Self:.healing Materials

Fundamentals, Design Strategies,and Applications

Edited bySwapan Kumar Ghosh

ffiWILEY­

VCH

WILEY-VCH Verlag GmbH & Co. KGaA

1

1.11.21.31.3.11.3.1.11.3.1.21.3.1.31.3.21.3.2.11.3.2.21.3.2.31.3.31.3.3.11.3.3.21.3.3.31.3.3.41.3.3.51.41.5

2

2.12.2

2.2.1

Contents

Preface xiList ofContributors xiii

Self-healing Materials: Fundamentals, Design Strategies, andApplications 1Swapan Kumar GhoshIntroduction 1Definition of Self-healing 1Design Strategies 2Release of Healing Agents 2Microcapsule Embedment 3Hollow Fiber Embedment 4Microvascular System 8Reversible Cross-links 9Diels-Alder (DA) and Retro-DA Reactions 10Ionomers 12Supramolecular Polymers 13Miscellaneous Technologies 17Electrohydrodynamics 17Conductivity 20Shape Memory Effect 21Nanoparticle Migrations 22Co-deposition 22Applications 23Concluding Remarks 25

Self-healing Polymers and Polymer Composites 29Ming Qiu Zhang, Min Zhi Rang and Tao YinIntroduction and the State of the Art 29Preparation and Characterization of the Self-healing Agent Consistingof Microencapsulated Epoxy and Latent Curing Agent 35Preparation of Epoxy-loaded Microcapsules and the Latent CuringAgent CuBr2(2-MeIm)4 35

VII Contents

2.2.22.2.32.3

2.3.12.3.22.3.32.4

2.4.12.4.22.4.32.5

3

3.13.23.2.13.2.23.33.3.13.3.23.3.33.3.43.3.53.3.63.43.53.6

4

4.14.24.2.14.2.24.2.34.34.3.14.3.24.3.34.3.4

4.3.4.1

Characterization ofthe Microencapsulated Epoxy 36Curing Kinetics of Epoxy Catalyzed by CuBr2 (2-MeIm)4 38Mechanical Performance and Fracture Toughness of Self-healingEpoxy 43Tensile Performance of Self-healing Epoxy 43Fracture Toughness of Self-healing Epoxy 43Fracture Toughness of Repaired Epoxy 45Evaluation of the Self-healing Woven Glass Fabric/EpoxyLaminates 49Tensile Performance of the Laminates 49Interlaminar Fracture Toughness Properties of the Laminates 51Self-healing of Impact Damage in the Laminates 57Conclusions 68

Self-Healing lonomers 73StephenJ. Kalista, Jr.Introduction 73Ionomer Background 74Morphology 75Ionomers Studied for Self-healing 78Self-healing of Ionomers 79Healing versus Self-healing 80Damage Modes 81Ballistic Self-healing Mechanism 83Is Self-healing an Ionic Phenomenon? (Part I) 84Is Self-healing an Ionic Phenomenon? (Part II) 86Self-healing Stimulus 88Other Ionomer Studies 89Self-healing Ionomer Composites 95Conclusions 97

Self-healing Anticorrosion Coatings 101Mikhail ZheludkevichIntroduction 101

Reflow-based and Self-sealing Coatings 103

Self-healing Bulk Composites 103

Coatings with Self-healing Ability based on the Reflow Effect 105

Self-sealing Protective Coatings 108

Self-healing Coating-based Active Corrosion Protection 109

Conductive Polymer Coatings 110

Active Anticorrosion Conversion Coatings 113Protective Coatings with Inhibitor-doped Matrix 119Self-healing Anticorrosion Coatings based on Nano-/Microcontainers of~orrosion Inhibitors 122Coatings with Micro-/Nanocarriers of Corrosion Inhibitors 123

4.3.4.24.4

5

5.15.25.2.15.2.1.15.2.1.25.2.1.35.2.1.45.2.25.2.35.2.3.15.2.3.25.35.3.15.3.25.3.35.3.45.3.4.15.3.4.25.3.4.35.3.4.45.3.4.55.3.55.3.65.3.75.3.85.45.4.15.4.25.4.2.15.4.2.25.4.35.4.45.4.55.55.6

Contents IVII

Coatings with Micro-INanocontainers ofCorrosion Inhibitors 128Conclusive Remarks and Outlook 133

Self-healing Processes in Concrete 141Erk Schlangen and ChristopherJosephIntroduction 141State of the Art 144Definition ofTerms 144Intelligent Materials 144Smart Materials 145Smart Structures 145Sensory Structures 146Autogenic Healing of Concrete 146Autonomic Healing ofConcrete 147Healing Agents 148Encapsulation Techniques 149Self-healing Research at Delft 152Introduction 152Description ofTest Setup for Healing of Early Age Cracks 152Description ofTested Variables 154Experimental Findings 155Influence of Compressive Stress 155Influence of Cement Type 156Influence ofAge When the First Crack is Produced 158Influence of Crack Width 159Influence of Relative Humidity 159Simulation ofCrack Healing 159Discussion on Early Age Crack Healing 163Measuring Permeability 164Self-healing ofCracked Concrete: A Bacterial Approach 165Self-healing Research at Cardiff 168Introduction 168Experimental Work 169Preliminary Investigations 169Experimental Procedure 172Results and Discussion 173Modeling the Self-healing Process 175Conclusions and Future Work 177A View to the Future 178Acknowledgments 179

