Download high strain rate behaviour of woven composite

HIGH STRAIN RATE
BEHAVIOUR OF WOVEN
COMPOSITE MATERIALS
RANA FOROUTAN
A thesis submitted to McGill University in partial fulfilment of the
requirements of the degree of Doctor of Philosophy
Department of Mechanical Engineering
McGill University, Montreal
December 2009
©Rana Foroutan, 2009
To my children
Ryan and Aiden
“Great things are not done by impulse,
but by a series of small things brought together.”
Vincent Van Gogh
ABSTRACT
The thesis centres on the dynamic behaviour of woven composite materials. The increase in
the use of these materials in the aerospace industry demands a reliable constitutive damage
model to predict their response under high strain rate loading. The current available models
do not include rate effects, are too complicated to use, are unable to provide a clear procedure
for characterizing the model parameters, or are unable to accurately predict response when
compared to experimental tests.
The present work focuses on developing a rate-dependant continuum damage mechanics
model which considers all in-plane damages, i.e., damage due to matrix cracks and fibre
failure in normal and shear directions. A physical treatment of growth of damage based on
the extensive experimental results is combined with the framework of continuum damage
mechanics models to form the foundation of the model for materials whose response is
governed by elastic deformation coupled with damage.
The developed model simulates the non-linear and rate dependant behaviour of woven
composite materials due to damage evolution. The model is implemented into a commercial
finite element code (ABAQUS) as a two-dimensional user material subroutine.
Detailed study of the behaviour of these materials under high velocity loading and several
experiments were required to establish material model parameters. Uniaxial tension as well
as bias extension shear tests were carried out at both static and dynamic rates on three
different woven composite materials.
A tensile version of the Hopkinson bar setup, in conjunction with the designed specimen and
specimen fixture, are used for testing laminated woven composite materials at high rates of
strain and a peak strain rate of up to 560/sec. Comparison of Hopkinson bar results with
results of tests performed at a quasi-static rate using a servo-hydraulic testing machine
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clearly show that strain rate has an effect on both the stress and the strain to failure for these
carbon woven composite materials. A higher stress is observed for dynamic tests, whereas
the strains at the maximum stress are higher for the static experiments. Also, an increase in
the initial undamaged elastic modulus is observed for both tensile and shear response with
increasing strain rate.
Finally, the simulation results presented in this work demonstrate that the model can well
predict the dynamic and the static response obtained from the experimental measurements.
Simulation results for higher than tested strain rates are also presented to illustrate the
response at these high rates. It is shown that the tensile strength increases with the increase in
loading rate, and also a strain softening phenomenon is observed after the maximum tensile
strength. The loading-unloading, as well as compression response of the materials are also
simulated to ensure the capability of the model to capture these responses.
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RESUME
Cette thèse se concentre sur le comportement dynamique des matériaux composites
tissés. L’augmentation de l’utilisation de ces matériaux dans l’industrie aéronautique
requiert le développement de modèles constitutifs fiables pour prédire leurs réponses
sous des conditions de chargement à haute vitesse de déformation. Actuellement, les
modèles disponibles n’incluent pas les effets de vitesse de déformation, sont compliqués
à utiliser ou ne sont pas capables de prédire avec précision le comportement
expérimental de ces matériaux.
Ce travail focalise sur le développement d’un modèle de mécanique continue de la
rupture tenant en compte la dépendance avec le taux de déformation. Ce modèle
considère tous les dommages dans le plan : dommages provoqués par la fissuration de la
matrice ainsi que ceux dus à la rupture de la fibre dans les directions normale et
tangentielle. Un traitement physique de la propagation du dommage, basé sur un vaste
ensemble de résultats expérimentaux, est combiné avec les modèles de mécanique
continu de la rupture afin de développer un modèle pour ces matériaux tissés, dont la
réponse comportementale est gouvernée par un couplage ente la déformation élastique et
la rupture.
Le modèle développé dans cette étude simule le comportement non-linéaire et dépendant
du taux de chargement des matériaux composites tissés provoqué par l’évolution des
dommages. Ce modèle est implémenté dans un logiciel commercial d’élément fini
(ABAQUS) en tant qu’un sous-programme définissant les propriétés bidimensionnel des
matériaux.
Une étude détaillée du comportement de ces matériaux, soumis à des chargements à
hautes vitesses, et plusieurs expériences ont été requises pour établir les paramètres du
modèle. Des tests en tension uni-axiale et en cisaillement à bi-extension ont été réalisés
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à la fois pour des cas statiques et dynamiques sur trois matériaux composites tissés
différents.
Une version en tension du dispositif à barres de Hopkinson, utilisant les spécimens et
fixations conçus, a été utilisée pour tester des laminés de matériaux composites tissés
dans des conditions de haut taux de déformation. Un taux maximum de déformation a
été obtenu à 560/sec. La comparaison des résultats obtenus avec les barres de Hokinson,
avec ceux obtenus avec des tests quasi-statique utilisant une machine de test servohydraulique, démontre clairement l’effet du taux de déformation sur les contraintes et
déformations à la rupture pour ces matériaux composite à fibres de carbone tissés. Des
contraintes plus importantes sont observées pour les tests dynamiques, tandis que les
déformations pour les contraintes maximales sont plus grandes pour les tests statiques.
De plus, une augmentation du module élastique initial non endommagé est observée
avec l’augmentation du taux de déformation, à la fois pour la réponse en tension et en
cisaillement.
Finalement, les résultats des simulations présentés dans ce travail démontrent que le
modèle peut prédire correctement la réponse statique et dynamique obtenue par les
mesures expérimentales. Les résultats des simulations, pour des taux de déformations
plus grands que ceux testés expérimentalement, sont aussi présentés pour illustrer la
réponse du matériau à ces haut taux de déformation. Cette étude montre que la résistance
en tension augmente avec l’augmentation du taux de chargement. De plus, un
phénomène d’adoucissement des déformations est aussi observé après la résistance
maximum en tension. Le chargement-déchargement, ainsi que la réponse en compression
de ces matériaux sont aussi simulés pour assurer la capacité du modèle à capturer ce type
de réponses.
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ACKNOWLEDGEMENTS
I would like to express my gratitude to all those who made it possible for me to complete this
dissertation. It is difficult to write this page after so many busy years, so many memories, and
so many people to remember. Many had a little word for me which kept me going. They are
numerous to name all, but I like to mention my supervisor Prof. James. A. Nemes who has
guided me through my life as a graduate student at McGill University with many fascinating
discussions. He challenged me when needed, encouraged me when necessary and provided
wisdom, insight and patience at all times. I was extremely fortunate to have him as my
supervisor and a great friend.
I certainly would like to extent my gratitude to Prof. Pascal Hubert, my co-supervisor, for his
kindness and friendly support as well as his invaluable suggestions to this research.
I would like to acknowledge the help and contribution provided by my research group at
CRIAQ, in particular, the project leader Prof. Augustin Gakwaya from Laval University, Mr.
Claude Boucher from Bombardier, Mr. Stephen Caufeild from Pratt and Whitney Canada,
Mr. Alain Cole and Mr. Mathieu Ruel from Bell Helicopter, and Mr. Dennis Nandlall from
DRDC Valcartier.
I would like to acknowledge the financial support of the Consortium for Research and
Innovation in Aerospace in Quebec (CRIAQ), the McGill University, and the generous
support of Dr. James. A. Nemes and Dr. Pascal Hubert.
I would like to thank my fellow students and the staff in the Department of Mechanical
Engineering at McGill University, in Particular, Mr. George Tewfik, Mr. Francois Di
Quinzio, Mr. Steve Kacani, and Dr. Lolei Khoun.
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I am truly grateful to my considerate father, Mostafa, my wonderful mother, Simin, and my
amazing brother, Shahram, without whose constant support, infinite love, encouragement and
care, throughout my life this work would not have been possible.
Finally, by no means the least, I specially would like to appreciate my loving and
understanding soul mate, Masoud Roshan Fekr. I like to acknowledge his patience, support
as well as the motivation he provided me for further studies. I also extend my appreciation to
my children, Ryan, who came to this world during this work, and allowed so much time to be
deprived from him while I was working and my second little one, who will be with us after
the completion of this thesis.
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TABLE OF CONTENTS
ABSTRACT ................................................................................................................. I RESUME .................................................................................................................. III ACKNOWLEDGEMENTS .............................................................................................. V TABLE OF CONTENTS .............................................................................................. VII LIST OF FIGURES ..................................................................................................... XI LIST OF TABLES.................................................................................................... XVII NOMENCLATURE ................................................................................................... XIX 1. INTRODUCTION ..................................................................................... 1 1.1 OVERVIEW ....................................................................................................... 1 1.2 OBJECTIVES AND STRATEGIES OF THIS RESEARCH ............................................. 4 1.3 STRUCTURE OF THIS THESIS ............................................................................. 5 2. LITERATURE REVIEW........................................................................... 7 2.1 CONTINUUM DAMAGE MECHANICS ................................................................... 9 2.1.1 INTERNAL STATE VARIABLES ........................................................................ 11 2.1.2 EFFECTIVE STRESS ....................................................................................... 15 2.1.3 STRAIN EQUIVALENCE HYPOTHESIS.............................................................. 18 2.2 DYNAMIC TENSILE EXPERIMENTS SETUPS ...................................................... 19 2.2.1 SPLIT HOPKINSON BAR SETUPS .................................................................... 21 2.3 STRAIN-RATE DEPENDANT EXPERIMENTS ...................................................... 25 2.4 REVIEW OF CDM MODELS FOR COMPOSITE MATERIALS ................................. 31 vii
3. EXPERIMENTAL SETUP ..................................................................... 41 3.1 INTRODUCTION ............................................................................................... 41 3.2 MATERIALS .................................................................................................... 42 3.3 DYNAMIC EXPERIMENTS ................................................................................. 45 3.3.1 HOPKINSON BAR SETUP AND THEORY ........................................................... 45 3.3.2 SPECIMENS AND FIXTURE DESIGNS ............................................................... 50 3.3.3 MEASUREMENT PROCEDURES ....................................................................... 56 3.4 STATIC EXPERIMENTS ..................................................................................... 62 3.4.1 UNIAXIAL TENSION TESTS .............................................................................. 63 3.4.2 BIAS EXTENSION SHEAR TESTS ....................................................................... 67 3.5 EXPERIMENTS SUMMARY ................................................................................ 72 4. EXPERIMENTAL RESULTS................................................................. 75 4.1 STATIC EXPERIMENTS..................................................................................... 75 4.1.1 TENSILE MODULUS AND TENSION TEST RESULTS ........................................... 76 4.1.2 SHEAR MODULUS AND BIAS EXTENSION TEST RESULTS................................... 85 4.2 DYNAMIC EXPERIMENTS ................................................................................. 91 4.2.1 EFFECTIVE GAGE LENGTH.............................................................................. 91 4.2.2 TENSION TEST RESULTS ................................................................................. 93 4.2.3 BIAS EXTENSION TEST RESULTS ..................................................................... 99 4.3 COMPARISON OF STATIC AND DYNAMIC RESULTS .......................................... 103 5. DEVELOPMENT OF A RATE DEPENDENT CONTINUUM DAMAGE
MODEL ................................................................................................. 109 5.1 INTRODUCTION ............................................................................................. 109 5.2 RATE DEPENDENT FOROUTAN-NEMES DAMAGE MODEL ................................ 114 viii
5.2.1 MODEL ASSUMPTIONS ................................................................................. 115 5.2.2 INTERNAL STATE VARIABLES ....................................................................... 116 5.2.3 EVOLUTION OF DAMAGE.............................................................................. 120 5.2.4 RATE SENSITIVE CONSTITUTIVE CDM MODEL ............................................. 121 5.3 MATERIAL CHARACTERIZATION ..................................................................... 126 5.3.1 UNDAMAGED STIFFNESS AND STRENGTH CONSTANTS................................. 127 5.3.2 STIFFNESS REDUCTION FUNCTIONS ............................................................ 133 5.3.3 DAMAGE EVOLUTION FUNCTIONS ............................................................... 134 5.3.4 STRESS RATE FUNCTIONS ........................................................................... 136 6. IMPLEMENTATION OF THE DAMAGE MODEL INTO A FINITE
ELEMENT CODE ................................................................................. 139 6.1 INTRODUCTION............................................................................................. 139 6.2 MODEL IMPLEMENTATION IN FINITE ELEMENT.............................................. 140 6.3 IMPLEMENTATION AND COMPARISON TO EXPERIMENTS ................................. 143 6.3.1 TENSILE TEST RESULTS ............................................................................... 143 6.3.2 BIAS EXTENSION SHEAR TEST RESULTS........................................................ 149 6.3.3 SIMULATION RESULTS FOR LOADING-UNLOADING-RELOADING .................... 154 6.4 SUMMARY .................................................................................................... 160 7. CONCLUSIONS ................................................................................... 161 7.1 SUMMARY AND CONCLUSIONS ....................................................................... 161 7.1.1 EXPERIMENTAL FINDINGS ........................................................................... 162 7.1.2 MATERIAL MODEL AND SIMULATION FINDINGS ............................................ 165 7.2 ORIGINALITY AND CONTRIBUTION TO KNOWLEDGE ........................................ 166 7.3 RECOMMENDED FUTURE WORK ..................................................................... 167 ix
BIBLIOGRAPHY ............................................................................................................... 169 APPENDIX A: SUBROUTINE FOR ABAQUS/EXPLICIT ......................................... 183 x
LIST OF FIGURES
Figure 1.1 Bird strike impact on wing leading edge (Oxford 2004) ....................... 3 Figure 2.1 Meso-definition of damage ................................................................... 13 Figure 2.2 Schematic of Hopkinson bar configuration using direct tensile pulse22 Figure 2.3 Various Hopkinson bar setups using indirect tensile pulse adapted
