modeling of composite laminates subjected to multi axial loadings
TRANSCRIPT
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
UMI MicroformCopyright
All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.
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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.
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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.
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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
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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
4.3.2 Axial-Hoop Stress Failure Envelope for [±85˚]S Laminate Made of E-
glass/MY750 Epoxy................................................................................................ 137
4.4 Evaluation of Model Predictive Capability..................................................... 141
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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
4.4.2 Biaxial Failure Envelope for [90˚/ ± 30˚/90˚]s Laminate Made of E-glass/
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
5.4.2 Carbon/epoxy Specimen under Shear to Axial Stress Ratio of 0.16/1 ....... 209
5.4.3 Carbon/epoxy Specimen under Shear to Axial Stress Ratio of 0.32/1 ....... 213
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
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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
5.5.1.3 Numerical predictions for S-glass/epoxy specimen for a shear to axial
stress ratio of 0.5/1.0........................................................................................... 237
5.5.1.4 Numerical predictions for S-glass/epoxy specimen for a shear to axial
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
5.5.2.3 Numerical predictions for carbon/epoxy specimen for a shear to axial
stress ratio of 0.32/1.0......................................................................................... 250
5.5.2.4 Numerical predictions for carbon/epoxy specimen for a shear to axial
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.3. S-glass specimen G3 under shear to axial stress ratio of 0.5/1.................... 283
Table D.4. S-glass specimen G4 under shear to axial stress ratio of 0.4/1.................... 284
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.7. Carbon specimen C2 under shear to axial stress ratio of 0.16/1.................. 287
Table D.8. Carbon specimen C3 under shear to axial stress ratio of 0.32/1.................. 288
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
Figure 4.5. Predicted responses for S]45[ o± E-glass/MY750 epoxy laminate under
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
Figure 4.13. Predicted stress-strain curves for s]55[ o± E-glass/MY750 epoxy laminate
under hoop stress, showing the effect of o-ring friction ................................................. 130
Figure 4.14. Predicted stress-strain curves for s]55[ o± E-glass/MY750 epoxy laminate
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
Figure 4.17. Initial and final failure envelopes for [ 55 ]S± o E-glass/MY750 epoxy
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
Figure 4.28. Stress-strain response for quasi-isotropic S]0/45/90[ ooo ± composite tubes
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
Figure 5.14. Shear stress-strain curves for S-glass/epoxy specimen G3 under shear stress
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 5.32. Strain gage lay-out for carbon/epoxy specimen C2 .................................. 210
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 5.36. Strain gage lay-out for carbon/epoxy specimen C3 .................................. 214
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 5.40. Strain gage lay-out for carbon/epoxy specimen C4 .................................. 217
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
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.................................................................................................................... 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
Figure 5.52. 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 0.5/1........................................................... 238
Figure 5.53. 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 0.5/1.......................................................................................................... 239
xxx
Figure 5.54. 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 4/1.............................................................. 241
Figure 5.55. 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 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
Figure 5.59. The predicted and experimental shear stress-strain curves for carbon/epoxy
laminate under SR = 0.16/1 ............................................................................................ 249
Figure 5.60. Comparison between the predictions and experimental data for hoop and
axial responses of carbon/epoxy laminate under SR = 0.32/1 ........................................ 251
Figure 5.61. The predicted and experimental shear stress-strain curves for carbon/epoxy
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
Figure 5.64. Comparison between the predicted and measured axial and hoop strains for
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
Transverse Compression
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
Transverse Compression
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
Longitudinal Stress (MPa)
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
Longitudinal Stress (MPa)
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
Longitudinal Stress (MPa)
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
Longitudinal Stress (MPa)
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
Transverse Stress (MPa)
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
Longitudinal Stress (MPa)
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
Longitudinal Stress (MPa)
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
Transverse Stress (MPa)
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
Transverse Stress (MPa)
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
Longitudinal Stress (MPa)
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
Longitudinal Stress (MPa)
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)
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
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
Figure 4.5. Predicted responses for S]45[ o± E-glass/MY750 epoxy laminate under
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
Figure 4.11. Predicted stress-strain curves for s]55[ o± E-glass/MY750 epoxy laminate
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)
Weeping of unlined tubes at 427±12MPa (Al Salehi et al., 1989)
kS = 15kT=25SER = 0.1SET = 1.0
μ=1.2 0.8 0.4 0.1
Numerical predictions
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
Weeping of unlined tubes at 427±12MPa (Al Salehi et al., 1989)
kS = 15kT = 25μ = 0.5SER = 0.1SET = 1.0
Axial Hoop
4%
2%
0%
Figure 4.13. Predicted stress-strain curves for s]55[ o± E-glass/MY750 epoxy laminate
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
Weeping of unlined tubes at 427±12MPa (Al Salehi et al., 1989)
kS = 15kT = 25μ = 0.5SER = 0.1SET = 1.0o-ring friction:2%Longitudinal stifness reduced for all laminae
Axial Hoop
Figure 4.14. Predicted stress-strain curves for s]55[ o± E-glass/MY750 epoxy laminate
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.