6 Self-healing ofSurface Cracks in Structural Ceramics 183Wataru Nakao, Koji Takahashi and Kotoji Ando

6.1 Introductiori, 1836.2 Fracture Manner of Ceramics 183

VIII I Contents

6.3 History 1856.4 Mechanism 1876.5 Composition and Structure 190

6.5.1 Composition 190

6.5.2 SiC Figuration _ 1926.5.3 Matrix 1936.6 Valid Conditions 1946.6.1 Atmosphere 1946.6.2 Temperature 1956.6.3 Stress 1986.7 Crack-healing Effect 2006.7.1 Crack-healing Effects on Fracture Probability 2006.7.2 Fatigue Strength 2026.7.3 Crack-healing Effects on Machining Efficiency 2046.8 New Structural Integrity Method 2076.8.1 Outline 2076.8.2 Theory 2076.8.3 Temperature Dependence of the Minimum Fracture Stress

Guaranteed 209

6.9 Advanced Self-crack Healing Ceramics 2126.9.1 Multicomposite 2126.9.2 SiC Nanoparticle Composites 213

7

7.17.27.2.17.2~2

7.2.37.2.3.17.2.3.27.2.47.2.4.17.2.4.2

7.2.4.37.2.5

7.2.5.17.2.5.27.2.5.37.37.3.1

Self-healing ofMetallic Materials: Self-healing ofCreep Cavity andFatigue Cavity/crack 219Norio ShinyaIntroduction 219Self-healing ofCreep Cavity in Heat Resisting Steels 220Creep Fracture Mechanism and Creep Cavity 221Sintering of Creep Cavity at Service Temperature 223Self-healing Mechanism of Creep Cavity 225Creep Cavity Growth Mechanism 225Self-healing Layer on Creep Cavity Surface 226Self-healing of Creep Cavity by B Segregation 227Segregation ofTrace Elements 227Self-healing of Creep Cavity by B Segregation onto Creep CavitySurface 229Effect of B Segregation on Creep Rupture Properties 234Self-healing of Creep Cavity by BN Precipitation on to Creep CavitySurface 234Precipitation of BN on Outer Free Surface by Heating in Vacuum 234Self-healing of Creep Cavity by BN Precipitation 234

,Effect ofBN Precipitation on Creep Rupture Properties 238Self-healing of Fatigue Damage 241Fatigue Damage Leading to Fracture 241

7.3.27.3.2.17.3.2.27.3.37.3.3.17.3.3.2

7.3.3.3

7.3.47.4

Contents IIXDelivery of Solute Atom to Damage Site 242Pipe Diffusion 242Solute-vacancy Complexes 243Self-healing Mechanism for Fatigue Cavity/Crack 243Closure of Fatigue Cavity/Crack by Deposition of Precipitate 244Closure of Fatigue Cavity/Crack by Volume Expansion withPrecipitation 244Replenishment of Strengthening Phase by Dynamic Precipitation onDislocation 244Effect of Self-healing on Fatigue Properties ofAl Alloy 246Summary and Remarks 247

8 Principles ofSelf-healing in Metals and Alloys: An Introduction 251Michele V. Manuel

8.1 Introduction 2518.2 Liquid-based Healing Mechanism 2528.2.1 Modeling ofa Liquid-assisted Self-healing Metal 2568.3 Healing in the Solid State: Precipitation-assisted Self-healing

Metals 2578.3.1 Basic Phenomena: Age (Precipitation) Hardening 2578.3.2 Self-healing in Aluminum Alloys 2588.3.3 Self-healing in Steels 2618.3.4 Modeling of Solid-state Healing - 2628.4 Conclusions 263

9 Modeling Self-healing of Fiber-reinforced Polymer-matrix Compositeswith Distributed Damage 267Ever). Barbero, Kevin). Ford,joan A. Mayugo

9.1 Introduction 2679.2 Damage Model 2689.2.1 Damage Variable 2689.2.2 Free-energy Potential 2699.2.3 Damage Evolution Equations 2709.3 Healing Model 2729.4 Damage and Plasticity Identification 2749.5 Healing Identification 2779.6 Damage and Healing Hardening 2799.7 Verification 280

Index 285

Preface

Scientists have altered the properties ofmaterials such as metals, alloys, polymers,and so on, to suit the ever changing needs of our society. As we entered intothe twenty-first century, search of advanced materials with crack avoidance andlong-term durability is on high priority. The challenge for material scientistsis therefore to develop new technologies that can produce novel materials withincreased safety, extended lifetime and no aftercare or a very less amount ofrepairing costs. To stimulate this interdisciplinary research in materials technology,the idea of compiling a book came to my mind in 2005. When I contacted one ofthe pioneer scientists in this field he remarked that it is too early to write a bookon such a topic. His opinion was right because the field of material science andtechnology is rapidly advancing and it would be worth to wait few more years toinclude the latest updates. Thus this book is complied when the field ofself-healingmaterials research is not matured enough as it is in its childhood.

The title Self-healing Materials itselfdescribes the context ofthis book. It intendsto provide its readers an upto date introduction of the field of self-healing ma­terials (broadly divided into four classes-metals, polymers, ceramics/concretes,and coatings) with the emphasis on synthesis, structure, property, and possibleapplications. Though this book is mainly devoted to the scientists and engineers inindustry and academia as its principle audience, it can also be recommended forgraduate courses.

This book with its nine chapters written by international experts gives a widecoverage ofmany rapidly advancing fields ofmaterial science and engineering. Theintroductory chapter addresses the definition, broad spectrum of strategies, andapplication potentials of self-healing materials. Chapter 2 summarizes the recentadvances in crack healing of polymers and polymer composites. Self-healing inmost common polymeric structures occurs through chemical reactions. However,in the case of ionic polymers or ionomers healing follows a different mechanism.This is the subject of Chapter 3. Corrosion causes severe damages to metals.Encapsulated corrosion inhibitors can be incorporated into coatings to provideself-healing capabilities in corrosion prevention of metallic substrates. This is dealtin Chapter 4. Ceramics are emerging as key materials for structural applications.Chapter 5 describes the self-healing capability ofceramic materials. Concrete is the

Self-healing Materials: Fundamentals, Design Strategies, and Applications. Edited by Swapan Kumar GhoshCopyright © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-31829-2

XII I Preface

most widely used man made materials for structural applications. The possibilityof introducing self-healing function in cements is the key subject of Chapter 6.Self-healing in metals is dealt in Chapter 7 while its subsequent Chapter 8 providesan insight of self-healing phenomenon in metallic alloys. The last chapter of thisbook describes the developments of a model to predict the effects of distributeddamages and its subsequent self-healing processes in fiber reinforced polymercomposites.

