches, (2) hollow fiber/mesoporous network approaches, and (3) thermally activated approaches. These concepts are illustrated in Fig. 22.1. The first two approaches involve the use of a liquid healing agent which fills the damage region to initiate healing, while the third approach requires a thermal stimulant to achieve healing in a solid state system. While the initial work has been with unreinforced polymers, a growing effort into the development of self-healing structural composites is ongoing in the research community. It is envisaged that as these materials become further refined, and as the price of producing them drops, they will begin to find use in a wide variety of composite applications.

22.2.1 Microcapsule approaches While experiments reporting the concept of self-repair had been reported in the early 1990s (see, for example, Wool, 1995; Dry and Sottos, 1993; Dry, 1996), the field of self-healing polymers and composites really began in 2001, when a team of researchers at the University of Illinois first published their work on the healing of an unreinforced thermoset epoxy by microencapsulated liquid healing agents (White et al., 2001). In this work, a liquid healing agent, dicyclopentadiene (DCPD), was first microencapsulated in a urea-formaldehyde shell. These microcapsules, with an average diameter of 220 µm, were dispersed throughout an epoxy matrix along with a suspended solid catalyst phase (Grubbs catalyst) and cured in place. Healing is triggered © 2008, Woodhead Publishing Limited

Self-healing composites Polymer Resin matrix system Polymer matrix

Catalyst

Hardner system

653

Hollow fiber

Capsule of liquid monomer

Crack plane

Monomer flows into crack

Contact with catalyst

(a) (b)

(c)

22.1 Self-healing concept using (a) embedded microcapsules, (b) embedded hollow fibers, (c) thermally remendable polymers (adopted with permission from Chen et al., 2002 and Kessler, 2007).

by crack propagation through the microcapsules, which then release the healing agent into the crack plane. Subsequent exposure of the healing agent to the chemical catalyst initiates polymerization and bonding of the crack faces. Experiments to quantify the effectiveness of self-healing using mode 1 fracture toughness experiments resulted in a healing efficiency of 75% (defined as the ratio of healed to virgin fracture toughness). These materials were further optimized (by varying the microcapsule size and loading of healing agent and catalyst) to recover as much as 90% of their virgin toughness (Brown et al., 2002). Figure 22.2 shows a fractured surface of such a selfhealing material; a ruptured microcapsule that has emptied its contents into the crackplane gives evidence of the crack induced microcapsule rupture. © 2008, Woodhead Publishing Limited

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22.2 Scanning electron microscope image of a ruptured microcapsule on a fracture plane of a self-healing epoxy.

In a similar strategy, Cho et al. (2006) reported on the development of a vinyl ester matrix material in which a PDMS-based healing agent is phase separated in the matrix during curing (through a reaction induced phase separation) creating a de facto encapsulated healing agent in the matrix. In this approach a tin-based catalyst is encapsulated separately and mixed with the vinyl ester matrix before it is cured in place with the PDMS healing agent. The phase separated healing agent approach, while limited to specific healing agent chemistries and not resulting in such high healing efficiencies, has the advantage of simplifying processing as the healing agent can be simply mixed into the polymer matrix. While the research reported on self-healing materials had initially been with neat-resin (unreinforced) systems, the self-healing concept holds exciting promise for fiber-reinforced composites. Matrix microcracks, which inevitably develop in FRPs, often lead to delaminations between plies which are difficult to detect and repair. A preliminary study addressing the feasibility of using the self-healing concept for healing interlaminar fracture damage in woven composite materials was first reported by Kessler et al. (2001). In this study the self-healing process was simulated by injecting respectively catalyzed and uncatalyzed monomer into the delaminations of double cantilever beam fracture specimens and measuring the healing efficiencies relative to the virgin fracture toughness. This study established that the healing agent system was capable of achieving reasonable levels of repair in composite materials and that the embedded catalyst remains active after composite curing and is still capable of triggering the ROMP polymerization of the healing agent. A follow-up paper (Kessler et al., 2003) demonstrated that autonomic healing © 2008, Woodhead Publishing Limited


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efficiencies of large scale delamination damage as high as 45% at room temperatures, and 80% efficiency at elevated temperatures could be obtained. This was the first and most successful demonstration of self-healing in a fiber-reinforced system published to date.

22.2.2 Hollow fiber/mesoporous network approaches A similar approach to the embedded microcapsule concept is to embed hollow fibers throughout the matrix which are filled with a single or multiple-part healing agent system. A key requirement for this concept is the ability to produce hollow fibers of sufficiently small size to combine with the structural reinforcing fibers (usually glass, carbon, or Kevlar). Hucker et al. (1999) developed a method of producing hollow borosilicate glass fibers from tubular performs with a high degree of hollowness and outer diameters from 20 to 100 µm. These hollow glass fibers were later added to traditional fiber reinforcements and cured into composite panels. The hollow fibers were subsequently filled with various resin systems (primarily two-part epoxy resin) using vacuum infusion and the fiber ends sealed to create self-healing systems (Trask and Bond, 2006; Pang and Bond 2005a, b). These composite panels were damaged by simulated impact and the residual flexural strengths of the composites were compared to undamaged control samples; substantial healing was demonstrated. Figure 22.3 shows some of the hollow glass fibers which are used to store the liquid healing agent. ~50 µm

22.3 Scanning electron microscope image of broken hollow fibers which can be manufactured into the composite to deliver their liquid healing agent into the crack plane (photo courtesy of Ian Bond, University of Bristol).


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While healing of delamination damage was not specifically investigated, one could easily imagine the incorporation of hollow glass fibers filled with appropriate resin healing agents stitched through the z-axis of the composite to simultaneously increase the virgin interlaminar fracture toughness (via through-thickness reinforcement) and provide for healing of eventual delamination damage when the fibers fracture and release their content into the crack plane. One advantage of using hollow fibers to store the healing agent is that there is potentially more healing agent available to be delivered to the crack plane than in the microcapsule based system. This allows for the possibility of having multiple healings provided that the healing liquid continues to be available at the damage region. Eventually, however, the liquid healing agent will all be used up or the hollow tubes will become clogged, unless some type of recirculation system is employed. Such systems have been suggested in several review articles (Kessler, 2007; Trask et al., 2007), but implementation is just in its infancy. These conceived systems are quite biomimetic, as they closely resemble healing found in the complex integrated microsystems found in nature. Two approaches to implementing the vascular network necessary to implement a circulatory system include the development of a large tubular network in a composite sandwich structure and a coating-substrate design where a three dimensional microvascular network in the epoxy substrate is used to deliver healing agent to cracks that develop in the outer coating (Toohey et al., 2007). Both systems offer approaches which open up new avenues for continuous delivery of healing liquids for multiple healing cycles.

22.2.3 Thermally activated healing While the healing approaches described thus far require an internal liquid healing agent to flow to the damage zone and polymerize in the crack planes, another approach to healing uses solid state material and external heat to facilitate repair. To date, the heat has been manually supplied to the damaged material, so these systems are less than fully autonomous. However, as damaged triggered heating strategies are developed (for example, through internal heating elements that sense the damage and respond with resistive heating at the damage zone) these materials hold significant promise. While the use of thermal treatment to mend or weld thermoplastics is well documented, the thermal healing of selected glassy thermosets has only recently been demonstrated. Thermally re-mendable A thermally re-mendable system specially designed by thermally reversible Diels-Alder (DA) cycloaddition of multi-furan and multi-maleimide is © 2008, Woodhead Publishing Limited


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described by Chen et al. (2002). This thermosetting material can be fractured (regenerating the furan and maleimide groups on the crack plane through retro-DA reactions), clamped together, and heated to again reform DA adducts across the crack phase. The advantage of this system is that it doesn’t require a separate liquid phase healing agent or suspended catalysts in the matrix and multiple crack healings are possible. However, an obvious disadvantage is the need for external heat application and immediate surface contact (pressure) to regenerate the broken covalent bonds across the crack plane. It also requires the custom synthesized matrix material which has not been used in conventional composite structures. Thermoset/thermoplastic blends Another thermally activated solid phase healing approach was recently demonstrated where a specially selected thermoplastic is mixed with a conventional epoxy matrix (Hayes et al., 2007). The thermoplastic, poly(bisphenol-A-co-epichlorohydrin), was carefully selected to have a similar solubility parameter to the epoxy matrix so that a homogeneous (not phase separated) system resulted upon cure of the epoxy blend. Then when the material is damaged, slight pressure is applied and the temperature raised (100–140°C) so that the thermoplastic polymer can gain enough molecular mobility to migrate across the crack plane, creating physical entanglements rather than covalent bonds. These systems have the advantage of allowing the use of standard thermosetting matrices and processing conditions but are also limited by their need for elevated temperatures and intimate crack surface contact to achieve repair.

22.3

Healing-agent development

22.3.1 Requirements for healing-agent system Since the majority of the self-healing research to date has focused on the first two approaches with a liquid healing agent, it is worth looking at the requirements for a successful healing-agent system. Several are listed below: • • •

Long shelf-life. The healing agent should be stored as a dormant liquid in the embedded microcapsules until the capsule is ruptured by an approaching microcrack. Low viscosity. Once released, the healing agent must be drawn into the crack by capillary action or crack closure. Reactivity. Once in the crack plane, the healing agent should react quickly at ambient temperature to bond the crack surfaces closed. The kinetics are a tricky variable, if the reaction takes place too quickly then the healing agent may polymerize before filling the entire crack, if the reaction


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•


is too slow the monomer may evaporate or diffuse away from the crack plane. Compatible adhesive. The polymerized healing agent must be a strong adhesive to the matrix and reinforcement.

Significant work on the development and optimization of various olefin metathesis based polymerization strategies showing substantial promise is discussed in the following sub-section.

22.3.2 Ring-opening metathesis polymerization One polymerization strategy that meets many of the requirements listed above is based on the ring-opening metathesis polymerization of strained cyclic olefins (such as DCPD). These polymerizations are made possible by the highly stable, functional group tolerant catalysts (such as the ruthenium based Grubbs catalysts) that have emerged within the last decade in the organic chemistry community. The importance of these remarkable catalysts led to the awarding of the 2005 Nobel Prize in Chemistry to Yves Chauvin, Robert H. Grubbs, and Richard R. Schrock. The initial ROMP healing agent in the seminal work in Nature by White et al. (2001), used the endo-isomer of DCPD as the liquid healing agent and Grubbs first generation ruthenium catalyst for the suspended chemical trigger. These materials worked well when relatively long healing times (~10–24 hrs) and high catalyst loadings (2–5 wt%) were employed. Additional healing agents which react much faster than the baseline endo-DCPD/1st generation Grubbs catalyst are now being investigated by several groups. One candidate is the exo-isomer of DCPD which has been shown to be much more reactive in ROMP than endo-DCPD primarily for steric reasons (Rule and Moore, 2002). A more readily available monomer with even higher reaction rates is 5-ethylidene-2-norbornene (ENB) (Lee et al., 2004). Mixtures of ENB with DCPD allows the materials engineer to control the reaction kinetics and ultimate thermo-mechanical properties of the healing agent system. Several bi-cyclic crosslinking agents have also been developed which can be blended at various concentration with ENB or DCPD in order to carefully control the crosslink density (and resulting Tg, modulus, and gel time) of the healing agent system (Sheng et al., 2007). In addition to modifying the healing agent, one can also choose alternate ROMP catalyst systems. While the first generation Grubbs catalyst provides good functional group tolerance and rapid reaction rates at room temperature, the second generation catalyst is even more robust to various functional groups and has a faster polymerization rate (though slower initiation rate) than the first generation system. Both systems are limited for practical applications because of their high cost due to the expensive ruthenium transition


Table 22.1 Key contributions in self-healing materials Healing approach

Material system

Contribution

Reference(s)

Embedded microcapsules

Epoxy matrix, UF microcapsules containing DCPD, suspended Grubbs catalyst. ,,

White et al., 2001

Epoxy matrix/woven carbon fiber composite with microencapsulated healing agent.

First successful demonstration of self-healing in a thermosetting polymer. Further refinement of material system to investigate the influence of healing time, catalyst and microcapsule loading, catalyst form, and damage mode. First successful demonstration of self-healing in a structural composite material.

Phase separated healing agent

Vinyl ester matrix with phase separated siloxane-based healing agent and encapsulated catalyst.

Self-healing demonstrated in a system where the healing agent is dispersed via phase separation.

Cho et al., 2006

Hollow fiber

Hollow glass fibers containing two part resin system embedded in an epoxy matrix composite.

Demonstration of self-healing functionality using embedded hollow glass fibers.

Trask et al., 2006, Pang and Bond, 2005a,b

Microvascular network

Epoxy coating with microvascular network containing DCPD healing agent.

Demonstration of repeated self-healing by using interconnected micro-channels.

Toohey et al., 2007

Thermally remendable

Specially designed matrix produced by thermally reversible Diels-Alder (DA) cycloaddition of multi-furan and multi-maleimide.

Thermosetting material demonstrated which can be fractured, clamped together, and heated to reform DA adducts across the crack phase. Multiple crack healings possible.

Chen et al., 2002, Chen et al., 2003

Self-healing composites 659


Brown et al., 2002, Brown et al., 2005, Rule et al., 2005, Jones et al., 2006, Mauldin et al., 2007, Kessler et al., 2003

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Healing approach

Material system

Contribution

Reference(s)

Thermoset/thermoplastic blend

Blend of a thermoplastic resin (poly(bisphenol-A-coepichlorohydrin) and a thermosetting epoxy (diglycidyl ether of bisphenol A (DGEBA)).

Thermally activated healing using system that can be processed using conventional thermosetting composite techniques.

Hayes et al., 2007





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metal core. One alternative that has recently been suggested is a tungsten (VI) chloride mixed with various co-activators (Rule, 2005). While this system is significantly less robust (functional group tolerant) than the ruthenium-based Grubbs catalysts, if appropriately protected through encapsulation or coatings, it may allow for the development of economically feasible self-healing matrix materials. Several different ROMP monomer healing agents and catalysts are listed in Table 22.2 along with appropriate references.

22.3.3 Cure kinetics vs. catalyst dissolution kinetics The issue of healing kinetics is a complicated one. If the polymerization kinetics of the healing-agent monomer is too fast then the monomer may gel and polymerize too quickly to completely fill the damage region. Likewise, the healing agent may contact the suspended catalyst phase and polymerize around it before the catalyst has fully dissolved into the liquid healing agent, quenching the polymerization of the entire healing-agent film. However, if the polymerization of the healing agent is too slow, then the volatile monomer may diffuse away from the crack plane before it has polymerized completely. In order for the healing to be most effective, the monomer should have enough time to thoroughly dissolve the embedded catalyst that it encounters in the crack before polymerization in order for the polymerized healing agent to most effectively bond the crack faces together. Thus, the dissolution kinetics of the healing agent and catalyst are coupled to the polymerization kinetics. For optimum healing, the dissolution kinetics should be as fast as possible. One way to do this is to decrease the size and or crystalline morphology of the embedded catalyst (Jones et al., 2006; Larin et al., 2006). However, if the crystal catalyst particles are too small, they may be dissolved by the host matrix during processing and not be available as an active species on the crack plane. Another approach is to modify the solubility of the healing agent system (by blending or modifying the structure of the monomer) in order to change the solubility parameter of the healing agent to more closely match that of the catalyst.