4.3.1 Axial-Hoop Stress Failure Envelope for [±55˚]S Laminate Made of E-
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
Numerical data; kS = 20
Numerical data; kS = 30
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 (Thick tubeswithout liner, Kaddour et al, 1997)
Experimental data (Thin tubes withliner, 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)
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
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
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 (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)
ks = 50SER = 0.1kT = 400 (tension)kT = 50 (compression)μ = 0.5
SR = 2/1
SR = -2/-1
Figure 4.17. Initial and final failure envelopes for [ 55 ]S± o E-glass/MY750 epoxy
laminate, showing good agreement between predictions and experimental results for the
selected set of model parameters
137
4.3.2 Axial-Hoop Stress Failure Envelope for [±85˚]S Laminate Made of E-
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.
4.4.1 Biaxial Failure Envelope for [90˚/ ± 30˚/90˚]s Laminate Made of E-glass/
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
4.4.2 Biaxial Failure Envelope for [90˚/ ± 30˚/90˚]s Laminate Made of E-glass/
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
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
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
Figure 4.28. Stress-strain response for quasi-isotropic S]0/45/90[ ooo ± composite tubes
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)
Numerical predictions
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
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
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
Final failure of 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)
G1 G2 Final failure of G1 Final failure of G2
Axial StrainHoop Strain
Figure 5.11. Comparison between the measured axial and hoop strains from G1 and G2
specimens
187
Final failure of G1, 356
Final failure of G2, 364
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
Figure 5.14. Shear stress-strain curves for S-glass/epoxy specimen G3 under shear stress
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
Hoop Strain Axial Strain
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
Axial StrainHoop Strain
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
5.4.2 Carbon/epoxy Specimen under Shear to Axial Stress Ratio of 0.16/1
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
Figure 5.32. Strain gage lay-out for carbon/epoxy specimen C2
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
5.4.3 Carbon/epoxy Specimen under Shear to Axial Stress Ratio of 0.32/1
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
Figure 5.36. Strain gage lay-out for carbon/epoxy specimen C3
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
Hoop Strain Axial Strain
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.