I hope the above mentioned chapters will deliver the readers useful informationon self-healing material developments. I am grateful to the contributing authorsof this book for their assistance to make this project a success. I would also liketo thank the whole Wiley-VCH team involved in this project. Though, last but notleast, I would like to dedicate this book to my wife Anjana and son Subhojit fortheir constant support and encouragement in this venture.

Swapan Kumar GhoshSeptember 2008

iXIII

List ofContributors

Kotoji AndoYokohama National UniversityDepartment of Energy & Safety Engineering79-5, Tokiwadai, Hodogaya-kuYokohama 240-8501Japan

EverJ. BarberoWest Virginia UniversityMechanical and Aerospace EngineeringMorgantown, WV 26506-6106USA

KevinJ. FordWest Virginia UniversityMechanical and Aerospace EngineeringMorgantown, WV 26506-6106USA

ChristopherJosephCardiff School of EngineeringQueen's BuildingsThe ParadeNewport RoadCardiff CF24 3AAUnited Kingdom

StephenJames Kalista,Jr.Washington and Lee UniversityDepartment of Physics and Engineering204 West Washington StreetLexington, VA 24450USA

swapan Kumar GhoshProCoat India Private LimitedKalayaninagar, Pune-411 014India

Michele V. ManuelUniversity of FloridaDepartment of Materials Science andEngineering152 Rhines HallP.O. Box 116400Gainesville, FL 32611-6400USA

Joan A. MayugoEscola Politecnica SuperiorUniversity de GironaCampus Montilvi, 17071 GironaSpain

Wataru NakaoYokohama National UniversityInterdisciplinary Research Center79-5, Tokiwadai, Hodogaya-ku,Yokohama, 240-8501,Japan

Min Zhi RongMaterials Science InstituteZhongshan University135# Xin-Gang-Xi Rd.Guangzhou 510275P. R. China

Self-healing Materials: Fundamentals, Design Strategies, and Applications. Edited by Swapan Kumar GhoshCopyright © 2009 WILEY·VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 978-3-527-31829-2

XIV I List ofContributors

Erik SchlangenDelft University ofTechnologyDepartment ofCivil Engineering andGeosciencesP.o. Box 50482600 GA DelftThe Netherlands

Norio ShinyaInnovative Materials Engineering Laboratory,Sengen Site,National Institute for Materials Science1-2-1, Sengen,Tsukuba, Ibaraki 305-0047Japan

Koji TakahashiYokohama National UniversityDivision of Materials Science andEngineering79-5, Tokiwadai, Hodogaya-kuYokohama, 240-8501Japan

Tao YinMaterials Science InstituteZhongshan University135# Xin-Gang-Xi Rd.Guangzhou 510275P. R. China

Ming Qiu ZhangMaterials Science InstituteZhongshan University135# Xin-Gang-Xi Rd.Guangzhou 510275P. R. China

Mikhail ZheludkevichDepartment of Ceramics and GlassEngineering, CICECO, University ofAveiro,Campus Universitario de Santiago,3810-193AveiroPortugal

9

Modeling Self..healing of Fiber-reinforced Polymer-matrixComposites with Distributed Damage

Ever). Barbero, Kevin). Ford andJoan A. Mayugo

9.1Introduction

Composite materials are formed by the combination of two or more distinct ma­terials to form a new material with enhanced properties [1]. Recently, self-healingpolymers and composites have been proposed. One system in particular in­corporates the use of ruthenium catalyst and urea-formaldehyde microcapsulesfilled with dicyclopentadiene (DCPD) [2]. Barbero and Ford [3] accomplishedself-healing of glass fiber-reinforced epoxy laminates by intralaminar dispersionof healing agent and catalyst. The healing agent, DCPD, is encapsulated andthen dispersed in the epoxy resin during hand layup. The catalyst is also encap­sulated and dispersed in a similar manner. Vacuum bagging technique is usedto consolidate the samples that are cured at room temperature. Then, experi­ments are performed to reveal damage, plasticity, and healing of laminates undercyclic load.

Barbero et al. [4] developed a continuous damage and healing mechanics (CDHM)model to predict the effects of damage and subsequent self-healing as a functionof load history. In Ref [4], damage and healing are represented in separatethermodynamic spaces. The damage portion of the model has been extensivelyidentified and verified with data available in the literature [5, Chapter 8]. Theself-healing portion of the previous model could not be identified or verifiedbecause of lack of experimental data for laminates undergoing distributed dam­age (e.g. microcracking). Until recently, data existed only for fracture toughnessrecovery due to healing of macrocracks [2, 6-8]. Then, Barbero and Ford [3]conducted a comprehensive experimental study to identifY the healing portionof the model. Their experimental evidence suggests that simplification of theearlier model is possible, thus motivating the model presented in this chapter.Specifically, a damage/healing model where both effects are described in a sin­gle thermodynamic space is presented, thus simplifying considerably the earliermodel.

2681 9 Modeling Self-healing ofFiber-reinforced Polymer-matrix Composites with Distributed Damage

The continuum damage-healing mechanics model proposed herein consists of acontinuum damage mechanics model extended to account for healing effects. Themodel must be identified with experimental data from unidirectional or cross-plylaminates. Therefore, a methodology for identification of the damage and healingparameters of the model is described. Once identified, the model is capable ofpredicting damage and healing evolution of other laminate stacking sequence(LSS). Prediction of damage, healing, damage hardening, and hardening recoveryupon healing are accomplished.

