Modeling of Composite Laminates Subjected to Multi Axial Loadings

MODELING OF COMPOSITE LAMINATES SUBJECTED TO MULTIAXIAL

LOADINGS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of the Ohio State University

By

Behrad Zand, M.S.

* * * * *

The Ohio State University

2007

Dissertation committee:

Professor William E. Wolfe, Advisor

Dr. Tarunjit S. Butalia, Coadvisor

Professor Harold W. Walker

Professor Edward Overman

Dr. Greg A. Schoeppner

Approved by

Advisor

Graduate Program of Civil Engineering

UMI Number: 3279788

32797882007

Copyright 2007 byZand, Behrad

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P.O. Box 1346 Ann Arbor, MI 48106-1346

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by ProQuest Information and Learning Company.

Copyright by

Behrad Zand

2007

ii

1 ABSTRACT

A nonlinear strain energy based failure model is proposed for fiber reinforced polymer

composites. A new strain energy based failure theory is developed to predict matrix

failure for a unidirectional laminate. It is shown that the proposed model complies with

energy conservation principles for elastic materials. A correction factor is incorporated

into the formulation to take into account the influence of transverse stress on the inplane

shear resistance. The maximum longitudinal strain criterion is adopted to predict fiber

failure. An incremental constitutive model is developed to predict stress-strain response

of the material under multiaxial loading, unloading, and reloading conditions. The failure

model is extended to multidirectional laminates, using classical lamination theory. An

empirical exponential stiffness reduction model is proposed to represent transverse and

shear moduli of the laminae embedded in a multidirectional laminate. Model parameters

are evaluated using experimental data from the literature. The proposed model is used to

predict stress-strain response and failure of unidirectional and multidirectional laminates

with various material properties and lay-ups. The predictions are shown to be in

agreement with available experimental data. Additional experimental data are obtained

by testing S-glass and carbon fiber specimens under combined axial and torsional loads.

The experimental observations show that the measured values from different strain gages

installed on the same specimen, as well as those installed on similar specimens tested

iii

under the same loading conditions are generally in agreement. For some cases the

measured strains from different strain gages installed on the same specimen were

somewhat different. The proposed model is shown to be capable of predicting stress-

strain responses as well as initial and final failures for the tested specimens.

iv

Dedicated to my wife and parents

v

2 ACKNOWLEDGMENTS

Earning a doctoral degree is a long journey that requires patience, dedication, and hard

work. I recall from the passed five years, the many nights that I stayed in the computer

lab at Hitchcock Hall, writing programs and reports, reading papers, entering

experimental data into spread sheets … . And this could not have been possible without

passion and believe in what I set out to accomplish as well as support from many people.

I take this opportunity to express my gratitude for the efforts and encouragements of

many people who made this dissertation possible. I would like to thank my advisor,

Professor Wolfe for his scientific guidance, financial support, and patience in correcting

my technical and stylistic mistakes over the passed five and half years. I am grateful to

Dr. Butalia for his enthusiasm and technical support during every step of this work and

other research projects that I was involved with. I would like to thank Dr. Schoeppner for

his encouragement and valuable technical comments; and Professor Walker for his

support. I wish to express my thanks to Professor Overman for his lectures, as well as the

discussions we had in and out of classes, which guided me throughout this work. I,

moreover, would like to thank Professor Henry Busby for his guidance and

encouragement.

vi

I am indebted to Laleh, my sweetheart wife, for her unconditional love, for always being

by my side, and for her patience. I wish to express my greatest thanks to my beloved

parents, who have instilled in me the value of hard work and knowledge. Without their

unconditional support I could never get this far. I am grateful to all who helped me

throughout the laboratory experiments, particularly Jay Hunter, James Howdyshell,

Katherine Walker, and Gursimran Singh. Finally, I would like to acknowledge Mr. Wei

Tu, my colleague, for his help and collaboration in this and many other projects that we

accomplished together over that passed five years.

vii

3 VITA

May 13, 1973 ............................................................................... Born – Tehran, Iran

1996 ....................................................................................... B.S., Civil Engineering,

Tehran University

1999 ......................................................................................... M.S., Civil Engineering

(Geotechnical)

Tehran University

2002 – 2004 ............................................................................ M.S. Civil Engineering

(Geotechnical)

The Ohio State University

Field of Study

Major .................................................................................... Civil Engineering

Minor ............................................................................. Applied Mathematics

viii

4 TABLE OF CONTENTS

Abstract ............................................................................................................................... ii

Acknowledgments............................................................................................................... v

Vita vii

Table of Contents.............................................................................................................viii

List of Tables .................................................................................................................. xvii

List of Figures .................................................................................................................. xix

List of symbols.............................................................................................................. xxxii

Chapter 1............................................................................................................................. 1

1 Introduction................................................................................................................. 1

Chapter 2............................................................................................................................. 3

2 Literature Review........................................................................................................ 3

2.1 Introduction......................................................................................................... 3

2.2 Composite Laminates.......................................................................................... 4

2.2.1 Fibers............................................................................................................... 4

2.2.2 Matrix.............................................................................................................. 5

2.2.3 Failure of Composite Laminates..................................................................... 5

ix

2.3 Failure theories.................................................................................................... 6

2.3.1 Phenomenological Theories............................................................................ 7

2.3.1.1 Non-interactive failure theories .............................................................. 8

2.3.1.2 Interactive non-mechanistic failure theories........................................... 9

2.3.2 Mechanistic Failure Theories........................................................................ 13

2.3.3 Damage Based Models ................................................................................. 15

2.3.4 Micromechanical Models.............................................................................. 16

2.3.4.1 Fracture mechanics based models......................................................... 16

2.3.4.2 Crack density based models.................................................................. 18

2.3.4.3 Microbuckling and kinking theory........................................................ 20

2.3.4.4 Finite element analysis.......................................................................... 21

2.3.4.5 Rule of mixtures.................................................................................... 22

2.3.4.6 Method of cells ..................................................................................... 23

2.4 Experimental Methods ...................................................................................... 24

2.4.1 Unidirectional Strength Testing.................................................................... 24

2.4.1.1 Unidirectional tension test .................................................................... 25

2.4.1.2 Unidirectional compression test............................................................ 25

2.4.1.3 Unidirectional shear test ....................................................................... 26

2.4.1.4 Bending test .......................................................................................... 26

2.4.2 Multi-directional strength testing.................................................................. 27

x

2.5 The World Wide Failure Exercise .................................................................... 28

2.6 Closing Remarks............................................................................................... 32

CHAPTER 3......................................................................................................................... 35

3 STRAIN ENERGY BASED MODEL................................................................................ 35

3.1 Introduction....................................................................................................... 35

3.2 The Original Strain Energy Based Model......................................................... 36

3.3 A Strain-Energy Based Model for Linear Elastic Composites ......................... 40

3.3.1 Constitutive model ........................................................................................ 40

3.3.2 Strain Energy Based Failure Model for Orthotropic Linear Elastic Materials

43

3.4 Failure Model for Unidirectional Fibrous Composites Under In-plane Loading

Condition....................................................................................................................... 48

3.4.1 Incremental Constitutive Law....................................................................... 49

3.4.2 Matrix Failure Criterion................................................................................ 52

3.4.2.1 Shear response ...................................................................................... 52

3.4.2.2 Shear strain energy in material with internal friction ........................... 57

3.4.2.3 Failure criterion..................................................................................... 58

3.4.2.4 Matrix failure modes............................................................................. 61

3.4.3 Fiber Failure Criterion .................................................................................. 62

3.4.4 Numerical Results......................................................................................... 65

xi

3.4.4.1 Effect of shape factors on matrix failure............................................... 69

3.4.4.2 Effect of LTA and LTm on matrix failure ............................................... 73

3.4.4.3 Effect of μ on matrix failure................................................................. 76

3.4.5 Comparison between Predictions and Experimental Data............................ 80

3.4.5.1 Longitudinal-transverse failure envelope for unidirectional E-

glass/MY750 epoxy laminate ............................................................................... 80

3.4.5.2 Transverse-shear failure envelope for unidirectional E-glass/LY556

epoxy laminate ...................................................................................................... 82

3.4.5.3 Biaxial failure envelope for unidirectional T300/914C epoxy laminate

under combined longitudinal and shear loading ................................................... 85

3.4.6 Unloading and Reloading ............................................................................. 88

3.4.6.1 Uniaxial unloading and reloading......................................................... 88

3.4.6.2 Unloading under combined axial and transverse loading ..................... 93

3.4.6.3 The effect of transverse unloading on shear strain ............................... 94

3.4.6.4 Unloading constitutive relations in matrix notation.............................. 96

3.5 The Strain Energy Failure Criterion for the Three-Dimensional Stress

Condition....................................................................................................................... 97

3.5.1 Notations ....................................................................................................... 97

3.5.2 Incremental Constitutive Law....................................................................... 99

xii

3.5.3 Strain Energy Based Failure Criterion Under Three-Dimensional Loading

102

3.6 Closing Remarks............................................................................................. 103

CHAPTER 4 ................................................................................................................... 105

4 MULTI DIRECTIONAL LAMINATES................................................................ 105

4.1 Introduction..................................................................................................... 105

4.2 Matrix Stiffness in Multi-Directional Laminates............................................ 106

4.2.1 Shear Response in a Multi-Directional Laminate....................................... 106

4.2.2 Exponential Stiffness Reduction Model ..................................................... 111

4.2.2.1 Stiffness reduction factor .................................................................... 112

4.2.2.2 Shear energy ratio ............................................................................... 113

4.2.3 Stiffness Reduction in Tensile Transverse Direction.................................. 120

4.2.4 Stiffness Reduction in Compressive Transverse Direction ........................ 124

4.3 Stiffness Reduction Parameter for Angle-Ply Laminates ............................... 132

4.3.1 Axial-Hoop Stress Failure Envelope for [±55˚]S Laminate Made of E-

glass/MY750 Epoxy................................................................................................ 132


glass/MY750 Epoxy................................................................................................ 137

4.4 Evaluation of Model Predictive Capability..................................................... 141

xiii

4.4.1 Biaxial Failure Envelope for [90˚/ ± 30˚/90˚]s Laminate Made of E-glass/

Epoxy Subject to Combined Axial and Torsional Loads........................................ 141


Epoxy Subject to Combined Axial and Hoop Stress .............................................. 146

4.4.3 Biaxial Failure Envelope for [90˚/ ± 45˚/0˚]s Quasi-Isotropic Laminate Made

of Carbon/Epoxy Subject to Combines Axial and Hoop Stress ............................. 148

4.4.4 Stress-Strain Curve For Quasi-Isotropic [90˚/ ± 45˚/0˚]s AS4/3501-6

Laminate Under Uniaxial Tension In the Hoop Direction...................................... 154

4.4.5 Stress-Strain Curve For Quasi-Isotropic [90˚/ ± 45˚/0˚]s AS4/3501-6

Laminate Under Hoop to Axial Stress Ratio of 2/1................................................ 155

4.4.6 Stress-Strain Curves for [±55º] E-glass/MY750 Epoxy Laminate under Hoop

to Axial Stress Ratio of 2/1..................................................................................... 158

4.4.7 Stress-Strain Curves for [0º/90º]S Cross-ply Laminate Made of E-

glass/MY750 Epoxy Under Uniaxial Stress in 90˚ Direction................................. 161

4.4.8 Stress-Strain Curve for [±85º]S Cross-ply Laminate Made of E-glass/Epoxy

Under Axial Stress .................................................................................................. 163

4.5 Closing Remarks............................................................................................. 165

CHAPTER 5 ................................................................................................................... 170

5 EXPERIMENTAL PROGRAM FOR MODEL VALIDATION ........................... 170

5.1 Introduction..................................................................................................... 170

xiv

5.2 Testing Procedures.......................................................................................... 171

5.2.1 Materials and Specimen Preparation .......................................................... 171

5.2.2 Testing Procedure ....................................................................................... 173

5.3 Test Results for S-glass/epoxy Tubes ............................................................. 178

5.3.1 S-glass/epoxy under Shear to Axial Stress Ratio of 0.2/1 .......................... 178

5.3.2 S-glass/epoxy Tube under Shear to Axial Stress Ratio of 0.5/1.0 .............. 188

5.3.3 S-glass/epoxy Tube under Shear to Axial Stress Ratio of 0.4/1.0 .............. 194

5.3.4 S-glass Tube under Shear to Axial Stress Ratio of 4/1............................... 198

5.3.5 Comparison between the Stress-Strain Curves Obtained Under Different

Stress Ratios............................................................................................................ 201

5.4 Test Results for Carbon/epoxy Tubes............................................................. 204

5.4.1 Carbon/epoxy Specimen under Shear to Axial Stress Ratio of 0.26/1 ....... 204



5.4.4 Carbon/epoxy Specimen under Axial Stress............................................... 217

5.4.5 Response of Carbon/epoxy Specimen under a Non-proportional Combination

of Axial Load and Torsion...................................................................................... 221

5.4.6 The Influence of Shear to Axial Stress Ratio on the Material’s Response. 225

5.5 Comparison between the Model Predictions and Experimental Data ............ 228

5.5.1 S-glass/epoxy Laminate .............................................................................. 229

xv

5.5.1.1 Numerical Predictions for S-glass/epoxy specimen for a shear to axial

stress ratio of 0.2/1.0 (tuned predictions)........................................................... 230

5.5.1.2 Numerical predictions for S-glass/epoxy specimen for a shear to axial

stress ratio of 0.4/1.0........................................................................................... 233


stress ratio of 0.5/1.0........................................................................................... 237


stress ratio of 4/1................................................................................................. 240

5.5.2 Carbon/Epoxy Specimen Laminate ............................................................ 243

5.5.2.1 Numerical predictions for carbon/epoxy specimen under shear to axial

stress ratio of 0.26/1.0 (tuned predictions)......................................................... 243

5.5.2.2 Numerical predictions for carbon/epoxy specimen for a shear to axial

stress ratio of 0.16/1.0......................................................................................... 247


stress ratio of 0.32/1.0......................................................................................... 250


stress ratio of 0.02/1............................................................................................ 253

5.5.2.5 Numerical predictions for carbon/epoxy specimen for a non-

proportional loading............................................................................................ 256

5.6 Closing Remarks............................................................................................. 259

xvi

6 SUMMARY AND CONCLUSIONS................................................................................. 261

A. APPENDIX A............................................................................................................. 265

B. APPENDIX B............................................................................................................. 268

C. APPENDIX C............................................................................................................. 277

D. APPENDIX D............................................................................................................. 280

LIST OF REFERENCES ....................................................................................................... 291

xvii

5 LIST OF TABLES

Table 2.1. The failure theories participated in the WWFE (Kaddour et al., 2004).......... 30

Table 2.2. Quantitative ranking of the participated theories (Hinton et al., 2004) .......... 31

Table 3.1. Mechanical properties of the unidirectional material systems (Soden et al.,

1998) ................................................................................................................................. 66

Table 4.1. Matrix and fiber failure indices for [90 / 30 / 90 ]S±o o o E-glass/epoxy laminate

loaded under hoop to axial stress ratio of -0.82/-1.......................................................... 147

Table 4.2. Mechanical properties of the unidirectional AS4/3401-6 (Soden et al., 1998)

......................................................................................................................................... 152

Table 4.3. Summary of the cases analyzed in Chapter 4 ............................................... 168

Table 4.4. Suggested experiments for evaluation of model parameters for a multi-

directional system made of a new material..................................................................... 169

Table 5.1. Summary of the specimens and loading conditions...................................... 177

Table 5.2. Material properties used in the analysis for unidirectional S-glass/epoxy

material system. The transverse and longitudinal moduli, as well as longitudinal tensile

failure strain were adjusted ............................................................................................. 232

Table 5.3. Material properties used for the numerical analysis. Transverse and

longitudinal moduli and the longitudinal failure strain were tuned to fit the numerical

predictions to the experimental data from C1................................................................. 244

Table D.1 S-glass specimen G1 under shear to axial stress ratio of 0.2/1..................... 281

xviii

Table D.2. S-glass specimen G2 under shear to axial stress ratio of 0.2/1.................... 282



Table D.5. S-glass specimen G5 under shear to axial stress ratio of 4/1....................... 285

Table D.6. Carbon specimen C1 under shear to axial stress ratio of 0.26/1.................. 286



Table D.9. Carbon specimen C4 under shear to axial stress ratio of 0.0/1.................... 289

Table D.10. Carbon specimen C5 under non-proportional loading............................... 290

xix

6 LIST OF FIGURES

Figure 2.1. Failure envelope for porous graphite obtained by parametric failure criterion

with five terms (Neals and Labossière, 1989; notations are changed). ............................ 12

Figure 2.2. Different fracture mechanisms of the matrix. (a) Transverse failure of the

matrix; (b) shear fracture of the matrix; (c) debonding of fiber and matrix; (d)

longitudinal fracture of the matrix (Berthelot, 1999)........................................................ 18

Figure 2.3. Classification of failure theories for composite laminates ............................ 34

Figure 3.1. Longitudinal stress-strain curve under biaxial loading .................................. 39

Figure 3.2. The influence of internal friction on in-plane shear strength of laminate ...... 55

Figure 3.3. The influence of a constant transverse stress on the shear stiffness............... 55

Figure 3.4. Influence of constant and variable transverse stresses on shear response of a

non-linear material ............................................................................................................ 56

Figure 3.5. (a) Failure mode interaction in the longitudinal-transverse energy plane; (b)

failure mode interaction in the transverse-shear energy plane.......................................... 64

Figure 3.6. Transverse and shear responses for a unidirectional E-glass/MY750 epoxy

laminate (Soden et al, 1998) ............................................................................................. 67

Figure 3.7. Transverse and shear responses for a unidirectional E-glass/LY556 epoxy

laminate (Soden et al, 1998) ............................................................................................. 68

xx

Figure 3.8. Transverse and shear responses for a unidirectional T300/BSL914C

carbon/epoxy laminate (Soden et al, 1998)....................................................................... 69

Figure 3.9. Effect of shape factors on longitudinal-shear failure envelope for

unidirectional E-glass/LY556 epoxy laminate.................................................................. 71

Figure 3.10. Longitudinal-shear failure envelope for unidirectional E-glass/LY556 epoxy

laminate, computed using various shape factors............................................................... 72

Figure 3.11. The effect of LTA on the shape of longitudinal-transverse failure envelope

for unidirectional E-glass/LY556 epoxy laminate............................................................ 74

Figure 3.12. The effect of LTm on the shape of the longitudinal-transverse failure

envelope for unidirectional E-glass/LY556 epoxy laminate ............................................ 75

Figure 3.13. Comparison between the influence of LTA and LTm on the failure envelope

of unidirectional E-glass/LY556 epoxy laminate ............................................................. 76

Figure 3.14. The effect of μ on geometry of the transverse-shear failure envelope for

unidirectional E-glass/LY556 epoxy laminate.................................................................. 78

Figure 3.15. The effect of shape factors on the transverse-shear failure envelope for

unidirectional E-glass/LY556 epoxy laminate with 5.0=μ . The dashed line is the

failure envelope using 0=μ and 1== TS mm ............................................................ 79

Figure 3.16. Biaxial experimental data for unidirectional E-glass/MY750 epoxy laminate

under combined longitudinal and transverse stress .......................................................... 81

xxi

Figure 3.17. Comparison between numerical longitudinal-transverse failure envelopes

and experimental data. Predictions of Butalia and Wolfe (2002), using the original the

original strain energy based theory, is presented with dashed line................................... 82

Figure 3.18. Biaxial experimental data for unidirectional E-glass/LY556 epoxy laminate

under combined transverse and shear loading .................................................................. 83

Figure 3.19. Transverse-shear failure envelopes for unidirectional E-glass/LY556 epoxy

laminate versus experimental data .................................................................................... 84

Figure 3.20. Biaxial experimental data for unidirectional T300/914C epoxy laminate

under combined longitudinal and shear loading ............................................................... 86

Figure 3.21. Longitudinal-shear failure envelopes for unidirectional T300/914C

carbon/epoxy laminate compared to experimental data.................................................... 87

Figure 3.22. Linear unloading with and without residual strain ....................................... 88

Figure 3.23. Residual strain growth during deformation.................................................. 91

Figure 3.24. Upper bound for unloading reloading modulus............................................ 92

Figure 3.25. The effect of transverse unloading on shear response.................................. 94

Figure 4.1. In-plane shear stress-strain curves for unidirectional E-glass/epoxy material

system from torsion tests on unidirectional laminates and the back-calculated response of

multi-directional laminates (Kaddour et al., 2003). ........................................................ 108

Figure 4.2. Predicted response compared with experimental data for S]45[ o± E-

glass/MY750 epoxy laminate under 1/1 −=SR ............................................................. 110

xxii

Figure 4.3. The influence of the wall thickness on the strength of S]45[ o± E-

glass/MY750 epoxy laminate test tubes under SR=1/-1 (Kaddour et al, 2003) ............. 111

Figure 4.4. Predicted responses for S]45[ o± E-glass/MY750 epoxy laminate under

1/1 −=SR using various values for Sk and 0.1=SER ................................................. 113


1/1 −=SR using various values for SER with 10=Sk ................................................ 115

Figure 4.6. Corrected loading path to account for specimen bulging............................. 117

Figure 4.7. The effect of the second order deformations................................................ 118

Figure 4.8. Instability of the stiffness matrix in the numerical analysis occurs at a stress

level about 15% over the observed final failure ............................................................. 119

Figure 4.9. Experimental and numerical stress-strain curves of [ 45 ]S± o angle-ply

laminate made of E-glass/MY750 epoxy under biaxial tension of 1/1SR = . The

numerical analyses show the effect of tensile transverse degradation factor on material

behavior........................................................................................................................... 123

Figure 4.10. Numerical predictions with 0.95 /1SR = versus experimental data .......... 124

Figure 4.11. Predicted stress-strain curves for s]55[ o± E-glass/MY750 epoxy laminate

under hoop stress, showing the effect of cTk ................................................................... 127

Figure 4.12. Predicted hoop and axial strains versus hoop stress curves for s]55[ o± E-

glass/MY750 epoxy laminate under internal pressure, showing the effect of μ ........... 128

xxiii


under hoop stress, showing the effect of o-ring friction ................................................. 130


under hoop stress. Material parameters and longitudinal stiffness were adjusted......... 131

Figure 4.15. Numerical and experimental final failure envelopes in the axial-hoop plane

for [ 55 ]S± o E-glass/MY750 epoxy laminate. Numerical predictions made using three

values of 40, 30, and 20 for Sk to demonstrate its effect on the failure envelope.......... 134

Figure 4.16. Initial and final failure envelopes for [ 55 ]S± o E-glass/MY750 epoxy

laminate, showing the effect of μ . The dashed lines are initial and solid lines are final

failure envelopes ............................................................................................................. 135


laminate, showing good agreement between predictions and experimental results for the

selected set of model parameters .................................................................................... 136

Figure 4.18. Axial versus hoop stress initial and final failure envelopes for [ 85 ]S± o E-

glass/MY750 epoxy laminate ......................................................................................... 138

Figure 4.19. Stress-strain curves for [ 85 ]S± o E-glass/MY750 epoxy laminate under

various hoop to axial stress ratios ................................................................................... 140

Figure 4.20. Biaxial initial and final failure envelopes under combined axial and shear

stress for [90 / 30 / 90 ]S±o o o laminate made of E-glass/epoxy material .......................... 143

xxiv

Figure 4.21. Initial and final failure envelopes under combined axial and shear stress for

[90 / 30 / 90 ]S±o o o laminate made of E-glass/epoxy material, showing the effect of cTk and

Sk .................................................................................................................................... 144

Figure 4.22. Biaxial failure envelopes for [90 / 30 / 90 ]S±o o o E-glass/epoxy laminate

before and after 15% increase in the stiffness of o90 plies ............................................ 145

Figure 4.23. Biaxial initial and final failure envelopes for [90 / 30 / 90 ]S±o o o laminate

made of E-glass /epoxy material..................................................................................... 147

Figure 4.24. Experimental data for biaxial failure of S]0/45/90[ ooo ± composite tubes

made of AS4/3501-6 carbon/epoxy laminate under combined pressure and axial load. 150

Figure 4.25. Transverse and in-plane shear responses for a unidirectional AS4/3401-6

laminate (Soden et al., 1998) .......................................................................................... 151

Figure 4.26. Biaxial failure envelopes for S]0/45/90[ ooo ± composite tubes made of

AS4/3501-6 carbon/epoxy laminate under combined pressure and axial load............... 153

Figure 4.27. Stress-strain response for quasi-isotropic S]0/45/90[ ooo ± composite tubes

made of AS4/3501-6 carbon/epoxy laminate under a hoop to axial stress ratio of 20/1 155


made of AS4/3501-6 carbon/epoxy laminate under hoop to axial stress ratio of 2/1..... 157

Figure 4.29. Numerical and experimental stress-strain curves for S]55[ o± angle-ply

laminate made of E-glass/MY750 epoxy for hoop stress to axial stress ratio of 2/1...... 160

xxv

Figure 4.30. Stress-strain curve of cross-ply E-glass/MY750 epoxy ]90/0[ oo laminate

under axial load in the y direction................................................................................. 162

Figure 4.31. The predicted and experimental stress-strain curve for S]85[ o± laminate

made of E-glass/MY750 epoxy under uniaxial loading in the axial direction ............... 164

Figure 5.1. The geometry of the specimens.................................................................... 172

Figure 5.2. End fixtures ................................................................................................. 175

Figure 5.3. A carbon specimen after end fixtures were bonded .................................... 175

Figure 5.4. A glass specimen installed on the load frame ............................................. 176

Figure 5.5. Strain gage layout for G1 specimen ............................................................ 180

Figure 5.6. Axial and hoop stress-strain curves for S-glass/epoxy specimen G1. Shear

stress to axial stress ratio was 0.2/1.0 ............................................................................. 181

Figure 5.7. Shear stress versus shear strain curves for S-glass/epoxy specimen G1 tested

under shear stress to axial stress ratio of 0.2/1.0............................................................. 182

Figure 5.8. Strain gage layout for G1 specimen ............................................................ 183

Figure 5.9. Axial and hoop strains from different strain gages versus axial stress for the

S-glass/epoxy specimen G2 under shear stress to axial stress ratio of 0.2/1.0 ............... 184

Figure 5.10. Shear stress-strain curves for S-glass/epoxy specimen G2 under shear stress

to axial stress ratio of 0.2/1.0 .......................................................................................... 185

Figure 5.11. Comparison between the measured axial and hoop strains from G1 and G2

specimens........................................................................................................................ 186

xxvi

Figure 5.12. Comparison between the measured shear strains from G1 and G2 specimens

......................................................................................................................................... 187

Figure 5.13. Strain gage lay-out for G3 ......................................................................... 189


to axial stress ratio of 0.5/1.0 .......................................................................................... 190

Figure 5.15. Axial and hoop stress-strain curves for S-glass/epoxy specimen G3 under

shear stress to axial stress ratio of 0.5/1.0....................................................................... 191

Figure 5.16. Measured strains from three strain gages aligned at 45− o with respect to the

axial direction of S-glass/epoxy specimen G3. Shear to axial stress ratio was 0.5/1.0 . 192

Figure 5.17. Specimen G3 after failure (strain gage numbering in the picture is different

than the current numbering)............................................................................................ 193

Figure 5.18. Strain gage lay-out for G4 ......................................................................... 195

Figure 5.19. Axial stress versus axial and hoop strains for the S-glass/epoxy specimen

G4 under shear stress to axial stress ratio of 0.4/1.0....................................................... 196

Figure 5.20. Shear stress-strain curves for the S-glass/epoxy specimen G4 under shear

stress to axial stress ratio of 0.4/1.0 ................................................................................ 197

Figure 5.21. Failure mode of specimen G4.................................................................... 197

Figure 5.22. Strain gage lay-out for S-glass specimen G5 ............................................ 198

Figure 5.23. Shear stress-strain curves for S-glass/epoxy specimen G5 under shear to

axial stress ratio of 4/1 .................................................................................................... 199

xxvii

Figure 5.24. Axial and hoop strains of S-glass/epoxy specimen G5 loaded under shear to

axial stress ratio of 4/1 .................................................................................................... 200

Figure 5.25. The effect of shear stress to axial stress ratio (SR) on axial and hoop strain

versus axial stress curves for S-glass/epoxy laminate .................................................... 202

Figure 5.26. The effect of shear stress to axial stress ratio (SR) on shear response of

]30/90/90/30/30/90[ oooooo −−+ S-glass/epoxy laminate.......................................... 203

Figure 5.27. Strain gage lay-out for carbon/epoxy specimen C1 .................................. 205

Figure 5.28. Axial and hoop strains versus axial stress from different strain gages for

carbon/epoxy specimen C1, tested under SR = 0.26/1 ................................................... 206

Figure 5.29. Comparison between the axial strains measured at the middle and bottom of

the test section for specimen C1 ..................................................................................... 207

Figure 5.30. Measured shear strains from the two rosette gages versus axial stress for

carbon/epoxy specimen C1 tested under SR = 0.26/1 .................................................... 207

Figure 5.31. Failure surface for carbon/epoxy specimen C1 tested under SR = 0.26/1 208


Figure 5.33. Measured axial and hoop strains from different strain gages versus axial

stress for carbon/epoxy specimen C2, tested under SR = 0.16/1.................................... 211

Figure 5.34. Shear stress-strain curves from the two rosette gages for carbon/epoxy

specimen C2 tested under SR = 0.16/1 ........................................................................... 212

Figure 5.35. Carbon/epoxy specimen C2 after failure under the stress ratio of 0.16/1 . 212

xxviii


Figure 5.37. Hoop and axial strains from three strain gages versus axial stress for

carbon/epoxy specimen tested under stress ratio of 0.32/1............................................. 215

Figure 5.38. Experimental shear stress-strain curve for carbon/epoxy specimen C3 tested

under stress ratio of 0.32/1.............................................................................................. 216

Figure 5.39. Carbon/epoxy specimen C3 after failure under the stress ratio of 0.32/1 . 216


Figure 5.41. Measured hoop and axial strains from different gages versus axial stress for

carbon/epoxy specimen tested under shear to axial stress ratio of 0.02/1 ...................... 219

Figure 5.42. Shear stress-strain curve for carbon/epoxy specimen tested under stress ratio

of 0.02/1 .......................................................................................................................... 220

Figure 5.43. The loading path for carbon/epoxy specimen C5...................................... 222

Figure 5.44. Hoop and axial strains from different strain gages versus axial stress for

carbon/epoxy specimen C5 ............................................................................................. 223

Figure 5.45. Shear stress-strain responses for carbon/epoxy specimen C5 ................... 224

Figure 5.46. The effect of stress ratio on the hoop and axial strains of carbon/epoxy

specimens........................................................................................................................ 226

Figure 5.47. The effect of the stress ratio of the shear stress-strain responses of

carbon/epoxy specimen................................................................................................... 227

xxix

Figure 5.48. Comparison between the tuned numerical predictions and experimental data

for [90 / 30 / 30 / 90 / 90 / 30 ]− + +o o o o o o S-glass/epoxy laminate under shear to axial stress

ratio of 0.2/1.................................................................................................................... 231

Figure 5.49. Comparison between predicted and experimental shear stress-strain curve


ratio of 0.2/1.................................................................................................................... 233

Figure 5.50. Comparison between the predictions and experimental hoop and axial

strains verses axial stress curves for [90 / 30 / 30 / 90 / 90 / 30 ]− + +o o o o o o S-glass/epoxy

laminate under shear to axial stress ratio of 0.4/1........................................................... 235

Figure 5.51. Comparison between the predictions and experimental shear stress-strain

curve for [90 / 30 / 30 / 90 / 90 / 30 ]− + +o o o o o o S-glass/epoxy laminate under shear to axial

stress ratio of 0.4/1.......................................................................................................... 236



laminate under shear to axial stress ratio of 0.5/1........................................................... 238



stress ratio of 0.5/1.......................................................................................................... 239

xxx



laminate under shear to axial stress ratio of 4/1.............................................................. 241



stress ratio of 4/1............................................................................................................. 242

Figure 5.56. Comparison between the tuned predictions and experimental data for hoop

and axial responses of carbon/epoxy laminate under SR = 0.26/1 ................................. 245

Figure 5.57. The predicted and experimental shear stress-strain curves for carbon/epoxy

laminate under stress ratio of 0.26/1 ............................................................................... 246

Figure 5.58. Comparison between the predictions and experimental data for hoop and

axial responses of carbon/epoxy laminate under SR = 0.16/1 ........................................ 248


laminate under SR = 0.16/1 ............................................................................................ 249


axial responses of carbon/epoxy laminate under SR = 0.32/1 ........................................ 251


laminate under SR = 0.32/1 ............................................................................................ 252

Figure 5.62. Comparison between the predicted and measured axial and hoop strains for

carbon/epoxy laminate under shear to axial stress ratio of 0.02/1.................................. 254

xxxi

Figure 5.63. Comparison between the predicted and measured shear strains for

carbon/epoxy laminate under shear to axial stress ratio of 0.02/1.................................. 255


carbon/epoxy laminate under non-proportional loading presented in Figure 5.43......... 257

Figure 5.65. Comparison between the predicted and measured shear stress-strain curves

for carbon/epoxy laminate under non-proportional loading presented in Figure 5.43 ... 258

Figure A.1. Calculation of strain increment for a given stress increment ..................... 267

Figure C.1. Specimen bulging due to internal pressure ................................................. 278

xxxii

7 LIST OF SYMBOLS

LA : Lamina strain energy at uniaxial longitudinal failure

TA : Lamina strain energy at uniaxial transverse failure

SA : Lamina strain energy at in-plane shear failure

LTA : Longitudinal-transverse failure strain energy

b : The correction factor due to transverse stress

C : Lamina stiffness matrix

ijc : Elements of C

LE : Longitudinal tangent modulus (fiber direction)

TE : Transverse tangent modulus

dTE : Reduced transverse tangent moduli

fE : Elastic modulus of fibers

mE : Matrix modulus

Lε : Longitudinal strain of lamina

Tε : Transverse strain of lamina

uLε : Uniaxial failure strain of lamina in longitudinal direction (tensile or compressive)

utLε : Uniaxial tensile failure strain of lamina in longitudinal direction

xxxiii

ucTε : Uniaxial compressive failure strain of lamina in transverse direction

ucLε : Uniaxial compressive failure strain of lamina in longitudinal direction (positive)

RLε : Residual strain of lamina after complete unloading in the longitudinal direction

uTε : Uniaxial failure strain of lamina in transverse direction (tensile or compressive)

RTε : Residual transverse strain of lamina after unloading

YTε : Transverse stain at the start of transverse stiffness reduction

FFI : Fiber failure index

G : Inplane tangent shear modulus

dG : Reduced inplane tangent shear modulus

γ : In-plane engineering shear strain of lamina

uγ : In-plane engineering failure shear strain of lamina

Rγ : Residual shear strain of lamina after unloading

Yγ : Shear strain at the start of shear stiffness reduction

I : Identity matrix

LTν , TLν : Lamina Poisson’s ratios in material directions

Sk : Inplane shear stiffness reduction factor

cTk : Transverse compressive stiffness reduction factor

tTk : Transverse tensile stiffness reduction factor

xxxiv

MFI : Matrix failure index

Lm : Longitudinal shape factor

Tm : Transverse shape factor

Sm : Shear shape factor

μ : Coefficient of internal friction between the fibers

LΠ : Longitudinal strain energy

TΠ : Transverse strain energy

SΠ : In-plane shear strain energy

S : Lamina compliance matrix

ijs : Elements of the compliance matrix

SER : Shear strain energy ratio at which shear stiffness reduction starts

SSER : Shear strain energy ratio

Lσ : Longitudinal stress

Tσ : Transverse stress

uLσ : Unidirectional longitudinal strength of lamina (tensile or compressive)

uTσ : Unidirectional transverse strength of lamina (tensile or compressive)

yx σσ , : The axial and hoop stresses in a composite tube

TER : Transverse strain energy ratio at which transverse stiffness reduction starts

xxxv

τ : In-plane shear stress (longitudinal-transverse plane)

0uτ : In-plane shear strength of lamina under zero transverse stress

uτ : In-plane shear strength of lamina

LU : Unloading reloading modulus in the longitudinal direction

TU : Unloading reloading modulus in the transverse direction

SU : Shear unloading reloading modulus

ξ : Curve parameter

Superscripts

t : Tensile

c : Compressive

u : Failure strength

Y : Initiation of stiffness reduction

1

CHAPTER 1

1 INTRODUCTION

The increasing need for advanced engineering materials such as fibrous composites in

many industries within the recent decades has made it indispensable to develop suitable

mathematical tools to predict the mechanical response of the new engineering materials.

