Delamination behaviour of composites - Repositório … behaviour of composites ... 2.6 Delamination tolerance in composites preliminary design 46 Limited 2.7 Cost-effective delamination

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itedDelamination behaviour of composites

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itedRelated titles:

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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 fibre-reinforced composite materials and structures. Materials development, analysisand prediction of structural behaviour and cost-effective design all have a bearingon the impact response of composites and this book brings together for the firsttime the most comprehensive and up-to-date research work from leadinginternational experts.

Mechanical testing of advanced fibre composites(ISBN 978-1-85573-312-1)Testing of composite materials can present complex problems but is essential inorder to ensure the reliable, safe and cost-effective performance of anyengineering structure. Mechanical testing of advanced fibre composites describesa wide range of test methods which can be applied to various types of advancedfibre composites. The book focuses on high modulus, high strength fibre/plasticcomposites and also covers highly anisotropic materials such as carbon, aramidand glass.

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itedDelamination

behaviour ofcomposites

Edited bySrinivasan Sridharan

Published by Woodhead Publishing and Maney Publishingon behalf of

The Institute of Materials, Minerals & Mining

CRC PressBoca Raton Boston New York Washington, DC

W O O D H E A D P U B L I S H I N G L I M I T E DCambridge England


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Woodhead Publishing Limited and Maney Publishing Limited on behalf ofThe Institute of Materials, Minerals & Mining

Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington,Cambridge CB21 6AH, Englandwww.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 LimitedThe authors have asserted their moral rights.

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Contents

Contributor contact details xvii

Introduction xxiii

S SRIDHARAN, Washington University in St. Louis USA

Part I Delamination as a mode of failure andtesting of delamination resistance

1 Fracture mechanics concepts, stress fields, strainenergy release rates, delamination initiation andgrowth criteria 3

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

1.1 Introduction 31.2 Fracture mechanics concepts 41.3 Delaminations 101.4 Future trends 231.5 Concluding remarks 241.6 References 25

2 Delamination in the context of composite structuraldesign 28

A RICCIO, C.I.R.A. (Centro Italiano Ricerche Aerospaziali –Italian Aerospace Research Centre), Italy

2.1 Introduction 282.2 Physical phenomena behind delamination onset 302.3 Physical phenomena behind delamination growth 362.4 Introduction to delamination management in composites

design 392.5 Impact induced delamination resistance in composites

preliminary design 41


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2.7 Cost-effective delamination management 552.8 References 60

3 Review of standard procedures for delaminationresistance testing 65

P DAVIES, IFREMER Centre de Brest, France

3.1 Introduction 653.2 Historical background 663.3 Mode I 673.4 Mode II 703.5 Mode III 743.6 Mixed mode I/II 763.7 Conclusion on fracture mechanics tests to measure

delamination resistance 793.8 Future trends 803.9 Conclusion 813.10 Sources of information and advice 813.11 Acknowledgements 813.12 References 81

4 Testing methods for dynamic interlaminar fracturetoughness of polymeric composites 87

C T SUN, Purdue University, USA

4.1 Introduction 874.2 Dynamic loading and crack propagation 904.3 Mode I loading with double cantilever beam (DCB) for

low crack velocity 934.4 High crack velocity with modified double cantileave

beam (DCB) and end notch flexure (ENF) 954.5 Mode I by wedge loading with hopkinson bar 1044.6 Acknowledgment 1154.7 References 115

5 Experimental characterization of interlaminar shearstrength 117

R GANESAN, Concordia University, Canada

5.1 Introduction 1175.2 Short beam shear test 1185.3 Double-notch shear test 1255.4 Arcan Test 133

Contentsvi


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5.6 References 1355.7 Appendix: Nomenclature 136

Part II Delamination: detection and characterization

6 Integrated and discontinuous piezoelectricsensor/actuator for delamination detection 141

P TAN and L TONG, University of Sydney, Australia

6.1 Introduction 1416.2 Typical patterns for piezoelectric (PZT) or-piezoelectric

fiber reinforced composite (PFRC) sensor/actuator 1436.3 Constitutive equations and modelling development for a

laminated beam with a single delamination andsurface-bonded with an integrated piezoelectricsensor/actuator (IPSA) 146

6.4 Parametric study 1496.5 Experimental verification 1576.6 Conclusions 1656.7 Acknowledgment 1656.8 References 1656.9 Appendix 167

7 Lamb wave-based quantitative identification ofdelamination in composite laminates 169

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

7.1 Introduction 1697.2 Lamb waves in composite laminates 1707.3 Lamb wave scattering by delamination 1777.4 Lamb wave-based damage identification for composite

structures 1807.5 Design of a diagnostic lamb wave signal 1817.6 Digital signal processing (DSP) 1827.7 Signal pre-processing and de-noising 1867.8 Digital damage fingerprints (DDF) 1877.9 Data fusion 1937.10 Sensor network for delamination identification 1987.11 Case studies: evaluation of delamination in composite

laminates 2027.12 Conclusion 211

Contents vii


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7.14 References 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 forMaterials Testing and Research, Switzerland

8.1 Introduction 2178.2 Acoustic emission (AE) analysis 2188.3 Acoustic emission analysis applied to investigation of

delaminations in fiber-reinforced, polymer-matrix (FRP) 2228.4 Acoustic emission monitoring of delaminations in

fiber-reinforcd, polymer matrix composite specimens 2238.5 Acoustic emission investigation of delaminations in

structural elements and structures 2538.6 Advantages and limitations for acoustic emission

delamination investigations 2678.7 Related nondestructive acoustic methods for delamination

investigations 2728.8 Summary and outlook 2728.9 Acknowledgments 2738.10 References 273