from (a) Ellwood et al. (1982), (b) Lindholm and Yeakley (1968), (c)
Kawata et al. (1980), (d) Harding et al. (1960), (e) Eskandari and
Nemes (2000), and (f) Li et al. (1993) ..................................................... 25 Figure 2.4 Stress-strain curves at dynamic and static strain rates in tension (a)
Glass Satin Woven Cloth/ Polyester (b) Carbon Plain Woven Cloth/
Epoxy (adapted from Kawata et al. 1981) .............................................. 27 Figure 2.5 Stress-strain curves for woven carbon fibre epoxy laminates at
different strain rates in tension (adapted from Harding et al. 1989) ... 28 Figure 2.6 Tensile stress-strain curves at different strain rates for (a) woven
aramid/polypropylene (b) woven polyethylene/ polyethylene (Rodriguez
et al. 1996 with permission of Elsevier) ................................................. 29 Figure 2.7 Stress-strain curves of glass woven fabric-reinforced composites
subjected to different strain rates (adapted from Sham et al. 2000) .... 30 Figure 3.1 Schematic and picture of different weaves of woven composite: (a)
plain weave, (b) 2×2 twill weave, (c) 8-harness satin weave ................. 43 Figure 3.2 Hopkinson bar setup ............................................................................ 45 Figure 3.3 Schematic representation of Hopkinson bar ....................................... 46 Figure 3.4 A specimen sandwiched between the input and output bars ............. 46 Figure 3.5 Schematic assembly of first specimen fixture design ......................... 51 Figure 3.6 Picture of the fixture with epoxy injection .......................................... 51 Figure 3.7 Schematic assembly of second fixture design (side and top views) .... 52 Figure 3.8 Schematic assembly of third fixture design ........................................ 53 xi
Figure 3.9 Schematic diagram of the modified fixture.......................................... 53 Figure 3.10 Picture of the modified fixture ........................................................... 54 Figure 3.11 Initial dimensions of the composite specimen in millimetres ........... 54 Figure 3.12 Failure in the specimens (a) [0/45/90/-45]s (b) woven [0]6 ................. 55 Figure 3.13 Specimen’s dimensions in millimetres ............................................... 55 Figure 3.14 The aluminum jig used to cut the specimens .................................... 56 Figure 3.15 Half Wheatstone bridge ...................................................................... 58 Figure 3.16 Typical recorded strain gage signals on the input and output bars . 59 Figure 3.17 Typical dynamic behaviour of a woven composite material .............. 60 Figure 3.18 Schematic comparison of actual versus effective gage length .......... 61 Figure 3.19 Typical behaviour using different gage lengths ................................ 61 Figure 3.20 Woven composite specimen with mounted strain gage ..................... 62 Figure 3.21 Servo-hydraulic MTS testing machine, with the specimen fixed in
place ......................................................................................................... 63 Figure 3.22 Correction for the extensometer arms length .................................... 64 Figure 3.23 Strain comparison using extensometer and strain gage ................... 65 Figure 3.24 Woven composite specimen with perpendicular strain gage ............ 66 Figure 3.25 Typical result to obtain the Poisson’s ratio........................................ 67 Figure 3.26 Loading direction and fibre direction coordinate systems ................ 69 Figure 3.27 Typical result for bias extension specimen to obtain Poisson’s ratio 71 Figure 4.1 Stress-strain curves for static tensile tests of material one (woven
carbon/epoxy prepreg [0º]6) in the two fibre directions .......................... 77 Figure 4.2 Stress-strain curves for static tensile tests of material two (woven
carbon/BMI prepreg [0º]6) in the two fibre directions ............................ 78 Figure 4.3 Stress-strain curves for static tensile tests of material three (woven
carbon/epoxy prepreg [0º]8) in two fibre directions ................................ 79 Figure 4.4 Average static tensile strengths and their variability for the three
materials in the two fibre directions....................................................... 81 Figure 4.5 Average static tensile modulus and their variability for the three
materials in the two fibre directions....................................................... 82 xii
Figure 4.6 Axial versus transverse strains for all the materials in the fibre
direction ................................................................................................... 83 Figure 4.7 Poisson’s ratio of material one in 45º specimens ................................. 85 Figure 4.8 Poisson’s ratio of material two in 45º specimens................................. 86 Figure 4.9 Poisson’s ratio of material three in 45º specimens .............................. 86 Figure 4.10 Shear stress -strain curves for static bias extension tests of material
one (woven carbon/epoxy prepreg [0º]6) in the 45º specimen ................. 88 Figure 4.11 Shear stress -strain curves for static bias extension tests of material
two (woven carbon/BMI prepreg [0º]6) in the 45º specimen ................... 88 Figure 4.12 Shear stress-strain curves for static bias extension tests of material
three (woven carbon/epoxy prepreg [0º]8) in the 45º specimen .............. 89 Figure 4.13 Comparison of shear stress-strain curves of the three materials .... 90 Figure 4.14 Typical comparison of different strain measurements ..................... 92 Figure 4.15 Stress-strain curves for dynamic tensile tests of material one in the
two fibre directions .................................................................................. 94 Figure 4.16 Stress-strain curves for dynamic tensile tests of material two in the
two fibre directions .................................................................................. 95 Figure 4.17 Stress-strain curves for dynamic tensile tests of material three in
the two fibre directions ........................................................................... 96 Figure 4.18 Average dynamic tensile strengths and their variability for the three
materials in the two fibre directions ...................................................... 98 Figure 4.19 Average dynamic strains at maximum stress and their variability
for the three materials in the two fibre directions................................. 99 Figure 4.20 Stress-strain curves for dynamic bias extension tests of material one
in the 45º specimen................................................................................ 100 Figure 4.21 Stress-strain curves for dynamic bias extension tests of material two
in the 45º specimen................................................................................ 101 Figure 4.22 Stress-strain curves for dynamic bias extension tests of material
three in the 45º specimen ...................................................................... 101 xiii
Figure 4.23 Average dynamic shear strength and strains at maximum stress
including their variability for the three materials in the 45º direction
................................................................................................................ 103 Figure 4.24 Comparison of static and dynamic behaviour in material one (a) 0º or
90º specimens, (b) 45º specimens .......................................................... 104 Figure 4.25 Comparison of static and dynamic behaviour in material two (a) 0º or
90º specimens, (b) 45º specimens .......................................................... 105 Figure 4.26 Comparison of static and dynamic behaviour in material three (a) 0º
or 90º specimens, (b) 45º specimens ...................................................... 106 Figure 5.1 Schematic view of the six damage variables in a composite element
(adapted from Dechaene et al. 2002) .................................................... 113 Figure 5.2 Typical stress-strain response showing zones of damage ................. 118 Figure 5.3 Dynamic stress-strain curve showing increment of stress due to
damage ................................................................................................... 122 Figure 5.4 Typical stress-strain response of a woven composite material ......... 128 Figure 5.5 Typical shear stress-strain response of a woven composite material
................................................................................................................ 129 Figure 6.1 Flowchart for implementation of the CDM model into ABAQUS .... 142 Figure 6.2 Comparison between the displacement-time profiles obtained from
dynamic test results and the equation 6.2 (a) material one, (b) material
two, (c) material three ........................................................................... 144 Figure 6.3 Two dimensional element, tensile simulation in ABAQUS .............. 146 Figure 6.4 Comparison of the ABAQUS simulation with the average dynamic
and static tensile test results (a) material one, (b) material two, (c)
material three ........................................................................................ 147 Figure 6.5 Two dimensional element, shear simulation in ABAQUS ................ 149 Figure 6.6 Comparison between the shear displacement-time profiles obtained
from dynamic test results and the equation 6.2 (a) material one, (b)
material two, (c) material three ............................................................ 150 xiv
Figure 6.7 Comparison of the ABAQUS simulation with the average dynamic
and static shear test results (a) material one, (b) material two, (c)
material three ........................................................................................ 152 Figure 6.8 Loading-unloading simulation results (a) material one, (b) material
two, (c) material three ........................................................................... 155 Figure 6.9 Loading-unloading-reloading simulation results (a) material one, (b)
material two, (c) material three............................................................ 157 Figure 6.10 Loading-unloading to compression-reloading simulation results (a)
material one, (b) material two, (c) material three ............................... 159 xv
xvi
LIST OF TABLES
Table 3.1 Summary of the materials used in this research .................................. 44 Table 3.2 Time to failure for the three woven composite materials ..................... 50 Table 3.3 Direction cosines .................................................................................... 69 Table 3.4 List of dynamic experiments performed................................................ 73 Table 3.5 List of static experiments performed .................................................... 74 Table 4.1 Summary of static tensile tests ............................................................. 80 Table 4.2 Poisson’s ratio measured in the fibre direction..................................... 83 Table 4.3 Summary of Poisson’s ratio and tensile modulus data from literature 84 Table 4.4 Poisson’s ratio measured in the 45º specimens ..................................... 87 Table 4.5 Summary of static bias extension tests ................................................. 90 Table 4.6 Summary of effective gage lengths ........................................................ 92 Table 4.7 Summary of dynamic tensile tests ........................................................ 97 Table 4.8 Summary of dynamic bias extension tests .......................................... 102 Table 4.9 Comparison of the average of static and dynamic results .................. 107 Table 5.1 Material constants present in the Foroutan-Nemes CDM model ...... 125 Table 5.2 Calculated threshold values for the three woven composite materials
................................................................................................................ 132 Table 5.3 Calculated constants for the three woven composite materials ......... 138 Table 6.1 Comparison of maximum dynamic tensile strength ........................... 148 Table 6.2 Comparison of maximum dynamic shear strength ............................. 154 xvii
xviii
NOMENCLATURE
ROMAN LETTERS
A0
cross sectional area of the specimen
Ai1 to Ai6
material rate dependant constants, (i = 1, 4) - fibre and 45 degree
direction
Ab
cross sectional area of the Hopkinson bars
Bi1 to Bi4
material damage constants, (i = 1, 4) - fibre and 45 degree direction
C0
undamaged stiffness tensor
C1 to C5
material constants
C(d )
damage dependant stiffness tensor
C ij
damaged elastic stiffness tensors
E0
initial undamaged modulus
E
damaged modulus
Eii
Tensile and shear modulus components (i= 1, 2, 4)
Eii0
initial undamaged tensile and shear modulus components (i= 1, 2, 4)
Eii0H
initial modulus components at high rates of strain (i= 1, 2, 4)
Eb
Young modulus of the Hopkinson bars
F0
nucleation function
Fi
instantaneous tensile load
GF
gage factor of strain gage
GL
short specimen gage length
xix
ΔL
displacement
LS
length of the striker tube
LSF
length of the specimen in between the two fixings
M (d )
fourth order damage effect tensor
P1 P2
,
applied load on each side of the specimen
Ri
resistors in Wheatstone bridge (i= 1, 2 ,3 and 4)
RG
nominal resistance of the strain gage
ΔR
strain-induced change in resistance
ΔR / R
fractional change in electrical resistance
S
shearing yield strength in the 12-plane
S
cross sectional area
S
effective area
Sd
damaged area due to micro cracks and void growth
VEX
excitation voltage of Wheatstone bridge
Vm
volume element
VO
output voltage of Wheatstone bridge
Wi1, Wi2,W13
material constants (i = 1, 4) - fibre and 45 degree direction
X
yield strengths under uniaxial loading in 11-direction
X
strength of the material associated with the damage mode
Y
yield strengths under uniaxial loading in 22-direction
Z11, Z12, Z4
strain rate dependant parameters for evolution of damage
cb
longitudinal wave velocity in the Hopkinson bars
xx
cS
longitudinal wave velocity in the specimen
d
scalar damage variable
d
second order damage tensor
dij
rate of damage, i = 1,2,4 - direction, and j = 1, 2 -damage type
Δ d ij
rate of damage in the incremental form
di
damage variables at their respective principle axes, i = 1,2,3
d ij
damage variables, i = 1,2,4 - direction, and j = 1, 2 -damage type
e
base of the natural logarithm
li , mi
direction cosines ( i = 1,2 ) in (1,2) coordinated system
l0
initial gage length of the short specimen
li
instantaneous gage lengths
m, m1, m2
shape of the damage growth curve in Weibull distribution
n
unit vector
t
time
Δt
time increments
ΔtI
duration time of the incident pulse
tf
time to failure
tS
transit time needed for the incident wave to travel through the
specimen once
u1 , u2
displacements at the ends of the short specimen in HB
u, v
displacement in x- and y-direction
xxi
GREEK LETTERS
Δ
displacement
Ωi
principle values of the damage tensor
α ,αi1
constants related to amount of damage ( i = 1,2,4 )
ε
strain
ε
strain tensor
ε
effective strain tensor
ε
strain rate
ε
average strain rate
Δε kl
strain increments in the fibre coordinate system ( k = 1,2,3 l = 1,2,3 )
ε 0,i
strains at initiation with damage equal to zero
ε act
actual strain
ε ext
extensometer strain
εf
nominal failure strain
ε I , ε R , εT
Incident, reflected, and transmitted strains measured
ε ij
strain components in the loading coordinate system
( i = x, y, z
j = x, y, z )
ε kl
strain components in the fibre coordinate system ( k = 1,2,3 l = 1,2,3 )
εm
strain relative to maximum stress
ε max,i
strains at zero stress with damage equal unity
εs
average strain in the specimen
γ 44
engineering shear strain
xxii
ν ij
Poisson's ratio i ≠ j
ρb
Density of the Hopkinson bars
ρS
density of the specimen
σ
Cauchy stress tensor
σ
effective stress tensor
σ
stress rate
σ01, σ02
scale parameters of the double Weibull distribution function
σ d
damaged stress rate
σ0
threshold stress
σ ii
threshold stress (i = 1, 4) - fibre and 45 degree direction
σ eng
engineering stress
σ ij
stress components in the loading coordinate system
0
( i = x, y, z
j = x, y, z )
σ kl
stress components in the fibre coordinate system ( k = 1,2,3 l = 1,2,3 )
Δσ kl
stress increments in the fibre coordinate system ( k = 1,2,3 l = 1,2,3 )
σmax
maximum stress
xxiii
xxiv
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
1
INTRODUCTION
1.1
Overview
Composite materials combine the desirable properties or behaviour of two or more
identifiable constituent materials in order to achieve improved strength, stiffness, and
toughness at reduced weight. Composites are gradually replacing traditional metallic
materials due to their high specific modulus, high specific strength, high stiffness to
weight ratio, and their capability of being tailored for a specific application. These
properties can be altered or modified by varying the constituents of the composite, i.e.,
the matrix and the fibres.