22.4

Application to healing of delamination damage in FRPs

22.4.1 Manufacturing issues While the production of self-healing composite polymers has been well studied (though, depending on the system there can be multiple complications), the presence of the reinforcing fiber and the processing to manufacture selfhealing laminated composite parts can raise several additional issues. In the © 2008, Woodhead Publishing Limited

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Table 22.2 ROMP healing agent systems Monomers

Expected benefit

References

Endo-DCPD

•

Baseline system

White et al., 2001 Kessler et al., 2003 Liu et al. 2007

• •

Faster polymerization kinetics Lower catalyst loadings

Rule et al., 2002 Larin et al., 2006 Mauldin et al., 2007

•

Much faster polymerization kinetics Lower catalyst loadings Improved low temperature performance

Lee et al., 2004

H H

Exo-DCPD

H H

ENB

• • Mixtures of ENB and DCPD

•

Faster polymerization kinetics with high temperature healing capability

Kessler et al., 2006 Liu et al., 2006 Larin et al., 2006 Lee et al., 2007

Mixtures of ENB and Crosslinking agents

•

Much faster polymerization kinetics Controlled levels of crosslink density and glass transition temperatures

Sheng et al., 2007

• •

Baseline system Good functional group tolerance

Schwab et al., 1996 White et al., 2001 Kessler et al., 2002 Jones et al., 2006

• •

Faster polymerization kinetics Excellent functional group tolerance

Scholl et al., 1999

•

Much cheaper than Ru-based catalysts

Rule, 2005

•

Catalysts 1st Generation Grubbs’ Catalyst PCy3 Cl Ru Ph

Cl PCy3

2nd Generation Grubbs’ Catalyst Mes N

N Mes Cl

Ru Cl

Ph PCy3

Tungsten (VI) Chloride based systems WCl6 + co-activator



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microencapsulated healing-agent systems, the hollow microcapsules and the suspended catalyst phase can alter the viscosity of the resin prepolymer, complicating compaction and consolidation. In addition, care must be taken not to rupture the hollow tubes or microcapsule during the composite manufacture process. Likewise, if a liquid composite molding process (such as resin transfer molding) is used, the fiber preform may selectively filter the larger microcapsules, retarding resin flow and resulting in a less than homogeneous distribution of the capsules through the matrix. Often the selfhealing composite must be specially designed with process changes and reinforcement architectures altered due to the self-healing functionality developed. Likewise the cure schedule of the matrix material must be compatible with the reactive self-healing agent or sold phase healing functionality. Typically, this limits the upper cure temperatures or the matrix resin system and excludes many of the high Tg materials, such as bismaleimides, which require a high temperature cure schedule. Reinforcement architecture Some reinforcement architectures are more appropriate for certain healing strategies than others. In high fiber volume fraction, unidirectional composite laminates there is little room in the matrix material to accommodate additional components in the matrix such as microcapsules, catalysts, or stitched hollow fibers. Woven composites, however, are particularly good candidates for the use of a self-healing polymer matrix because of the architecture of the reinforcement. Over recent years, woven fabrics have become an important form of reinforcement in aerospace composites. Weaving provides a network of reinforcing fibers that improves the interlaminar fracture toughness and impact resistance. These materials are good for self-healing functionality because the large resin-rich interstitial areas, formed by the interlacing of undulating warp and fill yarns, serve as natural sites for storage of microcapsules since their presence will not disrupt the inherent undulation of the fiber tows (as would occur for unidirectional fiber tows). Depending on the architecture of the weave and the fiber volume fraction, a large number of microcapsules can be stored in the interstitial regions without significantly changing the bulk material properties of the composite. Catalyst and microcapsule agglomeration One of the problems that has been found in characterizing the delamination fracture of self-healing laminated composites using the embedded microcapsule strategy is that the addition of microcapsules and catalyst cause the delamination © 2008, Woodhead Publishing Limited

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to sometimes propagate in an unstable manner. Also the virgin toughness of the self-healing system is reduced compared to similar systems without selfhealing functionality. This is likely due to the agglomeration of microcapsules and catalyst clusters which were formed during the processing of the composite and the resulting increased thickness of the interlaminar region in these systems (Kessler et al., 2003). If the catalyst and microcapsule agglomeration can be eliminated, then these issues should be significantly resolved. An additional suggestion is to investigate different dispersion methods for the catalyst, such as attaching the catalyst to the exterior of the embedded microcapsules or to incorporate the healing agent into the reinforcement phase directly. This latter suggestion is especially relevant for systems susceptible to delamination damage, because the delamination typically causes matrix/fiber debonding exposing the glass, carbon, or other fibers to the fracture surface. If the catalyst trigger were also available at this interface, it would increase its availability to polymerize the released healing agent.

22.4.2 Assessing repair efficiency There are many ways that healing efficiency can be characterized. They all, however, involve (1) measuring some virgin material property, (2) introducing damage into the material, (3) allowing the healing process to take place, and (4) measuring the same property of the healed material and comparing it to the virgin property and the residual property for a material that was damaged but not allowed to heal. For characterizing delamination damage in fiberreinforced composites, assessment of repair efficiency can be demonstrated on double cantilever beam (DCB) and width-tapered DCB fracture specimens in which a mid-plane delamination is introduced and then allowed to heal. The healing of other, more complex and/or coupled, damage modes can also be characterized by measuring properties such as the compression strength after impact. In the work previously mentioned (Kessler and White, 2001; Kessler et al. 2003), the effectiveness of healing interlaminar fracture damage on the macroscopic scale was assessed by using double cantilever beam (DCB) and width-tapered DCB specimens loaded in mode-I to compare the fracture toughness of the virgin material to the fracture toughness measured after crack closure and healing. A healing efficiency is defined as

η=

Healed K IC Virgin K IC

=

Healed G IC Virgin G IC

22.1

This definition for healing efficiency was chosen to match that used in the work related to healing of neat resin samples. Three different types of interlaminar fracture specimens were evaluated: (1) reference specimens in which the healing agent is manually catalyzed and then injected into the © 2008, Woodhead Publishing Limited


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delamination, (2) self-activated specimens where the catalyst is embedded in the polymer matrix and the healing agent is manually injected into the delamination, and (3) self-healing specimens in which microcapsules of the liquid healing agent and the catalyst are embedded into the polymer matrix and healing is autonomic. The first two types represent control experiments to verify that the polymerized healing agent is a suitable healing agent for the composite substrate and that the embedded chemical trigger is capable of effectively polymerizing the injected healing agent. These systems represent the upper bounds for what we might expect in the self-healing specimen. The three specimen types are illustrated in Fig. 22.4. Double cantilever beam (DCB) specimen The loading curve for a typical DCB reference specimen is shown in Fig. 22.5. A crack is propagated in the mid-plane of the virgin composite specimen. Then the healing agent is injected into the crack plane and the specimen is closed shut. After the healing agent has polymerized, it is retested. While the test is being performed, a video camera is recording the crack length on the side of the composite specimen. The crack length data is superimposed with the load displacement data in order to calculate the critical energy release rate of the sample as the crack propagated (both for the virgin and healed cases) from simultaneously recorded load, crack opening displacement, and crack length. This approach works relatively well when the crack (delamination)

Reference specimen

Carbon/epoxy DCPD and Grubbs’ catalyst

(a) Grubbs’ catalyst

Self-activated specimen DCPD (b) Microencapsulated DCPD Self-healing specimen Grubbs’ catalyst (c)

22.4 Three types of delamination specimens tested (reprinted with permission from Kessler et al., 2003).


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Delamination behaviour of composites h 60 A b

50

L

B: DCPD & catalyst injected

40

P (N)

ao

Virgin

C

E

30 Healed

D

20

10

0

0

10

20

30

40

50

δ (mm)

22.5 Typical double cantilever beam load-displacement curve for virgin and healed reference specimens: (A) Crack propagation commences for the virgin specimen ahead of the precrack. (B) Loading of the virgin specimen is completed and a mixture of DCPD monomer and Grubbs’ catalyst (1.38% wt.) is injected with a syringe into the delamination. After injection, the specimen is unloaded and the delamination is closed. (C) Crack propagation commences for the healed specimen. (D) The crack has propagated through the entire healed region. (E) Further loading creates a new ‘virgin’ crack ahead of the previously healed region. (Reprinted with permission from Kessler and White, 2001).

length can easily be measured. Where it is difficult to measure the crack length, width-tapered DCB specimens may be more appropriate. Width-tapered double-cantilever-beam (WTDCB) specimen In self-healing systems, measurement of the true crack length can be difficult since the crack surface may be continuously healing even as it is growing. Similarly, there may be regions of the crack plane where various levels of healing are achieved, further complicating crack length measurements. In these cases, where it is particularly difficult to measure the crack, it is desirable © 2008, Woodhead Publishing Limited


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that the energy release rate remain constant with crack length. By contouring the width of the standard DCB specimen, the energy release rate can be made independent of crack length. These width-tapered double-cantileverbeam (WTDCB) specimens (Fig. 22.6), first introduced by Mostovoy (1975), have been used by a number of investigators to measure Mode I interlaminar fracture toughness. The analysis of the width-tapered double-cantilever-beam specimen is based on linear elastic fracture mechanics and beam mechanics of a tapered beam such as the cantilevered beam configuration shown in Fig. 22.7. One can begin by expressing the energy release rate in terms of the sample compliance for displacement control dΠ  GI = –  = 1 P 2 dC dA  dA  δ 2

22.2

where Π is the potential energy of the specimen, A is the fracture area, P is the load, δ is the opening displacement, and C is the compliance (δ/P). If the width taper of the specimen is such that a/b = k, where k is a constant (see Fig. 22.7), then the change in fracture area is

d A = b ( a ) da = 1 d( a 2 ) 2k

22.3

where a is the crack length and b(a) is the width. Then combining Eqs. (22.2) and (22.3) gives

GI = kP 2 dC2 d( a )

22.4

2h

Loading blocks Teflon insert

22.6 Geometry of WTDCB specimen.


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b

a

P

h y x

22.7 Geometry of width-tapered cantilever beam.

From basic mechanics of materials the deflection of a beam in pure bending is governed by d2 y M( x ) = – EI dx 2

22.5

where y is the deflection of the beam at a distance x from the beam end, M is the bending moment (Px for the WTDCB specimen), E is the bending modulus of elasticity, I is the moment of inertia (bh3/12 for a rectangular beam), and h is the beam height (half thickness of the specimen). If one assumes perfect cantilever end conditions such that dy/dx = y = 0 at x = a and a constant tapered beam width, the solution to Equation 22.5 becomes y( x ) = – 6 Pk3 ( x 2 – 2 ax + a 2 ) Eh

22.6

The displacement at the end of the WTDCB specimen where the load is applied is then

δ = 2| y (0)| =

12 Pka 2 Eh 3

22.7

The compliance of the specimen is then 12 ka 2 C= δ = P Eh 3

22.8

If one takes the differential of (Equation 22.8) with respect to a2 and substitutes into Equation 22.4, the expression for energy release rate is © 2008, Woodhead Publishing Limited


12 P 2 k 2 GI = kP 2 dC2 = d( a ) Eh 3

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22.9

Note that Equation 22.9 is independent of crack length and only involves one unknown variable, P. The stress intensity factor can be expressed as GI E 3 = 2 Pk (1 – ν 2 ) (1 – ν 2 ) h 3

KI =

22.10

where ν is Poisson’s ratio. The inclusion of the term (1-ν2) presumes a plane strain condition. Thus, the healing efficiency for the WTDCB specimen by combining Equations 22.1 and 22.10 gives:

η=

Healed K IC Virgin K IC

=

PCHealed

22.11

PCVirgin

where PC is the critical load for crack propagation. Figure 22.8 shows a typical load displacement curve for a WTDCB selfhealing specimen. The virgin load is characterized by two large load drops as the crack propagates unstably across the laminate mid-plane upon reaching critical loads. The specimen was then clamped shut and allowed to heal at 80°C for several hours. The subsequent loading curve followed the initial 50 Virgin Healed

Load, P (N)

40

30

20

10

0 0

5

10 15 20 25 Crack-opening displacement, δ (mm)

30

22.8 Typical WTDCB loading curve for virgin and healed in situ specimens (healed at 80°C). (Reprinted with permission from Kessler et al., 2003).


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Table 22.3 Summary of healing of delamination damage (adapted from Kessler et al., 2003) Specimen type

No. of samples

KIC virgin avg. (MPa·m1/2)

KIC healed peak (MPa·m1/2)

ηmax avg. (%)

Reference

6

3.58 (0.22)*

3.82 (0.20)

107

Self-activated

6

2.54 (0.12)

2.09 (0.13)

82

Self-healing (@ room temp)

8

2.85 (0.22)

1.29 (0.25)

45

Self-healing (@ 80°C)

4

2.79 (0.30)

2.23 (0.18)

80

Notes: * ±1 standard deviation

slope of the virgin material until crack growth began through the healed region at a peak load of 33 N. The averages of all the healing efficiencies for the various control and self-healing WTDCB specimen are listed in Table 22.3 (Kessler et al., 2003). Other damage modes Another important step in the assessing the effectiveness of self-healing composites is the evaluation of other damage modes and mechanisms. The previous work has only focused on healing a large single crack or delamination in a laboratory specimen. However, in most applications for self-healing composites, the damage modes will include multiple, small-scale cracks, which may not coalesce into a single delamination. Similarly, different damage modes may respond differently to the self-healing concept. One test that can be used to evaluate the healing efficiency in these more complex damage modes is the compression after impact (CAI) test, described in detail in a SACMA standard SRM 2-94 (SACMA, 1994). The impact-induced microcracking will rupture the microcapsules, triggering the healing action. After sufficient time is allowed for the monomer to polymerize, the degree of healing can be determined by comparing the CAI strength with control samples. Also the ability to introduce, quantify, and heal small-scale microcracking (which if placed under repeated thermomechanical loading would eventually lead to large-scale damage such as delamination) is an important step forward in self-healing research. Healing the microcracks could delay or prevent largescale damage as opposed to healing the large-scale damage after it has formed.