Part III Analysis of delamination behaviour from tests

9 Experimental study of delamination in cross-plylaminates 281

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

9.1 Introduction 2819.2 Summary of current state 2829.3 Experimental methods for studying delaminations 2859.4 Fracture mechanics study of delamination in cross-ply

laminates 2869.5 Discussion and interpretation 3009.6 Structural elements or parts with

cross-ply laminates 3049.7 Summary and outlook 3059.8 Acknowledgments 3059.9 References 305

Contentsviii


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

10.1 Introduction 31010.2 Static mode II fracture characterization 31110.3 Dynamic mode II fracture characterization 32110.4 Conclusions 32410.5 Acknowledgements 32410.6 References 325

11 Interaction of matrix cracking and delamination 327

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

11.1 Introduction 32711.2 Mixed-mode cohesive damage model 33211.3 Continuum damage mechanics 33811.4 Conclusions 34111.5 References 342

12 Experimental studies of compression failure ofsandwich 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 Introduction 34412.2 Compression failure mechanism of debonded structures 34412.3 Compression failure of debonded sandwich columns 34612.4 Compression failure of debonded sandwich panels 35312.5 Acknowledgments 36212.6 References 362

Part IV Modelling delamination

13 Predicting progressive delamination via interfaceelements 367

S HALLETT, University of Bristol, UK

13.1 Introduction 36713.2 Background to the development of interface elements 36713.3 Numerical formulation of interface elements 36813.4 Applications 37313.5 Enhanced formulations 380

Contents ix


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ited13.6 Conclusions 382

13.7 Acknowledgements 38213.8 References 382

14 Competing cohesive layer models for prediction ofdelamination growth 387

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

14.1 Introduction 38714.2 UMAT (user material) model 38814.3 UEL (user supplied element) model 39114.4 Double cantilever problem 39414.5 UMAT model: details of the study and discussion of results 39414.6 UEL model: details of the study and discussion of results 40314.7 Delamination of composite laminates under impact 40714.8 Conclusion 42714.9 References 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 Introduction 42915.2 The cohesive approach 43115.3 Delamination failure in fiber reinforced composites 43215.4 Delamination failure in layered structures 44015.5 Summary and conclusions 45015.6 Acknowledgement 45115.7 References 452

16 Delamination in adhesively bonded joints 458

B R K BLACKMAN, Imperial College London, UK

16.1 Introduction 45816.2 Adhesive bonding of composites 45816.3 Fracture of adhesively bonded composite joints 46016.4 Future trends 47916.5 Sources of further information and advice 48016.6 References 481

Contentsx


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

17.1 Introduction and motivation 48517.2 Experimental data 48617.3 Damage mechanics models 48817.4 Simulation of delamination growth under fatigue loading

using cohesive elements: cohesive zone model approach 49017.5 Numerical representation of the cohesive zone model 49117.6 Constitutive model for high-cycle fatigue 49317.7 Examples 49817.8 Mode I loading 49817.9 Mode II loading 50217.10 Mixed-mode I and II loading 50417.11 Fatigue delamination on a skin-stiffener structure 50517.12 Conclusions 51017.13 Acknowledgments 51017.14 References 511

18 Single and multiple delamination in the presence ofnonlinear crack face mechanisms 514

R MASSABÒ, University of Genova, Italy

18.1 Introduction 51418.2 The cohesive- and bridged-crack models 51518.3 Characteristic length scales in delamination fracture 52818.4 Derivation of bridging traction laws 53518.5 Single and multiple delamination fracture 53918.6 Final remarks 55318.7 Acknowledgement 55518.8 References 555

Part V Analysis of structural performance in presenceof delamination and prevention/mitigation ofdelamination

19 Determination of delamination damage incomposites under impact loads 561

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

19.1 Introduction 56119.2 Composites failure modelling 563

Contents xi


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ited19.3 Delamination damage in low velocity impact 570

19.4 Delamination damage in high velocity impact 57619.5 Conclusions and future outlook 58319.6 References 584

20 Delamination buckling of composite cylindricalshells 586

A TAFRESHI, The University of Manchester, UK

20.1 Introduction 58620.2 Finite element analysis 58820.3 Validation study 59720.4 Results and discussion: analysis of delaminated

composite cylindrical shells under different types ofloadings 597

20.5 Conclusion 61420.6 References 616

21 Delamination failure under compression ofcomposite laminates and sandwich structures 618

S SRIDHARAN, Washington University in St. Louis, USA, Y LI, IntelCorporation, USA and El-Sayed, Caterpillar Inc., USA

21.1 Introduction 61821.2 Case study (1): composite laminate under longitudinal

compression 61921.3 Case study (2): dynamic delamination of an axially

compressed sandwich column 62821.4 Case study (3): two-dimensional delamination of

laminated plates 63521.5 Results and discussion 64421.6 Conclusion 64721.7 References 648

22 Self-healing composites 650

M R KESSLER, Iowa State University, USA

22.1 Introduction 65022.2 Self-healing concept 65222.3 Healing-agent development 65722.4 Application to healing of delamination damage in FRPs 66122.5 Conclusions 67022.6 References 671

Contentsxii


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H Y LIU, The University of Sydney, Australia andW YAN, Monash University, Australia

23.1 Introduction 67423.2 Z-pin bridging law 67523.3 Effect of Z-pin bridging on composite delamination 67723.4 Z-pin bridging under high loading rate 69323.5 Fatigue degradation on Z-pin bridging force 69923.6 Future trends 70323.7 References 704