1
INTRODUCTION Among the different types of composite materials, the class of woven fabric composites
has received attention in many structural applications because of the balanced in-plane
properties, manufacturability, and high impact resistance. Reinforcement in all directions
within a single layer, better impact resistance, better toughness and lower cost of
fabrication are some of the reasons that the woven fabric composites have been
recognized as more competitive than the unidirectional composites. Those properties
make the woven fabric composites attractive for structural applications. The ability to
simulate the material behaviour and to predict the survivability of structures becomes
more and more important at the stage of the structure design.
Woven composites are fabricated from strong and light fibres such as graphite, carbon,
aramid, glass, nylon or other advanced polymeric fibrous materials. The fibres form
yarns, which are woven in different textile architectures such as plain weave, twill, satin
or crowfoot. Similar architectures could be achieved if the yarns are braided. The textile
reinforcement is then impregnated by a resin and cured forming a lamina. It is also
possible to stack several lamina in a laminate or stitch several textile layers together
before impregnating and curing them. Generally, the fibres have transversely isotropic
elastic properties and the matrix material is non-linear isotropic.
Composite materials are being used in various fields including aerospace industries where
aircraft structures may undergo high strain-rate dynamic loading. Examples of these
loadings are bird strike or foreign object impact against wing leading edges, propellers,
and engine rotor containments, as shown in Figure 1.1. Several examples are illustrated
below where the aircraft structure experiences high strain rate impact loading.
Consider a turbofan engine, in which air passes through several stages of rotating fan
blades. If there is a flaw in the system, such as an unexpected obstruction, the fan blade
can break, spin off, and if not contained, damage other critical components. Therefore, if
the engine casing is made of the composite materials, it needs to be strong enough to
contain broken blades and also should be damage-tolerant to withstand the impact of a
loose blade-turned-projectile.
2
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Figure 1.1 Bird strike impact on wing leading edge (Oxford 2004)
A recent example of a damage tolerant engine casing is the crash of the US Airways
Flight 1549 making an emergency landing in the Hudson River. The plane took off from
New York's LaGuardia Airport, colliding with a flock of geese approximately 2,900 feet
above the ground. The most vulnerable part of the engine is the fan, which can be
damaged by an ingested bird. Pieces of broken blade can then rip through the rest of the
engine like shrapnel. Engines have been reinforced so that they can stay intact in their
casing in the event of such a strike, but they usually cannot be restarted once they are
damaged. Archie Dickey, an associate professor of aviation environmental science at
Embry-Riddle Aeronautical University's campus in Prescott, Arizona has said that hits
hard enough to cause a total failure are rare, only happening two or three times a year
worldwide, and the chance of it hitting both engines, is less than one percent.
3
INTRODUCTION It is true that aircraft hit thousands of birds every year which usually bounce off without
damaging the structure, nevertheless, it is important to consider such rare cases as well. In
the mentioned case, the engine casing could withstand the dynamic impact, and as a result
keep the rest of the plane intact from further damage. In the mentioned case, all the
passengers were saved after the pilot miraculously landed the plane in the Hudson River.
It is believed that composites can create lighter, more fuel-efficient engine casings that
might be stronger and safer than those made with aluminum or other traditional materials.
The use of composite materials for manufacturing many aerospace parts is an attractive,
viable option for manufacturers. This is because of the low-cost of manufacturing as well
as the added cost saving benefits of the weight reduction. However in order to safely use
these materials, a more thorough understanding of the behaviour of composite materials
under dynamic loadings coupled with damage is required.
While numerous studies on composite materials have shown that they exhibit ratedependant behaviour (Gilat et al. 2002; Harding and Welsh 1983; Kawata et al. 1981;
Rodriguez et al. 1996), most of the models available in the literature are not rate
dependant. Among the few models which include rate dependency, some consider the rate
effect only in the shear response (Marguet et al. 2007), and yet some models do not
consider shear damage in the model at all (Iannucci et al. 2001). Characterizing the
behaviour of woven composite materials under dynamic as well as static loading is
necessary for developing a reliable constitutive model that can predict the response of
these materials.
1.2
Objectives and strategies of this research
The general objective of this research is to develop a rate-dependant constitutive damage
model which is appropriate for various woven composite materials. A prerequisite for
such a model is to characterize the static and dynamic behaviour of these materials by
performing a series of experiments. It is essential to establish a methodology for the
4
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
determination of the damage parameters present in the model. The ultimate goal is to
implement the developed model into a material user subroutine of a commercial finite
element program that can be used in finite element simulations.
Specifically, this research focuses on the characterization and material modelling of three
woven composite materials, namely a 2×2 twill weave carbon/epoxy prepreg [0]6, a
carbon/BMI prepreg [0]6 with 8 harness satin weave, and a plain weave carbon/epoxy
prepreg [0]8.
All the materials considered in this thesis are balanced woven fabrics with a zero
direction stacking sequence. First, tensile and bias extension shear experiments are
performed on the coupon specimens to obtain the static and dynamic behaviour of the
materials. Then the Foroutan-Nemes rate dependant constitutive model is developed
which governs the mechanical response of the woven composite material. The model
includes damage variables that describe the mechanical effect of distributed micro
defects, as well as formulations regarding the evolution of damage in the system. The
third step is to develop a methodology for determining the material constants present in
the constitutive model. And finally, the developed constitutive model is implemented into
a material user subroutine of a commercial finite element program to solve boundary
value problems in finite element simulations.
1.3
Structure of this Thesis
A general introduction discussing the use of composite materials in the industry as well as
the need to develop a rate dependant constitutive damage model has been presented in
this chapter, Chapter 1.
An extensive literature review is presented in Chapter 2, including a review of continuum
damage mechanics. Different dynamic experimental set-ups as well as available ratedependant response of some composite materials are discussed. Also, a review of the
available continuum damage mechanics models for composite materials is presented.
5
INTRODUCTION In Chapter 3, a detailed description of the experimental procedure is explained. A
discussion about the theory behind the experiments and the related equations is presented
in this chapter. A brief discussion about the materials under study has also been described
here.
The experimental results are presented and discussed in Chapter 4.
The development of the rate-dependant continuum damage mechanics model is presented
in Chapter 5. The characterization of material constants needed to make use of the model
is explained in this chapter. Also, the methods to determine these constants from the
experimental data are described.
Numerical implementation and simulation procedures for justification of the material
model are discussed in Chapter 6. The simulation results and findings are also presented.
Finally, conclusions, comments, and recommendations for future work are presented in
Chapter 7.
6
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
2
Equation Section (Next)
LITERATURE REVIEW
Impact loading on composite materials often causes a substantial amount of damage. The
impact velocity, material properties, mass, and the geometry of the projectile are among
several parameters that govern the type of damage. The most influential parameter is the
impact velocity. Although low or intermediate impact velocities may not induce
penetration, they could initiate surface micro cracks, delamination, and perhaps fibre
fracture. However, high impact velocities, often called ballistic impact, may trigger
complete penetration and local delamination. The penetration of a blade or blade
fragment released from a failed rotor engine into the composite containment structure is
an example of a ballistic impact. These fragments are dispersed circumferentially in all
directions at very high velocities and could damage fluid lines, control systems hardware,
and airframes if not contained. Such incidents can affect flying performance in a number
7
LITERATURE REVIEW
of direct and indirect ways and can even lead to loss of aircraft and passenger fatalities.
Therefore, a better understanding of the behaviour of composites under high velocity
loading coupled with damage is required, particularly for the aerospace industries.
When a material is loaded beyond its maximum strength, it experiences failure. In
general, modelling the failure of materials is based on the following three approaches:
•
Failure criteria based on equivalent stress or strain
•
Fracture mechanics based on energy release rate
•
Continuum damage mechanics (CDM)
The failure criteria based on the strength of materials approach only specifies the
conditions under which failure occurs, but does not predict degradation in stiffness. A
failure criterion is used to determine whether failure has occurred. Examples of failure
criteria used by researchers are the Chang-Chang criterion (Chang and Chang 1987), the
Hashin criterion (Hashin 1980), and that of Lee (Lee 1980). It is assumed that when
damage occurs, the load carrying capacity of the material is lost. Depending on the type
of failure, the modulus of elasticity in that direction is set to zero and in the case of
delamination, the elastic modulus in the out of plane direction is set to zero. These models
ignore the damage evolution leading to failure as well as energy dissipation mechanisms
which are active during damage growth, and they have been mainly developed for
modelling damage in unidirectional layers of laminated composites. These models can be
used in low velocity loadings where the damage size is small.
The objective of fracture mechanics is to study the fundamental mechanisms of fracture
and to predict and control the process of fracture. The breakthrough in this field started at
the beginning of the 20th century. Griffith (1920) is believed to be among the pioneers in
this field in published papers related to the theory of fracture strength of glass. The theory
assumed that the fracture strength is limited by the existence of initial cracks, which were
subjected to an applied tensile stress. The basic concept behind his theory was the fact
that solids possess surface energy, and in order to propagate a crack, the corresponding
surface energy must be compensated through the externally added or internally released
8
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
energy. Griffith used Inglis (1913) solution, who had studied the stress analysis of a plate
containing an elliptical cavity loaded by applied stress based on elasticity theory. His
work is valuable in understanding the fundamental mechanism of fracture due to stress
concentration at some points near the cavity or crack. A fracture mechanics approach is
mainly used in composites when mechanics of macro cracks is under investigation. Many
researchers (Chamis et al. 1996; Minnetyan et al. 1992) use this approach to model the
progressive fracture in composite materials from initiation of crack to the final failure.
However, it is worth mentioning, the application of this approach requires a prior
knowledge of the crack specifications, e.g. the location of the crack.
In continuum damage mechanics, deformation and micro failure in the material occurs
simultaneously. A cracked composite body is treated as a homogeneous continuum
containing representative internal damage variables. On loading, the composites develop
multiple matrix cracks, which degrade the stiffness of the laminates and also give rise to
more severe forms of damage such as delamination, interfacial debonding, sliding, fibre
fracture etc. Continuum damage mechanics models can be used to predict stiffness
degradation in composite materials.
Continuum damage mechanics is also used to describe the strain softening behaviour of
brittle or quasi-brittle materials such as geologic materials, concrete and wood as well as
polymer composites. The aforementioned continuum damage mechanics is the adopted
approach for the development of the damage constitutive model presented in this thesis,
hence an extensive historical literature review of the knowledge and methodologies in this
field that are relevant to this work is provided in this section.
2.1
Continuum Damage Mechanics
The problems of idealized macro cracks are studied intensively in the framework of
conventional fracture mechanics. Fracture mechanics is now an established science and
has been a powerful tool in understanding the fracture of homogeneous solids and for
extending the traditional design philosophy to include more accurate and realistic
9
LITERATURE REVIEW
methods. However, in many more common cases, where defects are almost always at the
micro-scale, especially prior to loading, and have complex geometries and topologies, the
methodology employed in conventional fracture mechanics may lose its validity for
modelling the evolution of a micro-crack system. For example, in materials such as
ceramics, most composites, concrete, and rocks, the existing cracks are small and
certainly far from being in simple shapes, such as a penny shape or planar. Cavities, and
cracks with complex geometries can be assumed as damage. Accounting for every defect
and assessing their interactions as well as their influence on the integrity and failure of the
structure is not a task that can be approached using conventional fracture mechanics
theories, such as linear elastic fracture mechanics and elastic-plastic fracture mechanics.
For these cases, new theories and models needed to be developed to obtain a better
understanding of fractures.
Continuum damage mechanics (CDM) is a relatively new discipline that focuses on
predicting the effects of progressive degradation of mechanical properties (damage
evolution) on the macro-response and failure of solids. Damage mechanics deals with
fracture in a more general way and seems to be a good approach for dealing with damage
problems. Evolution of damage is related to nucleation and growth of micro-cracks and
other micro-defects. The CDM approach can predict the full range of the deterioration of
material, from the virgin material with no damage, to fully disintegrated material with full
damage for different failure modes.
CDM bridges the gap between traditional elasticity theory and fracture mechanics. The
CDM-description of a defect-free material is equivalent to the application of elasticity
theory. The CDM and elasticity theories will deviate as soon as micro defects start to
nucleate and grow with increased loading. At the other end of the spectrum, when the
growth of one or at most a few micro-cracks becomes the dominant aspect of the
deformation of the material, a CDM-description will no longer suffice and the fracture
mechanics theory is more suitable.
10
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
The CDM approach has been a subject of extensive research in the past decades. CDM
smooths out the irregularities in shape and inhomogeneity of spatial distribution of
microstructures of materials and thus make it possible to get a simple model that
considers effects from the micro level while making predictions on the macro level.
