Viscoelasticity and Reisdual stress

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V I S C O E L A S T I C C O N S T I T U T I V E M O D E L S FORE V A L U A T I O NOF R E S I D U A L S T R E S S E S INT H E R M O S E T C O M P O S I T E S D U R I N G C U R E

by

N I M A Z O B E I R Y

B . S c . ( C i v i lEngineering), University of Tehran, 1997

M . S c . ( C i v i lEngineering/Structural Engineering), Universi ty of Tehran, 1999

A T H E S I S S U B M I T T E D INP A R T I A L F U L F I L M E N T OFT H E R E Q U I R E M E N T S FOR THED E G R E E OF

D O C T O R OF P H I L O S O P H Y

i n

T H E F A C U L T YOF G R A D U A T E S T U D IE S

( C i v i l Engineering)

The University ofB r i t i shColumbia

July 2006

N i m a Z o b e i r y , 2006


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Abs t r ac tA particularly important aspect in the behaviour of thermoset matrix composite materials during themanufacturing process is the development of mechanical properties of the matr ix and the resulting buildup of stresses. The behaviour of the matrix is generally acknowledged to be viscoelastic, and as bothtemperatureand degreeof cure varywith time, the characterization and representation of the behaviour isboth critical and complex. Different approaches have been suggested for model ing this behaviour. Thecommon approaches that invoke the simple linear elastic cure hardening model have been shown toprovide good predictions but have not been studied for their accuracy and applicability. Moresophisticated representations of viscoelastic behaviour are the Prony series of M a x w e l l elementsimplemented in finite element codes in3Dhereditary integral forms.

In this thesis, different constitutive models are considered and their suitabili ty for representing thebehaviour of composite materials during cure is studied. The presented models provide the user with arangeof options depending on whether costsor accuracy of solutions are of primary concern.

For elastic hardening models, it is shown that the fu l l viscoelastic formulations can be progressivelysimplified, andthat thesesimplif ications are v a l i dfor the typicalcure cycles. It is shownthatin general i fthese models are properly calibrated they are v a l i d and efficient pseudo-viscoelastic models. It is alsonotedthat thesemodels are not always applicable and an efficient viscoelastic model is needed.

For suchcases, viscoelastic behaviour of the polymer isrepresentedusing a differential form approach. Itis shown that this form is equivalent to the more common integral form, but has significant benefits interms of extension to more general descriptions, easeof coding and implementation, and computer runtimes. This formulation is extended to composite materials, using an appropriate micromechanicalapproach, and to 3D behaviour with finite element implementation such that it can be used with anexisting code. Some important features are included, such as time-variabili ty of all material properties,methods for calculating polymer and fibre stresses, and considering thermoelastic effects. Several casestudies arepresented for verification/validationpurposesand to highlight various featuresof the models.

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TableofContents

Table o f Con ten t s

Abstract Tableof Contents "iListofTables . viiListofFigures ixListof Symbols xvi

Acknowledgements xxiChapter1. IntroductionandBackground 1

1.1. Processing andProcessModeling 11.2. Research Objectives andThesis Outline 4

Chapter2. Review of ConstitutiveModels 82.1. Mechanical Behaviouro fThermosetPolymers during Cure 82.2. Elastic Models 92.3. C H I L E Models 92.4. Viscoelastic Models 10

2.4.1. IntegralFormofViscoelasticity 122.4.2. Fractionalmodels 172.4.3. Differential FormofViscoelasticity 17

2.5. Summary 19

Chapter3. Pseudo-ViscoelasticModels 243.1. Review of Two Viscoelastic Models 24

3.1.1. Material1 243.1.2. MaterialII. 26

3.2. Pseudo-Viscoelastic M o d e l Derivation 283.2.1. SimplificationofViscoelasticFormulations 29

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3.2.2. Different FormsofPVE 353.3. Case Studies 37

3.3.1. Material1 373.3.2. MaterialII. 393.3.3. ComparisonofEfficiency 41

3.4. SummaryandDiscussion 42

Chapter4. DifferentialFormofViscoelasticityin ID 564.1. GoverningEquationsin ID 56

4.1.1. DifferentialEquationsforNon-Curing Materials 564.1.2. DifferentialEquationsforCuring Materials 594.1.3. NumericalSolutionofDifferentialEquations 654.1.4. ComparisonofDifferential Form andIntegralForm 66

4.2. Summary 67

Chapter5. Micromechanics 735.1. Fundamental Equations 735.2. Micromechanical M o d el forElastic Materials 745.3. Micromechanical M o d elforViscoelastic Materials 77

5.3.1. CorrespondencePrinciple 785.3.2. Virtual Material Characterization 805.3.3. MaterialProperties 845.3.4. Thermoelasticeffects 855.3.5. Summaryoftheproposedapproach 89

5.4. Numericalexamples 905.4.1. IsothermalCase 905.4.2. DifferentTemperaturesandDegreesofCure 92

Chapter6. DifferentialFormofViscoelasticityin 3D 1036.1. GoverningEquationsin3D 103

6.1.1. ChoiceofMaterialPropertiesandDevelopment of theDifferential Form 1036.1.2. Equationsfor ThermalExpansionandCureShrinkage 112

6.2. Finite Element Formulation 114

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6.2.1. NumericalSolutionofDifferentialEquations 1146.2.2. DevelopmentoftheFEFormulation 1166.2.3. Summaryand Discussion 120

6.3. ReverseMicromechanics 1236.3.1. ProposedMethod 123

6.4. Summary 125

Chapter7. Implementationand Verification 1267.1. Implementation inacode 126

7.1.1. CodeAIgorithmandPseudo Code 1277.1.2. Code Features 128

7.2. Verification Problems 1287.2.7. TransverselyIsotropicCases 1297.2.2. IsotropicCases 1317.2.3. ReverseMicromechanicsExamples 1357.2.4. Thermoelastic behaviour 1377.2.5. Summary 139

Chapter 8. Numerical Applications 1568.1. MaterialProperties 1578.2. SingleElement Cases 160

8.2.1. StandardCureCycles 1608.2.2. OtherCureCycles 165

8.3. More ComplexCases 1688.3.1. StandardCureCycles 1688.3.2. Non-StandardCureCycles 173

8.4. Summary 177

Chapter 9. Conclusions and FutureWork 2119.1. Summary 2119.2. Conclusions 2119.3. Futurework 2129.4. Contributions 214

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References 216

A p p e n d i x A . M a t h e m a t i c a No t e b o o k s 232

Ap pe nd i x B . Pse udo- Code for the D i f f eren t ia l F o r m I m p l e me n t a t i o n 245

A p p e n d i x C . A B A Q U S V is coe la s t i c M o d e l 251

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ListofTables

L i s t o f Tables

Table3.1 Relaxationtimesandweightfactors for3501-6resin fora =0.98 44Table3.2 Relaxationtimesandweightfactors for8551-7resin.. 44Table3.3Comparisonofru ntimesfor integral formV EandP V Efor a cured resin 44Table3.4 Comparisonofruntimesfor integral formV EandP V E for a curing resin in a one-hold cure

cycle 45Table3.5Comparisonof the features of the different models 45Table4.1Comparisonof Run-times for ThermoplasticMatrix(or constantdegreeofcurein athermoset

matrix) 69Table4.2Comparisonof Run-times for ThermosetMatrix(increasingdegreeof cure) 69Table4.3ComparisonofD F and IF models 70Table5.1 Materialproperties of fibre and resin for the numerical example 93Table5.2 Relaxationtimesandweightfactors used in the numerical example 93Table5.3 Relaxationtimesandweightfactors for different material properties in the example 93Table5.4 Values ofthe unrelaxed and relaxed moduli of the composite obtained from the V E

micromechanics in Mathematica 94Table7.1Comparisonof the features of the addedU M A T and the A B A Q U S built-inV Emodel 140Table7.2 Summary of examples to be analyzed 140Table8.1 Mechanical properties offibreand resin used forcasestudies 180Table8.2Thermaland cure shrinkage coefficients obtained for3501-6resin for thecasestudies 180Table8.3 Unrelaxed and relaxed values of different material properties obtained for AS4/3501-6 180

Table8.4 Relaxationtimesand correspondingweightfactors for different material properties ofAS4/3501-6 180

Table8.5 Thermo-chemical properties of AS4/3501-6 181Table8.6 Properties of the tooling materials used in thecasestudies 181Table8.7 Summary ofsingle-elementcasesto be analyzed 181

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ListofTables

Table8.8Resultsofspring-inanglei ndegreesfor anL-shaped partondifferent tool ing, analyzed withthe Differential Form ofViscoelasticityand Pseudo-Viscoelasticity 182

Table8.9Comparisonofruntimes inastandardcurecyclefor anL-shaped composite ondifferent too lingl 82Table8.10Spring-in and runtimesfromtheanalysis onaunidirectional partshown in Figure 8.29 182Table 8.11 Spring- in from theanalysis onthepartshown in Figure 8.29 wi th different lay-ups 182Table 8.12 Spring-inresultsfor aC-shaped composite part underacurecycle withapostcureheat-up183Table 8.13 Spring-inresultsfor aC-shaped composite part underacurecycle with secondary cureof a

partially cured material 183Table 8.14 Results of spring-inforcycles intheform of secondary cureof partially curedparts 183