24 Delamination suppression at ply drops by plychamfering 706

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

24.1 Introduction 70624.2 Behaviour of tapered composites with

ply drops 70724.3 Methods of chamfering plies 71124.4 Results of ply chamfering 71124.5 Summary and conclusions 71924.6 References 720

25 Influence of resin on delamination 721

S MALL, Air Force Institute of Technology, USA

25.1 Introduction 72125.2 Resin toughness versus composite toughness 72225.3 Resin toughness effects on different modes 72525.4 Resin effects on cyclic delamination behaviour 72925.5 Temperature considerations 73325.6 Effects of interleafing and other methods 73525.7 Summary 73725.8 References 739

Index 741

Contents xiii


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Introduction

S. Sridharan*Department of Mechanical,

Aerospace and StructuralEngineering

Washington University in St. LouisSt. Louis, MO 63130USA

E-mail: [email protected]

Chapter 1

I. S. Raju* and T. K. O’BrienNASA-Langley Research Center

Hampton, VA 23861USA


Chapter 2

A. RiccioC I R A (Centro Italiano Ricerche

Aerospaziali – Italian AerospaceResearch Centre)

Via Maiorise, S/N81043Capua (Caserta)Italy


Chapter 3

P. DaviesMaterials and Structures Group(DOP/DCB/ERT/MS)IFREMER Centre de BrestBP7029280 PlouzanéFrance


Chapter 4

C. SunSchool of Aeronautics and

AstronauticsPurdue UniversityWest Lafayette, IN 47907USA


(* = main contact)

Contributor contact details


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R. GanesanConcordia Centre for CompositesDepartment of Mechanical and

Industrial EngineeringConcordia UniversityRoom EV 4 - 211, 1515 StCatherine WestMontrealQuebec H3G 2W1Canada


Chapter 6

P. Tan?1 and L. Tong*School of AerospaceMechanical and Mechatronic

EngineeringUniversity of SydneyNSW 2006Australia

E-mail: [email protected]@aeromech.usyd.edu.au

Chapter 7

Z. SuThe Department of Mechanical

EngineeringThe Hong Kong Polytechnic

UniversityHung HomKowloonHong Kong

E-mail: [email protected]@aeromech.usyd.edu.au;

L. Ye*Centre for Advanced MaterialsTechnology (CAMT)School of AerospaceMechanical and Mechatronic

Engineering (AMME)The University of SydneyNSW 2006Australia


Chapter 8

J. Bohse*BAM – Federal Institute forMaterials Research and TestingDivision V.6 Mechanical Behaviour

of PolymersUnter den Eichen 87D-12205 BerlinGermany


A. J. BrunnerLaboratory for Mechanical Systems

EngineeringEmpa – Swiss Federal Laboratories

for Materials Testing andResearch

Ueberlandstrasse 129CH-8600 DuebendorfSwitzerland


Contributor contact detailsxvi

1 Ping Tan is currently working as a research scientist at the Australian Defence Scienceand Technology Organisation


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A. J. BrunnerLaboratory for Mechanical Systems

EngineeringEmpa – Swiss Federal Laboratories

for Materials Testing andResearch

Ueberlandstrasse 129CH-8600 DuebendorfSwitzerland


Chapters 10 and 11

Marcelo Francisco de SousaFerreira de MouraDepartamento de Engenharia

Mecânica e Gestão IndustrialFaculdade de Engenharia daUniversidade do Porto

Rua Dr. Roberto Frias s/n, 4200-465 PortoPortugal


Chapter 12

F. Avilés*Centro de Investigación Científica

de Yucatán, A.C.Unidad de MaterialesCalle 43 # 103Col. Chuburná de HidalgoC.P. 97200.MéridaYucatánMéxico

L. CarlssonDepartment of Mechanical

EngineeringFlorida Atlantic UniversityBoca Raton, FL 33431USA

E-mail: [email protected]@fau.edu

Chapter 13

S. HallettAdvance Composites Centre for

Innovation and ScienceUniversity of BristolQueens BuildingUniversity WalkBristol BS8 1TRUK


Chapters 14 and 21

S. Sridharan*Department of Mechanical,Aerospace and Structural

EngineeringWashington University in St. LouisSt. Louis, MO 63130USA

E-mail : [email protected]

Y. LiIntel CorporationChandler, AZ 85224USA


Contributor contact details xvii


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R. Yu*E.T.S Ingenieros de Caminos,Canales y PuertosUniversidad de Castilla-La Mancha13071 Ciudad RealSpain


A. PandolfiDipartimento di Ingegneria

StrutturalePolitecnico di MilanoPiazza Leonardo da Vinci 3220133 MilanoItaly


Chapter 16

B. BlackmanDepartment of Mechanical

EngineeringImperial College LondonLondon SW7 2AZUK


Chapter 17

P. Camanho*DEMEGIFaculdade de EngenhariaUniversidade do Porto

Rua Dr. Roberto Frias4200-465 PortoPortugal


A. Turon and J. CostaAMADE

Polytechnic SchoolUniversity of Girona

Campus Montilivii s/n17071 GironaSpain

Chapter 18

Roberta MassabòDepartment of Civil,

Environmental and ArchitecturalEngineering

University of GenovaVia Montallegro, 116145, GenovaItaly


Chapter 19

Alastair F. Johnson and NathalieToso-Pentecôte

German Aerospace Center (DLR)Institute of Structures and Design

Pfaffenwaldring 38-4070569 StuttgartGermany


Contributor contact detailsxviii


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A. TafreshiSchool of Mechanical

Aerospace and Civil EngineeringThe University of ManchesterP.O. Box 88Sackville StreetManchester M60 1QDUK