Kachanov (1958) and Rabotnov (1963) were the first to develop the concept of a scalar
continuous damage variable to represent damage progression during creep of metals.
They considered the ratio of void areas to total area in a cross section of a body as a
measure of damage. Although their work was not directly in the field of CDM, the
successful introduction of a separate internal state field variable, which provides a
qualitative measure of the effects that randomly distributed micro-defects exert on the
macro parameters of a structure, sparked the development of the concepts and theories
which later became continuum damage mechanics (Chaboche 1988a; 1988b; Lemaitre
and Chaboche 1990). The earliest research on applications of CDM can be found in
works of Krajcinovic (Krajcinovic 1983; Krajcinovic 1984; Krajcinovic and Fonseka
1981; Krajcinovic and Silva 1982), Chalant (Chalant and Remy 1983), and Resende
(Resende and Martin 1984). The CDM approach was later implemented for composite
materials by Talreja (1985a).
2.1.1 Internal State Variables
The type of internal state variables (damage variables) used in a CDM model greatly
affects its success. The damage variables, besides being capable of representing the
complex micro defect features, should be simple enough to make the model applicable to
engineering applications. The area density definition of damage proposed by Kachanov
and developed by Rabotnov is widely accepted in continuum damage mechanics (see
Lemaitre and Chaboche 1990). Although Kachanov's approach is reasonable, it may be
difficult to calculate the effective area of a cross-section directly from the crack geometry
and distribution while taking stress concentration and interaction into account. It is known
that the presence of a crack or defect causes a perturbation in the stress/strain and thus
strain energy density in a certain volume around the crack or defect. That is to say, the
effect of a crack on the strain energy density perturbation is volumetric in nature.
11
LITERATURE REVIEW
Murakami (1983) defined damage as a micro structural change that results in a
deterioration of material properties. It is therefore possible to define the macroscopic
damage variables through a volumetric consideration to describe the mechanical state of
the material. There are numerous ways to define the damage variable as discussed in
Lemaitre (1996).
Depending on the material, damage can be represented as a scalar, where a single variable
can represent the complete damage state of a volume element of the material, as a vector,
where the overall damage can be characterized by components in each co-ordinate
direction, or as a higher order tensor.
In composite materials, damage is usually related to crack density, however its definition
and representation varies from model to model. For example, the pioneers in the field of
CDM (Kachanov 1958; Rabotnov 1963), used the void area density in a cross-sectional
area of a bar as the scalar damage variable for a one dimensional problem. Some other
earliest proposed damage theories involved scalar damage variables based on the net
reduction of load-bearing area due to micro-defect cavity growth. A scalar damage
parameter was employed to describe the damage in composite materials by Pickett et al.
(1990). They mentioned that the scalar representation was to simplify the model in the
interest of computational efficiency.
To explain a common damage definition, consider a volume element at macro scale, large
enough to contain many defects, and small enough to be considered as a material point of
mechanics of continua (Lemaitre 1996; Matzenmiller et al. 1995). Let S be the cross
sectional area of this volume element, V m , defined by unit vector n , before any loading
has occurred. This area becomes the effective area, S , after loading due to formation of
micro cracks and void growth with an area of S d as illustrated in Figure 2.1. In the case
of isotropy, that is when micro-cracks and cavities are distributed and oriented uniformly,
12
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
n
S
Sd
Vm
Figure 2.1 Meso-definition of damage
there is no directionality involved, so the area no longer depends on n , hence the damage
becomes an isotropic damage parameter represented by the following equation
d=
Sd S − S
=
S
S
(2.1)
It’s worth mentioning that, the scalar damage variable, d , is defined in such a way that d
equal to zero implies the initial undamaged state of the material, and when d is equal to
one, it refers to the final failed state of the material, hence;
0 ≤ d ≤1
(2.2)
Many models can be derived for damage, the difficulty being the choice of the analytic
expression and its identification from experiments in each particular case of damage.
Among several different methods, damage may come from the variation of Young’s
modulus through elasticity law coupled with damage (Li et al. 1998; Zou et al. 2003).
Relating damage to the influence it has on the stiffness is one of the most commonly used
formulations.
d = 1−
E
E0
(2.3)
13
LITERATURE REVIEW
where E0 and E are the initial undamaged and current damaged stiffness, respectively.
Li et al. (1998) also used a scalar damage parameter, however, its influence was related to
the elastic constants of all directions. Budiansky and O'Connell (1976) assumed all cracks
are elliptical and have the same aspect ratio, and derived an analytical expression for the
effective elastic moduli as a function of the crack density parameter. When damage is
isotropic or in other words it is randomly distributed, it can be represented through a
scalar damage parameter. However, anisotropic damage requires a higher order tensor
representation of damage.
Due to the inevitable isotropic nature of a scalar damage variable, sometimes a more
elaborate means to characterize the anisotropic damage observed experimentally in some
materials is needed. In that case, more general states of damage can be captured using a
second rank tensor. Second-rank tensor representation of damage was first proposed by
Vakulenko and Kachanov (1971) and later developed by many other researchers
(Kachanov 1980; Murakami and Ohno 1981). As such, a variety of symmetric second
rank tensors were proposed for this purpose. A general formulation of this second order
damage tensor, d , shown in principle directions n i , has the form
3
d = ∑ Ωi n i ⊗ n i
(i = 1, 2,3; no summation)
(2.4)
i =1
where Ω i are the principle values of the damage tensor in a spectral representation.
Importantly, if d is formulated such that Ω i = 1 /(1 − d ) for i = 1, 2, 3 , the resulting
damage tensor provides a three dimensional generalization of the Kachanov-Rabotnov
damage theory. The tensor d can be experimentally determined, and can be used to
formulate a damage relation between the Cauchy and the effective stress tensors (Chow
and Wang 1987). In general, some existing damage theories (Lemaitre and Chaboche
1990) use tensor variables to describe the state of the material (Allen et al. 1987).
Nevertheless, as noted by Krajcinovic (1989), the mathematical complexity of the higher
ranked tensors make them inexpedient for practical purposes.
14
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
In all the models damage is explained as a variable degrading a property of the original
undamaged material. The most used type of variable for isotropic damage modelling is
the scalar damage variable. Scalar variables mainly impose some constraints on the
evolution of elastic properties. It is worth mentioning that isotropic damage does not
mean that the original material is isotropic, it only indicates that the original isotropy or
anisotropy of the material is preserved. In other words, in theory, it is possible for an
orthotropic material to stay orthotropic after going through an isotropic damage process.
In this work the state of damage at a point is described by a limited number of scalar
damage variables, each representing a particular type of damage. They will be defined by
the effect they have on the mechanical properties, in particular by the reduction in the
elastic moduli.
2.1.2 Effective Stress
The appearance of discontinuities leads directly to the concept of effective stress, that is,
the stress calculated over the section which effectively resists the forces. If the stress
reaches a certain value, damage starts to develop. This stress level is the damage
threshold stress. After this, and due to presence of damage in the cross section, the
resisting area is diminished and the effective stress will be acting on the resisting section.
The mapping from the Cauchy stress tensor σ , to the effective stress tensor σ based on
the area and effective area becomes straightforward as explained by Chow and Wang
(1987), that is;
σ=σ
S
σ
=
S 1− d
(2.5)
This is known as the Kachanov-Rabotnov damage theory, which was originally
developed in one dimension, and formed the basis for the development of CDM.
The effective stress concept states that any deformation behaviour, whether uniaxial,
multi-axial, elastic, plastic or viscoplastic, of a damaged material is represented by the
15
LITERATURE REVIEW
constitutive laws of the virgin material in which the applied stress σ is replaced by the
effective stress σ .
The deformation of damaged material depends not only on the damage parameters (net
area reduction, for example), but also on their spatial arrangement. This implies that in
the case of a second order damage tensor, the Cauchy stress tensor, σ , should be mapped
to the effective stress tensor, σ , by a fourth order tensor.
The fourth order damage effect tensor M( d ) , derived from the damage tensor , was
developed for this purpose. The damage effect tensor provides a linear mapping of the
stress tensor to the effective stress tensor, resulting in a redistribution of stresses over the
reduced remaining area, and thus higher local stresses.
σ = M (d ) σ
(2.6)
In the case of isotropic material behaviour, M( d ) should reduce to the scalar
representation of the damage parameter as illustrated in the right hand side of equation
(2.5).
There are many proposed formulations for the damage effect tensor, M( d ) . Chow and
Wang (1987) proposed an alternate form of the tensor M( d ) that was later discussed by
Ju (1989) in the principle coordinate system. The 6x6 matrix representation of the damage
effect tensor is as follows:
16
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
⎡ 1
⎢1 − d
1
⎢
⎢
⎢ 0
⎢
⎢
⎢ 0
M (d ) = ⎢⎢
⎢ 0
⎢
⎢
⎢ 0
⎢
⎢
⎢ 0
⎢⎣
0
1
1 − d2
0
0
0
0
0
0
0
0
0
1
1 − d3
1
0
0
0
0
0
0
0
0
0
(1 − d 2 )(1 − d 3 )
1
(1 − d1 )(1 − d 3 )
0
⎤
⎥
⎥
⎥
0
⎥
⎥
⎥
0
⎥
⎥
⎥
0
⎥
⎥
⎥
0
⎥
⎥
⎥
1
⎥
(1 − d1 )(1 − d 2 ) ⎥⎦
0
(2.7)
In this formulation, d1 , d2 , and d3 are the damage variables at their respective principle
axes and the effective stress tensor is used in its reduced format. It is important to note
that the formulation in this equation does not depend on a priori knowledge of the
principle stress directions. A special vector form of this formulation uses M( d ) as a 3x3
matrix of the damage variables represented in the principal form as follows:
⎡ 1
⎢
⎢1 − d1
⎢
M (d ) = ⎢ 0
⎢
⎢
⎢ 0
⎣
0
1
1 − d2
0
⎤
0 ⎥
⎥
⎥
0 ⎥
⎥
1 ⎥
⎥
1 − d3 ⎦
(2.8)
where di ( i = 1,2,3 ) are the damage variables in their respective principle axes, and the
effective stress vector is given as σ = [σ 1 σ 2
σ 3 ] . Note that the formulation does
T
reduce to isotropic material damage in the case of d1 = d2 = d3 = d . The damage effect
matrix of equation (2.8) must be modified when the principle axes of the stresses are not
known, and this restricts its use in general applications.
17
LITERATURE REVIEW
Several other forms of the fourth order damage tensor M( d ) were discussed by Betten
(1983) and Chow and Lu (1989). It is worth nothing that in each case the tensor M(d )
was defined by some seemingly arbitrary assumption aimed at making the effective stress
tensor and researchers have used different forms which depend on the choice and form of
the damage parameters.
2.1.3 Strain Equivalence Hypothesis
A hypothesis of strain equivalence for isotropic damage was proposed by Lemaitre in
which the Cauchy stress was replaced by the effective stress in the constitutive equation.
In the hypothesis he states that (Lemaitre 1996):
“Any strain constitutive equation for a damaged material may be derived in the same way
as for a virgin material except that the usual stress is replaced by the effective stress.”
For solving the damage problems, one must establish that the strain tensor is the same in
the effective damaged and undamaged state, that is
ε =ε
(2.9)
Combining the damage effect tensor, M( d ) , with the undamaged stiffness tensor, C 0 ,
results in a damage dependant stiffness tensor, C (d ) . The stress-strain relationship in a
damaged material uses the effective stress and the damaged elastic stiffness tensor. A
general constitutive relationship for a CDM model is represented as
σ = C (d ) ε
(2.10)
It is worth mentioning that the damaged elastic stiffness tensor, C (d ) , is not guaranteed
to be symmetric based on this principle. This elastic CDM model can be expanded to
18
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
consider the strain rate effects in the material behaviour (Armero and Oller 2000;
Chaboche 1988a; 1988b; Dechaene et al. 2002; Nemes and Speciel 1996).
σ = C (d , d ) ε + C (d ) ε
(2.11)
where the over dot represents the material time derivative.
2.2
Dynamic Tensile Experiments Setups
The increasing use of composite materials in structures subjected to static and dynamic
loading demands the full characterization of their mechanical properties under a wide
range of strain rates. Unlike metals, which do not exhibit significant rate dependency at
room temperature for rates below 1000/sec, composite materials may have this behaviour
at much lower rates. The low rate tests are usually performed using hydraulic testing
machines. On the other hand, various methods have been developed to characterize
material behaviour at intermediate and high rate loading.
Reliable data on the dynamic properties of composite materials is sparse, which could be
due to the difficulties associated with the experimental procedures required in obtaining
the dynamic response. Moreover, most of the results available in the literature are
incomplete and as a result, cannot contribute to developing a reliable constitutive model
for predicting the response of woven composite materials under dynamic loading. A
review on the various methods used for characterizing the dynamic response of composite
materials is provided in this section.
The experimental method used to determine the dynamic properties depends on the range
of strain rates concerned and on the load case. Some of the earliest systems are the
Charpy pendulum (Harris et al. 1971; Hayes and Adams 1982), with fixtures adapted to
perform tensile tests or to other test configurations, and Izod impact testing (Hancox
1971). These testing methods, in which a notched beam specimen is subjected to impact
loading, were originally developed for use with isotropic materials, to obtain fracture
energy. Even though high strain deformation can be achieved near the notches,
19
LITERATURE REVIEW
fundamental formulation of strain rate effects on the material properties is not feasible
due to the complex stress and strain fields as well as combined modes of failure involved
in these methods.