22.5

Conclusions

Self-healing composites possess great potential for solving some of the most limiting problems of polymeric structural materials: microcracking and hidden © 2008, Woodhead Publishing Limited


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damage. Microcracks are the precursors to structural failure and the ability to heal them will endow structures with longer lifetimes and less maintenance. Filling microcracks will also mitigate the deleterious effects of environmentally assisted degradation such as moisture swelling. This chapter has summarized the various strategies for self-healing materials and focused on the microcapsule based approach for healing of delamination damage in structural composites. The potential benefits are quite high; however, this technology is still in its infancy and several practical limitations need to be overcome in order for the benefits to be fully realized.

22.6

References

Bent A and Hagood N W (1997) ‘Piezoelectric fiber composites with interdigitated electrodes’, Journal of Intelligent Material Systems and Structures, 8(11), 903–919. Brown E N, Sottos N R and White S R (2002) ‘Fracture testing of a self-healing polymer composite’, Experimental Mechanics, 42(4), 372–379. Brown E N, White S R and Sottos N R (2005) ‘Retardation and repair of fatigue cracks in a microcapsule toughened epoxy composite – Part II: In situ self-healing’, Composites Science and Technology, 65, 2474–2480. Chen X, Dam M A, Ono K, Mal A, Shen H, Nutt S R, Sheran K and Wudl F A (2002) ‘Thermally re-mendable cross-linked polymeric material’, Science, 295, 1698– 1702. Chen X, Wudl F, Mal A K, Shen H and Nutt S R (2003) ‘New thermally remendable highly cross-linked polymeric materials’, Macromolecules, 36, 1802–1807. Cho S H, Andersson H M, White S R, Sottos N R and Braun P V (2006) ‘Polydimethylsiloxane-based self-healing materials’, Advanced Materials, 18, 997– 1000. Dry C (1996) ‘Procedures developed for self-repair of polymeric matrix composite materials’, Comp Struct, 35, 263–269. Dry C and Sottos N (1993) ‘Smart structures and materials’ in Smart Materials (ed. Varandan, VK) Vol 1916, 438 (SPIE Proceedings, SPIE, Bellingham, WA, 1993). Hayes S A, Jones F R, Marshiya K and Zhang W (2007) ‘A self-healing thermosetting composite material’, Composites Part A: Applied Science and Manufacturing, 38(4), 1116–1120. Hucker M, Bond I, Foreman A and Hudd J (1999) ‘Optimisation of hollow glass fibres and their composites’, Adv. Compos. Lett., 1999, 8(4), 181–189. Jones A S, Rule J D, Moore J S, White S R and Sottos N R (2006) ‘Catalyst morphology and dissolution kinetics of self-healing polymers’, Chemistry of Materials, 18, 1312– 1317. Kessler M R (2007) ‘Self-healing: a new paradigm in materials design’, Proceedings of the Institution of Mechanical Engineers. Part G, Journal of Aerospace Engineering, 221(4), 479–495. Kessler M R and White S R (2001) ‘Self-activated healing of delamination damage in woven composites’, Composites Part A: Applied Science and Manufacturing, 32(5), 683–699. Kessler M R, Sottos N R and White S R (2003) ‘Self-healing structural composite materials’, Composites Part A: Applied Science and Manufacturing, 34(8), 743–753.


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Kessler M R, Larin G E and Bernklau N (2006) ‘Cure characterization and viscosity development of ring-opening metathesis polymerized resins’, Journal of Thermal Analysis and Calorimetry, 85(1), 7–12. Larin G E, Kessler M R, Bernklau N and DiCesare J C (2006) ‘Rheokinetics of ringopening metathesis polymerization of norbornene based monomers intended for selfhealing applications’, Polymer Engineering and Science, 46(12), 1804–1811. Lee J K, Hong S J, Liu X and Yoon S H (2004) ‘Characterization of dicyclopentadiene and 5-ethylidene-2-norbornene as self-healing agents for polymer composite and its microcapsules’ Macromol. Res., 12(5), 478–483. Lee J K, Liu X, Yoon S H and Kessler M R (2007) ‘Thermal analysis of ring opening metathesis polymerized healing agents’, Journal of Polymer Science Part B: Polymer Physics, 45, 1771–1780. Liu X, Lee J K, Yoon S H and Kessler M R (2006) ‘Characterization of diene monomers as healing agents for autonomic damage repair’, Journal of Applied Polymer Science, 101, 1266–1272. Liu X, Sheng X, Kessler M R and Lee J K (2007) ‘Isothermal cure characterization of dicyclopentadiene: the glass transition temperature and conversion’, Journal of Thermal Analysis and Calorimetry, 87(2), 435–457. Mauldin T C, Rule J D, Sottos N R, White S R and Moore J S (2007) ‘Self-healing kinetics and the stereoisomers of dicyclopentadiene’, Journal of the Royal Society Interface, 4(13), 389–393. Mostovoy S (1975) ‘Fracture mechanics for structural adhesive bonds’, 10th Interim Report, LR27614-10, Contract F 33615-75-C-5224, AFML. Pang J W C and Bond I P (2005a) ‘Bleeding composites’ – damage detection and selfrepair using a biomimetic approach’, Composites Part A: Applied Science and Manufacturing, 36, 183–188. Pang J W C and Bond I P (2005b) ‘A hollow fibre reinforced polymer composite encompassing self-healing and enhanced damage visibility’, Composites Science and Technology, 65, 1791–1799. Rule J D and Moore J S (2002) ‘ROMP Reactivity of endo- and exo-Dicyclopentadiene’ Macromolecules, 35, 7878–7882. Rule J D (2005) Polymer Chemistry for Improved Self-Healing Composite Materials. PhD Thesis, University of Illinois at Urbana-Champaign, Urbana, IL. Rule J D, Brown E N, Sottos, N R, White S R and Moore J S (2005) ‘Wax-Protected Catalyst Microspheres for Efficient Self-Healing Materials’, Advanced Materials, 17, 205–208. SACMA Recommended Test Method SRM 2-94, Compression After Impact Properties of Oriented Fiber-Resin Composites, Suppliers of Advanced Composite Materials Association, Arlington, VA (originally issued April 1989, revised 1994). Scholl M, Ding S, Lee C W and Grubbs R H (1999) ‘Synthesis and activity of a new generation of ruthenium-based olefin metathesis catalysts coordinated with 1,3 dimesityl4,5-dihydroimidazol-2-ylidene ligands’ Org. Lett., 1, 953–956. Schwab P, Grubbs R H and Ziller J W (1996) ‘Synthesis and applications of RuCl2(=CHR’)(PR3)2: the influence of the alkylidene moiety on metathesis activity’, J. Am. Chem. Soc., 118, 100–110. Sheng X, Kessler M R and Lee J K (2007) ‘The influence of cross-linking agents on ringopening metathesis polymerized thermosets’, Journal of Thermal Analysis and Calorimetry, 89(2), 459–464.



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Toohey K, Lewis J A, Moore J S, White S R and Sottos N R (2007) ‘Self-healing materials with microvascular networks’, Nature Materials, 6, 581–585. Trask R S and Bond I P (2006) ‘Biomimetic self-healing of advanced composite structures using hollow glass fibres’ Smart Materials and Structures, 15, 704–710. Trask R S, Williams H R and Bond I P (2007) ‘Self-healing polymer composites: mimicking nature to enhance performance’, Bioinsp. Biomim. 2, P1-P9. White S R, Sottos N R, Geubelle P H, Moore J S, Kessler M R, Sriram S R, Brown E N and Viswanathan S (2001) ‘Autonomic healing of polymer composites’, Nature, 409, 794–97. Wool R P (1995), Polymer Interfaces: Structure and Strength Ch 12 445–479 (Hanser Gardner, Cincinnati, OH).


23 Z-pin bridging in composite delamination H-Y L I U, The University of Sydney, Australia and W Y A N, Monash University, Australia

23.1

Introduction

Traditional fibre composites are manufactured by stacking together a number of plies, in which the fibres are orientated to provide in-plane reinforcement for the composite. A direct consequence of this process is that no fibres are positioned across the laminate thickness. Interlaminar delamination becomes the most common failure mode in composite laminates. A successful solution to this problem is to provide through-thickness reinforcement to the laminated composites because bridging by reinforcing fibres in the laminate thickness provides direct closure tractions to the delamination crack faces. Over the last decade, many techniques have been developed to enhance the strength of the composite laminates in the thickness direction, or z-direction. Among them, a novel approach, so-called z-pinning, has been developed (Freitas et al., 1994). In this technique, short pins initially contained in foam are inserted into the composite through a combination of heat and pressure compacting the foam. The pins are made of titanium alloy or fibrous carbon composite and account for 0.5% to 5% of the volume content of the laminate. The size of the pins is typically between 0.1 and 1.0 mm in diameter (Cartie, 2000). Liu, Yan and their co-workers in CAMT have been investigating z-pin bridging in composite delamination since 2001, which includes the experimental studies on the z-pin bridging law, mode I delamination of z-pinned composite laminates, loading rate and fatigue behaviour of z-pinned composite; and theoretical studies of mode I and mode II delamination (Liu et al., 2006). These investigations are reviewed in this chapter. Experimental study on zpin pull tests to establish bridging law is first presented in Section 23.2. Section 23.3 focuses on the theoretical investigation and finite element analyses of mode I, mode II delamination and buckling of z-pinned laminates. The loading rate effect on z-pin bridging law and mode I delamination from experimental study is discussed in Section 23.4. Fatigue degradation on z-pin bridging force is presented in Section 23.5. 674 © 2008, Woodhead Publishing Limited


23.2

675

Z-pin bridging law

When a delamination crack opens, a z-pin experiences an axial tension. As a reaction, the z-pin provides a closure bridging force to the laminates against delamination growth. This process can be described by the load-displacement curve of z-pin pullout. The functional relationship between the bridging force and associated crack opening during pullout is called the bridging law. To understand the effect of z-pin on composite delamination, the z-pin bridging law is one of the most essential issues to study. In this section, the test method of z-pin pullout is introduced and a closed-form bridging law, which can be used for computer simulation of z-pinned composite delamination, will be suggested.

23.2.1 Z-pin pullout test Z-pin pullout test is used to determine directly the relationship between the bridging force and pullout displacement (Dai et al., 2004). The test set-up for pullout of a 3 × 3 z-pins sample is shown in Fig. 23.1. The z-pins are vertically inserted into the central areas of two composite prepregs by an ultrasonic insertion machine before curing. Generally, an insertion machine is composed of a standard power supply and a sequence of transducer, signal booster and insertion horn. The inserting force comes from the ultrasonic vibration of the horn, which can be controlled through the power supply. Different types of ultrasonic insertion machines have been developed by Aztex Inc (Cartie, 2000). The prepreg was 40 mm long and 20 mm wide. A

z-pins

Laminates

23.1 Illustration of experimental configuration for 3 × 3 z-pins pull-out tests.


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thermally insulated film with a thickness of 10 µm is inserted between the upper and lower laminates to avoid any adhesive bonding between them. Two T-shaped tabs (20 mm long and 15 mm wide) are glued to the top and bottom surfaces of the laminates and are firmly secured in the testing machine. Load-displacement curves can be recorded until the pins are completely pulled out. Since the displacements recorded by the machine also include the deformations of the two T-shaped tabs, a separate tensile test on the tabs should be done to measure their load-displacement curve. The measured displacement of the z-pinned sample will be modified by taking away the deformation of the tabs from the total displacement. Z-pin pullout tests with different pin sizes were carried out (Dai et al., 2004). The difference between the experimental results from a similar set of samples is reasonably small. Additionally, all the measured load-displacement curves shows very similar patterns which can be well represented by the curve shown in Fig. 23.2, in which, Ps is the bridging force and δ is the displacement of a single z-pin. The whole process consists of three stages. In the first stage, 0 ≤ δ ≤ δ1, the interface between the z-pin and the laminates is fully bonded. The bridging force is caused by the elastic deformation of the z-pin. With the load increasing, the interfacial shear stress between z-pin and laminates exceeds the interfacial shear strength. Debonding starts and propagates. Hence, in the second stage,δ1 ≤ δ ≤ δ2, the bridging force is caused by both the elastic deformation (in the bonded region) and interfacial friction (in the debonded region). After the interface has fully debonded, the z-pin is pulled out from the laminates. The bridging force in this stage, δ2 < δ ≤ h, is completely caused by the interfacial friction, in which h is halflength of a z-pin and is equal to the thickness of one cantilever beam. The four parameters, maximum debonding force Pd, maximum frictional pull-

Pull-out force Ps

(δ 1 , P d )

( δ 2 , Pf)

0

Displacement δ

h

23.2 Load-displacement curve of z-pin pullout.



677

out force Pf and their corresponding displacements δ1 and δ2 can be determined by the pullout test.

23.2.2 Recommended bridging laws Based on the experimental results from z-pin pullout tests, it is suggested that the functional relationship between the bridging force and z-pin displacement can be best characterized by following tri-linear law δ  δ 1 Pd  δ – δ δ – δ1 Ps =  2 Pd + Pf δ δ δ – 2 1 2 – δ1  P + δ 2 – δ P  f h – δ 2 f

(0 ≤ δ ≤ δ 1 ) (δ 1 ≤ δ ≤ δ 2 )

23.1

(δ 2 < δ ≤ h)

In some cases, if there is no (or very weak) initial bonding between the pin and the laminates, the bridging law can be simplified to a bi-linear function determined by two parameters: maximum load and corresponding displacement. Hence, Equation 23.1 is reduced to the first and third expressions with Pd (= Pf) and δ1 (= δ2), which is the case of frictional pullout so that δ P 0 ≤ δ ≤ δ2 δ f Ps =  2 Pf  Pf + (δ – δ ) δ 2 ≤ δ ≤ h h – δ2 2 

23.2

This pullout model is totally determined by the peak bridging force, Pf, its corresponding pullout displacement, δ2, and the ultimate pullout displacement, h, equal to half-thickness of the DCB.

23.3

Effect of z-pin bridging on composite delamination

To study the effect of z-pinning in composite delamination growth, a theoretical model based on beam theory was developed by Liu et al. (2003). In this model, the z-pin bridging law given in previous section was adopted to simulate the effect of z-pinning on delamination resistance. The experimental observation by Liu et al. (2007) shows that during a quasi-static delamination (load rate is less than 10 mm/min), the bending resistance from z-pins is very small compared to their pullout resistance, only the bridging force of z-pins was incorporated in the model. In this section, analytical solution of mode I delamination of z-pinned laminates by this model will be introduced and verified. © 2008, Woodhead Publishing Limited

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Numerical simulation always plays an important role in the study of delamination of composite laminates. The finite element method has been applied to model the delamination in z-pinned composite laminates under different loading conditions. Yan et al. (2003) first investigated the mode I delamination problem. They then considered the mode II delamination due to shear force (Yan et al., 2004). Recently, Yan and Liu (2006) further analyzed the influence of z-pinning on the buckling of composite laminates under edge-wise compression. Their numerical models and simulation results are also discussed in this section.

