Delamination behaviour of composites · 2 Delamination in the context of composite structural design 28 A RICCIO, C.I.R.A. (Centro Italiano Ricerche Aerospaziali – Italian Aerospace
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Multi-scale modelling of composite material systems(ISBN 978-1-85573-936-9)Predictive modelling provides the opportunity both to understand better howcomposites behave in different conditions and to develop materials with enhancedperformance for particular industrial applications. This important book focuses onthe fundamental understanding of composite materials at the microscopic scale,from designing micro-structural features, to the predictive equations of thefunctional behaviour of the structure for a specific end-application. Chaptersdiscuss stress and temperature-related behavioural phenomena based onknowledge 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 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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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
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ited7.13 Acknowledgement 211
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
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
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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
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ited17 Delamination propagation under cyclic loading 485
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
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ited23 Z-pin bridging in composite delamination 674
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
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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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310
10.1 Introduction
The application of composite materials in the aircraft and automobile industrieshas led to an increase of research into the fracture behaviour of composites.One of the most significant mechanical properties of fibre reinforced polymercomposites is its resistance to delamination onset and propagation. It isknown that delamination can induce significant stiffness reduction leadingto premature failures. Delamination can be viewed as a crack propagationphenomenon, thus justifying a typical application of fracture mechanicsconcepts. In this context, the interlaminar fracture characterization ofcomposites acquires remarkable relevancy. There are several tests proposedin the literature in order to measure the interlaminar strain energies releaserates in mode I, mode II and mixed mode I/II. Whilst mode I has alreadybeen extensively studied and the Double Cantilever Test (DCB) test isuniversally accepted, mode II is not so well studied, which can be explainedby some difficulties inherent to experimental tests. Moreover, in many realsituations delaminations propagate predominantly in mode II, as is the caseof composite plates under low velocity impact (Choi and Chang, 1992). Thisgives relevancy to the determination of toughness propagation values insteadof the initiation ones commonly considered in design. Some non-negligibledifferences can be achieved considering the R-curve effects (de Morais andPereira, 2007). These issues make the fracture characterization in mode II anactual and fundamental research topic. However, problems related to unstablecrack growth and to crack monitoring during propagation preclude a rigorousmeasurement of GIIc. In fact, in the mode II fracture characterization teststhe crack tends to close due to the applied load, which hinders a clearvisualization of its tip. In addition, the classical data reduction schemes,based on beam theory analysis and compliance calibration, require crackmonitoring during propagation. On the other hand, a quite extensive FractureProcess Zone (FPZ) ahead of crack tip exists under mode II loading. Thisnon-negligible FPZ affects the measured toughness as a non-negligible amount
10Interlaminar mode II fracture
characterization
M. F. S. F. d e M O U R A, Faculdade de Engenharia daUniversidade do Porto, Portugal
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Interlaminar mode II fracture characterization 311
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itedof 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 theselected data reduction scheme. To overcome these difficulties a new datareduction scheme based on crack equivalent concepts and depending only onthe specimen compliance is presented in the next section. The main objectiveof the proposed methodology is to increase the accuracy of experimentalmode II fracture tests on the GIIc measurements. In fact, a rigorous monitoringof the crack length during propagation is one of the complexities of thesetests.
10.2 Static mode II fracture characterization
There are three fundamental experimental tests used to measure GIIc. Themost popular one is the End Notched Flexure (ENF), which was developedfor wood fracture characterization (Barrett and Foschi, 1977). The test consistson a pre-cracked specimen under three point bending loading (see Fig. 10.1).