The model presented herein is likely to work, with modifications, for otherhealing processes such as geological rock densification [9], self-healing of concrete[10, 11], self-healing ofceramic materials [12, 13], bone remodeling, wounded skinregeneration [14-16], and compaction ofcrushed rock salt [9].

9.2Damage Model

In this chapter, damage is represented by its effects on residual stiffness andstrength. No attempt is made to identify actual microstructural modes ofdamage.However, various components of the model, namely, the damage variable adoptedand several simplifying assumptions alje rooted in experimental evidence. Themodel is composed ofthree main ingredients, the damage variable, the free-energypotential, and the damage evolution equations.

9.2.1Damage Variable

Damage represents distributed, irreversible phenomena that cause stiffness andstrength reductions. The choice of damage variable has a direct impact on thenumber of material parameters required to describe the phenomena and theaccuracy of the model predictions. In this work, damage is represented bya second-order tensor D. For convenience, the integrity tensor is defined asQ = ,Jf=l), where I is the second-order identity tensor.

Experimental evidence suggests that damage in the form of microcracks, delam­ination, fiber break, and so on, occur in planes parallel to the principal materialdirections. Therefore, it is assumed that the principal directions of damage nl, nz,

n3 coincide with the material orientations Xl, Xz, X3. In this case, the damage tensorcan be described by its three eigenvalues dl , dz, d3 • Such simplification allows us tothink ofthe eigenvalues as a representation offictitious, equivalent system ofcracksthat represent damage in the longitudinal, transverse, and thickness directions,respectively.

The model is developed taking two configurations into account: damaged andeffective. In the damaged (actual) configuration, the material is subjected tonominal stress and undergoes damage D, which results in reduced stiffness C(D).The effective configuration is a fictitious configuration where an increased effective

9.2 Damage Model 1269stress (J acts upon a fictitious material having undamaged (virgin) elastic stiffnessC. According to the energy equivalence hypothesis ([5, Chapter 8]), the elasticenergy in the actual and effective configurations is identical.

The transformation of stress and strain between effective and damaged configu­rations is accomplished by

(9.1)

(9.2)

where an over-bar indicates that the quantity is evaluated in the effective config­uration, the superscript e denotes quantities in the elastic domain, and M is thedamage e.ffect tensor defined as

1Mijkl = 2(QikQjl + QilQjk)

The stress-strain relationship in the effective configuration is simply that of alinearly elastic material with virgin properties, given by

(9.3)

The constitutive equation in the damaged configuration is obtained by substitut­ing Equations 9.3 into Equations 9.1,

(Yij = MijkWkl = MijkICklrs£~s'

(Yij = MijkICklrsMrstus~u'

(Yij = CijklSkl'

e M-1- e M-1-S-sij = ijklsk/ = ijkl klrs(Y rs,

e M-1-S M-1S" = "kl klrs "kl(Ytu,l:J l:J l:J

sij = SijkWkl

(9.4)

where by virtue of M being symmetric and using the energy equivalence hypothesis,the constitutive tensors C and S in the damaged domain are symmetric tensorsgiven by

C M C M 5 M -l-S M-lijkl = ijkl klrs rstu; ijkl = ijkl klrs rstu

9.2.2Free-energy Potential

(9.5)

(9.6)

The constitutive equations are derived from thermodynamic principles. TheHelmholtz free energy includes the elastic energy and additional terms to representthe evolution of the internal parameters as follows:

t = ~(" ,I" D) - et [ctexp (:t)]-S[1, exp (;,) ]

where €, €p are the elastic and plastic strain tensors, respectively; p, 1> are thehardening variables, and ct, c1, d{, 0., are material parameters used to adjust themodel to experimental data.

(9.8)

(9.10)

(9.9)

(9.7)

270 I 9 Modeling Selfhealing ofFiber-reinforced Polymer-matrix Composites with Distributed Damage

The following thermodynamic state laws can be obtained by satisfying theClausius-Duhem inequality [17], thus> assuring nonnegative dissipation

(Jij = aa~ = Cijkl(Skl - sfl) = CijklSklsij

a1fr 1 p aCklpq p 1 e aCklpq eYij = - aDij = -"2(Skl - ski) aDij (Spq - Spq) = -"2ck1 aDij cpq

y(8) = - ~~ ~ c; [exp ( :;) - 1]RW) = - ~~ ~ ef [exp ( ~) - 1]

where (J', Y, y, and R are the thermodynamic forces associated with the internalstate variables s, D, p, and o.

The thermodynamic forces can be written explicitly in terms ofstress [5, Appendix2] as

9.2.3Damage Evolution Equations

The evolution ofthe internal variables is defined as follows. First, damage initiationis controlled by a damage function, such as

(9.12)

where Yo is the damage threshold and defines the hardening function. For planestress, and taking into account the symmetry ofthe thermodynamic force tensor Y,the damage surface is given by Equation 9.12 with parameters AI, A2, BJ, B2,]I,]2.

The thermodynamic force tensor is assumed to be separable into a componenty N arising from normal strains and a component yS = Y - yN arising from shearstrains

(9.13)

9.2 Damage Model 1271with m = 1, 2, 3. The values of Em are the values of the three normal strains in theprincipal material directions. The three values Am represent the relation betweendamage thresholds in uniaxial compression and uniaxial extension in the principalmaterial directions. The coefficients in the diagonal, fourth order, positive definitetensors A, B, J are calculated from available experimental data for unidirectionalor cross-ply laminates as explained in Ref. [5, Chapter 9], while the parameters aredetermined by using experimental in-plane shear stress-strain data.