Such mathematical models should be able to predict the strength of the material with any

geometry and loading condition. A composite laminate can fail in various failure modes,

each governed by a different failure mechanism. Additionally, environmental conditions

such as exposure to moisture and elevated temperatures can influence their mechanical

response and failure strength. These factors render challenges in predicting their

behavior. Several approaches, such as phenomenological, micromechanical, and fracture

mechanics approaches are used by the researchers to predicting mechanical response and

strength of composite laminates. A broad literature review is presented in Chapter 2 to

discuss these approaches.

A state-of-the-art World Wide Failure Exercise (WWFE) launched by Hinton and Soden

(1998) and Soden, Hinton, and Kaddour (1998) to evaluate the performance of selected

2

failure models by judging their predictions against reliable experimental data. The cases

studied included a wide range of laminate lay-ups and loading conditions (Soden et al.,

1998). The strain-energy based model proposed by Sandhu (1974) at Wright-Patterson

Air Force Based, Ohio, was among the failure models invited to the competition. The

current work, which is the continuation of the previous work accomplished by this author

(Zand, 2004), is aimed at improving this failure model.

In Chapter 3 a strain-energy based failure theory is developed for unidirectional fibrous

laminates. This failure theory complies with energy conservation principle for elastic and

inelastic material. The failure theory is extended to include unloading-reloading cycles

and also to 3-dimensional stress conditions. In Chapter 4 the theory is extended to

multidirectional laminates. Classical lamination theory (Reddy, 2003) is employed to

proceed from a unidirectional laminate to any multidirectional lay-up. A failure mode

dependent post initial failure model is developed to predict failure progression beyond the

initial failure. In Chapter 5 an experimental program, developed to produce new set of

experimental data is presented. Ten tubular specimens, five S-glass and five carbon

reinforced polymer, are tested under combined axial load and torsion, using material

testing facilities at the Civil Engineering Department of the Ohio State University.

Model predictions are compared to the experimental data to validate the proposed model.

Presented in Chapter 6 are summary of the work, and proposed future developments.

3

CHAPTER 2

2 LITERATURE REVIEW

2.1 Introduction

One challenge to fully exploiting the potential of advanced engineering materials such as

composites is our ability to predict their mechanical response. In this chapter, a review of

the literature of failure theories for fibrous composite laminates is conducted. The focus

of this effort is to overview the existing approaches and categorize them without getting

into details of each theory. Comprehensive reviews on this topic have already been

published by a number of researchers including Rowlands (1985), Soni and Pagano

(1987), Beaumont (1989), Kyriakides et al. (1995), Echaabi et al. (1996), Okoli and

Abdul-Latif (2002) that can be referred to for further information. While most of the

models are applicable to three dimensional loading conditions, this study is limited to in-

plane loadings.

4

2.2 Composite Laminates

A composite lamina is the basic structural component of a composite laminate. A lamina

is made of an array of fibers embedded in an adhesive matrix. The array of fibers can be

unidirectional, woven, or even three dimensional. The strength of a composite lamina is

highly directional dependent. The focus of the study in this work is multidirectional

fibrous laminates made by stacking several unidirectional laminae in different

orientations.

2.2.1 Fibers

Natural fibers such as straw and sisal were the earliest materials used by human beings

for reinforcement of mud, brick, etc. At this time, a wide variety of fiber types are

available for composite laminates, including natural fibers, synthetic organic fibers, and

synthetic non-organic fibers (Rosen and Norris, 1988). Examples of synthetic organic

fibers are nylon, polyester, polypropylene, and aramids. Fibers typically possess a linear

stress-strain curve with a brittle failure (Rosen and Norris, 1988). Their major physical

characteristics are low unit weight, low stiffness, and high strength. They have limited

industrial application due to their low stiffness. Synthetic non-organic fibers typically

have high strength, high stiffness and relatively low cost, which make them favorable for

many industries such as automotive and aerospace. The most common fiber types of this

category are glass fibers and carbon fibers. Glass fibers exhibit good resistance against

5

harsh environmental conditions and they have a low cost, while carbon fibers have

excellent strength and thermal properties, but are more expensive.

2.2.2 Matrix

Typical matrix materials are metals, ceramics, and polymers. Metallic matrices are

characterized by their high strength and relatively high ductility. Ceramic matrices have

high stiffness (generally stiffer then metals) and moderate strength. Polymeric materials

on the other hand have low strength and stiffness, but high ductility and low density.

Polymers are the most commonly used matrix materials, because compared to metal and

ceramic matrix composites, polymer composites are less costly to manufacture (Rosen

and Norris, 1988).

2.2.3 Failure of Composite Laminates

A unidirectional fibrous laminate may fail due to fiber rupture, matrix cracking, or

delamination. For a typical fiber based composite under vast majority of loading

conditions, the stress level at which matrix cracking initiates is lower than the ultimate

strength due to fiber rupture. An embedded lamina with a cracked matrix can usually

sustain additional load in the fiber direction. Numerical simulations as well as

experimental observations have shown (Gosse, 2001) that free plastic flow cannot take

place in a reinforced matrix, thus matrix failure in a unidirectional system is relatively

6

brittle. During failure of a multidirectional laminate, plies with different orientation

typically fail at different stress levels. The lowest stress level at which some of the plies

fail (usually matrix failure) is called initial failure and the sequence of failures that occur

between the initial and ultimate failures are called intermediate failures.

As a result of such a complicated failure procedure, the failure surface of multi-

directional laminates typically exhibits a complex geometry in stress space. The strength

of laminates can be influenced by various factors such as history and direction of applied

loads, environmental condition (temperature, pressure, humidity, etc), and residual

thermal stresses induced during the fabrication and curing processes. The choice of an

appropriate failure model depends on the application and circumstances in which the

composite structure is to function. Echaabi et al. (1996) stated that no failure model had

been proposed by 1996 that could take all these factors into account.

2.3 Failure theories

The literature contains two major approaches for failure prediction of composite

laminates: macroscale and microscale approaches. The first approach treats a composite

lamina as a continuous homogeneous orthotropic medium. The focus is to find a

mathematical model to predict the instantaneous moduli and failures of each lamina on

either an empirical or a mechanistic basis. Classical lamination theory (Reddy, 2003) is

7

typically used to proceed from lamina to a multidirectional laminate. The empirical

approach may or may not account for some associated physical phenomena such as the

fibrous structure of the material and variety of failure modes. The micromechanical

approach takes into account the heterogeneous structure of a laminate on a microscopic

scale. However, the output of such a model is the mechanical response of the laminate on

a macroscopic scale. A developing approach is nano-scale modeling based on molecular

dynamics. In this approach computational models are built to predict material thermo-

mechanical response by studying interaction between the molecules (Gates et al., 2005

and Buryachenko et al., 2005). Such models are outside the scope of this work and will

not be reviewed.

Since conducting enough tests to capture all possible loading and lay-up configurations is

too cost prohibitive and impractical, the experimental approach is primarily a method of

validation for failure theories, and thus it is not considered as an independent approach.

2.3.1 Phenomenological Theories

Phenomenological theories are those theories that neglect the inhomogeneous structure of

composites by employing a non-mechanistic based failure criterion. These models can be

divided into two categories. The first category includes those theories that neglect

interaction between transverse, longitudinal, and shear deformations under multiaxial

loadings. That is, failure in any of the material directions (longitudinal, transverse, or

8

shear) is assumed to be dependent on the stress and/or strain in that direction only. These

models can readily determine the failure mode and are simple, but they neglect the

interaction between different directions (Hart-Smith, 1993). In the second category are

those failure theories that consider the interaction between longitudinal, transverse, and

shear failure modes. The phenomenological theories are unable distinguish between the

different failure modes on a physical basis. Mathematically, phenomenological models

are suitable hyper-surfaces that interpolate or extrapolate experimental data. In 1979

Cowin presented a review on historical development of intractive phenomenological

models from Hankinson (1921) to Tsai and Wu (1971).

2.3.1.1 Non-interactive failure theories

Well known examples of non-interactive failure theories are maximum stress and

maximum strain theories. The model parameters are evaluated through uniaxial strength

tests, conducted on unidirectional laminates. The maximum stress and maximum strain

failure criteria are the simplest and probably the most commonly used failure theories

(Echaabi et al., 1996). The popularity of these theories is mainly due to their simplicity

rather than their accuracy or rationality (Rowland, 1985). Jenkins (1920) was the first to

extend the ‘maximum normal of principal stresses theory’ from isotropic to orthotropic

material. More information on applicability and validity of these models can be found in

the work of Petit and Waddoups (1969), Hart-Smith (1991, 1992, 1998a, 1998b),

Toombes et al. (1985), Swanson et al. (1986, 1987, and 1992), and Echaabi et al., (1996).

9

2.3.1.2 Interactive non-mechanistic failure theories

An interactive failure criterion is a mathematical function of selected state variables to

interpolate between the experimental data points, with no or minor respect for the

physical mechanism of failure. Historically, Hankinson (1921) was among the pioneers

who proposed a failure criterion for orthotropic material to predict failure of wood.

Eventually, more advanced theories were developed in the literature to predict failure of

composite laminates, such as polynomial and tensorial criteria. Polynomial criteria are a

hyper-surface expressed by a quadratic or higher order relationship between the selected

state variables (usually stresses) at failure. The polynomial coefficients are determined

by regressing the function to experimental data. Theoretically, the accuracy of such a

model can be increased by increasing the number of polynomial constants. However,

having too many constants is not desirable due to the difficulty of evaluating them.

Polynomial criteria are a special case of the more general category of tensorial criteria.

Tensorial criteria were first proposed by Gol’denblat and Kopnov (1965) and used by

Ashkenazi (1965) and Malmeister (1985) for failure predictions of composite laminates.

Furthermore, the well-known Tsai-Wu quadratic criteria are special forms of tensorial

criteria (Liu and Tsai, 1998, Tsai and Wu, 1971). Many authors such as Marin (1957),

Malmeister (1966), Chamis (1967), Fischer (1967), Hoffman (1967), Tsai and Wu

(1971), and Cowin (1979) proposed various forms of quadratic polynomial criteria,

which are different in longitudinal-transverse interaction coefficient. Cubical forms of

polynomial criteria were developed by Tennyson et al. (1978, 1985) and Jiang and

10

Tennyson (1989). The advantage of tensorial form over polynomial form is that the

former is invariant under rotation of coordinate axis. Tensorial and polynomial criterion

can also be written in the strain space, instead of stress space. However, stress based

formulations have been proven to provide a better agreement to experimental data

(Echaabi et al., 1996).

The most commonly used failure model of this type is the quadratic Tsai-Wu failure

model. A study conducted by Hinton, Kaddour, and Soden (2002a and 2002b) showed

good performance of this model compared to other models available. However, accuracy

of the biaxial and multiaxial predictions may vary from one loading combination to

another. For example, fitting experimental data in one quadrant of the coordinate system

may lead to unrealistic predictions in other quadrants (Echaabi et al, 1996). A piecewise

quadratic failure criteria was proposed by Yeh and Kim (1994), which can lead to similar

degrees of accuracy in all quadrants. Griffith and Baldwin (1962) offered a

phenomenological failure model that is an extension to orthotropic material of the

VonMises failure criterion (maximum distortional energy). This theory had been

developed for material with identical compressive and tensile strengths, and needed only

one reference strength to identify the yield surface. Other VonMises type theories such

as developed by Marin (1957), Ashkenazi (1965), and Chamis (1976) possess the general

form of VonMises theory and reduce to it for isotropic materials. Unlike the above

mentioned theories, in Griffith’s failure theory the yielding of the material is independent

of hydrostatic pressure (Rowland, 1985). Thus, this theory is suitable for ductile

orthotropic materials such as rolled metal sheets.

11

The strain-energy based failure theory suggested by Sandhu (1974) is another

phenomenological model developed to predict failure of composite laminates. In the

previously mentioned models the failure of the material is expressed as a function of

either stresses or strains. In the strain-energy based model, on the other hand, the failure

surface is expressed in terms of the deformation energies which are defined by the entire

stress-strain curve. This characteristic is useful when dealing with nonlinear material

(Wolfe and Butalia, 1998). The strain-energy based failure theory is similar to another

nonlinear model had been developed by Petit and Waddoups in 1969. The strain energy

based failure theory is the subject of interest in this work and a detailed review of the

model will be subsequently presented.

Parametric failure theories are another category of phenomenological models first offered

for anisotropic metals by Budiansky (1984). Later, Neale and Labossiere (1989)

extended this criterion to composite laminates. In the parametric formulation, the stress

state at failure is expressed by a set of trigonometrical functions in the spherical

coordinate system. The capability of the model to reproduce a set of experimental data

depends on the number of terms in the expansion. The minimum number of experimental

data points needed to evaluate the input parameters is twice the number of terms, and

they have to be obtained through multi-axial strength tests. As stated by Neals and

Labossiere (1989) model parameters can be evaluated by solving a linear system which is

an advantage compared to tensorial and polynomial forms, where a system of quadratic

or higher order equations must be solved. Figure 2.1 presents an example of a parametric

failure envelope in longitudinal-transverse plane.

12

Regression-based models disregard the fact that the failure process in fibrous composite

laminates has various modes, each being governed by a different mechanism. Over the

past several decades, there has been an increasing tendency among the researchers to

develop and utilize failure theories capable of distinguishing between the different failure

modes by having at least two distinct failure criteria for matrix and fiber (Hashin ,1980,

Hahn et al. 1982, and Hart-Smith, 1993). Such failure theories are called mechanistic

failure theories and are discussed in the next section.

Figure 2.1. Failure envelope for porous graphite obtained by parametric failure criterion

with five terms (Neals and Labossière, 1989; notations are changed).

13

2.3.2 Mechanistic Failure Theories

Mechanistic failure theories, recognize the fibrous structure of composite laminates to

distinguish failure modes. These failure theories include two or more different failure

rules, each representing a different failure mechanism (such as matrix cracking and fiber

rupture). The strength of the lamina is determined by the first reached criterion.

Hashin and Rotem (1973) and Rotem and Hashin (1975) suggested a mechanistic failure

theory, in which four separate failure criteria are combined to identify four distinct failure

modes. Hashin (1980) presented a methodology to determine the plane of failure in the

matrix. He proposed a quadratic stress criterion for matrix failure. The failure plane was

assumed to be parallel to the longitudinal direction of the lamina, which could be

uniquely specified by an angle. The failure plane was the one that corresponds to the

lowest safety factor. Sanders and Grand (1982) originated another failure mode

dependent model at British Aerospace (BAe) with three failure criteria to identify three

different failure modes, namely, initial failure, final failure, and delamination. Their

theory was later advanced by Edge (1989, 1998).

Puck and Schneider (1969) and Puck and Schurmann (1998, 2002) presented a failure

theory that included two failure criteria to distinguish between fiber and matrix failures.

The first criterion was a maximum fiber stress criterion to describe the longitudinal

failure of lamina due to fiber rupture. The inter-fiber failure of lamina (matrix cracking)

and the corresponding failure surface were predicted using a methodology similar to the

14

one proposed by Hill (1982). The inter-fiber strength of the material was assumed to

have a frictional nature. The post initial failure action in this model was assumed to be

failure mode dependent. Predictions presented by Puck and Schurmann (2002) showed

good agreement with experimental data for a wide range of material types, loading

conditions, and lay-ups (Hinton, M.J., Kaddour, A.S. and Soden, 2002). Cuntze and

Freund (2004a and 2004b) developed a failure theory which was similar to Puck’s theory.

Rotem and Hashin (1975) and Rotem and Nelson (1981) presented another mechanistic

theory after modifying the model presented by Hashin and Rotem (1973). The failure

model consisted of a fiber failure and a matrix failure criterion. Fiber failure was

assumed to occur due to longitudinal stress only and a quadratic polynomial function was

suggested to predict matrix failure. Later, Christensen (1988) offered a similar failure

model with separate failure criteria for matrix and fiber. Based upon finite elasticity

concepts, Christensen made an analogy between his suggested strength parameters and

those of the Tsai-Wu model.

Other examples of mechanistic theories are the ones proposed by Zinoviev et al. (1998,

2002) and Bogetti et al. (2004a and 2004b). Zinoviev et al. employed a maximum stress

failure criterion together with a failure mode dependent post initial failure constitutive

model. Their proposed approach can identify six different cases based on the failure

mode and direction of the applied stresses to take the appropriate post initial failure

action. Bogetti et al. developed a three-dimentional maximum strain failure criterion and

a nonlinear constitutive model. Their model can identify several different failure modes,

including through the thickness failure, and select the post initial failure action

15

accordingly. Although, both the above mentioned models utilize a simple non-interactive

failure criterion they are categorized under mechanistic theories because of their well-

structured post initial failure models. The literature contains several other mechanistic

models that are less commonly used such as proposed by Christensen (1997) and Feng

(1991). Echaabi and Francois (1997) presented a comparison between some of the

mechanistic theories.

2.3.3 Damage Based Models

Damage based models are extensions to orthotropic material of classical damage

mechanics (Kachonov, 1985). The damage based theories quantify the extent of damage

on an average sense in a strained material. For the one-dimensional problem, the

effective cross-sectional area of damaged material is assumed to be:

)1( dAAeffective −=

Where, d , which varies from zero for intact material to one for completely disintegrated

material, represents the amount of damage. For a three dimensional problem the damage

parameters are directionally dependent because even isotropic materials can become

anisotropic after some damage. Damage is expressed at a point in an anisotropic material

using tensor notation and tensorial functions. The work of Betten (1992) contains details

on the application of tensor functions in continuum damage mechanics. The constitutive

16

rule for the undamaged (effective) cross-section of material is typically assumed to be the

same as that of intact material, while the stress is computed with respect to the actual

cross section. The damage tensor is related to the elastic strain tensor through a

constitutive model. Hayakawa and Murakami (1997) investigated experimentally the

existence of a damage surface and a corresponding normality rule. Zhao and Yu (2000)

developed a damage based theory for orthotropic composites with ellipsoidal inclusions.

Special cases of the model were presented for materials with circular voids, needle voids,

and cracks by assigning zero stiffness to the inclusions. This model included the effect of

material micro-structure to describe the stress-strain behavior at a macroscale. Recent

examples of a damage based models for composite laminates are those proposed by

Iannucci and Ankersen (2006) and Paepegem (2006b).

2.3.4 Micromechanical Models

2.3.4.1 Fracture mechanics based models

Fracture mechanics is the study of the initiation and propagation of micro-cracks due to

thermo-mechanical loads. Material micro-cracking processes initiate from a local

imperfection, in the fiber, matrix, or at their interface, and eventually propagate to form

larger cracks (Berthelot, 1999). Microcracking begins well before any detectible change

to the macro-scale properties of the laminate.

17

During the failure procedure, as the density of microcracks increases, they join together

to form larger discontinuities, known as macrocracks. The typical elementary cracking

mechanisms are fiber fracture, transverse matrix fracture, longitudinal matrix fracture,

shear matrix fracture, and fiber-matrix interface debonding. These mechanisms are

illustrated in Figure 2.2. Final failure of the laminate can occur due to the propagation

and accumulation of various types of cracks. Crack propagation in any material has five

stages as (Gotsis et al., 1998a): 1- initiation, 2- growth, 3- accumulation, 4- stable

propagation, and 5- unstable propagation and collapse. After the third stage, the effect of

damage propagation is reflected in the global structural response of the laminate in

various manners, such as decrease in the stiffness and reduction of natural frequency or

buckling resistance.

Fracture mechanics based models examine the stress distribution in the vicinity of pre-

existing microcrack tips and other imperfections such as fiber breakage, matrix fiber

debonding, etc. A simple method of idealization is to replace the lamina by a

homogeneous orthotropic material (Sih and Ogawa, 1982; Parhizgar et al, 1982). More

advanced theories can account for additional effects such as the spatial distribution of the

microscopic defects, bridging effect of fibers, and interaction between the cracks. Some

examples of such models can be found in the work of Waddoups et al. (1971), Mileiko et

al. (1982), Whitney (1988), Muju at al. (1998), and Oguni and Ravichandran (1999 and

2001). A recent example of fiber bridging in laminates was presented by Huang (2004a

and 2004b).

18

Figure 2.2. Different fracture mechanisms of the matrix. (a) Transverse failure of the

matrix; (b) shear fracture of the matrix; (c) debonding of fiber and matrix; (d)

longitudinal fracture of the matrix (Berthelot, 1999).

2.3.4.2 Crack density based models

Crack density based models can be considered as the interface between the

micromechanics and macromechanics failure theories (McCartney, 2002). As mentioned

in the previous section, prediction of stress distribution in a cracked material is the first

step towards a microscale analysis. Hedgepeth’s (1961) shear lag model (SLM) was one

of the earliest techniques to study the stress distribution at around a crack tip in a fibrous

material. Later, more variations of the shear lag theory were proposed to take into

account additional factors such as material nonlinearity, fiber-matrix debonding,

19

contribution of the matrix stiffness along the fiber direction, yielding, and splitting of the

matrix. Recent developments in shear lag models can be observed in the works of Landis

et al. (2000), Huang (2002), He et al. (2003), Xia, et al. (2002), Banks-Sills et al. (2003),

and Roberts et al. (2003). Further information on the historical development of shear lag

models can be found in the review paper of Rossettos and Godfrey (1998). In addition,

Okabe, et al. (2001) and Okabe and Takeda (2002) introduced the statistical concepts of

the spatial crack distribution to advance the theory.

The shear-lag theory is a subcategory of a more general category, called crack density

based models. These models predict the crack density versus applied load, using a stress

transfer function to satisfy both the equilibrium and the stress boundary conditions

(McCartney, 2002). Once a transfer function is established, the solution of the problem

can be obtained by minimization of the complementary energy of the system. Such a

solution relates material moduli to its crack density. The solution, in general, does not

satisfy continuity equations, and thus results in a lower-bound estimation. Hashin (1987)

and Nairn (1989) developed such solutions for S]90/0[ oo laminates. Nairn and Hu

(1992) established similar solutions for cross-ply laminates. Moreover, Nairn (1995)

presented a displacement-based upper bound solution to the problem.

McCartney (1998) developed another analytical method that predicts the stress

distribution as well as the displacement field for cross-ply laminates. Later on, in 2002,

he extended his model to predict the response and failure of angle-ply and [0º/90º/±45º]

20

quasi isotropic laminates under biaxial loading. He incorporated a maximum fiber strain

failure criterion into his model to predict material failure.

2.3.4.3 Microbuckling and kinking theory

Experimental observations have shown that the compressive strength of fibrous

composites is considerably lower then their tensile strength, Kyriakides et al. (1995).

Microbuckling and kinking theories have been proposed to describe this difference.

Microbuckling refers to the buckling of the fibers due to the lack of lateral support, and

kinking refers to the compressive instability caused by misalignment or buckling of the

fibers. The compressive failure typically starts with microbuckling that leads to kinking

failure. The literature contains many studies on the microbuckling and kinking of fibers,

including experimental, numerical, and theoretical investigations. A recent example is

the model presented by Yerramalli and Waas (2003) to predict the failure of fibrous

composite plates under a combination of compression and shear stresses. This theory

was validated by comparing its predictions against experimental data for glass and carbon

fiber reinforced polymer composites. Yerramalli and Waas (2003), and Niu and Talreja

(2000) have presented brief reviews on this topic, which can be referred to for further

information. Schultheisz and Waas (1996a and 1996b) published state-of-the-art reviews

on this topic. Longitudinal splitting is another failure mode that can occur under

compressive stress. Besides the fibrous material, this failure mode is observed in other

brittle materials, such as rocks and some ceramics. The compressive splitting in fibrous

composite laminates was first reported by Piggott (1981).

21

2.3.4.4 Finite element analysis

Due to their complex nature the vast majority of micromechanical models cannot be

solved analytically. Numerical methods offer flexible, cost effective means for solving

these models. In this respect the finite element approach is a numerical technique for

solving a failure model, rather than being an independent approach. One example of such

analysis is the work of Gosse (2001), in which finite element analysis was employed to

distribute applied loads between matrix and fibers. A strain invariant failure criterion

was introduced to check matrix failure, and the maximum shear strain failure criterion,

suggested by Hart-Smith (2001), was utilized to check fiber failure. Some recent

developments in finite element based models were presented by Huang (2002) and Xia

(2002).

Mayesa and Hansenb (2004a and 2004b) developed a non-linear finite element based

model to predict failure progression. In their approach, known as muli-continuum theory

(MCT), a representative cell is analyzed using finite element method to compute micro-

scale stress and strain distributions as well as macroscale composite moduli of the cell.

This model takes into account the nonhomogeneous structure of the material. Multi-scale

analysis is another developing approach to study a composite structure. The aim of this

method is to predict initiation and accumulation of damage at a microscopic level and its

influence on the global behavior of the structure. Having experimental data, a ‘top-down

trace’ analysis from macroscale (laminate) down to the microscale (fibers and matrix) is

employed to estimate material properties. The calculated material properties are used to

22

analyze a composite structure with any given lay-up and geometry. This process is called

‘up-ward integrated’ (Gotsis et al. 1998b). At each level, a different tool may be used to

conduct the analysis. At microscopic level, a finite element analysis or an analytical

method can be used. The classical lamination theory is typically used to advance from a

single ply to a multidirectional laminate. Finally, a macroscale structural analysis is

conducted, using finite element methods, to predict the global behavior of a composite

structure.

2.3.4.5 Rule of mixtures

In this method the overall thermo-mechanical properties of a multi-phase composite is

estimated using the properties of the ingredients as well as their arrangement. A simple

application of rule of mixtures can be seen below, where the longitudinal modulus of a

fibrous composite is expressed as a function of fiber and matrix moduli and their volume

fractions (Reddy, 2003):

(1 )L f f f mE V E V E= ⋅ + − ⋅

In this example fE and mE are the moduli of fiber and matrix, respectively, and fV is

fiber volume fraction. LE is the overall longitudinal modulus of the composite. To

derive the above formula it was assumed that fiber and matrix strains were equal in the

longitudinal direction. Several variations of rule of mixture approximation have been

proposed. Suresh and Mortensen (1998) reviewed these methods. A rule of mixtures

23

approximation combined with a failure criterion for each ingredient can be used to

compute the instantaneous moduli in a composite material and predict its progressive

failure. The failure model proposed by Huang (2004a and 2004b) is one example of such

models.

2.3.4.6 Method of cells

Method of cells (MOC) is a micromechanical theory that predicts mechanical response of

a unidirectional laminate, regarding volume fraction and mechanical properties of

constituents. Aboudi (1987 and 1989) developed a set of constitutive equations for a

composite lamina by dividing the material up into cubical cells, each consisting of a few

(usually two or four) subcells. Each subcell was assumed to be a linear elastic element

with linear shape functions. The closed form solution was obtained by satisfying

equilibrium equations together with continuity of displacements and tractions at

interfaces between adjacent subcells. The continuity of the stresses and deformations

was satisfied in an average sense. The outcomes of the model included effective moduli,

strength, fatigue strength, and the thermal expansion coefficient on macroscopic scale.

The model was generalized by Paley and Aboudi (1992) and Pahr and Arnold (2002) to

include curing residual thermal stresses, and material non-linearity.

24

2.4 Experimental Methods

Advanced test methods have been developed to measure strength and other physical,

mechanical, environmental, and chemical properties of composite materials. Some of

these test methods have been standardized and are listed in ASTM D 4762-04. In the

subsequent sections some, but not all, of these standard test methods will be reviewed.

Available testing methods include uniaxial, bending, and multiaxial tests. Flat,

cylindrical, three point, or four point beam specimens are used in the tests. The drawback

of flat and beam shaped samples is stress singularities that occur at free edges of the

multidirectional laminates. Ting and Chou (1982) and Yin (1999) developed elasticity-

based solutions for assessment of stress concentration near a free edge. Cole et al (1974)

compared advantages and drawbacks of flat and cylindrical samples for biaxial strength

test.

2.4.1 Unidirectional Strength Testing

Four types of tests are commonly used to obtain unidirectional strengths of composite

laminates, namely unidirectional tension and compression tests, shear test, and bending

test. The test specimen may posses a unidirectional or multidirectional lay-up. The

measured strength of anisotropic material is sensitive to specimen configuration (Adsit,

1988).

25

2.4.1.1 Unidirectional tension test

Flat specimens are commonly used for this test. The standard test method for flat

samples of high modulus fiber polymer matrix composites is described in ASTM D 3039.

This standard recommends straight sided long strip specimens with or without end tabs.

End tabs are strongly recommended for unidirectional, or unidirectionally dominated

composites to prevent any breakage close to grips. When a structure is to be made using

the filament winding method, filament wound test specimens are better representatives of

material properties than other type of specimens such as prepreg (Adsit, 1988). Standard

testing method for transverse tensile strength of cylindrical specimens under internal

pressure is furnished in ASTM D 5450.

2.4.1.2 Unidirectional compression test

ASTM D 3410 describes standard compression test method for a high-modulus fiber

composite with a polymer matrix. The test specimen is a straight sided strip of the

material with a rectangular cross-section with or without end tabs. In this test method,

the compressive stress is applied through the fixtures to the specimen by side friction

action. Since the results can be extensively influenced by the gripping method efforts

such as described by Adsin (1983) have been dedicated to minimize this effect by

developing standard fixtures. The test method described in ASTM D 3410 was originally

developed for unidirectional laminates, but it can be used for multidirectional laminates.

The length of the test section must be relatively short to minimize structural buckling.

However, a too short gage length is not desirable because it may not be statistically

26

representative of the material. The test results can be influenced by stress non-

uniformities along the specimen width due to the gripping effect.

2.4.1.3 Unidirectional shear test

The measured shear stress-strain curve of unidirectional laminates is sensitive to the

testing method (Swanson et al., 1985). There are five different standard test methods to

obtain shear properties and interlaminar shear resistance of composites., including

standard notched specimen, ±45° tensile (ASTM D 3518), 10+ o off-axis, double notched

(Iosipescu), and torsion of tubular sample (ASTM D 5448). Among these methods the

last two methods have led to more consistent results. Swanson et al. (1985) compared the

Iosipescu and torsion test methods and concluded that they both lead to similar outcomes.

Due to stress localization at free edges and notch tips that may occur in Iosipescu

specimens (Herakovich and Bergner, 1980) rotation test on cylindrical specimens usually

leads to more consistent results.

2.4.1.4 Bending test

Three or four point bending tests are used to measure bending strength and stiffness of

composite laminates. Conduction of a bending test on a beam specimen is relatively

easy, but interpretation of the data may need additional effort because compressive and

tensile moduli are different (Adsit, 1988), and stress localization at the free edges of the

specimen can influence the test results for multidirectional laminates. Three or four point

27

bending tests can also be used to measure interlaminar shear strength of composites

(Adsit, 1988).

2.4.2 Multi-directional strength testing

Multidirectional strength tests can be conducted on flat and cylindrical specimens. Thin-

walled tubular specimens under internal or external pressures, axial load and torsion are

the most commonly used multiaxial test method (Rowland, 1985, Whitney et al., 1973,

Guess, 1980, Ikegami et al., 1982, and Lee et al., 1999). Loading of tubular specimens

requires extreme care to ensure that the end grips do not cause non-uniform stress fields

in the test section. The axial load and torsion are imparted into the specimen through end

fixtures and the internal or external pressure is applied using a hydraulic system. The two

ends of the tubular specimens are typically reinforced using buildups to suppress

premature failure near the end fixtures where stress localizations are present. To provide

a relatively uniform stress distribution across the test section, the region between the test

section and the reinforced ends is tapered.

Toombes et al. (1985) and Swanson and Christoforou (1986, 1987, and 1988) developed

a testing method for biaxial testing of composite tubes under combined internal pressures

and compression, that was proven to be very effective (Cohen, 2002). This test method

was advanced by Cohen (2002) and used to produce experimental data under combined

axial, hoop, and shear stress. Recently advanced test devices have been developed to test

28

flat composite specimens under combined in-plane and out-of-plane multiaxial loading

conditions. For details see Hine et al. (2005) and Welsh et al. (2006).

2.5 The World Wide Failure Exercise

Hinton and Soden (1998) launched a competition, known as The World Wide Failure

Exercise (WWFE), to evaluate and compare predictive capabilities of selected failure

theories. All the participants were provided with the material properties of four

unidirectional material systems and were asked to make predictions for fourteen uniaxial

and biaxial loading cases for unidirectional and multidirectional lay-ups (Part A). The

cases were selected to cover a wide range of materials and lay-ups (Soden et al., 1998).

Part A papers were published in a special issue of Composites Science and Technology

journal (Vol. 58, No. 7, 1998). After all Part A papers were submitted, the experimental

data were disclosed to the participants (Soden et al., 2002) and they were asked to

evaluate their predictions and, if needed, tune their models. This was called Part B of the

exercise (Hinton et al., 2002a). The failure theories were ranked on the basis of their

performance (Hinton et al., 2002b). Part B papers and the experimental data were

released in 2002 in another special issue of Composites Science and Technology (Vol 62,

No 12-13).

29

In 2004 the exercise was extended to Part C to accommodate more theories in the

competition (Kaddour et al., 2004). Part C predictive cases and experimental data were

the same as those used in Parts A and B, and the new theories were ranked in the same

manner (Hinton et al., 2004). Table 2.1 shows the failure theories that participated in the

failure exercise (Kaddour et al., 2004), and Table 2.2 presents the overall ranking of the

theories (Hinton et al., 2004). As can be seen in the tables some of the participants

presented more than one theory.

30

Contributor Analysis type

Thermal stress Micromechanics Failure criteria

Chamis Linear Yes (a) Yes Micromechanics based Eckold Linear No No BS4994 (British design code) Edge Nonlinear Yes No Grant-Sanders model

Hart-Smith Linear (b) No Yes Max. strain, generalized Tresca, and 10% rule

McCartney Linear Yes No Fracture mechanics Puck Nonlinear Yes (c) Yes Puck’s model Rotem Nonlinear Yes Yes Rotem model Sun Linear Yes No Rotem-Hashin model

Sun Nonlinear Yes No Plasticity theory based on Hill’s model

Tsai Linear Yes (d) Yes Tsai-Wu quadratic criterion Wolfe Nonlinear No No SEB failure model Zinoviev Linear No No Max. stress criterion Bogetti Nonlinear No No Max. strain criterion Mayes Nonlinear (a) Yes Multi-continuum theory Cuntze Nonlinear Yes Yes Failure mode theory Huang Nonlinear Yes (a)(e) Yes Bridging model (a) Not in all cases (b) Occasionally initial moduli were replaced by secant moduli (c) Only part of thermal residual strain was considered (d) Tsai assumed a certain amount of moisture existed in the laminates to cancel the thermal stress (e) Huang attempted to account for the microthermal stresses generated in the laminate

Table 2.1. The failure theories participated in the WWFE (Kaddour et al., 2004)

31

Rank Paper

1 Cuntze – B

2 Zinoviev

3 Bogetti

4 Puck

5 Cuntze

6 Tsai – Part B

7 Mayes – B

8 Wolfe – Part B

9 Edge – Part B

10 Tsai – Part A

11 Sun(Linear)

12 Edge – Part A

13 Huang – B

14 Huang

15 Mayes

16 Wolfe – Part A

17 Hart-Smith (3)

18 Chamis (2)

19 Rotem

20 Hart-Smith (1)

21 Hart-Smith (2)

22 McCartney - Part B

23 Sun (Non-Linear)

24 Eckold

25 McCartney

Table 2.2. Quantitative ranking of the participated theories (Hinton et al., 2004)

32

2.6 Closing Remarks

In this chapter the literature was reviewed for the failure models pertinent to fibrous

composite materials. Figure 2.3 presents a classification of the failure theories reviewed

in this chapter. Macroscale models replace each lamina with a homogeneous orthotropic

material and attempt to simulate mechanical behavior on an average sense.