a δ, P
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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itedUnstable crack propagation constitutes one of the disadvantages of the ENF
test. Another possibility is the End Loaded Split (ELS) test which is based oncantilever beam geometry (see Fig. 10.1). Although the ELS test involvesmore complexities during experiments relatively to the ENF test, it providesa larger range of crack length where the crack propagates stably. In fact, theENF test requires a0/L>0.7 to obtain stable crack propagation (Carlsson etal., 1986), whereas in the ELS test a0/L>0.55 is sufficient (Wang and Vu-Khanh, 1996). However, both of these tests present a common difficultyrelated to the crack length measurement during the experimental test. Differentmethods 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 specialdisplacement gage was developed for direct measurement of the relativeshear slip between crack surfaces of the ENF specimen. The test was performedunder constant crack shear displacement rate, which guarantees stable crackpropagation. Yoshihara et al. (Yoshihara and Ohta, 2000) recommended theuse 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.(Tanaka et al., 1995) concluded that to extend the stabilized crack propagationrange in the ENF test, the test should be done under a condition of controlledCSD. Although the CSD method provides the measurement of the mode IItoughness without crack length monitoring, this method requires a servovalve-controlled testing machine and the testing procedure is more complicatedthan that under the loading point displacement condition. Alternatively theFour Point End Notched Flexure test (4ENF) (Fig. 10.1) can be used toevaluate the mode II R-curve. This test does not require crack monitoring butinvolves a more sophisticated setup and larger friction effects were observed(Shuecker and Davidson, 2000). In the following, a summary of the classicalreduction 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, applieddisplacement and crack length (P-δ-a) are registered in order to calculate thecritical strain energy release rate using the Irwin-Kies equation (Kanninenand Popelar, 1985)
G PB
dCdaIIc
2 =
2 10.1
where B is the specimen width and C = δ/P the compliance. In the ENF andELS tests a cubic relationship between the compliance (C) and the measuredcrack length a is usually assumed (Davies et al., 2001)
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itedC = D + ma3 10.2
where D and m are constants. GIIc is then obtained from
GP m a
BIIc
2 2
= 3
210.3
For the 4ENF test a linear relationship (Yoshihara, 2004) between thecompliance (C) and the measured crack length a is used
C = D + ma 10.4
being D and m the respective coefficients. It should be noted that relationsC = f(a) given by Equations 10.2 and 10.4 are based on the beam theoryapproach, as it will be shown in the next sub-section. GIIc is given by
G PB
mIIc
2 =
210.5
The three tests require the calibration of the compliance in function of thecrack length. This can be done by measurement of crack length duringpropagation or, alternatively, considering several specimens with differentinitial cracks lengths to establish the compliance–crack length relation, whichis 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 CorrectedBeam Theory (CBT)
Ga P
B h EIIcI
2 2
2 31
= 9( + 0.42 )
1∆
610.6
where E1 is the axial modulus and ∆I a crack length correction to account forshear deformation
∆ I1
13
2
= 11
3 – 21 +
hEG
ΓΓ( )
10.7
with
Γ = 1.18 1 2
13
E EG
10.8
where E2 and G13 are the transverse and shear moduli, respectively. In theELS case a similar expression is proposed (Wang and Williams, 1992)
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itedFor the 4ENF test the beam theory leads to the following equation (Silva,
2006)
C dE I
d a d L d L = 24
(18 – 20 + 60 – 6 )1
2 2 10.10
where I is the second moment of area and d represents the distance betweeneach support and its nearest loading actuator (Fig. 10.1). Using Equation[10.1] GIIc can be obtained from
G P dE B hIIc
2 2
12 3 = 9
1610.11
In summary, the application of beam theory to ENF and ELS tests involvesthe crack length, which does not occur in the 4ENF test. However, it shouldbe 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, theENF and ELS tests emerge as the most appropriate to fracture characterizationof composites in mode II. In this context, a new data reduction scheme, notdepending on the crack length measurements, is proposed in the followingsection for these experimental tests.
10.2.2 Compliance based beam method (CBBM)
In order to overcome the difficulties associated to classical data reductionschemes a new method is proposed. The method is based on crack equivalentconcept and depends only on the specimen compliance. The application ofthe method to ENF and ELS tests is described in the following.