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ListofFigures

L i s t of F ig ur es

Figure 1.1 Integratedsub-model approach inprocessmodeling (adapted from Johnstonet al., 2001) 7Figure 2.1 A typical temperature/pressurecycleusedtocurethermosetpolymers in an autoclave 21Figure 2.2 Joining of molecules as aresultof curing in athermosetpolymer ; 21Figure 2.3 Polymer changes form as itgoesthroughthe curecycle : 21Figure 2.4 Undeformed and deformed shapesin a typical U-shaped composite partafter cure 22Figure 2.5 Schematic oftime-temperaturesuperposition 22Figure 2.6 A M a x w e l l element 22Figure 2.7 A K e l v i n element 22Figure 2.8 A generalized M a x w e l l element 23Figure 2.9 Different levels of constitutive modeling 23Figure 3.1 Comparison of shift factors for 8551-7 resin using two different equations(with ot=0.99) 46Figure 3.2 Comparison ofreduced timesfor P V E and V E models for 3501-6 resin (witha=0.99) for

different valuesof cool-down rates 46Figure 3.3 Comparison o f moduli for PV E and VE models for 3501-6 resin (witha=0.99)for different

valuesof cool-down rates 47Figure 3.4 Schematic of a typical variation of shift factor as a function o ftime during a hold 47Figure 3.5 Temperatureand degreeofcurefor the curecycleusedfor3501-6resin 48Figure 3.6 Comparison of two definitions of modulus, basedonstoragemodulus and relaxation modulus,

for 3501-6resin 48Figure 3.7 Comparison of two definitions of modulus, basedonstoragemodulus and relaxation modulus,

for 8551-7resin 49

Figure 3.8 Elastic modulus profile based on a variable time Pseudo-Viscoelastic model for the one- curecycle in Figure 3.5 (3501-6 resin) 49

Figure 3.9 Comparison ofstress profiles for variable time Pseudo-Viscoelastic and Viscoelastic modelsfor the curecycle in Figure 3.5 (3501-6resin) 50

Figure 3.10 Temperatureand degreeo fcureprofile for the two-hold cycleusedin the casestudy for3501-6resin 50

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ListofFigures

Figure 3.11 Elastic modulus profiles for different constantfrequency Pseudo-Viscoelastic models for thecure cycle in Figure 3.10 (3501-6 resin) 51

Figure 3.12 Comparison ofstressprofiles from the constantfrequency Pseudo-Viscoelastic models andthe Viscoelastic model for the cure cycle in Figure 3.10 (3501-6 resin) 51

Figure 3.13 Residualstress as a function of frequency for the constant-frequency P V E model, the curecyclein Figure 3.10 (3501-6 resin). A l s o shown is the V E model predict ion 52

Figure 3.14 Predicted residual stress for variable time Pseudo-Viscoelastic and Viscoelastic models fordifferent coolingratesfor a one-hold cycle with T h o i d = 1 8 0C (3501-6 resin) 52

Figure3.15 Predicted residual stressfor variable time Pseudo-Viscoelastic and Viscoelastic models fordifferent coolingrates, m, for a one-hold cycle with T h o i d = 1 6 0 C (3501-6 resin) v .. 53

Figure 3.16 Predicted residual stressforconstanttime Pseudo-Viscoelastic and Viscoelastic models fordifferent coolingrates, m , (and corresponding times) for a one-hold cycle with T h o i d = 1 8 0 C (3501-6 resin) 53

Figure3.17 Predicted residual stressforconstanttime Pseudo-Viscoelast ic and Viscoelastic models fordifferent coolingrates (and corresponding times) for a one-hold cycle with T h o i d = 1 6 0 C (3501-6resin) 54

Figure3.18 Var iable time Pseudo-Viscoelast ic and Viscoelasticstresspredictions for a widerangeofdifferent cure cycles (8551-7 resin) 54

Figure 3.19 Constant frequency Pseudo-Viscoelastic and Viscoelasticstresspredictions for a wide rangeo f different cure cycles (8551-7 resin) 55

Figure 4.1 (a) M a x w e ll element (b) Generalized M a x w e l l model 71Figure 4.2 Cure cycles used to compare IF and DFstresspredictions 71Figure 4.3 Comparison of predicted stresshistories using IF and DFapproaches. Notethat the two

approachesgive identicalresultsfor each caseandthereforecannotbe distinguished 72Figure 5.1 ConcentricCylinder Assembly ( C C A ) model 95Figure 5.2 Schematic illustration of the micromechanical approach 95

Figure5.3 V e r t i c a l shift factor for thermorheologically complex behaviour of 3502 resin 95Figure 5.4 Flowchart of the micromechanical analysis used 97Figure 5.5 Comparison of the value of composite modulus from different methods: micromechanics,

Pronyseries fit to micromechanics, and Prony serieswit h the resin weight factors 98Figure 5.6 Comparison of the value of composite Poisson's ratio from different methods:

micromechanics, Prony seriesfit to micromechanics, and Prony serieswith the resin weight factors98

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ListofFigures

Figure5.7Comparisonof the value of composite modulus from different methods: micromechanics,Pronyseriesfit to micromechanics, and Pronyserieswith the resinweightfactors 99

Figure5.8 Comparisonof the value of composite modulus from different methods: micromechanics,Pronyseriesfit to micromechanics, and Pronyserieswith the resinweightfactors 99

Figure5.9 Comparisonof the value of composite modulus from different methods: micromechanics,Pronyseriesfit to micromechanics, and Pronyserieswith the resinweightfactors 100

Figure5.10Comparison ofthe value of composite Poisson's ratio from different methods:micromechanics, Pronyseriesfit to micromechanics, and Pronyserieswith the resinweightfactorslOO

Figure5.11 Comparisonof the value of composite coefficient ofthermalexpansion different methods:micromechanics and Pronyseriesfit to micromechanics 101

Figure5.12 Comparisonof the value of composite coefficient ofthermalexpansion different methods:micromechanics and Pronyseriesfit to micromechanics 101

Figure5.13 Comparisonof the values of composite modulus for different shift factors 102Figure5.14Comparisonof the values of composite modulus for different shift factors 102Figure7.1 Schematic form of the relationshipbetweenthe D F code and an available commercial code 141Figure7.2Algorithmof the DifferentialFormcode for a given timestep 142Figure7.3 Specifications for Example 1 143Figure7.4 Comparisonof an as predicted by the DifferentialForm U M A Tand analytically for Example

1 144Figure7.5 Comparisonof c2 2 as predicted by the DifferentialForm U M A Tand analytically for Example

1 144Figure7.6 Comparisonof au as predicted by the DifferentialForm U M A Tand analytically for Example

2 145Figure7.7 Comparisonof


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Listof Figures

Figure 7.11 Comparison of theexact cure shrinkageand the simplified linear model, used in Example 5147Figure 7.12 Temperature cycle and the resulting degreeo fcure used in Example 5 148

Figure 7.13 Comparison ofstressesas predicted by the Differential Form U M A T and the previouslyverified IDcode in Example 5 148

Figure 7.14 Geometry of Example 6 149Figure 7.15 Standard mesh used in Example 6 150Figure 7.16 Comparison of radialstressesat different timest=0.1 to 10 as predicted by the Differential

Form U M A T (dots)and analytically (lines) using astandard mesh in Example 6 150Figure 7.17 Refined mesh, usedfor Example 6 151Figure 7.18 Comparison of radial stressesat different timest=0.1 to 10 as predicted by the Differential

Form U M A T (dots) and analytically (lines) for a refined mesh in Example 6 151Figure 7.19 Comparison of matrix stressesas predicted by the Differential Form and analytically for

Example 7 152Figure 7.20 Comparison of matrix stressesas predicted by the Differential Form and analytically for

Example 7 152Figure 7.21 Comparison o f matrix stresses as predicted by the Differential Form and analytically for

Example 8 153Figure 7.22 Comparison of matrix stresses as predicted by the Differential Form and analytically for

Example 8 153Figure 7.23 Comparison ofstrains as predicted by the Differential Form U M A T and analytically for

Example 9 154Figure 7.24 Comparison ofstrainsas predicted by the Differential Form U M A T and analytically for

Example 9 154Figure 7.25 Comparison ofstrains as predicted with and without thermoelastic effects by the Differential

Form U M A T for Example 9 155Figure 7.26 Comparison ofstrains as predicted with and without thermoelastic effects by the Differential

Form U M A T for Example 9 155Figure 8.1 Evolution of the volume changefor 3501-6resin in acurecycle,usedto find thermal and cure

shrinkageproperties 184Figure 8.2 Temperature cycle and the resulting degreeo fcureused in Case 1 to Case 6 184Figure 8.3 Comparison of


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Listof Figures

Figure 8.4 Comparison of


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ListofFigures

Figure 8.26 Distribution of longitudinalstressesfor an L-shaped unidirectional composite on a convexsteelmould from Pseudo-Viscoelasticity 197

Figure 8.27 Distribution oftransversestresses for an L-shaped unidirectional composite on a convex steelmould from Differential Form of Viscoelast icity 197

Figure 8.28 Distribution oftransversestressesfor an L-shaped unidirectional composite on a convex steelmould from Pseudo-Viscoelasticity 198

Figure 8.29 FEmeshof an L-shaped AS4/3501-6parton an aluminium tool usedin the case studies... 198Figure 8.30 Deformed part after tool removal in acurecycle, along with the undeformed shape 199Figure 8.31 Evolutiono f temperatureand residual stressesat the point shown in Figure 8.29 199Figure 8.32 Evolution ofmoments in an L-shaped composite partfor a cross-section passing through the

point shown in Figure 8.29 200Figure 8.33 Evolution of axial forces i n an L-shaped composite partfor a cross-section passing through

the point shown in Figure 8.29 200Figure 8.34 Distribution of finalaxial residual stressesthroughthe thickness for a cross-section passing

throughthe point shown in Figure 8.29 201Figure 8.35 Distribution ofmomentsalong the length of the partshown in Figure 8.29 201Figure 8.36 Evolut ion of composite and matrix stresses in the fibre direction at the point shown in Figure