Phenomenological models employ an interpolating function, usually in stress space, to

describe the failure surface of a lamina. The mechanistic models attempt to identify the

failure mode on a physical basis and take a post-initial failure action based on the failure

mode. Micromechanical based models study the microstructure of the lamina and

interactions between its constituents to evaluate stress distribution in the material. Most

of these models can take into account residual thermal stresses induced during curing due

to the difference between thermal expansion coefficients of fiber and matrix.

All of the presented theories had limited success in predicting the behavior of fibrous

composites, because the failure processes in composite materials are complex in nature.

In spite of their high potential, the micromechanical models have not been advanced

enough to be used for practical applications. Macroscale theories are more time efficient

compare to microscale approach, but they face challenges in prediction of the post initial

failure response of matrix dominated lay-ups. In order to validate failure theories,

researchers have developed experimental methods to measure uniaxial and multiaxial

strengths of composite laminates. Currently, advanced experimental techniques and

33

equipment are available for multiaxial testing of tubular and flat specimens. In spite of

this, the literature contains limited published biaxial and multiaxial experimental data.

34

Figure 2.3. Classification of failure theories for composite laminates

Failure Theories

Macro-scale approach

Micro-scale approach

Phenomenological Theories

Mechanistic theories

Non-interactive theories

Interactive theories

Damage based theories

Fracture mechanicsbased theories

Micro-buckling and kinking theories

Rule of Mixtures

Micro and multi-scale finite element

analysis

Crack density based theories

Method of cells

35

CHAPTER 3

3 STRAIN ENERGY BASED MODEL

3.1 Introduction

The strain energy based failure theory originally developed by Sandhu, (1974 and 1976)

was modified and extended by Wolfe and Butalia (1998), and Butalia and Wolfe (2002)

as a part of the World Wide Failure Exercise (WWFE) organized by Hinton and Soden

(1998) and Hinton et al. (2002a). The performance of the strain energy based theory was

generally good compared to the other participating theories (Hinton at al. 2002b) but the

presence of discontinuities in the failure envelope for certain loading conditions provided

a motive to undertaking the current work, in which a modified strain energy based failure

theory for a general unidirectional orthotropic material is developed based on the

mechanics of deformation. In this Chapter, a failure model with the emphasis on stress-

path independent failure prediction is developed for a linear elastic orthotropic material.

Then, the improved model is extended to a general non-linear inelastic orthotropic

material with internal friction. The new failure model combined with quadratic spline

36

interpolation functions to map nonlinear stress-strain curves and an incremental solution

algorithm are employed to predict the mechanical response and failure of unidirectional

composite laminates. Model predictions are obtained for several unidirectional laminates

and loading cases to study the sensitivity of the predictions to model parameters. The

numerical predictions are compared to experimental data collected by Soden et al. (2002)

for the WWFE and to the predictions of Butalia and Wolfe (2002). The mechanical

properties of the unidirectional material systems studied herein are taken from Soden et

al. (1998). Finally, the proposed model is extended to include unloading and reloading

under combined loading and also to the 3-dimentional loading case.

3.2 The Original Strain Energy Based Model

Sandhu (1974) suggested that the longitudinal, transverse, and shear strain energies can

be treated as independent parameters to measure the extent of damage in a stressed

composite material. His model for failure in an orthotropic nonlinear composite under

plane stress condition was given as:

1=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⋅

⋅

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⋅

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⋅

⋅

∫

∫

∫

∫

∫

∫∗∗∗

S

u

T

uT

T

L

uL

L

mm

TT

TT

m

LL

LL

d

d

d

d

d

d

τ

τ

ε

ε

ε

ε

γτ

γτ

εσ

εσ

εσ

εσ

(3.1)

37

Where, LL dεσ ⋅ , TT dεσ ⋅ , and γτ d⋅ are the incremental longitudinal, transverse, and

shear strain energies, respectively. The superscripts ∗ and u indicate the induced and

failure conditions. Each energy ratio in this equation (longitudinal, transverse, and shear)

declares one possible failure mode. The shape factors, Lm , Tm , and Sm are material

parameters that define the interaction between the three failure modes. For a general

orthotropic material the shape factors can be different in compression and tension.

However, based on limited experimental data Sandhu (1974) proposed Lm = Tm = Sm =1

in both compression and tension. Sandhu considered the failure to be longitudinal when

the longitudinal strain energy ratio exceeded a critical value (the critical LSER), which he

took to be 1.0 . Subsequent analyses by Butalia and Wolfe (2002) showed that a better

agreement between predictions and experimental data was obtained when this ratio was

5.0 . The failure was assumed to be a matrix failure (transverse or shear) if the

longitudinal strain energy ratio was less than the critical value at failure point. The

matrix failure mode was taken to be transverse when the transverse stress was tensile or

the shear stress was zero, and shear failure otherwise.

An incremental non-linear constitutive model along with the classical lamination theory

were employed to distribute the applied loads between the laminae. Sandhu’s model

required seven stress-strain curves as inputs, specifically longitudinal tension and

compression, transverse tension and compression, in-plane shear, and finally LTν versus

longitudinal tensile and compressive strains. Each curve was established by a quadratic

polynomial interpolation through the data points. Therefore, all tangent moduli (slope of

38

the curves) were continuous functions of longitudinal, transverse, or shear strains.

Sandhu (1974) introduced the concept of ‘Equivalent strains’ to take into account the

coupling between longitudinal and transverse deformations under incremental biaxial

loadings, as:

( )LTLT

LeqL dd

ddσσν

εε⋅−

=1

, ( )TLTL

TeqT dd

ddσσν

εε⋅−

=1

(3.2)

Under a biaxial longitudinal-transverse loading, due to the interaction between the two

deformations, the stress-strain curves do not follow uniaxial stress-strain curves (Figure

3.1). The equivalent strains are the remaining parts of longitudinal and transverse strains

after the coupling deformation is excluded through a linear superposition. Thus, during

any deformation the equivalent strains follow the uniaxial stress-strain curves. Sandhu’s

theory assumes that the compliance coefficients in the longitudinal and transverse

directions depend only on the corresponding equivalent strains. For example, the

longitudinal modulus is a function of longitudinal equivalent strain, but is independent of

the equivalent transverse and shear strains.

Sandhu (1974) combined this failure theory with a failure mode dependent post initial

failure model to predict the mechanical response and failure progression of unidirectional

and multidirectional fibrous composite laminates. Hinton et al. (2002b) compared the

predictions made by Wolfe and Butalia (1998), and Butalia and Wolfe (2002) using this

model with experimental data. The predictions show relatively good agreement with the

39

experimental data for most cases. However, the longitudinal-transverse failure envelope

for a unidirectional lamina exhibited discontinuities and other features not in agreement

with the experimental data. Zand (2004) showed the discontinuities are due to the

interaction between the longitudinal and transverse deformations.

Figure 3.1. Longitudinal stress-strain curve under biaxial loading

40

3.3 A Strain-Energy Based Model for Linear Elastic Composites

3.3.1 Constitutive model

In this section we assume the material to be studied is orthotropic linear elastic prior to

failure. Using the contracted notation (Lekhnitskii, 1950) the stress-strain relationship

for a linear elastic material with orthotropic symmetry is given as (Ting, 1996):

σSε ⋅= (3.3)

T654321 ],,,,,[ εεεεεε=ε (3.4)

T654321 ],,,,,[ σσσσσσ=σ (3.5)

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

66

55

44

333231

232221

131211

000000000000000000000000

ss

ssssssssss

S (3.6)

Where, S is the elastic compliance matrix,

111 εε = , 222 εε = , 333 εε =

23234 2 γεε == , 31315 2 γεε == , 12126 2 γεε == (3.7)

41

And,

111 σσ = , 222 σσ = , 333 σσ =

23234 τσσ == , 31315 τσσ == , 12126 τσσ == (3.8)

In elastic materials the stress path independency of the strain energy necessitates the

compliance and stiffness matrices be symmetric (Ting, 1996). Furthermore, the

compliance matrix must be positive definite because the strain energy must be positive

for any given deformation:

021 T ≥⋅⋅=Π σSσ (3.9)

Since the compliance matrix is positive definite, it is always reversible and Equation (3.3)

can be written in the following alternative form:

εCσ ⋅= (3.10)

ICSSC =⋅=⋅ (3.11)

C is the elastic stiffness matrix, which also must be positive definite. For a general

anisotropic material with no planes of symmetry, for the elastic compliance matrix to be

symmetric, the presence of the factor 2 in Equations (3.7)4,5,6 and its absence in the

Equations (3.8)4,5,6 are necessary (Ting, 1996). For an orthotropic material under plane

42

stress conditions ( 0543 === σσσ and 054 == εε ) the above formulations can be

simplified to:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

τσσ

γεε

T

L

T

L

sssss

66

2221

1211

0000

(3.12)

LEs /111 = , TEs /122 = , Gs /166 =

TTLLLT EEss //2112 νν −=−== (3.13)

Where, LE , TE , and G are the elastic longitudinal, transverse, and in-plane shear

moduli, ijs are components of the compliance matrix, and LTν and LTν are the major and

minor Poisson’s ratios, respectively. The subscripts L, T, and S denote longitudinal,

transverse, and in-plane shear directions, respectively. The subscript T (italic) should not

be confused with superscript T that is used as matrix transpose operator. The in-plane

stresses can be written in terms of in-plane strains:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

γεε

τσσ

T

L

T

L

ccccc

66

2221

1211

0000

(3.14)

Where, ijc are components of stiffness matrix.

43

3.3.2 Strain Energy Based Failure Model for Orthotropic Linear Elastic Materials

Using the elements of the compliance matrix, the strain energy of a material loaded from

zero to the stress state TTL ],,[ ∗∗∗ τσσ along an arbitrary loading-path Γ is given as:

( )

( )

( )∫

∫

∫

Γ

Γ

Γ

=Π

+=Π

+=Π

ττ

σσσ

σσσ

ds

dsds

dsds

S

TLTT

TLLL

66

2212

1211

(3.15)

The total strain energy is the sum of the longitudinal, transverse, and shear components:

STL Π+Π+Π=Π (3.16)

For elastic materials, the total strain energy must be stress-path independent as it is in the

above formulation. However, the longitudinal and transverse energy components, as

defined in equations (3.15)1 and (3.15)2 depend on the stress-path. Therefore, the failure

criterion presented in Equation (3.1) leads to a stress-path dependent failure prediction

that is not consistent with the assumption of the material being elastic. A new strain

energy based criterion is proposed, leading to a stress-path independent failure prediction

for elastic materials. In the proposed model, the energy components that emerge in the

formulation are stress path independent, and thereby the failure criterion is stress path

44

independent. To continue, it is more convenient to express Γ in a parametric form, in

which ξ is a curve parameter that is equal to zero at the beginning of the load path (zero

stress state) and ∗ξ at the end point of the load path:

)(ξσ LL = , )(ξσ TT = , )(ξτ S= (3.17)

With ],0[ ∗∈ ξξ

Therefore:

( ) ∫

∫∫∗

∗∗

⋅′⋅+⋅=Π

⋅′⋅+⋅′⋅=Π

ξ

ξξ

ξξξσ

ξξξξξξ

012

2*11

001211

)()(21

)()()()(

dTLss

dTLsdLLs

LL

L

(3.18)

By defining

( )2*112

1LLL s σ⋅=Π (3.19)

∫∗

⋅′⋅=Πξ

ξξξ0

12 )()( dTLsLT (3.20)

One can write:

LTLLL Π+Π=Π (3.21)

45

Where, LLΠ is the portion of the longitudinal strain energy induced by the longitudinal

stress, and LTΠ is the portion of the longitudinal strain energy induced by the transverse

stress. LTΠ is a measure of the strain energy produced by the coupling between

longitudinal and transverse deformations. Here, S is used to denote the shear stress path

in parametric form and it should not be confused with ijs , or S , the components of the

compliance matrix and the compliance matrix, respectively. Similarly:

TLTTT Π+Π=Π (3.22)

( )2*222

1TTT s σ⋅=Π (3.23)

∫∗

⋅′⋅=Πξ

ξξξ0

12 )()( dLTsTL (3.24)

( )266

066

066 2

1)()( ∗⋅=⋅′⋅=⋅=Π ∫∫∗∗

τξξξττξτ

sdSSsdsS (3.25)

In the above relationships, TLΠ is the transverse work caused by longitudinal stress, and

TTΠ is the transverse work caused by transverse stress. LLΠ , TTΠ , and SΠ are stress-

path independent because their values are functions of the final state of the stress.

However, LTΠ and TLΠ in the above formulation are, in general, stress-path dependent.

Since the total strain energy is stress-path independent TLLT Π+Π must also be so. This

is, in fact, straightforward to be shown:

46

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⋅′⋅+⋅′⋅=Π+Π ∫∫

∗∗ ξξ

ξξξξξξ00

12LTTL )()()()( dTLdLTs

[ ]∫∗

⋅⋅′+′⋅=ξ

ξξξξξ0

12 )()()()( dLTLTs

[ ] ∗∗ ⋅⋅=⋅′⋅= ∫∗

TLsdLTs σσξξξξ

120

12 )()(

(3.26)

The fact that TLLT Π+Π is stress-path independent in any elastic material makes it an

intuitive variable for a strain energy based failure criterion. Therefore, the following

strain energy based failure criterion, resulting a stress-path independent failure prediction

for any linear elastic material, was suggested by Zand (2004):

( )STTLTLL m

S

Sm

TT

TT

m

LT

couplecouple

m

LL

LL

AAAAFI ⎥

⎦

⎤⎢⎣

⎡Π+⎥

⎦

⎤⎢⎣

⎡Π+

ΠΠ+⎥

⎦

⎤⎢⎣

⎡Π=

2sgn (3.27)

, where TLLTcouple Π+Π=Π

and ⎩⎨⎧

<Π−≥Π

=Π0 if 10 if 1

)sgn(couple

couplecouple

In the above equations, LLA , TTA , and SA are failure strain energies (areas under stress-

strain curves at failure) under uniaxial longitudinal, transverse, and shear stresses,

47

respectively; and FI is a failure index. Compared to Equation (3.1), the above Equation

requires eight new parameters, LTm and LTA , where LTm are shape factors and LTA are

the coupling failure strain energies (the values can be different in each quadrant of

longitudinal-transverse plane). These additional parameters are to be evaluated through

experimental data. In the absence of sufficient experimental data, LTA can be expressed

as a function of LLA and TTA , following the below analogy (Zand, 2004):

( )211

221 u

LuL

uLLL

sA σεσ =⋅=

( )222

221 u

TuT

uTTT

sA σεσ =⋅=

( )266

221 uuu

Ss

A τγτ =⋅=

Suggesting:

uT

uLTLLT

sAA σσ ⋅==212

TLLTTTLL AA νν ⋅⋅⋅= (3.28)

For a linear material the strain energy based failure criterion does not have any advantage

over stress-based failure models. However, for nonlinear material the strain energy gives

a better measure of the extent of damage accumulation, because it includes material

48

nonlinearity (Wolfe and Butalia, 1998). In the next section the model will be extended to

nonlinear inelastic material with internal friction, with application for unidirectional

fibrous laminates.

3.4 Failure Model for Unidirectional Fibrous Composites Under In-plane Loading

Condition

In this section a failure model is presented for fiber reinforced polymer (FRP)

composites. As previously mentioned, experimental observations have shown that a

fibrous lamina can fail in any of several different modes. The strain energy based failure

criterion developed in the previous section is used to predict the initial failure of matrix

material in the transverse and shear modes. Since the failure strain of the matrix material

in polymer matrix composites is considerably larger than the failure strain of fibers

(typical values are 4% versus 1 to 2%, respectively), failure of the matrix is not expected

in the direction of the fibers. Fiber failure is predicted using the maximum strain failure

criterion in the longitudinal direction of the material. Thus, the model presented in this

work uses separate failure criteria to distinguish between different failure modes. In this

chapter only unidirectional laminates are studied. In Chapter four the failure model is

extended to multidirectional laminates by developing a failure mode dependent post

initial failure constitutive law.

49

Unidirectional fibrous materials are in general inelastic and nonlinear. Thus, in order to

extend the model developed in the previous section to such materials, a non-linear

constitutive law based on the incremental constitutive law suggested by Sandhu (1974) is

proposed.

3.4.1 Incremental Constitutive Law

Assuming the material does not undergo any unloading, that is all ΠΠ /d are positive

throughout the stress-path Γ , a nonlinear stress-strain curve for an orthotropic material

can be expressed by rewriting Equation (2) in incremental form:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

τσσ

γεε

ddd

sssss

ddd

T

L

T

L

66

2221

1211

0000

(3.29)

Where as before:

LEs /111 = , TEs /122 = , Gs /166 =

LLT Es /12 ν−= , TTL Es /21 ν−= (3.30)

The moduli are the instantaneous slopes of the stress-strain curves (tangent moduli)

which are in general functions of stress, strain, and loading history. Due to its

50

demonstrated success in predicting the stress-strain behavior (Hinton et al., 2002b) of

fibrous laminates, we adopt Sandhu’s (1974) notion that the longitudinal and transverse

moduli are each a function of the corresponding equivalent strain:

( )LTLT

LeqL dd

ddσσν

εε⋅−

=1

, ( )TLTL

TeqT dd

ddσσν

εε⋅−

=1

(3.31)

As previously mentioned, the equivalent strains represent the longitudinal and transverse

strains after the interaction between them is removed. To visualize the model’s

predictions, imagine that the longitudinal stress versus the longitudinal equivalent strain

curve induced by any combined loading will follow the longitudinal stress-strain curve

under uniaxial loading, and similarly for the transverse stress and strain. Since there is a

one-to-one correspondence between induced stresses and equivalent strains, each

modulus can also be written as a function of the corresponding stress. The use of

equivalent strains in a numerical scheme is particularly advantageous when the loading is

proportional, because in that case the equivalent strains can be computed using strain

increments only:

( )Bdd

LT

LeqL ⋅−

=ν

εε

1, ( )B

ddTL

TeqT /1 ν

εε

−=

With constBL

T ==σσ

51

Further, it is assumed that the compliance matrix is symmetric, that is 2112 ss = . For a

general inelastic material the compliance and stiffness matrices are non-symmetric,

because as previously mentioned this kind of symmetry is a necessary but not sufficient

condition for the strain energy to be path independent. If the material is inelastic, the

stiffness and compliance matrices can be nonsymmetrical. The major Poisson’s

ratio, LTν , is assumed to be a function of the longitudinal equivalent strain, and

LTLTTL EE /⋅=νν . Five unidirectional stress-strain and two Poisson’s ratio curves are

required to evaluate the compliance coefficients at any stress level. These curves are the

tensile and compressive responses of the lamina in the longitudinal and transverse

directions, the in-plane shear response, and the major Poisson’s ratio against longitudinal

tensile and compressive stress.

In the absence of unloading, there is always a one-to-one correspondence between the

longitudinal or transverse equivalent strains and the corresponding induced stress and the

tangent modulus. In this model, each stress-strain curve is established by a quadratic

spline interpolation. The constitutive law presented in Equations (3.29) through (3.31)

are implicit because in general the Poisson’s ratios are functions of longitudinal and

transverse equivalent strains.

52

3.4.2 Matrix Failure Criterion

The objective is development of a strain energy based failure criterion that is suitable for

matrix failure prediction of nonlinear composites. First, a model is developed for shear

response of the material to include internal friction between the fibers, assuming no

interaction between the shear and other failure modes exists. This failure model, then, is

integrated into Equation (3.27) to predict failure of the material under combined loading.

3.4.2.1 Shear response

Experimental observations have shown that the in-plane shear strength (or interlaminar

shear strength under a 3D stress state) can increase with compressive transverse stress for

relatively low to moderate levels of transverse stress (Swanson et al., 1987). In 1980,

Hashin proposed a Mohr type criterion for the in-plane shear strength:

Tu

Tu μστστ −= 0)( (3.32)

Where μ represents the internal friction coefficient, and 0uτ is the uniaxial shear strength

under zero transverse (normal) load. Puck and Schürmann (1998) and Cuntze and Freund

(2004) used a similar criterion in their models, and their predictions were validated

during the WWFE by Hinton et al. (2002b and 2004). Since any deformation involving

friction is irreversible, μ must be zero for an elastic material.

53

The concept of internal friction is demonstrated in Figure 3.2 for a unidirectional

composite coupon under two dimensional loading conditions. The shear strength term in

Equation (3.27) is modified to include the effect of internal friction on the in-plane shear

strength. In this model the failure plane under in-plane loading is assumed to be

perpendicular to the fiber direction, as presented in Figure 3.2. Additionally, it is

assumed that the internal friction is generated as a result of random contacts between the

adjacent fibers, allowing the fibers to carry a portion of in-plain shear stress. Under a

constant transverse stress of 0<= constTσ , the maximum shear resistance of the

material during deformation is limited to Tf μστ −= . According to this assumption

transverse stress influences the shear strength but the shear failure strain remain

unaffected, as depicted in Figure 3.3. The dashed line shows the upper limit for the shear

stress when the compressive transverse stress remains constant during the loading. The

shear stress-strain curve is established by assuming that the amount of shear taken by the

fibers is proportional to the shear stress taken by the matrix. Therefore, the shear stress-

strain curve is corrected as below:

)()(0

66

Tbds

dσ

τγγ

⋅= (3.33)

With

01)( uT

Tbτμσ

σ −= (3.34)

Gs /1066 = (3.30)3

54

One advantage of this method of establishing the shear stress-strain curve is that it can be

easily extended to cases when the transverse stress is variable, or the material is

unloaded. For any elastic material or a material with no internal friction 1=b . For a

material with internal friction 66s can be written as below:

bs

Gbs

066

661

=⋅

= (3.35)

Figure 3.4 presents the influence of compressive transverse stress on the shear response

of a nonlinear material. Curve (a) shows a nonlinear shear stress-strain curve under zero

transverse stress, curve (b) shows shear response of the same material during a

proportional loading of 75.01=Tστ , and curve (c) is the shear response under a

constant transverse stress equal to 75% of the uniaxial shear strength. All the three

curves terminate at the same failure strain because the shear failure strain was assumed to

be unaffected by the transverse stress. The computed stress-strain curve under a constant

transverse stress is stiffer than the one under proportional loading.

55

Figure 3.2. The influence of internal friction on in-plane shear strength of laminate

Figure 3.3. The influence of a constant transverse stress on the shear stiffness

56

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Shear strain / Failure Shear Strain

Shea

r Str

ess

/ Uni

axia

l She

ar S

tren

gth

(a) : −σT = 0

(c): -σT = 0.75 τou

(Constant)

(b) : −σT = 0.75 τ(Proportional)

Figure 3.4. Influence of constant and variable transverse stresses on shear response of a

non-linear material

In this model the longitudinal stress is assumed to not influence the shear response of the

material. There is experimental evidence suggesting that the shear strength slightly

increases with longitudinal tension (Soden et al, 2002), however more data would be

required to incorporate this effect into the proposed model. To the best of this author’s

knowledge the literature of composite laminates does not contain any major experimental

57

effort aimed at studying the effect of the transverse or longitudinal stress on the shear

behavior. This signifies the need for additional experimental investigations to validate

the current model for shear response or develop more robust approaches.

3.4.2.2 Shear strain energy in material with internal friction

The shear strain energy in a material with internal friction is:

∫∫∗

⋅′⋅⋅=

⋅⋅=Π

Γ

ξ

ξξξξττ

0

6666

))(()()(

TbdSSs

bds

S

)

∫∗

⋅′⋅⋅=

ξ

ξξξξ

0

66

)(~)()(~

b

dSSs (3.36)

In the above equation and in the rest of this work the ^ symbol indicates the value is

expressed as a function of stress and ~ indicates that it is expressed in terms of the

parameter ξ . According to this equation, the shear strain energy for a material with

internal friction is stress-path dependent, unless 1=b , which is the case for elastic

materials. The shear strain energy ratio is:

S

S

AbSSER

⋅Π

= ∗ (3.37)

,where, ∗b is the value of )b( Tσ at ∗= TT σσ .

58

3.4.2.3 Failure criterion

For a general nonlinear material with the constitutive law defined in Equations (3.29) to

(3.31) the strain energies in the longitudinal and transverse directions are as below:

∫Γ

⋅⋅=Π LL ds σσ11LL (3.38)

∫ ⋅⋅=ΠT

TT dsσ

σσ0

22TT (3.39)

∫Γ

⋅⋅+⋅⋅=Π )( 2112couple TLLT dsds σσσσ (3.40)

Since 11s is function of Lσ only, (3.37) can be written as:

∫∫∫∗∗∗

⋅=⋅⋅=⋅′⋅⋅=ΠL

eqLLLL ddsdLLs

εξξ

εσσσξξξ00

110

11LL )()(~ ) (3.41)

Similarly:

∫∫∗∗

⋅⋅=⋅′⋅⋅=ΠT

TT dsdTTsσξ

σσξξξ0

220

22TT )()(~ ) (3.42)

∫∫∗∗

⋅′⋅⋅+⋅′⋅⋅=Πξξ

ξξξξξξ0

210

12couple )()(~)()(~ dLTsdTLs (3.43)

59

For a general inelastic material 2112 ss ≠ . However, the above equation can be simplified

for the special case when 2112 ss = :

[ ] ∫∫∗∗

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅−⋅⋅=′⋅=Π ∗∗∗

ξξ

ξξ

ξξσσξξξ0

1212

012couple

~)()()()(.~ d

dsd

TLsdTLs TL (3.44)

Having the incremental constitutive law and all the components of the strain energy

defined, the strain energy based failure criterion can be written as:

( )STTLTLL m

S

Sm

TT

TT

m

LT

couplecouple

m

LL

LL

AbAAAMFI ⎥

⎦

⎤⎢⎣

⎡⋅

Π+⎥

⎦

⎤⎢⎣

⎡Π+

ΠΠ+⎥

⎦

⎤⎢⎣

⎡Π= ∗2

sgn (3.45)

The above equation can be used for either the prediction of the matrix failure, or

computation of the safety factor for the matrix failure. The failure strain energy for

longitudinal-transverse interaction, LTA , can be computed using Equation (3.28), with the

average Poisson’s ratios:

TLLTTTLL AA νν ⋅⋅⋅=LTA (0.28)

The average values of the Poisson’s ratios are computed by integrating LTν versus

uniaxial longitudinal strain and TLν versus uniaxial transverse strain.

60

Among the above identities, LLΠ , and TTΠ are stress-path independent for any material,

whereas coupleΠ , and SΠ are, in general, stress-path dependent. In a nonlinear elastic

material all the energy components must be stress-path independent, therefore there are

restrictions on the material properties. In order for coupleΠ to be stress-path independent

for a general elastic material under any loading condition a potential field ),( TL σσφ must

exist such that:

LT

TL

s

s

σσφ

σσφ

12

21

=∂∂

=∂∂

(3.46)

The above equations have as a solution (can be easily detected):

TLes σσλ

λφ ⋅⋅=

0

(3.47)

where:

TLesss σσλ ⋅⋅⋅== 02112 (3.48)

Thus, Poisson’s ratios become a function of both longitudinal and transverse stresses:

TL

TL

eEs

eEs

TTTL

LLLTσσλ

σσλ

σν

σν⋅⋅

⋅⋅

⋅⋅−=

⋅⋅−=

)(

)(0

0

(3.49)

61

In the above equations, 0s and λ are integration constants. If (3.48) holds true for a

particular material, the potential field for that material φ exists and coupledΠ can be

written as:

φσσφσ

σφσσσσ ddddsdsd L

LT

TLTTLcouple =

∂∂

+∂∂

=⋅⋅+⋅⋅=Π 2112

λσσφ

0

),( sTLcouple −=Π ∗∗

One special case is when 0=α , for which 12s and 21s are constants:

constEEss TTLLLT =−=−== //2112 νν (3.50)

Furthermore, for an elastic material 1=b since SΠ must be stress-path independent.

Since this model is not limited to elastic materials, the actual values obtained from

experiments for LTν , 12s , and μ could be used.

3.4.2.4 Matrix failure modes

Each energy ratio in Equation (3.45) indicates the relative amounts of damage

accumulation with respect to the capacity of the material in that direction. Thus, each

energy ratio represents a potential failure mode. The shape factors (exponents) define the

interaction between the failure modes in the energy space. In a typical fiber reinforced

62

composite, a uniaxial longitudinal deformation cannot cause matrix failure because the

failure strain of the matrix material is higher than that of the fibers. However, the

longitudinal damage accumulation can influence the strength of the matrix in other

directions, as described by the energy based failure criterion. Figure 3.5 presents the

interaction between the failure modes in the longitudinal-transverse and transverse-shear

energy planes. In this model four different matrix failure modes are defined, namely:

tensile failure, compressive failure, shear failure, and combined compressive-shear

failure:

1- 0>Tσ : Matrix tensile failure

2- 0 ,0 ≠= τσ T : Matrix shear failure

3- 0 ,0 =< τσ T : Matrix compressive failure

4- 0 ,0 ≠< τσ T : Matrix compressive-shear failure

3.4.3 Fiber Failure Criterion

The use of the maximum longitudinal strain criterion for fiber failure prediction has been

shown to be in good agreement with available experimental data (Hart-Smith, 2001).

Thus, the criteria adopted for fiber failure are:

63

utL

LFFIεε

= if 0>Lε

ucL

LFFIε

ε−= if 0<Lε ( uc

Lε has positive sign) (3.51)

Fiber failure occurs when FFI reaches unity. The compressive or tensile failure modes

are determined based on the direction of the longitudinal strain. The maximum

longitudinal strain criterion does not include the influence of transverse and shear stresses

on the longitudinal strength of material, however, the longitudinal strength is typically

several times higher than the transverse or shear strengths and this effect can be

neglected.

64

(a)

(b)

Figure 3.5. (a) Failure mode interaction in the longitudinal-transverse energy plane; (b)

failure mode interaction in the transverse-shear energy plane

65

3.4.4 Numerical Results

In this section the influence of the shape factors, μ , and LTA on the geometry of the

failure envelope under biaxial loading condition is studied. The results presented are

obtained through series of numerical analyses. Additional analyses are also presented for

model evaluation and comparison purposes. In these analyses longitudinal-transverse,

longitudinal-shear, and transverse shear failure envelopes obtained for three material

types and the numerical failure envelops are compared to available experimental data to

evaluate the accuracy of the model. The predictions are also compared to the predictions

made by Butalia and Wolfe (2002) using the original strain energy based model (Sandhu,

1973). The mechanical properties for these material systems (with unidirectional lay-up)

were furnished by Soden et al., (1998). The experimental data are taken from Soden et

al., (2002). Table 3.1 presents the mechanical properties of the material systems used in

the analyses. Figures 3.6 to 3.8 present transverse tensile and compressive and shear

stress-strain curves for the unidirectional laminates. Details of the computer program and

numerical scheme developed to implement this model are given in Appendix A.

According to previous discussions, for non-elastic material the shape of failure envelopes

are in general non-unique and load-path dependent. Failure envelopes presented in this

section are obtained by proportionally loading the laminates until failure.

66

Fiber Type Silenka E-Glass 1200 tex

E-glass 21xK43 Gevetex

T300

Matrix MY750/HY917 /DY063 epoxy

LY556/HT907 /DY0 epoxy

BSL914C epoxy

Specification Filament Winding

Filament Winding

Filament Winding

Fiber volume fraction

0.6

0.62

0.6

Longitudinal modulus

45.6 (GPa)

53.48 (GPa)

138 (GPa)

Major Poisson’s ratio, LTν

0.278

0.278

0.28

Longitudinal tensile strength

1280 (MPa)

1140 (MPa)

1500 (MPa)

Longitudinal compressive strength

800 (MPa)

570 (MPa)

900 (MPa)

Longitudinal tensile failure strain

2.807%

2.132%

1.087%

Longitudinal compressive failure strain

1.754%

1.065%

0.652%

Transverse Modulus

16.2 (GPa)

17.7 (GPa)

11 (GPa)

Transverse tensile strength

40 (MPa)

35 (MPa)

27 (MPa)

Transverse compressive strength

145 (MPa)

114 (MPa)

200 (MPa)

Transverse tensile failure strain

0.246%

0.197%

0.245%

Transverse compressive failure strain

1.2%

0.644%

1.818%

Initial in-plane shear modulus

5.83 (GPa)

5.83 (GPa)

5.5 (GPa)

In-plane shear strength

73 (MPa)

72 (MPa)

80 (MPa)

In-plane shear failure strain

4.0%

3.8%

4.0%

Table 3.1. Mechanical properties of the unidirectional material systems (Soden et al.,

1998)

67

0

50

100

150

200

0.0% 1.0% 2.0% 3.0% 4.0% 5.0%

Strain

Stre

ss (M

Pa)

.

Transverse Tension

Transverse Compression

In-plane Shear

Figure 3.6. Transverse and shear responses for a unidirectional E-glass/MY750 epoxy

laminate (Soden et al, 1998)

68

0

50

100

150

200

0.0% 1.0% 2.0% 3.0% 4.0% 5.0%

Strain

Stre

ss (M

Pa)

Transverse Tensile


in_plane Shear

Figure 3.7. Transverse and shear responses for a unidirectional E-glass/LY556 epoxy

laminate (Soden et al, 1998)

69

0

50

100

150

200

250

0.0% 1.0% 2.0% 3.0% 4.0% 5.0%

Strain

Stre

ss (M

Pa)

Transverse Tensile


In-plane Shear

Figure 3.8. Transverse and shear responses for a unidirectional T300/BSL914C

carbon/epoxy laminate (Soden et al, 1998)

3.4.4.1 Effect of shape factors on matrix failure

In order to demonstrate the effect of the shape factors on the matrix failure of a

unidirectional laminate, the longitudinal-shear failure envelope is selected because other

model parameters such as μ and LTA do not affect the failure envelope in this plane.

70

Figure 3.9 presents the effect of transverse and shear shape factors, LLm and Sm , on the

matrix failure envelope of a unidirectional E-glass/LY556 epoxy laminate. Each failure

data point is computed by proportionally loading the laminate along a line designated by

a constant longitudinal to shear stress ratio. The figure presents several failure envelopes

for the shape factors ranging from 0.5 to 10. Because of the symmetry of the failure

envelope with respect to the horizontal axis only half of the envelope is presented. It can

be seen that as the shape factors increase from 0.5 to infinity, the failure envelope in the

half plane changes from a triangle to a rectangle. For any value of shape factors the shear

strength of the material is always lower than the uniaxial shear strength upon the

existence of a positive or negative longitudinal stress.

Figure 3.10 presents the same failure envelope for the case when the longitudinal and

shear shape factors are not equal. Four failure envelops are presented in this figure with

each of LLm and Sm being equal to 0.5 and 1.0. As expected the two envelops with one

of the shape factors being equal to 1.0 and the other one 0.5 intersect. For all the failure

envelops presented in this section the matrix failure modes are shear failure throughout

the envelope except at the vicinity of the intersection with horizontal axis, where the

failure mode is fiber failure. This is because 0≠τ and 0=Tσ .