23.3.1 Analytical solution of mode I delamination of z-pinned laminates The Double-Cantilever-Beam (DCB) is a standard geometry to experimentally study mode I delamination toughness of composite laminates. A DCB specimen is shown in Fig. 23.3, in which the beams are reinforced by nc column and nr row pins. dc and dr represent the spacing between adjacent z-pin columns and rows, respectively. The initial crack is created in the laminate mid-plane with length a0 and the distance of the nearest pin column to the crack tip is ap . Due to symmetry only one single beam is considered in the model. The delamination length is named as Lc and F is the load corresponding to the applied displacement, ∆, at x = 0. The bridging force caused by the ith column of z-pins (i = 1, 2 … nc) at location xi, is added to the beam as an external force, Pi. From a generalized beam theory, the differential equation of the deflection curve is given by EIz″ = M(x)

23.3

where EI is the flexural rigidity of the laminated beam and M is the bending moment. Before the delamination tip reaches the first column of pins, 0 ≤ Lc F dr

2h

2∆

w

F

a0

ap

dc

z

y x

23.3 Schematic of a Double-Cantilever-Beam (DCB) test for z-pinned composite laminate.



679

≤ x1, there is no bridging force on the beam and the total bending moment is: (0 ≤ x ≤ Lc)

M(x) = Fx

23.4

The solution of Eq. (23.3) for this initial stage becomes

z = ∆3 ( x 3 – 3 xL2c + 2 L3c ) 2 Lc

(0 ≤ x ≤ Lc)

23.5

where ∆ is the applied displacement at the loading end, that is, ∆ = z(0). The fracture energy method is used as the delamination criterion (Anderson, 1995). The strain energy release rate is calculated by G I = 1 ∂U w ∂Lc

23.6

in which w is the width of the laminated beam; U is total strain energy in the bent beams and is

U= 1 EI

∫

Lc

M 2 ( x ) dx

23.7

0

From Equations 23.4, 23.6 and 23.7, we obtain GI =

F 2 L2c wEI

23.8

If the energy release rate of the bent beams is greater than a critical intrinsic toughness of the unpinned DCB, GIC, the delamination will propagate. After the crack has passed the first column of pins, Lc > x1, the pins start to provide closure forces to the opening crack. The differential equation of the deflection curve in each interval is given by

(0 ≤ x ≤ x1 ) Fx EIz ′′ =   Fx – P1 ( z )( x – x1 ) ( x1 ≤ x ≤ Lc )

23.9

The boundary conditions in each interval are z(x1)+ = z(x1)–, z′(x1)+ = z′(x1)–, z′(Lc) = z(Lc) = 0,

23.10

In Equation 23.9, P1 is the total bridging force from the first column of z-pins at the location x = x1, that is, P1 = nr × Ps. Here Ps is the bridging force from a single pin, which is a function of the flexural displacement z(x1) given by the bridging law, Equation 23.1 in Section 23.2.2. Thus, it is mathematically difficult to obtain a closed-form solution of Equation 23.9. Instead, an iteration method is used to obtain a numerical solution. In the iterative calculation, the displacement, ∆, and the crack length, Lc, are added step by step by a tiny increment. In the first step, a tiny increase in the crack length is given by


680


Lc = x1 + dLc

23.11

Since the increment dLc is very small, the displacement of the first pin is very small. Thus, the bridging force of the z-pin can be neglected in Equation 23.9. According to Equation 23.5, the displacement of the first pin can be approximated by

z ( x1 ) = ∆3 ( x13 – 3 L2c x1 + 2 L3c ) 2 Lc

23.12

Substituting the obtained displacement, z(x1), into the pin bridging law, the bridging force, P1, can be obtained. Increasing the applied displacement, ∆ = ∆ + d∆ and applying the bridging force, P1 from the previous step to Equations 23.9 and 23.10, the deflection of the beam under this applied displacement, ∆ = ∆ + d∆, can be approximated by  1 Fx 3 + Cx + EI∆ (0 ≤ x ≤ x1 )  EIz =  6 1 1  6 Fx 3 – 6 P1 ( x – x1 ) 3 + Cx + EI∆ ( x1 ≤ x ≤ Lc ) 

23.13

in which F=

3 EI∆ P1 ( Lc – x1 ) 2 (2 Lc + x1 ) + 2 L3c L3c

C1 =

FL2c P1 ( Lc – x 1 ) 2 – 2 2

23.14

23.15

The new displacement of the first column of pins, z(x1), can be obtained from Equation 23.13. Substituting z(x1) into the bridging law, a new bridging force at this step can be obtained. This bridging force is subsequently applied to calculate the deflection of the beam in the next step. If the number of columns of the bridging z-pins is nc when the delamination crack length is Lc, the differential equation of the deflection curve in each interval becomes  (0 ≤ x ≤ x1 )  Fx i  EIz ′′ =  Fx – Σ Pj ( z )( x – x j ) ( x i ≤ x ≤ x i+1 , i = 1, 2 … n c – 1) j =1  nc  Fx – Σ P ( z )( x – x ) ( x ≤ x ≤ L ) j j nc c  j =1 23.16 Following the above iteration method, the solution at this step can be solved as © 2008, Woodhead Publishing Limited


681

1 3 (0 ≤ x ≤ x 1 )  6 Fx + C x + EI∆ i  EIz =  1 Fx 3 – Σ 1 Pj ( x – x j ) 3 + C x + EI∆ ( x i ≤ x ≤ x i+1 , i = 1, 2 … n c – 1) j =1 6 6  Nc  1 Fx 3 – Σ 1 P ( x – x ) 3 + C x + EI∆ ( x ≤ x ≤ L ) j j nc c  6 j =1 6

23.17 in which, nc

C = – 1 FL2c + Σ 1 Pj ( Lc – x j ) 2 j =1 2 2

23.18

 nc  F = 1 3  Σ Pj ( Lc – x j ) 2 (2 Lc + x j ) + 6 EI∆  =1 j 2 Lc  

23.19

Then, the displacement of the ith column of pins is: EIz ( x i ) =

Fx i3 i –1 Pj – Σ ( x i – x j ) 3 + C x i + EI∆ j =1 6 6

i = 1, 2, … nc [23.20]

Adding the applied displacement step by step and using the solved displacement to calculate the current bridging force, Pi in Equation 23.19, a new set of displacement z(xi) can be obtained. Combining Equations 23.4, 23.6, 23.7 and 23.16, the strain energy release rate at the current applied displacement can also be obtained. The above process is repeated until the energy release rate is large enough to cause the delamination crack to grow.

23.3.2 Verification of the model with measured bridging laws To verify the recommended bridging law, experiments on z-pinned DCB mode I delamination were carried out in the CAMT (Liu et al., 2006). In those tests, z-pins were made of carbon fibre (T300) reinforced BMI resin with a diameter of 0.28 mm. The laminated beams were made of carbon fibre (IMS) reinforced epoxy (924) with dimensions: 150 mm in length, 20 mm in width and 1.5 mm in thickness. The z-pins (6 rows × 7 columns) were vertically inserted into the beams by an ultrasonic insertion machine before curing. A thermally insulated film with a length of 50 mm was inserted between the upper and lower beams to create an initial crack between the two beams. Three samples were tested in an Instron 5567 universal machine at crosshead speed of 1 mm/min. Load-displacement traces were recorded until the delamination crack propagated to the whole length. In all samples, the first z-pin was located 5 mm away from the initial delamination tip and © 2008, Woodhead Publishing Limited

682


the length of pinned region was 25 mm. In the simulation, the parameters, Pd, Pf, δ1 and δ2, in the bridging law given by Equation 23.1, were measured by previous z-pin pullout tests with the same materials of z-pins and the laminates. As shown in Fig. 23.4, the simulation results agree well with the experimental data. The only difference is that the simulated displacement can reach a larger value than that of the measured displacement. This may be caused by the dynamic effect during crack propagation which was not considered in the theoretical model. In the tests, when the delamination crack had passed the pinned area, the crack propagated rapidly and became unstable leading to the conclusion of the test.

23.3.3 Simulation of mode I delamination growth in z-pinned composite laminates by the finite element method To simulate the delamination growth in a z-pinned DCB test by using the finite element method, a bi-linear z-pin pullout model Equation 23.2 was applied to describe the pullout process of individual pins. This pullout model is completely determined by the peak bridging force, Pa, its corresponding pullout displacement, δa, and the ultimate pull-out displacement, h, equal to half-thickness of the DCB (see Fig. 23.3). The purpose of the delamination simulation is to quantify the effect of zpinning on the through-thickness reinforcement of the DCB sample. Therefore, it is not necessary to simulate the complicated z-pin pulling out process and the pin effect can be simulated by distributed springs along the thickness of the beam at the same location, which is schematically shown in Fig. 23.5. 100

Load (N)

80 60 40

v = 1 mm/min d = 0.28 mm

20 0 0

10

20 Displacement (mm)

30

40

23.4 Load-displacement curve of DCB mode I delamination. Dotted curve shows simulated results and solid curves are experimental results from three tests.



683

Composite

h Springs

δ Pin

Crack face

P (a)

(b)

23.5 Schematically illustrating FE simulation of the effect of z-pin by distributed nonlinear springs in FE model: (a) a section of z-pinned composite; (b) FE model for this section with distributed springs.

Note that a section of the z-pinned laminate with a pin pulling-out is given in Fig. 23.5(a) and its FE model in Fig. 23.5(b). Several identical non-linear springs are arranged on the FE nodes, which are highlighted by black dots in Fig. 23.5(b). The peak bridging force per unit width, ps, of a non-linear spring in plane stress is determined by ps =

Pa n r , wn s

23.21

where ns is the number of identical springs in one column used to represent a column of pins. The energy release rate from linear elastic fracture mechanics was applied to quantify the delamination toughness in the present study. According to linear elastic fracture mechanics (Anderson, 1995), the energy release rate is equal to a contour integral with the integrating path starting from the lower crack surface and ending at the upper crack surface. Two contours, Γ1 and Γ2, are shown in Fig. 23.6. Contour Γ1 includes only the composite around the crack-tip without the springs, i.e., z-pins. During crack growth, the calculated energy release rate, G, is equal to the intrinsic toughness of the unpinned composite, GIC. Contour Γ2 includes all the springs, that is, all the effects of the z-pins. The calculated energy release rate, GR, based on this contour, now © 2008, Woodhead Publishing Limited

684


dS Springs

z

COD

x Crack tip

Γ1

Γ2

23.6 Integral contours for calculating energy release rates: Γ1: contour excluding springs (z-pins); Γ2: contour including all the springs (z-pins).

represents the total energy release rate, which includes the energy dissipation due to the creation of new crack surfaces and the energy dissipation due to the z-pins. The FE package ABAQUS (2001) was applied in the simulation, which adopts a domain integral method to numerically calculate the contour integral based on the divergence theorem. This method has been proved to be quite effective in the sense that accurate contour integral estimates are usually obtained even with quite coarse meshes because the integral is taken over a domain of elements surrounding the crack front. Errors in local solution parameters have less effect on the domain integrated value, i.e., the energy release rate. Therefore, it is not necessary to simulate the stress singularity near the crack-tip. Fine mesh has been carefully chosen and justified through the comparisons in Fig. 23.7 and Fig. 23.8. In total, there are about 13 700 ordinary four-node bi-linear plane stress elements used in the FE model. It takes about 16 hours in a Compaq ES45 supercomputer with one CPU to finish a crack growth simulation. To simulate the delamination crack growth, the crack opening displacement (COD) criterion was applied. The relation between the energy release rate, G, and the COD is (Suo, 1990; Poursartip et al., 1998) COD = 4(2)1/4


( a11 a 22 )1/4  2 a12 + a 66 +  2a  π 11

a 22  a11 

1/4

G r , 23.22


685

100

80

F (N)

60

40

20 FE results with Pa = 15 N Experimental data 0 0

5

10

15 2∆ (mm)

20

25

30

23.7 A plot of the applied force and opening displacement at the loaded ends of a DCB during delamination growth.

2000

GR Unpinned sample Gc

1800 1600

G (J/m2)

1400 1200 1000 800 600 400 200 0 0

10

20

30

40

∆a (mm)

23.8 Calculated energy release rates as a function of crack growth with GIC = 265 J/m2.


50

686


where a11, a22, a12 and a66 are determined by material elastic constants. In the case of plane stress studied here, they are a11 = 1 , a 22 = 1 , E1 E2 ν 21 ν 12 a12 = – = – , a 66 = 1 E2 E1 G12

23.23

The r in Equation 23.23 represents the distance from the crack-tip, which is illustrated in Fig. 23.6. The material constants and geometrical parameters used in the FE calculations are based on the tests by Cartie and Partridge (1999), which are summarized in Tables 23.1 and 23.2. E1 and E2 in Table 23.1 are Young’s moduli in x- and z-direction, respectively. ν12 is Poisson coefficient, which characterizes compression in z-direction due to tension along x-direction. µ12 is shear modulus for planes parallel to the coordinates xOz. The values of the parameters to describe the DCB and z-pinning are listed in Table 23.2. The FE output of the reaction force of the DCB versus opening displacement at the load-points for Pa = 15 N is shown by the solid line in Fig. 23.7. It shows that the force increases linearly with opening displacement at the initial stage, which corresponds to the linear deformation of the DCB before crack growth. The force drops immediately after the crack has started to propagate. If there were no z-pins, the force would continue to decrease. However, owing to the closure tractions exerted by the z-pins, the applied force increases gradually over a wide range of crack growth. The maximum value reaches ~80 N, which is much larger than the maximum force of the unpinned specimen. This means that the delamination toughness of the DCB has been greatly improved by the z-pins. At the final stage, the load drops again because all the pins have been pulled out. The black dots are the experimentally measured data from Cartie and Partridge (1999), and compared to the FE results using Pa = 15 N, very good agreement is obtained. Table 23.1 The material constants of the composite laminate E1 (GPa)

E2 (GPa)

ν12

µ12 (GPa)

165

11

0.3

38

Table 23.2 Values of parameters to describe the DCB and z-pinning in mode I FE simulation h (mm) w (mm) L (mm) a0 (mm) ap (mm) δa (mm) Pa (N)

nc

nr

dc(mm)

1.5

8

5

3.5

20

150


50

5

0.1

15


687

The energy release rate is a suitable parameter to quantify the toughness of a structure. Here, the energy release rate versus crack growth, or GRcurve, obtained from the FE analysis was calculated using a Γ2 type contour in Fig. 23.6 and represented in Fig. 23.8 by the solid line. The dotted line is the FE result of the energy release rate for the unpinned sample, which is constant during crack growth and is equal to GIC = 265 J/m2. The dashed line is the energy release rate, Gc, derived from the z-pinned DCB based on a Γ1 type contour in Fig. 23.6. Gc should represent the energy release rate due solely to the creation of new crack surfaces during delamination crack growth, that is, Gc = GIC. Figure 23.8 clearly confirms this prediction. Comparing the solid and dotted lines in Fig. 23.8, it is shown that the crack-resistance GRcurve of the z-pinned laminate is overall much larger than the unpinned laminate except at initial crack growth, where the z-pins have not yet started to function. The maximum GR is ~1900 J/m2, which is ~7 times the unpinned DCB. Hence, z-pinning is a very effective technique to improve the mode I delamination toughness of composite laminates. To understand the contribution of the parameters on z-pin enhanced toughness of DCB, a parametric study is carried out. For example, Fig. 23.9 shows the effect of the normalized pullout model parameter, Pa /GICh. It can be seen that the total energy release rate or crack-resistance increases as the peak pullout force, Pa, increases. For Pa /GIC h = 37.7, the maximum normalized crack-resistance, GR /GIC, is about 7 but this becomes >13 for Pa /GIC h = 14

Pa / GIC h = 37.7 Pa / GIC h = 50.3 Pa / GIC h = 61.9 Pa / GIC h = 75.7

12

GR /GIC

10

8 6

4 2 0 0

5

10

15 ∆a/h

20

25

23.9 Influence of normalized pullout model parameter, Pa /GICh, on normalized delamination toughness, GR /GIC, during crack growth with nc = 8, δa /h = 0.0667 and dc /h = 2.33.