Drop weight, pendulum and explosively driven hammers can be employed to use
unnotched specimens to obtain a uniform state of stress (Adams and Adams 1989;
Armenakas and Sciammarella 1973; Lifshitz 1976; Rotem and Lifshitz 1971). These
testing methods work more or less with the same principle, in which, a hammer driven by
gravity, a spring or an explosive force hits an anvil connected, directly or indirectly, to
one end of the specimen, as a result loading the specimen. Load in the specimen is usually
measured by means of a load cell connected to the non-impacted side of the specimen.
For strain measurements, however, different techniques have been used. Adam and Adam
used pendulum impact tests, with strain rates of approximately 50/sec to test
unidirectional graphite/epoxy and glass/epoxy under tensile loading. Lifshitz and his
coworkers used a drop weight impact machine for angle ply specimens of glass/epoxy.
Both tests need instrumentation of the specimen for strain measurements. Armenakas and
Sciammarella used an explosively driven hammer to apply the load, and high speed
photography and a Moire technique to measure the strain field. However their strain
measurement was not uniform along the longitudinal and transverse axis, with as much as
100% difference. Therefore, they had to average their strain results to obtain the stressstrain curves. The non-uniformity of strain results is likely to hold true for the drop
weight and the pendulum techniques due to the similarity of the methods. Moreover,
these techniques are not free of stress wave reflection in the hammer as well as in the
specimen. These waves are superimposed on the stress-time response of the specimen and
create difficulties in interpreting the experimental results.
The various experimental methods show that tests at high strain rates are quite difficult to
perform and that they are not easy to interpret. The most popular method is the use of
split Hopkinson bars, because they permit a uniaxial homogenous state of stress at very
high strain rate. But this method is complex and needs to be suited to the range of
materials to be studied.
20
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
2.2.1 Split Hopkinson Bar Setups
Kolsky (1949) was the pioneer of the split Hopkinson bar apparatus; he introduced the
compression version in which he used detonators to launch compressive waves. Over the
last several decades, shear, torsion and tensile versions of the apparatus have been
developed. Nowadays, an axial impact from a striker bar fired by an air gun is used to
achieve high strain rate loading. The Hopkinson bar apparatus is capable of achieving
strain rates between 100/sec and 10000/sec, as compared to the quasi-static loading rate
usually taken to be 0.001/sec.
Despite the well established methods of the Hopkinson bar, at least for conventional
materials, not much data on the high rate mechanical response of composite materials is
available. Even though compression testing is the simplest to perform, for several
reasons, tensile testing is preferred for obtaining the high rate response of composite
materials. The compression failure is not as simple and well defined as tensile failure, and
the compression of composite in the fibre direction is usually of little interest. This is due
to the fact that failure occurs as a result of matrix cracking and interfacial debonding
rather than by any type of fibre breakage. It is shown by many researchers that, even in
the static strain rates, composite materials have different behaviour in compression and
tension.
Researchers have developed different tensile versions of the Hopkinson bar for testing
composite materials. Generation of a tensile pulse as well as specimen gripping are the
most noted difficulties in the tensile Hopkinson bar testing technique. Unlike metals,
which can be fastened easily by the use of threaded ends, composite specimens, which are
usually in the form of flat strips, cannot be threaded. In addition, to produce high strain
rates, Hopkinson bar testing requires a short gage length in the specimens; hence the
specimens cannot be designed according to standards for uniaxial static testing of
composite materials (ASTM 1989).
21
LITERATURE REVIEW
Several arrangements have been described in the literature for tensile testing. In general,
tensile versions of the Hopkinson bar use two approaches for applying tensile pulse to the
specimen. In one approach, the input bar is subjected to a direct tensile pulse which is in
turn transmitted to the specimen. In the other approach, the input bar is subjected to an
initial compressive pulse, which is subsequently transformed into a tensile pulse.
APPROACH ONE, DIRECT TENSILE PULSE
In this approach a clamp/load reaction assembly was used to pre-stress a portion of the
input bar (Albertini and Montagnani 1974; Staab and Gilat 1991). Initially the clamp was
tightened and a direct tension is applied at the end of the bar through a system of cables
pulleys and hydraulic pump, which generates an input load stored in that section of the
bar. On releasing the clamp, a tensile wave of half the magnitude of the stored force
propagates towards the specimen as the tensile load, and a released wave propagates in
the opposite direction (Gilat et al. 2006) (Figure 2.2).
specimen
clamp
applied
load
Figure 2.2 Schematic of Hopkinson bar configuration using direct tensile pulse
APPROACH TWO, INDIRECT TENSILE PULSE
In the second approach, the researchers used a normal compressive version of Hopkinson
bar in order to produce tension. This was first suggested by Nicholas (1981), and was
later used by other researchers (Al-Mousawi et al. 1997; Ellwood et al. 1982; Peroni and
Peroni 2008). In this design, a collar made of the same material as the set-up bars is
placed over the specimen and firmly fitted between the incident and transmitted bars
(Figure 2.3a). When the input bar is struck by a striker, the input compressive wave is
transmitted almost entirely through the collar to the output bar, with little effect on the
specimen. On reaching the free end of the transmitted bar, it is reflected back as a tensile
22
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
pulse. This tensile pulse loads the specimen, since the collar is unable to sustain any
tensile load.
Lindholm and Yeakley (1968) also used a compressive Hopkinson bar setup to obtain
tensile behaviour. They fitted a special ‘hat’ shaped tensile specimen between a solid
cylindrical incident bar and a tubular transmitter bar (Figure 2.3b). The applied loading
and deformation of the specimen are derived from the strain-time measurements on the
radial surface of the two bars using diametrically opposed strain gages. The internal wave
propagation is neglected in this analysis. They investigated the effect of the unusual
geometry of the ‘hat-shaped’ specimen by comparing its static stress strain behaviour
with that of round tensile test specimens. Even though they found good agreement in their
results, later Lindholm et al. (1971) have reported that the effect of the ‘hat shaped’
geometry of the specimen was the only source of error for the drop in the strength of an
aluminum alloy.
Kawata et al. (1980) developed a bar-to-block setup (Figure 2.3c), which consists of a
hammer (rotating disc or pendulum type), impact block, specimen and output bar. The
impact block is impacted by the hammer and the specimen is deformed. They believe that
this is a simpler method as compared to the bar to bar method.
In the other designs, the compression version of Hopkinson bar is significantly modified
to produce tension in the specimen. The first design was developed by Harding et al.
(1960) where the input loading bar is a hollow tube within which the output bar slides
freely. The specimen connects the two bars at the yoke. Harding and Welsh (1983)
modified this model some years later to include an instrumented input bar preceding the
specimen (Figure 2.3d). Leblanc and Lassila (1993) used the same concept, however
changed the hollow compression loading tube to a split compression tube in order to
allow easy access to the incident and the transmitter bars during the installation of the
sample grip assembly. Similar concept was used by Hauser (1966) and later by Eskandari
and Nemes (2000) where instead of using the tube, two transmitter side bars was used to
transform the compressive pulse into a tensile pulse (Figure 2.3e).
23
LITERATURE REVIEW
specimen
striker
bar
collar
(a)
specimen
striker
bar
(b)
applied
load
impact
block
specimen
output bar
(c)
specimen
transmitter bars
transfer connection
input bar
specimen
striker
(d)
(e)
specimen
impact
block
(f)
24
striker
tube
thin part thick part
of input bar
damping
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Figure 2.3 Various Hopkinson bar setups using indirect tensile pulse adapted from
(a) Ellwood et al. (1982), (b) Lindholm and Yeakley (1968), (c) Kawata et al. (1980),
(d) Harding et al. (1960), (e) Eskandari and Nemes (2000), and (f) Li et al. (1993)
Some researchers have used a short striker tube around the input bar (Huh et al. 2002;
Lataillade et al. 1995; Li et al. 1993). The striker loaded by a hammer or a gas gun
impacts the end of the thicker part of the input bar. This produces a tensile pulse in the
input bar which travels to the specimen (Figure 2.3f).
The apparatus used in this research project is similar to those which use a striker tube
around the input bar. This is the apparatus available at McGill University and its
specifications will be explained in chapter three of this thesis. In that, a striker tube which
is loaded with a gas gun impacts an anvil, which in turn produces a tensile pulse that
loads the specimen in tension.
2.3
Strain-Rate Dependant Experiments
Traditionally, composite materials have been assumed to have a linear elastic behaviour.
However, results from the literature introduce doubts concerning this assumption. The
stress-strain curves obtained by many researchers have shown significant non-linearity. In
particular, at high strain rates the non-linear relationship between strains and stresses is
more evident.
A number of researchers have reported high strain-rate behaviour of composites, mainly
noting that the failure stress and mechanism are strain-rate dependent. Abrate (1991;
1994; 2001) has presented a comprehensive review of the literature on impact on
composite materials. As the rate of loading is increased, there is less time for damage to
develop. Hence, the amount of accumulated damage at a particular strain level decreases
as strain rate increases, and at high strain rates the material can withstand higher load and
failure strain.
25
LITERATURE REVIEW
Harding and Welsh (1983) performed impact tensile tests on both GRFP (Glass
Reinforced Fibre Plastic) and CFRP (Carbon Reinforced Fibre Plastic), which are
unidirectional composites. They found strain rate sensitivity in the experiments performed
at the strain rate range of about 5E-5/sec, 7/sec, and 450/sec for CFRP and 1E-4/sec,
23/sec, and 870/sec for GFRP.
Newill and Vinson (1993) tested various polymer matrix composites. The strain rates
involved varied from static values up to 2000/sec and they found that strain rate effects
are significant in general for all composite materials, or at least one should assume so
until proven otherwise by numerous tests and subsequent analysis.
Staab and Gilat (1995) explained the rate sensitivity of angle ply glass-epoxy laminates
and in a later paper Gilat et al., (2002) showed the significant effect of strain rate on the
response of carbon/epoxy composites. They conducted tensile tests on the same type of
specimen over a wide strain rate range with tests performed at 5E-5/sec, 1/sec, and
400/sec. They showed that both unidirectional and angle ply composites are strain rate
dependent. The shear results from the tests with [ ±45°]S specimen orientation also show
considerable strain rate dependence (Goldberg et al. 2003). In general, with increasing
strain rates, a stiffer material response is observed, that is, the stress-strain curve is
steeper for higher strain rates. It is observed that, at all strain rates, the stress-strain curves
have an initial linear response, a rounded transition to inelastic response followed by an
inelastic range with a nearly constant hardening rate.
Weeks and Sun (1998) and Jadhav et al.(2003) studied the rate dependent behaviour of
angle ply composites of AS4-PEEK and graphite-epoxy, respectively at various strain
rates of about 1E-5/sec for static and a moderate to dynamic strain rate range of 100/sec
to 1000/sec. Rate dependency is confirmed in angle ply composites from their results.
Their experiments were conducted with a servo-hydraulic testing machine and a tensile
split Hopkinson pressure bar setup.
26
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Barre et al. (1996) have conducted experiments on five materials all reinforced with
glass. They showed strain rate has a significant effect on tensile modulus and tensile
strength, and also, noted significant discrepancies in the papers they reviewed. For a
given material, some studies report an increase in tensile strength, some find negligible
changes, while others find a decrease. Most of the available test data do not provide a
complete characterization of the material under investigation.
Kawata et al. (1981), Welsh and Harding (1985), Harding et al. (1989), Rodriguez et al.
(1996), and Sham et al. (2000) studied the tensile behavior of woven composites.
Excluding Kawata et al. who performed their dynamic tensile experiments using the one
bar method of block-to-bar with a rotating disc or pendulum as the loading equipment, the
other researchers used a tensile version of the Hopkinson bar apparatus for the dynamic
experiments. The results from Kawata et al. show (Figure 2.4) that glass woven
cloth/polyester and carbon woven cloth/epoxy composites are strain-rate sensitive. It can
be seen that at the static strain rate the stress-strain curve is linear but at dynamic rates the
behavior becomes non-linear and also the peak tensile stress increases.
80
Dynamic
70
50
Stress [Kg/mm 2 ]
Stress [Kg/mm 2 ]
60
40
Dynamic
30
20
Static
10
60
50
40
30
20
Static
10
0
0
0
5
Strain %
10
15
0
5
Strain %
10
Figure 2.4 Stress-strain curves at dynamic and static strain rates in tension (a) Glass
Satin Woven Cloth/ Polyester (b) Carbon Plain Woven Cloth/ Epoxy (adapted from
Kawata et al. 1981)
27
LITERATURE REVIEW
Harding and Welsh (1983; 1985) used a tensile Hopkinson pressure bar apparatus to
characterize the response of composite materials. They obtained tensile stress strain
curves at strain-rates of about 1E-4/sec, 100/sec, and 1000/sec for polyester resin
specimens reinforced with satin-weave fabric using carbon, Kevlar and glass fibres. For
all three composites, they observed rate-dependence of the tensile modulus. Harding et al.
(1989) performed experiments on woven carbon fibre epoxy laminates at quasi-static and
dynamic rates (9E-4/sec, 290/sec), and found that the laminates were strain-rate sensitive
as illustrated in Figure 2.5. The stress-strain curve results show significant increase in the
tensile failure strength with strain rate.
600
Impact test
~ 290/sec
Stress [MPa]
500
400
300
Quasi-Static test
~ 9*10 -4/sec
200
100
0
0.0
0.5
1.0
1.5
Strain %
Figure 2.5 Stress-strain curves for woven carbon fibre epoxy laminates at different
strain rates in tension (adapted from Harding et al. 1989)
Rodriguez et al. (1996) performed tensile tests at three strain rates of approximately
0.001/sec, 1/sec, and 1000/sec on aramid and polyethylene woven fabric composites
using a tensile Hopkinson bar and conventional testing machine. Some of their
experimental results are shown in Figure 2.6. Strain-rate sensitivity is apparent from these
results as well.