71

0

40

80

120

-600 -300 0 300 600 900 1200

Longitudinal Stress (MPa)

Shea

r Str

ess

(MPa

)

m =

1.0

0.5

0.7

2.0 10.0

mLL = mS = m

Figure 3.9. Effect of shape factors on longitudinal-shear failure envelope for

unidirectional E-glass/LY556 epoxy laminate

72

0

40

80

120

-600 -300 0 300 600 900 1200


Shea

r Str

ess

(MPa

)

mLL=mS = 0.5

mLL=mS= 1.0

mLL= 1.0mS= 0.5

mLL= 0.5mS= 1.0

Figure 3.10. Longitudinal-shear failure envelope for unidirectional E-glass/LY556 epoxy

laminate, computed using various shape factors

73

3.4.4.2 Effect of LTA and LTm on matrix failure

To demonstrate the effect of LTA on the matrix failure of a unidirectional laminate, the

longitudinal-transverse plane is selected because other model parameters such as shape

factors and μ do not affect this failure envelopes. Depicted in Figure 3.11 are three

failure envelope using different values for LTA . The failure envelope presented by a

solid line is obtained using the default value as computed from Equation (3.28), and the

other envelopes are obtained using defaultLT AA 5.0= and defaultLT AA 2= in all the

quadrants. In all the envelopes the vertical cut-off at the two ends is due to the fiber

failure in the longitudinal direction. Increasing LTA will cause a rise in the biaxial

strength of the laminate in the second and fourth quadrants and a decrease in the first and

third quadrants. This is because coupleΠ is positive in the first and third quadrants and

negative in the other quadrants. For all the failure envelopes the shape factors are taken

to be equal to one.

Figure 3.12 shows the effect of LTm on the shape of the failure envelope. Four failure

envelopes are presented in this figure using LTm equal to 0.5, 1, 2, and 10 at all the

quadrants. As these shape factors increase to infinity, the second term in Equation (3.45)

vanishes which leads to a smoother failure envelope. As shown in Figure 3.13,

increasing these shape factors has an effect on the failure envelope similar to increasing

LTA because in both cases the second term in Equation (3.28) decreases.

74

-180

-140

-100

-60

-20

20

60

100

-600 -300 0 300 600 900 1200


Tran

sver

se S

tres

s (M

Pa)

ALT = Adefult ALT = 2Adefult ALT = 0.5 AdefultALT = Adefault ALT = 2.0xAdefault ALT = 0.5xAdefault

Figure 3.11. The effect of LTA on the shape of longitudinal-transverse failure envelope

for unidirectional E-glass/LY556 epoxy laminate

75

-180

-140

-100

-60

-20

20

60

100

-600 -300 0 300 600 900 1200


Tran

sver

se S

tres

s (M

Pa)

m = 0.5 m = 1.0 m = 2.0 m = 10.0 mLT = 1.0 mLT =2.0mLT = 0.5

mLL = mTT = mS = 1 0

mLT = 10.0

Figure 3.12. The effect of LTm on the shape of the longitudinal-transverse failure

envelope for unidirectional E-glass/LY556 epoxy laminate

76

-180

-140

-100

-60

-20

20

60

100

-600 -300 0 300 600 900 1200


Tran

sver

se S

tres

s (M

Pa)

r m = 10.0 ALT = 4.0Adefault ALT = 4Adefault mLT = 10 ALT = Adefault

mLT=1

Figure 3.13. Comparison between the influence of LTA and LTm on the failure envelope

of unidirectional E-glass/LY556 epoxy laminate

3.4.4.3 Effect of μ on matrix failure

Figure 3.14 presents transverse-shear failure envelopes for unidirectional E-glass/LY556

epoxy laminate using different values of μ . For 0=μ , the shear strength always

decreases with increasing compressive transverse stress, but when 0>μ friction

increases with compressive transverse stress, so does the shear strength when the failure

77

is dominated by shear. When the transverse stress is tensile, an increase in the transverse

stress decreases the shear strength, as the tensile stress reduces the residual compressive

stress in the matrix, generated as a result of contraction of the matrix around the fibers

during curing.

Figure 3.15 shows the effect of the shape factors on the failure envelope when 5.0=μ .

The solid lines are failure envelopes obtained using different values of transverse and

shear shape factors. For comparison purposes, the figure also presents a failure envelope

obtained using 0=μ and 1S TTm m= = (dashed line).

78

0

40

80

120

-120 -80 -40 0 40

Transverse Stress (MPa)

Shea

r Str

ess

(MPa

)μ =

0.2 0.0

0.4 0.6

1.0

Figure 3.14. The effect of μ on geometry of the transverse-shear failure envelope for

unidirectional E-glass/LY556 epoxy laminate

79

0

40

80

120

-120 -80 -40 0 40


Shea

r Str

ess

(MPa

)

μ = 0.5mS = mT = m

m =

0.5

1.0

2.0

0.7

μ = 0.0m = 1.0

Figure 3.15. The effect of shape factors on the transverse-shear failure envelope for

unidirectional E-glass/LY556 epoxy laminate with 5.0=μ . The dashed line is the

failure envelope using 0=μ and 1== TS mm

80

3.4.5 Comparison between Predictions and Experimental Data

3.4.5.1 Longitudinal-transverse failure envelope for unidirectional E-glass/MY750

epoxy laminate

These test results were obtained by Al-Khalil et al. (1996) through testing nearly

circumferentially wound tubes with liners under combined axial load and internal

pressure. Figure 3.16 presents biaxial experimental data together with the uniaxial

strengths from Table 3.1. The biaxial data were corrected for bulging of the gage section

due to the application of internal pressure. Depicted in Figure 3.17 are the computed

failure envelopes for a unidirectional E-glass/MY750 epoxy laminate in the longitudinal-

transverse plane using the current model and the one presented by Butalia and Wolfe

(2002) using the original strain energy based model superimposed on the test data. The

shape factors were assumed to be unity in both analyses. In the current analysis LTA is

taken to be equal to the default value computed from equation (3.28). Figure 3.17 shows

a remarkable agreement between predictions of the current model and experimental data.

The predictions of Butalia and Wolfe were also in a good agreement with the

experimental data. Nonetheless, their failure envelope exhibited a bulge in the first

quadrant that although the experimental data do not cover this region such a significant

increase in the strength of the laminate is unexpected (Hinton et al., 2002b) and difficult

to explain. Hence, the new predictions are likely to be more realistic. The good

agreement between experimental data and numerical predictions confirms the

appropriateness of the selected values for shape factors and LTA .

81

-200

-100

0

100

200

-1000 -500 0 500 1000 1500


Tran

sver

se S

tres

s (M

Pa)

.

Experimental data (Al-Khalil et al, 1996) Uniaxial strengths from Table 3.1

Figure 3.16. Biaxial experimental data for unidirectional E-glass/MY750 epoxy laminate

under combined longitudinal and transverse stress

82

-200

-100

0

100

200

300

-1000 -500 0 500 1000 1500


Tran

sver

se S

tres

s (M

Pa)

.

The current model Butalia amd Wolfe, 2002 Experimental data (Al-Khalil et al, 1996)

Figure 3.17. Comparison between numerical longitudinal-transverse failure envelopes

and experimental data. Predictions of Butalia and Wolfe (2002), using the original the

original strain energy based theory, is presented with dashed line

3.4.5.2 Transverse-shear failure envelope for unidirectional E-glass/LY556 epoxy

laminate

The computed transverse-shear failure envelope for this laminate was presented in

Section 3.4.4.3. In this section the predictions are compared to experimental data and

also to the prediction of Butalia and Wolfe (2002). The experimental data were provided

83

by Hütter et al. (1974) by testing filament wound tubes under combinations of axial load

and torsion. The tubes were 60 mm internal diameter, 2 mm thick with the fiber volume

fraction of 62% and were cured at 100˚C and 150˚C, each for two hours. These data are

not quite in agreement with the uniaxial transverse and shear strengths presented in Table

3.1 (see Figure 3.18). Thus, the unidirectional stress-strain curves used for the analysis

are scaled along the vertical axis (stress axis) to match the experimental values of Hütter

et al.

0

40

80

120

-150 -100 -50 0 50


Shea

r Str

ess

(MPa

) .

Experimental data (Hutter et al, 1974) Uniaxial strengths (Soden et al., 1998)

Figure 3.18. Biaxial experimental data for unidirectional E-glass/LY556 epoxy laminate

under combined transverse and shear loading

84

Figure 3.19 shows a comparison between the experimental data, the new failure envelope,

and the failure envelope of Butalia and Wolfe (2002). In the new analysis 6.0=μ and

0.1== TS mm .

0

40

80

120

-150 -100 -50 0 50


Shea

r Str

ess

(MPa

) .

The current model Butalia and Wolfe, 2002

Experimental data (Hutter et al, 1974) Unidirectional strengths from Table 3.1

Figure 3.19. Transverse-shear failure envelopes for unidirectional E-glass/LY556 epoxy

laminate versus experimental data

85

3.4.5.3 Biaxial failure envelope for unidirectional T300/914C epoxy laminate under

combined longitudinal and shear loading

The experimental results that we use in this section for model evaluation were originally

obtained by Schelling and Aoki (1975-1980 and1992) as cited by Soden et al. (2002).

The test specimens were in the form of axially wound tubes made of prepreg

carbon/epoxy material, and tested under combined axial load and torsion. The tubes had

internal diameter of 32 mm, thickness of 1.9 to 2.3 mm, and fiber volume fraction of 0.56.

These experimental data together with the uniaxial strengths from Table 3.1 are presented

in Figure 3.20. Depicted in the figure are three sets of experimental data obtained by

Schelling and Aoki using similar test specimens. The measured uniaxial shear strengths

from experimental data sets 1 and 3 are not in agreement and the reason for such apparent

scatter was reported by Soden et al. (2002) to be unknown. The trend of the experimental

data under biaxial loading suggests that higher observed values are correct (Set 3). Since

similar material types and testing methods were used for all the specimens, this difference

could have been caused by defect in the material used for Set 1 specimens or structural

buckling of these specimens under pure torsion. Unfortunately no further information

was available on the thicknesses of Set 1 specimens. The experimental data suggests an

increase in the material shear strength at moderate levels of axial tension. Due to the

scatter in the experimental data this observation is not conclusive and more experimental

data are required to firmly establish the effect of longitudinal stress on the shear strength.

Therefore, no effort is dedicated to include this effect in the current model.

86

0

40

80

120

160

-1200 -800 -400 0 400 800 1200 1600


Shea

r Str

ess

(MPa

) .

Set 1 (D= 32 mm, th= 1.9-2.3 mm) Set 2 (D= 32 mm, th= 1.9-2.3 mm)

Set 3 (D= 32 mm, th= 2.2 mm) Uniaxial strengths (Soden et al., 1998)

Biaxial experimental data (Schelling and Aoki ,1992, 1975-1980)

Figure 3.20. Biaxial experimental data for unidirectional T300/914C epoxy laminate

under combined longitudinal and shear loading

Figure 3.21 presents the computed failure envelope compared to the experimental data.

The longitudinal shape factor is taken to be 1.0. In order to improve the predictions, the

uniaxial shear strength is increased from 80 MPa to 90 MPa and the uniaxial compressive

strength is reduced from 900 MPa to 850 MPa in the longitudinal direction. Since the

experimental data are not consistent, it is not possible to comment on the goodness of fit.

87

0

40

80

120

160

-1200 -800 -400 0 400 800 1200 1600


Shea

r Str

ess

(MPa

) .

Set 1 (D= 32 mm, th= 1.9-2.3 mm) Set 2 (D= 32 mm, th= 1.9-2.3 mm)Set 3 (D= 32 mm, th= 2.2 mm) Uniaxial strengths (Soden et al., 1998)Numerical failure envelope

Biaxial experimental data from Schelling and Aoki (1992, 1975-1980)

Longitudinal shape factor is equal to 0.5 in compression

Figure 3.21. Longitudinal-shear failure envelopes for unidirectional T300/914C

carbon/epoxy laminate compared to experimental data

88

3.4.6 Unloading and Reloading

3.4.6.1 Uniaxial unloading and reloading

The unloading and reloading in the longitudinal (fiber) direction is elastic, because

material mechanical behavior is dominated by that of strong fibers which are

predominantly elastic material. Therefore, longitudinal unloading and reloading moduli

are equal to the loading modulus at any given stress (equivalent strain) level. Unloading

and reloading in the matrix dominated direction (transverse and shear) is assumed to be

linear (Figure 3.22).

Figure 3.22. Linear unloading with and without residual strain

89

There are experimental evidences that unloading of the material is associated with some

residual strain particularly in the shear direction (Paepegem et al., 2006a). The residual

strains occur due to imperfect crack closure. When a crack opens under tensile or shear

forces, debris of failed material and dust particles can get into cracks, causing imperfect

crack closure. In addition to that, imperfect crack closure can occur because of plastic

flow of material at crack tips that leads to mismatch between the two sides of a crack.

In a numerical code the residual strain can be easily included with almost no additional

effort. However, due to the lack of experimental data, it is not possible to establish a

mathematical function to express residual strain as material deforms. Therefore, in this

work the residual strains are assumed to be zero. Transverse tensile response of the

material is usually linear, consequently this method of unloading is more or less elastic in

that direction, because unloading modulus is the same as loading modulus. Material

response is slightly nonlinear for transverse compression and highly nonlinear for shear.

The unloading moduli of the material in the transverse and shear directions are computed

as:

)( RT

eqT

IT

TUεε

σ−

= (3.52)

)( RI

I

SUγγ

τ−

= (3.53)

90

Where RTε and Rγ are the residual strains and there are assumed to be zero in this work,

and TU and SU are unloading/reloading moduli in the transverse and shear directions,

respectively. The superscript I denotes conditions at the beginning of unloading. In

general the residual strains can be expressed in terms of the maximum experienced

(equivalent) strains in the corresponding direction:

( )eqTT

RT εε Ρ= (3.54)

( )SSRT εε Ρ= (3.55)

Where, TΡ and SΡ are increasing functions of strains to be established through

experimental data, and both are zero during unloading or reloading, as shown in Figure

3.23:

⎩⎨⎧

<≡Ρ=>Ρ

max

max

)( if 0)( if 0

εεεε

(3.56)

Here, max)(ε is the absolute value of the maximum experienced (equivalent) strain in the

corresponding direction.

91

Figure 3.23. Residual strain growth during deformation

The above formulation can be written in incremental form. For the ease of writing L and

T subscripts are eliminated:

εερε dd R ⋅= )( (3.57)

Where ε

ερddΡ

=)( . As demonstrated in Figure 3.24, since the unloading/reloading

modulus cannot increase as the strain increases, an upper bound can be established for

)(ερ :

92

)()(1)(0

εεερ

ε UE

ddU

−≤⇒≤ (3.58)

For example if )0(EU = for any strain level:

0

)(1E

E ερ −= (3.59)

0ER σεε −= (3.60)

Figure 3.24. Upper bound for unloading reloading modulus

93

3.4.6.2 Unloading under combined axial and transverse loading

The cross compliance coefficients 12s and 21s along with 11s are assumed to be the same

for loading and unloading. Starting from a deformed configuration ],[ IT

IL σσ and ],[ I

TIL εε

in the longitudinal-transverse plane, the material undergoes complete unloading in the

transverse direction, while the longitudinal stress remains unchanged. The new strains

are computed as below:

⎥⎦

⎤⎢⎣

⎡−⎥

⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡

ΔΔ

ITTT

L

Usss

σεε 0

/112

1211

And

eqT

IT

TUεσ

=

Therefore:

eqT

ITT

ITT

IT

ILL

ILL s

εεεεε

σεεεε

−=Δ−=

⋅+=Δ−= 12

94

3.4.6.3 The effect of transverse unloading on shear strain

For a material with internal friction under combined shear and transverse compressive

load, if the transverse compressive stress decreases the shear strain is expected to increase.

This is demonstrated in Figure 3.25. The solid curve (a) shows shear stress-strain curves

of a material with 0>μ under the presence of a compressive transverse stress and the

dashed line (b) shows the uniaxial shear response.

Figure 3.25. The effect of transverse unloading on shear response

95

The material is first loaded to the stress state of ITσ and Iτ where 0<I

Tσ . Shear strain

of the material at this point is denoted by Iγ which is smaller than the shear strain under

an equal uniaxial ( 0=Tσ ) shear load. The material is, then, unloaded in the transverse

direction. During transverse unloading the shear strain is expected to increase to IIγ (the

shear strain under uniaxial shear stress). It must be noted that increase in compressive

transverse stress will not decreases material shear strain, although it increases the shear

modulus. In order to formulate the increase in the shear strain due to transverse

unloading, the shear strain increment is assumed to be proportional to compressive stress

increment:

TIT

dd σσζγ ⋅−= (3.61)

Where,

),0( IT

Iτσ

γγζ=

−= (3.62)

Iτ is the shear stress and ITσ is the transverse stress at the start of unloading. The minus

sign on the right hand side of the Equation (3.61) is because shear strain and transverse

stress increments are positive while transverse stress is negative (compressive). ζ is

equal to the difference between shear strains under combined transverse and shear stress

and an equal uniaxial shear stress.

96

3.4.6.4 Unloading constitutive relations in matrix notation

Based on the discussion in the Sections 3.4.6.1 to 3.4.6.3 the incremental unloading

constitutive law can be written as:

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

τσσ

γεε

ddd

ssssss

ddd

T

L

T

L

6626

2221

1211

000

(3.63)

With,

IT

eqTs

σε

=22 (3.52)

( )I

Its

τ

γτσ ,0

66== (3.64)

IT

sσζ

=26 (3.65)

As mentioned previously the subscript I denotes the condition at which the unloading

begins. The compliance matrix is non-symmetric because the loading and unloading

moduli are assumed to be different. For an elastic material, in addition to the conditions

described in Section 3.4.2.3, the unloading moduli must be equal to the loading moduli,

and hence the loading and unloading compliance coefficients are essentially the same.

97

3.5 The Strain Energy Failure Criterion for the Three-Dimensional Stress

Condition

Realistic prediction of mechanical response of a structure under actual combinations of

service loads may require a 3-dimensional stress analysis. One common example is

stress localization at the vicinity of a free edge in multidirectional laminates. In such

condition, the failure of the material cannot be accurately predicted without considering

out-of-plane stress and strain components. The strain energy based failure criterion and

the constitutive law developed in Section 3.4 are extended to 3-dimensional loading

condition.

3.5.1 Notations

In this section the use of contracted notation is more convenient than the notation used in

the previous sections. In this notation the subscripts 1, 2, and 3 denote the material

directions of an orthotropic material and subscripts 4, 5, and 6 denote 2-3, 1-3, and 1-2

shear planes. Transformation between standard tensorial and compact notations can be

accomplished by replacing subscripts ij (or ks ) with α or β using the following roles

(Ting, 1996):

98

6 21or 125 31or 134 32or 233 33 2 22 1 11 or or

↔↔↔↔↔↔↔ βαksij

(3.66)

Similarly, the transformation between the notation used in the previous sections for 2-

dimensional loading condition and contracted notation can be accomplished by replacing

subscripts α or β with L , T , or S using the following roles:

S

LSTL

6 T 2

1 or , , or

↔↔↔↔βα

(3.67)

For a fiber reinforced lamina, subscripts 1,2, and 3 denote longitudinal, transverse, and

normal to the plane directions.

99

3.5.2 Incremental Constitutive Law

For a general orthotropic material the incremental constitutive law is:

σSε dd ⋅= (3.68)

Tddddddd ],,,,,[ 654321 εεεεεε=ε (3.69)

Tddddddd ],,,,,[ 654321 σσσσσσ=σ (3.70)

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

66

55

44

333231

232221

131211

000000000000000000000000

ss

ssssssssss

S (3.71)

For a nonlinear stress-strain curve, as material deforms, the tangent moduli of the

material changes. In this model, under multi-axial loadings the tangent modulus in each

direction is assumed to be a function of the stress in the same direction:

( ) ]6,,1[ ,Eα L∈= ασ ααE (3.72)

( )αα

αα σααEs s1

== (3.73)

100

Material response can be different in axial compression and tension, thus nine uniaxial

stress-strain curves are required to establish the tangent moduli. These curves are six

uniaxial tensile and compressive curves in the axial directions 1, 2, and 3, and three

uniaxial shear curves in shear planes 4, 5, and 6. The cross elements of the compliance

matrix are taken to be:

βαβαν

α

αβαβ ≠∈−= and ]3,2,1[, ,

Es (3.74)

For a fiber reinforced lamina the longitudinal direction (direction 1) is the dominant

direction and typically 21ν and 31ν are of the order of about 10-1 compared to 12ν and

13ν . Thus, it is assumed that:

( )( )11313

11212

νν

σνσν

==

(3.75)

Similarly, 23ν is taken to be a function of 2σ :

( )22323 ν σν = (3.76)

The reason that the transverse stress is selected but not the normal stress is that it plays a

more significant role in the over all behavior of a structure made of composite shell.

Therefore, six uniaxial Poisson’s ratio verses axial stress (or strain) curves are required to

101

establish the values of Poisson’s ratios under any combination of multiaxial loading.

These curves are 23ν versus axial tensile and compressive stress along direction 2, and

12ν and 13ν versus axial tensile and compressive stress along direction 1. Further, it is

assumed that at any state of deformation βααβ ss = . Therefore:

)(ν)(E)(E

),(ν

)(ν)(E)(E

),(ν

)(ν)(E)(E

),(ν

11311

33313131

11211

22212121

32322

33323232

σσσ

σσν

σσσ

σσν

σσσ

σσν

⋅==

⋅==

⋅==

(3.77)

In the previous sections it was mentioned that for the material to be elastic there are some

constrains on the cross members of the compliance matrix. Similar restrictions are

applied to the cross coefficients for any elastic material:

βαβαβααβ σσλαββααβ ≠∈⋅== ⋅⋅ and ]3,2,1[, ,0 esss (3.78)

Where, 0αβs and αβλ are integration constants.

102

3.5.3 Strain Energy Based Failure Criterion Under Three-Dimensional Loading

Suppose that the material is loaded along the stress path Γ from zero to stress state ∗σ .

As before, the stress path can be presented in parametric form, using the curve parameter

ξ :

]6,,1[ ),( K∈= αξσ αα L (3.79)

With ],0[ ∗∈ ξξ

The strain energy based failure criterion under 3-dimensional stress, using contracted

notation, is:

231312

23

322323

13

311313

12

211212

6

1

mmmm

AAAAFI

Π+Π⋅+

Π+Π⋅+

Π+Π⋅+⎥

⎦

⎤⎢⎣

⎡Π= ∑

=

sssα αα

αααα

(3.80)

Where

( ) βαβαβααβαβ ≠=Π+Π= and ]3,2,1[, ,sgns (3.81)

]6,,1[, ,.~0

L∈⋅′⋅=⋅⋅=Π ∫ ∫Γ

∗

βαξσσξ

βααββααβαβ dLLsds (3.82)

103

ααA ’s are the areas under uniaxial stress-strain curves and αβA ’s ( βα ≠ ) are computed

as:

βαβανν βααβββαααβ ≠∈⋅⋅⋅= and ]3,2,1[, ,A AA (3.83)

In the above relation, αβν and βαν are average values of the Poisson’s ratios under

uniaxial loading:

βαβασ

σν

να

σ

σ

ααβ

αβ

α

β ≠∈=

∫= and ]3,2,1[, ,

.

00

u

u

d

(3.84)

3.6 Closing Remarks

In this section a strain energy based failure model was developed to predict static

mechanical response and failure of a general brittle orthotropic material. Emphases to

distinguish between nonlinear elastic and inelastic deformations were made. To apply

the failure model to fiber reinforced composites, it was combined with maximum fiber

strain failure criterion. A computer program was developed based on the model that was

used to predict biaxial failure envelopes for three unidirectional laminates. Comparisons

104

between available experimental data and numerical results showed a good agreement

between the two. However, the existing experimental data was not sufficient to validate

all the features of the model, and it was concluded that more experimental data are

required to evaluate model parameters with higher accuracy and also to evaluate the

performance of the model in longitudinal-shear plane.

105

CHAPTER 4

4 MULTI DIRECTIONAL LAMINATES

4.1 Introduction

In this chapter a failure model for multi-directional laminates is proposed and model

predictions are validated by comparing them to experimental data. Classical lamination

theory (Reddy, 2004) with instantaneous moduli is employed to distribute in-plane load

increments between laminae during loading. The strain energy based theory developed in

Chapter 3 is utilized to predict matrix and fiber failures for each lamina. In this work the

ultimate (final) failure is generally considered to be the first fiber failure in one or more

laminae. Since in a multidirectional laminate fibers are arranged in multiple directions,

the maximum strength of a laminate can be higher than the stress level at the first fiber

failure, but this difference is only marginal, and for practical purposes the first fiber

failure should be considered as the ultimate failure. In this Chapter a post initial failure

model is developed for shear and transverse responses.

106

The experimental data used to develop the model and evaluate its parameters are taken

from the literature. The vast majority of these data were produced by testing tubular

specimens under combinations of axial loads and internal or external pressures. In all the

numerical analyses, the material is assumed to be planar and the curvature of the test

specimens is neglected. The global x direction in the numerical models is taken to be

the axial direction of the tubes and the global y direction is taken to be the hoop

direction. The first part of this chapter is devoted to the development of the model and

the evaluation of model parameters using experimental data. In Section 4.4 blind

predictions are made using the established model parameters and the predictions are

judged against experimental data. In one case, besides the blind predictions, tuned

predictions are also presented, i.e. the model parameters are adjusted to fit the predictions

to the experimental data. This last kind of prediction illustrates the model’s abilities to

reproduce a set of experimental data rather than the predictive capability of the model.

4.2 Matrix Stiffness in Multi-Directional Laminates

4.2.1 Shear Response in a Multi-Directional Laminate

Theoretically, in a [ 45 ]± o laminate under pure in-plane shear both the +45˚ and -45˚ plies

undergo pure shear deformation. Thus, this combination of lay-up and loading condition

107

is suitable for studying the stiffness characteristic of each lamina in a multi-directional

laminate. Figure 4.1, taken from Kaddour et al. (2003), presents shear stress-shear strain

curve for a single 90˚ unidirectional tube under torsion with back-calculated curves from

experimental responses of [ 45 ]± o tubes under various loading conditions. The back

calculated responses in this figure were obtained using classical lamination theory. The

shear stress-strain curves extracted from the response of [ 45 ]± o tubes under various hoop

to axial stress ratios are different from the shear response of the 90˚ tube. For example,

the back-calculated response under hoop stress to axial stress ratio of 1/ 1− (SR=1/-1) is

softer than the response of 90˚ tube but the failure occurs at a higher strength.

Previous work done by this author (Zand, 2004) revealed that reducing the transverse and

shear stiffness to zero upon matrix failure is unrealistic and leads to an under estimation

of the final strength. While this inaccuracy may be minor for fiber dominated lay ups in

which matrix stiffness does not play an important role, it can be significant in angle-ply

laminates under certain loading conditions. Figure 4.2 shows an example for such

conditions where the assumption of brittle matrix failure leads to unrealistic results. In

this analysis the shape factors and μ are taken to be one (see Chapter 3) and 0.5

respectively (although for this particular analysis, outcomes are not sensitive to changes

in μ or the shape factors because the transverse and longitudinal stresses are very small).

108

Figure 4.1. In-plane shear stress-strain curves for unidirectional E-glass/epoxy material

system from torsion tests on unidirectional laminates and the back-calculated response of

multi-directional laminates (Kaddour et al., 2003).

The experimental data presented in Figure 4.2 are the results of the work reported by

Kaddour et al. (2003) at the University of Manchester Institute of Research and

Technology (UMIST). They tested tubular specimens with and without liners, and no

significant difference between the measured strengths was found, confirming the failure

109

mode to be shear failure of the matrix. The measured final strengths exhibited

dependency in the wall thickness, as can be seen in Figure 4.3. It is believed that this

dependency was due to the structural buckling of the test tube under axial compression

(Kaddour et al., 2003). For wall thicknesses larger than 3 mm (radius to thickness ratios

smaller than about 30) the strength was found to be independent of the thickness. Thus,

the selected experimental result for this study is one of the stress-strain curves obtained

by testing a 5.9 mm thick tube which failed at a hoop stress of 94.8 MPa, hoop strain of

9.9%, and axial strain of -11.2%. The hoop stresses were calculated using thick tube

theory (Kaddour et al., 1998), and the values presented are the computed stresses and

measured strains at the internal wall.

The material properties and stress-strain curves used in the analysis were given in Table

3.1 and Figure 3.6. To implement the model, a computer code was developed for multi-

directional laminates with nonlinear behavior. At each solution step, the program applies

a load increment (including axial and shear loads, and bending moments) to a rectangular

element with unit length. Assuming the stress distribution is uniform across the edges of

the element, the stiffness matrix from the previous step is used to compute strain

increments. The strain increments are applied to each lamina, computing the

corresponding stress increment and updating the stiffness matrix using the model

developed in the previous chapter. A global unbalanced load vector is computed for the

laminate by integrating the stresses through the thickness using two Gaussian integration

points for each lamina. At each step, failure (matrix or fiber) of the lamiae is checked

and upon failure the load step size is divided by two and re-applied. This procedure is

110

repeated until the initial or ultimate failure point is determined within the desired

accuracy. Details of the solution algorithm are given in Appendix B.

0

20

40

60

80

100

120

-16% -12% -8% -4% 0% 4% 8% 12% 16%

Strain (%)

Hoo

p St

ress

(MPa

)

Numerical data with brittle matrix failure Experimental data (Lined tube, Kaddour et al, 2003)

HoopAxial

Figure 4.2. Predicted response compared with experimental data for S]45[ o± E-

glass/MY750 epoxy laminate under 1/1 −=SR

111

Figure 4.3. The influence of the wall thickness on the strength of S]45[ o± E-

glass/MY750 epoxy laminate test tubes under SR=1/-1 (Kaddour et al, 2003)

4.2.2 Exponential Stiffness Reduction Model

Exponential stiffness reduction models have been used to predict progressive failure of

composite laminates beyond initial failure by a number of researchers (Rotem, 1998). In

the proposed study the following exponential model is investigated:

( )exp loading unloading & reloading

YSd

S

G kG

U

γ γ⎧ ⎡ ⎤⋅ − −⎪ ⎣ ⎦= ⎨⎪⎩

(4.1)

112

Where, G is the shear modulus, k is a material parameter that dictates the rate of

stiffness reduction, and the subscript S refers to the in-plane shear direction. The

superscript Y refers to the material condition at the start of stiffness reduction, and dG is

the material modulus after reduction. As seen in Figure 3.1, stiffness reduction can start

before the initial failure, thus to model actual behavior, the transverse and shear strain

energy ratios at which degradation begins are taken to be material properties established

through experimentation.

4.2.2.1 Stiffness reduction factor

The stiffness reduction factor, Sk , is expected to be a function of material type, laminate

lay up, and curing processes. The effect of curing residual thermal stresses and interface

stress localization are embedded in the reduction factor Sk . Figure 4.4 demonstrates the

effect of Sk on the response of S]45[ o± E-glass/MY750 epoxy laminate under shear

loading. For the material system in Figure 4.4, a good fit between the numerical and

experimental responses is obtained when a relatively low value (10.0) is assigned to Sk ,

indicating that the shear stiffness reduction is relatively slow. In these analyses stiffness

degradation was assumed to begin upon the matrix failure, when the shear energy ratio

(SER) reached 1.0.

113

0

20

40

60

80

100

120

-20% -16% -12% -8% -4% 0% 4% 8% 12% 16% 20%

Strain

Hoo

p St

ress

(MPa

)

Numerical Data; ks = 1000 Numerical Data; ks = 50

Numerical Data; ks = 20 Numerical Data; ks = 10

Experimental data (Lined tube, Kaddour et al, 2003)

Ks =

2050

1000

10

HoopAxial

Matrix failure (SER = 1)


1/1 −=SR using various values for Sk and 0.1=SER

4.2.2.2 Shear energy ratio

In multi-directional laminates the residual thermal stresses generated during curing

combined with stress localization at ply interfaces causes crack growth to begin at a

lower energy level than laminates with unidirectional lay-ups. Thus, an appropriate value

for SER (shear strain energy ratio at the start of stiffness reduction) can be less than one.

114

In this section the influence of SER on the stress-strain curves is studied. Figure 4.5

presents the material response for three different value of SER , (1.0, 0.5, and 0). The

shear stiffness reduction factor, Sk , is taken to be 10 in all the three analyses. The initial

part of all the three curves is in good agreement with the experimental data. The curve

with 5.0=SER predicts the final strength with a somewhat better accuracy than the other

two values while on 0.0=SER reproduces the middle part of the curve more correctly.

For practical design purposes predicting material response at service strain levels is more

useful. Thus, a relatively low SER (less than 0.5) seems to be an appropriate value

because it leads to a better prediction of the initial and middle parts of the shear stress-

strain curve. In all the foregoing analyses, final failure occurred due to instability of the

laminate stiffness matrix, causing deformations to grow boundlessly.

115

0

20

40

60

80

100

120

-20% -16% -12% -8% -4% 0% 4% 8% 12% 16% 20%

Strain (%)

Hoo

p St

ress

(MPa

)

Numerical Data; ks = 10; SER = 1.0 Numerical Data; ks = 10; SER = 0.5

Numerical Data; ks = 10; SER = 0.0 Experimental data (Lined tube, Kaddour et al, 2003)

SER = 1.0 0.50.0

HoopAxial

E-glass/epoxy [+45/-45]s laminateunder σy/σx = 1/-1


1/1 −=SR using various values for SER with 10=Sk

When the material fails at relatively high strains, the second order effects such as bulging

of the test section and change in the fiber directions due to shear deformation (scissoring

effect) become significant. For a tubular specimen tested under internal pressure, as the

specimen bulges, an additional axial load is induced in the specimens (for details of

testing method see Kaddour et al., 2003). Furthermore, bulging causes diameter change

116

at the middle of the test section, which in turn increases hoop stress. These additional

axial and hoop stresses can be estimated knowing the values of hoop strain and the

internal pressure. The experimental data points and loading path have been corrected for

the specimen bulging. The corrected loading path can be seen in Figure 4.6. More details

of the correction methodologies can be found in Appendix C. Additional analyses are

performed to study the second order effects, in which fiber reorientation is accomplished

by updating orientation angles of the plies at each solution step and the effect of the

specimen bulging is included by using the corrected loading-path of Figure 4.6.

Presented in Figure 4.7 are three predicted stress-strain curves, all obtained using

1.0=SER and 15=Sk . The experimental data are corrected to include the increase in

the hoop stress due to specimen bulging. For the first curve both bulging and fiber

reorientation are included and SER and Sk are evaluated by trying different values until

a satisfactory fit to the experimental data was achieved. For the second curve only the

bulging effect is included and for the third one no second order effect is included. As

demonstrated in the figure, the bulging effect only slightly increases the predicted

strengths. Clearly, the reorientation of the fibers can significantly change the trend of the

stress-strain curves and the ultimate strength.

The predicted curve with fiber reorientation included exhibits an increase in the shear

stiffness at about 4% of axial strain. This increase, which occurs because the fibers

117

gradually reorient towards the principal stress directions, is known as the scissoring

effect. The occurrence of fiber scissoring has been confirmed through experimental

observations (Hinton et al., 2002).