30

688


75.7. The contribution of the z-pins on the enhanced toughness is a manifestation of the work dissipated in pulling out the pins. That is the work area under the curve of the pullout force P versus pullout displacement δ. Thus, by increasing Pa the work area and hence the z-pin pullout work is also increased. This in turn improves the delamination toughness of a z-pinned structure. These conclusions are consistent with the study on cohesive failure (Williams and Hadavinia, 2002). Practically, higher pullout peak force Pa can be achieved by improving the z-pinning technique to gain stronger bonding between pins and laminate. Figure 23.10 shows the influence of the number of z-pin columns on the energy release rate of z-pinned laminates. Here five cases are considered for nc = 1 to 8. The number of z-pin columns represents the size of the z-pinned zone in the crack growth direction for a given column spacing, dc. Therefore, it is not surprising to see that the enhanced toughness GR /GIC covers a longer delaminated distance for higher number of z-pin columns. Fig. 23.10 also shows that the maximum crack-resistance GR increases rapidly from nc = 1 to 4. This observation indicates that interaction between pin columns can also enhance the delamination toughness of the composite laminate. With nc increasing continuously from 4 to 8, this effect does not increase at the same rate. For example, GR /GIC are almost identical for nc = 6 and 8. Hence, it is expected that further increasing nc beyond 8 would not lead to any improvement in GR /GIC. In sum, there is a limit to the enhanced toughness by simply 12

nc = 1 nc = 2 nc = 4 nc = 6 nc = 8

10

GR /GIC

8

6

4

2

0 0

5

10

15 ∆a/ h

20

25

30

23.10 Influence of number of z-pin columns, nc, on normalized delamination toughness, GR /GIC, during crack growth with Pa /GICh = 61.9, δa /h = 0.0667 and dc /h = 2.33.



689

increasing the number of the z-pin columns. However, Fig. 23.10 also indicates that a steady toughening state can be reached when the number of the z-pin columns is over 8.

23.3.4 Simulation of mode II delamination growth in z-pinned composite laminates by the finite element method Since the End-Notched-Flexure (ENF) beam is commonly accepted as a standard test to evaluate the delamination toughness of composite laminates subjected to mode II failure, the FE analysis was carried out to examine the ENF test on z-pinned laminates (Yan et al., 2004). Figure 23.11 shows a schematic of the ENF test geometry for z-pin reinforced composite laminate. Z-pins are inserted in the laminates along the z direction. An initial crack with length a0 was created in the mid-plane of the laminate. The distance of the nearest column of z-pins to the crack-tip is ap. The distribution of the zpins can be described by four parameters: nc, nr, dc and dr. With increasing applied force 2P at the mid-point of the beam, a delamination crack will grow along the mid-plane. It has been proven that truly mode II fracture can be obtained in ENF tests for the unpinned laminates (Carlsson and Gillespie, 1989). In the case of z-pinned laminates, mode II fracture will still dominate the delamination process. Figure 23.12 is a schematic of the pullout process of a z-pin caused by mode II delamination. During delamination the crack faces move relative to each other along the crack growth direction with a displacement 2δ at the location of the pin. Consequently, the pin is forced to pull out of the laminate. A bi-linear function between the tensile force T and the pullout displacement 2δ

dc a0

2P

ap

z

y

2h

Laminates

x

Crack

z-pins 2L

23.11 Illustration of a mode II delamination test for z-pin reinforced laminate: end-notched-flexure (ENF) geometry.


690

Delamination behaviour of composites Pin

Composite

T

Crack surface

2δ Pin

23.12 Schematic of the pullout of a z-pin under mode II delamination.

 Ta δ , 0 ≤ δ ≤ δa δ T=  a Ta  Ta – (δ – δ a ), δ a ≤ δ ≤ h δa h – 

23.24

The delamination toughness of the z-pinned ENF beam is quantified by the energy release rate, G, which is similar to the mode I delamination study. The crack grows when the shear stress at a specified distance ahead of the crack-tip reaches a critical value under mode II loading condition. The relation between the energy release rate, G or GII, and the shear stress, τ12, ahead of the crack-tip is (Suo, 1990; Poursartip et al., 1998)

τ 12 =

1/8 1  a11   0.5 + 2 a12 + a 66  4 a11 a 22  2π a11  a 22  

–1/4

G II , r

23.25

where a11, a22, a12 and a66 are defined by Equation 23.23. Figures 23.13(a) and (b) show the crack-resistance GR versus crack growth ∆a curves computed from the FE analysis. In Fig. 23.13(a), three FE curves are illustrated with different values of Ta while keeping a constant δa of 0.01 mm; and in Fig. 23.13(b), three different values of δa are chosen for the FE curves with the same Ta of 180 N. The dotted curves represent the results for unpinned ENF samples. Here, the effect of z-pinning is clearly demonstrated



691

2200

GR with Ta = 200 N GR with Ta = 180 N GR with Ta = 160 N Experimental data Unpinned sample

2000 1800

GII (J/m2)

1600 1400 1200 1000 800 600 400 0

5

10

15

20

25

∆a (mm) (a) 2000 1800 1600

GII (J/m2)

1400

GR with δa = 0.01 mm GR with δa = 0.016 mm GR with δa = 0.02 mm Experimental data Unpinned sample

1200 1000 800 600 400

0

5

10

15

20

25

∆a (mm) (b)

23.13 (a) Predicted crack-resistance GR curves of z-pinned ENF laminates versus crack growth ∆a for different values of Ta while keeping constant δa of 0.01mm, compared to experimental data and an unpinned sample. (GIIC = 700 J/m2); (b) Predicted crack-resistance GR curves of z-pinned ENF laminates versus crack growth ∆a for different values of δa while keeping constant Ta of 180 N, compared to experimental data and an unpinned sample. (GIIC = 700 J/m2).


692


in all the five cases with different Ta and δa. For example, as shown in Fig. 23.13(a) a maximum GR of 2030 J/m2 can be achieved in the case of Ta = 200 N and δa = 0.01 mm. This value is 2.9 times that of the unpinned value. Therefore, z-pinning is an effective technique to improve mode II delamination toughness of composite laminates. The z-pinning parametric influence on the mode II delamination toughness was also examined in Yan et al. (2004).

23.3.5 Buckling of z-pinned composite laminates under edge-wise compression In practice, buckling accompanying delamination under edge-wise compression is a typical failure mode in laminated composites. Compression after impact becomes a routine test either to evaluate the impact damage or to investigate the influence of delamination on buckling strength, or damage tolerance. Generally, two kinds of buckling modes have been observed in practice: local buckling and global buckling. In a case of local buckling, the delamination planes will open and z-pins will provide resistance against the opening, as shown in Fig. 23.14 of the half of the simulation model. The numerical study was aimed quantifying the influence of z-pinning on the local buckling strength (Yan and Liu, 2006). The z-pin bridging law is defined by Equation 23.2 and the effect of an individual pin is modelled by a set of nonlinear springs as in the mode I delamination study. Figure 23.15 shows the compressive force per unit width, 2P, plotted against applied compressive displacement, U, for a given imperfection di = 0.08 mm, dc = 1.75 mm and da = 20 mm. Both unpinned sample and z-pinned specimen are analysed. In the initial stage, the compressive force increases linearly with displacement. The unpinned sample bifurcates first and no more compressive force can be applied, which leads to buckling of the structure. In contrast, the z-pinned specimen can continuously support an increasing force linearly over the bifurcation point of the unpinned sample until the force reaches a higher value. In the post-buckling state, it collapses shedding all the additional load carried due to the z-pinning. Figure 23.15

L

dm

dc U (P ) h z

di Z-pins

da

23.14 Numerical model for local buckling analysis.


x

Compressive force per unit width, 2P, (kN/mm)


693

1.5

1.0

0.5

Unpinned sample Z-pinned specimen 0.0 0.0

0.2 0.4 Compressive displacement, U, (mm)

0.6

23.15 Compressive force plotted against compressive displacement for unpinned sample and z-pinned specimens with di = 0.08 mm, dc = 1.75 mm and da = 20 mm for both cases.

shows that both compressive force and displacement drop rapidly during unstable collapse. This theoretical result is obtained based on static analysis without considering the dynamic effect. In practice, only the compressive force will drop precipitously. We are concerned here with the maximum compressive force, 2Pcr, just before rapid buckling, which is obtained accurately from the numerical calculation. Simulations also indicate that, to obtain 2Pcr, simple static analysis gives exactly identical result as Riks’ method. Figure 23.15 clearly shows that the critical compressive force is increased from 0.84 to 1.28 kN/mm caused by z-pinning. This increase is in good agreement with available experimental results (Zhang et al., 2003). Further parametric study can be found in Yan and Liu (2006).

23.4

Z-pin bridging under high loading rate

With rapid expansion of the application of composites, composite structures often face rather complex in-service conditions, one of which is the high rate of loading. During mode I delamination, a reinforcing z-pin provides a closure stress to the opening crack. The efficiency of z-pin reinforcement is strongly dependent on the bridging mechanisms. When the loading rate is high, a high-rate shear/friction between the pin and the laminates may cause a significant change in the z-pin pullout behaviour, which changes the z-pin bridging mechanism accordingly. Furthermore during the delamination crack opening, the embedded z-pins can also provide resistant moments to the bent © 2008, Woodhead Publishing Limited

694


beams when the z-pin’s bending stiffness is high. As we mentioned in the last section, when the load rate is low, the effect of bending of z-pin on the delamination growth can be ignored. However, at a high loading rate, the cross-head displacement of the beam is increased at a very high rate. The delamination crack may not propagate at the same rate due to the resistance imposed by the z-pins. At a certain crack length and a certain applied displacement, the curvature of the beam under a high loading rate would be different from that under a low loading rate. It means that z-pins under different loading rates may suffer differing bending deformation before being pulled out. Consequently, the resistance they offer to the bending deformation of the beam would depend on the loading rate. In this section, the effects of loading rate on the fracture load of z-pinned DCB mode I delamination will be discussed based on the experimental observations by Liu et al. (2007).

23.4.1 Experimental procedure DCB mode I delamination test was carried out. The laminated beams were made of carbon fibre (IMS) reinforced epoxy (924) (unidirectional) with dimensions: 150 mm in length, 20 mm in width and 1.5 mm in thickness. The pultruded T300/bismaleimide pins were vertically inserted into the beams by an ultrasonic insertion machine before curing (Liu et al., 2007; Cartie, 2000). A thermally insulated film with a length of 50 mm was inserted between the upper and lower beams to create an initial crack between them. Two T-shaped tabs were glued to the top and bottom surfaces of the laminates and were firmly gripped for testing in an Instron 5567 universal machine at cross-head speeds (V) of 1 and 100 mm/min, respectively. Load-displacement traces were recorded until the delamination crack propagated to the right end of the beams. Crack growth was recorded by a video camera with a microscope. In all samples, the first column of z-pins was located at 5 mm away from the initial delamination tip and the length of pinned region was 25 mm. Two types of samples were tested, which were (1) big-pin reinforced DCB with an areal density D = 2%; (2) small-pin reinforced DCB with an areal density D = 2%. Three samples were tested in each case. The results given in the following sections were only average values. To understand the z-pin bridging force under different loading rates, a series of z-pin pullout tests was carried out. The test samples for the pullout tests are the same as those shown in Fig. 23.1. The z-pins and prepregs were made of the same materials as were used for the DCB samples. The prepreg was 40 mm long and 20 mm wide. A thermally insulated film with a thickness of 10 µm was inserted between the upper and lower laminates to avoid any adhesive bonding between them. The thickness of the sample was 3 mm. Two T-shaped tabs were glued to the top and bottom surfaces of the laminates and were firmly secured in an Instron 5567 testing machine at © 2008, Woodhead Publishing Limited


695

cross-head speeds of 1 mm/min and 100 mm/min, respectively. Loaddisplacement curves were recorded until the pins were completely pulled out. Three samples were tested in each case. The results were given by the average values.

23.4.2 Experimental results and discussions

200

200

150

150

100

d = 0.51 mm, D = 2% v = 100 mm/min v = 1 mm/min

50 0

Load (N)

Load (N)

Figure 23.16(a) shows the load-crack length curves of big-pin reinforced DCB delamination, in which the z-pin diameter, d, is 0.51 mm and the areal density, D, is 2% (8 columns × 6 rows). During the tests, the delamination crack growth was slowed down by the reinforcing pins. When the crack reached the pins, the opening force caused the pins debonding from the laminates and then all pins were pullout with friction. It is shown that at the higher loading rate, a larger applied load was needed to propagate the delamination crack. The results of the small-pin reinforced DCB tests are shown in Fig. 23.16(b), where the laminated beams were reinforced by 15 columns × 12 rows pins (d = 0.28 mm), that is D = 2%. It can be seen that there is a significant degradation on the fracture load when the loading rate is higher. This is different from the results of the big-pin reinforced DCB. A significant difference from big-pin samples is that z-pin pullout was not observed in the small-pin reinforced DCB tests. All pins broke when the delamination crack reach them. To understand the effect of loading rate on the delamination resistance, zpin pullouts on both big-pin and small-pin samples were conducted. Figure 23.17(a) shows the load-displacement curve of the big-pin pullout test. It is shown that with increasing displacement, the bridging force increased until it reached a maximum value, Pmax. At this value, the pins were debonded

100

d = 0.28 mm, D = 2% v = 100 mm/min v = 1 mm/min

50 0

50

60 70 80 Crack length (mm) (a)

90

50


90

23.16 Load-crack length curve of z-pinned DCB mode I delamination test, in which (a) d = 0.51 mm, D = 2% and (b) d = 0.28 mm, D = 2%; v = 1 mm/min and 100 mm/min, respectively.