E-mail:[email protected]

Chapter 22

M. KesslerIowa State UniversityMaterials Science and Engineering2220 Hoover HallAmes, IA 50011-2300USAE-mail: [email protected]

Chapter 23

H. Liu*Centre for Advanced Materials

TechnologySchool of AerospaceMechanical and Mechatronic

EngineeringThe University of SydneySydneyNSW 2006Australia


Contributor contact details xix

W. YanDepartment of Mechanical

EngineeringMonash UniversityMelbourneVictoria 3800Australia

E-mail:[email protected]

Chapter 24

M. R. WisnomUniversity of BristolAdvanced Composites Centre for

Innovation and ScienceQueens BuildingUniversity WalkBristol BS8 1TRUK

E-mail: M. [email protected]

Chapter 25

S. MallAir Force Institute of TechnologyAFIT/ENYBldg. 6402950 Hobson WayWright-Patterson AFB, OH 45433USA


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


Introduction


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Introductionxxii

Laminated composites are becoming the preferred material system in a varietyof industrial applications, such as aeronautical and aerospace structures, shiphulls in naval engineering, automotive structural parts, micro-electro-mechanical systems as 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 resistancein comparison to conventional metallic materials and potential for structuraltailoring are some of the factors that have contributed to the advancement oflaminated composites. Their increased use has underlined the need forunderstanding their modes of failure and evolving technologies for the continualenhancement of their performance.

The principal mode of failure of layered composites is the separationalong the interfaces of the layers, viz. delamination. This type of failure isinduced by interlaminar tension and shear that develop due to a variety offactors such as: Free edge effects, structural discontinuities, localizeddisturbances during manufacture and in working condition, such as impactof falling objects, drilling during manufacture, moisture and temperaturevariations and internal failure mechanisms such as matrix cracking. Hiddenfrom superficial visual inspection, delamination lies often buried betweenthe layers, and can begin to grow in response to an appropriate mode ofloading, drastically reducing the stiffness of the structure and thus the life ofthe structure. The delamination growth often occurs in conjunction withother modes of failure, particularly matrix cracking.

A study of composite delamination, as does any technological discipline,has two complementary aspects: An in depth understanding of the phenomenonby analysis and experimentation and the development of strategies foreffectively dealing with the problem. These in turn lead to a number ofspecific topics that we need to consider in the present context. These compriseof:

1. An understanding of the basic principles that govern the initiation ofdelamination, its growth and its potential interaction with other modesof failure of composites. This is the theme of the first chapter, but severalauthors return to this theme in their own respective contributions.

2. The determination of material parameters that govern delaminationinitiation and growth by appropriate testing. These must necessarily beinterfacial strength parameters which govern interlaminar fracture initiationand interlaminar fracture toughness parameters, viz. critical strain energyrelease rates that must govern interlaminar crack growth. The book containsseveral valuable contributions from leading international authorities inthe field of testing of composites.

3. Development of analytical tools : What are the methodologies one mayemploy to assess the possibility of delamination onset and growth under


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Introduction xxiii

typical loading scenarios? This may be approached from the points ofview of fracture mechanics, damage mechanics, cohesive modelingapproach and approaches which draw from and combine these. In particular,the cohesive modeling approach has proven to be a powerful and versatiletool in that when embedded in a nonlinear finite element analysis, it cantrace the two-dimensional delamination growth without user interference,is robust from the point of view of numerical convergence, and canpotentially account for a variety of interfacial failure mechanisms. Thissubject is discussed thoroughly in several authoritative contributions.

4. Detection of delamination: Ability to diagnose the presence of delaminationand to be able to capture in graphical terms the extent of delaminationdamage is a desideratum towards which the composite industry iscontinuing to make progress. Several nondestructive evaluation toolsare available and have been used with varying degrees of success. Acousticemission, Lamb-wave and Piezo-electric technologies are discussed inthe context of delamination detection in the present work.

5. Prevention of delamination: Several techniques of either inhibitingdelamination or altogether suppressing it are available. The book containsa section treating the following techniques of delamination prevention/inhibition: ‘Self-healing’ composites which internally exude adhesivematerial as soon as crack advances thus effectively arresting the crack;Z-pin bridging in which fibers are introduced across the interlaminarsurfaces, liable to delaminate, artfully tapering off discontinuities whichare sources of potential delamination and the use of toughened epoxies.

6. Delamination driven structural failure: Certain loading scenarios cancause delamination growth if there is some preexisting delamination inthe 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 ofdelamination and dynamic loading in the presence of in-plane compression.Impact loading and any form of dynamic loading in the presence ofsignificant compressive stress in sandwich structures are known to triggerdelamination failure which is abrupt and total. These aspects have beendiscussed in several contributions.

The book has been divided into several sections to address the issuesmentioned in the foregoing. It has been a pleasure to work with a number ofauthors of international standing and reputation who had spent a great dealof effort in developing their respective chapters. The references cited at theend of each chapter should supplement and corroborate the concepts developedin the chapter. We hope that researchers and engineers who are concerned toapply state of the art technologies to composite structural analysis, designand evaluation of risk of failure will find this book useful and a valuablesource of insight.


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327

11.1 Introduction

Laminated composite materials have high strength-to-weight and stiffness-to-weight ratios. They can be considered as laminar systems with weakinterfaces. Consequently, they are very susceptible to interlaminar damagedesignated by delamination. In the presence of delaminations the materialstiffness and, consequently, of the associated structure, can be drasticallyreduced which can lead to its catastrophic failure. Moreover, delamination isan internal damage and is not easily detected, which increases the associatedrisks.