0

20

40

60

80

100

120

-120 -100 -80 -60 -40 -20 0

Axial Stress (MPa)

Hoo

p St

ress

(MPa

)

Target load path

Corrected loadpath

Figure 4.6. Corrected loading path to account for specimen bulging

118

Final Failure

0

20

40

60

80

100

120

-16% -12% -8% -4% 0% 4% 8% 12% 16%

Strain

Hoo

p St

ress

(MPa

)

Bulging and fiber reorientation are considered Bulging is considered

Bulging and fiber reorientation are not considered Experimental data (Lined tube, Kaddour et al, 2003)

Ks = 15SER = 0.1

HoopAxial

Figure 4.7. The effect of the second order deformations

Kaddour et al. (2003) reported that the failure mode for this case was localized fiber

failure followed by a drop in pressure that occurred at the hoop strain of about 10%. That

the failure mode was localized fiber failure instead of a failure surface extending across

the entire test section confirms that stress distribution across the test section was not

uniform at such a high level of hoop strain. In the numerical analysis stress distribution

119

was assumed to be uniform and, as shown in Figure 4.8, the final failure occurred due to

the instability of the stiffness matrix, causing the hoop and axial deflections to go to

infinity.

0

20

40

60

80

100

120

140

-30% -20% -10% 0% 10% 20% 30%

Strain

Hoo

p St

ress

(M

Pa)

numerical predictions

Experimental data (Lined tube, Kaddour et al, 2003)

Ks = 15SER = 0.1The experimental data and loading path corrected for bulging effect

HoopAxial

Figure 4.8. Instability of the stiffness matrix in the numerical analysis occurs at a stress

level about 15% over the observed final failure

120

4.2.3 Stiffness Reduction in Tensile Transverse Direction

The same degradation model that was developed for shear response will be used for

transverse stiffness degradation:

( )exp loading unloading & reloading

eq YT T T Td

T

T

E kE

U

ε ε⎧ ⎡ ⎤⋅ − −⎪ ⎣ ⎦= ⎨⎪⎩

(4.2)

The subscript T refers to the transverse direction. Since transverse compressive and

tensile responses are in general different, different model parameters can be used

depending on the sign of transverse stress.

The experimental data presented in this section are the result of extensive studies of Reid

et al. (1995) which were cited by Soden et al. (2002). The data were obtained by testing

filament wound tubes made of E-glass/MY750 epoxy under combined axial load and

internal pressure. The tube wall consisted of four layers of E-glass/epoxy oriented at,

[45˚/-45˚/45˚/-45˚], the nominal thickness was 1 mm. These specimens had fiber volume

fraction of 0.55, as determined by burn-off test. The stress-strain curve presented herein

is one of the curves reported by Reid et al. (1995), for which the final hoop stress was

444 MPa and the corresponding hoop and axial strains were 2.47% and 2.17%

respectively. Readouts from several strain gages installed at four different specimens

tested under the same loading conditions varied by up to 22%, reported by Soden et al.

121

(2002). Although theoretically the hoop and axial strains are expected to be identical

(due to symmetry in loading and geometry), the measured hoop strains were larger than

axial strains. The failure strength of unlined tubes (weeping strength) under the same

loading conditions was measured by Soden et al. (1993) to be about 216 MPa. This

strength is consistent with the observations reported by Reid et al. (1995) that cracking

started at hoop stresses of 50 to 70 MPa with an increasing growth rate up to 200 MPa.

For the presented stress-strain curve, the hoop stress at which the maximum strains were

recorded was 419 MPa at which the strain gage failed. The failure strength of similar

tubes under the same loading conditions was 502±35 MPa, reported by Soden et al.

(1993).

Analysis results along with experimental data can be seen in Figure 4.9. Since the test

tubes had a fiber volume fraction of 0.55 instead of 0.6 as in Table 3.1 the input

longitudinal stiffness and strengths of the material were corrected accordingly:

uf

uL

mfffL EVEVE

εε =

−+⋅= )1( (4.3)

Where, fE and mE are the moduli of fiber and matrix and fV is the fiber volume

fraction. All the shape factors used in the analyses were taken to be one (see Chapter 3).

Since shear deformation is almost zero under this loading condition, shear properties of

the material (e.g. μ , Sk , SER ) do not play a role. Figure 4.9 includes four numerical

122

stress-strain curves obtained by assigning four different values to tTk . In all the analyses

the transverse stiffness reduction was assumed to start at the initial failure, that is the

transverse strain energy ratio at which reduction started (TER ) was assumed to be one.

The predicted initial failure occurs at about 63 MPa that is consistent with the

experimental observations. The measured weeping strength is about three times bigger

than the computed initial strength. Weeping of the tubes occurs when the cracks are

extended throughout the thickness and they are wide enough make the material

permeable to the liquid. The stress level at which the matrix failure of the last plies occur

(63 MPa in this example) can be interpreted as a conservative lower bound for the

weeping strength, but the proposed model does not include any criteria to predict

weeping strength.

Comparison between numerical and experimental results (Figure 4.9) indicates that

higher values of the reduction factor ( 200Tk = and 1000 ) lead to more realistic

predictions. This shows that after matrix failure, the transverse tensile modulus of the

material lowers relatively fast. As mentioned previously the experimental hoop and axial

strains do not coincide. This could be due to non-uniform stress distribution across the

test section, which in turn had resulted in a lower average hoop stress across the test

section than the assumed uniform value (the confining effect of the end reinforcement).

To investigate this possibility, a new set of analyses were performed with hoop to axial

stress ratio of 0.95 and different values of Tk and TER . The best results, presented in

Figure 4.10, were obtained when Tk and TER were equal to 400 and 1.0 respectively. It

123

is apparent that the predicted stress-strain curves and final strength are in a remarkable

agreement with the experimental data. The initial failure is predicted to occur at a hoop

stress of 63 MPa which is in agreement with the experimental observations mentioned

earlier. Since in the new analyses, hoop and axial stresses were not equal the plies under

went shear deformation. The values of Sk and SER were taken to be 15 and 0.1

respectively.

Crack initiation (Reid et al, 1995)

Weeping strength (Soden et al., 1993)

Final failure of tubes with 0.55 fiber volume fraction (Reid et al.,

1995)

0

200

400

600

800

0% 1% 2% 3% 4%

Strain

Hoo

p St

ress

(MPa

)

Experimental data for axial strain (Lined tubes, Reid et al, 1995)Experimental data for hoop strain (Lined tubes, Reid et al, 1995)

kT=10

50

200

1000

Figure 4.9. Experimental and numerical stress-strain curves of [ 45 ]S± o angle-ply

laminate made of E-glass/MY750 epoxy under biaxial tension of 1/1SR = . The

numerical analyses show the effect of tensile transverse degradation factor on material

behavior

124

Crack initiation (Reid et al, 1995)

Weeping strength (Soden et al., 1993)

Final failure of tubes with 0.55 fiber volume fraction (Reid et al.,

1995)

0

200

400

600

0% 1% 2% 3% 4%

Strain (Axial and Hoop)

Axi

al S

tres

s (M

Pa)

Experimental data for axial strain (Lined tubes, Reid et al, 1995)Experimental data for hoop strain (Lined tubes, Reid et al, 1995)Numerical predictions

Model parameters:kT = 400 (tension)TER = 1.0kS = 15SER = 0.1

Figure 4.10. Numerical predictions with 0.95 /1SR = versus experimental data

4.2.4 Stiffness Reduction in Compressive Transverse Direction

Structural buckling of tubes makes it challenging to produce experimental data under

compressive axial load and/or external pressure. Increasing the thickness of the tubes in

the gage section can suppress structural buckling but it introduces uncertainty in

125

interpretation of the data because there is no direct way to determine the stress

distribution across the thickness of a thick multi-directional laminate. In order to study

transverse behavior of laminae in a multi-directional laminate an indirect method is

chosen, that is the response of s]55[ o± laminate internal pressure. The experimental

stress-strain data for this case is taken from the work of Al-Khalil (1990), who tested

s]55[ o± tubes made of E-glass/MY750 epoxy under internal pressure. The axial load

generated by internal pressure was minimized by installation of tie-rods between the end

fittings.

Using classical lamination theory, it can be shown that for this loading case and lay-up all

the laminae undergo equal amounts of compressive deformation in their local transverse

directions. Although the applied stress is tensile, because laminae transverse stiffness is

considerably lower than longitudinal stiffness the tensile load is taken by the fibers

oriented in o55± directions, inducing a tensile component in x direction. Thus, laminae

transverse stresses must be compressive to cancel the tensile component from the

longitudinal directions.

The measured failure strength of the material was 595 MPa, which occurred at the hoop

strain of 8.8% and the axial strain of -10.9%. This strength is consistent with the values

reported by Soden et al (1989 and 1993) and Kaddour and Soden (1996). Soden et al.

(1989) and Al-Salehi et al. (1989) tested unlined tubes under the same loading condition

to measure weeping strength. Soden et al. (1989) reported two values of 362 MPa and

126

410 MPa for the weeping strength, while Al-Salehi et al. (1989) reported 427±12 MPa.

Equation 4.2 was used to reduce the transverse compressive stiffness of the material

beyond the initial failure. The shape factors were taken to be one and reorientation of

fibers due to shear deformation was accounted for. Other model parameters used in these

analyses were: 15Sk = , 0.1SER = , 0.1μ = , and 1.0TER = . The effect of cTk ( c

superscript stands for compression) on the stress-strain curves is shown in Figure 4.11.

It is apparent that using lower values of cTk resulted in a better agreement between the

experimental data and numerical predictions. Both numerical and experimental data

confirm that although the material is loaded in the hoop direction, axial strain is larger

than hoop strain.

127

0

200

400

600

800

-15.0% -10.0% -5.0% 0.0% 5.0% 10.0% 15.0%

Strain

Hoo

p St

ress

(MPa

)

Experimental data (Lined tubes, Al-Khalil, 1990)

Weeping of unlined tubes at 427±12MPa (Al Salehi et al., 1989)

kS = 15SER = 0.1SET = 1.0

kT=25

50

200

1000

Numerical predictions

Axial Hoop


under hoop stress, showing the effect of cTk

The next set of analyses is performed to demonstrate the effect of μ . The results are

presented in Figure 4.12. This parameter only influences shear stiffnesses of the laminae,

which are small compared to the longitudinal stiffnesses, and therefore they do not

significantly influence the behavior of the laminate. Since shear stiffnesses of the

128

laminae contributes more in the resultant axial stiffness than in the hoop stiffness the

effect of μ is higher on the axial strain than hoop strain. Since the transverse stress is

compressive for all the plies, laminate stiffness increases with increasing μ . The initial

failure occurs at hoop stresses of 267, 285, 310, and 328 MPa for μ equal to 0.1, 0.4,

0.8, and 1.2, respectively. These values are lower than the observed weeping strength of

427 MPa, reported by Al-Salehi (1989) and Soden et al. (1998).

0

200

400

600

800

-15.0% -10.0% -5.0% 0.0% 5.0% 10.0% 15.0%

Strain

Hoo

p St

ress

(MPa

)

Experimental data (Lined tubes, Al-Khalil, 1990)


kS = 15kT=25SER = 0.1SET = 1.0

μ=1.2 0.8 0.4 0.1


Axial Hoop

Figure 4.12. Predicted hoop and axial strains versus hoop stress curves for s]55[ o± E-

glass/MY750 epoxy laminate under internal pressure, showing the effect of μ

129

In the test method used by Al-khalil (1990) to produce the current experimental data, o-

rings were used at the two ends of the specimen to seal the end fixtures against the

specimen internal wall. Figure 4.13 presents the effect of o-ring friction on the response

of the laminate. The results are presented for two frictional values 2% and 4% of the

hoop stress. It is apparent that as axial stress increases both hoop and axial curves

become stiffer. Application of frictional forces more than 4% would decrease the

agreement between the experimental and numerical data.

The results presented in Figures 4.12 to 4.13 shows that the predicted response in the

hoop direction is stiffer than experimental response. This indicates that the longitudinal

modulus used in the analysis was higher than the actual value. In the next analysis,

presented in Figure 4.14, the longitudinal stiffness of the material is decreased by 10%.

O-ring friction values of 0, 2, and 4% are tried and the best agreement between the

numerical predictions and experimental data was achieved with 2% friction (the other

two are not plotted). The new stress-strain curves are in a good agreement with the

experimental data. The computed axial strains are slightly higher than the experimental

values at high levels of axial strains. The predicted final failure mode is fiber failure that

is in agreement with the experimental observations.

130

0

200

400

600

800

-15% -10% -5% 0% 5% 10% 15%

Strain

Hoo

p St

ress

(MPa

)

Experimental data (Lined tubes, Al-Khalil, 1990) Numerical data; no o-ring friction

Numerical data; 2% o-ring friction Numerical data; 4% o-ring friction


kS = 15kT = 25μ = 0.5SER = 0.1SET = 1.0

Axial Hoop

4%

2%

0%


under hoop stress, showing the effect of o-ring friction

131

0

200

400

600

800

-15.0% -10.0% -5.0% 0.0% 5.0% 10.0%

Strain

Hoo

p St

ress

(MPa

)

Experimental data (Lined tubes, Al-Khalil, 1990) Numerical data


kS = 15kT = 25μ = 0.5SER = 0.1SET = 1.0o-ring friction:2%Longitudinal stifness reduced for all laminae

Axial Hoop


under hoop stress. Material parameters and longitudinal stiffness were adjusted

132

4.3 Stiffness Reduction Parameter for Angle-Ply Laminates

The stiffness reduction coefficients for an [ ]θ± angle-ply laminate are expected to be

highly dependent on θ . As θ decreases from 45o to 0o , the stiffness reduction

coefficients should increase as laminate behavior becomes more like the behavior of a

unidirectional laminate. In the previous sections the coefficients were estimated using

experimental stress-strain curves of [ 45 ]S± o and [ 55 ]S± o laminates. In this section axial

versus hoop stress failure envelopes are developed for [ 55 ]S± o and [ 85 ]S± o laminates

made of E-glass/MY750 epoxy. Model parameters for these lay-ups were evaluated by

fitting numerical predictions to the available experimental data.


glass/MY750 Epoxy

The experimental data for E-glass/epoxy specimens were obtained by Soden et al (1989

and 1993) and Kaddour and Soden (1996) by testing tubular specimens under combined

axial load and internal pressures. Most of the tubes tested without a liner failed due to

weeping or jetting of oil. Details of the testing methods and end fixtures have been

published by Kaddour et al. (1998). Most of the specimens tested under compression had

the inside diameter to the wall thickness ratio of 5 and the fiber volume fraction of 0.68.

Soden et al. (2002) reported the compressive strength of unidirectional material system

133

with the fiber volume fraction of 0.7 to be 1150 MPa. Thus, compressive strength of

1150 MPa instead of 800 MPa given in Table 3.1 was used in the analyses. The tubes for

which the failure data are presented herein failed by rupture rather than by structural

buckling. The presented axial and hoop strengths are theoretical maximum stresses

calculated by Kaddour et al. (1998) using orthotropic elastic thick plate theory that

occurred at the inner wall.

Several analyses were performed with different values of sk , SER , cTk until a

satisfactory agreement between measured and computed values was achieved. On the

bases on the results presented in the previous sections, the relatively large value of 400

was assigned to tTk . In all the analyses, fiber reorientation (scissoring effect) was

included, but test section bulging was neglected. The axial versus hoop stress was

assumed to be proportional during loading. Figure 4.15 presents experimental data with

some of the analytical final failure envelopes obtained by varying Sk . Figure 4.16

presents the effect of μ on initial and final failure envelopes. In this figure, three initial

(dashed) and three final (solid) failure envelopes are presented for μ equal to 0.0, 0.5,

and 1.0.

The shear stiffness reduction factor, Sk , affects the failure envelopes primarily in the first

quadrant, whereas μ affects the entire initial failure envelope and the final failure

envelope in the second and third quadrant. Several dozen analyses were performed, each

with a different set of material parameters to find a satisfactory fit between the

134

experimental and numerical results. From the results plotted in Figure 4.17 it can be seen

that a good fit is achieved when 50T

cSk k= = , 0.1SER = , 400t

Tk = , and 0.5μ = are

selected.

-1200

-800

-400

0

400

800

1200

-800 -400 0 400 800

Axial Stress (Mpa)

Hoo

p St

ress

(MPa

)

Numerical data; kS = 50



Experimental data (Thick tubeswithout liner, Kaddour et al, 1997)

Experimental data (Thin tubeswith liner, Soden et al, 1989,1993)

Experimental data (Thick tubeswith liner, Soden et al, 1989,1993)

Experimental data (Thin tubeswithout liner, Soden et al,1989,1993)

Experimental data (Thick tubeswithout liner, Soden et al, 1989,1993)

SER = 0.1kT = 400 (tensile)kT = 25 (compressive)μ = 0.5

kS = 20

kS = 30

kS = 50

Figure 4.15. Numerical and experimental final failure envelopes in the axial-hoop plane

for [ 55 ]S± o E-glass/MY750 epoxy laminate. Numerical predictions made using three

values of 40, 30, and 20 for Sk to demonstrate its effect on the failure envelope

135

-1200

-800

-400

0

400

800

1200

-800 -400 0 400 800

Axial Stress (Mpa)

Hoo

p St

ress

(MPa

)


Experimental data (Thin tubes withliner, Soden et al, 1989, 1993)

Experimental data (Thick tubeswith liner, Soden et al, 1989, 1993)



Model Parameters:kS = 40SER = 0.1kT = 400 (tensile)kT = 40 (compressive)

Final Failure μ = 0.0 0.5 1.0

Initial Failure μ = 0.0 0.5 1.0


laminate, showing the effect of μ . The dashed lines are initial and solid lines are final

failure envelopes

Predictions for final strength envelope are in good agreement with the test results for the

lined tubes. However, the numerical predictions in the third quadrant are less accurate

because they are obtained using a 2-dimensional theory assuming the geometry is planar,

which is a realistic assumption for thin tubes but may not be as accurate for thick tubes.

136

On the basis of linear lamination theory, this lay-up has its maximum strength at

1/ 2SR = , that is the stress ratio for a pressure vessel with end caps. This is consistent

with the experimental observation in the third quadrant and the model predictions. On

the other hand, the experimental data in the first quadrant shows a maximum strength at a

SR higher than 2. Figure 4.17 shows that the predicted initial failure strengths define a

lower bound for the tubes tested without a liner.

-1200

-800

-400

0

400

800

1200

-800 -400 0 400 800

Axial Stress (Mpa)

Hoo

p St

ress

(MPa

)

Predicted final fialure

Predicted initial failure


Experimental data (Thin tubeswith liner, Soden et al, 1989,1993)

Experimental data (Thick tubeswith liner, Soden et al, 1989,1993)



ks = 50SER = 0.1kT = 400 (tension)kT = 50 (compression)μ = 0.5

SR = 2/1

SR = -2/-1


laminate, showing good agreement between predictions and experimental results for the

selected set of model parameters

137


glass/MY750 Epoxy

These test data for this loading condition were obtained by Al-Khalil (1990) by testing

[ 85 ]S± o wound tubes (with a liner) under combined axial load and internal pressure. The

specimens had wall thickness of 0.95 or 1.2 mm fiber volume fraction of 0.6.

Relatively large values of 300Sk = , 0.8SER = , and 600tTk = were selected for the

stiffness reduction parameters and μ was assigned to be to 0.5. Figure 4.18 presents a

comparison between the predicted and experimental final failure envelopes. Also

presented is the predicted initial failure envelope for this laminate. The experimental

data are provided for the second quadrant only, where all the laminae are under tensile

stresses in both transverse and longitudinal directions. In this quadrant, agreement

between experimental and numerical data is independent of material compressive

properties such as cTk . The figure shows good agreement between the experimental data

and model predictions.

138

-1200

-800

-400

0

400

800

1200

1600

-400 -300 -200 -100 0 100 200 300 400

Axial Stress (Mpa)

Hoo

p St

ress

(MPa

)

Predicted final failuredata

Predicted initial failruedata

Experimental data (Al-Khalil, 1990)

kS = 300SER = 0.5kT = 600 (tension)kT = 600 (compression)μ = 0.8

Figure 4.18. Axial versus hoop stress initial and final failure envelopes for [ 85 ]S± o E-

glass/MY750 epoxy laminate

The values of Sk and tTk for this lay-up are both significantly higher than those for

[ 55 ]S± o lay-up. The stiffness reduction is greater for [ 85 ]S± o lay-up as compared to

[ 55 ]S± o lay-up. This is apparent in Figure 4.18, as unlike Figure 4.17 here the computed

initial and final failure envelopes are relatively close. The value of SER for this laminate

139

is also higher than the value obtained for [ 55 ]S± o laminate (0.8 versus 0.1). These

observations are consistent with the concept of the stiffness reduction theory for multi-

directional laminates mentioned earlier in Section 4.2.

Figure 4.19 shows several stress-strain curves computed for different tensile hoop to

compressive axial stress ratios ranging from 1/ 0 to 1.5 / 1− . The hoop response is linear

at any stress ratio but the axial response which is dominated by matrix stiffness is

nonlinear at stress ratios larger than 10 / 1− . The final strength is highly influenced by

the stress ratio, as under uniaxial hoop stress it is about 1150 MPa and as the stress ratio

increases it reduces quickly.

140

0

200

400

600

800

1000

1200

1400

-3.0% -2.0% -1.0% 0.0% 1.0% 2.0% 3.0%

Axial and Hoop Strains

Hoo

p St

ress

(MPa

)

Hoop StrainAxial Strain

Fiber Failure

Matrix Failure

S.R. = 1.5/-1

4/-1

7/-1

10.5/-1 32/-1 1/0

kT = 600kS = 300TER = 1.0SER = 0.8

Figure 4.19. Stress-strain curves for [ 85 ]S± o E-glass/MY750 epoxy laminate under

various hoop to axial stress ratios

141

4.4 Evaluation of Model Predictive Capability

In this section the predictive capability of the proposed failure theory is evaluated by

comparing model predictions with experimental data from the literature. Two types of

predictions are presented in this section: 1- blind predictions, and 2- tuned predictions.

Blind predictions are presented for every case studied in this section in which model

parameters evaluated in the foregoing sections are used. For selected cases model

parameters are further tuned to get the best fit between the experimental data and model

predictions. Blind predictions represent the accuracy of the model when no or little

experimental data are available to evaluate model parameters under multi-axial loadings.

The tuned predictions show how well the proposed model can reproduce an existing set

of experimental data.


Epoxy Subject to Combined Axial and Torsional Loads

In this section the biaxial failure envelope for a [90 / 30 / 90 ]S±o o o laminate made of E-

glass/epoxy (E-glass/LY556/HT907/DY063) under combined axial and shear stress is

computed. Experimental data for this case, as cited by Soden et al. 2002, were originally

obtained by Hütter et al. (1974). The laminate was not quasi-isotropic, as the o90 layers

formed 17.2% and o30± layers formed 82.9% of the total thickness. The material

142

properties for a unidirectional system were given in Table 3.1. Transverse-shear shape

factors used in the numerical analysis were taken to be 0.6 as evaluated in Chapter 3 and

μ was assumed to be 0.5. Based on the results from the foregoing sections, 400tTk = ,

25cTk = , 15Sk = , 0.1SER = , and 1.0TER = were selected. Figure 4.20 presents a

comparison between the predicted and experimental failure envelopes. Also presented in

this figure is the predicted initial failure envelope, although there are no experimental

data available for the initial failure. It is apparent that the experimental and predicted

failure envelopes are in good agreement in most regions. In the first quadrant the model

over predicts the measured shear strengths for shear to axial stress ratios larger than about

6/5. The experimental data has some scatter in the first quadrant particularly for the axial

stress ratios larger than 21 . Thus, it is possible that the measured strengths for the stress

ratios larger than 6/5 are influenced by the boundary conditions or other experimental

factors not included in the analysis. In the second quadrant the predictions are generally

conservative and while they are close to the experimental data, they do not follow the

trend of the data.

Figure 4.21 presents the effect of cTk and Sk on the final failure envelope. The initial

failure envelope is insensitive to stiffness reduction factors and all the initial failure

envelopes coincide. Compared to [ 55 ]S± o angle-ply laminates previously studied (Figure

4.15), the final strengths of this laminate are less sensitive to stiffness reduction factors,

because the final strengths for the current lay-up are fiber-dominated.

143

0

200

400

600

-800 -400 0 400 800

Axial Stress (MPa)

shea

r Str

ess

(MPa

)Experimental data (Hutter et al, 1974) Numerical initial failrue

Numerical final strength

Model Parameters:kS = 15SER = 0.1kT = 25 (compression)kT = 400 (tension)TER = 1.0μ = 0.5

Figure 4.20. Biaxial initial and final failure envelopes under combined axial and shear

stress for [90 / 30 / 90 ]S±o o o laminate made of E-glass/epoxy material

144

0

200

400

600

-800 -400 0 400 800

Axial Stress (MPa)

shea

r Str

ess

(MPa

)

Experimental data (Hutter et al, 1974)

Model Parameters:kS = var.SER = 0.1kT = 400 (compression)kT = var. (tension)TER = 1.0μ = 0.5

kT = kS = 200

kT = kS = 50kS = 15kT = 25

kT = kS = 10

Figure 4.21. Initial and final failure envelopes under combined axial and shear stress for

[90 / 30 / 90 ]S±o o o laminate made of E-glass/epoxy material, showing the effect of cTk and

Sk

In the subsequent analysis, the parameters and material initial moduli were altered to

improve the fit of the model to the experimental data. The final results are depicted in

Figure 4.22. The outcomes are significantly improved after the stiffnesses of the 90o

plies were increased by 15% and the stiffness reduction factors of 30± o plies were

increased. The current results suggest that the stiffness reduction rate of the outer layers

145

can be lower than other layers. This is physically sound because unlike the interfaces

between different laminae, at the free outer layers of the laminate there is no stress

localization. More investigation would help to confirm this observation.

0

200

400

600

-800 -400 0 400 800

Axial Stress (Mpa)

shea

r Str

ess

(MPa

)

Experimental data (Hutter et al, 1974) Predicted initial failrue

Predicted final strength using parameter set I Predicted final strength using parameter set II

SER = 0.1kT = 400 (tension)TER = 1.0μ = 0.5

Parameter set I:kS = 28 and 15kT = 40 and 25 (compression)15% increase in Stiffness of 90 deg plies

Parameter set II:kS = 15kT = 25 (compression)

Final failure w/ Parameter set I

Final failure w/ Parameter set II

(pre tuning) (post tuning)

Figure 4.22. Biaxial failure envelopes for [90 / 30 / 90 ]S±o o o E-glass/epoxy laminate

before and after 15% increase in the stiffness of o90 plies

146


Epoxy Subject to Combined Axial and Hoop Stress

In this section the biaxial failure envelope for [90 / 30 / 90 ]S±o o o E-glass/epoxy laminate

in the axial-hoop stress plane is computed. The experimental data for this case were

originally obtained by Hütter et al. (1974) and cited by Soden et al. (2002). The data

were produced by testing lined filament wound tubes under combined internal or external

pressures and axial forces. Material type and testing conditions were the same as those

explained in the previous section. The experimental data under axial compression is

reported to be affected by structural buckling of the specimens (Soden et al., 2002).

The predictions were made using model parameters from the previous section and the

results for initial and final strengths are presented in Figure 4.23. The predicted final

strengths are fairly realistic in the first quadrant. The experimental data shows some

scatter in the first quadrant for hoop to axial stress ratios of one. In this region the

numerical predictions seem to be at the upper bound of the measured strengths. From the

figure, it is apparent that as compressive stresses increase the predicted strengths diverge

from the measured values, probably due to structural buckling of the tubes under

compressive stresses. As mentioned previously, Soden et al. (2002) reported that the

experimental data under axial compression had been influenced by structural buckling.

147

-800

-400

0

400

800

-800 -400 0 400 800

Axial Stress (MPa)

Hoo

p St

ress

(MPa

)

Experimental data (Hutter et al, 1974) Predicted final failure envelope

Predicted initial failure envelope

Model Parameter:SER = 0.1kS = 28, 15TER = 1.0kT = 400 (tension)kT = 40, 25 (compression) μ = 0.5

Figure 4.23. Biaxial initial and final failure envelopes for [90 / 30 / 90 ]S±o o o laminate

made of E-glass /epoxy material

Ply Fiber failure index Matrix failure index 90o 1.00 0.96 30+ o 1.00 0.96 30− o 1.00 0.96

Table 4.1. Matrix and fiber failure indices for [90 / 30 / 90 ]S±o o o E-glass/epoxy laminate

loaded under hoop to axial stress ratio of -0.82/-1

For one data point in the third quadrant the predicted initial and final strengths coincide.

148

Here the first failure mode is fiber failure. This data point was computed under hoop

stress to axial stress ratio of -0.82/-1. Table 4.1 shows the ply failure indices for the

laminate at failure under this load proportion. It is interesting that for this particular

loading condition all the fiber and matrix failure indices are close to one.

4.4.3 Biaxial Failure Envelope for [90˚/ ± 45˚/0˚]s Quasi-Isotropic Laminate Made

of Carbon/Epoxy Subject to Combines Axial and Hoop Stress

Failure of quasi-isotropic S]0/45/90[ ooo ± composite tubes made of AS4/3501-6

carbon/epoxy under combined axial and hoop stresses was studied by Swanson and

Nelson (1986), Swanson and Christoforou (1986), Swanson and Colvin (1989), and

Colvin and Swanson (1993). The experimental data are presented in Figure 4.24.

Specimens with various wall thickness to diameter ratios were tested. The measured

mean strengths were 637 MPa for thick tubes and 375 MPa for thin specimens. Soden et

al. (2002) stated structural buckling of the thinner tubes as a possible explanation of this

difference in the finals strengths. The reported failure mode in the tension-tension

quadrant was fiber failure and in compressive-compressive quadrant was structural

buckling. Responses of the specimens tested under internal pressure and tensile axial

loads were fiber dominated (linear) with a small change in the slope after initial failure.

All the specimens had a plastic liner to hold pressure after matrix cracking. The

experimental data were not furnished for the second quadrant, however the envelope

should be symmetric with respect to the yx σσ = line.

149

The material stress-strain curves and other properties used in the analyses were taken

from Soden et al. (1998) and they are presented in Figure 4.25 and Table 4.2. The

transverse compressive and particularly shear responses are non-linear and the

longitudinal tensile response was slightly nonlinear. Figure 4.26 presents computed

initial and final failure envelopes using two sets of model parameters plotted with the

experimental data. The model parameters set I are typical values evaluated in the

previous sections for multi-directional laminates, while in the second set, large values are

assigned to k s. The latter analysis is presented to show the sensitivity of model

predictions to the stiffness reduction factors for this lay-up. Since the behavior of the

laminate is fiber dominated the two failure envelopes are close, with the first one being

closes to the experimental data.

150

-1200

-800

-400

0

400

800

1200

-1200 -800 -400 0 400 800 1200

Axial Stress (Mpa)

Hoo

p St

ress

(Mpa

)

Experimental data(Swanson and Nelson,1986)

Experimental data(Swanson and Trask, 1989)

Experimental data(Swanson and Christoforou,1986)

Experimental data (Colvinand Swanson, 1993)

Experimental data(Swanson and Colvin, 1989)

Figure 4.24. Experimental data for biaxial failure of S]0/45/90[ ooo ± composite tubes

made of AS4/3501-6 carbon/epoxy laminate under combined pressure and axial load

In the first quadrant the predictions are in good agreement with experimental data. In the

fourth quadrant the failure envelope overestimates the measured strengths. As the

compressive stress in the hoop direction reduces, the amount of over-estimation decreases

and better agreement between measured and predicted strengths is apparent. The

proposed model can predict the measured strengths of thick tubes under axial

151

compression. The disagreement between the predictions and the experimental data in the

third quadrant is believed to be due to structural buckling of the test tubes under axial and

circumferential compression. In this quadrant the experimental data are likely to be

erroneous because they are not symmetric with respect to the x yσ σ= line, while

theoretically they are supposed to be so. The predicted failure modes in the first quadrant

were fiber failure which is in agreement with the experimental observations.

0

50

100

150

200

250

0.000 0.005 0.010 0.015 0.020 0.025

Strain (%)

Stre

ss (M

Pa)

Transverse Tensile

Transverse Compressive

Shear

Figure 4.25. Transverse and in-plane shear responses for a unidirectional AS4/3401-6

laminate (Soden et al., 1998)

152

Fiber Type AS4

Matrix 3501-6 epoxy

Specification Prepeg

Fiber volume fraction 0.6

Longitudinal modulus 126 (GPa)

Major Poisson’s ratio, LTν 0.28

Longitudinal tensile strength 1950 (MPa)

Longitudinal compressive strength 1480 (MPa)

Longitudinal tensile failure strain 1.38%

Longitudinal compressive failure strain 1.175%

Transverse modulus 11 (GPa)

Transverse tensile strength 48 (MPa)

Transverse compressive strength 200 (KPa)

Transverse tensile failure strain 0.436%

Transverse compressive failure strain 2.0%

Initial in-plane shear modulus 6.6 (GPa)

In-plane shear strength 79 (MPa)

In-plane shear failure strain 2.0%

Table 4.2. Mechanical properties of the unidirectional AS4/3401-6 (Soden et al., 1998)

153

-1200

-800

-400

0

400

800

1200

-1200 -800 -400 0 400 800 1200

Axial Stress (Mpa)

Hoo

p St

ress

(Mpa

)

Experimental data (Swanson and Nelson, 1986) Experimental data (Swanson and Trask, 1989)

Experimental data (Swanson and Christoforou, 1986) Experimental data (Colvin and Swanson, 1993)

Experimental data (Swanson and Colvin, 1989) Numerical predictions using parameter set I

Numerical predictions using parameter set II Predicted initial failure

Parameter Set IkS = 28SER = 0.1kT = 400 (tension)kT = 40 (compression)TER = 1.0μ = 0.5

Parameter Set IIkS = 1000kT = 1000 (tension & compresion)SER = TER = 1.0μ = 0.5

Figure 4.26. Biaxial failure envelopes for S]0/45/90[ ooo ± composite tubes made of

AS4/3501-6 carbon/epoxy laminate under combined pressure and axial load

154

4.4.4 Stress-Strain Curve For Quasi-Isotropic [90˚/ ± 45˚/0˚]s AS4/3501-6 Laminate

Under Uniaxial Tension In the Hoop Direction

In this section the stress-strain response of the quasi-isotropic AS4 laminate (same lay-up

and material system as the one studied in the previous section) is studied under uniaxial

stress in the hoop direction. The experimental data were produced by Christoforou

(1984) and later published by Swanson and Christoforou (1987). Due to the friction

induced by the O-rings, the actual stress ratio was estimated to be 20/1 instead of 1/0

(Soden et al. 2002). The final failure occurred at the hoop stress of 718 MPa and axial

and hoop strains of -0.36% and 1.45%, respectively. The mean strength of similar tubes

tested under similar conditions was reported by Soden et al. (2002) to be 42713 ± MPa,

which is in agreement with 718 MPa.

Presented in Figure 4.27 are the predicted axial and hoop responses for this loading case

along with the experimental data. The predicted and measured values are in a remarkable

agreement. The model parameters used in this analysis are the same as those used in the

previous section. The model predicts the matrix failure of o0 plies to occur at the hoop

stress of 230 MPa and that for o30± plies at the hoop stress of 416 MPa. The latter is

very close to the hoop level of 400 MPa at which a small reduction was observed in slope

of the axial strain versus hoop stress curve (Soden et al. 2002). The final strength was

predicted to be 732 MPa that is very close to the experimental values of 718 MPa. The

predicted failure mode is in agreement with the observed failure mode (fiber failure).