696


400

Pmax

σmax (MPa)

Pullout force (N)

500

300 200

200

100

100 0

0 0

0.5 1 1.5 Displacement (mm) (a)

2

1 100 Pullout rate (mm/min) (b)

23.17 (a) Load-displacement curve of 3 × 3 big-pin pullout; (b) Maximum bridging stress of z-pin pullout test, in which, d = 0.51 mm, v = 1 mm/min and 100 mm/min, respectively.

from the laminates and then pulled out from the laminates with increasing displacement. At this stage, the bridging force was caused by the interfacial friction between the z-pins and the laminates. The results of the maximum frictional bridging stresses calculated from the bridging forces for different pullout rates are given in Fig. 23.17(b), in which, σmax is the average stress given by a single pin. Clearly, at the higher rate, the bridging stress is higher. Since the bridging stress is the most dominant parameter in z-pin bridging, it explains why the fracture load of the DCB increases with loading rate increasing as shown in Fig. 23.16(a). Figure 23.18(a) is the load-displacement curve of a small-pin pullout test. In contrast to the big-pin pullout shown in Fig. 23.17(a), there was a fast stress-drop upon reaching the maximum force which indicated pin debonding (Dai et al., 2004). After that, the pins were pulled out against friction. The maximum debonding force and the maximum frictional pullout force are defined as Pd and Pf, respectively. Figure 23.18(b) shows the results of σd and σf of a single pin calculated from Pd and Pf by pullout tests at different loading rates. It is clear that at a higher rate, the debonding stress is reduced but the frictional stress increased. However, in small-pin reinforced DCB tests, all pins broke before being pullout. Since the pins rupture when the crack passes them, the total bridging force of the pins was determined by the debonding force (σd). The frictional stress (σf) cannot affect the fracture load of the DCB. As shown in Fig. 23.18(b), before debonding, the pins can provide a higher bridging force at a lower loading rate. Therefore, it is expected that a higher fracture load is needed for delamination growth at a lower loading rate which is consistent with the results shown in Fig. 23.16(b). Furthermore, besides the bridging force, the resistant bending moment from the z-pins can also provide resistance to delamination growth, especially when the stiffness and the density of the pins are high enough. At the higher © 2008, Woodhead Publishing Limited

Z-pin bridging in composite delamination 800

Pd

200

Pf

100 0 0

σd

σf

600

300

σmax (MPa)

Pullout force (N)

400

697

0.5 1 1.5 Displacement (mm) (a)

2

400 200 0 1 100 Pullout rate (mm/min) (b)

23.18 (a) Schematic illustration of load-displacement curve of 3 × 3 small-pin pullout, in which, Pd represents the maximum debonding force and Pf represents the maximum frictional force after debonding, respectively. (b) Maximum bridging stresses of z-pin pullout test, in which d = 0.28 mm, v = 1 mm/min and 100 mm/min, respectively.

loading rate, the cross-head displacement of the beam increased very quickly, which was 100 times higher than that at the lower loading rate. However, the delamination crack could not always propagate at the same rate due to the resistance imposed by the z-pins (Liu et al., 2007). Therefore, at a certain crack length, the radius of curvature of the beams under the higher loading rate is smaller than that under the lower loading rate. It means that z-pins under a high loading rate suffered more severe bending during pullout. As a result, the reinforcing pins provide a higher resistance to the bent beam to delay the delamination. A higher applied load is hence required for further crack growth. In addition, when the pins are bent, its embedded length applies an additional pressure to the laminates as a reaction to the bending. This pressure increases the ‘snubbing’ friction against pin pullout which, consequently, causes an increase in the fracture resistance. Figures 23.19(a) and 23.19(b) shows the optical photomicrographs of the ends of z-pins that were pulled out after the DCB tests. An interesting fact is that when the loading rate was low, the z-pin was pulled out without any obvious damage. However, when the loading rate was high, the z-pin was pulled out accompanied by a number of splits along the length of the pin, which could be seen as evidence of considerable shearing during bending of the beams. It should be noted that these observed results are highly repeatable. The effect of bending under high loading rate can also be shown by the comparison of the results of fracture loads of the big-pin and small-pin samples. If we compare the two ‘hollow triangle’ curves (v = 1 mm/min) in Figs 23.16(a) and 23.16(b), which are for the big-pin sample and small-pin sample, respectively, we can find that the fracture load of the small-pin sample is higher than that of the big-pin sample with the same density of z© 2008, Woodhead Publishing Limited

698

Delamination behaviour of composites 100 µm

(a) 100 µm

(b)

23.19 Optical photomicrographs of z-pins after DCB delamination tests in which d = 0.51 mm, D = 2%: (a) z-pin end, v = 1 mm/min; and (b) z-pin end, v =100 mm/min. Splitting along length of pin due to shear failure is apparent at the high loading rate.

pins when the loading rate is 1 mm/min. However, when the loading rate is high, as discussed in above paragraphs, the bending resistance from the bigpins and the ‘snubbing’ friction become more prominent than that at the low loading rate. Therefore, the big-pin reinforced DCB has higher delamination growth resistance. In contrast, because of their small inertial moment of cross-section, the small pins cannot provide much resistant moment to the DCB. Moreover, when the crack has passed the small pins, they are virtually all ruptured and hence cannot provide further frictional resistance to the pullout. Hence, at the high loading rate, as shown by the ‘solid triangle’ © 2008, Woodhead Publishing Limited


699

curves in Figures 23.16(a) and 23.16(b), the fracture load of the small-pin reinforced DCB drops below that of the big-pin sample. Despite the above observations and results, two questions remain unanswered. What causes the breakage of the z-pins in the high density small-pin reinforced samples? Why does the maximum friction stress increase with increasing loading rate? Could the latter be caused by the viscoelastic nature of the composite interface? With a high loading rate, this may increase the temperature during frictional sliding, which in turn changes the features of both the matrix resin and the interface between the z-pin and the laminates, and consequently, the bridging stress. Clearly, further studies on the effects of temperature and rate are needed to understand the physical mechanisms responsible for these experimental results. It should also be noted that in this study only two rates were considered: 1 and 100 mm/min. Under these rates, the kinetic energy effect on crackopening was small compared to that of the strain energy and could be ignored. At much higher loading rates, considerable increase of kinetic energy may weaken the effect of interfacial friction on the delamination behaviour and accelerate the delamination growth (Sridhar et al., 2002). In contrast, a very low pullout rate may cause stick-slips during z-pin pullout (Povirk and Needleman, 1993), which may induce extra resistance to the delamination.

23.5

Fatigue degradation on z-pin bridging force

In many structural applications composite materials must withstand cyclic or fluctuating loads rather than static loads. Therefore, a need exists to characterize the bridging mechanics of z-pins under fatigue loading. In this section, a very recent experimental study on z-pin degradation under cyclic fatigue is introduced (Zhang et al., 2007).

23.5.1 Specimen preparation Z-pin pullout tests were carried out to study the bridging force degradation under cyclic fatigue. The z-pinned laminate specimens used to study the fatigue bridging properties were carbon/epoxy prepreg. The prepreg stack was debulked by vacuum bagging and then reinforced using pultruded carbon/ bismaleimide z-pins manufactured by Axtex Inc (Waltham, MA). The z-pins were driven into the prepreg using a hand-held ultrasonically actuated horn in a process described by Freitas et al., (1994). The laminate specimens were 40 mm long by 20 mm wide, and a 10 mm × 10 mm region in the centre was reinforced with z-pins arranged in a square grid pattern. After z-pinning, the prepreg was consolidated and cured in an autoclave at an overpressure of 500 kPa and temperature of 115 °C for 0.5 h and then at 750 kPa and 180 °C © 2008, Woodhead Publishing Limited

700


for 1.0 h. The cured specimens were then bonded to two T-shaped loading tabs, see Fig. 23.1. The fatigue test matrix is presented in Table 23.3. The effect of pin size on the fatigue bridging properties was examined by reinforcing the laminate with small (0.28 mm) or large (0.51 mm) diameter pins to the same volume fraction (2%). The spacing between the small pins was varied to determine the influence of pin content on the fatigue properties. Tests were performed on specimens with pin volume contents of 0.5% (4 × 4) and 2.0% (7 × 7).

23.5.2 Fatigue test procedure Before fatigue testing, a series of monotonic load tests was performed to measure the static bridging forces. These tests were conducted at a crosshead speed of 0.5 mm/min using a 1 kN load capacity Instron 5567 machine. The fatigue testing was performed in the displacement control mode by applying a triangular load waveform with a frequency of 0.5 Hz on the specimens. The maximum displacement, δmax were 50% or 80% of the debonding displacement, δ1, which was measured in the monotonic pull-out test. The minimum displacement, δmin, applied in the fatigue tests was set at 10% of δmax (i.e., displacement ratio R = 10). The fatigue tests were performed over a specific number of load cycles, and then the specimen was pulled to failure under monotonic loading to measure the residual debonding force, Pd . The degradation of the friction pull-out force, Pf, due to cyclic loading was measured using a novel test procedure. A static load was applied to the specimen until the load dropped due to debonding of the pins (Dai et al., 2004). The static load was removed from the specimen at a certain load, Pfs, and corresponding displacement, δs. A fatigue load was then applied having maximum and minimum displacements specified by the conditions:

δmax = δs + 0.5 δ1

23.26a

δmin = δs + 0.05 δ1

23.26b

The reduction in the frictional force was measured with increasing number of load cycles to monitor the degradation of the friction pull-out load. Table 23.3 Parameters of pullout specimens Sample type

Number of pin

Z-pinned area(mm2)

Diameter of pin (mm)

1 2 4

4×4 7×7 4×4

10 × 10 10 × 10 10 × 10

0.28 0.28 0.51



701

23.5.3 Test results Figure 23.20 shows the residual debonding force (Pd/Pdo) degradation with increasing number of load cycles for the two pin sizes, and presumably this is due to the accumulation of fatigue-induced damage (e.g., micro-cracks) along the bond-line between the pins and the laminates. Under a fixed fatigue displacement ratio, δmax/δ1 = 0.5, Pd decreases slowly with the number of load cycles up to ~103 for both pin sizes. However, with further cyclic loading, the maximum debonding force for the large diameter pins (Type 4) decreases more rapidly than the small diameter pins (Type 1). This suggests that fatigue damage accumulates more rapidly along the bond-line of the large diameter z-pin. Compared to a small diameter pin, there is a larger interfacial area between a large diameter pin and the laminates, which may contain more and larger existing flaws. During cyclic fatigue, these flaws propagate to form large interfacial cracks along the bond-line and cause pin debonding. This may be the reason why the debonding force of a large diameter pin degrades more rapidly. Furthermore, the degradation rate of the debonding force increases with displacement ratio. As shown in Table 23.4, when the displacement ratio was increased to δmax/δ1 = 0.8, it took only thousands of cycles for the degradation to become significant. Figure 23.21 shows the reduction in the friction pull-out force with increasing number of load cycles for three specimens containing an array of 7 × 7 small diameter pins (type 2). The friction pullout force (Pfs) and corresponding displacement (δs) at which the static test was stopped to begin fatigue test for 1.2

1

Pd/Pd0

0.8

0.6

0.4

0.2

Type 4, δmax /δ1 = 0.5 Type 1, δmax /δ1 = 0.5

0 0

1

2

3

4 Log (N)

5

6

7

23.20 Residual debonding force degradation with numbers of cycles for different z-pin sizes.


702

Delamination behaviour of composites Table 23.4 Residual maximum debonding force for big pins Sample type

δmax/δ1

N

Pd/Pd0

4 4 4 4 4 4 4 4

0.5 0.5 0.5 0.5 0.8 0.8 0.8 0.8

2000 5000 18 500 30 000 2000 5000 18 500 30 000

0.9501 0.9329 0.7927 0.6181 0.8934 0.7353 0.6477 0.6263

1.2 1

Pf/Pfs

0.8 0.6 0.4 Sample 2–10 Sample 2–9 Sample 2–1

0.2 0 0

1

2 Log (N)

3

4

23.21 Frictional force degradation with number of cycles increasing. Table 23.5 Parameters for frictional force degradation test Sample number

Pfs (N)

ds (mm)

2–1 2–2 2–10

908 736 856

0.665 0.606 0.643

three specimens are shown in Table 23.5. The frictional force at certain loading cycles (Pf) is normalized to the friction pullout force under monotonic loading (Pfs). Compared to the debonding force (Pd), the frictional pullout force decreases more rapidly with load cycles. For example, the residual friction force dropped by 20% of the original value after only a few load cycles. In comparison, it took more than 60 000 cycles to degrade the debonding force by 20%. This suggests that fatigue-induced damage accumulates more © 2008, Woodhead Publishing Limited


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rapidly once the pins have fully debonded, when they are free to slide back and forth against the walls of the hole. This sliding action will wear the surfaces of the pin and the hole. Figure 23.22 compares the surface of z-pins pulled out under monotonic and cyclic loads. When the pin was pulled out under monotonic loading the surface was relatively smooth and the fibres were coated with the resin. After cyclic loading, the pin texture was coarser because most of the resin had been worn out from the surface and any remaining resin contained fine-scale hackles that are indicative of shear cracking. This suggests that under cyclic fatigue the pin surface is worn out by micro-cracking of the resin generated by cyclic frictional shear stresses along the bond-line. This may cause gradual thinning of the pins and reduces their resistance against pull-out.

23.6

Future trends

It has been shown that the poor interlaminar fracture toughness of laminated composites can be significantly and effectively improved by z-pinning. Investigations in recent years have covered a wide range of issues, which include z-pin bridging mechanism, bending effect of z-pin, mode I, mode II and mixed mode delamination with z-pinning, prediction of in-plane properties of z-pinned laminated composite, z-pinned laminates buckling and fatigue. The major challenges would be the applications of this new technique, especially, in structural-scale level. Our future research will focus on the following aspects: (a) The reduction of the in-plane properties by through-thickness z-pinning or stitching in the insertion process is always a major concern in the application of z-pinning technique. This reduction is caused by either the breakage of fibres during the insertion or the in-plane fibre waving that forms a resin-rich zone surrounding the pin and reduces the fibre

(a)

(b)

23.22 Scanning electron micrographs of a single pin after pullout following (a) 1 load cycle and (b) 30000 load cycles.


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volume fraction. When the z-pinning is applied to the structural-scale engineering components under a complex loading condition, the prediction of potential degradation in in-plane properties is essential in the analysis and design to ensure the reliability and integrity of this technique. (b) In the previous modelling and finite element analyses, the z-pins were assumed perfectly aligned in the through-thickness direction. However, in many applications, in the composite structures the z-fibre® pins were inserted at a specific angle to form a truss, such as X-CorTM and KCorTM sandwich panels (see http://www.aztex-z-fiber.com). These novel composite structures provide an optimized combination of the mechanical properties in three dimensions but bring complexity to the analysis. Therefore, theoretical modelling and finite element analysis will be developed to predict the performance of these new composite structures which may be subjected to tension, compression, bending, shearing and buckling.