28
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
(a)
(b)
Figure 2.6 Tensile stress-strain curves at different strain rates for (a) woven
aramid/polypropylene (b) woven polyethylene/ polyethylene (Rodriguez et al. 1996
with permission of Elsevier)
Sham et al. (2000) studied the tensile behaviour of glass woven fabric reinforced
composites. Figure 2.7 shows the stress-strain curves at various strain rates ranging from
about 1E-4/sec to 900/sec. According to their observation, the tensile strength and failure
strain of GFRP increased with increasing strain rate due to the brittle-ductile transition of
glass fibre, and also the strain rate sensitivity was higher in the warp direction than in the
weft direction.
Unlike the tensile response, not much data is available on the comparison of in-plane
static and dynamic shear response of woven composite materials. However, it is worth
mentioning that it is well understood in the literature that the behaviour in shear is
strongly nonlinear and irreversible as was mentioned in many papers (Johnson et al.
2001; Marguet et al. 2007). This could be from the strain rate dependant results available
for unidirectional [45°] specimens tested in static and dynamic rates. Many researchers
(including Hosur et al. 2003b; Naik et al. 2007; Riendeau and Nemes 1996; Werner and
Dharan 1986) have investigated the interlaminar shear response of composite materials.
29
LITERATURE REVIEW
800
Stress [MPa]
900/sec
600
600/sec
400
0.18/sec
1.67E-4/sec
200
0
0%
1%
2%
3%
4%
5%
Strain %
Figure 2.7 Stress-strain curves of glass woven fabric-reinforced composites
subjected to different strain rates (adapted from Sham et al. 2000)
Werner and Dharan (1986) obtained the interlaminar shear as well as transverse shear
response of woven graphite epoxy laminates using split Hopkinson bar. They concluded
that in transverse shear, the strain rate effect is significant but not so in interlaminar shear
response.
Chiem and Liu (1987) have obtained the shear response of woven carbon-epoxy
composite materials using a torsional split Hopkinson bar. The strain rates varies from
6E-4 to 2E-3/sec for quasi static loading and from 2623/sec to 5410/sec for dynamic
loading. In another paper (Chiem and Liu 1988) they compared the tensile and shear
strength of these materials subjected to very high strain rates from 612-1368/sec for
tensile and 1021-5410/sec for the shear loading. They concluded that these materials are
strain rate dependent in both tension and shear, which according to them can be described
by a power law function.
Hsiao et al. (1999) conducted dynamic and static experiments from quasi static to
1200/sec on [45°] unidirectional off-axis carbon epoxy laminates specimen. They
30
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
investigated the strain rate effect on the in-plane shear behaviour of the unidirectional
composite and from their results concluded that the shear stress-strain curves show high
nonlinearity with a plateau region at a stress level that increases with increasing strain
rate.
Hou and Ruiz (2000) performed tensile and in-plane shear tests at different strain rates of
4/sec and 600/sec on woven CFRP T300/914. Their in-plane shear experiments were
conducted with a split Hopkinson pressure bar on 5-ply [±45°] specimen. From their
results, they concluded that the shear response is strain rate dependent; however, they
observed linear elastic behaviour for the specimens under tension.
From the above discussions it is evident that the advanced composite materials are strain
rate dependent. The effect of strain rate is clearly distinct and significant as observed
from the various stress-strain responses at different strain rates. It is worth noting that not
much data is available in literature for characterizing the behaviour of carbon or graphite
woven composite materials. However, in all the configurations tested, higher stiffness is
observed with increasing strain rate. Since a rate-dependent constitutive relation is
required for adequately modelling the failure in composites, in particular woven fabric
composites which is the material used for this study, a review of the available CDM
models is presented in the next section.
2.4
Review of CDM Models for Composite Materials
The general theory of continuum damage mechanics is applied to the specific problem of
fibre reinforced composites. Continuum damage mechanics is developed on the concept
of material science and continuum mechanics. It is based on the thermodynamics of
irreversible process, internal state variable theory and consideration of physical material
response. Decisions must be made regarding the mathematical approach to formulate the
damage parameters, as well as experimental approaches to characterize these parameters.
Lemaitre (1996) postulated the strain equivalence principle. Alternative theories were
postulated under this scenario; such as the elastic energy equivalence concept, the free
31
LITERATURE REVIEW
energy function, and damage energy release rate. In any case, CDM introduces the
effective stress tensor, to replace the traditional Cauchy stress tensor, in stress-strain
relationships for solving boundary value problems. In general, the development of CDM
models involves three primary issues: damage variables, damage evolution laws and the
constitutive equations which are present in most CDM models.
Since damage presents itself in the form of cracks in the material, it is a function of crack
density. As explained before, there are numerous ways to model the damage variables.
Due to different modes of damage in composite material, i.e. matrix cracking parallel to
the fibres, fibre breakage, and fibre matrix debonding, each damage variable can
represent a specific type of damage in the models. In the CDM models available in
literature, different assumptions exist for defining the damage variables (Woo and Li
1993).
The term damage evolution is commonly used to designate a non-linear deformation
process during which two different phenomena take place, namely, nucleation of micro
cracks and growth of the existing micro-cracks. The evolution equations of damage in the
CDM models are perhaps the most arbitrary part of the model development. The
evolution equations very much depend on how the defined damaged variables behave in a
particular form of material. Various forms and logic are used by researchers which will be
explained in more detail.
Talreja (1985a; b) was one of the first researchers who used CDM for modelling
composite materials. He assigned different damage parameters to the fibre and matrix
directions in unidirectional lamina (Talreja 1989). The model was used to predict the
stiffness reduction in angle ply laminates, however no detail was provided for the damage
growth. The model was modified later to include the rate effects for predicting the
viscoelastic behaviour of cross-ply laminates with transverse cracks (Kumar and Talreja
2003).
32
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
A simple CDM model was developed by Pickett et al. (1990). A number of assumptions
were incorporated to have a computationally efficient model; however, this limited the
capability of the model. Their model had linear damage development, i.e. a linear
modulus reduction with a scalar representation of damage. Among other works, CDM
models of Aboudi (1987) and Harris et al. (1988) also used a linear stiffness reduction
with the damage parameters.
Matzenmiller et al. (1995) constructed a CDM model based on the work of Talreja
(1985a). The model consists of three damage parameters, two of which are related to the
in-plane lamina directions, and one is associated with the effect of damage on shear. Their
damage growth model was based on a Weibull distribution of strength. Williams and
Vaziri (1995) implemented the model into a finite element code. Each damage growth
parameter has the form
⎡
1 ⎛ ε
d = 1 − exp ⎢- ⎜
⎢ me ⎜⎝ ε f
⎣
⎞
⎟⎟
⎠
m⎤
⎡ 1 ⎛ Eε ⎞m ⎤
⎥ = 1 − exp ⎢- ⎜
⎟ ⎥
⎥
me
X
⎝
⎠ ⎥⎦
⎢
⎣
⎦
(2.12)
where E, ε f, and X are the elastic modulus, nominal failure strain, and strength of the
material associated with the damage mode respectively, e is the base of the natural
logarithm, and m defines the shape of the damage growth curve. The higher the m value,
the more abrupt the descending branch of the stress-strain curve beyond the nominal peak
stress and hence more instantaneous is the failure. The low m value represents a more
gradual strain-softening response.
The assumptions behind the formulation of their model are (Nandlall et al. 1998): (i) a
quasi-isotropic laminated composite can be represented as an axisymmetric homogeneous
continuum, irrespective of the damage state, (ii) applying linear elasticity provided the
damage state does not change, (iii) damage is the only cause of non-linear behaviour of
the composite lamina, that is no nonlinear elastic or plastic deformation of the
constituents exist, (iv) damage takes the form of distributed disk-like cracks oriented
33
LITERATURE REVIEW
parallel or normal to the principal material directions and as a result, symmetry of the
lamina is preserved, and (v) The state of damage present in the material is characterized
by a set of directionally-dependent damage parameters. Their work was further improved
by Floyd (2004), who among other things included a technique that explicitly accounts
for the energy dissipation by single elements. However, even though both works found
that rate dependency is an important factor, it was not considered in their model.
Wang and Xia (1997) used a bimodal Weibull distribution to describe the strength
distribution of fibres and their damage growth parameter has the form of
⎡ ⎛ Eε ⎞m1 ⎛ Eε ⎞m2 ⎤
d = 1 − exp ⎢ − ⎜
⎟ −⎜
⎟ ⎥
σ
⎢ ⎝ σ 01 ⎠
⎝ 02 ⎠ ⎥⎦
⎣
(2.13)
where m1, m2 and σ01, σ02 are the shape and scale parameters of the double Weibull
distribution function. Wang and Xia (2000) assumed the unidirectional composite as a
coated fibre bundle which, consists of a number of parallel coated fibres. Every coated
fibre has the same cross-sectional area and length. They developed a one-dimensional
damage constitutive equation for unidirectional composites. The model included strainrate effects in a double Weibull distribution and has the form,
⎡ ⎛ E ε ε ⎞m1(ε ) ⎛ E ε ε ⎞m 2 (ε ) ⎤
( )
( )
⎥
σ = E ( ε ) ε exp ⎢− ⎜⎜
−⎜
⎟⎟
⎜ σ ( ε ) ⎟⎟
⎢ ⎝ σ 01 ( ε ) ⎠
⎥
⎝ 02
⎠
⎣
⎦
(2.14)
It was found that with increasing strain rate, there is a tendency to increase Young’s
modulus (E), maximum stress (σmax), and strain relative to maximum stress (εm).
Therefore E, m1, m2, σ01, and σ02 are all function of strain rate.
Several researchers have been concerned with the development of CDM laws for
composites failure using finite element codes. Ladeveze and his coworkers (Ladeveze
1994; Ladeveze and Le Dantec 1992) developed a CDM model for unidirectional ply
34
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
composite materials. In their model, two scalar damage variables are considered; one
normal to the fibre, that is 11- or 22-direction and the other in the shear 12-directions. The
total strain is the summation of the elastic and plastic strains. They assume that no plastic
deformation occurs in the fibre direction.
Johnson et al. (2001) then extended Ladeveze’s model to fabric-reinforced composites
under in-plane loads. Their model contains elastic damage in fibre directions and elasticplastic damage for inelastic shear effect. Neither of the models is rate-dependent;
however, the delamination failure envelope is incorporated in their model. They defined
three damage parameters; associated to damage in the two principal directions and inplane shear. Their damage parameters are in terms of the damage evolution function,
which is a linear or log linear function of damage energy release rates. The plastic strains
are associated only with the matrix dominated in-plane shear response, and the total strain
is the summation of the elastic and plastic strains.
Randles and Nemes (1992) and Nemes and Speciel (1996) developed a rate-dependent
CDM model for thick composites subjected to high strain-rate deformation. It is worth
noting that their model was shown to lead to unique and stable solutions and was mesh
insensitive for a realistic range of damage propagation time constants. The model was
used to simulate cantilever bending tests on graphite/epoxy at high strain rates. They did
not show any comparison with real test data.
Later, Eskandari (1998) developed a three-dimensional rate-dependent continuum
damage model. The model considers a quasi-isotropic and laminate-based model in which
the damage is assumed isotropic; i.e. the original isotropy or anisotropy is preserved. In
these models, even though they seem to include strain rate effects in their formulation, the
effect of various loading rates on the initial undamaged elastic coefficients is not
considered.
Dechaene et al. (2002) developed a constitutive damage model for woven glass fibre/
epoxy composite materials in which strain-rate effects are included into the model using a
35
LITERATURE REVIEW
damage lag methodology. In this CDM model the effect of damage in the warp and weft
are considered individually. One interesting innovation in their model is including rate
effects in the evolution of damage equations.
Their model is a stress rate formulation consisting of two parts. The first term is the
elastic change in stress with reduced elastic modulus due to damage, and the second term
is the drop in stress due to growth of damage resulting from matrix cracks as well as fibre
failure (Iannucci et al. 2001). The latter term is a function of damage and rate of damage.
σ = E ε − σ d (σ , d , d )
(2.15)
Dechaene et al. proposed the rate of damage as an increasing function of stress and
damage, which consists of the two stages of damage development: nucleation and growth.
Since the rate of damage is expressed as a function of stress and damage, the damage lags
behind the stress. In their model each fabric warp or weft layer is treated as an equivalent
unidirectional layer, i.e., a 0/90 cross-ply composite, even though, the behaviour of 0/90
cross-ply composites can be different from that of the woven fabric composite. They also
ignored the rate-dependent shear damage, which is reported in the literature by many
researchers (Gilat et al. 2002). Quantitative comparisons to experiment results were not
included.
Iannucci (2006) continued the work of Dechaene et al. and proposed an in-plane failure
model for a thin woven carbon composite. His approach is based on an unconventional
thermodynamic maximum energy dissipation approach, which controls the damage
evolution and hence energy dissipation per second, rather than damage. In the rate
dependant model he assumes that behaviour of the woven carbon composite under tensile
loading is linear until failure occurs in fibre bundles, that is, no damage is assumed for
matrix cracks. Hence, in his model, the initial undamaged modulus is assumed unaffected by an increase in strain rates. However, many experimental results in the
literature including Gilalt et al. (2002) and Harding et al. (1989) have shown the initial
undamaged modulus is affected by increasing the strain rate from static to dynamic. For
36
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
the non-linear shear evolution response, the model predicts the failure at much higher
strains than expected, and the use of a limited number of constants to change the shape of
the curves, was blamed for the poor shear response.