On the other hand, the plastic strain evolution is modeled by classical plasticityformulation [5, Chapter 9] and an associate flow rule. For the yield surface, athree-dimensional Tsai-Wu criterion shape is chosen due to its ability to representdifferent behavior among the different load paths in stress space. Plastic strainsand damage effects are coupled by formulating the plasticity model in effectivestress space. Therefore, the yield surface is a function ofthe thermodynamic forces0', R in the effective configuration as follows:

(9.14)

where i = 1, 2, 6, Ro is the yield stress, and R is defined by the hardening law.The coefficients Ii, I ij are obtained from the strength properties of unidirectionalor cross-ply laminates in terms of the lamina strength values as follows:

(9.15)

The parameters Fit, Fie, and Fi are the effective strength values. That is, thestrength values in effective configuration. They are defined as

- F2tF2t =-;Q 2t

- F2eF2e =-;

Q 2e

- G13Fs = Fs-'

G* '13

(9.16)

where the parameters Fit and Fie (with i = 1, 2, 3), and Fi (with i = 4, 5, 6) are thestrength values in tension, compression, in-plane shear, and out-of-plane shear fora composite lamina. Gt and Gi (with i = 12, 13) are the damaged shear modulusand the undamaged shear modulus, respectively. These values are tabulated in theliterature or they can be easily obtained following standardized test methods [3, 18].

272/ 9 Modeling Self-healing ofFiher-reinforced Polymer-matrix Composites with Distrihuted Damage

The evolution of internal variables is defined by flow rules

;;~. = 1P agP• . . P agP

Col) A a-' P =).. -O'ij aR

. . d agd . . d agdD.. -)..-· 0=)..- (9.17)

I) - aYij' ay

in terms of plastic strain and damage multipliers ).,P, ).,d. These are found by usinga return mapping algorithm [5, Chapter 9].

9.3Healing Model

Healing represents the extent of repair of distributed damage. Similar to damage,healing is represented by second-order tensor H. Since healing can only heal exist­ing damage, healing is represented by a diagonal tensor with principal directionsaligned with those ofthe damage tensor D. The principal values hI, hz, h3 representthe area recovery normal to the principal directions, which are aligned with thematerial directions Xl, XZ, X3 in the material coordinate system. As a consequenceof the particular definition of damage and healing used herein, the healing modelpresented uses the same th~rmodynamicspace for damage and healing. That is,both damage and healing are defined in the space of thermodynamic damageforces Y.

In Refs [7, 19], the crack-healing efficiency is defined as the percentage recoveryof fracture toughness measured by tapered double cantilever beam (TDCB). Inthis chapter, it is postulated that the healing tensor is proportional to the damagetensor

(9.18)

The proportionality constant is the efficiency ofthe healing system. Furthermore,healing represents recovery of damage. Therefore, the principal values of thehealed-damage tensor are given by

(9.19)

Experimentally, it is possible to measure the following (Figure 9.1):• the virgin moduli Gi from the initial slope of the first cycle of

loading;G the damaged moduli Gi

d from the unloading portion of thefirst cycle; and

e the healed moduli G? from the loading portion of the secondcycle (after healing).

(9.20)

9.3 Healing Model 1273

(j

Ii,Ii -I~ G12IiIiI iI iI iI"'!

IIIIIII

Fig. 9.1 Definition of measurable, relevant variables.

From these data, the efficiency is defined here as

G~ - G~I I

rJi = Gi - G~I

For composites reinforced by strong fibers, the self-healing system is incapableof healing fiber damage, which results in "'1 = o. Once the self-healing polymer isreleased, it travels by capillary action and penetrates all the microcracks regardlessof orientation, and thus "'2 = "'3 = .".

Once the material damages, higher levels of thermodynamic damage force Yarerequired to produce more damage. Higher thermodynamic damage force requireshigher stress and correspondingly higher applied strain. This process, which iscalled damage hardening, is represented in the model by a state variable, thehardening parameter 8, which is a monotonically increasing function that providesa threshold below which no further damage can occur. A secondary effect ofhealing is to reduce the hardening threshold. That is, after healing, further damagecan occur below the previous damage hardening threshold because some of thedamage has been repaired. The model proposed herein captures this behavior bythe following reduction in the damage hardening parameter

(9.21)

When the material is completely repaired, one has." = 1, 8 = 0, y = 0 in thehardening law and the threshold for damage returns to the value for the virginmaterial. Thus, the material will begin to damage at the same threshold load asthat of the virgin material. When the healing agent is exhausted, ." = 0 and.thematerial is just a damaging material with 8h = 8; Y i- 0 in the damage hardening

2741 9 Modeling Self-healing ofFiber-reinforced Polymer-matrix Composites with Distributed Damage

law, while the threshold for damage continues to increase, thus representing thenormal hardening-behavior ofa purely damaging material.

9.4Damage and Plasticity Identification

For damage without healing, the damage parameters are AI, A2, B1, B2, ] I, ]2, Yo,ct, c~. Plasticity entails additional parameters as follows:!i,!ij, Ro, cr, q. All theseparameters depend on the following material properties, which are experimentallydetermined using standard testing methods:

• stiffness values (EI, E2 , en)• strength values (Flt' FIe, F2e , F2t, F6 )

• critical damage values (dlt , dIe, d2t)• damaged shear moduli at imminent failure (et2)• in-plane shear plastic threshold (FtP

)

• in-plane shear damage threshold (FtD )

• plastic strain (yt) as a function oftotal applied strain (Y6)

Cyclic shear stress-strain tests are used to obtain the nonlinear damagingbehavior (J6(Y6), as shown in Figure 9.2. The loading modulus is measured withina range of strain specified by the standard. The unloading modulus is measuredover the entire unloading portion ofthe data.