11.1.1 Damage mechanism

In the majority of real applications delamination does not occur alone. It isknown that matrix cracking inside layers and delamination are usuallyassociated and constitute a typical damage mechanism of composites (Takedaet al., 1982; Joshi and Sun, 1985), especially when structures are submittedto bending loads. In fact, although the phenomenon can occur under tensileloading it acquires a remarkable importance under bending loads, e.g. lowvelocity impact (Choi et al., 1991a). It is commonly accepted that there is astrong interaction between matrix cracking inside layers and delaminationbetween 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 originate 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 twodifferently oriented layers it is unable to easily penetrate the other layer, thuspropagating as a delamination. On the other hand, two adjacent laminaehaving different fibre angles induce extensional and bending stiffness mismatchwhich, combined with the low strength of the matrix, make composite materialsvery sensitive to delamination at those interfaces. In general, the delamination

11Interaction of matrix cracking and

delamination

M F S F de M O U R A, Faculdade de Engenharia daUniversidade do Porto, Portugal

WH-Delamination-11 6/5/08, 2:36 PM327

Delamination behaviour of composites328

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itedresistance of a given interface decreases with the increase of its stiffness

mismatching degree (Liu and Malvern, 1987). During delamination propagationextensive micro-matrix cracks are generated in the adjacent layers.

It is worth noting that this complex interaction damage mechanism canoccur in different forms depending on the bending stiffness of the structure.Flexible components respond primarily in a flexural mode, inducing significanttensile stresses at farther layers from the loaded surface. Therefore, the crackslocated in these outermost layers are vertical and caused by bending effects(Choi and Chang, 1992) and their initiation and growth occur in an almostmode I fracture process. These cracks will generate a delamination along theupper adjacent interface, which will interact with other matrix cracks of theneighbour upper ply, leading to delamination on the second interface and soon (see Fig. 11.1). Delamination patterns in flexible laminates present afrustum-conical shape, where delaminations’ size increases towards the lowerface of the laminate (Levin, 1991). A different failure mechanism occurs forstiffer laminates. In this case, only small deflections take place and damageinitiates near to the loaded surface as a result of contact forces. Shear cracksdevelop near to the impact indentation and propagate through the upper plyup to the neighbour interface degenerating in a delamination. This delaminationextends from the loaded region until it also deflects into a lower ply by shearand inclined cracks (see Fig. 11.2). Further delamination growth in stifflaminates occurs in a barrel shape region by growth of delaminations aroundthe midplane (Olsson et al., 2000).

Matrix bending cracks are a predominant mode I fracture process governedby transverse normal tensile stresses whilst matrix shear cracks are governedby interlaminar shear and transverse normal tensile stresses (Choi et al.,

11.1 Schematic representation of damage initiation mechanism inflexible laminates.


Interaction of matrix cracking and delamination 329

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

11.1.2 Classical prediction methodologies

Two main approaches have been used in order to simulate the damagemechanism described above. One is based on strength of materials conceptswhere materials are assumed to be free of defects. However, in many situationsthe problem of stress concentrations nearby to a notch or a flaw leads tomesh dependency in numerical approaches. To overcome this problem thestresses obtained analytically or numerically are used in a point stress oraverage stress criteria (Whitney and Nuismer, 1974) in order to evaluate theoccurrence of failure. In the point stress criterion the stresses are evaluatedat a characteristic distance whereas in the average stress criterion they areaveraged over a distance. An example is given by Choi et al. (1991b) wherequadratic average stress criteria were used to predict matrix cracking when

n

n

n

niY S

σ σ222

232

+ = 1

11.1

and delamination when

DS S Ya

n

ni

n

ni

n

n

σ σ σ232 +1

13+1

222

2

+ + = 1

11.2

11.2 Schematic representation of damage initiation mechanism instiff laminates.



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itedThe 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 pliesof the nth interface. Y and Si are the in situ ply transverse and shear interlaminarstrengths, respectively, and Da is an empirical constant which was determinedfrom experiments using a fitting procedure. The stress components are averagedwithin the ply thickness

nij

n n

t

ijhdz

n

σ σ = 1–1∫ 11.3

where tn and tn–1 correspond to the position of the upper and lower interfacesof the nth ply, h is the ply thickness and z the axis normal to the plate. Thesecriteria 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; thiscriterion is applied to the lower adjacent interface if a shear crack waspredicted and to the upper one if a bending crack was found.

Unlike what happens in strength of materials based approaches, the fracturemechanics approach assumes the presence of an inherent defect in the material.The majority of the proposed works are based on the concepts of strainenergy release rate. It is usually assumed that damage propagation occurswhen the strain energy at the crack front is equal to the critical strain energyrelease rate, which is a material property. The strain energy release rates arecommonly obtained by using the Virtual Crack Closure Technique (VCCT)(Krueger, 2002). Considering a two dimensional problem (see Fig. 11.3), thestrain energies (GI and GII) can be calculated by the product of the relativedisplacements at the ‘opened point’ (nodes l1 and l2) and the loads at the‘closed point’ (node i)

GB a

Y vi iI = 12

∆ ∆

GB a

X ui lII = 12

∆ ∆ 11.4

being B∆a the area of the new surface created by an increment of crackpropagation (see Fig. 11.3). It should be assured that self similar propagationoccurs and an adequate refined mesh should be used. Liu et al. (1993) usedthe VCCT to obtain the strains energy release rates and a linear mixed-modecriterion

GG

GG

I

Ic

II

IIc + = 1 11.5



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to simulate the propagation of a small initial delamination crack introducedafter the occurrence of the initial matrix cracking failure. GIc and GIIc representthe interlaminar critical strain energy release rates in modes I and II,respectively. A similar approach was followed by Zou et al. (2002) where amixed-mode delamination growth is considered when

GG

GG

GG

I

Ic

II

IIc

III

IIIc+ + = 1

α β γ11.6

and α, β and γ are mixed-mode fracture parameters determined from materialtests.