8.29 202

Figure 8.37 Evolut ion of composite and matrixstressesperpendicular to the fibre direction (in-plane) atthe point shown in Figure 8.29 202Figure 8.38 Distribution ofmomentsalong the length of the part shown in Figure 8.29for two analyses

with and without thermoelasticeffects 203Figure 8.39 FEmeshof a C-shaped AS4/3501-6parton an aluminium toolusedin the casestudies 204Figure 8.40 Cure cycle applied to a C-shaped composite shown in Figure 8.29 204Figure 8.41 Evolut ion oftemperatureand residual stressesat the point for a C-shaped partshown in

Figure 8.29 205Figure 8.42 Distribution ofmomentsalong the web o f the C-shaped partbefore tool removal, as in Figure

8.40 205Figure 8.43 Distribution ofmomentsalong the web of the C-shaped partshown in Figure 8.29 at the end

o fpostcureheat-up 206Figure 8.44 Evolut ion of composite and matrix stressesin the fibre direction for a C-shaped composite at

the point shown in Figure 8.29 underthe curecycle in Figure 8.40 206

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Listof Figures

Figure 8.45 Evolution of composite and matrix stressesperpendicular to the fibre direction for a C-shapedcomposite at the point shown in Figure 8.29 underthe curecycle in Figure 8.40 207

Figure 8.46 Cure cycle applied to a C-shaped composite shown in Figure 8.29 207Figure 8.47 Evolutionoftemperatureand residual stressesat the point for a C-shaped part shown in

Figure 8.29 208Figure 8.48 Distribution ofmomentsalong the web of the C-shaped partbefore tool removal, curecycle

shown in Figure 8.46 208Figure 8.49 Distribution ofmomentsalong the web of the C-shaped partat the end of cycle,curecycle

shown in Figure 8.46 209Figure 8.50 Evolution of composite and matrix stressesin the fibre direction for a C-shaped composite at

the point shown in Figure 8.39 209Figure 8.51 Evolution of composite and matrix stressesperpendicular to the fibre direct ion for a C-shaped

composite at the point shown in Figure 8.39 210Figure 8.52 Different cure cycles in the form ofsecondary cureof partially cured parts 210

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ListofSymbolsParameterused to define the shift factor for 3501-6 resinParameterused to define the shift factor for 3501-6 resinV e r t i c a l shift factorHorizontal Shift factorMaterial stiffnessThermalexpansion coefficientLongitudinalcoefficient of thermal expansionTransverse coefficient of thermal expansionTransverse coefficient of thermal expansionFiber coefficient o fthermal expansionGlassycoefficient of thermal expansionMatrix coefficient o fthermal expansionRubberycoefficient of thermal expansionThermalexpansion coefficient in the radial directionThermalexpansion coefficient in the circumferential directionCreep complianceMaterial stiffness matrixMaterial stiffness matrix for fibres

Materialstiffness matrix for matrix material

Material stiffness matrix, for a viscoelastic materialInitialvalue ofcreepcomplianceYoung's modulusLaplacetransformed modulusLongitudinal You ng' s modulus in an orthotropic mediaTransverse Young's modulus in an orthotropic mediaTransverse Young's modulus in an orthotropic media

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j

ListofSymbols

E Relaxed Young' s modulusEm Young'smodulus of the matrix

E, Young'smodulus of the fibreEr RelaxedYoung' s modulusEu UnrelaxedYou ng' s modulusF L o a d vectorG Shear ModulusG , 2 In-plane shear modulus in an orthotropic media

In-plane shear modulus in an orthotropic mediaG 2 3 Transverse shear modulus in an orthotropic mediaG Storage shear ModulusG" Loss shear ModulusSi Weightfactors, to define bulk modulusI Number ofM a x w e l lelements to define relaxation modulusk Springstiffness in an analogue element, e.g. M a x w e l l elementk Springstiffness rate,for thermoelastic effectsk, Springstiffness of any of the M a x w e l l elements in a modelk" Unrelaxed modulusK B u l k modulusKT G l o b a l stiffness matrixK2 Plane strain bulk modulus in a transversely-isotropic mediaK

gGlassybulk modulusRubberybulk modulus

m Cool-downratein a cure cyclen TimestepnumberN NumberofM a x w e llelements to define relaxation modulusN NumberofM a x w e l l elements for material property pP Anumber, associated with one the material propertiesP Denotingany material property

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List ofSymbols

t Time, current timet' Dummy time (for time integration)tx Ameasureof the variable time, used in the pseudo-viscoelastic modelt2 A parameter,used to evaluate the variable time in the pseudo-viscoelastic modelte Variable time, used in the pseudo-viscoelastic modeltf Time at the onseto fcool-downin a cure cycleT TemperatureT Reference temperatureT Glass transitiontemperatureTg0 Glass transitiontemperatureof the monomerTa> Glass transitiontemperatureof the f u l l y cured polymeru Displacement vectorvf Fibre volume fractionV Volumeof the compositeVj Volumeof the fibresV Volumeof the matrix

m

wj Weight factors, to define relaxation modulusw Weight factors, to define relaxation modulusa Degree of curea0 Reference degreeof cureaf Valueof thedegreeof cure at the onseto fcool-downin a cure cycleP A small number, used to evaluate the variable time in the pseudo-viscoelastic model

Forcevector incrementAF_f Force vector increment, from thermo-chemical strainsAF_a Force vector increment, from internalstressesAs Strain incrementASf Free strain incrementAd Change of angle in a compositepart(spring-in)

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List ofSymbols

At Time incrementAT Temperature incrementA C T Stress increment

StrainStrainrateLaplace transformed strainVolume-averaged strain of the composite

G Deviatoricstrain vectorH Strain, intensor form

K Volumetricstrain vectorVolume-averagedstrain of the fibres

IL Volume-averagedstrain of the matrixStrain,associated withmaterial property pFree strain, associated withmaterial property p

s'c Thermo-chemical strain,o,a Total strainr Modulus,in Laplacespace

Dashpot viscosity in an analogue element, e.g. M a x w e l l elementn, Dashpot viscosity of any of the M a x w el l elementsin a modeli material property in a transversely-isotropic materials

Poisson's ratioVX2 MajorPoisson's ratio in an orthotropic media

"13 MajorPoisson's ratio in an orthotropic mediaMajorPoisson's ratio

(9 Initialangle in a compositepartcr Stressfj Stressratea Laplacetransformed stress

Volume-averagedstress o f the composite

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ListofSymbols

af Volume-averagedstress o f the fibres_ G Deviatoricstressvectorcr. Stressin anyof the M a x w e l l elementsinamodelov Stress, intensor formq_K Volumetricstressvectora Volume-averagedstressof the matrixtn CTapl StressinM a x w e ll element i, associated withmaterial property pT Dummy time (for time integration)r 0 Reference relaxation timeT Equivalent relaxation time,forthermoelastic effectsT CI Relaxat ion times, associated with shearbehaviourr, Relaxat ion times,todefine relaxation modulusrKi Relaxationtimes, associated withbulk behaviourTP Peak relaxation time, usedfor3501-6 resinr . Relaxationtimes, associated withmaterial property pTA Relaxa tion times,todefine relaxation modulusco Frequency ofadynamictest Reduced time,atcurrent time (t)

Reduced time,atdummy time( r )

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Acknowledgements

A c k n o w l e d g e m e n t sI would l i k e to acknowledge several individuals who truly made this work possible. First, Iwould l i k e tothank my supervisors Dr. Reza V a z i r i and Dr. Anoush Poursartip. I really appreciate the guidance andhelp they gave me during the course of my PhD.

Many thanks to several past andpresent U B C Composites Group Members for their friendship and help.Especially, Iwould l i k e to thank Mr. Ahamed Arafath and Mr .A m i r Osooly for our fruitful discussionsand their assistance on many occasions during the course of my degree.

I truly value the friendship of several close friends that I found in Vancouver. I appreciate theirenthusiasm, support, and encouragements during the highs and lows of my degree.

Last but not least, Iwould l i k e to thank my mother, father, and brother who were with me every stepo fthe way and their love and affection was always what gave me the strength to continue. I thank H i v a , whoin the past several months was always by my side and her words of kindness and support were myconstant source ofmotivation.

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Chapter1:Introduction andBackground

Chapter 1. INTRODUCTIONA N D B A C K G R O U N D1.1. PROCESSING A N DPROCESS M O D E L I N GFibre reinforced polymer matrix composite materials are increasingly finding new applications inindustry. The growing popularity ofthese materials has been due to their advantages to other availablematerials, e.g. their high strength to weight ratio, high stiffness to weight ratio, and durabili ty. At thesametime, in usingthesematerials in the industry it has been critical to optimize their manufacturing forthe purpose o f reducing the cost of production, an important and sometimes prohibitive proposition,

which i f done properly can work as anadvantageforthese materials. A l s o , it has been essential to have aproper understanding of the manufacturing process to be able to control the final shapeof the material.For this reason, modeling of the manufacturing process of composite materials or process modeling hasgained popularity as a tool able to meet the above demands, in addition to its ability to predict residualstresses generated in the material dur ing the manufacturing, an important factor in designing compositestructures.

Process model ing has been the topic of many research works in the literature. One of the first modelspresented is the wor k done by Hahn and Pagano (1975).Thiswork issimplified in several aspects, e.g. ituses a linear elastic constitutive model and assumes a stress-free state prior to cool-down. Since thiswork, many studies have been carried out on this topic, which have taken various complexities of theprocess into account and have used different approaches.