23.7

References

ABAQUS (2001) Version 6.2. Providence, RI: HKS Inc. Anderson T L (1995), Fracture mechanics: Fundamentals and applications, Second Edition, CRC Press, Boca Raton, FL. Carlsson L A and Gillespie J W Jr (1989), ‘Mode-II interlaminar fracture of composites’, in Friedrich K, Application of fracture mechanics to composite materials, Elsevier, Amsterdam, 113–157. Cartie D D R (2000), Effect of z-fibres on the delamination behaviour of carbon fibre/ epoxy laminates, PhD thesis, Cranfield University, UK. Cartie D D R and Partridge I K (1999), ‘Delamination behaviour of z-pinned laminates’, Proceedings of the 12th international conference on composite materials, ICCM12, Paris 5–9 July. Dai S C, Yan W, Liu H Y and Mai Y W (2004), ‘Investigation on z-pin bridging law by z-pin pullout test’, Comp Sci Tech, 64, 2451–2457. Freitas G, Magee C, Dardzinski P and Fusco T (1994), ‘Fiber insertion process for improved damage tolerance in aircraft laminates’, J Adv Mater, 25, 36–43. Liu H-Y, Yan W and Mai Y-W (2003), ‘Z-fibre bridging force in composite delamination’, in: Blackman B R K, Pavan A, Williams J G, editors. Fracture of polymers, composites and adhesives II, ESIS Publication 32, Elsevier, Amsterdam 491–502. Liu H-Y, Yan W and Mai Y-W (2006), ‘Z-pin bridging in composite laminates and some related problems’, Australian Journal of Mechanical Engineering, 3, 11–19. Liu H-Y, Yan W, Yu X-Y and Mai Y-W (2007), ‘Experimental study on effect of loading rate on mode I delamination of z-pin reinforced laminates’, Comp Sci Tech, 67, 1294– 1301. Poursartip A, Gambone A, Ferguson S and Fernlund G (1998), ‘In-situ SEM measurements of crack tip displacements in composite laminates to determine local G in mode I and II’, Eng Frac Mech, 60, 173–185. Povirk G L and Needleman A (1993), Finite element simulations of fibre pull-out, Journal of Engineering Materials and Technology, 115, 286–291.



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Sridhar N, Massabo R and Cox B N (2002), Delamination dynamics in through-thickness reinforced laminates with application to DCB specimen, International Journal of Fracture, 118(2), 119–144. Suo Z (1990), ‘Delamination specimens for orthotropic materials’, ASME J Appl Mech, 57, 627–634. Williams J G and Hadavinia H (2002), ‘Analytical solutions for cohesive zone models’, J Mech Phys Solids, 50, 809–825. Yan W, Liu H-Y (2006), ‘Influence of Z-pinning on the buckling of composite laminates under edge-wise compression’, Key Engineering Materials, 312, 127–132. Yan W, Liu H-Y and Mai Y-W (2003), ‘Numerical study on the mode I delamination toughness of z-pinned laminates’, Comp Sci and Tech, 63, 1481–1493. Yan W, Liu H-Y and Mai Y-W (2004), ‘Mode II delamination toughness of z-pinned laminates’, Comp Sci and Tech, 64, 1937–1945. Zhang A, Liu H-Y, Mouritz A P and Mai Y-W (2007), ‘Experimental study and computer simulation on degradation of Z-pin reinforcement under cycle fatigue’, Composites, Part A (in press). Zhang X, Hounslow L and Grassi M (2003), ‘Improvement of low-velocity impact and compression-after-impact performance by z-fibre pinning’, in Proceedings of the 14th international conference on composite materials, San Diego, CA, 14-18 July 2003.


24 Delamination suppression at ply drops by ply chamfering M R W I S N O M and B K H A N, University of Bristol, UK

24.1

Introduction

Changes in thickness in composite structures are usually accomplished by dropping off plies within the layup. Similar arrangements are used both for terminating prepreg plies and dry blankets in resin infused parts. Ply drops can be important in initiating delamination, as they form a discontinuity, leading to very high localised stresses. The load carried in the discontinuous ply has to be transferred into the adjacent continuous plies, resulting in high interlaminar shear stresses. These are accompanied by interlaminar direct stresses that may be either tensile or compressive depending on the geometry and loading configuration. Analysis of such a sharp discontinuity gives rise to a stress singularity, similar to that occurring at a crack, and so fracture mechanics needs to be applied. In addition to stresses caused by the discontinuity, there are less localised interlaminar stresses due to the tapered geometry. There are several possible approaches to reducing the risk of delamination. The resin toughness can be increased (Chapter 25) or through thickness reinforcement can be introduced (Chapter 23). This chapter discusses suppression of delamination by reducing the source of the discontinuity at the ply drops by chamfering the ends of the plies to remove the step. The behaviour of tapered composites with ply drops is first described, initially for the basic case of unidirectional composites, and then for laminates. Static and fatigue response are both considered, and the factors affecting strength discussed. Procedures for chamfering plies in cured laminates and uncured material are described, and a number of experimental results presented summarising the effect of ply chamfering on strength. It is shown that this technique can completely suppress delamination, resulting in substantial increases in both static and fatigue strength of tapered composites.



24.2

707

Behaviour of tapered composites with ply drops

24.2.1 Static behaviour of unidirectional composites Typical behaviour is illustrated by tensile tests carried out on tapered unidirectional E-glass/913 epoxy specimens with dropped plies [1]. The configurations tested were symmetric, with a taper at both ends so that the gripped ends were both the full thickness to avoid failures at the tabs. The samples had eight continuous plies and two dropped plies at the middle of the thick section. Figure 24.1 shows a typical specimen schematically. The first damage was normally a crack in the triangular resin pocket ahead of the terminated plies. This appeared suddenly at a stress of approximately 30–60% of the ultimate failure stress. Subsequently delamination initiated from the end of the discontinuous ply, usually both above and below it, propagating into the thick section. A small amount up to 2 mm of stable delamination was generally observed with no significant change in the slope of the load-deflection response. This then suddenly propagated along the length of the specimen, accompanied by a sharp drop in load. Delamination was reasonably uniform across the width, although there was a tendency for it to start at the edge. The delamination surface generally followed the interface between continuous and discontinuous plies. Delamination into the thin section was not observed in any of these tests, which all had relatively shallow taper angles of 4–6°. Delamination stresses were relatively high. Specimens with eight continuous and two dropped plies failed at a thin section stress of 1209 MPa, close to the manufacturer’s quoted fibre direction tensile strength of 1310 MPa. Tests on similar sized untapered specimens with plies cut across the full width at the same location as the ply drops exhibited very similar failure mechanisms, showing that the discontinuity associated with the terminating plies is the critical factor controlling delamination [1]. Specimens with dropped plies are much more susceptible to delamination when loaded in compression than in tension. This is shown, for example, by tapered panels of unidirectional carbon fibre-epoxy with 16 continuous plies Dropped plies

Delamination

Continuous plies

Resin pockets

24.1 Schematic view of typical delaminated unidirectional tapered composite.


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and a block of four plies dropped together at the mid-plane. Identical specimens failed at a thin section stress of 1739 MPa in tension, but only 558 MPa in compression [2]. Failure mechanisms in compression were found to be similar to those in tension [3], except that no damage in the matrix pocket ahead of the drop was observed in compression. A small amount of stable delamination generally occurred on both sides of the dropped plies into the thick section, followed by sudden propagation. This caused bowing out of the outer plies, producing some delamination into the thin section, and in many cases was accompanied by complete fracture of the specimen.

24.2.2 Static behaviour of laminates Tests on glass and carbon fibre laminates with 0° and 45° fibre orientations containing dropped or cut plies showed similar overall behaviour to unidirectional ones [4–6]. For specimens where the discontinuous plies were adjacent to 45° rather than 0° plies, delamination was observed to step down through matrix cracks in the 45° plies until reaching a 0° ply rather than remaining at the interface between the continuous and discontinuous plies, as seen in unidirectional specimens [6]. This is illustrated schematically in Fig. 24.2 for tension tests on the thick section layup (–452/+452/04)(04/+452/ –452)(–452/+452/04)(04/+452/–452) with the second of the four blocks dropped in a single step. In these latter tests, in addition to the catastrophic thick section delamination, some stable delamination into the thin section was observed first. A single delamination initiated above the matrix crack at the resin pocket, and stepped down through the thickness to the 0° interface closest to the mid-plane, as shown in Fig. 24.3. It did not give rise to a significant drop in load, but a small change in stiffness, accompanied by bending of the specimen. This thin section delamination is believed to be driven by the straightening out of the plies as the ones above and below the discontinuous plies separate. The tests where this was observed had severe discontinuities, with eight plies dropped together (about 1 mm of material), and steep taper angles (about + 0 0 ± + 0 0 ±

To tabs

5 5

24.2 Thick section delamination in 0/45 laminate, crack lengths in millimetres (not to scale).


+ 0 + 0 0 ±


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+ 0

0 ± + 0 0 ±

7 3

25

+ 0 + 0 0 ±

24.3 Thin section delamination in 0/45 laminate, crack lengths in millimetres (not to scale).

20°), both much larger than the previously mentioned tests where thin section delamination was not observed. Specimens have a tendency to delaminate first from the free edge, but this was not significantly more marked in laminates compared with unidirectional specimens. Although high interlaminar stresses may be present locally near the edge of a laminate, the strain energy release rate due to delamination from the dropped plies was much larger than that due to delamination from the free edge. The delamination was therefore driven primarily by the dropped plies. However, this may not be the case in other layups which are more susceptible to free edge effects. Here the presence of dropped plies may cause edge delamination to initiate at a lower load than in a constant thickness specimen. This may particularly apply to 90° dropped plies which have a low stiffness, but also very low Poisson ratio and hence greater tendency to provoke free edge delamination.

24.2.3 Fatigue behaviour Failure mechanisms observed under tension fatigue loading were very similar to those under static loading [4–6]. In most cases the matrix in the resin pocket ahead of the terminated ply cracked immediately, and delamination above and below started to propagate into the thick section straight away. The delamination rate was approximately constant. The specimens with the severe drop that delaminated into the thin section under static loading also showed similar crack propagation in fatigue. Different specimen geometries and materials were found to behave quite similarly when loaded at similar fractions of their static delamination stress. Glass fibre specimens were also found to be susceptible to fibre fatigue, especially at high stress levels. This could be detected from broken fibres visible at the edge of the specimen in the region above and below the end of the terminating plies after some delamination had already occurred into the thick section. In some cases delamination then propagated into the thin section from areas of fibre damage above and below the drop. For carbon fibre specimens no fibre fatigue was observed. © 2008, Woodhead Publishing Limited

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Failure mechanisms in compression fatigue were similar to those under static compression loading [7]. However, the delamination tended to be much less symmetric above and below the dropped plies. In contrast to tension fatigue, under compression the delamination growth rate was not constant, but accelerated. Initially a single crack on one side of the dropped plies tended to dominate. Then as a second crack on the other side started to grow, the delamination rate increased sharply. Delamination growth in compression is particularly critical because eventually it can lead to unstable failure due to buckling.

24.2.4 Factors affecting strength Several plies terminated together were found to delaminate at a lower stress than a single ply. This is illustrated by results for cut ply glass fibre-epoxy laminates with eight continuous plies. Specimens with two cut plies failed at a thick section stress of 773 MPa, compared with 1154 MPa for similar specimens with only one cut ply [8]. Tapered carbon fibre specimens with 0° plies dropped singly and ±45° plies dropped in pairs were stronger than ones with 0° and ±45° plies dropped in groups of two and four plies respectively [4]. Multiple ply drops interleaved between continuous plies should therefore be used rather than dropping blocks of plies together in order to avoid delamination. Internal ply drops with continuous material on both sides are similarly less prone to delamination than external drops on the surface where there is only one interface for load transfer. The latter specimens with only single plies dropped together were stronger despite a taper angle of about 8° compared with about 4° for the others, showing that the angle is not the key parameter. In fact net section tensile strength was achieved despite the 8° taper [4]. However, when the taper angle is too steep, delamination into the thin section may occur. Tapered specimens were found to be stronger in tension than equivalent constant thickness ones with cut plies. The geometry of the taper induces through thickness compressive stresses at the ends of the discontinuous plies that are not present in the constant thickness specimens. Compression tends to increase the interlaminar shear strength and effective mode II fracture energy, inhibiting delamination [9]. Terminated plies of 0° orientation are much more susceptible to delamination than ±45° dropped plies. For example in specimens containing both 0° and ±45° dropped plies, delamination occurred first at a pair of 0° plies dropped together [4]. This happened despite there being a double pair of ±45° plies (i.e. four plies) dropped together nearer to the thin section where the stress was also higher. Where there are multiple drops of the same type, delamination can be expected to occur at the drop nearest the thin section. Multiple drops behave independently where there are continuous interleaving © 2008, Woodhead Publishing Limited


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plies, or where the spacing between them exceeds a certain minimum distance [10]. Changing the angle of the ply drop relative to the loading direction whilst maintaining ply orientations was not found to have a significant effect for delamination into the thick section, but did affect thin section delamination, a perpendicular drop being the worst case [11].

24.3

Methods of chamfering plies

24.3.1 Machining of cured composites with external ply drops Chamfering plies reduces the stiffness discontinuity which causes high interlaminar shear stresses and energy release rates at a ply drop. The more gradual transition also reduces the peel stresses arising due to the offset between continuous and discontinuous plies. Ply drops on the surface of a part can be chamfered by machining after cure. This technique was applied very carefully in a study on ultimate tensile strength of unidirectional carbon-epoxy using specimens tapered symmetrically from 3 mm down to 1 mm [12]. Composite plates were cured between flat metal plates with fill-in plies placed on the outside of the release cloth to produce a constant overall thickness, as shown in Fig. 24.4. This technique produces good consolidation without the need for special tooling. The tapered fill-in pieces matched the geometry of the stepped plate very closely, except that they were 1 mm shorter. This was done to give a gap of 0.5 mm between the ends of the plies when the fill-in pieces and the plate were put together for curing. After curing, the plates were cut into 10 mm wide specimens using a diamond wheel. The tapers on each specimen were then individually machined using the side of a 10 mm diameter tungsten carbide end mill. Each taper was machined from the thick section towards the thin section, with the rotation of the cutter such that at the surface it was always moving towards the thin section. This was done in order to produce a good surface finish and avoid any tendency for the cutter to pick up fibres. Great care was taken when approaching the thin section to avoid any undercutting whilst at the same time not leaving a step at the end of the taper. This method was very time consuming, but produced high quality specimens.