ENF test
Following strength of materials analysis, the strain energy of the specimendue to bending and including shear effects is
UM
E Idx
GBdy dx
Lf
f
L
h
h
= 2
+ 20
2 2
0
2
–
2
13∫ ∫ ∫ τ 10.12
where Mf is the bending moment and
τ = 32
1 – 2
2
VA
yc
i
i i
10.13
where Ai, ci and Vi represent, respectively, the cross-section area, half-thicknessof the beam and the transverse load of the i segment (0 ≤ x ≤ a, a ≤ x ≤ L or
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itedL ≤ x ≤ 2L). From the Castigliano theorem, the displacement at the loading
point for a crack length a is
δ = = (3 + 2 )8
+ 3
10
3 3
313
dUdP
P a LE Bh
PLG Bhf
10.14
Since the flexural modulus of the specimen plays a fundamental role on theP-δ relationship, it can be calculated from Equation 10.14 using the initialcompliance C0 and the initial crack length a0
Ea L
BhC
LG Bhf =
3 + 2
8–
310
03 3
3 013
–1
10.15
This procedure takes into account the variableness of the material propertiesbetween different specimens and several effects that are not included inbeam theory, e.g., stress concentration near the crack tip and contact betweenthe two arms. In fact, these phenomena affect the specimen behavior andconsequently the P-δ curve, even in the elastic regime. Using this methodologytheir influence are accounted for through the calculated flexural modulus.On the other hand, it is known that, during propagation, there is a regionahead of crack tip (Fracture Process Zone), where materials undergo propertiesdegradation by different ways, e.g., micro-cracking, fibre bridging and inelasticprocesses. These phenomena affect the material compliance and should beaccounted for in the mode II tests. Consequently, during crack propagationa correction of the real crack length is considered in the equation of compliance(10.14) to include the FPZ effect
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itedThis data reduction scheme presents several advantages. Using this
methodology crack measurements are unnecessary. Experimentally, it is onlynecessary 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 modulusis calculated from the initial compliance and initial crack length, thus avoidingthe influence of specimen variability on the results. The unique materialproperty needed in this approach is G13. However, its effect on the measuredGIIc was verified to be negligible (de Moura et al., 2006), which means thata typical value can be used rendering unnecessary to measure it.
ELS test
Following a procedure similar to the one described for the ENF test, theapplied P-δ relationship is
δ = = (3 + )2
+ 3
5
3 3
31 13
dUdP
P a LBh E
PLBhG
10.20
In order to include the root rotation effects at clamping and the details ofcrack tip stresses or strains not included in the beam theory, an effectivebeam length (Lef) can be achieved. In fact, considering in Equation 10.20 theinitial crack length (a0) and the initial compliance (C0) experimentallymeasured, it can be written
Ca
Bh E
L
Bh EL
BhG003
31
ef3
31
ef
13 –
3
2 =
2 +
35
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
Ca a
Bh E
L
Bh EL
BhG –
3( + )2
= 2
+ 3
5FPZ
3
31
ef3
31
ef
13
∆10.22
Combining Equations 10.22 and 10.21, the equivalent crack length can begiven by
a a a C CBh E
aeq FPZ 0
31
03
1/3
= + = ( – )2
3 + ∆
10.23
GIIc can now be obtained from
GP a
B h EIIc
2eq2
2 31
= 9
410.24
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itedFollowing this procedure GIIc can be obtained without crack measurement
during propagation which can be considered an important advantage. Equation10.24 only depends on applied load and displacement during crack growth.Additionally, the influence of root rotation at the clamping point and singularityeffects at the crack tip are accounted for, through initial compliance C0.During propagation, the effect of FPZ on the compliance is also includedusing this methodology. In this case (ELS test) it is necessary to measure thelongitudinal 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 ELStests were performed. A cohesive mixed-mode damage model based on interfacefinite elements was considered to simulate damage initiation and propagation.A constitutive relation between the vectors of stresses (σ) and relativedisplacements (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 inputteddata parameters [8, 9]. Damage onset is predicted using a quadratic stresscriterion
σσ
σσ
σσ σ
σσ
σσ σ
I
,I
2II
, II
2III
, III
2
I
II
, II
2III
, III
2
I
+ + = 1 if 0
+ = 1 if 0
u u u
u u
≥
≤10.25
where σi, (i = I, II, III) represent the stresses in each mode. Crack propagationwas simulated by a linear energy criterion
GG
GG
GG
I
Ic
II
IIc
III
IIIc + + = 1 10.26
Basically, it is assumed that the area under the minor triangle of Fig. 10.2 isthe energy released in each mode, which is compared to the respective criticalfracture energy represented by the bigger triangle. The subscripts o and urefer to the onset and ultimate relative displacement and the subscript mapplies to the mixed-mode case. More details about this model are presentedin de Moura et al. (2006).