155

Matrix failure of 0 deg plies

Matrix failure of +30/-30 deg plies

Fiber failrue of 90 deg plies

0

200

400

600

800

-1.0% 0.0% 1.0% 2.0% 3.0%

Strain

Hoo

p St

ress

(MPa

)

Predicted axial strain Predicted hoop strain

Experimental data (Christoforou, 1984) Predicted iIntermediate failures


made of AS4/3501-6 carbon/epoxy laminate under a hoop to axial stress ratio of 20/1

4.4.5 Stress-Strain Curve For Quasi-Isotropic [90˚/ ± 45˚/0˚]s AS4/3501-6 Laminate

Under Hoop to Axial Stress Ratio of 2/1

The material type and lay-up are the same as those studied in Sections 4.4.3 and 4.4.4.

The experimental data used to validate model predictions is the result of the work

156

accomplished by Trask (1987), who tested two specimens under this loading proportion.

Failure hoop strengths of the specimens were 857 and 847 MPa. The stress-strain data

presented herein are extracted by Soden et al. (2002) from the graphs presented by Trask.

Model parameters and material properties used in the analysis were the same as those

presented in Sections 4.4.3. Figure 4.28 presents a comparison between the predicted

and experimental stress-strain curves under this loading condition. The predictions and

experimental data are in good agreement. As marked on the figure, the first predicted

intermediate failure is matrix failure of o0 plies, followed by matrix failure of o30±

plies, matrix failure of o90 plies, and finally fiber failure o90 plies. The predicted

response is linear both in the hoop and axial directions. The experimental data shows a

sudden increase in the hoop strain at the hoop stress of about 450 MPa that is not

predicted by the model. The predicted final failure occurs at the hoop stress of 869 MPa

that is close to the measured values of 852 MPa from the average of the two tested

specimens. The experimental data indicates a very slight decrease in the slope of hoop

strain curve after the predicted initial failure, which confirms that the predicted initial

failure is where the matrix cracking of the specimen started.

157

Fiber failure of 90 deg plies

Matrix failure of 0 deg plies

Matrix failure of +30/-30 deg plies

Matrix failrue of 90 deg plies

0

200

400

600

800

1000

0.0% 0.5% 1.0% 1.5% 2.0% 2.5%

Strain

Hoo

p St

ress

(MPa

)

Experimental data (Christoforou, 1984) Predicted axial strain

Predicted hoop strain Predicted intermediate failures


made of AS4/3501-6 carbon/epoxy laminate under hoop to axial stress ratio of 2/1

158

4.4.6 Stress-Strain Curves for [±55º] E-glass/MY750 Epoxy Laminate under Hoop

to Axial Stress Ratio of 2/1

The material type and lay-up of this laminate are the same as those of the laminate

analyzed in Section 4.2.4. The test data for this loading condition were obtained by Al-

Khalil (1990), who tested [ 55 ]± o filament wound tubes with liner and end caps under

combined axial load and internal pressure. The final failure occurred due to fiber fracture

at a hoop stress of 668 MPa. The measured failure hoop strain was 2.5% and the

measured axial failure strain ranged from 3 to 4.2%. Similar experiments under the same

loading conditions were conducted by Soden et al. (1989) and Al-Kalil (1990) measuring

the final failure hoop stress to be 684 and 692 respectively. Soden et al. (1989) tested

one unlined tube under the same loading condition. Failure occurred due to oil weeping

at a hoop stress of 280 MPa.

Figure 4.29 presents a comparison of the numerical predictions using the material

properties and model parameters of Section 4.4.2 and the experimental data. The

presented axial strains are from the strain gage that recorded the highest value (4.2%) at

failure. Both the numerical predictions and the experimental data are corrected for

bulging effects. The agreement between the predictions and the experimental data is

good for the hoop strain. The predicted final strength of 776 MPa overestimates the

measured value of 692 MPa reported by Al-Khalil (1990). The model can predict the

initial part of the axial strain response with acceptable accuracy. Both the predicted and

159

experimental axial responses that are initially linear, exhibit a strong non-linearity after

the predicted initial failure. This confirms that the predicted initial strength is realistic.

However, the model cannot predict the significant reduction in the axial stiffness after the

weeping strength. The experimental data show that the axial and hoop strains intersect at

hoop stress of about 400MPa that is not predicted by the model.

Although, Soden et al. (2002) reported that the axial strains measured under this loading

condition by several different researchers are somewhat scattered, the difference between

predictions and experimental data is compelling. The disagreement is believed to be due

to a phenomenon not included in the model. It should be noted that none of the theories

included in the WWFE were able to predict the trend of the axial strain beyond the

weeping strength (Hinton et al., 2002b). That the predicted and experimental curves

diverged after observed weeping strength combined with the scatter in the measured axial

strains strongly suggests crack opening as a reason for this disagreement. For example,

formation of transverse cracks at the two ends of the test section, where the stress

localization exists, can decrease the effective cross-section of the specimens and increase

the actual axial stress across the test section.

160

Weeping strength; Soden et al. (1989)

Weeping strength; Kaddour et al. (1996)

Final failure; Kaddour et al. (1996)

Final failure; Al-Khalil (1990)

0

200

400

600

800

1000

0.0% 1.0% 2.0% 3.0% 4.0% 5.0%Strain

Hoo

p St

ress

(Mpa

)

Experimental data (Al-Khalil, 1990) Numerical predictions Predicted initial failure

Axial strain

Hoop strain

Corrections made for bulging effectskS = 20SER = 0.1kT = 400kT = 20 TER = 1.0μ = 0.5

Fiber failure

Figure 4.29. Numerical and experimental stress-strain curves for S]55[ o± angle-ply

laminate made of E-glass/MY750 epoxy for hoop stress to axial stress ratio of 2/1

161

4.4.7 Stress-Strain Curves for [0º/90º]S Cross-ply Laminate Made of E-

glass/MY750 Epoxy Under Uniaxial Stress in 90˚ Direction

Soden et al (2002) reported experimental results conducted to determine the behavior of

coupon specimens of [90˚/0˚/0˚/90˚] laminate under uniaxial tension. Five specimens

were tested under uniaxial load in the direction of the 0˚ plies. The measured mean

strength was 590 MPa and the failure occurred due to fiber failure of o90 plies. The

mean failure strain in the direction of loading was 2.69%. In the transverse direction

failure strain was -0.13%. Particular attention was paid to monitor the onset of the

cracking and it was determined that the cracking began at a stress level of 117.5 MPa.

Weeping strength of filament wound tubular specimens made of similar material system

was measured by Eckold (1995) to be 400 MPa.

Figure 4.30 presents the predicted stress-strain curves under this loading condition along

with the experimental data. The final failure strength for the presented experimental

stress-strain curves is 609 MPa. The material properties are the same as those evaluated

in Section 4.2.4 for a similar lay-up. Two intermediate failures were predicted at axial

stresses of 78 and 406 MPa, corresponding to matrix failure in the o0 plies and o90 plies,

respectively. The predicted initial failure occurs at about 70% of the stress level at which

the onset of cracking was observed. The second predicted intermediate failure matches

the measured weeping strength. The experimental data shows some nonlinearity after the

onset of cracking which is predicted by the model. The predicted responses in the

162

x direction is somewhat stiffer than the experimental responses after the weeping

strength. The predicted final strength is 672 MPa due to the fiber failure of 90o plies.

The difference between the predicted and measured final strength (672 MPa versus 602

MPa, respectively) may be due to the edge effects in the coupon specimens that can

reduce their strength.

Weeping Strength of tubular sample, Eckold

(1995)

Cracking of coupon specimens, Hinton

(1997)

0

200

400

600

800

-1.0% 0.0% 1.0% 2.0% 3.0% 4.0%εx and εy

σ y (M

Pa)

Experimental data of coupon specimens, Hinton (1997)


Predicted intermediate failures

Material Parameters: kS = 15SER = 0.1kT = 25 (compression)kT = 400 (tension)TER = 1.0μ = 0.5

Failure of 90 deg plies

Figure 4.30. Stress-strain curve of cross-ply E-glass/MY750 epoxy ]90/0[ oo laminate

under axial load in the y direction

163

4.4.8 Stress-Strain Curve for [±85º]S Cross-ply Laminate Made of E-glass/Epoxy

Under Axial Stress

The experimental data for this case were presented by Al-Khalil (1990). Experimental

data and numerical predictions are given in Figure 4.31. Model parameters used for the

numerical analysis were the same as those presented in Section 4.3.2 for a similar

material and lay-up. The measured hoop strains from three strain gages installed on the

specimen ranged from 0.3% to 0.6%. The hoop strains were slightly non-linear, where as

the axial response was linear. Since the numerical values of the experimental data were

not available, the graph presented by Al-Khalil was scaled and superimposed on the

predictions. The agreement between the experimental and theoretical data is good in the

axial direction. For the hoop direction the experimental data are scattered and the

predictions are consistent with the strain gage that measured the highest values.

However, the model does not predict the non-linearity that can be seen in the hoop

direction. This nonlinearity indicates that the Poisson’s ratio is not constant and it

decreases as the material deforms. The proposed model can take into account this

nonlinearity; however, the difference between the predicted and observed responses is not

large enough to justify the additional effort.

164

Figure 4.31. The predicted and experimental stress-strain curve for S]85[ o± laminate

made of E-glass/MY750 epoxy under uniaxial loading in the axial direction

165

4.5 Closing Remarks

In this chapter a new strain energy based model was developed to predict mechanical

response, initial failure, and ultimate failure of multi-directional fibrous composite

laminates. A stiffness reduction methodology was incorporated into the model to

characterize failure progression beyond the initial failure. Material parameters were

evaluated by fitting model predictions to experimental data from the literature. These

model parameters were then used to make predictions for several other cases and the

predictions were compared to experimental data, showing good agreement between the

two. It was shown that unlike a unidirectional laminate, shear and compressive

transverse responses of embedded laminae in a multidirectional laminate are relatively

ductile and an embedded lamina can sustain increased load in those directions after

matrix failure. Table 4.3 presents a summary of the cases analyzed in this chapter along

with the model parameters, which can be used for similar material systems and lay-ups.

With one exception (Case 9 from Table 4.3), the experimental data used in this chapter

were produced by testing the material under biaxial load in axial and hoop directions.

Furthermore, the data presented in the table are based on two material systems only. The

next chapter is devoted to an experimental program developed to evaluate model

parameters for two other material systems under combinations of axial and in-plane shear

stress. This information will expand the existing data base for the selection of model

parameters.

166

The predicted initial failures were close to but generally lower than the experimentally

observed matrix failure initiation (as indicated by a change in the slope of experimental

stress-strain curves). The stress level at which the last predicted matrix failure occurred

was often a conservative lower bound (as low as 30%) for weeping strength.

Furthermore, it was observed that for fiber dominated responses presented in figures 4.27

to 4.30 the predicted final strengths were slightly higher than the measured values with

the experimental stress-strain curves being slightly softer than the predicted curves

beyond initial failure. Thus, it seems appropriate to decrease the longitudinal stiffness of

the laminae up to 5% after matrix failure.

In the proposed model, the effect of residual stresses is embedded in the stiffness

reduction parameters, Sk , SER , tTk , c

Tk , and TER . It is possible to include this effect

explicitly, knowing the curing conditions and thermo-mechanical properties of the

material. An explicit analysis to evaluate the amount of these stresses and strains (for

example Hyer and Cohen, 1984 and Hahn and Pagano, 1975) can be conducted and the

corresponding residual strain-energies can be computed. This is expected to improve the

predicted initial failures as well as post initial failure behavior, and it can be a topic for

another study. In doing so, the main technical challenge is to express material moduli as

a function of temperature and time.

Presented in Table 4.4 is a suggested experimental program to evaluate model parameters

for a multidirectional laminate made of a new material. The first five tests are used to

167

determine uniaxial stress-strain curves and Poisson’s ratios for a unidirectional material

system. Because of material variability some replicates may be required. However, since

uniaxial tests are conducted using flat specimens and conventional load frames,

additional tests can be carried out with a reasonable cost. The remaining three tests are

needed to determine stiffness reduction parameters. For fiber dominated lay-ups such as

quasi-isotropic, these parameters are not very influential and thus the last three tests can

be eliminated. For other material parameters, including shape factors and μ , the

suggested valued from Table 4.3 can be used. Furthermore, LTA can be evaluated using

Equation (3.28).

168

Case Material Lay up Loading conditions

Shape factors

tTk c

Tk Sk SER TER μ

1 E-glass/MY750 ±45 / 1/ 1y xσ σ = − 1 - - 15 0.1 - -

2 / 1/1y xσ σ = 1 400 - 15 0.1 1.0 -

3 [0/90] / 1/ 0y xσ σ = 1 400 25 15 0.1 1.0 0.5

4 ±55 / 1/ 0y xσ σ = 1 400 25 15 0.1 1.0 0.5

5 / 2 /1y xσ σ = 1 400 25 15 0.1 1.0 0.5

6 Failure envelope 1 400 50 50 0.1 1.0 0.5

7 ±85 Failure envelope 1 600 600 300 0.5 1.0 0.5

8 / 0 /1y xσ σ = 1 600 600 300 0.5 1.0 0.5

9 [90/±30]S Failure envelope 1 400 40 28 0.1 1.0 0.5

10 Failure envelope 1 400 40 28 0.1 1.0 0.5

11 As4 carbon/epoxy [90/±45/90] Failure envelope 1 400 40 28 0.1 1.0 0.5

12 / 1/ 0y xσ σ = 1 400 40 28 0.1 1.0 0.5

13 / 2 /1y xσ σ = 1 400 40 28 0.1 1.0 0.5

Table 4.3. Summary of the cases analyzed in Chapter 4

169

Test No.

Specimen lay-up

Loading Conditions Measured Quantities

Model Parameters

1 Unidirectional Uniaxial tensile loading in longitudinal direction

Longitudinal stress and strain, transverse strain

tLLA ,

( )tL LE ε ,

( )tL Lν ε

2 Unidirectional Uniaxial compressive loading in longitudinal direction

Longitudinal stress and strain, transverse strain

cLLA ,

( )cL LE ε ,

( )cL Lν ε

3 Unidirectional Uniaxial tensile loading in transverse direction

Transverse stress and strain, longitudinal strain

tTTA ,

( )tT TE ε ,

( )tT Tν ε

4 Unidirectional Uniaxial compressive loading in transverse direction

Transverse stress and strain, longitudinal strain

cTTA ,

( )cT TE ε ,

( )cT Tν ε

5 Unidirectional In-plane shear test on tubular or double notched specimen

Shear stress and strain

SA

( )G γ

6 Multi-directional

Thin wall tubular specimen under tension

Axial stress, axial and hoop strains

tTk , c

Tk , Sk

7 Multi-directional

Thin wall tubular specimen under torsion or combined axial load and internal pressure

Applied stresses, axial, hoop and shear strains

tTk , c

Tk , Sk

8 Multi-directional

Thick wall tubular specimen under axial compression

Compressive stress, axial and hoop strains

tTk , c

Tk , Sk

Table 4.4. Required experiments for evaluation of model parameters for a multi-

directional system made of a new material

170

CHAPTER 5

5 EXPERIMENTAL PROGRAM FOR MODEL VALIDATION

5.1 Introduction

In Chapter 4 a failure theory was developed for multidirectional laminates. Model

predictions were compared to experimental data taken from the literature. For all the

cases studied, with one exception, the laminates were loaded under combinations of axial

and hoop stresses (or combined xσ - yσ for flat specimens). The objective of the

experimental program developed and presented herein is to validate model predictions

when the material is loaded under combined axial and inplane shear stresses. Five S-

glass/epoxy and five carbon/epoxy specimens were tested under combinations of shear

and axial stresses and the stress-strain responses are presented. Then, the proposed

model is used to predict material response under the same loading conditions used for the

tests and the predictions are compared to the experimental data. Appendix D includes the

produced experimental data in numerical format.

171

5.2 Testing Procedures

5.2.1 Materials and Specimen Preparation

The material systems used in the study were S-glass reinforced polymer (S2- 284

GSM/Bryte BT250E-1) and carbon reinforced polymer (34-600/Newport NCT301). The

lay-ups were [90 / 30 / 30 / 90 / 90 / 30 ]− + +o o o o o o for glass sections and 3[ 30 / 30 ]+ −o o for

carbon sections. Five tubular specimens from each lay-up (total of ten specimens) were

tested under combined axial and shear stress.

The specimens were manufactured by Innovative Composite Engineering (ICE). The

selected geometry for the tubular specimens, shown in Figure 5.1, was the one

recommended by Swanson et al. (1988). The specimens had total length of 12.6” (32 cm)

with a 2.2” (5.59 cm) long test section and an internal diameter of 3.02” (7.671 cm). The

nominal wall thickness at the test section was 0.045” (1.14 mm) and 0.06” (1.52 mm) for

carbon and glass specimens, respectively. Both ends of each specimen were reinforced

within the gripping area and tapers were provided between the reinforced regions and the

test section to minimize stress localization. The geometry of the specimens is shown in

Figure 5.1.

To fabricate the specimens, 63” long tubes were made from prepreg material system

using rolling technique (ASM Handbook, 2003) and partially cured. Then, the fiber glass

172

build-ups (7781 Woven/Hexcel F155-200) were added to the tubes at a uniform thickness

throughout and underwent a final cure at 125 Co . Each tube was sliced into five 12.6”

long specimens. The middle part of each specimen was machined, leaving the glass built

ups and tapers per the drawing presented in Figure 5.1.

Figure 5.1. The geometry of the specimens

173

5.2.2 Testing Procedure

The tests were conducted following the testing procedure developed by Cohen (2002).

Each specimen was tested under combinations of axial tension and torsion. The loading

device was an MTS biaxial load frame with servo-hydraulic control system. In order to

impart the axial and torsional loads into the specimen, internal and external end fittings

were designed and made in the machine shop of the Civil Engineering Department at The

Ohio State University (Figure 5.2). The fixtures were designed to sustain combined axial

and tortional loads equivalent to the maximum capacity of the load frame without any

permanent deformation. The end fixtures were bonded to the specimens using 3M DP

810 adhesive and oven cured at 50 Co for three hours and at room temperature for 24

hours prior to testing. Figure 5.3 shows a carbon tube after the end fixtures were installed.

After testing, each specimen, with the attached fixtures, was placing in a 300 Co oven to

burn off the adhesive. The fixtures were retrieved using a low capacity load frame,

cleaned, and reused.

The specimens were loaded under axial and torsional loads simultaneously using the load

control mode. Data from the axial and torsional load cells as well as the strain gages

were collected at an acquisition rate of 5 Hz. The axial stress in the test section was

computed by dividing the axial load by the cross-sectional area of the specimen at the test

section. The shear stress was computed using thin wall tube theory (Boresi et al., 1993),

as:

174

tAT

m ⋅⋅=

2τ (5.1)

Where: T is the applied torque, t is specimen wall thickness, and mA is the area

encompassed by the section:

2)5.0( tRA im ⋅+⋅= π (5.2)

In the above equation iR is the internal diameter. Axial, hoop, and shear strains were

measured using five to ten strain gages installed on each specimens (Figure 5.4).

Rectangular rosette gages and unidirectional strain gages were used in this study. For

selected specimens (G1, G2, and C1) strain gages were installed on the opposite sides to

detect any possible bending. For all the specimens, strain gages were installed on the

middle and bottom of the gage sections. The axial and hoop strains were measured using

strain gages aligned in those directions and shear strains were measured as:

( 45) ( 45)γ ε ε− += − (5.3)

Where, ( 45)ε − and ( 45)ε + were measured strains along 45− o and 45+ o directions relative

to the tube axis.

175

Figure 5.2. End fixtures

Figure 5.3. A carbon specimen after end fixtures were bonded

176

Figure 5.4. A glass specimen installed on the load frame

Different strain gage layouts were used in this experimental program. In some of the

tests, strain gages were installed at the opposite sides of the specimen to detect bending

deformation. In other tests, strain gages were installed both at the middle and the bottom

of the test section to compare the measured strains at the two locations, and determine the

confining effect of the end reinforcement. In one case, besides concentric factory layout

rosette gages, a hand layout rosette gage was installed on the specimen. The hand lay out

gage covered a considerably wider area (3 versus 0.8 cm span); and the intent was to

compare the average shear strains measured throughout a wide span with those measured

at a point.

177

Table 5.1 lists the S-glass and carbon specimens tested in this study and the loading

conditions for each specimen. The selection of shear to axial stress ratios was primarily

limited by the capacity of the load frame. For example, for the carbon tubes the

maximum shear to axial stress ratio was estimated to be about 0.35, and therefore ratios

higher than 0.32 were not tried. One of the S-glass specimens was tested under a high

shear to axial stress ratio of 4/1 and, as expected, the maximum torque capacity of the

load frame was reached before failure.

Label Fiber type

Shear to axial stress ratio

Problems encountered

G1 S-glass 0.2/1 G2 S-glass 0.2/1 G3 S-glass 0.5/1

G4 S-glass 0.4/1

G5 S-glass 4/1 Torsional control channel saturation (max. capacity of load cell reached)

C1 Carbon 0.26/1.0 Some strain gage failure C2 Carbon 0.16/1.0 Some strain gage failure C3 Carbon 0.32/1.0 Some strain gage failure C4 Carbon 0.0/1.0 Premature failure of the glass built ups

C5 Carbon Non-proportional

Table 5.1. Summary of the specimens and loading conditions

178

5.3 Test Results for S-glass/epoxy Tubes

5.3.1 S-glass/epoxy under Shear to Axial Stress Ratio of 0.2/1

Two S-glass specimens (G1 and G2) were tested under a shear to axial stress ratio of

0.2/1.0. Figure 5.5 presents the strain gage lay-out for the first specimen (G1). Two

rosette gages (strain gages 5, 6, and 7 and 8, 9, and 10) were installed on the opposite

sides of the test section to detect bending due to the eccentricity in axial load. The axial

strains were measured and recorded by the strain gages 1, 4, 6, and 9, and the hoop

strains were measured by gage 2 (gage 3 did not work). The shear strains were calculated

from rosette gages 1 and 2 using Equation (5.3).

Figure 5.6 presents recorded axial and hoop strains from different strain gages versus the

measured axial stress. The axial strains measured from four different strain gages,

including the two (gages 6 and 9) installed on the opposite sides, were in agreement. A

partial failure occurred at an axial stress of about 310 MPa, which induced a sudden

increase in the measured axial and hoop strains. Figure 5.7 presents the measured shear

strains from the two rosette gages versus measured shear stress from the torsional load

cell. It is apparent that the two curves are very close. A sudden increase recorded by

both the gages at a shear stress of 65 MPa was most likely the result of a local fiber

failure. Although the ultimate failure occurred at a higher strength, strain gages 7 and 10

failed at this point and thus the measurement of shear strains could not be continued.

179

That the strain gages installed on the opposite sides of the specimen recorded very close

values of axial, hoop, and shear strains is an indication of uniform stress distribution

around the circumference of the specimen.

The stress-strain curves presented in Figure 5.6 indicate that the axial response becomes

softer at axial stress of about 170 MPa, most likely because of matrix failure. At about

the same load level, the specimen started to generate clicking sounds. The shear stress-

strain curve presented in Figure 5.6 exhibits highly nonlinear behavior. At a shear stress

of about 40 MPa and an axial stress of 200 MPa, the shear stiffness increased. The final

failure was recorded at axial and shear stresses of 356 and 77.7 MPa, respectively which

was the result of fiber failure.

180

Figure 5.5. Strain gage layout for G1 specimen

2.0 cm

1

2

4

5

6

7

3

8 9

10

45o 45o

Side 1

Rosette 1 Rosette 2

Side 2

181

0

200

400

600

-1.0% 0.0% 1.0% 2.0% 3.0%

Axial and Hoop Strain

Axi

al S

tres

s (M

pa)

Gage1 Gage 4 Gage 6 Gage 9 Gage 2

Axial StrainHoop Strain

Figure 5.6. Axial and hoop stress-strain curves for S-glass/epoxy specimen G1. Shear

stress to axial stress ratio was 0.2/1.0

182

0

40

80

120

0.00% 0.20% 0.40% 0.60% 0.80%

Shear Strain

Shea

r Str

ess

(MPa

)

Rosette 1 Rosette 2

Rosette 1

Rosette 2

Figure 5.7. Shear stress versus shear strain curves for S-glass/epoxy specimen G1 tested

under shear stress to axial stress ratio of 0.2/1.0

The second S-glass specimen (G2) was tested under similar loading conditions as used in

specimen G1 to verify that the test results are reproducible. Figure 5.8 presents the strain

gage lay-out for this specimen. Two rosette and two unidirectional gages were installed

on the opposite sides of the test section. Unfortunately, gage 1 did not respond during the

test and thus the shear strain was calculated from rosette gage 2 only. Depicted in Figure

183

5.9 are the measured axial and hoop strains from the different strain gages versus the

axial stress. It is apparent that both the axial and hoop strains measured from the

different strain gages are in agreement, indicating a uniform strain distribution around the

specimen’s circumference. Figure 5.10 presents the shear strain-axial stress curve. The

curve is nonlinear after an axial stress of about 200 MPa, which is a result of matrix

failure. Final failure occurred because of fiber failure at axial and shear stresses of 364

and 79.3 MPa, respectively.

Figure 5.8. Strain gage layout for G1 specimen

5.58 cm

7

1 2

3 4

56

45o 45o

Side 1

Rosette 1 Rosette 2

Side 2

8

184

0

200

400

600

-0.8% 0.0% 0.8% 1.6% 2.4% 3.2%

Strain

Axi

al S

tres

s (M

Pa)

Gage2 Gage 5 Gage 7 Gage 8

Gage 8

Gage 7

Hoop Axial

Gage 2

Gage 5

Figure 5.9. Axial and hoop strains from different strain gages versus axial stress for the

S-glass/epoxy specimen G2 under shear stress to axial stress ratio of 0.2/1.0

185

0

200

400

600

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Axi

al S

tres

s (M

Pa)

Rosette 2


to axial stress ratio of 0.2/1.0

Figure 5.11 presents a comparison between the measured axial and hoop strains for the

specimens G1 and G2. The presented strains for each specimen are average values from

the different strain gages. The measured strains from the two tests are in agreement. The

two axial strains versus axial stress curves essentially coincide. The measured hoop

186

strains from the second test are slightly larger than the measured strains from the first

test. The measured final strengths from the first and second test are 356 and 364 MPa,

respectively.

Final failure of G1, 356


0

100

200

300

400

-0.8% 0.0% 0.8% 1.6% 2.4% 3.2%Strain

Axi

al S

tres

s (M

Pa)

G1 G2 Final failure of G1 Final failure of G2


Figure 5.11. Comparison between the measured axial and hoop strains from G1 and G2

specimens

187



0

100

200

300

400

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Axi

al S

tres

s (M

Pa)

G1 G2 Final failure of G1 Final failure of G2

Figure 5.12. Comparison between the measured shear strains from G1 and G2 specimens

Figure 5.12 presents a comparison between the two shear strain versus axial stress curves

from the two tested specimens. The measured strains for the specimen G2 can be seen to

be up to 160% of the measured values for specimen G1. A part of this difference the

variability in material properties and testing conditions. However, the fact that the

measured hoop and axial strains were in agreement, implies that some of this difference

188

may be from measurement error. The shear strains were not measured directly; rather

they were computed by subtraction of two relatively close and small values that could

have increased the error.

5.3.2 S-glass/epoxy Tube under Shear to Axial Stress Ratio of 0.5/1.0

The S-glass specimen G3 was tested under a shear to axial stress ratio of 0.5/1.0.

Presented in Figure 5.13 is the lay-out of the strain gages installed on this specimen.

Figure 5.14 shows that shear strains calculated at the two gage locations are identical.

Figure 5.15 presents the responses of the two axial (gages 2 and 5) and one hoop (gage 8)

strain gages. Figure 5.16 shows a comparison between the measured strain from the three

45− o strain gages (gages 1, 4, and 6). The presented curves in the above mentioned

figures indicate that the axial and shear stress distributions were essentially uniform

across the test section. The stress-strain curves presented in Figure 5.15 shows a slope

change at an axial stress of about 120 MPa, which was a result of initial failure. Final

failure occurred at an axial stress of 310 MPa and shear stress of 159.2 MPa, due to fiber

failure of 30+ o plies and delamination of 90o plies at the inside of the tube. Figure 5.17

shows pictures from outside and inside of this specimen after failure.

189

Figure 5.13. Strain gage lay-out for G3

5.58 cm

12

3

45

Rosette 2Rosette 1

7

8

62.0 cm

190

0

40

80

120

160

200

0.0% 0.5% 1.0% 1.5% 2.0% 2.5%

Shear Strain

Shea

r Str

ess

(MPa

)

Rosette 1 Rosette 2

Rosette 1

Rosette 2


to axial stress ratio of 0.5/1.0

191

0

100

200

300

400

-0.8% -0.4% 0.0% 0.4% 0.8% 1.2% 1.6% 2.0%

Axial Strain

Axi

al S

tres

s (M

pa)

Gage2 Gage 5 Gage 8

Gage 5

Gage 2

Hoop Strain Axial Strain

Figure 5.15. Axial and hoop stress-strain curves for S-glass/epoxy specimen G3 under

shear stress to axial stress ratio of 0.5/1.0

192

0

100

200

300

400

-0.80% -0.60% -0.40% -0.20% 0.00%

Strain

Axi

al S

tres

s (M

pa)

Gage 1 Gage 4 Gage 6

Gage 1

Gage 4

Gage 6

Figure 5.16. Measured strains from three strain gages aligned at 45− o with respect to the

axial direction of S-glass/epoxy specimen G3. Shear to axial stress ratio was 0.5/1.0

193

Figure 5.17. Specimen G3 after failure (strain gage numbering in the picture is different

than the current numbering)

194

5.3.3 S-glass/epoxy Tube under Shear to Axial Stress Ratio of 0.4/1.0

S-glass specimen G4 was tested under a shear to axial stress ratio of 0.4/1.0. Figure 5.18

presents the strain gage lay-out for this specimen. One rosette gage was installed at the

mid-section and two unidirectional gages, aligned in the hoop direction, were installed at

the middle and bottom of the test section. Since the previous measurements confirmed

that the shear and axial strains were uniform, only one rosette gage was used to measure

the strain field. Figures 5.19 and 5.20 present the measured strains versus axial stress.

The measured hoop strains from gage 4 were larger those from gage 5 for axial stresses

larger than about 200 MPa. This difference was most likely due to variations in the stress

distribution close to the tapers. The shear response was more or less linear, while the

axial and hoop responses exhibited slight nonlinearity. The final failure occurred due to

fiber breakage at axial and shear stresses of 332 and 138.7 MPa, respectively. During the

failure, a large crack formed at the middle of the test section and extended toward the

sides at an angle of about 30+ o with respect to the axial direction. Figure 5.21 shows the

specimen during the failure. Just before the failure the upper and lower tapers were

detached from the test section.

195

Figure 5.18. Strain gage lay-out for G4

5.58 cm

12

3

5

4

Rosette 1

2.0 cm

196

0

100

200

300

400

-0.8% 0.0% 0.8% 1.6% 2.4%

Strain

Axi

al S

tres

s (M

Pa)

Gage 2 Gage 4 Gage 5


Gage 5 Gage 4

Figure 5.19. Axial stress versus axial and hoop strains for the S-glass/epoxy specimen

G4 under shear stress to axial stress ratio of 0.4/1.0

197

0

40

80

120

160

200

0.0% 0.5% 1.0% 1.5%Shear Strain

Shea

r Str

ess

(MPa

)

Rosette 1

Figure 5.20. Shear stress-strain curves for the S-glass/epoxy specimen G4 under shear

stress to axial stress ratio of 0.4/1.0

Figure 5.21. Failure mode of specimen G4

198

5.3.4 S-glass Tube under Shear to Axial Stress Ratio of 4/1

S-glass specimen G5 was tested under shear stress to axial stress ratio of 4/1. Two

rosette gages were used to measure shear strains during the test, as shown in Figure 5.22.

The hoop and axial strains were also measured using uniaxial gages installed in the axial

and hoop directions. In this test the maximum torsional capacity of the load cell was

reached before the occurrence of final failure.

Figure 5.22. Strain gage lay-out for S-glass specimen G5

5.58 cm

12

3

4

6

Rosette 2Rosette 1

5

199

0

45

90

135

180

0.0% 1.0% 2.0% 3.0% 4.0%

Shear Strain

Shea

r Str

ess

(MPa

)

Rosette 1 Rosette 2

Rosette 1

Rosette 2

Figure 5.23. Shear stress-strain curves for S-glass/epoxy specimen G5 under shear to

axial stress ratio of 4/1

Depicted in Figure 5.23 are the shear stress-strain curves from the two rosette gages. It is

apparent that the two curves are very close. Figure 5.24 presents measured axial and

hoop strains during the test. Although an axial stress was applied to the specimen, the

axial and hoop strains remained very small during the test. As will be discussed

subsequently, the proposed model predicts very small axial and hoop strains under the

200

stress ratio of 4/1, because interaction between shear and axial deformations cancels out

the axial strain. This observation was first made during preliminary numerical analysis

performed in prior to the test. In fact, the current stress ratio was selected to investigate

the prediction.

0

45

90

135

180

-0.08% -0.06% -0.04% -0.02% 0.00% 0.02% 0.04%

Strain

Shea

r Str

ess

(MPa

)

Gage 5 Gage 2

Hoop strain(gage 5) Axial strain

(gage 2)

Figure 5.24. Axial and hoop strains of S-glass/epoxy specimen G5 loaded under shear to

axial stress ratio of 4/1

201

5.3.5 Comparison between the Stress-Strain Curves Obtained Under Different

Stress Ratios

In this section the stress-strain curves presented in the Section 5.3.1 to 5.3.4 are compared

to give an insight into the effect of the ratio of shear to axial stress on the material’s

behavior. Figure 5.25 shows the effect of stress ratio (SR) on the axial and hoop strain

versus axial stress curves. Four axial and four hoop strain curves are presented in this

figure, among which two pair of curves were obtained for SR=0.2/1, one pair for

SR=0.4/1, and one pair for SR=0.5/1. Whenever more than one strain gage was

available, the presented data were computed by averaging the responses of all the

available strain gages. It is apparent that the axial stiffness of the material slightly

increases and the strength slightly decreases with increasing stress ratio. This increase in

the axial stiffness is because of interaction between shear and axial deformations of the

material (the lay-up is nonbalanced and nonsymmetric). The axial response exhibits

more nonlinearity under lower stress ratios.

The hoop strain versus axial stress curves under different SRs are very close except the

one from specimen G2. Depicted in Figure 5.26 are shear stress-shear strain curves of

the laminate under shear to axial stress ratios of 0.2/1 (two curves), 0.4/1, 0.5/1, and 4/1.

It is apparent that as the stress ratio increases the shear stiffness decreases. At lower

stress ratios the curves exhibit strong nonlinearity. The three curves obtained for SR =

0.2/1 and 0.4/1 show somewhat increase in the slope after initial failure. As mentioned

earlier, this increase is a result of interaction between axial and shear deformations. It

202

will be shown in the subsequent sections that the proposed model can predict this

increase. The other two curves are slightly nonlinear with decreasing slopes after initial

failure.

0

100

200

300

400

-0.8% 0.0% 0.8% 1.6% 2.4% 3.2%

Strain

Axi

al S

tres

s (M

Pa)

SR = 0.2/1 (G1) SR = 0.2/1 (G2) SR = 0.4/1 SR = 0.5/1


Figure 5.25. The effect of shear stress to axial stress ratio (SR) on axial and hoop strain

versus axial stress curves for S-glass/epoxy laminate

203

0

40

80

120

160

200

0.0% 0.6% 1.2% 1.8% 2.4% 3.0%

Shear Strain

Shea

r Str

ess

(MPa

)SR = 0.2/1 (G1) SR = 0.2/1 (G2) SR = 0.4/1

SR = 0.5/1 SR = 4/1

Max capacity of load cell reachedTest stopped

Figure 5.26. The effect of shear stress to axial stress ratio (SR) on shear response of

]30/90/90/30/30/90[ oooooo −−+ S-glass/epoxy laminate

204

5.4 Test Results for Carbon/epoxy Tubes

5.4.1 Carbon/epoxy Specimen under Shear to Axial Stress Ratio of 0.26/1

The first carbon/epoxy specimen was tested under a shear to axial stress ratio of 0.26/1.