24.3.2 Chamfering of uncured prepreg Chamfering prepreg before cure has similar benefits to the geometry of the ply drop as machining. It completely eliminates the formation of resin pockets © 2008, Woodhead Publishing Limited

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Porous cloth Non-porous cloth (a) Cure of stepped plate with fill-in plies

Tooling

(b) Machining direction to achieve small taper

24.4 Manufacturing of tapered plates machined after cure.

where matrix cracks occur and which are the initiation sites for delamination. It can be applied to both internal and external ply drops. A technique has been developed that involves machining a chamfer on the uncured ply edges by abrading them with medium grit-sized abrasive media prior to laying up. The chamfering machine comprises a rotating abrasive wheel mounted on a swivelling head and a traversing bed, as shown schematically in Fig. 24.5. The prepreg ply is gripped firmly in a special fixture on the bed and the wheel is traversed across the width at a very shallow angle. To avoid debris contaminating the prepreg, the abrasion is done with the backing films still in place and a vacuum extractor is placed close to the abrading wheel to pick up all debris. For a given diameter the rotation speed of the abrasive wheel and the traversing feed rates are precisely controlled so that the localised temperature of the prepreg is not allowed to build up. Depending on the orientation of the fibres with respect to the traversing direction, the abrasive wheel head is swivelled such that the abrasive action is in the direction of the fibres. By the action of the abrasion the prepreg thickness is reduced from 0.125 mm to only 0.03 mm at the tip, with a tapered length of 2.75 mm, as shown in Fig. 24.6. With 0.25 mm plies the taper could be extended to about 4.25 mm. There is no need to chamfer 90° © 2008, Woodhead Publishing Limited


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Traversing bed for mounting ply Abrasive wheel Fibre direction for 0 deg plies Ply

Traverse direction (a) Arrangement for 0 deg plies

Abrasive wheel

Traversing bed for mounting ply Fibre orientation for 45 deg plies

Ply

Traverse direction (b) Arrangement for 45 deg plies

24.5 Schematic of setup for chamfering uncured plies.

plies, as the fibres naturally move during the cure to form a gradual taper. The process can be fully mechanised and could relatively easily replace the normal ply cutting process using knives. It could be mounted on an automated tape laying or tow placement machine to place chamfered material directly onto the part during lay up. The same technique can be used to put a chamfer on dry fibre materials. The principle is essentially the same except that the dry fibres need to be coated with a binder to keep them together. A revised method based on vacuum suction could be adopted to grip the fibres with a protective backing film to prevent any contamination.

24.4

Results of ply chamfering

24.4.1 Static strength of unidirectional composites Tensile strength of carbon-epoxy specimens machined after curing Unidirectional XAS/913 carbon-epoxy tapered specimens with surface plies machined after curing as described in Section 24.3.1 were tested in tension © 2008, Woodhead Publishing Limited

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Glass pre-preg

Chamfered length, 2.75 mm

Chamfered edge

24.6 Chamfered edge on uncured ply.

[12]. For comparison, specimens with constant thickness of 1 mm were also tested. These were 250 mm long, with 60 mm tabs bonded at each end. Delamination from the discontinuous plies was effectively suppressed, and the mean tensile strength from the tapered specimens was 2410 MPa, 14% higher than the value of 2120 MPa obtained with the uniform ones. There was no overlap between the ranges of results; the highest value from the uniform specimens was below the lowest strength from the tapered tests. The tensile strength is 21% higher than the value of 1990 MPa quoted on the manufacturer’s data sheet. Although the results show a considerable increase in tensile strength as a result of the tapering, it is possible that the true ultimate strength could still be underestimated since some splitting still occurred at the specimen edge, propagating back to the grips, and in most cases there was some fibre fracture at the end tabs. Tensile strength of carbon-epoxy specimens machined before curing Tapered specimens were made from AS4/8552 prepreg with two continuous plies and five dropped plies, with a single dropped ply at the centre close to the thin end, and two pairs of plies dropped symmetrically, further back, as shown in Fig. 24.7 [13]. Specimens were symmetrical, with the same tapering arrangement at both ends. A stagger of 5 mm was used between the successive dropped plies. Specimens were manufactured with the plies chamfered before curing, and © 2008, Woodhead Publishing Limited


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Chamfered configuration

Straight cut configuration

24.7 Tapered carbon-epoxy specimens with two continuous and five dropped plies.

with the normal knife cutting procedure using the same stagger of 5 mm. Panels were cured between flat plates, using a similar ply fill-in technique to that shown in Fig. 24.4. The dimensions of the specimens were 10 mm wide, 180 mm long and with a thin section gauge length of 30 mm. The thicknesses at the thin and thick sections were 0.5 and 1.75 mm, respectively. The normal dropped ply specimens failed suddenly at 1873 MPa, with delamination and fibre failure both occurring. The chamfered specimens failed at 2146 MPa, an increase of 15%, and there was no delamination at the ply drops. Some splitting occurred from the specimen edge prior to fibre fracture, as observed with the specimens machined after curing. Tensile strength of glass-epoxy specimens machined before curing Tests were carried out on unidirectional E-glass/913 prepreg specimens with four dropped plies and six continuous plies as shown in Fig. 24.8, with both chamfered and straight cut plies. Specimens were 10 mm wide, 160 mm long, with a thin section length of 30 mm. The thicknesses were 0.5 mm and 1.25 mm in the thin and thick sections. Specimens without chamfers failed by delamination initiating at the ply drop closest to the thin section, and propagating into the thick section [10]. The thin section stress at failure was 1092 MPa. When the chamfered specimens were tested, no visible delamination around the ply drops could be seen before final fibre failure at 1402 MPa, 29% higher than that achieved without ply chamfering [13]. This is also higher than the manufacturer’s quoted tensile strength of 1310 MPa. © 2008, Woodhead Publishing Limited

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Delamination behaviour of composites Stagger distance

Chamfered dropped ply specimen

Resin pockets

Specimen with straight cut plies

24.8 Tapered glass-epoxy specimens with four continuous and six dropped plies.

Compressive strength of glass-epoxy specimens machined before curing Specimens of the same E-glass/913 prepreg were manufactured with 32 continuous plies and four plies dropped together symmetrically at the centre of the layup. Compression tests can be susceptible to buckling if the specimens are too long. To avoid this problem only a single taper was made in order to keep the gauge length down to 20 mm. Specimens were 10 mm wide, tapering from 4.5 mm to 4 mm thickness. Specimens without tapering showed a small amount of stable delamination which then propagated suddenly into the thick section at 636 MPa [3]. In contrast, no delamination could be seen in the chamfered specimens before catastrophic failure occurred in the thin section close to the start of the taper. Post failure observation showed compressive failure with no evidence of delamination running into the thick section. The strength based on the thin section was 1130 MPa, 78% higher than the unchamfered case [13], and also much higher than the manufacturer’s quoted untapered compressive strength of 750 MPa.

24.4.2 Static strength of laminates Quasi-isotropic laminates of IM7/8552 carbon-epoxy with a layup of [45/90/ -45/0]2s in the thin section were manufactured [13]. To make the taper, a © 2008, Woodhead Publishing Limited


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whole block of eight plies was added at the middle of the two continuous blocks to give a thick section layup of [45/90/-45/0]3s, as shown in Fig. 24.9. All the plies in the dropped block were individually chamfered before curing except the 90 plies. A tapered length of 2.5 mm was achieved from a starting ply thickness of 0.125 mm, as before. All the plies were arranged symmetrically in pairs across the line of symmetry with a stagger equal to the chamfered length except for the first central pair, where a stagger of an extra 2 mm was introduced (Fig. 24.9). The specimens were manufactured using the plyinfill technique to give a waisted section in the middle. For comparison, another configuration with unchamfered plies dropped together was made. The dimensions of the specimens were 10 mm wide and 160 mm long with a thin section length of 30 mm. The thicknesses were 2 mm and 3 mm in the thin Extra 2 mm stagger

+45 Chamfered configuration 90 –45 0

Straight cut configuration

24.9 Tapered quasi-isotropic carbon-epoxy with two continuous and one dropped ply blocks.


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and thick sections respectively. Additionally, a set of 2 mm thick unwaisted samples were made without any dropped ply blocks. To avoid grip failures on these constant section specimens, glass-epoxy end tabs were bonded at either end, giving a gauge length of 60 mm in the middle. The specimens with straight cut dropped plies failed at the transition between the thick and thin sections at a stress of 610 MPa. The chamfered ply specimens also failed suddenly at the transition, but at a much higher stress of 860 MPa, an increase of 41%. The constant thickness samples failed by fibre failure within the gauge section with no obvious indication of premature delamination, although it is believed that free edge effects had an influence. The average strength of the specimens without ply drops was 911 MPa, only 6% higher, showing that the chamfering was effective in retaining almost the full untapered strength.

24.4.3 Fatigue strength Fatigue tests were carried out on symmetrically tapered E glass/913 specimens with two plies dropped together and eight continuous ones. The test frequency for a maximum stress of 70% of the static strength was fixed at 4 Hz. For tests at other stress levels, the frequency was altered to keep the stress rate the same. The samples were periodically monitored for any visible sign of damage. Unchamfered specimens failed by delamination into the thick section 1400 1200

Max stress (MPa)

1000 800 600 400 Cycles to complete failure of chamfered specimen First fibre failure in chamfered specimen First fibre failure in specimens with normal cut plies

200 0 103

104

105 Number of cycles

106

107

24.10 Comparison of fatigue test results for chamfered and straight cut glass-913 specimens.



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from the ends of the dropped plies [5]. In contrast the chamfered specimens did not show any signs of delamination except for a single isolated case [13]. A plot of maximum stress versus number of cycles at which fibre damage was first observed is shown in Fig. 24.10 for both chamfered and unchamfered specimens. The best-fit line for the chamfered specimens lies well above that for the normal tapered specimens, with about 26% higher stress at 3 × 104 cycles, and an order of magnitude increase in cycles to failure. Data for complete failure of the chamfered specimens is also shown.

24.5

Summary and conclusions

Composites with dropped plies are susceptible to delamination, driven primarily by shear due to load transfer from the discontinuous plies, which can cause delamination into the thick section. Where taper angles are steep, delamination can also occur into the thin section, driven by straightening out of the curved plies. Dropping of unidirectional plies has the largest effect. The thickness of material dropped together is the most significant parameter affecting delamination, and large reductions in strength can occur, especially in compression. The discontinuities due to dropped plies can be removed by machining either before or after cure. A technique has been presented for chamfering uncured plies with a traversing abrasive wheel that is very effective, and could be integrated into composite manufacturing equipment. Chamfering plies has been demonstrated to completely suppress delamination, and give substantial increases in static and fatigue strength. The tensile strength of unidirectional carbon-epoxy with plies dropped individually increased by about 15%, with similar increases whether they were machining before or after cure. The tensile strength of glass-epoxy with two plies dropped together increased by 29% with chamfering. Even larger increases were found in compression, with a 78% increase in strength for specimens with four plies dropped together. In several cases the measured strengths of chamfered tapered specimens exceeded the manufacturer’s quoted values for untapered material. Similar trends were seen in laminates, with a 41% increase in tensile strength of quasi-isotropic carbon-epoxy with a single block of eight plies dropped together. Fatigue tests on chamfered unidirectional glass-epoxy showed similar increases in strength to those under static loading, translating into an order of magnitude increase in fatigue life. Ply chamfering has been shown to be a remarkably effective technique to prevent delamination and to substantially increase static and fatigue strength. Using thicker plies and larger ply drops can significantly reduce manufacturing costs but can be detrimental to performance. Ply chamfering is a promising approach to improving manufacture of thick composites whilst avoiding the risk of delamination. © 2008, Woodhead Publishing Limited

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24.6


References

1. Wisnom, M.R., Delamination in tapered unidirectional glass fibre-epoxy under static tension loading, AIAA Structural Dynamics and Materials Conference, Baltimore, MD, USA, pp 1162–1172, April 1991. 2. Cui, W., Wisnom, M.R. and Jones, M.I., New model to predict the static strength of tapered laminates, Composites, 26: 141–146, 1995. 3. Wisnom, M.R., Prediction of delamination in tapered unidirectional glass fibre epoxy with dropped plies under static tension and compression, AGARD meeting on Debonding/delamination of Composites, Patras, Greece, May 1992, AGARD CP530 pp 25/1–7. 4. Wisnom, M.R., Jones, M.I. and Cui, W., Failure of tapered composites under static and fatigue tension loading, AIAA Journal, 33: 911–918, 1995. 5. Wisnom, M.R., Jones, M.I. and Cui, W., Delamination in composites with terminating internal plies under tension fatigue loading, Composite Materials: Fatigue and Fracture, Vol. 5, pp 486–508, ASTM STP 1230, 1995. 6. Wisnom, M.R., Dixon, R. and Hill, G., Delamination in asymmetrically tapered composites loaded in tension, Composite Structures, 35: 309–322, 1996. 7. Wisnom, M.R., Jones, M. and Cui, W., Delamination of unidirectional glass fibre epoxy with terminating plies under compression fatigue loading, Fibre Reinforced Composites Conference, Newcastle, pp 2/1–9, March 1994. 8. Cui, W., Wisnom, M.R. and Jones, M., An experimental and analytical study of delamination of unidirectional specimens with cut central plies, Journal of Reinforced Plastics and Composites, 13: 722–739, 1994. 9. Cui, W., Wisnom, M.R. and Jones, M.I., Effect of through thickness tensile and compressive stresses on delamination propagation fracture energy, Journal of Composites Technology and Research, 16: 329–335, 1994. 10. Cui, W., Wisnom, M.R. and Jones, M.I., Effect of step spacing on delamination of tapered laminates, Composites Science and Technology, 52: 39–46, 1994. 11. Wisnom, M.R. and Gustafsson, M., Delamination in tapered laminates under tension with ply drops skewed at an angle to the load, Paper 016, Proceedings of the American Society for Composites 17th Technical Conference, Purdue University, October 2002. 12. Wisnom, M.R. and Maheri, M.R., Tensile strength of unidirectional carbon fibreepoxy from tapered specimens, 2nd European Conf. on Composites Testing and Standardisation, Hamburg, September 1994, pp 239–247. 13. Khan, B., Potter, K. and M.R. Wisnom, Suppression of Delamination at Ply Drops in Tapered Composites by Ply Chamfering, Journal of Composite Materials, 40: 157–174, 2006.


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