Three-dimensional approaches (Figs 10.3 and 10.4) were carried out toinclude all the effects that can influence the measured GIIc. The interfaceelements were placed at the mid-plane of the specimens to simulate damageprogression. Very refined meshes were considered in the region of interestcorresponding to crack initiation and growth. The specimens’ geometry and
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itedσ 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
10.2 Pure and mixed-mode damage model.
10.3 The mesh used for the ENF test: global view and detail of therefined mesh at the region of crack initiation and growth.
material properties and 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 predominanceof mode II along the specimens’ width, although some spurious mode IIIexists at the specimens edges (de Moura et al., 2006 and Silva et al., 2007).
10.4 The mesh used for the ELS test.
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Appropriate values of critical strain energy release rates were considered foreach of the three modes, respectively (see Table 10.3). Consequently, theefficacy of the proposed data reduction scheme can be evaluated by itscapacity to reproduce the inputted GIIc from the P-δ results obtainednumerically.
The application of the CBBM is performed by three main steps. The firstone is the measurement of the initial compliance C0 from the initial slope ofthe P-δ curves (Figs (10.5) or (10.7)). This parameter is then used to estimatethe flexural modulus in the ENF test (Equation 10.15). The next step is the
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itedevaluation of the equivalent crack length (Equations 10.17 or 10.23) in 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 criticalstrain energy release rate in mode II (GIIc). These plateau values are comparedwith the reference value (Figs (10.6) and (10.8)), which represents the inputtedGIIc. The excellent agreement obtained in both cases demonstrates the
GII (CBBM)
GIIc (Reference value)
GII
(N/m
m)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.075 80 85 90 95 100
aeq (mm)
10.6 R-curve of the ENF specimen.
P (
N)
70
60
50
40
30
20
10
00 2 4 6 8 10 12 14 16
δ (mm)
10.7 P-δ curve of the ELS specimen.
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effectiveness of the CBBM as a suitable data reduction scheme to determineGIIc, without crack length monitoring during propagation. As the ENF test ismuch simpler to execute than the ELS one, it can be concluded that using theCBBM, the ENF test is the most suitable for the determination of GIIc and itshould 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 thefocus of several authors in the recent years. The dynamic fracturecharacterization of composites is not easy to perform. In fact, it is experimentallydifficult to induce high speed delamination growth in a simple and controlledmanner (Guo and Sun, 1998). However, the determination of dynamic fracturetoughness of composites is of fundamental importance in the prediction ofthe dynamic delamination propagation in composite structures. In addition,it is known that the impact delamination is mainly governed by mode IIfracture (Wang and Vu-Khanh, 1991). However, there are several unclearphenomena related to dynamic crack propagation. One of the most importantissues is related to the influence of rate effects on the propagation of dynamiccracks. An example of this occurrence is the dynamic delamination propagationoccurring in composites submitted to low velocity impact. In this case, rateeffects in the FPZ can interact with the well known rate-dependency ofpolymers leading to a very complicated phenomenon. In addition, Kumarand Narayanan (1993) verified that when glass fibre reinforced epoxy laminatesare impacted, the total delamination area between the various plies multipliedby the quasi-static energy release rate exceeds the energy of the impacting
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itedmass. This suggests that under high crack speeds, delamination propagates
at lower toughness which leads to larger damaged areas. In order to explainthis behaviour, Maikuma et al. (1990) suggest that the calculation of criticalstrain energy release rate should account for the kinetic energy (Ekin) inEquation 10.26
G PB
dCda
dEBdaIIc
2kin=
2 – 10.27
The kinetic energy expression can be obtained from
E B hdw x
dtdx
Lt
kin0
2
= 12
2 ( )ρ ∫
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 quasi-static approach may provide an adequate approximation to the dynamicproblem if the contribution of kinetic energy is small.