The stress and fracture mechanics based criteria present some disadvantages.The stress based methods present mesh dependency during numerical analysisdue to stress singularities. On the other hand, the point/average stress criteriarequire the definition of a critical dimension which depends on the materialand stacking sequence (Tan, 1989), and do not have a physically powerfultheoretical foundation. Fracture mechanics approach relies on the definitionof an initial flaw or crack length. However, in many structural applicationsthe locus of damage initiation is not obvious. On the other hand, the stress-based methods behave well at predicting delamination onset, and fracturemechanics has already demonstrated its accuracy in the delaminationpropagation modelling. In order to overcome the referred drawbacks andexploit the usefulness of the described advantages, cohesive damage modelsand continuum damage mechanics emerge as suitable options. Thesemethodologies combine aspects of stress based analysis to model damage

y

Thickness = B

vl2l2

ul2

ul1

vl1

Yi

iXi

l1

a ∆a ∆a

x

11.3 Scheme of the VCCT.



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itedinitiation and fracture mechanics to deal with damage propagation. Thus, it

is not necessary to take into consideration an initial defect and mesh dependencyproblems are minimized.

11.2 Mixed-mode cohesive damage model

Cohesive damage models are frequently used to simulate damage onset andgrowth. They are usually based on a softening relationship between stressesand relative displacements between crack faces, thus simulating a gradualdegradation of material properties. They do not depend on a predefinedinitial flaw unlike conventional fracture mechanics approaches. Typically,stress based and energetic fracture mechanics criteria are used to simulatedamage initiation and growth, respectively. Usually cohesive damage modelsare 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 solidelements. Those elements are placed at the planes where damage is prone tooccur which, in several structural applications, can be difficult to identify apriori. However, an important characteristic of delamination is that itspropagation is restricted to a well defined plane corresponding to the interfacebetween two differently oriented layers, thus leading to a typical applicationof cohesive methods. Taking this into consideration, a cohesive mixed-modedamage model based on interface finite elements is presented.

The formulation is based on the constitutive relationship between stresseson the crack plane and the corresponding relative displacements

σ = Eδr 11.7

where δr is the vector of relative displacements between homologous points,and E a diagonal matrix containing the penalty parameter e introduced bythe user. Its values must be quite high in order to hold together and preventinterpenetration of the element faces. Following a considerable number ofnumerical simulations (Gonçalves et al., 2000), it was found that e = 107

N/mm3 produced converged results and avoided numerical problems duringthe non-linear procedure.

The interface finite element includes a damage model to simulate damageonset and growth. Equation 11.7 is only valid before damage initiation. Theconsidered damage model combines aspects of strength-based analysis andfracture mechanics. It is based on a softening process between stresses andinterfacial relative displacements and includes a mixed-mode formulation.After peak stress the material softens progressively or, in other words, undergoesdamage (see Fig. 11.4). To avoid the singularity at the crack tip and itseffects, a gradual rather than a sudden degradation, which would result inmesh-dependency, is considered. It is assumed that failure occurs gradually



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as energy is dissipated in a cohesive zone behind the crack tip. This isequivalent to the consideration of a ‘Fracture Process Zone’, defined as theregion in which the material undergoes softening deterioration by differentways, 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 theinterfacial stress reaches the respective strength σu,i (see Fig. 11.4). Thesoftening relationship can be written as

� = (I – D)E�r 11.8

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

diu i i o i

i u i o i=

( – )

( – ), ,

, ,

δ δ σδ δ σ 11.9

where δo,i and δu,i are, respectively, the onset and ultimate relative displacementsof 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 respectivecritical strain energy release rate

Gic u i u i = 12 , ,σ δ 11.10

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

σ i

σ u,i

σ um,i

σ om,i

Gi

δo,i

Gic

Pure modemodel

i = I, II, III

Mixed-modemodel

δ i

δum,i δu,i

11.4 Pure and mixed-mode damage model.



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itedabove (see Fig. 11.4). Damage initiation is predicted by using a quadratic

stress criterion

σσ

σσ

σσ σ

σσ

σσ σ

I

,I

2II

,II

2III

,III

2

1

II

,II

2III

,III

2

1

+ + = 1 if 0

+ = 1 if 0

u u u

u u

≥

≤11.11

assuming that normal compressive stresses do not promote damage.Considering Equation 11.7, the first equation 1.11 can be rewritten in functionof relative displacements

δδ

δδ

δδ

om

o

om

o

om

o

,I

,I

2,II

,II

2, III

, III

2

+ + = 1

11.12

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

δ δ δ δm = + + I2

II2

III2 11.13

and mixed-mode ratios

β δδi

i = I

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 δ δ δβ β

δ δ β δ δ β δ δοom o oo o o o o o

=1 + +

( ) + ( ) +( ),I ,II ,IIIII2

III2

,II ,III2

II ,I ,III2

III ,I ,II2

11.15

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

δ β δβ β

om ii om

,

II2

III2

= 1 + +

11.16

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

GG

GG

GG

I

Ic

II

IIc

III

IIIc + + = 1 11.17



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itedThe released energy in each mode at complete failure can be obtained from

the area of the minor triangle of Fig. 11.4.