Due to the complexity of process modeling, composite materials have generally been modelled using the

'integrated sub-model' approach. Based on this approach, a complex process model is divided intoseveral simpler sub-models, whichcan be studied more or less independently, as shown schematically inFigure 1.1, presented by Johnston (2001) and used in the development o f the composite process model ingcode, C O M P R O . According to the schematic of the sub-model approach, the modeling procedure


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includesseveral modules, each one responsible for one aspect of the material behaviour. The modulethatisof interest i n this work is the stressmodule, i.e. the module responsible for calculation of the residual

stresses and deformations. In most of this thesis we restrict our attention to this module in the processmodel. This approach was first applied to process modeling by Loos and Springer (1983). Since then,many other researchers have used this methodology more comprehensively and for different processes.Some examples for autoclave processing are Bogetti andGillespie(1991, 1992), White and Hahn (1992a,1992b), Kiasat (2000), Johnston et al. (2001), K i m et al. (2002), Zhu et a l. (2003), Wijskamp et al .(2003),Antonucci et al. (2006), and C l i f f o r d et al. (2006) and for other processing methods are K i m and

White(1998) for filamentwinding, Y a n get el. (2004a) for map-mould process, and Douven et al. (1995)forinjection moulding.

In earlier works on process modeling , the researchers modeled only the f i n a l cool-down section of thecuring process. This was due to the fact that (according to A d o l f and M a r t i n , 1996) the majority of thestudies assumed that the only source of residual stress in the material was thermal strains (e.g. G r i f f i n ,1983; Loos and Springer, 1983). For this reason, a stress-free temperature was considered and it was

assumed that stresses generated by cooling down from this temperature. Some examples, using elasticconstitutive models include Flaggs and Crossman (1981), Hahn (1984); Hahn and Pagano (1975), andwithviscoelastic constitutive models are Weitsman (1980) and Harper and Weitsman (1981). The recentworks model the entire cure cycle for the calculation ofstresses and take into account other sources ofresidual stress generation, such as cure shrinkage strains or tool-part interaction,thus making for morecomplex and accurate models. More information on the sources of residual stress can be found inJohnston etal. (2001).

Another aspect o f the models is their dimensionality. The majority of the studies have used one-dimensional models in their analyses (e.g. Loos and Springer, 1983; Prasatya et al., 2001; D j o k i c et al.,2001; A d o l f and M a r t i n , 1996; Lange et a l . , 1995, 1997). Some others have assumed thatthe effect of one


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o f the directions does not significantly change the behaviour of the material and modeled the material by2-D analyses (e.g. Bogetti and G i l l e s p i e , 1991, 1992; Johnston et al., 2001). At the same time, morerecent studies have modeled the material in 3-D (e.g. Zhu et al, 2003; Poon et al., 1998) to be able toanalyze compositeparts of any shape. These models offer abetter representation of the geometry and thestateofstressin the material and therefore are more desirable. It should be noted, however,thatdue to thecomplexity of the constitutive models employed in the available research worksthese approaches havenormallybeen very time-consuming.

A n important consideration in process modeling is the method used to solve the relevant equations overthe domain of interest. Some people have used Laminated Plate Theory (L PT ) based models for stressanalysis. LPT-based models are simpler methods and, as clear from their names, are based on the theoryo f thin plates. This approach can be used to model different lay-ups and material behaviour, but is notable to model complexshapesor boundary conditions. Hahn and Pagano (1975) applied this approach tocalculate the stressesin cool-downonly.Bogetti andGillespie (1992), however, modeled the whole curecycle for this purpose. Some other examples of this approach are Loos and Springer (1983) and White

and Hahn (1992). Finite difference has also been used to analyze fa i r l y complex cases (Bogetti andG i l l e s p i e , 1992). The method o f choice for most studies done recently on composi te materials, however,is the finite elements (FE) method. This method enables modeling of more complexcases and is notlimited to the assumptions made by L P T . Several researchers have used elastic F E analysis to calculatethe cool-downstresses(e.g. G r i f f i n , 1983; Stango and Wang, 1984). However, modeling the whole curecycle has been more limited in the literature. This is perhaps due to the complexity and requirement oflarger computation of FE-based methods. Some examples of this approach include Bogett i (1989),Johnston et al . (2001), and Svanberg and Holmberg (2004a, b).

Different aspectsof residualstresscalculation in process modeling were discussed above and some of thecomplexitiesof the problem were mentioned.Takingsuch complexities of the behaviour into account, we


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Chapter1 :Introduction and Background

need a suitable approach to properly simulate the responseof composite materials during processing. Thisunderlines the importance of the accuracy of the employed models and their ability to take into account

different complexities of the process in a comprehensive fashion.

In addition, process models usually involve significant numerical runs, consistent with the size andcomplexity of the industrially relevant composite parts. This makes efficiency of the model a crucialfactor in controlling the modeling costs.

1.2. R E S E A R C H O B J E C T I V E S AN D THESI S O U T L I N E

The objective o f this thesis is to develop accurate yet more efficient approaches for modeling residualstresses inthermoset polymer matrix composite materials during cure. This is accomplished by studyingthe available modeling options and further expandingthesecapabilities by:

Studying simpler constitutive models available in the literature, as efficient models

Development of a moreaccurateconstitutive modelthat is also efficient

Improving upon other featuresavailable in existing constitutive models

Implementing the constitutive models in a functional formwithFinite Element method

Evaluating different models studied for their benefits, limitations, andcosts

I f theseobjectives are achieved, a variety of constitutive models for composite materials during cure w i l lbe available. This w i l l provide different modeling options in cost vs. accuracy decisions in processmodeling of composites. It would also offer new modeling features not currently available in the

literature.

Basedontheseobjectives, this thesis is organized as follows:

Chapter 2 - Reviewof Constitutive models:


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Chapter1:Introduction and Background

A review of constitutive models available in the literature for polymers and composites is presented.Elastic, instantaneously elastic, and viscoelastic models are discussed.

Chapter 3- Pseudo-Viscoelastic Model:

Instantaneously elastic models are studied as efficient and simple models. The bound of validity andaccuracy ofthesemodels is discussed and the concept of pseudo-viscoelastic model is introduced.

Chapter 4Differential Form of Viscoelasticity in ID:

A viscoelastic approach based onsolving governing differential equations is introduced and the equationsfor polymer materials in one-dimensional form are derived. The effects o ftemperatureanddegreeof cureon material behaviour are considered. A l s o ,a comparisonwithother models in the literature is conducted.

Chapter 5 - Micromechanics:

Micromechanics is used to combine the polymer and fibre mechanical properties in order to obtain thecomposite properties. The micromechanical approach employed in this thesis for viscoelastic materials isdescribed.

Chapter 6 Differential Form of Viscoelasticity in3D:

Having established the constitutive equations i n ID along with a micromechanical model, the equationsfora three-dimensional stateo fstressare derived. The relevant finite element equations are developed.

Chapter 7 -Implementation and Verification:

The procedure to implement the derived formulation in a code and the coding itself are discussed. Some

simple examples to verify the implementation are presented.Chapter 8 - Numerical Applications:


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Chapter1:IntroductionandBackground '

Somecasestudies are presented to show the capability of different constitutive models, theirvalidity,andtheirefficiency. Firstsome simple examples are analyzed to show the behaviour of the models moreclearly.These are then confirmed by looking at more complexcases.

AppendixAMathematica Notebooks:

The Mathematicanotebooks, used to obtain analytical solutions for anumberof examples are presented.

Appendix B -Pseudo Codefor theDifferential Form Implementation:

Thedetailed algorithm for implementation of this constitutive model in a code is presented.

Appendix C - ABAQUS Viscoelastic Model:

The viscoelastic model available in the commercial code ABAQUS is discussed. It is also compared tothe model implemented in this thesis.


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Chapter I: Introduction and Background


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Chapter2:Reviewof Constitutivemodels

Chapter 2. REVIEW OF CONSTITUTIVE MODELSA suitable consti tutive model for the material is an important part of any process model. In this chapter,constitutive models used in the literature in residualstressmodeling for polymers and polymer compositeduringcure are reviewed.A l s o , since some of the details related tothesemodels are needed for the futurechapters, some background information is also presented.

First, the mechanical behaviour o f the polymer during a curing process is discussed. Then, a review ofdifferent models used in the literature to model the polymer and/or composite materials is presented.

2.1. M E C H A N I C A L B E H A V I O U R O F T H E R M O S E T P O L Y M E R S DURING C U R EA s mentioned in the previous chapter, polymers are taken through a cur ing process with elevatedtemperature and pressure, e.g. Figure 2.1 for an autoclave. As a result, the moleculesj o i nand form a 3-Dnetwork, i.e. they cure, as shown schematically in Figure 2.2. Duringprocessing, the material transformsfrom a l i q u i d state at the beginning of the process to a gel type material (gelation) as cure advances andsubsequently to a s o l id state (vitrification) by the end of the cure; Figure 2.3 shows these phases in atypicalcure cycle.

A s a result of changes taking place in the polymer , mechanical properties of polymers vary as a functiono f temperature and the advancement of cure. In addition,there are ample indications in the literature thatpolymers show viscoelastic behaviour, which is even more pronounced at higher temperatures. There arenumerous studies in the literature that show the viscoelastic properties of polymers; examples of someprominent works are Tobolsky (1958), Leaderman (1958), and Ferry (1980).

Taking into consideration the complexities of the materials, a proper constitutive model is needed inresidual stressmodeling of the composite. As discussed in the previous chapter, such a modelneedsto beaccurate in taking into account different aspects of the material behaviour and also efficient, due to thesizeof the problems normally encountered in realworld applications.


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Chapter 2:Reviewof Constitutivemodels

Different models in the literature have been used to describe the behaviour ofthermoset polymers duringcure. A review of the common models available in the literature ispresentedbelow.