Wang and Vu-Khanh (1995) have suggested that the dynamic fracturebehaviour of materials depends on the balance between the energy releasedby the structure over a unit area of crack propagation (G) and the materialresistance (R), which can be viewed as the energy dissipated in creating thefracture surface. When unstable crack growth occurs, the difference G-R isconverted into kinetic energy. If G increases with crack growth the crackspeed also increases because more energy is available. Crack arrest willoccur when G becomes lower than R and, consequently, no kinetic energy isavailable for crack growth. Thus, it can be affirmed that fracture stabilitydepends on the variations of the strain energy release rate and the materialsresistance during crack growth. On the other hand, the fracture resistance ofpolymer composites is generally sensitive to loading rate. Under impact loador during rapid delamination growth, the strain rate at the crack tip can bevery high and the material toughness significantly reduced. The fracturesurface exhibited ductile tearing and large scale plastic deformation of thematrix. The dynamic fracture surface in the initiation exhibits less plasticdeformation; during propagation even less deformation is observed. It wasalso verified that plastic zone size at the crack tip diminishes with increasingrate. Consequently, the decrease in mode II interlaminar fracture toughnessis attributed to a transition from ductile to brittle matrix dominated failurewith increasing rate.
The decreasing trend of toughness with increase of crack speed was alsoobserved by Kumar and Kishore (1998). The authors used a combination ofnumerical and experimental techniques on the DCB specimens to carry outdynamic interlaminar toughness measurements of unidirectional glass fibreepoxy laminate. They observed a sharp decrease of dynamic toughness values
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itedrelatively to the quasi-static ones. In fact, they measured dynamic toughness
initiation values of 90–230 N/m2 against quasi-static values of 344–478N/m2. Propagation values of 0–50 N/m2 were obtained for crack speed rangingbetween 622–1016 m/s.
The majority of the experimental studies consider unidirectional laminates.Lambros and Rosakis (1997) performed an experimental investigation ofdynamic crack initiation and growth in unidirectional fibre-reinforcedpolymeric-matrix thick composite plates. Edge-notched plates were impactedin a one-point bend configuration using a drop-weight tower. Using an opticalmethod the authors carried out a real-time visualization of dynamic fractureinitiation and growth for crack speeds up to 900 m/s. They verified that theelastic constants of the used material are rate sensitive and the measuredfracture toughness values are close to those typical of epoxies. This wasconsidered consistent, because in unidirectional lay-ups crack initiation andgrowth occurs in the matrix.
Tsai et al. (2001) used a modified ENF specimen to determine the modeII dominated dynamic delamination fracture toughness of fiber compositesat high crack propagation speeds. A strip of adhesive film with higher toughnesswas placed at the tip of interlaminar crack created during laminate lay-up.The objective was to delay the onset of crack extension and produce crackpropagation at high speeds (700 m/s). Sixteen pure aluminium conductivelines were put on the specimen edge side using the vapour deposition technique,to carry out crack speed measurements. The authors concluded that the modeII dynamic energy release rate of unidirectional S2/8553 glass/epoxy compositeseems 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 thepre-crack from the mid-plane to 1/3 of the ENF specimen thickness ofunidirectional AS4/3501-6 carbon/epoxy laminates. The maximum inducedcrack speed produced was 1100 m/s. They found that that the critical dynamicenergy release rate is not affected by the crack speed and lies within thescatter range of the respective static values.
For numerical simulations of the dynamic crack propagation the cohesivedamage models emerge as the most promising tools. The major difficulty isthe incorporation of the rate-dependent effects in the constitutive laws.Corigliano et al. (2003) developed a cohesive crack model with a rate-dependent exponential interface law to simulate the nucleation and propagationof cracks subjected to mode I dynamic loading. The model is able to simulatethe rate-dependent effects on the dynamic debonding process in composites.The authors concluded that the softening process occurs under larger relativedisplacements in comparison to rate-independent models. They verified thatthe type of rate-dependency can affect dynamic crack processes, namely thetime to rupture and fracture energy. They also state that these effects diminishwhen inertial terms become dominant.