The strain gage lay-out for this specimen is shown in Figure 5.27. Two rosette and four

unidirectional gages were installed on the opposite sides of the specimen to measure

shear, axial, and hoop strains. Figure 5.28 presents measured axial and hoop strains from

different strain gages versus axial stress. The axial gages 2 and 7 installed at the mid-

section on the opposite sides of the tube are in agreement, so are the axial gages 5 and 10

installed on the opposite sides at the bottom of the section. However, the data show that

the axial strains are slightly higher at the middle as compared to the bottom of the test

section (Figure 5.29). Gage 7 failed at the axial stress of about 400 MPa. This figure

also shows the measured hoop strains from the two gages 4 and 9 installed on the

opposite sides at the middle of the section. The measured hoop strains were initially in

agreement, but they diverge above the axial stress of 180 MPa, with values recorded by

gage 4 being higher than those recorded by gage 9. Although no hoop stress was applied,

the measured hoop strains were larger than the axial strains, because the material stiffness

was higher in the axial direction than in the hoop direction.

205

Figure 5.27. Strain gage lay-out for carbon/epoxy specimen C1

Figure 5.30 presents measured shear strains versus shear stress from the two rosette

gages. The measured strains were initially identical up to the shear stress of 35 MPa,

after which the readouts from the two rosette gages deviate. The shear stress of 40 MPa

corresponds to the axial stress of 180 MPa at which the measured hoop strains from

gages 4 and 9 started to deviate (Figure 5.30). The measured strains from rosette gage 2

decrease beyond shear stress of about 60 MPa. At shear stress of 105 MPa, which

corresponds to the axial stress of 400 MPa at which gage 2 failed (Figure 5.28), the

2.0 cm

94

5

1

2

3

10

8 7

6

45o 45o

Rosette 1 Rosette 2

5.58 cm

Side 1 Side 2

206

measured shear strains exhibits a rapid increase. These observations suggest that a crack

was forming in the area where the rosette gage 2 and unidirectional gage 9 were installed

that caused the unexpected behavior.

0

200

400

600

800

-4.0% -3.0% -2.0% -1.0% 0.0% 1.0% 2.0%

Strain

Axi

al S

tres

s (M

pa)

Gage 2 Gage 5 Gage 7 Gage 10 Gage 4 Gage 9

Gage 4

Gage 9

Gage 7

Gage 2

Gage 5

Gage 10

HoopAxial

Figure 5.28. Axial and hoop strains versus axial stress from different strain gages for

carbon/epoxy specimen C1, tested under SR = 0.26/1

207

0

200

400

600

800

0.0% 0.4% 0.8% 1.2% 1.6% 2.0%Axial strain

Axi

al S

tres

s (M

pa)

Average of 2 and 7 Average of 5 and 10

Bottom

Middle

Figure 5.29. Comparison between the axial strains measured at the middle and bottom of

the test section for specimen C1

0

40

80

120

160

200

0.0% 0.2% 0.4% 0.6% 0.8% 1.0%Shear Strain

Shea

r Str

ess

(MPa

)

Rosette 1 Rosette 2

Rosette 2

Rosette 1

Figure 5.30. Measured shear strains from the two rosette gages versus axial stress for

carbon/epoxy specimen C1 tested under SR = 0.26/1

208

The measured axial and shear stresses at final failure were 599 and 155 MPa,

respectively. The failure was not catastrophic and during failure a 30− o crack, initiated at

the bottom of the test section, eventually grew towards the middle of the specimen. After

the test was completed the test section was cut from the rest of the specimen and split into

two parts to view the failure surface (Figure 5.31). The initial failure is likely to have

occurred at an axial stress lower than 400 MPa, because at about 400 MPa gage 7 failed

and gages 5 and 9 recorded a sudden change (Figure 5.28). The data used in the

subsequent analyses in this chapter and those presented in Appendix D are from rosette 1

for shear, average of gages 2, 5, and 10 for axial, and average of gages 4 and 9 for hoop

strains.

Figure 5.31. Failure surface for carbon/epoxy specimen C1 tested under SR = 0.26/1

209


The strain gage lay-out for this specimen is presented in Figure 5.32. Since the data from

the previous tests (G1, G2, and C1) confirmed that the recorded values from strain gages

installed on the opposite sides of the specimen were in agreement, the strain gages were

installed only on one side. Two rosette and two unidirectional gages were installed at the

middle and bottom of the test section. Figure 5.33 presents the recorded hoop and axial

strains at the middle and bottom of the test section versus axial stress. Figure 5.34 shows

the measured shear strains at the middle and bottom versus axial stress. It can be seen

that the rosette gage 1 and strain gage 2 (which is a part of rosette 1) failed at axial and

shear stresses of about 220 and 35 MPa, respectively. The axial strains from gages 2 and

5 were in agreement before the failure of strain gage 2. The shear and hoop strains were

larger at the middle of the test section than the bottom, likely because to the confining

effect of the end reinforcements. Due to the failure of rosette gage 1, the data used in

subsequent analysis (Section 5.5.2) and those presented in Appendix IV are from Gage 5

and rosette 2.

The final failure occurred at axial and shear stresses of 600 and 99.4 MPa, respectively.

The axial stress-strain curves were linear. A post test examination showed a crack in the

composite tube immediately under the position of rosette gage 1. Therefore, the failure

of rosette gage 1 is likely to have happened when the growing crack reached the gage,

and thus the initial failure probably occurred at an axial stress between 250 to 400 MPa.

210


5.58 cm

12

3

8

7

Rosette 1

2.0 cm 4

56

Rosette 2

211

0

200

400

600

800

-3.0% -2.0% -1.0% 0.0% 1.0% 2.0%

Strain

Axi

al S

tres

s (M

pa)

Gage 2 Gage 5 Gage 7 Gage 8

Gage 5

Gage 2

Strain gage failure

Gage 8

Gage 7

Hoop

Axial

Figure 5.33. Measured axial and hoop strains from different strain gages versus axial

stress for carbon/epoxy specimen C2, tested under SR = 0.16/1

212

0

40

80

120

0.00% 0.10% 0.20% 0.30% 0.40% 0.50%

Shear Strain

Shea

r Str

ess

(MPa

)

Rosette 1 Rosette 2

Rosette 1

Rosette 2

Figure 5.34. Shear stress-strain curves from the two rosette gages for carbon/epoxy

specimen C2 tested under SR = 0.16/1

Figure 5.35. Carbon/epoxy specimen C2 after failure under the stress ratio of 0.16/1

213


The strain gage lay-out for carbon specimen C3 is presented in Figure 5.36. This

specimen was tested under a shear to axial stress ratio of 0.32/1, which was about the

maximum stress ratio that could fail the specimen regarding the capacity of the rotational

load cell. One rosette and one unidirectional gage were installed at the middle of the test

section and two unidirectional gages were installed at the bottom of the test section.

The axial and hoop strain versus axial stress curves for this specimen are presented in

Figure 5.37. Axial strains measured by gages 2 and 4 were identical before the failure of

gage 4 at an axial stress of about 400 MPa, after which no data was recorded by this gage.

Date from strain gage 6 was not recorded. The axial and hoop responses exhibited

nonlinearity after the axial stress of about 250 MPa. Figure 5.38 shows the shear stress-

shear strain curve, which is also nonlinear. Upon the final failure the axial and shear

stresses were 587 and 185 MPa, respectively. Figure 5.39 presents a picture of the

specimen after failure.

214


5.58 cm

12

3

5

4

Rosette 1

2.0 cm

6

215

0

200

400

600

800

-4.0% -3.0% -2.0% -1.0% 0.0% 1.0% 2.0% 3.0%

Strain

Axi

al S

tres

s (M

pa)

Gage2 Gage 4 Gage 5

Gage 2

Failure of strain gage 4

Gage 5


Figure 5.37. Hoop and axial strains from three strain gages versus axial stress for

carbon/epoxy specimen tested under stress ratio of 0.32/1

216

0

60

120

180

240

0.00% 0.20% 0.40% 0.60% 0.80%

Shear Strain

Shea

r Str

ess

(MPa

)Rosette 1

Figure 5.38. Experimental shear stress-strain curve for carbon/epoxy specimen C3 tested

under stress ratio of 0.32/1

Figure 5.39. Carbon/epoxy specimen C3 after failure under the stress ratio of 0.32/1

217

5.4.4 Carbon/epoxy Specimen under Axial Stress

In this test a very low shear to axial stress ratio of 0.02/1 was applied to the specimen.

The intent was to produce data for the response of the material under axial stress.

However, a small shear stress was also applied to investigate the effect of the axial stress

on the shear stress-strain response. The strain gage lay-out is shown in Figure 5.40. Two

rosette and two unidirectional gages were used to measure strains at the middle and

bottom of the test section.


5.58 cm

12

3

7

Rosette 1

2.0 cm

8

45

6

Rosette 2

218

Figure 5.41 presents hoop and axial strains from four strain gages versus the axial stress.

The measured axial strains from gage 2 are about 6% larger than those from gage 5. This

difference was because of the effect of the end reinforcement. Measured hoop strains

from gage 7 are larger than those form gage 8, most likely because of the confining effect

of the end reinforcement. The hoop and axial responses became slightly nonlinear above

an axial stress of 200 MPa. The shear stress-strain responses are presented in Figure

5.42. The measured shear strains from rosette 1 show an unusual trend. This could have

been caused by the gage not being properly attached to the specimen. Thus, the data

presented in Appendix D and those used in the subsequent analyses (Section 5.5.2) are

from rosette 2. However, due to the end restrains, the measured strains from this gage

would be expected to be somewhat smaller than the shear strains at the middle of the

section.

The test was stopped upon the premature failure of the top tapers at axial and shear

stresses of 557 and 11.2 MPa, respectively. Inspection of the specimen after the test did

not show any visible crack within the test section.

219

0

200

400

600

800

-2.4% -1.8% -1.2% -0.6% 0.0% 0.6% 1.2% 1.8%Strain

Axi

al S

tres

s (M

pa)

Gage2 Gage 5 Gage 7 Gage 8

Gage 7

Gage 2

Gage 5Gage 8

Hoop strain Axial strain

Figure 5.41. Measured hoop and axial strains from different gages versus axial stress for

carbon/epoxy specimen tested under shear to axial stress ratio of 0.02/1

220

0

4

8

12

16

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)

Rosette 1 Rosette 2

Rosette 1

Rosette 2

Figure 5.42. Shear stress-strain curve for carbon/epoxy specimen tested under stress ratio

of 0.02/1

221

5.4.5 Response of Carbon/epoxy Specimen under a Non-proportional Combination

of Axial Load and Torsion

Carbon epoxy specimen C5 was tested under the bilinear stress-path presented in Figure

5.43. The strain gage lay-out was similar to the one presented in Figure 5.40. Figure

5.44 shows the axial and hoop strains from different strain gages versus the axial stress.

The axial strain from gages 2 and 5 are about 13% different, with the strains from gage 2

(middle) being larger. The unidirectional gage 7 installed at the middle of the test section

did not calibrate and is not included. Thus, the presented hoop strains are recorded

values from gage 8 installed at the bottom of the test section. Based on the data presented

in the previous sections (Figures 5.28, 5.33, and 5.41), the mid-section hoop strains

would be expected to be up to 30% higher. Figure 5.45 presents the shear stress-strain

curves from the two rosette gages. The shear stress-strain curve at the middle of the test

section (rosette 1) is linear, while data from rosette 2 exhibits some nonlinear

characteristics. As before, the shear strains at the middle of the test section were larger

than those at the bottom. The change in the loading proportion did not appear to have

any significant effect on the axial and shear stress-strain responses, while the hoop strain-

axial stress curve exhibited some softening after increasing the shear stress ratio.

222

0

50

100

150

200

250

0 200 400 600 800Axial Stress (Mpa)

Shea

r Str

ess

(Mpa

)

(513 MPa, 96 MPa)

(521.6 MPa, 157 MPa)

Figure 5.43. The loading path for carbon/epoxy specimen C5

223

0

200

400

600

800

-2.4% -1.2% 0.0% 1.2% 2.4%

Strain

Axi

al S

tres

s (M

pa)

Gage2 Gage 5 Gage 8

Rosette 8 Rosette 5

Rosette 2

Change in loading proportion

Hoop strain Axial strain

Figure 5.44. Hoop and axial strains from different strain gages versus axial stress for

carbon/epoxy specimen C5

224

0

40

80

120

160

200

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)

Rosette 1 Rosette 2

Rosette 2

Rosette 1

Change in loading proportion

Figure 5.45. Shear stress-strain responses for carbon/epoxy specimen C5

225

5.4.6 The Influence of Shear to Axial Stress Ratio on the Material’s Response

In this section the stress strain curves from the specimens loaded under proportional load

(C1 to C4) are plotted on the same graph to demonstrate the effect of stress ratio on the

stress-strain behavior of the material. Figure 5.46 shows the axial and hoop strains versus

axial stress under various shear to axial stress ratios. It is apparent that the initial moduli

decrease with increasing stress ratio, however, this increase is not significant.

Furthermore, it can be seen that for the stress ratios between 0.16/1 to 0.32/1, the stress

ratio did not affect the final strength. The specimen tested under the stress ratio of 0.02/1

experienced a premature failure of the tapers. Thus, the final strength under this stress

ratio could not be measured.

Figure 5.47 presents the effect of stress ratio on the shear stress-strain response for this

laminate. It can be seen that the initial shear modulus of the material increases with the

shear stress ratio. The curve produced under the stress ratio of 0.16/1 is highly nonlinear

and it shows a behavior very different that the other curves. Furthermore, the shear

stress-strain curve under SR = 0.26/1 exhibits an increase in slope at the shear stress of

about 60 MPa. Such phenomenon is not seen in any other shear stress strain curves. In

general the shear stress-shear strain curves exhibit more variability as compared to axial

and hoop responses. Unlike the S-glass specimens, for carbon specimens no clear trend

can be observed between the shear stress ratio and the shear stress-shear strain curves

when shear strains exceeds 2%. This kind of behavior is not out of normal for composite

laminates with angle-ply lay-ups (Soden et al., 2002). A part of the differences between

226

the stress-strain curves presented in Figure 5.47 is a result of different shear stress ratios.

The second factor is believed to be variability in material physical properties and

manufacturing processes of the specimens.

0

200

400

600

800

-4.8% -3.6% -2.4% -1.2% 0.0% 1.2% 2.4%Strain

Axi

al S

tres

s (M

Pa)

SR = 0/1 SR = 0.16/1 SR = 0.26/1 SR = 0.32/1

Axial Strain

Hoop Strain

Premature fialure of tapers

Figure 5.46. The effect of stress ratio on the hoop and axial strains of carbon/epoxy

specimens

227

0

60

120

180

240

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)SR = 0/1 SR = 0.16/1 SR = 0.26/1 SR = 0.32/1

Figure 5.47. The effect of the stress ratio of the shear stress-strain responses of

carbon/epoxy specimen

228

5.5 Comparison between the Model Predictions and Experimental Data

In this section the proposed failure model is used to predict mechanical responses for the

material systems and loading conditions studied in the previous sections. Material

properties for S-glass/epoxy and carbon/epoxy systems, including initial moduli and

failure stresses and strains, were provided by the manufacturer. Since nonlinear shear

stress-strain curves were not available, shear stress-strain curves from similar material

systems (E-glass/MY750 epoxy and T300/epoxy) were used in the analyses. For each

material system the first set of experimental data were used to tune material moduli and

failure strains (tuned predictions) and the tuned material properties were used to make

predictions for the remaining cases. Based on the recommendations from the previous

chapter, the stiffness degradation parameters were selected as:

300.1

400

401.0

S

tTcT

kSERk

kTER

==

=

==

for [90 / 30 / 30 / 90 / 90 / 30 ]− + +o o o o o o lay-up and

200.1

400

301.0

S

tTcT

kSERk

kTER

==

=

==

229

For [ 30 / 30 ]n+ −o o lay-up. In Chapter 4 it was shown that up to 5% reduction in the

longitudinal stiffness of the material upon initial failure would improve the predictions.

In the new analyses the model was modified to include this reduction.

5.5.1 S-glass/epoxy Laminate

With the nonsymmetric lay-up of [90 / 30 / 30 / 90 / 90 / 30 ]− + +o o o o o o , interaction between

membrane and bending deformation occurs. The relationship between strains and

resultant loads for a rectangular element of the laminate is written as (Reddy, 2003):

x x

y y

xy

xx

yy

xyxy

NN

N

MM

M

εε

γκκ

κ

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎡ ⎤⎪ ⎪ ⎪ ⎪= ⋅⎨ ⎬ ⎨ ⎬⎢ ⎥

⎣ ⎦⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭⎩ ⎭

A BB D

(5.4)

In the above equation N and M are the resultant membrane forces and bending moments

across a unit length of the material, ε and γ are the axial and shear strains at the mid-

plane of the element, and κ are the flexural strains (curvatures of the element due to

bending). A is the extensional stiffness matrix, D is the bending stiffness matrix and

B is the extensional-bending stiffness matrix. B is a zero matrix for symmetric lay-ups

and nonzero for nonsymmetric lay-ups, as is the case in the glass laminates. The tubular

230

geometry and the rigid boundary conditions at the two ends of the specimens should have

minimized bending deformations. Thus, the boundary conditions of the model were set

such that no bending could occur ( 0x y xyκ κ κ= = = ). Thus, bending moments were

induced in the element to keep the bending strains zero at every step during the analyses.

5.5.1.1 Numerical Predictions for S-glass/epoxy specimen for a shear to axial stress

ratio of 0.2/1.0 (tuned predictions)

Two S-glass specimens, G1 and G2, were tested under shear to axial stress ratio of 0.2/1.

These experimental data were used to evaluate material properties, including initial

moduli and failure strains. The shear stress-strain curve that was used in the analyses was

the same as the one presented in Figure 3.6. Longitudinal and transverse stress-strain

curves were assumed to be linear in both tension and compression.

Figure 5.48 presented comparisons between tuned numerical predictions and

experimental data for axial and hoop strains versus axial stress. The longitudinal and

transverse moduli of the material were adjusted to fit the predicted initial slopes of axial

and hoop strain curves to the experimental data. As previously mentioned, the

experimental hoop strain versus axial stress from G1 is expected to be more accurate, and

therefore this curve was selected as the baseline. The stiffness degradation parameters

were selected based on the recommendations from Chapter 4. The longitudinal tensile

failure strain of the material was also adjusted to fit the predicted final failure to the

measured strength. The adjusted material properties are given in Table 5.2.

231

Figure 5.49 presents a comparison between predicted and experimental shear stress-strain

curves for specimens G1 and G2. The predicted response is in remarkable agreement

with the experimental response of G1 up to the shear stress of 40 MPa. At higher levels

of shear stress the predicted curve hold between the two experimental curved from G1

and G2. Similar to the experimental curve, the predicted curve exhibits somewhat

increase in the slope before final failure.

Final failure from G1, 356

Final failure from G2, 364

0

100

200

300

400

-0.8% 0.0% 0.8% 1.6% 2.4% 3.2%Strain

Axi

al S

tres

s (M

Pa)

Experimental data from G1 Experimental data from G2

Final failure from G1 Final failure from G2



Figure 5.48. Comparison between the tuned numerical predictions and experimental data


ratio of 0.2/1

232













In-plane shear strength 72 (MPa)


Table 5.2. Material properties used in the analysis for unidirectional S-glass/epoxy

material system. The transverse and longitudinal moduli, as well as longitudinal tensile

failure strain were adjusted

233

Final failure from G1, 78 MPa

Final failure from G2

0

20

40

60

80

100

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)

Experimental data from G1 Experimental data from G2 Final failure from G1

Final failure from G2 Numerical predictions

79 MPa

Figure 5.49. Comparison between predicted and experimental shear stress-strain curve


ratio of 0.2/1

5.5.1.2 Numerical predictions for S-glass/epoxy specimen for a shear to axial stress

ratio of 0.4/1.0

The model parameters and material properties used in this analysis were the same as

those from the previous section. Thus, these predictions reflect the predictive capabilities

of the model. Figure 5.50 presents a comparison between the predicted and experimental

234

axial and hoop strain-axial stress curves. Both the predicted axial and hoop responses

show good agreement to the experimental data. The predicted axial response slightly

deviates from the experimental curve beyond the axial stress of about 180 MPa, while the

predicted and experimental hoop responses are in good agreement throughout. The

predicted final strength was 328 MPa compared to the measured value of 332 MPa. The

predicted final failure occurred because of the fiber failure of 30+ o plies. The initial

failure was predicted to occur because of matrix failure of 90o plies at the axial stress of

45.9 MPa. Since the experimental response linear in hoop and axial directions it is not

possible to identify the point of initial failure.

Figure 5.51 presents the predicted and experimental shear stress-strain curves. The two

curves are in a good agreement before the shear stress of 80 MPa. Beyond this shear

stress level the experimental curve exhibits a slope increase, which is not predicted by the

model. As previously mentioned this increase in the slope occurred when the shear to

axial stress ratio was less than 0.4/1 (Figure 5.26), and it was because of interaction

between shear and axial deformations. The proposed model predicted this increase for

shear stress ratio of 0.2/1 (Figure 5.49).

235

Measured final strength (332 MPa)

0

100

200

300

400

-0.8% 0.0% 0.8% 1.6% 2.4%Strain

Axi

al S

tres

s (M

Pa)

Experimental data Measured final strengthNumerical predictions Predicted initial failrue


Predicted initial fialure(45.9 MPa)



laminate under shear to axial stress ratio of 0.4/1

236

Measured final failure (τ=139 MPa)

0

40

80

120

160

0.0% 0.6% 1.2% 1.8% 2.4%

Shear Strain

Shea

r Str

ess

(MPa

)

Experimental data Numerical predictions Predicted initial failure

Predicted initial failure (t = 19.2 MPa)



stress ratio of 0.4/1

237


ratio of 0.5/1.0

Depicted in Figure 5.52 are the predicted and experimental axial and hoop stress-axial

strain curves. The predicted response of the hoop strains is in a remarkable agreement

with the experimental data. The predicted axial strains are in agreement with the

experimental data up to an axial stress of 100 MPa, after which the predicted strains are

slightly higher that the measured values. The predicted final strength was 299 MPa,

which is very close to the measured strength of 310 MPa. Both the experimental and

predicted axial responses exhibit some nonlinearity after the initial failure. The predicted

initial failure was 46 MPa, and it occurred because of matrix failure in the 90o plies.

Based on the change in the slope of axial stress-strain curve, the experimental initial

failure is estimated to have occurred at an axial stress of about 60 MPa. The predicted

final failure was due to the fiber failure of 30+ o plies, that is consistent with the

experimental observations described in Section 5.3.2.

Figure 5.53 presented a comparison between the experimental and predicted shear stress-

strain curves for this laminate. The two curves are in a good agreement and both exhibit

non-linear characteristics.

238

Measured final strength (310 Mpa)

0

100

200

300

400

-0.8% 0.0% 0.8% 1.6% 2.4%Strain

Axi

al S

tres

s (M

Pa)

Experimental data Measured final failure

Numerical presictions Predicted initial fialure


Predicted initial failure (46 MPa)



laminate under shear to axial stress ratio of 0.5/1

239

Measured final strength

0

40

80

120

160

200

0.0% 0.6% 1.2% 1.8% 2.4% 3.0%

Shear Strain

Shea

r Str

ess

(MPa

)


Predicted initial failure(τ = 23.6 MPa)



stress ratio of 0.5/1

240


ratio of 4/1

As previously mentioned the maximum capacity of the tortional load cell (2200 N.m) was

reached during this test. In order to protect the load cell the test was hold and no

experimental data could be collected beyond. The experimental observations up to load

cell capacity showed that the axial and hoop strains under this stress ratio are very close

to zero. Although the axial stress is not zero, because of the interaction between shear

and axial deformations, the axial strains remained very small during loading. Figure 5.54

presents comparisons between experimental and predicted axial and hoop responses. The

predicted axial and hoop strains are very small and in good agreement with the

experimental values. It should be noted that whenever values are relatively close to zero,

absolute error provides a better measure for error than relative error. For this case the

maximum absolute error in the predicted axial strains was 0.022% and that for hoop

strains was 0.016%.

Figure 5.55 provides a comparison between the experimental and predicted shear stress-

strain curves. The two curves are in good agreement. The predicted initial failure occurs

at the shear stress of 50 MPa. The predicted final strength is 220 MPa and the failure

mode is fiber breakage of the interior 90o ply. Since the predicted curve follows the

nonlinear pattern of the experimental data

241

0

60

120

180

240

-0.12% -0.08% -0.04% 0.00% 0.04% 0.08% 0.12%Strain

Shea

r Str

ess

(MPa

)

Experimental data Numerical predictions Predicted initial failrue

Axial Strain

Hoop Strain

Predicted initial fialure(45 9 MP )



laminate under shear to axial stress ratio of 4/1

242

0

60

120

180

240

0.0% 1.0% 2.0% 3.0% 4.0%

Shear Strain

Shea

r Str

ess

(MPa

)



Maximum capacity of the load frame reached



stress ratio of 4/1

243

5.5.2 Carbon/Epoxy Specimen Laminate

The analysis procedure is the same as the one used for S-glass/epoxy laminates. The test

data for specimen C1 were used to determine initial moduli and longitudinal failure

strains for the material. These material properties along with the stiffness degradation

parameters from the previous chapter were used in the remaining analyses and the results

are compared to the experimental data from C2 to C5.

5.5.2.1 Numerical predictions for carbon/epoxy specimen under shear to axial stress

ratio of 0.26/1.0 (tuned predictions)

Figure 5.56 presents the tuned model predictions for this loading case. The material

properties used in this analysis are presented in Table 5.3. The longitudinal failure strain

and the longitudinal and transverse moduli are adjusted values, while the other material

properties are the same as those provided by the manufacturer. The experimental strains

were calculated by taking the average of two strain gages in the hoop direction and three

strain gages in the axial direction. A good agreement between the tuned predictions and

the experimental data shows the capability of the model to reproduce an existing set of

experimental data. The predicted initial failure was at an axial stress of 377 MPa, while

is in agreement with the experimental observations of Section 5.4.1. The model predicts

the nonlinear trend of the stress-strain curves both before and after the initial failure.

244













In-plane shear strength 75.4 (MPa)


Table 5.3. Material properties used for the numerical analysis. Transverse and

longitudinal moduli and the longitudinal failure strain were tuned to fit the numerical

predictions to the experimental data from C1

245

Measured final failure (599 Mpa)

0

200

400

600

800

-3.2% -2.4% -1.6% -0.8% 0.0% 0.8% 1.6% 2.4%

Strain

Axi

al S

tres

s (M

Pa)

Experimental data Measured strength

Numerical predictions Predicted initial failure


Figure 5.56. Comparison between the tuned predictions and experimental data for hoop

and axial responses of carbon/epoxy laminate under SR = 0.26/1

Figure 5.57 presents the predicted shear stress-strain curve compared to the experimental

curve. Since rosette 2 failed at a low axial stress levels, the presented experimental data

are from rosette 1. The initial slopes of the two curves are about the same, but they

deviate at shear stress of about 20 MPa. The model did not predict any increase in the

slope of the shear stress-strain curve. The predicted failure strain was 0.58% while the

246

measured value was only 0.4%. The predicted final failure mode was fiber breakage of

30− o plies that is in agreement with the experimental observations (Figure 5.31). The

variability in the experimental data, discussed in Section 5.4.6, prohibits further

assessments on the performance of the model.

Final failure at 155 MPa

0

40

80

120

160

200

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)


Numerical predictions Predicted initial failrue


laminate under stress ratio of 0.26/1

247

5.5.2.2 Numerical predictions for carbon/epoxy specimen for a shear to axial stress ratio

of 0.16/1.0

The material properties determined in the previous section were used to make prediction

for this case. Figure 5.58 presents comparisons between the predicted axial and hoop

responses and the experimental data. The experimental hoop strains are the average

values from the two strain gages (gages 7 and 8 in Figure 5.32). The experimental axial

strains are from one strain gage because the second gage (gage 2 in Figure 5.32) failed at

the axial stress of about 400 MPa. The predicted and experimental stress-strain curves

are in good agreement. The predicted strength is about 7% higher than the measured

value (643 MPa versus 600 MPa). The predicted initial failure was at the axial stress of

390 MPa that is in the same range at which strain gage 2 failed. Depicted in Figure 5.59

are the predicted and experimental shear stress-strain curves. The predictions are in good

agreement with the experimental data up to the shear stress at which initial failure was

predicted. After this stress level the experimental data exhibited an unusual trend, as the

shear moduli first increased dramatically and then decreased. That the predicted initial

failure coincides with the starting point of this unexpected behavior suggests that a

growing crack was passing through the strain gage. The opening of the crack under the

axial shear stress could have caused an apparent increase in the measured strains.

248

Experimental final strength ( 600 Mpa)

0

200

400

600

800

-3.2% -2.4% -1.6% -0.8% 0.0% 0.8% 1.6% 2.4%

Strain

Axi

al S

tres

s (M

Pa)

Experimental data from G2 Final failure from G2

Numerical predictions Predicted initial failre



axial responses of carbon/epoxy laminate under SR = 0.16/1

249

Measured final strength (99.4 Mpa)

0

40

80

120

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)


Numerical predictions Predicted initial failure


laminate under SR = 0.16/1

250


of 0.32/1.0

The material properties determined in the previous sections were used in this analysis.

Figure 5.60 presents comparisons between the predicted axial and hoop responses and the

experimental data. The experimental hoop and shear strains were measured using one

unidirectional and one rosette gage, as shown in Figure 5.36. Since one of the axial

gages (gage 4 in Figure 5.36) failed well before the final failure, the experimental axial

strains are from the strain gage 2 only. The predicted axial and hoop strains were about

15% lower than the measured values. The predicted strength (axial) of 576 MPa was

about 6.6% lower than the measured value of 617 MPa. The predicted initial failure was

at the axial stress of 365 MPa that is close to the axial stress of 400 MPa at which strain

gage 4 failed. The predicted hoop and axial stress-strain responses follow the nonlinear

trend of the experimental data.

Figure 5.61 presents the predicted and experimental shear stress-strain curves. It is

apparent that the predicted shear strains are larger than the measured values. The

predicted shear response was linear, whereas the measured response exhibited

nonlinearity.

251

Measured final strength (618 Mpa)

0

200

400

600

800

-4.6% -3.4% -2.2% -1.0% 0.2% 1.4%Strain

Axi

al S

tres

s (M

Pa)

Experimental data Measured final failure

Numerical presictions Predicted initial fialure


Predicted initial failureat axial stress of 365 MPa


axial responses of carbon/epoxy laminate under SR = 0.32/1

252

Measured final strength (184.4

Mpa)

0

60

120

180

240

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)


Predicted initial failureat shear stress of 116 MPa)


laminate under SR = 0.32/1

253


of 0.02/1

Figure 5.62 presents the predicted axial and hoop strains with the experimental data under

the shear to axial stress ratio of 0.02/1. As mentioned previously the experimental data

terminate before the final failure because premature failure at the tapers. The figure

shows a good agreement between the predicted axial and hoop strains up to the axial

stress for which experimental data are presented. The predicted axial stress versus axial

strain curve can follow the nonlinear trend of the experimental data. The initial failure is

predicted to occur at axial stress of 422 MPa. The predicted final failure occurred at the

axial and shear stresses of 777 and 15.5 MPa. Since the test had to be stopped after the

failure of tapers, the final strength could not be determined experimentally. Comparison

between predicted and measured shear strains is presented in Figure 5.63. The initial

slope of the curve is predicted accurately; however, the two curves deviate after a shear

stress of about 3 MPa. The presented shear strains were measured at the bottom of the

test section. The shear strains at the middle of the test section were expected to be

somewhat higher (Section 5.4.4). Furthermore, the predicted shear stress-strain curve is

linear, while the experimental data curve is slightly nonlinear.

254

Failure of tapers

0

200

400

600

800

1000

-4.8% -3.6% -2.4% -1.2% 0.0% 1.2% 2.4%Strain

Axi

al S

tres

s (M

Pa)

Experimental data Failure of tapersNumerical predictions Predicted initial failrue


Predicted initial fialure(422 MP )


carbon/epoxy laminate under shear to axial stress ratio of 0.02/1

255

Failure of tapers

0

4

8

12

16

20

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)


Predicted initial failure (8.5 MPa)

Figure 5.63. Comparison between the predicted and measured shear strains for

carbon/epoxy laminate under shear to axial stress ratio of 0.02/1

256

5.5.2.5 Numerical predictions for carbon/epoxy specimen for a non-proportional

loading

The loading path for this case was the bilinear path presented in Figure 5.43. Figure 5.64

presents the predicted axial and hoop strains with the experimental data. The figure

shows a remarkable agreement between the predicted and measured axial strains. The

predicted hoop strains can be seen to be higher than the measured values. Recall from

Section 5.4.5 that the hoop strains were measured at the bottom of the test section

(because rosette gage installed at the mid-section failed at a low strain), and the measured

values were expected to be influenced by the confining effect of the end reinforcements.

The actual hoop strains at the middle of the test section would be expected to be up to

20% higher that the presented values (Figures 5.28, 5.33, and 5.41). This can explain the

difference between the predicted and measured hoop strains. Figure 5.65 presents

excellent agreement between predicted and measured shear stress-strain curves.

The initial failure is predicted to occur at axial stress of 363 MPa, before the change in

the shear to axial stress proportions. The predicted final failure occurred at the axial and

shear stresses of 513 and 151 MPa, respectively, which are very close to the measured

values of 521 and 157 MPa, respectively.

257

Experimental final failure

0

200

400

600

800

-3.2% -2.4% -1.6% -0.8% 0.0% 0.8% 1.6%Strain

Shea

r Str

ess

(MPa

)

Experimental data Experimental final failureNumerical predictions Predicted initial failrue


Predicted initial fialure(363 MPa)


carbon/epoxy laminate under non-proportional loading presented in Figure 5.43

258

0

60

120

180

240

0.0% 0.2% 0.4% 0.6% 0.8%

Shear Strain

Shea

r Str

ess

(MPa

)

Experimental data Numerical predictionsPredicted initial failure Experimental final failure


Change in the shear to axial stress ratio

Figure 5.65. Comparison between the predicted and measured shear stress-strain curves

for carbon/epoxy laminate under non-proportional loading presented in Figure 5.43

259

5.6 Closing Remarks

Five S-glass/epoxy and five carbon/epoxy tubular specimens were tested under combined

axial force and torsion and the results were presented. The tests were conducted under

load control mode and the applied axial loads and torques were measured during the test.

Thin wall tube theory was utilized to compute axial and shear stresses across the test

section. Axial, hoop, and shear strains were measured during each test using several

strain gages installed on the outside of the specimens. Five glass specimens and four

carbon specimens were tested under proportional combinations of axial and shear stress

and the remaining carbon specimen was tested under a non-proportional loading. The

end fixtures and gripping method used in this testing program was found to be very

effective and it is highly recommended for future studies. For future studies, it is

suggested that the length of the test section and tapers be increased to minimize the

confining effect of the end reinforcements.