10 +--i'-l--t---y--t--t---y---j'F--/------"f------------7<----. --------1

60..,.-------------------------------,

co40+-------F---FJf--7--+cF------F----f-----I-----------I--_Ia..~(J)(J)

~ 30 +----+-:.H---;F--Jr+---F--1~_U'--------JtL--F---___+_---------_F_--_It).....euQ).cC/) 20 +-~'_+-----I--fi'----F----H----F_I__+_J'-----/--------j'------------------___/_----_____I

50 +--------------------:71""'..-----:7""''----------+-1

7653 4Shear strain (%)

2

O-{L--j~~~~.ar-~....._I'~-.-r--r-l---.-_....,t!~:;-,._,_-.-+_r_,__._,____+__,___.__,___,__Jo

Fig. 9.2 Shear stress-strain behavior of unidirectional, neatspecimen (no self-healing system). Loss of stiffness andaccumulation of plastic strain are evident [3].

9.4 Damage and Plasticity Identification 1275

32.521.510.5O+-....-,.--j=ko"==1=='---.--.....---.---+----'-----,---,----.---l--.---.---..----.,........+--,----.--.----.--+-....-...--.--,----J

o

- 4';!2.~

c'03....

3+--en0

:;::;eneela. 2

1

6.-------------------------------.

5+--------------------------~=!..---l

Maximum applied strain (%)

Fig. 9.3 Plastic strain versus applied strain for unidirectional,neat specimen (no self-healing system). Threshold plasticstrain (Le. yield strain) is evident [3].

Unrecoverable (plastic) strain can be observed upon unloading, but only after athreshold value of stress (Le. the yield strength) or strain (i.e. the yield strain) isreached during loading (Figure 9.3).

Even though plastic strain is accumulated, initially the unloading modulus re­mains unchanged and equal to the loading modulus. For the unloading modulus tochange, that is, to decrease below the value of the initial loading modulus, damagemust appear. Note that the word "unloading" is added for emphasis and becausethe reduction in modulus is first detected during unloading of the specimen. But,of course, the modulus reduction is permanent. A reduction in the unloadingmodulus with respect to the initial loading modulus can be observed only after athreshold value ofstress (Le. the damage threshold stress) or strain (i.e. the damagethreshold strain) is reached during loading (Figure 9.4).

The threshold stress F6EP for appearance ofunrecoverable. (plastic) strain (Le. theyield strength) and the threshold stress F6ED for appearance ofirreversible damageare read from the loading portion of the (}6(Y6) curve (Figure 9.2) with the aid ofFigures 9.3 and 9.4 (see also [5, chapter 9]). The unrecoverable (plastic) strain rt asa function of the applied total strain Y6 is read for each cycle after full unloadingand reported in Figure 9.3. The slope ofthe unloading curves provides the damagedelastic modulus Cf2 as a function of total applied strain Y6 (Figure 9.4).

Existence of damage and a damage threshold are demonstrated by the fact thatmeasured unloading modulus is less than the loading modulus after the damagethreshold has been reached (Figure 9.4). No loss of stiffness occurs when the

2761 9 Modeling Self-healing ofFiber-reinforced Polymer-matrix Composites with Distributed Damage

5D Unloading

Unloading trend4.5 • Cycle 1 loading- • No damage

4

coa..(!)--- 3.5 DamageC/)::J threshold = 0.95::J"'C0E 3....coQ)..cen

2.5

32.521.50.5

2+--,...-,.--,,.........,--+--r---r-___r_-.---I--r--,...-~.____I____r--r---r-___r__+__,___.__.___.,_j___,___r____r__r__I

oMax applied strain (%)

Fig. 9.4 Shear modulus versus applied strain of unidirec­tional, neat specimen (no self-healing system). Thresholddamage strain is evident [3].

applied strain is less than the threshold. After the threshold is reached, the lossof modulus is proportional to the applied strain. Since careful visual inspectionafter each loading cycle does not reveal appearance of any macrocrack, the loss ofmodulus is attributed and modeled as distributed damage.

Also noticeable in Figure 9.3 is the accumulation ofunrecoverable (plastic) strain.Although the physical, microstructural, and morphological mechanisms leading toplasticity in polymers are different than those leading to plasticity in metals, from aphenomenological and modeling point ofview, unrecoverable deformations can bemodeled with plasticity theory as long as the plastic strains are not associated to areduction in the unloading modulus. The reduction in unloading modulus, whichoccurs independent ofthe plastic strain, can be accounted for by continuum damagemechanics. Each of these two phenomena has different thresholds for initiationand evolve with different rates. They are, however, coupled by the redistributionof stress that both phenomena induce. In the model, this is taken into accountby formulating the plasticity model in terms of effective stress computed by thedamage model [4].

Shear tests reveal marked nonlinearity (Figure 9.2) reaching almost total lossof tangent stiffness prior to failure, which occurs at large values of shear strain.Unloading secant stiffness reveals marked loss of stiffness due to damage, whichworsens during cyclic reloading (Figure 9.4). Also, unloading reveals significantplastic strains accumulating during cyclic reloading (Figure 9.3).

9.5 Healing Identification 1277

The standard test method ASTM-D-3039 is used to determine EI , E2 , lJ12, Fit, F2t•

The standard test method SACMA-SRM-IR-94 is used to determine FIc , F2c . Thestandard test method ASTM-D-5379 is used to determine C12 , F6 . The configurationof ASTM-D-5379 is used to determine C12 as a function of damage, healing, andnumber of cycles. Also yt, FF, F~D are found using the same test configuration.The identification procedure linking the damage and plasticity parameters to themeasured material properties is described in Ref. [5, Chapter 9].