2.2. E L A S T I C M O D E L SIn earlier process models, some researchers used purely elastic models to describe the behaviour of thematerial (e.g. Harper and Weitsman, 1981; Loos and Springer, 1983; Stango and Wang, 1984; Nel son andCairns, 1989). Alt hough a purely geometric approach, the work by Nelson and Cairns is noteworthy here.They showedthat when a curved compositepart undergoesa temperaturechange it would deform due tothe difference between through-thickness and in-plane properties. This phenomenon, shown in Figure 2.4,iscalled "spring-in".

For this simple geometry, they showed thatthe spring-in can be calculated as follows:

A6=G(CTEe - CTER ) AT1 + CTERAT (2.1)

where A6 is the change in the included angle, 9 is the original angle, CTEg and CTER are the laminate

thermal expansion coeff icients in the circumferential and radial directions, and AT is the temperaturechange. By assuming astress-free temperature in a cure cycle, the residual deformations are calculated bycoolingdown from this temperatureand takingadvantage o fthe above equation.

Elasticmodels offer good insight in general, but are not sophisticated enough tocapturethe complexity ofthe problem or give quantitatively good results in realcases.

2.3. C H I L E M O D E L SSome researchers have used C H I L E models (this term was coined by Johnston et al., 1996, to denoteCure Hardening Instantaneously Linear Elas tic) to analyse the polymer as it goes through the curingprocess (e.g. Bogetti and G i l l e s p i e , 1992; Lange et al. , 1995; Johnston et al., 2001; Fernlund, 2002a,


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200b, 2003; Svanberg and Holmberg, 2004a, b; Antonucci et al, 2006). These models are essentiallyelastic, in which the modulus of elasticity changes as a function oftemperature and degree of cure. Inother words:

In which, a is the stress, E' is the instantaneous elastic modulus, s is the strain, t is the current time, xisthe time integration variable, T is the temperature, and a is the degreeof cure.

C H I L E models have been popular in the literature due to the ease of their characterization, and finiteelement implementation.

A s mentioned, polymers generally exhibit viscoelastic (V E) behaviour. Considering the behaviour of thematerial it is expected that only using a V E formulation for polymers provide accurate prediction fordeformations and stresses. However, the accuracy of the C H I L E model has not been comprehensivelyinvestigated before nor have there been enough experimental studies done in the literature to verify theresults of either of the models. In general, the C H I L E constitutive models for thermoset polymers havebeen shown to provide good predictions in analysing the behaviour of composite materials (Fernlund etal . , 2002a, 2002b, 2003). Nevertheless, the fundamental validity and applicabili ty ofthese models havenot been proven. A n investigation of this issue, by Zobeiry et al. (2006), w i l lbe presented in detail in this

thesis.

2.4. VISCOELASTIC MODELSIt is known that polymers in general show viscoelastic behaviour (e.g. Ferry, 1980). This behaviour isespecially pronounced for partially cured polymers at high temperatures in a cure cycle. For this reason,

Aa = E'(T,a)Ae (2.2)O r in integral form:

(2.3)

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Chapter2 :Reviewof Constitutivemodels

the trend of the literature for residual stress model ing of polymers and polymer composites has beentowards employing V E constitutive equations (e.g. Douven et al., 1995; K i m and White, 1996; A d o l f andM a r t i n , 1996; Poon et al ., 1998; Wiersma et al., 1998; Prasatya et al., 2001;O'Brien et al. , 2001; Zhu, eta l . , 2003).

T he modeling of V E materials with Finite Element Method (FEM)spans around four decades. Theresearch on this area has not been limited to only polymers or polymer composites. For a review ofmodelingV E materials using F E M ,the readercan refer to Zocher et a l . (1997).

Early works on composite materials, such as Morland and Lee (1960), established the fundamentalequations for composite V E behaviour with temperature change. Most V E works on composites in thepastfocused on the behaviour of the materialwith or withoutchangeoftemperature and sometimes withenvironmental effects. Some examples o f the earlier works are Schapery (1974), Flaggs and Crossman(1981), Nakamura et a l. (1988), and L i n and Hwang (1989) and some recentstudies include Chen et al.(2000),Y i et al. (2002), and Ernst et al . (2003). However , it was not until recentlythatprocess modelingo f composites considered V E behaviour (taking the effects of cure into account as well), perhaps due tothe complexity of the analysis. The analyses of composites during cure startedwithsuch works as M a r t i nand A d o l f(1990) and White and Hahn (1992a, 1992b). Since then, other researchershave performed V Emodeling of composites during processing. Some important references are Douven et al. (1995), A d o l fand M a r t i n (1996), K o k a n (1997), K i m and White (1997, 1998), Lange et al . (1997), White and K i m(1998), Poon et a l. (1998), Wiersma et al . (1998), Park (2000), Prasatya et al. (2001), O'Brien et al.(2001), D j o k i c et al. (2001), K i m et al. (2002), Zhu, et al. (2003), Wijskamp et al . (2003), Laht inen(2003), Svanberg and Holmberg (2004a, b), Y a n get a l .(2004a), and C l i f f o r d et al . (2006).

Most of these studies on composite materials have assumed a linear V E behaviour. In some works,however, the behaviour of materials has been considered to be nonlinear (e.g. Qiaoet al., 1994; Chen eta l . , 2001; Y i et al., 2002) andthere is experimental evidence to support this (Schapery, 1969;Qiaoet al.,


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1994; X i a oet al., 1994; X i aet al., 1998). This assumption results in a complex constitutive model,whichcan be very time consuming both in computer implementation and runtime. In addition, most ofthese

models have not modeled the composite behaviour during cure. Alternatively,a linear V E model is easierwhenmodeling the behaviour of composite materials during cure.

In the following sections, a review of different mathematical representations of the V E behaviour used inthe literature w i l l be presented.

2.4.1. IntegralForm ofViscoelasticityIn almost all V E models in the literature, the constitutive equations have been based on the integral formo f viscoelasticity, the so-called Integral Form (IF). This includes almost al l the references given abovethatuse V E models for composite process modeling.

Todescribe the behaviour of a linear V E material in 1-D with IF, the Bolt zmann superposition principle(see e.g. Ferry, 1980) is used. As a result, the constitutive equation for such a material under isothermalconditionscan be written as follows:

Inwhich, E is the relaxation modulus. This equation can be used in homogeneous, non-ageing materials,withno change in temperature. Alternatively,in 3-D we can write:

where,CJkl is the material stiffness tensor.

Under non-isothermal conditions, the best way to incorporate the temperature changes in the constitutiveequation is by using time-temperature superposition (see e.g. Ferry). This principle, used in almost all ofthe recent research works on viscoelastici ty, is based on the ideathatthe effects o ftemperature and time

(2.4)

(2.5)

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on the polymers are similar. In other words, the behaviour of the material, for instance, at increasedtemperature is equivalent to that after a longer time with the original temperature. In effect, a change intemperaturecan be replaced by a shift in time. Considering a relaxation modulus vs. time graph, this ideacan be shown more clear ly by showing the effect of different temperatures, T,on the modulus,which is ashift in time marked by the shift factor ar, as seen in Figure 2.5. In this figure, E" is the unrelaxed

modulus, and E" is the relaxed modulus.

Thisprinciple has been extensively used in the literature, especially for the characterization of materials.The behaviour of polymers can span through 10, 15, or 20decadesof time (Tschoegl, 2002b), which canmake the characterization difficult or impractical. However, by applying increased or decreasedtemperature, the behaviour of material for different time scales can be characterized. Examples of thisapproach are ample, but some include Yee and Takemori (1982), A d o l f and M a r t i n(1990), Douven et al.(1995), Simon et al. (2000), O'Brien et al. (2001), Sane and Knauss (2001b), Ernst et al. (2003),Svanberg and Holmberg (2004a, b), M e l o and Radford (2004), and Hojjati et al. (2004). There are cases,such as Plazek (1991, 1996) and He et al . (2004), where time-temperature superposit ion has not beenused. In this reference, creep and recovery data have been combined to construct the entire creep curve.A l s o , Heymans (2003) has dealt withthis using hierarchical models.

Mostrecent studies on V E residualstresseshave applied this established principle to combine the degreeo f cure with time and temperature (time-temperature-cure superposition). However , the proof ofvalidityo f this approach does not seem to be present in the literature. Some researchers (e.g. A d o l f and M a r t i n ,1996; Lange et al., 1997) have not taken advantage of time-temperature-cure superposition and havemade other assumptions in their analyses.

The constitutive equation in 1-D for a polymer in a cure cycle is:

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cr(f)= \E(t-T,T,a)dr (2.6)

Another important assumption made on the behaviour of polymers in most studies isthat the material isthermorheologically simple (see e.g. Schapery, 1974). This means that the constitutive model presentedabove w i l l become:

Put differently, this assumption in effect means that only a horizontal shift in modulus vs. time graph issufficient to superpose the material properties under different temperatures, which also means that theunrelaxed and relaxedmoduli are constant with temperature and degree and cure, as shown in Figure 2.5.Many studies in the literature have used this assumption on different materials (e.g. K i m and White, 1996,Prasatya et al., 2001) . For a review of literature on the effects o f temperature (and pressure) onthermorheologically simple polymers, one can refer to Tschoegl et al. (2002b). It should also be notedthat there are studies in the literature showing thermorheologically complex behaviour for the resinmaterial. Even though most of the work in this thesis w i l l be done on thermorheologically simple, spmediscussionon the other types of materials w i l lbe presented, e.g. refer to Sections 4.1.2 and 5.3.4.

The conventional and most commonly used method of representing the V E material behaviour is to uselinear spring and dashpot elements. By combining springs and dashpots virtually unlimited forms ofmaterial behaviour can be modelled. These rheological analogue models have been widely used in theliterature to model the curing behaviour o f polymer resins. Examples of simple forms ofthese elementsare given in Figure 2.6, M a x w e llelement, and Figure 2.7, K e l v i n element.