The energy based CDM model of Iannucci and Willows (2006) extends the previously
published models (Dechaene et al. 2002; Iannucci 2006) to include evolution of damage
as a function of strain, rather than damage energy rate. In the incremental form, the rate of
damage related to the warp and weft fibre fracture case (i = 1, 2) is represented by
Δdi =
⎡ ε 0,i ⎤
ε max,i
⎢
⎥ Δε ii
(ε max,i − ε 0,i ) ⎢ ε ii2 ⎥
⎣
⎦
(2.16)
where ε 0,i and ε max,i are the strains at initiation with damage equal to zero, and at zero
stress with damage equal to unity, respectively. Their model includes compression failure,
and the approach uses a bilinear stress-strain law for the material response. However,
again a linear behaviour until failure is assumed. An interface delamination model is also
defined; however a prior knowledge of the propagation path is required to place the
interface elements where delamination is expected.
Marguet et al. (2007) have also developed a rate dependent constitutive model. Their
damage model is coupled with viscoplasticity, that is, they split the total strain into an
elastic part and a viscoplastic part. Their model assumes a brittle rupture of fibres under
tensile loading, and hence does not consider the nonlinear strain rate effects in the normal
direction. They do not have a real material response comparison with their model; hence a
“pseudo experimental data” from their proposed model is generated. A material
parameters identification method is developed which is based on a pattern search
optimization algorithm, where the solution obtained is expected to give a satisfying
behaviour to the model.
To the author’s knowledge, there is no reliable accepted constitutive model to predict the
detailed mechanical response of woven composite materials. Most of the models are not
37
LITERATURE REVIEW
rate dependant, and some models include damage only in the shear direction (Johnson et
al. 2001; Marguet et al. 2007). Some of the models, that include rate dependency, assume
damage only in the strain softening part of the response (Eskandari 1998; Iannucci and
Ankersen 2006; Nemes and Speciel 1996). Some consider damage mainly in the shear
direction (Marguet et al. 2007), while others do not include shear damage in their model
(Dechaene et al. 2002). Overall, to summarize, the available models do not include rate
effects, are too complicated to understand, or are unable to provide a clear procedure for
characterizing the model parameters, are unable to accurately match experimental tests, or
require too much computational power, and so on.
Consequently, as was mentioned in Chapter 1 in the objective section, a constitutive
model is needed to predict the response of woven composite materials under high strain
rate loading. To do so, the damage strain rate dependant model proposed by Dechaene et
al. needs to be modified to include the shear damage besides providing a quantitative
comparison to experimental results, which are performed in this research. Moreover, a
methodology must be also established for determination of the damage parameters. In
addition, numerical models are required to simulate the high velocity experiments using
the material model. This is a complex problem that requires a thorough understanding of
composite behaviour under dynamic loadings. It is believed that this research will prove
directly useful for aerospace industries.
Given the above theoretical framework, a system to solve problems can be established by
means of material science, continuum mechanics and finite element computational
methods using the following steps:
1. Perform experiments to obtain the behaviour of the material under different
loading conditions,
2. Develop mechanical variables (internal damage state variables) describing the
mechanical effects of distributed micro-defects,
3. Develop equations to govern the mechanical response of the damaged material
(constitutive equation) generally by empirical means,
38
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
4. Formulate equations to describe the evolution of damage variables (evolution
equations), and
5. Use the above equations to solve boundary value problems, commonly with
FEA tools.
39
LITERATURE REVIEW
40
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
3
Equation Section (Next)
EXPERIMENTAL SETUP
3.1
Introduction
As discussed in chapter two, composite material structures are more and more often
subjected to static and dynamic loading. At room temperature, composite materials, in
general, exhibit rate dependency at much lower rates compared to metals (Eskandari
1998). Therefore, tests need to be conducted over a wide range of strain rates to properly
characterize the mechanical properties of these materials.
This chapter discusses the experimental procedure and the tests used to determine the
mechanical properties of the materials used in this study, which are woven composite
materials used in aerospace applications.
41
EXPERIMENTAL SETUP
3.2
Materials
The aerospace need for lightweight, high performance vehicles subjects materials to
extreme service. Textile composites offer unique combinations of properties that cannot
be obtained using conventional tape laminates. The primary advantages of textile
composites are their high speed preform manufacturing as well as their increased damage
tolerance due to the tow interlacing. Also, there is the potential for large reductions in part
count due to the ability to create complex preforms. They can be mass produced and are
cost-effective as compared to conventional tape laminates. Textile composites are being
used in applications ranging from prostheses for amputees to shrouds to capture debris
from a failed engine.
A number of textile manufacturing techniques are available to make fibre preforms.
Braiding, weaving and knitting are the dominant forms of textile manufacturing
techniques. They all share the characteristic that fibre tows are interlaced to create a
preform that is impregnated with resin to make a composite. Woven fabric composites are
a two dimensional class of textile composites where the warp and fill fibre tows are
woven into each other to form a layer.
Textile composites have complex microstructures characterized by tow undulation and
interlacing. The tow consists of thousands of fibres. The woven structure is characterized
by the orthogonal interlacing of two sets of tows called the warp and the fill tows. The fill
tows, also called weft, run perpendicular to the direction of the warp tows which are
along the fabric direction. Plain weave, twill weave, and satin weave are the dominant
forms of woven architectures and are shown in Figure 3.1. In all the cases, the tows have
both the undulated and straight regions except for the case of the plain weave in which the
entire length of the tow of both fill and warp tows is undulated. But in the case of other
weaves, the tows have some straight region before starting to undulate.
42
HIGH STRAIN
IN RATE BEHA
AVIOUR OF WOVEN
O
COMPO
OSITE MATERIIALS
(a)
(
(b)
(c)
F
Figure
3.1 Schematic
S
a picture of differentt weaves of woven
and
w
comp
posite: (a) plain
p
weave, (b)
( 2×2 twilll weave, (c)) 8-harness satin
s
weavee
T plain weeave is the simplest forrm where warp
The
w
and weeft yarns arre interlacedd in a
reegular sequeence of ‘one over and onne under’ inn each directtion as show
wn in Figure 3.1a.
A
Altering
this sequence will
w result in a differeent structuree. The plainn weave haas the
m
maximum
fab
bric stabilityy and firmneess with minnimum yarnn slippage. The
T pattern gives
unniform stren
ngth in two directions
d
w
when
yarn sizze and countt are similarr in warp and fill.
T
This
weave type is beliieved to bee most resisstant to in-pplane shear and is therrefore
coonsidered to
o be a stiff weave.
T twill weeave is moree pliable thaan the plain weave andd drapes withh less resisttance.
The
H
Hence,
these structures are
a better foor compoundd curves thaan plain weaave structurees. At
thhe same timee they mainttain more fabbric stabilityy than a four or eight harrness satin weave.
w
T weave pattern
The
p
is chharacterized by a diagonnal rib creatted by a ‘tw
wo over andd two
unnder’ sequen
nce. As show
wn in Figure 3.1b, the first
f
weft paasses over warps
w
1 and 2 and
unnder warps 3 and 4; thee next weft passes
p
over warps
w
2 andd 3 and undeer warps 4 and
a 5;
annd so on. Tw
will weaves feel
f generallly tighter, orr more closelly woven, thhan plain weaaves.
43
EXPERIMENTAL SETUP
A satin weave structure is one of the easiest fabrics to use and it is ideal for laying up
contoured surfaces with minimal distortions. These weave patterns are most pliable and
can comply with complex contours and spherical shapes. The ‘harness’ number (eg. 4harness, 5-harness, 8-harness, etc.) indicates the number of tows passed over or under
before the tow repeats the pattern. For example, in an 8-harness satin weave, one warp
yarn is carried over seven then under one weft yarn as shown in Figure 3.1c. This weave
is less stable than plain or twill weaves. Also, anti-symmetry and coupling effects are
present in these laminates and they generate undesired effects such as warpage and
temperature sensitivity (Ishikawa and Tsu-Wei 1982).
It is important to note that the mechanical properties of woven fabrics are governed by
weave parameters such as weave architecture, yarn size, yarn spacing length, fibre
volume fraction as well as by laminate parameters such as stacking orientation and
overall fibre volume fraction. Woven fabrics in general have good dimensional stability
in both the warp and fill directions but low in-plane shear stiffness.
Three types of materials, with different characteristics, are used in this research. They are
summarized in Table 3.1. The first composite material consists of 6 plies of carbon/epoxy
prepreg with a 2×2 twill weave as shown in Figure 3.1b. The second material also
consists of 6 plies and it is a carbon/Bismaleimide (BMI) composite with 8 harness satin
weave as illustrated in Figure 3.1c. The third material as shown in Figure 3.1a, is a plain
weave carbon/epoxy composite with 8 prepreg plies. All these materials can be
considered as balanced woven fabrics with a zero direction staking sequence.
Table 3.1 Summary of the materials used in this research
Description
Type of weave
Staking sequence
Thickness
Material-1
carbon/epoxy prepreg
2×2 twill
[0°]6
1.8 mm
Material-2
carbon/BMI prepreg
8 harness satin
[0°]6
2.3 mm
Material-3
carbon/epoxy prepreg
Plain
[0°]8
1.6 mm
44
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
3.3
Dynamic experiments
To perform dynamic experiments, a tensile version of Hopkinson bar was used. This
apparatus was designed and fabricated at McGill University. It can produce an average
strain rate of up to 500/sec on a woven composite material with a gage length of 10mm.
3.3.1
Hopkinson Bar Setup and Theory
In this configuration of tensile Hopkinson bar, the high strain rate loading is achieved by
an axial impact from a striker bar fired by an air gun. Figure 3.2 shows a picture of the
Hopkinson bar and Figure 3.3 schematically represents the test set-up in which the
specimen is held between two pressure bars and is loaded by a single traveling pulse in
tension. The experimental set-up works as following: the striker, which is accelerated by
the gas gun, hits the anvil, hence, generating a tensile pulse through the input bar towards
the specimen located between input and output bars. On reaching the specimen, a part of
this tensile incident pulse is reflected as a compressive pulse, and the rest is transmitted to
the output bar.
Figure 3.2 Hopkinson bar setup
45
EXPERIMENTAL SETUP
Using the well established elementary linear elastic wave propagation theory, the stress,
strain, and strain-rate versus time are calculated from the records of strain gages on the
input and output bars. Based on this theory, the strain in the specimen is directly
proportional to the time integral of the reflected pulse and stress is directly proportional to
the amplitude of the transmitted pulse (Al-Mousawi et al. 1997).
Fixings
Output Bar
Input Bar
Pressure
reading
P
Anvil
Specimen
Gas Gun
Strain Gauges
Transmitted
Pulse
Striker
Incident Pulse
Reflected Pulse
Figure 3.3 Schematic representation of Hopkinson bar
Taking a short specimen of initial length, l0 , (Figure 3.4) sandwiched between the two
bars, and taking the displacements at the ends of the specimens as u1 and u2 , the average
strain in the specimen can be expressed as
l0
output bar
input bar
εI
εR
εT
u2
P2
u1
P1
Figure 3.4 A specimen sandwiched between the input and output bars
46
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
εS =
u1 − u2
l0
(3.1)
where the displacements, in terms of incident, reflected and transmitted strains, are given
by
t
t
t
u1 = cb ∫ ε I dt + (−cb ) ∫ ε R dt = cb ∫ (ε I − ε R )dt
0
0
(3.2)
0
t
u2 = cb ∫ εT dt
(3.3)
0
Here, cb is the elastic wave velocity in the bars. The applied load on each side of the
specimen can be expressed as
P1 = Eb Ab (ε I + ε R )
(3.4)
P2 = Eb AbεT
(3.5)
where Eb and Ab are the Young’s modulus and cross sectional area of the bars,
respectively. It is assumed that the short specimen is in equilibrium during the course of
deformation, that is the stress across the specimen is constant, therefore
P1 ≅ P2
ε I + ε R ≅ εT
or
(3.6)
Hence, the average stress, strain, and strain rate in the specimen are obtained based on
hypothesis of equilibrium using the following expressions:
σS =
P1 + P2 1 Eb Ab (ε I + ε R + ε T ) Eb ( Ab )
=
≅
εT
2 A0
2
A0
A0
(3.7)
εS =
u1 − u2 cb
=
l0
l0
(3.8)
εS =
cb (ε I − ε R − ε T ) cb
≅ εR
l0
l0
t
∫0
(ε I − ε R − ε T ) dt ≅
cb
l0
t
∫0 ε R dt
(3.9)
47
EXPERIMENTAL SETUP
Where,
ε I , ε R , εT
Incident, reflected, and transmitted strains measured
Eb , Ab
Young’s modulus and cross sectional area of the bars
l0 , A0
Gage length and cross sectional area of the specimen
cb
Longitudinal wave velocity in the bars
Eb ρ b
To be able to use the one-dimensional theory of elastic wave propagation, the input and
the output bars must remain elastic while the wave propagates throughout the test. The
bars should be long enough so as to satisfy the one-dimensional theory, and also to avoid
the overlap of the incident and reflected signals at the input bar strain gage. Besides, the
bars must have enough mechanical strength not to deform plastically. The length to
diameter ratio of the bars has to be greater than 20 (Follansbee 1978) to satisfy the one
dimensional wave propagation which requires at least 10 bar diameter to dampen the end
effects due to non-uniformities at the striker/incident bar interface. In this Hopkinson bar
setup, the input and output bars are both 19.05 mm in diameter fabricated from high-yield
steel material. The input bar is 4.88 m long and the output bar is 2.44 m long and they can
move horizontally without any restriction. The striker tube is made of the same material
as the bars and its length determines the duration time of the incident pulse, as expressed
by
Δt I =
2 LS
cb
(3.10)
where LS is the length of the striker tube. The strain gages attached on the bars, at equal
distances from each end of the bars, monitor the incident, reflected and transmitted
pulses. It is worth mentioning that the strain rate varies during the test, and the stress,
strain and strain rate versus time is calculated using equation (3.7) to (3.9). Although, it is
theoretically possible to increase the average strain rate of the test by increasing the
pressure in the gas gun, however the increase in the pressure has little effect on the strain
rates calculated.