Gi um i um i = 12 , ,σ δ 11.18

Considering Equations 11.7, 11.13 and 11.14, the energies (Equations [11.10and 11.18]) can be written in function of relative displacements. Substitutinginto Equation 11.17, it can be obtained

δβ βδ

β βum

ome G G G =

2 (1 + + ) 1 + + II2

III2

Ic

II2

IIc

III2

IIIc

11.19

which corresponds to the mixed-mode displacement at failure. The ultimaterelative displacements in each mode, δum,i, can be obtained from Equations11.13, 11.14 and 11.19

δ β δβ β

um ii um

,

II2

III2

= 1 + +

11.20

The damage parameter for each mode can be obtained substituting δom,i andδum,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 acohesive damage model (de Moura and Gonçalves, 2004). Circular clampedplates of 50 mm diameter were tested and damage, identified by X-raymethod, was constituted by:

• a longitudinal long crack parallel to fibres direction in the outermostgroup of equally oriented layers and caused by bending loading;

• an extensive delamination located at the distal interface between differentlyoriented layers relatively to the loaded surface; the delamination has acharacteristic two-lobed shape with its major axis oriented on the directionof the lower adjacent ply.

In the numerical model only half plate was considered due to geometricaland material symmetrical conditions. With the aim to numerically simulatethis damage mechanism, interface finite elements were placed at the criticalinterface in order to simulate the observed delamination, and at the verticalsymmetry plane of the used mesh. The objective of these vertical elementswas to model the onset and growth of the longitudinal bending crack in theoutermost group of equally oriented layers which was observed to be theinitial damage (see Fig. 11.5). This crack induces delamination in the adjacentinterface as it can be seen in Fig. 11.6. This damage mechanism occurs in aprogressive way, i.e., the growth of the vertical crack is associated with anincreasing delamination. Numerical results agreed with the experiments.



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The shape of the delamination was accurately predicted (see Fig. 11.7). Thedelamination and crack lengths in function of the maximum load were alsoin agreement (see Figs 11.8 and 11.10), although some non-negligibledifferences were noted in the delaminations width (see Fig. 11.9). The globaltrend was captured for all cases.

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.



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ExperimentNumerical

1600 1800 2000 2200 2400 2600Pmax (N)

l (m

m)

50

40

30

20

10

0

11.8 Delamination length (l) in function of maximum load.

ExperimentNumerical

1600 1800 2000 2200 2400 2600Pmax (N)

W (

mm

)

50

40

30

20

10

0

11.9 Delamination width (W) in function of maximum load.

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

ExperimentNumerical

1600 1800 2000 2200 2400 2600Pmax (N)

l c (

mm

)

50

40

30

20

10

0



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itedAlthough promising results were obtained with this approach it should be

emphasized that it cannot be considered an adequate prediction model formore general applications. In fact, in laminates with several different orientedlayers it is not suitable to include numerous vertical interface elements in alllayers to predict matrix cracking in any layer. The alternative is to considerthat solid elements can also include a damage model in order to simulatedamage inside layers. This can be done using continuum damage mechanicswhich is discussed in the next section.

11.3 Continuum damage mechanics

The classical approaches based on strength of materials usually assume thatonce matrix cracking arises, a sudden loss of material properties occur,which is generally denominated by ply discount models (Hwang and Sun,1989). However, it is known that in presence of matrix cracking, the compositedoes not loose its load carrying capacity immediately. In fact, damage can beconsidered as the progressive weakening mechanism which occurs in materialsprior to failure. It can be constituted by micro-cracking, voids nucleation andgrowth, and several inelastic processes that deteriorate the material. Theanalysis of cumulative damage is fundamental in life prediction of componentsand structures under loading. Tan (1991) proposed a progressive damagemodel relating the material elastic properties with internal state variablesDi

T 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 materialproperties are abruptly reduced according to the respective residual strengthexperimentally observed. Each damage mode is predicted by the subsequentexpressions:

Fibre tensile fracture

E D Ed T d d11 1 11 12 13 = ; = = 0ν ν 11.21

Fibre compressive fracture

E D Ed C d d11 1 11 12 13 = ; = = 0ν ν 11.22

Matrix tensile failure

E D E E D E

G D G G D G G D G

d T d T d d

d T d T d T22 2 22 33 2 33 12 23

12 6 12 13 6 13 23 6 23

= ; = ; = = 0

= ; = ; =

ν ν11.23

Matrix compressive failure or shear cracking

E D E E D E

G D G G D G G D G

d C d C d d

d C d C d C22 2 22 33 2 33 12 23

12 6 12 13 6 13 23 6 23

= ; = ; = = 0

= ; = ; =

ν ν11.24



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itedThe author (Tan, 1991 and Tan and Perez, 1993) obtained good agreement

with experimental results considering D D D DT T T C1 2 6 1 = 0.07, = = 0.2, = 0.14

and D DC C2 6 = = 0.4. Although this type of models consider a residual strength

in accordance with the physical reality they are mesh dependent duringnumerical analysis.

To avoid the sudden loss of material properties and mesh dependency, thecontinuum damage models combining strength of materials and fracturemechanics concepts are an appealing alternative. Material damage is simulatedby introducing damage variables into the constitutive equations (Lemaitreand Chaboche, 1985). After the matrix cracking initiation is predicted, agradual softening post-failure analysis is also performed by appropriatelyreducing the material properties within the elements where matrix crackingonset occurred.