Figure 5.40. Strain gage lay-out for carbon/epoxy specimen C4
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
Numerical predictions
Axial StrainHoop Strain
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
232
Longitudinal modulus 44 (GPa)
Major Poisson’s ratio, LTν 0.28
Longitudinal tensile strength 946 (MPa)
Longitudinal compressive strength 748 (MPa)
Longitudinal tensile failure strain 2.15%
Longitudinal compressive failure strain 1.17%
Transverse modulus 11 (GPa)
Transverse tensile strength 30 (MPa)
Transverse compressive strength 132 (KPa)
Transverse tensile failure strain 0.27%
Transverse compressive failure strain 1.2%
Initial in-plane shear modulus 5.83 (GPa)
In-plane shear strength 72 (MPa)
In-plane shear failure strain 3.7%
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
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
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
Axial StrainHoop Strain
Predicted initial fialure(45.9 MPa)
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
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)
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
237
5.5.1.3 Numerical predictions for S-glass/epoxy specimen for a shear to axial stress
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
Axial StrainHoop Strain
Predicted initial failure (46 MPa)
Figure 5.52. 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 0.5/1
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
)
Experimental data Numerical predictions Predicted initial failure
Predicted initial failure(τ = 23.6 MPa)
Figure 5.53. 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 0.5/1
240
5.5.1.4 Numerical predictions for S-glass/epoxy specimen for a shear to axial stress
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 )
Figure 5.54. 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 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
)
Experimental data Numerical predictions Predicted initial failure
Predicted initial failure (50 MPa)
Maximum capacity of the load frame reached
Figure 5.55. 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 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
Longitudinal modulus 150 (GPa)
Major Poisson’s ratio, LTν 0.322
Longitudinal tensile strength 1350 (MPa)
Longitudinal compressive strength 1125 (MPa)
Longitudinal tensile failure strain 0.9%
Longitudinal compressive failure strain 0.75%
Transverse modulus 10 (GPa)
Transverse tensile strength 39 (MPa)
Transverse compressive strength 200 (KPa)
Transverse tensile failure strain 0.39%
Transverse compressive failure strain 2.0%
Initial in-plane shear modulus 4.4 (GPa)
In-plane shear strength 75.4 (MPa)
In-plane shear failure strain 4.0%
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
Axial StrainHoop Strain
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
)
Experimental data Measured strength
Numerical predictions Predicted initial failrue
Figure 5.57. The predicted and experimental shear stress-strain curves for carbon/epoxy
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 StrainHoop Strain
Figure 5.58. Comparison between the predictions and experimental data for hoop and
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
)
Experimental data Measured strength
Numerical predictions Predicted initial failure
Figure 5.59. The predicted and experimental shear stress-strain curves for carbon/epoxy
laminate under SR = 0.16/1
250
5.5.2.3 Numerical predictions for carbon/epoxy specimen for a shear to axial stress ratio
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
Axial StrainHoop Strain
Predicted initial failureat axial stress of 365 MPa
Figure 5.60. Comparison between the predictions and experimental data for hoop and
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
)
Experimental data Numerical predictions Predicted initial failure
Predicted initial failureat shear stress of 116 MPa)
Figure 5.61. The predicted and experimental shear stress-strain curves for carbon/epoxy
laminate under SR = 0.32/1
253
5.5.2.4 Numerical predictions for carbon/epoxy specimen for a shear to axial stress ratio
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
Axial StrainHoop Strain
Predicted initial fialure(422 MP )
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
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
)
Experimental data Numerical predictions Predicted initial failure
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
Axial StrainHoop Strain
Predicted initial fialure(363 MPa)
Figure 5.64. Comparison between the predicted and measured axial and hoop strains for
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
Predicted initial failure (68 MPa)
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:
This data block has a format similar to the one above.
272
Transverse compressive:
This data block has a format similar to the one above.
In-plane shear:
This data block has a format similar to the one above.
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:
This data block has a format similar to the one above.
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%
Table D.3. S-glass specimen G3 under shear to axial stress ratio of 0.5/1
284
σx τxy εx εy γ Mpa Mpa 0.00 0.00 0.00% 0.00% 0.00%
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%
Table D.4. S-glass specimen G4 under shear to axial stress ratio of 0.4/1
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
σx τxy εx εy γ Mpa Mpa 0.00 0.00 0.00% 0.00% 0.00%
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
σx τxy εx εy γ Mpa Mpa 0.00 0.00 0.00% 0.00% 0.00%
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
Table D.7. Carbon specimen C2 under shear to axial stress ratio of 0.16/1
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%
Table D.8. Carbon specimen C3 under shear to axial stress ratio of 0.32/1
289
σx τxy εx εy γ Mpa Mpa 0.00 0.00 0.00% 0.00% 0.00%
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
Table D.9. Carbon specimen C4 under shear to axial stress ratio of 0.0/1
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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