(2.7)

In this equation, and are called the reduced time and are calculated as follows:

(2.8)

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In most available V E models (e.g. Ki m and White, 1996; Prasatya et al., 2001), a number of M a x w e l lelements in parallel are used to model the behaviour of the material, i.e. Figure 2.8. The relaxationmodulus of such a material is of the following form, i.e. Prony series:

E(t)= Er +(EU -Er)JTwfe^ (2.9);=1In which, wt are the weight factors, and r are the relaxation times and N is the number of M a x w e l lelements.

Another form of modulus commonly used in the literature called the power law (e.g. D j o k i c , 2001), is asfollows:

E(t) = Er +(EU-Er)e~^y' (2.10)

2.4.1.1. Integration techniques

L o o k i n g at the IF constitutive model, clearly an integration scheme is needed in order to calculate thestresses from strains. Harper (1981), White and Hahn (1992a, 1992b), and Klasztomy and W i l c z y n s k i(2004) simply used a direct integration between time zero and the current time, / . This method, whenapplied to the hereditary integral shown above, would require a re-calculation of this integral at everypoint in time from time zero. Evi dently , this approach, i f applied to process modeling, woul d be very timeconsuming and impractical for realworld examples.

For this reason, some researchershave developed other, mostly recursive, integration techniques. Intheseapproaches, knowledge o f the values of the parameters(e.g. stresses) in only one or two previous steps isrequired for the calculation of their new values, i.e. at current time, during the analysis. This eliminatesthe need for the history dependent integration. The most common ofthese approaches was developed byTaylor et al. (1970) and has been used in numerous works, such as L i n and Hwang (1989) and Zhu et al.(2001).

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Zak (1967) and White (1968) presented two other integration methods. According to Hammerand andKapania(1999), the integration method developed byTayloret al. is more accurate than thesetwo, due tothe way the integrals are evaluated in the approach. Even though these methods are much more efficientthan the direct integration approaches mentioned above, they are s t i l l very time-consuming. Moreimportantly, looking at these formulations and considering the complexity ofthese approaches it is clearthattheir implementation in a finite element code is very difficult.

Another approach was used by Zocher et al. (1997) and also similarly by K a l i s k e and Rothert (1997),PoonandAhmad (1998), andKiasat(2000). The application of this method, sometimes called the internalvariable approach, in different studies has been reviewed by Johnson (1999). In this approach, someinternal variables, say stresses, are chosen and their values are stored at each time step. These values arethen used to calculate the stressesin the next time step.Accordingto Poon et al. (1998), this approach iseasier to implement in an available finite element code, since itonlyrequires the manipulation of the codeat the integration point l e v e l , unlike other established methods such as by Y i et al. (1995) and Ki m andWhite(1997),which 'requires manipulation of the finite element equations at the global l e v e l ' . It is noted

that not all models in this category have written the equations i n this way; for example, Zocher et al.appear to have developed the formulation such that it requires manipulation at the global l e v e l . Eventhough these techniques are more efficient than the others mentioned above, thereare simplificationsinvolved in most derivations. For example, in these works the reduced time, instead of real time, isdiscretized. This would require assumptions on the form of the shift factor. This w i l l be discussed morelater on.

It is important to notethat these methods have been developed for polymers modeled withProny series.According to Simo and Hughes (1998) itappears that similar method for power law or other forms ofmodulus have not been developed,which wouldrequire the use of historydependentintegration.

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2.4.2. FractionalmodelsA n alternative approach to V E modeling as reported in the literature is to use fractional models (e.g.Alcoutlabi and Martinez-Vega, 1998; Schmidt and G a u l , 2002). Althoughthese models have not beenextensively applied to process modeling their popularity has been on the rise. These models are based onfractional calculus and they are able to model the behaviour of materials using fewer material parameters.Therefore, the characterizat ion o f the model parameters is easier. However, it is acknowledged bySchmidt and G a u lthat such methods, howeverelegant, are not numerically efficient in implementation.

2.4.3. DifferentialForm ofViscoelasticityAnother approach to viscoelastic modeling is to use the governing differential equations as theconstitutive model; this approach w i l lbe called the Differential Form (DF) of V E . This form of V E is infact fundamentally the same as the IF, with a different mathematical representation. A n example oftheseequations for a M a x w e ll element (Figure 2.6) is as follows:

+ (2-11)k nwhere, k and n are the spring stiffness and the dashpotviscosity, respectively.

Itappears that this form of viscoelastic modeling has been used for model ing the f l o wof polymers in thefield o f rheology (see, e.g. B i r d et al., 1987 and Macosko, 1994). However, studying these models isoutside the scope of this thesis, limited to modeling V E materials in a s o l i d mechanics framework, andthey w i l lnot be discussed here.

It is noted that when the number of spring/dashpot elements in the analog model are increased to betterrepresent the real behaviour of material, the order of the differential equation grows and can makeimplementation of this formulation very difficult or even impract ical . For the widely-used model

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consisting of N M a x w e l l elements in parallel , e.g. Figure 2.8, the differential equation takes thefollowing general form:

da d'a d"a , ds , d's , d"sana +a, +... +a, - +...+a = bns + b, +... +h - +... +b (2.12)0 ' dt ' dt' " dt" 0 1 dt ' dt' " dt"

where, all a. and b are constants, with i = l,...,n.

Even thoughthereare examples of the DFpresentin the literature, this form of constitutive modeling hasnot been widely used for calculation ofstresses in V E solids; the rapidly increasing complexi ty of theabove equation (as for instance used by Wiersma et al., 1998 and Mokeyev,2001) is the apparent reason.Zienkiewicz (1968), Bazant (1972, 1974a, 1974b), and Carpenter (1972) first looked at DF for analysis ofthe behaviour of viscoelastic materials. Some more recent references for general use of the DF includeJurkiewiez(1999), Idesman et al. (2000, 2001), Mesquita et al. (2001), and Mesquita andCoda (2003).

Application of this approach for residual stress modeling of composite materials during processing iseven more limited. The studies thatcouldbe found in the literature are thoseby E l l y i n et al. (e.g. X i aandE l l y i n , 1998; Chen et al., 2001) on non-curing composites, K o k a n (1997) on the filament windingprocess, and Wiersma et al. (1998). Another model, by Zobeiry et al. (Submitted), w i l lbe discussed in thecoming sections.

E l l y i net al. analyzed the material with a multiple-Kelvin nonlinear V E model in cool-down, assuming astress-free temperature. Wiersma et al. modeled the material properties with a simple combination of asingle M a x w e l l element and a spring. K o k a n analyzed a composite cylinder under cure. He assumedmultiple K e l v i n elements for representation of the transverse behaviour of the material. The materialparametersin this study were assumed to bedependenton thedegreeof cure, but not on temperature.

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2.4.3.1. SolutiontechniquesIn a similar way to the IF, a solution to the constitutive model is required to relate stresses and strains ateach point in time. Since the equations are in the form of differential equations the solutions available inthe literature for ordinary differential equations can be used (a review ofthese solutions can be found inButcher,2000).

Bazant (1972) and Carpenter (1972) used Runge-Kutta solutions in their V E analysis (see e.g. Butcher,1996 , 2003). However, the method of choice for the majority of studies on DF has been the finitedifference method (see Hughes, 1987). The studies mentioned above have also used different finitedifference schemes, e.g. K o k a n , and Mesquita and Coda used a backward Euler method, whileZ i e n k i e w i c z ,Bazant, Jurkiewiez, and Idesman et al . used an exponential form ofsolution.

In general, the finite difference schemes, no matter how different they may seem, in effect update thestress based on the values ofstress at the previous time step. This is in fact very similar to the internalvariable approach used in some IF studies, as mentioned in Section 2.4.1.1. These approaches are easierto implement in an available code, since they operate at the integration point l e v e l (Poon et al., 1998).A l s o ,due to thenatureof finite difference schemes, only knowledge o f variables at the previous time-stepis needed, unlike the prevalent IF methods, such as Taylor et al. (1970), where knowledge of variablesfromthe previous two timestepsis required.

2.5. SUMMARYDifferent constitutive models avai lable in the literature for modeling the behaviour of composite materialsduring cure were discussed in this chapter. The simplest formulations, i.e. elastic models, are not complexenough to provide accurate results for our applications.

T he simplest modelthat seemsto beadequatein process model ing is the C H I L E model. It was mentionedthatthe C H I L E models have been shown to provide good predictions, and are fast and easy to implement.

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iChapter2:ReviewofConstitutivemodels

In addition, characterization of material behaviour for input in the codes using this approach, such asC O M P R O (Johnston, 2001), is fai r ly easy and runtimes are short even on personal computers. However,the formulations based on this model have not been studied carefully for their general validity andaccuracy. As a result, there is a need to reconcile what we might c a l l the pragmatic and economicalC H I L E approach with the fundamentally sound viscoelastic approach. This can prove valuable, whenefficiency of the model is important. For this reason, a study of this model and its bounds of accuracy andvalidity w i l lbe performed in the next chapter.

A t the same time, when a more accurate model is needed or when an analysis falls outside the bounds ofvalidity of the C H I L E models, a viscoelastic formulation is needed. Most models available in theliterature are very time consuming and complex for implementation in finite element codes. Differentialform of viscoelasticity has a simpler form than the integral form commonly used in residual stressmodelingof composite materials during processing. In addition,thesemodels can be easily implementedin FE codes, since they only need manipulations at the integration point l e v e l .There is an apparentlack ofmodels based on differential form ofviscoelasticityin this f i e l d ,but due to their promise of efficiency and

simple form they seem to be the best choice for residual stress modeling of composite materials duringcure, especially when accuracy of the model is of importance.