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itedIn summary, dynamic fracture toughness characterization of composite
materials has been the centre of attention of several authors with no apparentconsensus on the results. Although the majority of the studies point to adecrease of the fracture toughness with increasing load rate there is nounanimity about this topic. Some authors observed the opposite trend(Corigliano et al., 2003) and others detected no remarkable influence ofcrack speed on toughness (Tsai et al., 2001). Although some of thesediscrepancies can eventually be explained by different behaviour of the testedmaterials, and the attained crack speed values, it is obvious that more profoundstudies about the subject are necessary.
10.4 Conclusions
Interlaminar fracture characterization of composites in mode II acquiresspecial relevancy namely under transverse loading such as low velocityimpact. Up to now there is no standardized test in order to measure thecritical strain energy release rate in mode II. Due to their simplicity, the ENFand ELS tests become the principal candidates to standardization. However,they present a common difficulty associated with crack monitoring duringpropagation which is fundamental to obtaining the R-curves, following theclassical data reduction schemes. To surmount these difficulties a new datareduction scheme based on specimen compliance is proposed. The methoddoes not require crack length measurement during propagation, and accountsfor the effects of the quite extensive FPZ on the measured critical strainenergy release rate. Numerical simulations of the ENF and ELS testsdemonstrated the adequacy and suitability of the proposed method to obtainthe mode II R-curves of composites. Due to its simplicity the ENF test isproposed for standardization.
Little work has been done on dynamic fracture of composite materials,namely under mode II loading. This is due to experimental difficulties relatedto inducing high crack speeds in a monitored way. Although the majority ofthe published works point to a decrease of the dynamic toughness withincrease of crack speed, it appears that dynamic toughness can be similar tothe 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 unsafestructural design can occur if the quasi-static values of toughness are used ina dynamically loaded structure.
10.5 Acknowledgements
The author thanks Professors Alfredo B. de Morais (UA, Portugal), JoséMorais (UTAD, Portugal) and Manuel Silva for their valorous collaboration,advices and discussion about the matters included in this chapter. The author
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itedalso thanks the Portuguese Foundation for Science and Technology for
supporting part of the work here presented, through the research projectPOCI/EME/56567/2004.
10.6 References
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Carlsson LA, Gillespie JW, Pipes RB (1986), ‘On the analysis and design of the endnotched 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 laminatedcomposites resulting from low-velocity point impact’, J Compos Mater, 26: 2134–2169.
Corigliano A, Mariani S, Pandolfi A (2003), ‘Numerical modelling of rate-dependentdebonding processes in composites’, Compos. Struct., 61: 39–50.
Davidson BD, Sun X (2005), ‘Effects of friction, geometry and fixture compliance on theperceived 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 PolymersAdhesives and Composites, Amsterdam, London, New York: Elsevier; 307–334.
de Morais AB, Pereira AB (2007), ‘Application of the effective crack method to mode Iand mode II interlaminar fracture of carbon/epoxy unidirectional laminates’, CompositesPart A: Applied Science and Manufacturing, 38: 785–794.
de Moura MFSF, Silva MAL, de Morais AB, Morais JJL (2006), ‘Equivalent crack basedmode 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. CompositeMaterials: Fatigue and Fracture, ASTM STP 1110, Vol. 3. Philadelphia PA: ASTM,p. 210–225.
Kumar P, Kishore NN (1998), ‘Initiation and propagation toughness of delaminationcrack uncder an impact load’, J. Mech. Phys. Solids, 46: 1773–1787.
Lambros J, Rosakis AJ (1997), ‘Dynamic crack initiation and growth in thick unidirectionalgraphite/epoxy plates’, Composites Science & Technology, 57: 55–65.
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Silva MAL (2006), ‘Estudo das Propriedades de Fractura em Modo II e em Modo III daMadeira de Pinus pinaster Ait.’, Master Thesis, FEUP, Porto.
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Wang H, Vu-Khanh T (1996), ‘Use of end-loaded-split (ELS) test to study stable fracturebehaviour of composites under mode II loading’, Compos. Struct., 36: 71–79.
Wang Y, Williams JG (1992), ‘Corrections for Mode II Fracture Toughness Specimens ofComposite Materials’, Composites Science & Technology, 43: 251–256.