Ladevèze and Le Dantec (1992) developed a continuum damage mechanicsformulation for orthotropic materials to account for ply degradation.Considering a damaged layer in a state of plane stress the strain-stressrelationship take the general form

� = S� 11.25

where � and � are vectors of elastic strain and stress, respectively. In a localsystem associated with orthotropy axes it can be written

� = (σ11, σ22, σ12)T; � = (ε11, ε22, 2ε12)

T 11.26

and S the compliance matrix

S =

1(1 – )

– 0

– 1(1 – )

0

0 0 1(1 – )

1 1

12

1

12

1 2 2

12 12

E d E

E E d

G d

υ

υ

11.27

The damage parameters (d1, d2 and d12) define the damage state for the threetypes of stress loading and varies between 0 (undamaged state) and 1 (completeloss of stiffness). No additional parameter is used to simulate degradation inPoisson’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 beshown from Equations 11.25, 11.26 and 11.27 that –ε22/ε11 = υ12(1 – d1);similarly it can be easily demonstrated that υ21 is affected by (1 – d2). Inorder to establish the evolution of damage parameters in function of thedamage growth the concept of strain energy density ϕ is used

ϕ = 1

2T� �S 11.28



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itedThe theory also includes the concept of ‘thermodynamic forces’, Y1, Y2, Y12,

associated with the internal damage variables d1, d2, d12. Those parametersare also considered as driving forces for damage development and are definedby

Yd

= ∂∂ϕ

11.29

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

and 11.29 it follows

YE d

YE d

YG d1

112

1 12 2

222

2 22 12

122

12 122 =

2 (1 – ); =

2 (1 – ); =

2 (1 – )

σ σ σ

11.30

In the absence of fibre breakage d1 is zero throughout the load history andthe longitudinal modulus does not degrade; therefore only Y2 and Y12 drivingforces should be considered to model matrix cracking. A linear combination

Y Y bY = + 12 2 11.31

where b is a material constant, can be used to account for coupling betweentransverse tension and shear effects. To avoid healing phenomena the maximumvalue of Y up to the current time t is defined as

Y t Y bYt

( ) = max ( + )12 2τ ≤11.32

Experimental results for carbon-epoxy laminates showed that damageparameters can be written as

dY Y

Yd

Y Y

Y12

0

C2

0

C

= –

; = – ˆ ˆ ′

′11.33

where Y0, YC, ′Y0 and ′YC are damage evolution parameters. They are determinedexperimentally 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 agreementwas obtained with the respective experimental σ–ε curves.

An approach similar to the one used in the cohesive damage model describedin Section 11.2 is proposed by other authors (Crisfield et al., 1997; Pinhoet al., 2006; de Moura and Chousal, 2006). In this case there is a softeningrelationship between stresses and strains instead of between stresses andrelative displacements considered in the cohesive model. Consequently, inthis case a characteristic length lc must be introduced to transform the relativedisplacement into an equivalent strain. (see Fig. 11.4)



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itedG lic u i u i c = 1

2 , ,σ ε 11.34

This parameter was considered to be equal to the length of influence of aGauss point in the given direction and physically can be regarded as thedimension at which the material acts homogeneously. The stress–strain relationcan be written considering an equation similar to Equation 11.8

� = (I – D)C� 11.35

being C the stiffness matrix of the undamaged material in the orthotropicdirections. Assuming that matrix cracking occur in mixed-mode I+II thedamage model described in Section 11.2 can be adopted. The damage parameteris calculated by an expression similar to Equation 11.9 but considering strainsinstead of relative displacements. A linear softening law is also used. Theproperties are smoothly reduced due to the energy released at the FPZ. Thematerial properties at a given Gauss point are degraded according to theassumed criterion. This leads to load redistribution for the neighbouringpoints, thus simulating a gradual propagation process.

In summary, it can be affirmed that these methods allow simulating damageinside solid finite elements used to model composite layers and can be usedto simulate matrix cracking phenomenon. A gradual degradation of propertiesinstead of a sudden one avoids the singularity effects and minimizes theconsequent mesh sensitivity.

11.4 Conclusions

Matrix crack and delamination are intrinsically associated in compositematerials, namely under bending loads. The interaction between these twomodes of damage constitutes a complex damage mechanism that has notbeen addressed in a realistic level. Such interaction is fundamental to beconsidered in a failure model prediction because one mode may initiate theother and they may intensify each other. Two different models come out todeal with the referred damage modes. Mixed-mode cohesive damage modelsjoin the positive arguments of stress based and fracture mechanics criteriaovercoming their inherent difficulties. These models have being used withsuccess to simulate delamination initiation and growth. They are usuallybased on interface finite elements including a softening relationship betweenstresses and relative displacements. The continuum damage mechanics isbeing applied on the simulation of matrix cracking. These models are basedon the introduction of damage parameters into the constitutive equations inorder to simulate material damage. These damage parameters increase smoothlywith growing damage, leading to a slow degradation of material propertiesinstead of an abrupt one which is not realistic and originate mesh dependencies.



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itedIn summary, cohesive and continuum damage models are actually the

most prominent numerical tools in order to simulate matrix cracking inducingdelamination damage mechanism of composites. However, it should berecognized that an accurate methodology addressing all the realistic issuesof this complex damage mechanism is still lacking. The solution points tothe development of a numerical tool incorporating the two kinds of models.The two methods (cohesive elements and damage mechanics) can coexist. Infact, both models can be implemented via user subroutines in commercialsoftware. When the selected damage criterion in a solid element is satisfiedthe element fails simulating matrix cracking. This induces important relativedisplacements at the adjacent interfaces leading to delamination initiationand propagation according to the damage criterion of the cohesive elements.Some aspects should require special attention like the influence of stressconcentration 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 ableto accurately model such particularity. Some efforts should also be dedicatedto the development stress and fracture criteria adequate to the specificities ofthis damage mechanism.

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