Study of different models insubsequent chapters w i l l demonstrate a range of constitutive relations thatca n be used in model ing and demonstrates that there is a continuum of trade-off of investment versusaccuracy as we go from a f u l l viscoelastic approach to the simplest elastic approach. A schematicrelationship between different models, i.e. elastic, C H I L E , and V E , in general terms is illustrated in

Figure2.9.

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Chapter2:ReviewofConstitutivemodels

2 3 4 5 6C u r e cycl e ti me (hour s)

-Autoclavetemperature Autoclavepressure|

Figure2.1 Atypical temperature/pressure cycle usedtocure thermoset polymersin anautoclave

\ \ . ,s 1 / 1 /_

CureFigure2.2Joiningofmoleculesas aresultofcuringin athermoset polymer

200

100 150 200 250Time(min)

Figure2.3Schematicof changes in the state ofapolymeras it goesthroughatypical cure cycle

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L U -I

Dash=InitialshapeSolid=FinalshapeFigure2.4Undeformedanddeformedshapes in atypical U-shapedcompositepart after cure

log(time)

Figure2.5Schematicoftime-temperature superposition

1 k W W -Figure2.6 AMaxwellelement

D M

Figure2.7 AKelvinelement

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A / W V - ,

WW- 1Tin K

Figure2.8 A generalizedMaxwell element

Figure2.9 Differentlevelsof constitutive modeling

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Chapter3:Pseudo-Viscoelastic Model

Chapter3. PSEUDO-VISCOELASTIC M O D E L SInthis chapter we compare the C H I L E approach to the 'gold standard' of the viscoelastic approach,discuss why and when C H I L Ecan work well, and determine the bounds of its validity and accuracy. Inorderto do this, itwillbe shown that there are approximations that allow thefullviscoelastic approachtobe simplified to aform of C H I L E model, and thattheseapproximations are appropriate for the typicalcure cycle that a thermoset polymer undergoes. First, a summary of the viscoelastic models for twodifferent materialswillbe presented.Then,the necessary equationswillbe derived and finally,somecasestudieswillbe presented.Thisstudy can also be found in the reference byZobeiryet el. (Submitted).

3.1. REVIEW OFT w o VISCOELASTIC MODELSIn this section, we present a summary of two similar viscoelastic models that have recently beendeveloped forcuringthermoset polymers. These formulations are based on the works ofKimand White(1996)andPrasatyaetal.(2001).

3.1.1. MaterialIRecallthe integralformofaviscoelastic constitutive model as follows:

Where e"""ands'c are the total strain and freethermo-chemicalstrains, respectively.

Kimand White proposed aV Econstitutive model for3501-6epoxy resin that is formally identical to thisequation, in which the relaxation modulus is expressed as:

0(3.1)

( , )=ET(a)+[Eu(a)-ET( ) ] W(a)exp (3.2)

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where E and E" are the relaxed and unrelaxed modulus, respectively, and Wa and ra are the weight

factors and discrete stress relaxation times for the co'h M a x w e l l element (in an assembly of / parallelM a x w e l l elements), respectively.

Based on the experimental results for this material, it is assumed that Eand E" are constants andindependent of the degree of cure and temperature. A l s o , the weight factors are assumed to be constantthroughout the cure process. The relaxation times, however, change with the degree of cure according tothe following equation:

log(rff(a)) =log(rff(a0)) +[/ '(a ) - (a - a)logfX)] (3.3)where, a 0 is the reference degree of cure (0.98 in this case) and:

f\a) = -9.3694 + 0.6089a + 9.13 47a2 (3.4)

(3.5)

where, rp is the peak relaxation time shown in Table 3.1 along with other relaxation times and weight

factors for the reference degree of cure.

The reduced time, cf, is calculated as follows:

(3.6)

where, aT is the shift function calculatedfrom the following equation:

log(a r) = c,(a)T + c 2 (a) (3.7)

i n which:

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c,(a) = -ax exp(-^)-a2 (3.8)a-\

c2(a) =-T0C](a) (3.9)

In these equations, for 3501-6 epoxy resin, al=lA /C , a2 =0.0712/C, and T is the reference

temperature (in this case T =3 0 C ) .

3.1.2. MaterialIISimon, M c K e n n a, andco-workers (Simonetal., 2000andPrasatyaetal. 2001) have presented asimilarV E model forthebulk behaviour ofHexcel8551-7 epoxy resin. The form of the bulk relaxat ion modulusisas follows:

K(,T,t) = Kr+[ K g-K r] g exp (3.10)

where, K is thebulk modulus, K andKg are therubbery andglassy values of the bulk modulus, both

o f which areassumed to beindependent oftemperature anddegree o fcure, T,is the i'h relaxation time

and gt is the corresponding weight factor, such that _^g ,=1.0. Both parameters areassumed to be

constant throughout thecure cycle.The values fortheseparameters arelisted in Table 3.2.

Inthis model,theshift factor, aTa , isalso usedtoaccount for thedependence of the relaxation timer, on

temperature and degree of cure and isevaluatedas follows:

T-T T(a)-T

i n which, C andTm areconstants andTg is theglass transition temperature taken to be afunction of the

degree of cureasfollows:

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T.-To Aa (3.12)Tg-Tga 1-(1-A)

where Tg0 is the Tg of the monomer,Tai is the Tg of the f u l l y cured material, and X is a materialconstant.

The major difference between this model and the previous one lies in the form of the shift factors(Equation (3.7) vs. Equation (3.11)). For the purpose of deriving the equations in the next section,however, it is desirable torepresent the shift factor in the form of Equation (3.7). It can be shownthat thedifference between these two forms in representing the results of the experiments is minimal (Thiscomparison is only made to show that the two material models are similar and the same derivationprocedure can be used for both materials; the actual formulation w i l l not be changed). The graph that isused in Prasatya et al. to derive the shift factor parametersis reproduced hereand shown in Figure 3.1. Itis clearthat Equation (3.7) is a good fit to the results, and thusthe same approximation approach w i l l bev a l i dfor both materials. For this material the parameters for a f u l l y cured resin are cx(a) = -0.2298 andT0 = 1 6 0 C .

There is another difference between the two models. In the reference paper, the stressis calculated fromthe followingformulation:

OO 7

o- = -\K(t-t',T,a)dt' (3.13)v ^ r (3.15)

3.2.1. SimplificationofViscoelasticFormulationsW e w i l l nowdetermine the conditions under whichtheV E formulation simplifiesto a correspondingP V E formulation forawiderangeo f conventional cure cycles.

In a viscoelastic material, rewriting Equation (3.1)here in a more convenient form, stress can becalculatedasfollows:


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Chapter3:Pseudo-ViscoelasticModel

valueof the relaxation modulus, E ,at a suitable and as yet undetermined time, say te, that is appropriate

for the temperature and degree of cure at time r , i.e. T(r) and a(x). According to this definition, we

canwrite:

E ' (T) = E(te,T(r),a(T)) . (3.19)Invoking the time-temperature superposit ion and notingthat in this case r represents a fixed instant of

time and not a dummy integration variable, the reduced time becomes [- = - , and the relaxationJ0ar(r) aT(r)

modulus can be expressed as:

aT(r)FromEquations (3.18) - (3.20), we have:

E(te,T,a) =E(- -) (3.20)

aT(r)Thisequation canonly be v a l i d i f the arguments of E are equal, i.e

(-4-r)= (3.21)

aT(r)SubstitutingEquation (3.17) here, we obtain:

(3.22)

J ^ J f - J f - J f L ( 3 .23)ar( T) oa r oaT taT

Inorder to f i n d suitable values of te, simplifications in the V E formulat ion are made by consider ing twodistinct temperature regimes within the cure cycle: the cool-down, in which the temperature varies

(reduces) linearly with time, and the hold, in which the temperature is constant. These temperature

regimes are characteristic of al l conventional curecycles.

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Chapter 3: Pseudo-Viscoelastic Model

3.2.1.1. Simplification during cool-downThe temperature-timerelationship in the cool-downsegmentcan be written in a generic form as follows:

T(t') =-m(t'-p) +q (3.24)

i nwhich,m is the coolingrate, t' is the time, and p and q are constants.

N o w ,using Equation (3.7) for the shift factor we can write:

aT = w r + c > w = ex p( C | r + C 2 ) (3.25)log(e)

B y substituting these two equations into Equat ion (3.23), assuming that the degree of cure is constantduring the cool-down (a v a l i d assumption), say af, performing the integral, and simplifying, we obtain:

ar(r)

where

1 1aT(r) aT(t) (3.26)

t ] = - - ^ L (3.27)c^(af)mNotethat c ,(a) is a negative number. No w, since the shift factor changesexponentially, its f i na l value is

muchgreaterthan its value at any other instant during the cool-down. In other words, - - andaT(t) aT(r)

consequently, teat l.

The assumption that the shift factor at t is muchgreaterthan thatat r is particularlytruewhen t is theduration of the cure cycle and the f i n a l residual stress is being calculated. When r approaches t in thecool-down regime the error resulting from such an assumption increases. However, this w i l l not cause asignificant error in the calculations,becausewhen r approaches t, the value of tends to zero and

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the relaxation modulus at time te w i l l be equal to the unrelaxed modulus. Thus, an error in evaluating theshift factor w i l l not affect the results significantly, as the modulus w i l l almost be equal to the unrelaxedmodulus.