9.5Healing Identification

For healing modeling, all that is required is experimental determination of thehealing efficiency as a function of damage. Under shear loading C12, the amountotdamage dl in the fiber direction is negligible when compared to the amount ofdamage d2 transverse to the fibers. Therefore, the change in the (unloading) shearmodulus due to both damage and healing is given by

d -C12 = G 12(l - d2 + h2)

and the healing efficiency can be calculated as

(9.22)

C12 - Ct21]2 = d (9.23)

G12 - G 12

Taking into account that the induced damage d2 is a function of the appliedstrain, it is possible to represent the efficiency as a function of damage with apolynomial as follows:

1]2 = 1 + a d2 + b d~ (9.24)

as shown in Figure 9.5.Twenty-two unidirectional samples containing self-healing system were loaded

in shear with one and one-half cycles consisting ofloading, unloading, followed by48 h ofhealing time, and reloading [3]. Each specimen was loaded to a unique valueofmaximum applied shear strain in the range 0.5 -4.0% with roughly equal numberof specimens loaded up to 0.5, 1.0, ... , 4.0% at intervals of 0.5%. A yield strainthreshold value of0.43% was found for the specimens with self-healing system [3].

Recovery was measured for each level ofstrain and from it the healing is calculatedusing Equation 9.23. The tests used to characterize efficiency consist oftwo loadingcycles separated by healing. For the particular situation of just one reloading afterhealing, such as in these tests, efficiency is a function of strain as well as damage.However, for the more general case of multiple loading cycles, the amount ofdamage is a more appropriate independent variable to define the efficiency function.

2781 9 Modeling Self-healing ofFiber-reinforced Polymer-matrix Composites with Distributed Damage

Y= -1.6148x2 + 0.1051x+ 1 _• R2 = 0.3115

•

~••

• ~ •• ~

• '"'

• •

1.6

1.4

i=::-1.2

>.0c 1.0Q)

'(3

~ 0.8Q)

OJ.~(ij 0.6Q)

:r:0.4

0.2

0.00.0 0.1 0.2 0.3 0.4

Damage d

0.5 0.6 0.7

Fig. 9.5 Healing efficiency versus transverse damage.

Damage is a state variable; that is, damage describes univocally the state ofaccu­mulateddamage regardless of the path followed to reach such damage. The totalstrain applied during multiple loading cycles is not a state variable because it canbe achieved by various combinations ofstrain applied on each cycle. Each differentcombination would yield, in general, a different amount ofdamage. Therefore, eventhough applied strain is measured in the experiments, the amount ofinduced dam­age, not strain, is used to define the efficiency function. Shear tests are performed

y= 0.125In(x) + 0.3514 ~R2 =0.3085

--

•-- • --

• • .".....~. ••~ • •./' ------.--

¢>

• -~

--

1.0

0.9

0.8

0.7

Q) 0.6OJctS 0.5EctS0 0.4

0.3

0.2

0.1

0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Maximum applied shear strain (%)

Fig. 9.6 Owing to hardening, increasingly large amounts ofstrain must be applied in order to produce more damage.

9.6 Damage and Healing Hardening 1279at strain levels of 0.25 .. .4%. The healing efficiency from all specimens at eachstrain level (or damage level) are used to fit Equation 9.24, as shown in Figure 9.6.

9.6Damage and Healing Hardening

Damage is represented by a state variable that accounts for the history of damagealong the material principal directions (1, fiber-; 2, transverse-; 3, thickness direc­tion). Once a certain level of damage is present, it takes higher stress (or strain) toproduce additional damage (Figure 9.6). In this case, it is said that the material hard­ens. Damage hardening is represented by a hardening function, which is a functionof the hardening variable o. The effect of the hardening function is to enlarge thedamage surface (Equation 9.12) that limits the stress space where damage does notoccur. Damage hardening is represented in Figure 9.6 by a logarithmic function

d2 = a' In(y) + b' (9.25)

as shown in Figure 9.5. It can be seen that additional strain must be applied inorder to increase the amount ofdamage, thus hardening takes place.

Healing has two effects. First, it reverses some or all of the damage so that thestiffness of the material is recovered. At the same time, it resets the hardeningthreshold to a lower value, so that new damage can occur upon reloading atlower stress than would otherwise be necessary to cause additional damage onan unhealed material. This is merely a computational description of experimentalobservations. Such behavior can be interpreted as follows. The healed materialcan be damaged by reopening of the healed cracks, by creating new cracks, or bya combination thereof. In any case, the hardening function must be reset uponhealing to be able to represent correctly the observed behavior.

In order to update hardening due to healing, first it is necessary to calculate howmuch of the damage can be healed. Since the self-healing system can only healmatrix damage, the damage that can be healed in the fiber direction is zero. Thetotal damage that can be healed is then calculated as the sum of the damage in thetwo directions that can be healed

(9.26)

Next, the ratio ofdamage that can be healed in each direction to the total damageare calculated as

(9.27)

By taking the healing efficiency into account, the amount ofhardening recoveredfrom healing in each direction can be calculated as

(9.28)

280 I 9 Modeling Self-healing ofFiber-reinforced Polymer-matrix Composites with Distributed Damage

Then, the overall hardening recovered from healing is calculated as

(9.29)

The amount of hardening recovered fJ- due to healing depends on the amountof healing that occurs in each direction. If the damage and healing phenomenaare dominant in one direction, then that direction's healing efficiency will controlthe amount of recovery of hardening. Finally, the damage hardening parameter isupdated as

(9.30)

9.7Verification

The damage-healing model was identified using experimental data from a [0/90]symmetric as explained in Sections 9.4-9.6. Material properties are shown inTable 9.1.

ANSYS is compiled with a user subroutine implementing the damage-healingmodel. Finite element analysis is then used to represent the behavior ofthe samplematerials. Model prediction and experimental data from a [0/90] specimen notused in the material characterization study and a quasi-isotropic [0/90/45/ - 45]5laminate are presented here for verification.