For most of the specimens more than one strain gage was used to measure each of axial,

hoop, and shear strains. The shear strains were computed by subtracting the measured

strains along o45+ and o45− directions, while the axial and hoop strains were measured

directly. Unlike the glass specimens, for carbon specimens, the shape of shear stress-

shear strain curves did not show consistent trends as the shear stress ratio was changed

form 0.16/1 to 0.36/1. The observed disagreements in the shear response were most

likely because of variability in manufacturing processes and material properties.

Measured strains from different gages for glass specimens were, in general, identical.

260

Two of the glass specimens were tested under the same shear to axial stress ratios, and it

was found that the results were reproducible. The observed differences between the shear

strains of the two specimens was concluded to be due to the lower accuracy in the

measurement of the shear strains combined with variability in material properties.

The proposed model was used to predict the response of the material under loading and

boundary conditions similar to those used in the tests. The predictions showed good

agreement with the experimental data. The predicted axial and hoop strains, as well as

the final strengths were in agreement with measured values. The predicted initial failures

were also in agreement with the experimental values as indicated by change in the slope

of the stress-strain curves or failure of some strain gages, whenever such observations

could be made. The accuracy of the predictions was higher for axial and hoop strains

than for the shear strains. The predicted shear strains for carbon specimens sometimes

had disagreement with the experimental data. However, the observed variability in the

shear response of carbon specimens made it difficult to evaluate model performance.

261

CHAPTER 6

6 SUMMARY AND CONCLUSIONS

In this work a nonlinear failure model was proposed for fiber reinforced polymer

composites. Chapter 1 was introduction. In Chapter 2 the literature was reviewed for the

existing failure theories and experimental methods for composite laminates, and the

failure theories were classified. In Chapter 3, a new strain energy based failure theory

was developed for a unidirectional laminate. The strain energy based failure criterion

proposed by Sandhu (1972) was studied and it was modified on the basis of energy

conservation principles. A correction factor was incorporated into the model to take into

account the influence of transverse stress on the shear resistance. This strain energy

based criterion combined with a maximum longitudinal strain criterion was used to

predict matrix and fiber failures, respectively. The incremental constitutive model

proposed by Sandhu (1972) was modified to include the interaction between the shear

and transverse deformations, and used to predict nonlinear stress-strain response of the

material.

In Chapter 4 the failure model was extended to multidirectional laminates. Classical

lamination theory was employed to establish the relationship between resultant loads and

262

strains for a multidirectional laminate, and determine induced stress and strain fields for

each lamina. An empirically based exponential model was developed to reduce laminae

transverse and shear moduli after the strain energy ratios exceeded certain levels during

loading. These strain energy ratio levels, as well as other model parameters were

evaluated using experimental data from the literature. The transverse tensile stiffness

was found to quickly reduce to zero upon matrix failure. The transverse compressive and

shear stiffness reduction rates were found to be relatively slow. Good agreements

between experimental and predicted stress-strain curves were obtained when the shear

stiffness reduction was started in prior to the initial failure (at a shear strain energy ratio

lower than one). The model with the tuned parameters was used to predict stress-strain

responses and failure envelopes for several laminates with different lay-ups and material

properties (blind predictions). The predictions were shown to be in good agreement with

experimental data for most of the cases.

In Chapter 5 biaxial experimental test results, conducted by the author, were presented

for five S-glass and five carbon fiber reinforced polymer specimens. The specimens had

tubular geometry with end reinforcements and were tested under combined torsion and

axial loads in load control mode. Strain gages were installed on each specimen

measuring axial, hoop, and shear strains during the tests. The S-glass sections were

tested under four different proportional combinations of shear and axial stresses. The

measured strains from different strain gages installed on the same specimen were

generally very close.

263

Four carbon specimens were tested under proportional loadings and the fifth specimen

was tested under a bilinear loading path. For some cases, the measured strains from

different strain gages installed on the same specimen exhibited some discrepancies. In

general, it was observed that the hoop and shear strains at the bottom of the test section

were lower than those measured at the middle. This was believed to be a result of

confining effect of the tapers. The measured shear stress-strain curves from different

specimens exhibited some variability. The predicted hoop and axial strains, as well as the

initial and final strengths, using the proposed model were shown to be in good agreement

with the experimental data. The predicted shear strains were up to 30% different than the

measured values. However, regarding the variability in experimental shear stress-shear

strain curves the accuracy of the model to predict the shear response of carbon specimens

could not be evaluated.

Based on the comparisons presented in Chapters 4 and 5, it can be concluded that the

model predictions are in general realistic (blind prediction). The agreement between

model predictions and experimental data were better for unidirectional laminates and

laminates with three directions of fiber orientations, than for angle-ply laminates.

Furthermore, the model is remarkably capable to reproduce an existing set of

experimental data, either a failure envelope or a stress-strain response (tuned prediction).

The below improvements can be proposed for future studies:

- A strain based failure criterion can be added to the model to predict weeping

strength,

264

- Using a physically based method, thermal residual stresses induced in

multidirectional laminates during curing process can be evaluated and

incorporating into the failure analysis. This will provide an insight into the post

initial failure response of the material, and is expected to improve the predictions.

Furthermore, it was concluded (Chapter 5) that the length of the tubular specimens can be

increased to minimize the confining effect of the end reinforcements and produce a more

uniform stress field across the test section.

265

A. APPENDIX A

Nonlinear lamina analysis

266

The computer programs developed during this study were written in C++ language. In

order to analyze a lamina under combined loadings, several classes of functions have

been developed. Class lam_moduli contains four functions that can calculate tangent

longitudinal, transverse, and shear moduli, as well as the Poisson’s ratios of a lamina

given the plane strain field. The method proposed in Chapter 3 is used to calculate the

moduli and Poisson’s ratios. These functions employ several other functions under

o_SScurve class to map nonlinear uniaxial stress-strain curves and compute the areas

encompassed by each. As mentioned in Chapter 3, each uniaxial stress-strain curve

(program inputs) is mapped using quadratic spline interpolation. Class SEBlam contains

functions which can compute lamina stiffness matrix, strain increments for a given stress

increment, and the failure mode, if failure occurs. The function comp_de() is developed

to calculate the strain increment of a lamina for a given strain increment. The flowchart

of this function is shown in Figure A.1. This function uses the proposed failure criteria to

check for fiber and matrix failures, and determine the failure mode. The calculated strain

increments and moduli, as well as the failure mode are stored in temporary variables.

The function returns the failure mode (failure mode is zero before failure occurs). Also

included in this class, is another function called comp_ds() which computes stress

increments that correspond to a given strain increment. The computational algorithm

developed for this function is very similar to the one presented in Figure A.1. After each

load step is applied and if no failure is detected the calculated stresses, strains, and

moduli are updated by calling update(). If matrix or fiber failure occurs in the last load

step, the step size is decreased and re-applied. This procedure is continued until the

failure strength is calculated with a given accuracy.

267

Figure A.1. Calculation of strain increment for a given stress increment

Compute strain increments [dε1] using moduli from previous step

Call MODULI function to compute moduli based on strains from previous step

Compute strain increments [dε2] using moduli from previous step

Read stress increments

Compute strain increment as: [dε]=0.5([dε1]+ [dε2])

Store [dε] and moduli in temporary variables

Call MODULI function to compute moduli based on based on [dε]

Check fiber failure Check matrix

failure

No

Return 1

Yes

Yes

Return 0

No

Determine matrix failure mode Store failure mode

in a temporary variable

Return failure mode

Store failure mode (1) in a temporary variable

268

B. APPENDIX B

Nonlinear laminate analysis

269

This computer program was developed for the analysis of multi-directional laminates

with nonlinear behavior, using the proposed model. Material properties, lay-up, ply

thicknesses, model parameters, and loading conditions are defined in an input file. At

each solution step, the program applies one load increment (including resultant axial and

shear loads as well as bending moments) to a rectangular element with unit length.

Assuming the stress distribution is uniform across the edges of the element, the stiffness

matrix from the previous step is used to compute strain increments:

x x

y y

xy

xx

yy

xyxy

NN

N

MM

M

εε

γκκ

κ

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎡ ⎤⎪ ⎪ ⎪ ⎪= ⋅⎨ ⎬ ⎨ ⎬⎢ ⎥

⎣ ⎦⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎩ ⎭⎩ ⎭

A BB D

The computed strain increments are imposed to each lamina to evaluate the

corresponding stress increments using comp_ds() function described in Appendix A. A

global unbalanced load vector is computed for the laminate by integrating the stresses

through the thickness using two Gaussian integration points for each lamina. The

unbalanced force is applied to the laminate and new strain increments are computed. The

iterations are continued until the norm of the unbalance load vector is less than the value

specified by the user.

270

At each step, failure (matrix or fiber) of all the lamiae are checked and upon failure of

one or more laminae the load step size is divided by two and re-applied. This procedure

is repeated until the initial or ultimate failure point is determined with a desired accuracy.

Below is the format of the input file:

Model Parameter Block

Line 1: Np

Line 2: tLm c

Lm tTm c

Tm Sm Sm ttLTm tc

LTm ctLTm cc

LTm μ

M

Line (N+1): tLm c

Lm tTm c

Tm Sm Sm ttLTm tc

LTm ctLTm cc

LTm μ

Description:

This block includes the shape factors and the parameter μ . More than one parameter set

can be defined in this block. Np is the number of parameter sets.

Stress-Strain Data Block

Line 1: SSN

Description:

271

Uniaxial stress-strain curves and Poisson’s ratio versus longitudinal stress curves are

defined for all the material systems intended for the analysis. SSN is the number of

material systems.

Longitudinal tensile:

Line 2: ,data lN oS

Line 3: oε oσ 1ε 1σ K

M

Line (N+2): K 1−Nε 1−Nσ

Description:

This section includes stress-strain curve of the material system under uniaxial tensile load

in the longitudinal direction. ,data lN is the number of points to define the stress-strain

curve, oS is the initial slope (modulus), iε is strain, and iσ is the corresponding stress.

If the modulus is constant 0=oS .

Longitudinal compressive:

This data block has a format similar to the one above.

Transverse tensile:


272

Transverse compressive:


In-plane shear:


Major Poisson’s ratio in tension:

Line 1: ,dataN ν oS

Line 2: oε oLT )(ν 1ε 1)( LTν K

M

Line (N+1): K Nε 1)( −NLTν

Description:

This section includes input data for the major Poisson’s ratio versus longitudinal tensile

stress. ,dataN ν is the number of data points along the stress-strain curve, oS is the initial

slope of the curve, iε is strain, and iLT )(ν is the corresponding value of Poisson’s ratio.

Longitudinal-transverse Poisson’s ratio in compression:


273

Laminate Lay-up and Properties Block

Lay-up:

Line 1: N

Line 2: 1θ 2θ K Nθ

Line 3: 1t 2t K Nt

Description:

N is the number of the plies. iθ and it are the orientation angle and the thickness of the

i’th ply, respectively.

Lamina material system number:

Line 4: 1..nm Nnmnm .... 2L

Description:

. .im n is the number that corresponds to the material type for the i’th ply, where

0 . .i SSm n N< ≤ for [1, ]i N∈

Lamina parameter set number:

Line 6: 1P NPP L2

274

Description:

iP is the number that corresponds to the SEP parameter set for the i’th ply, and

0 i pP N< ≤ for [1, ]i N∈

Stiffness reduction parameters:

Line 7: ( )1tTk ( ) ( )N

tT

tT kk L2

Line 8: ( )1cTk ( ) ( )N

cT

cT kk L2

Line 9: ( )1Sk ( ) ( )NSS kk L2

Line 10: 1χ Nχχ L2

Line 11: 1TER 2 NTER TERL

Line 12: 1SER 2 NSER SERL

Description:

( )tTk : Transverse tensile stiffness reduction factor

( )cTk : Transverse compressive stiffness reduction factor

( )Sk : Inplane shear stiffness reduction factor

χ : Longitudinal stiffness reduction ratio after matrix failure

TER : Transverse energy ratio at which transverse stiffness reductions starts

SER : Shear energy ratio at which shear stiffness reductions starts

275

Command Block

L command:

Line 1: L prn

Line 2: xNδ yNδ xyNδ xMδ yMδ xyMδ

Description:

This command defines a proportional loading. prn is the number of load steps and

x

y

xy

x

y

xy

NN

N

MM

M

δδ

δ

δδ

δ

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

is the resultant load increment to be applied at each step. The analysis will be stopped in

the event of final failure.

E command:

Line 1: E datan plane failure lδ

Description:

Calculates data points for a failure envelope. datan is the number of data points to be

generated in each quadrant. plane defines the plane for the failure envelope as:

276

1plane = : Xσ versus yσ failure envelope

2plane = : Xσ versus xyτ failure envelope

3plane = : yσ versus xyτ failure envelope

And finally lδ is the magnitude (length) of the load vector increment to be used in the

analysis

Command P:

Line 1: P sN

Line 2: Nsteps Nx Ny Nz Mx My Mz

.

.

.

Line N: Nsteps Nx Ny Nz Mx My Mz

Description:

This command is used to introduce a non-proportional load.

sN : Number linear segments that define the load path

Nsteps: Number of steps for each load segment

Parameters Block

Line 1: computational ε maxε

277

C. APPENDIX C

Correction of the axial and hoop stresses for effect of specimen bulging

278

As shown in Figure C.1, when a tubular specimen is subjected to internal pressure the

specimen bulges and the diameter at the middle part of the specimen increases. This

increase can considerably increase the actual hoop stresses, when the hoop strains are

large. A method similar to the one proposed by Kaddour et al. (2003) is adopted herein

to correct the hoop stresses and loading path.

Figure C.1. Specimen bulging due to internal pressure

279

The increase in specimen perimeter can be estimated as:

2 ( ) (1 )hl l R R lπ εΔ + = ⋅ ⋅ + Δ = ⋅ +

Where, R and l are the initial radius and perimeter of the specimen before the

deformation and hε is the measured hoop strain at the middle of the specimen. Thus,

2 . (1 ) (1 )2

hh

RR R Rπ ε επ

⋅ ⋅ ++ Δ = = ⋅ +

⋅

,(1 )( ) (1 )

2 2h

h mid h hRR R

t tρ ερσ σ ε⋅ ⋅ +⋅ + Δ

= = = ⋅ +⋅ ⋅

Where, ,h midσ is the corrected value of the hoop stress at the middle of the section, hσ is

the hoop strain before the correction, ρ is the magnitude of the internal pressure, and t

is the wall thickness.

280

D. APPENDIX D

Experimental data

281

σx τxy εx εy γ Mpa Mpa

0 0 0.000% 0.000% 0.000% 9.52 2.13 0.053% -0.012% 0.014%

18.95 4.11 0.103% -0.024% 0.029% 28.31 6.08 0.154% -0.036% 0.046% 37.40 8.06 0.204% -0.048% 0.062% 46.52 10.05 0.256% -0.060% 0.078% 55.54 12.04 0.308% -0.071% 0.094% 64.71 14.05 0.361% -0.084% 0.110% 73.73 16.02 0.415% -0.095% 0.126% 82.93 18.01 0.469% -0.107% 0.140% 92.03 19.99 0.524% -0.119% 0.155% 101.17 21.97 0.580% -0.132% 0.170% 110.27 23.98 0.636% -0.144% 0.184% 119.33 25.96 0.693% -0.156% 0.197% 128.38 27.96 0.750% -0.168% 0.210% 137.47 29.93 0.808% -0.180% 0.223% 146.62 31.94 0.867% -0.192% 0.236% 155.64 33.93 0.927% -0.205% 0.248% 164.79 35.92 0.989% -0.217% 0.260% 173.83 37.89 1.052% -0.228% 0.270% 182.97 39.89 1.117% -0.241% 0.280% 192.11 41.87 1.184% -0.253% 0.288% 201.29 43.88 1.253% -0.265% 0.293% 210.34 45.85 1.326% -0.276% 0.297% 219.43 47.84 1.400% -0.287% 0.300% 228.57 49.84 1.477% -0.298% 0.302% 237.55 51.82 1.554% -0.308% 0.301% 246.68 53.81 1.632% -0.319% 0.302% 255.79 55.80 1.710% -0.330% 0.303% 264.95 57.78 1.790% -0.344% 0.303% 274.11 59.79 1.871% -0.357% 0.301% 283.15 61.77 1.951% -0.368% 0.300% 292.26 63.77 2.032% -0.380% 0.302% 295.72 64.52 2.064% -0.385% 0.302% 321.90 70.23 2.418% -0.438% 0.342% 356.00 77.70 NA NA NA

Table D.1 S-glass specimen G1 under shear to axial stress ratio of 0.2/1

282

σx τxy εx εy γ Mpa Mpa 0.00 0.00 0.00% 0.00% 0.00% 9.35 1.98 0.05% -0.01% 0.02%

21.37 4.48 0.11% -0.03% 0.04% 30.63 6.46 0.16% -0.04% 0.06% 42.31 8.99 0.22% -0.06% 0.08% 51.44 10.99 0.27% -0.07% 0.10% 63.09 13.55 0.34% -0.09% 0.12% 72.22 15.54 0.39% -0.10% 0.14% 83.95 18.07 0.46% -0.12% 0.17% 93.03 20.05 0.51% -0.13% 0.19% 113.82 24.61 0.64% -0.17% 0.23% 125.49 27.18 0.72% -0.18% 0.25% 134.62 29.16 0.78% -0.20% 0.27% 146.19 31.71 0.86% -0.22% 0.29% 155.27 33.70 0.92% -0.23% 0.31% 166.96 36.26 1.01% -0.25% 0.33% 176.11 38.25 1.08% -0.26% 0.35% 187.77 40.80 1.17% -0.28% 0.37% 196.95 42.81 1.24% -0.29% 0.38% 208.53 45.37 1.33% -0.31% 0.39% 217.59 47.37 1.41% -0.33% 0.40% 229.33 49.89 1.51% -0.35% 0.41% 238.48 51.89 1.59% -0.37% 0.42% 250.06 54.44 1.69% -0.40% 0.44% 259.10 56.42 1.77% -0.41% 0.45% 270.90 58.99 1.87% -0.43% 0.46% 279.95 60.98 1.95% -0.45% 0.47% 291.63 63.53 2.06% -0.47% 0.48% 298.93 65.10 2.13% -0.48% 0.48% 312.77 68.08 2.27% -0.51% 0.49% 321.68 70.05 2.36% -0.53% 0.50% 333.25 72.62 2.47% -0.55% 0.51% 342.38 74.59 2.55% -0.57% 0.51% 353.95 77.15 2.66% -0.59% 0.52% 357.18 77.85 2.81% -0.59% 0.52%

364 79.3 NA NA NA

Table D.2. S-glass specimen G2 under shear to axial stress ratio of 0.2/1

283

σx τxy εx εy γ Mpa Mpa 0.00 0.00 0.00% 0.00% 0.00%

10.67 5.46 0.05% -0.01% 0.06% 21.14 10.90 0.09% -0.02% 0.12% 31.74 16.32 0.14% -0.03% 0.18% 42.19 21.75 0.18% -0.04% 0.24% 52.66 27.21 0.23% -0.06% 0.30% 63.21 32.60 0.28% -0.07% 0.36% 73.74 38.02 0.33% -0.08% 0.43% 84.20 43.45 0.37% -0.09% 0.49% 100.04 51.60 0.45% -0.11% 0.59% 110.45 57.03 0.50% -0.12% 0.66% 121.05 62.46 0.55% -0.14% 0.73% 131.53 67.87 0.60% -0.15% 0.80% 142.00 73.30 0.66% -0.16% 0.88% 152.62 78.74 0.72% -0.18% 0.95% 163.01 84.15 0.79% -0.19% 1.03% 173.66 89.59 0.86% -0.20% 1.11% 184.35 95.02 0.92% -0.21% 1.18% 195.00 100.45 0.99% -0.23% 1.26% 205.69 105.87 1.06% -0.24% 1.34% 216.33 111.29 1.13% -0.25% 1.42% 226.97 116.73 1.20% -0.27% 1.49% 232.31 119.45 1.24% -0.27% 1.53% 237.63 122.15 1.27% -0.28% 1.57% 243.01 124.86 1.31% -0.29% 1.61% 248.25 127.58 1.34% -0.29% 1.65% 253.65 130.29 1.38% -0.30% 1.69% 258.94 133.00 1.41% -0.31% 1.73% 264.31 135.72 1.45% -0.31% 1.77% 269.63 138.43 1.48% -0.32% 1.81% 274.91 141.14 1.52% -0.33% 1.85% 280.17 143.85 1.55% -0.33% 1.89% 285.43 146.55 1.59% -0.34% 1.93% 290.68 149.26 1.62% -0.35% 1.97% 295.94 151.97 1.66% -0.35% 2.01% 301.20 154.67 1.69% -0.36% 2.05% 306.46 157.38 1.73% -0.37% 2.09% 309.97 159.19 1.83% -0.37% 2.11%


284


12.36 4.95 0.05% -0.01% 0.05% 23.50 9.65 0.11% -0.02% 0.09% 34.72 14.34 0.16% -0.03% 0.14% 46.03 19.06 0.22% -0.05% 0.19% 51.61 21.41 0.25% -0.05% 0.21% 62.76 26.12 0.31% -0.07% 0.26% 73.97 30.82 0.37% -0.08% 0.31% 85.24 35.54 0.43% -0.09% 0.36% 96.46 40.22 0.49% -0.11% 0.41% 107.62 44.94 0.55% -0.12% 0.46% 118.79 49.64 0.61% -0.14% 0.51% 130.02 54.37 0.67% -0.15% 0.56% 141.19 59.08 0.73% -0.16% 0.61% 151.83 63.54 0.79% -0.18% 0.66% 163.02 68.23 0.86% -0.19% 0.70% 174.29 72.94 0.92% -0.21% 0.75% 185.51 77.63 0.99% -0.22% 0.80% 196.77 82.34 1.05% -0.24% 0.85% 207.81 87.04 1.12% -0.25% 0.90% 219.06 91.77 1.18% -0.27% 0.95% 230.26 96.46 1.25% -0.28% 0.99% 235.82 98.81 1.28% -0.29% 1.02% 241.42 101.16 1.32% -0.30% 1.04% 247.07 103.50 1.35% -0.30% 1.06% 252.65 105.87 1.39% -0.31% 1.08% 258.25 108.20 1.42% -0.32% 1.11% 263.89 110.56 1.46% -0.33% 1.13% 269.48 112.92 1.49% -0.34% 1.15% 275.18 115.25 1.53% -0.34% 1.17% 280.76 117.60 1.56% -0.35% 1.19% 286.39 119.95 1.60% -0.36% 1.21% 291.85 122.28 1.64% -0.37% 1.23% 296.90 124.40 1.67% -0.38% 1.24% 302.51 126.74 1.71% -0.39% 1.27% 308.22 129.07 1.75% -0.40% 1.30% 331.70 138.63 1.90% -0.52% 1.38%


285

σx τxy εx εy γ Mpa Mpa 0.00 0.00 0.00% 0.00% 0.00% 1.63 4.58 0.00% 0.00% 0.04% 2.38 8.72 0.00% 0.00% 0.10% 3.38 12.88 0.00% 0.00% 0.15% 3.86 17.07 0.00% 0.00% 0.21% 4.43 21.25 0.00% -0.01% 0.27% 5.19 25.41 0.00% -0.01% 0.33% 6.14 29.58 0.00% -0.01% 0.39% 7.16 33.74 0.00% -0.01% 0.45% 8.12 37.94 0.00% -0.01% 0.51% 9.06 42.08 0.00% -0.01% 0.57% 9.89 46.29 0.00% -0.01% 0.63%

10.90 50.45 -0.01% -0.01% 0.70% 11.84 54.61 -0.01% -0.02% 0.76% 12.91 58.78 -0.01% -0.02% 0.82% 14.04 62.97 -0.01% -0.02% 0.89% 15.02 67.11 -0.01% -0.02% 0.96% 15.97 71.28 -0.01% -0.02% 1.02% 17.02 75.48 -0.01% -0.02% 1.09% 18.09 79.63 -0.01% -0.02% 1.16% 19.07 83.81 -0.01% -0.03% 1.23% 19.99 87.99 -0.01% -0.03% 1.30% 21.30 92.17 -0.01% -0.03% 1.37% 22.42 96.32 -0.01% -0.03% 1.44% 23.60 100.48 -0.01% -0.03% 1.52% 24.52 104.65 -0.01% -0.03% 1.59% 25.57 108.84 -0.01% -0.04% 1.67% 26.80 113.00 -0.01% -0.04% 1.74% 28.27 117.16 -0.01% -0.04% 1.82% 29.36 121.32 -0.01% -0.04% 1.89% 30.62 125.49 -0.01% -0.04% 1.97% 31.61 129.65 -0.01% -0.05% 2.05% 32.74 133.81 -0.01% -0.05% 2.13% 34.05 137.96 -0.01% -0.05% 2.21% 35.19 142.11 -0.01% -0.05% 2.29% 36.68 146.25 -0.01% -0.06% 2.37% 38.16 150.39 -0.01% -0.06% 2.45% 39.43 154.48 -0.01% -0.06% 2.53% 40.39 156.82 0.00% -0.07% 2.58%

Table D.5. S-glass specimen G5 under shear to axial stress ratio of 4/1

286


23.54 7.19 0.05% -0.08% 0.02% 47.26 13.21 0.10% -0.16% 0.04% 70.63 19.25 0.15% -0.25% 0.06% 94.04 25.30 0.19% -0.33% 0.07% 117.65 31.35 0.25% -0.43% 0.09% 141.23 37.39 0.30% -0.52% 0.11% 164.49 43.41 0.35% -0.62% 0.12% 176.24 46.46 0.38% -0.66% 0.13% 199.82 52.50 0.43% -0.76% 0.15% 211.62 55.50 0.46% -0.82% 0.16% 234.95 61.55 0.52% -0.92% 0.17% 258.37 67.63 0.58% -1.02% 0.19% 270.14 70.62 0.61% -1.06% 0.20% 281.83 73.65 0.64% -1.12% 0.20% 293.72 76.67 0.67% -1.18% 0.21% 305.45 79.68 0.70% -1.24% 0.22% 317.20 82.70 0.73% -1.30% 0.22% 328.91 85.71 0.77% -1.36% 0.23% 340.63 88.75 0.80% -1.42% 0.24% 352.25 91.78 0.83% -1.45% 0.25% 363.97 94.80 0.86% -1.51% 0.25% 375.80 97.81 0.90% -1.58% 0.26% 387.49 100.84 0.93% -1.64% 0.27% 399.25 103.85 0.96% -1.71% 0.27% 410.96 106.90 1.00% -1.78% 0.28% 422.68 109.90 1.03% -1.85% 0.29% 434.35 112.92 1.07% -1.92% 0.29% 446.22 115.95 1.12% -1.99% 0.30% 457.92 118.95 1.16% -2.06% 0.31% 469.61 122.01 1.20% -2.13% 0.31% 481.30 125.01 1.23% -2.20% 0.32% 493.14 128.04 1.27% -2.26% 0.33% 504.88 131.05 1.31% -2.33% 0.33% 516.54 134.06 1.34% -2.40% 0.34% 528.29 137.11 1.39% -2.47% 0.35% 540.06 140.08 1.43% -2.54% 0.36% 551.79 143.13 1.44% -2.68% 0.37% 563.64 146.12 1.47% -2.71% 0.37% 575.34 149.12 1.50% -2.73% 0.38% 587.15 152.16 1.54% -2.76% 0.39% 598.25 155.01 1.56% -2.78% 0.40%

Table D.6. Carbon specimen C1 under shear to axial stress ratio of 0.26/1

287


17.74 2.98 0.04% -0.06% 0.01% 41.64 6.84 0.08% -0.13% 0.02% 65.17 10.74 0.13% -0.20% 0.04% 88.66 14.65 0.18% -0.27% 0.05% 112.20 18.52 0.22% -0.35% 0.06% 141.57 23.37 0.28% -0.44% 0.08% 165.06 27.27 0.33% -0.52% 0.09% 188.46 31.17 0.39% -0.61% 0.10% 211.98 35.08 0.44% -0.69% 0.12% 235.54 38.93 0.49% -0.79% 0.13% 258.96 42.85 0.54% -0.88% 0.14% 288.31 47.73 0.61% -1.01% 0.15% 311.86 51.61 0.67% -1.12% 0.17% 335.31 55.52 0.73% -1.23% 0.18% 352.97 58.45 0.77% -1.31% 0.19% 364.81 60.38 0.81% -1.37% 0.19% 376.35 62.34 0.84% -1.43% 0.20% 388.32 64.26 0.88% -1.49% 0.21% 399.95 66.23 0.91% -1.56% 0.22% 417.65 69.11 0.96% -1.65% 0.24% 429.32 71.08 1.00% -1.72% 0.25% 441.12 73.02 1.03% -1.78% 0.32% 452.84 74.98 1.07% -1.85% 0.35% 464.61 76.94 1.10% -1.91% 0.37% 476.41 78.86 1.14% -1.97% 0.38% 488.05 80.81 1.18% -2.02% 0.40% 499.90 82.74 1.22% -2.08% 0.41% 511.62 84.72 1.26% -2.14% 0.43% 523.34 86.63 1.30% -2.18% 0.44% 535.05 88.60 1.33% -2.20% 0.44% 546.91 90.55 1.37% -2.26% 0.42% 558.55 92.48 1.41% -2.33% NA 570.32 94.42 1.45% -2.40% NA 582.10 96.37 1.48% -2.47% NA 593.81 98.33 1.52% -2.54% NA 600.29 99.38 1.53% -2.58% NA


288

σx τxy εx εy γ Mpa Mpa 0.00 0.00 0.00% 0.00% 0.00% 8.69 3.41 0.02% -0.04% 0.01%

28.09 9.30 0.07% -0.11% 0.02% 46.98 15.12 0.12% -0.18% 0.03% 65.88 20.93 0.17% -0.26% 0.05% 84.61 26.78 0.21% -0.33% 0.06% 103.44 32.61 0.26% -0.40% 0.07% 126.91 39.88 0.32% -0.49% 0.09% 145.83 45.69 0.37% -0.57% 0.11% 164.55 51.51 0.42% -0.65% 0.12% 183.35 57.34 0.47% -0.73% 0.14% 202.21 63.15 0.52% -0.81% 0.15% 221.00 68.97 0.58% -0.90% 0.16% 239.68 74.80 0.63% -0.99% 0.18% 258.49 80.62 0.68% -1.09% 0.20% 277.37 86.44 0.74% -1.19% 0.21% 296.14 92.26 0.80% -1.29% 0.23% 315.01 98.05 0.86% -1.40% 0.24% 333.81 103.87 0.91% -1.52% 0.26% 352.55 109.71 0.98% -1.63% 0.28% 371.34 115.55 1.04% -1.76% 0.30% 390.15 121.38 1.10% -1.88% 0.31% 408.90 127.17 1.17% -2.01% 0.33% 427.79 132.98 1.23% -2.15% 0.35% 446.52 138.79 1.30% -2.29% 0.37% 465.28 144.62 1.37% -2.43% 0.39% 484.24 150.43 1.44% -2.57% 0.41% 502.99 156.27 1.52% -2.73% 0.44% 521.74 162.03 1.59% -2.89% 0.47% 540.52 167.82 1.66% -3.05% 0.49% 559.32 173.65 1.73% -3.20% 0.51% 578.18 179.34 1.80% -3.36% 0.53% 596.98 185.08 1.87% -3.52% 0.56% 601.22 186.38 1.89% -3.56% 0.56% 617.40 191.11 1.95% -3.70% 0.58%


289


25.35 0.47 0.05% -0.08% 0.00% 48.47 0.96 0.09% -0.15% 0.01% 71.57 1.48 0.14% -0.23% 0.01% 94.81 1.98 0.19% -0.31% 0.01% 117.96 2.48 0.24% -0.40% 0.01% 141.10 2.96 0.29% -0.48% 0.02% 152.61 3.23 0.31% -0.53% 0.02% 164.14 3.46 0.34% -0.57% 0.02% 175.88 3.73 0.36% -0.62% 0.02% 187.30 3.96 0.39% -0.67% 0.02% 198.97 4.22 0.42% -0.72% 0.02% 210.45 4.46 0.45% -0.77% 0.03% 222.10 4.72 0.47% -0.82% 0.03% 233.68 4.99 0.50% -0.87% 0.03% 245.27 5.20 0.53% -0.92% 0.03% 256.82 5.49 0.56% -0.96% 0.04% 268.41 5.72 0.59% -1.02% 0.04% 280.06 6.00 0.61% -1.07% 0.04% 291.51 6.21 0.64% -1.12% 0.04% 303.24 6.48 0.67% -1.17% 0.04% 314.68 6.73 0.70% -1.23% 0.05% 326.43 6.97 0.73% -1.28% 0.05% 337.90 7.22 0.76% -1.33% 0.05% 349.59 7.49 0.80% -1.39% 0.05% 361.01 7.72 0.83% -1.44% 0.06% 372.72 7.99 0.86% -1.49% 0.06% 384.10 8.22 0.89% -1.54% 0.06% 395.89 8.50 0.92% -1.60% 0.06% 407.45 8.72 0.96% -1.65% 0.07% 419.01 8.99 0.99% -1.71% 0.07% 430.45 9.23 1.02% -1.76% 0.07% 442.19 9.49 1.06% -1.82% 0.07% 453.68 9.72 1.09% -1.87% 0.08% 465.39 10.12 1.12% -1.93% 0.08% 476.84 10.36 1.16% -1.98% 0.08% 488.44 10.61 1.19% -2.02% 0.09% 505.77 10.85 1.24% -2.08% 0.09% 517.46 11.08 1.28% NA NA 557.00 11.15 NA NA NA


290

σx τxy εx εy γ Mpa Mpa 0.0 0.0 0.00% 0.00% 0.00% 22.4 5.3 0.04% -0.06% 0.02% 46.1 9.6 0.09% -0.12% 0.03% 69.5 13.9 0.14% -0.18% 0.05% 93.0 18.3 0.19% -0.23% 0.07%

116.5 22.6 0.24% -0.29% 0.08% 140.0 27.0 0.29% -0.35% 0.10% 163.4 31.3 0.35% -0.42% 0.12% 186.8 35.7 0.41% -0.48% 0.13% 210.3 40.0 0.47% -0.55% 0.15% 233.8 44.4 0.53% -0.62% 0.17% 257.3 48.7 0.59% -0.69% 0.19% 280.8 53.1 0.66% -0.76% 0.20% 304.2 57.4 0.72% -0.84% 0.22% 327.7 61.7 0.79% -0.92% 0.24% 351.2 66.1 0.86% -1.01% 0.25% 374.7 70.4 0.93% -1.10% 0.27% 398.2 74.8 1.00% -1.19% 0.29% 421.7 79.1 1.08% -1.29% 0.30% 433.3 81.3 1.11% -1.34% 0.31% 445.2 83.5 1.15% -1.39% 0.32% 456.8 85.7 1.19% -1.45% 0.33% 468.7 87.8 1.23% -1.50% 0.34% 480.3 90.0 1.27% -1.56% 0.35% 492.1 92.2 1.31% -1.61% 0.35% 506.1 94.7 1.36% -1.68% 0.36% 513.0 96.0 1.38% -1.72% 0.37% 513.8 102.2 1.39% -1.73% 0.39% 514.7 108.3 1.39% -1.74% 0.42% 515.5 114.4 1.39% -1.75% 0.44% 516.4 120.6 1.39% -1.77% 0.47% 517.3 126.7 1.39% -1.78% 0.49% 518.1 132.8 1.40% -1.79% 0.51% 519.0 139.0 1.40% -1.80% 0.54% 519.8 145.1 1.40% -1.81% 0.56% 520.7 151.2 1.40% -1.83% 0.59% 521.6 157.4 1.40% -1.84% 0.61%

Table D.10. Carbon specimen C5 under non-proportional loading

291

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Modeling of Composite Laminates Subjected to Multi Axial Loadings

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