Thus using the following simple equation w i l l y i e l dstresses thatare equivalent to those predicted by thefu l l V E model for the cool-down:

T o show the effect o f this simplification, the reduced times and moduli from the P V E and fu l l V Emethods are compared for a cure temperature of 1 8 0 C , a * l , and cool-down to room temperature atthree different cooling rates, namely for m = \, 3, and 5 C / m i n . The results for and modulidevelopment are plotted versus r (for t equal to the time at the end of the cure cycle) in Figure 3.2 andFigure 3.3, respectively. It can be seen that the reduced times obtained from the two methods are veryclose.Some errors occur, as r approaches t. But, as explained before and shown in Figure 3.3, this errori n evaluating the reduced time does not result in a large difference in the predicted modulus. Thesefigures clearly indicate thatby using the P V E method in cool -down the predicted stressesshould closelymatchthosecalculated by the far more computationally intensive V E model. It is also noted thatthe slightwaviness in Figure 3.3 is due to discrete relaxation spectrum of the modulus, i.e. the relaxation modulusconsists of several M a x w e l l elements thatrelax in different time frames andcause the waviness.

A n important observation isthat the value of te is a function of the cooling rate m .Thismeansthat for

any given curecycle, the appropriate time should be calculated and used.

3.2.1.2. Simplification duringtemperatureholdO n the hold,Equation (3.23) can be written as:

(3.28)

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J _ _ _ , f _ _ _ , / f _ _ + , f _ _ (3.29)aT(r) iar \ ar J aT

where, tf denotes thetime at the end o fthe hold andbeginningo fthe cool-down regime. Thesecond

integral pertainingto thecool-downhas already been evaluated in the previous section. Therefore,wemerely needtos i m p l i f y thefirst integral.

Wenotethatduringthehold thevariationo fshift factor as afunction time is at the l i m i t either theformshown in Figure 3.4,orconstant withtime. To evaluate theintegral,whichis thearea under thecurve,we

defineatime t2, priortowhich decreaseslinearlywithtime, and thereafter it isconstant:aT

1 =P+ ( - /*) ? (3.30)aT(t2) ar(r) aT(tf)

The form o f theabove equation captures both limits of thevariationo f , i f(5 ischosen to be anaTappropriately small number. Generally,theobserved behaviour iscloserto thecurve in Figure 3.4, where

^ 1 1after time t2, wehave .aT{t') ar(r)

The area under the curve can thenbefoundas:/ f ^ = i ( , 2 _ r ) | _ ^ + _ ^ ] + ( , / - ? 2 ) _ ^ (3.3DaT 2 V6 7 r r ) aT(t2)j aT(tf)

In the cases where is notconstant past t2, there w i l l be some error inevaluating the integral.ar

However, this erroris notsignificant since thefirst term isdominant. Now, using Equations (3.29)and(3.31) we have:

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' - - V ' > f i + ? V ( < , - . ) ? , < " 2 >V aT(t2) aT(tf)Thisgivesus thevalue o ftimetoevaluate themodulus during the hold. Wenotethat /? ischosento be a

small number, but itshouldnot be sosmallthattheapproximately linear form of between times rarand t2 isundermined.Avalue of 0.1 hasbeen observedtowork w e l l .

It shouldbenoted thatinorder to use theabove equations it isassumed thatthedegreeofcure isknownat every time stepduringthehold. These values areneeded inorder tocompute t2 from Equation (3.30).A s a result, oneneeds to perform the thermo-chemical analysison the model first before theresidualstressanalysis.

I f the valueso fthe degreeofcure cannot becalculated beforehand, onecan assume t2 = T and thereforeuseasimpler form of the above equation as follows:

0.33)

In this equation, only theknowledgeof thedegree ofcure at time / / ; i.e. theonset o fcool-down,is

sufficient.

3.2.1.3. SummaryofequationsHere,asummary o f the P V E equations ispresented. Thestressesarecalculatedas follows:

er0 = )E(te,T(T),a(T)) dT (3.34)

Wherethereare two regimestocalculate te, during cool-down and duringtemperature hold.

Duringcool-down:

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r . = / , = - ^ f (3.35)c,{af)m

Where af is thedegreeof cureattheonseto fcool-downand m is thecool-downrate.

Duringtemperaturehold:

, # =I ( , 2- r) 1+ r M + (, - + , ( 3.36)ar(tf)Where tf is the timeatthe onseto fcool-down and t2 is atime, definedas f o l l o w s :

1 =P{+(\-p)-^ (3.37)aT(t2) aT(r) ar(tf)P ischosento be asmall number.

I fthe values of the degreeof cure cannot bedetermined beforehand, thef o l l o w i n g canbeused duringthetemperaturehold:

3.2.2. Different Forms ofPVEThe above simplification assumesthat the materialisfu l l y characterized viscoelastically,but thatwe wishto be computationally efficient. From now on, this simplification procedure w i l l becalledthe'variabletime' P V E method, where theinstant o f timeatwhichtheelastic modulus iscalculated varies throughoutthe cure cycle.This value of time clearly comes fromadirect simplification of the V E constitutive model.

Alternatively,if one wishesto beeven more efficient and simple,asingle, constant time canbedefinedatwhich themodulus is calculated at allpoints in thecure cycle. Onecould then save significantlyoncharacterization costs. Thiscanalsobedone usingaconstant value o ffrequency. Theincentivefor thelatter is to be able to use the results of c y c l i c loading tests, such as D M A (Dynamic Mechanical

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Analyzer), used for material characterization. These additionallysimplified procedures w i l l be called the'constant time' or 'constant frequency' P V E method, notingthatthe added simplification comes at a cost

o f reduced accuracy.

Thebestestimate for this constant value of time is the one thatapplies to the cool-down, as calculated byEquation (3.27). If use of a constant frequency is desired wenotefrom Ferry (1980) that:

G(t) =G'{(o)- 0.40G"(0.40


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Chapter3 :Pseudo-Viscoelastic Model

It should be noted that if Equation (3.27) is not used, and an intuitive or arbitrary value of frequency isused instead, then we have the classical C H I L E model availabel in the literature (e.g. Johnston et al.2001).

3.3. CASE STUDIEST o evaluate the validity of the simplification procedure introduced in the previous section and thedifferent forms of P V E some case studies are presented here.For s i m p l i c i t y , a fu l l y constrained block ofpolymer undergoing a given cure cycle is considered. Neglecting the influence of external mechanicalloads (e.g. autoclave pressure) the induced strain w i l l consist only of thermal and cure shrinkagecontributions. The advancement of cure must be taken into account using an appropriate thermo-chemicalmodel. To perform the analysis, a simple F O R T A N code was written. This program analyses a 1-Dstructure (i.e. a bar) using a P V E and/or a fu l l V E formulation as it undergoes the cure cycle andcalculates the residual stress at each instant. It is noted that the bar is held at fixed length and the endeffects are not examined.

3.3.1. M a t e r i a lIThetemperatureprofile for a conventional one-hold cure cycle and the resultingdegreeof cure are showni n Figure 3.5. The thermo-chemical model is based on the work by Lee et al . (1982). The time historieso f the elastic modulus of the P V E , and stressresulting from a variable time P V E and V E are presented inFigure 3.8 and Figure 3.9, respectively. The agreement between the stress predictions of variable-timeP V E and the fu l l V E methods is excellent. It is noted that the relaxed modulus is assumed to bepresentthroughout the cure cycle and hence it also exists before gelation. This is necessary, as the material isassumed to be a viscoelastic s o l i d ,which has a non-zero relaxed modulus.

The temperature profile for a two-hold cure cycle and the resulting degree of cure are shown in Figure3.10. A constant frequency P V E analysis at frequencies of 0.1Hz, 0.001Hz, and 0.0001Hz was

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performed and the results are presented in Figure 3.11 and Figure 3.12. A l s o shown in Figure 3.12 arethe stressespredicted by the V E model.Clearly an accurate prediction results i f the selected frequency issomewhere between 0.001Hz and 0.0001Hz .

To investigate the effect o f frequency on residualstressprediction, the P V E predictions are plotted versusfrequency in Figure 3.13 and compared with the unique (frequency independent) value of the stressobtained from the V E model. A frequency of around 2.7 x l O ^ T / z yields a residual stress thatcloselymatches the V E predicted stress. Use of Equations (3.27) and (3.40) gives 9.0 x 10"4Hz, which leads to aP V Eprediction of only about 1MPa higher than the V E prediction,thussuggesting thatEquation (3.40)is a good estimate of the appropriate frequency to characterize a material. It should be noted that thesefrequencies are quite low, andthushigher frequencies, as used in a large number ofcases in the literature,w i l l result in slightly higher stresses. This is perhaps unavoidable since the required frequencies areexperimentallyimpractical.

To evaluate the effect of the cooling rate on the residual stresses, several one-hold cure cycles withvarying cooling rateshave been analyzed. Figure 3.14 and Figure 3.15 show the variable time P V E andV E predicted residual stress as a function of cooling rate for these cycles at two different holdtemperatures. Note that in each case the temperature hold continues until complete cure prior to c o o l -down. In all cases, very goodagreementis obtained between the V E and P V E results. Furthermore, theseresults show that the effect ofcooling rate on the residual stress is only significant at very lowratesofcooling (between 0.1 to1 C / m i n ) .

Constant time PV E predictions for thesecase studies are presented in Figure 3.16 and Figure 3.17. Theappropriate time for each cooling rateis calculated using Equation (3.40). While the correlation betweenthese P V E predictions and the V E predictions is not as good,witha maximum error of 8%, it is s t i l l quiteacceptable. This suggests that when we use an appropriately calculated constant time to evaluate themodulus in a P V E model , a reasonable estimate o f residual stresses is obtained. Note that the biggest

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contributor to error is the stress b u i l d up duringhold. In fact, the stressesgenerated during cool-down areobserved to match closely.

3.3.2. Material IIIn Prasatya et al. (2001), V E analyses were performed for a wide range of cure cycles to obtain a largevariationin predicted residualstress:This set o

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Viscoelasticity and Reisdual stress