Shear tests of a single [0/90]5 specimen was preformed. Quasi-static tests ofthe specimen loaded to 2.25% strain, unloaded, healed, and loaded again areshown in Figure 9.7. It can be seen that the computational model tracks thedamaging Stress-strain behavior very well. Furthermore, the healing efficiency forthis particular specimen was calculated using Equation 9.23 and used in the model.Comparison between experimental data and model prediction for the second(healed) loading of the specimen is shown also in Figure 9.7, where it can be seen

Table 9.1 Material properties of unidirectional composite.

Material Without Standard With Standardproperty self-healing deviation self-healing deviation

El (MPa) 34784 2185.89 30571 4185E2 (MPa) 13469 587.32 8699 829

U12 0.255 0.032 0.251 0.035G12 (MPa) 3043 439.74 2547 207Flt (MPa) 592.3 29.32 397 66Flc (MPa) 459.1 43.66 232 59Flt (MPa) 68.86 9.17 45 10

F2c (MPa) 109.5 9.25 109 9F6 (MPa) 49.87 3.39 38 2

9.7 Verification 1281

3.02.51.0 1.5 2.0

Shear strain (%)

-0- Experimental data+-------D'''--_.........'-------------j first loading

~ Experimental datahealed loading

~ SCDHM predictionfirst loading

.__ SCDHM predictionhealed loading

70

60

«l 50c..6en 40enQ)...+-'en 30...coQ)..cCf) 20

10

00.0 0.5

Fig. 9.7 Comparison betwe.en predicted response and exper­imental data for the first loading (damaging) and secondloading (after healing) of a [0/90]s laminate.

that the computational model tracks reasonably well the damaging stress-strainbehavior after healing.

Shear tests of a quasi-isotropic [0/90/45/ - 45]5 laminate were preformed.Damage-healing tests ofthree specimens loaded to 1.5% strain and four specimensloaded to 2.25% strain were conducted. The loading shear stress-strain data ofeach specimen is then fitted with the following equation:

0'6 = a" + b" exp(-k"Y6) (9.31)

The parameters a", b", kif of all the specimens loaded to the same strain level(say 2.25%) are then averaged. Comparison of model predictions with the first(damaging) loading up to 2.25% strain is shown in Figure 9.8. The damagemodel predicts the actual damaging behavior very well. This is notable becausethe model parameters were adjusted with an entirely different set of samples,which shows significant variability (see Table 9.1 and [3, Table 2]). Comparisonof model predictions with the second loading (after healing) of the same setof four samples is shown in Figure 9.8. Again, the accuracy of the model isremarkable.

In summary, microcapsules were fabricated in the same manner outlined inthe literature. Grubbs' first generation ruthenium catalyst was encapsulated in thesame manner outlined in the literature. Fiber-reinforced laminates were fabricatedwith the self-healing system dispersed within the laminae. Tests were conductedto quantify the damage, plasticity, and healing parameters in the self-healingcomputational model. Additional tests on samples not used in the parameteridentification were performed in order to verify the predictive capabilities of the

2821 9 Modeling Selfhealing of Fiber-reinforced Polymer-matrix Composites with Distributed Damage

3.53.02.51.5 2.0Shear strain (%)

1.0

+-~~~---,-w;r:...~~~~---.P~~~~~---I ... SCDHM predictionfirst loading

-0- Experimental dataF--~~~~~~~~~~~~-1 first loading

-<>- Experimental datahealed loading

..... SCDHM predictionhealed loading

30

25

(ij 2011.

6-(J)(J)

15Q)....-(J)....coQ)

10..cCf)

5

00.0 0.5

Fig. 9.8 Comparison between predicted response and exper­imental data for the first loading (damaging) and secondloading (after healing) of a [0/90/45/ - 45]5 laminate.

proposed model. It is observed that the proposed computational model tracks wellthe loss of stiffness due to damage, damage hardening, healing recovery, healinghardening, and damaging stress-strain behavior after healing.

References

Barbero, E.J. (1999) Introductionto Composite Materials Design. Tay­lor & Francis, Philadelphia, PA.

2 White, S., Sottos, N., Geubelle, P.,Moore, J., Kessler, M., Sriram, S.,Brown, E. and Viswanathan, S.(2001) "Autonomic healing ofpolymer composites". Nature, 409794-97. (Erratum Nature, 415 (6873),817).

3 Barbero, E.J. and Ford, K.J. (2007)"Characterization of self-healingfiber-reinforced polymer-matrixcomposite with distributed dam­age". Journal ofAdvanced Mate­rials (SAMPEj, 39 (4), 20-27

4 Barbero, E.J., Greco, F. andLonetti, P. (2005) "Continuumdamage-healing mechanics withapplication to self-healing com­posites". International Journal ofDamage Mechanics, 14 (1), 51-81.

5 Barbero, E.J. (2007) Finite ElementAnalysis of Composite Materials, Tay­lor & Francis, Boca Raton, FL.

6 White, S., Sottos, N., Guebelle, P.,Moore, J., Kessler, M., Sriram, S.,Brown, E. and Viswanathan, S.(2001) "Autonomic healing of poly­mer composites". Nature, 409 (6822),794-97.

7 Brown, E., Sottos, N. and White, S.(2002) "Fracture testing of aself-healing polymer composite". Ex­perimental Mechanics, 42 (4), 372-79.

8 Kessler, M. and White, S. (2001)"Self-activated healing of delamina­tion damage in woven composites".Composites Part A Applied Scienceand Manufacturing, 32 (5), 683-99.

9 Miao, S., Wang, M.L. and Schreyer,H.L. (1995) "Constitutive modelsfor healing of materials with appli­cation to compaction of crushed

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