48
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Furthermore, for the split Hopkinson bar results to be taken as valid, there should at least
be three to four, exactly π, reflections of the stress wave within the specimen prior to
failure (Davies and Hunter 1963; Gama et al. 2004). This directly relates to the
assumption of uniform stress field mentioned earlier. Let the wave velocity in the
specimen be
cS =
ES / ρ S
(3.11)
where ρ S is an approximate value for the density of the carbon woven composite
material and is taken to be 1600 kg/m3 and E is the Young modulus of the material. If LSF
represents the length of the specimen between the two fixings, then the transit time, tS ,
that is needed for the incident wave to travel through the specimen once is given by
tS =
LSF
cS
(3.12)
The time to failure, t f , should then be greater than the time required to obtain a uniform
state of stress within the specimen, i.e.
t f ≥ π t S = π ( LSF / cS )
(3.13)
where π is the number of times the pulse is reflected back and forth. Using this principle,
the minimum time to failure for the three materials is given in Table 3.2 and it shows that
the dynamic tests are valid because all the calculated values of π tS are less than the time
to failure of the experimental values taken from the shortest duration tests.
For more comprehensive references on the theory and governing equations of Split
Hopkinson bar set-up one can refer to Kawata et al. (1981), Harding and Welsh (1983),
Staab and Gilat (1991), Al-Mousawi et al. (1997), Huh et al. (2002), Gama et al. (2004),
and Yang and Shim (2005).
49
EXPERIMENTAL SETUP
Table 3.2 Time to failure for the three woven composite materials
Max. specimen
Length between
fixings (m)
(LSF)
Specimen
wave velocity
(m/s)
cS = E S / ρ S
Transit time
(μs)
tS = LSF cS
(μs)
Time to
failure
(μs)
t f ≥ π tS
[0°]6
0.03
6282
5
15
55
[90°]6
0.03
6094
5
15
56
[45°]6
0.03
1785
17
53
62
[0°]6
0.03
6456
5
15
62
[90°]6
0.03
6425
5
15
68
[45°]6
0.03
1962
15
48
57
[0°]8
0.03
6543
5
14
59
[90°]8
0.03
6455
5
15
57
[45°]8
0.03
1762
17
54
78
Description
Material 1
Material 2
Material 3
π tS
3.3.2 Specimens and Fixture Designs
As was explained in section 3.3.1, in a Hopkinson bar set up the specimen needs to be
placed between the input and the output bars. To achieve that, a special fixture had to be
designed which will be screwed to the bars while holding the specimen in between. There
are two basic conditions that the specimen fixtures have to satisfy. Firstly, the gripping
force should be large enough to prevent the specimen from slipping out of the fixture and
secondly, change in the mechanical impedance of the fixtures must be as small as
possible to prevent disturbance in the wave propagation. To satisfy the latter condition the
material and the diameter of the fixture must be the same as the bars which is 19.05 mm.
Rodriguez et al. (1996) and Gilat et al. (2002) have also used fixtures that maintain the
cylindrical geometry of the Hopkinson bars. It is worth noting that for consistency, the
same fixture is used for both static and dynamic tensile tests. Below is a description of all
the designs that were considered.
50
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
The first design was adapted from Eskandari (1998) in which the specimen is fixed in a
hollow cylindrical fixture by injecting epoxy resin through the holes as shown
schematically in Figure 3.5. A collar was used in between the fixture to hold and align the
specimen in position. The whole setup was left alone for 24 hours so that the resin would
set.
epoxy injection hole
connected
to the bar
epoxy
specimen
fixture
connected
to the bar
Figure 3.5 Schematic assembly of first specimen fixture design
Even though this was a good fixture design for Eskandari’s work, it was not adequate for
this research. This could be due to the fact that the materials used in this research have a
higher strength compared to those in Eskandari’s work. The epoxy resin could not keep
the specimen in the fixture, and during the experiment, at a loading below the tensile
strength of the woven composite material, the composite sample would slip out of the
fixture as can be seen in Figure 3.6.
slipped out
of fixture
Figure 3.6 Picture of the fixture with epoxy injection
51
EXPERIMENTAL SETUP
The next design was to sandwich the specimen directly between a half cylinder and a
plate, as is presented schematically in Figure 3.7. This design was unsuccessful because
the screws could not withstand the dynamic pulse applied to them during the experiment.
When the tensile pulse reaches the fixture, the plate on top of the composite sample is
stationary, placing a shear loading on the screws. As a result, all eight screws were cut off
as soon as the dynamic pulse went through them.
screws
plate
specimen
half cylinder
Figure 3.7 Schematic assembly of second fixture design (side and top views)
The third design was to sandwich the composite specimen between two half cylindrical
pieces that fits into a hollow cylinder. The two pieces have rough-teeth shaped grooves
machined on their flat surfaces. On tightening the six screws on the hollow cylinder wall,
the two gripping pieces are pushed towards each other and hence the composite specimen
sandwiched tightly in the fixture as shown in the Figure 3.8. The specimen must be
aligned with the fixtures to avoid eccentric loading.
52
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
screws
connected
to the bar
teeth-shaped
gripping pieces
Figure 3.8 Schematic assembly of third fixture design
Although this design was initially successful, after several dynamic tests, the teeth shaped
surfaces were damaged. Also the loads that were transferred from the screws were not
applied at the edge of the gripping pieces which resulted in their deformation. The design
was then modified to use eight screws so that a better distributed load is applied on the
griping pieces which have the composite sample sandwiched in between them. Also the
teeth shaped gripping surfaces are replaced by two gripping pieces, one with a surface
that has a smooth finish and the other with its surface machined in such a manner that the
specimen can be fitted inside as illustrated in Figure 3.9. The presence of a barrier at the
tip of this piece prevents the specimen from sliding under tensile loading. The picture of
the modified fixture is shown in Figure 3.10. To avoid eccentric loading, it is essential
that the specimen be aligned with the fixtures.
screws
connected
to the bar
gripping pieces
gripping piece for
fitting the specimen
Figure 3.9 Schematic diagram of the modified fixture
53
EXPERIMENTAL SETUP
specimen fitted in
the gripping piece
Figure 3.10 Picture of the modified fixture
The key points for designing the shape and size of the specimen are the gage length and
the cross section of the specimen. The specimen gage length must be short enough to
allow stress equilibrium to be reached in the dynamic tests and also to achieve sufficiently
high strain rates. Also, the specimens used in all the tests should have the same geometry
so as to avoid any uncertainties related to size effect. The initial dog bone specimen’s
geometry had the dimensions shown in Figure 3.11. This geometry was acceptable for
dynamic tensile tests on [0/45/90/-45]s unidirectional graphite/ epoxy laminated
composite, as all the specimen failed in the middle of the gage length as shown in Figure
3.12a. However, when the same geometry was used for the woven composite materials in
this study, they mostly failed at either side of the transition area as shown in Figure 3.12b.
This is believed to be mainly due to stress concentration at that point.
Figure 3.11 Initial dimensions of the composite specimen in millimetres
54
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
(a)
(b)
Figure 3.12 Failure in the specimens (a) [0/45/90/-45]s (b) woven [0]6
Consequently, after trying different modifications a dog-bone specimen was designed
with the shape and dimensions shown in Figure 3.13. The specimen width is designed to
minimize stress concentration at its ends by selecting a smooth radius of curvature in the
transition region. Other researchers such as Gilat et al.(2002) and Rodriguez et al. (1996)
have also used a high radius of curvature in the transition region of the dog-bone
specimen. Also the width at the gage section must be large enough to be a good
representative of the woven composite material. The chosen size would ensure the
presence of two rows of undamaged weaves in the gage section.
Figure 3.13 Specimen’s dimensions in millimetres
The specimens were cut from the plate using a CNC milling machine. As the composite
plate thickness is very small, a special jig was designed to minimize damage to the
specimen while being cut. The woven composite sheet was sandwiched between two
aluminum plates which are firmly fixed to the base aluminum plate by means of screws.
55
EXPERIMENTAL SETUP
The screws are positioned at the two ends of each specimen as shown in the Figure 3.14.
Diamond-like coated end mill suitable for cutting carbon composite laminates was used to
cut the dog-bone specimen out of the composite plate.
Specimens in (0°), (90°), and (45°) directions were prepared from plates of the three
materials discussed in section 3.2. The (0°) and the (90°) specimens were used to obtain
static and dynamic tensile behaviour in the normal directions, whereas the (45°)
specimens are used in biased extension shear tests to obtain the in-plane shear properties.
screws
aluminum plates
dog-bone specimen
Figure 3.14 The aluminum jig used to cut the specimens
3.3.3 Measurement Procedures
As explained in section 3.3.1, strain gages are attached on the input and output bars of the
Hopkinson bar setup to monitor the incident, reflected and transmitted pulses during the
experiment. These 350-Ω strain gages are of type EA-06-062AQ-LE-350 (Measurement
Group Inc) and have a gage length of 1.58mm. Their gage factor, GF, is equal to 2.105;
56
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
this is their sensitivity to strain expressed quantitatively, and is defined as the ratio of
fractional change in electrical resistance to the fractional change in length (strain) that is
GF =
ΔR / R Δ R / R
=
ε
ΔL / L
(3.14)
It is commonly known that the strain gage transforms the strain applied to it into a
proportional change of resistance, and this small change in resistance can be measured
using a Wheatstone bridge with a voltage or current excitation source (Dally et al. 1993;
Nilsson and Riedel 1996). A general Wheatstone bridge consists of four resistive arms
with an excitation voltage, VEX, that is applied across the bridge at terminals A and B as
shown in Figure 3.15. The output voltage of the bridge, VO, is measured at the terminals
C and D. The bridge is considered balanced when no current flows across C-D and VO is
zero. Using Kirchhoff’s law for electrical circuits, it can be shown that the balanced
circuit requires R1 R2 = R3 R4 . In general the output voltage VO is given by
⎡ R3
R2 ⎤
VO = ⎢
−
⎥ VEX
R
+
R
R
+
R
4
1
2⎦
⎣ 3
(3.15)
In order to use the bridge for strain gage measurements 1, 2 or 4 of the resistors in the
bridge are replaced by strain gages. Figure 3.15 illustrates a half bridge configuration
used in the lab experiment with two 350-Ω dummy resistors in place of R1 and R2 and two
active strain gages in place of R3 and R4. These active strain gages are actually attached
on opposite sides of the surface of the bar, and the deformation of the strain gage
increases the resistance to (RG +ΔR) which also accounts for the bending. Any changes in
the strain gages resistance will unbalance the bridge and produce a nonzero output
voltage. If the nominal resistance of the strain gage is designated as RG, and the straininduced change in resistance as ΔR, then by simplifying equation (3.15) and substituting
in equation (3.14), the strain experienced by the stain gage during the test can be written
as
ε =−
2VO
ΔR
=−
GF ⋅ RG
GF ⋅VEX
(3.16)
57
EXPERIMENTAL SETUP
A
R1
+
−
VEX
C
R4 = RG + ΔR
−
VO
+
D
R3 = RG + ΔR
R2
B
Figure 3.15 Half Wheatstone bridge
The bridge is connected to a power supply and an oscilloscope. The power supply
provides the required excitation voltage and the output voltage signals are recorded using
a Nicolet Pro 40 digital oscilloscope. The oscilloscope simultaneously records the
incident, reflected, and transmitted pulses versus time. The shapes of these pulses depend
on the mechanical response of the specimen used. Figure 3.16 shows typical oscilloscope
signals recorded in the experiment. It shows that most of the input pulse is reflected back
on reaching the specimen, which results in a small transmitted signal for the samples
being tested. This is due to the fact that the Hopkinson bar setup used is a bit strong for
the material being used, nevertheless, as will be explained in chapter 4, very good results
were obtained using this Hopkinson bar setup.
58
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
20
Incident+Reflected
Transmitted
Recorded signals [mV]
15
10
5
0
-5
-10
-15
-20
0
200
400
600
800
Time [μsec]
1000
1200
Figure 3.16 Typical recorded strain gage signals on the input and output bars
The voltage recordings of the input bar consists of the incident and reflected pulses
separated by the time it takes for the pulse to travel, and the voltage recording of the
output bar is the transmitted pulse. These can be converted to strain experienced by each
strain gage using equation (3.16). The stress, strain and strain rate in the specimen can be
calculated using equations (3.7) to (3.9) and a typical stress versus strain curve is shown
Figure 3.17.
It is worth mentioning that in composite material characterization, strain measurements
are always challenging. The dynamic tests, as mentioned above, are performed using a
tensile version of the Hopkinson bar, and the dynamic strains are calculated using
displacement of the bars (equation(3.7)-(3.9)). The primary difficulty arises because a
smooth transition is required between the specimen gage length and the grips to avoid
failure due to stress concentrations at the specimen ends, and so the specimen gage length
is no longer well-defined. Due to the fact that the Hopkinson bar data analysis only
provides data on the relative displacement between the input and the output bars, an
effective gage length must generally be used (Davis 2004).
59
EXPERIMENTAL SETUP
800
Stress [MPa]
700
600
500
400
300
200
100
0
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
Strain
Figure 3.17 Typical dynamic behaviour of a woven composite material
Researchers have used different methods to obtain the effective gage length (Ellwood et
al. 1982; Harding et al. 1960; Li and Ramesh 2007; Verleysen and Degrieck 2004). They
have shown that the effective gage length was different from both the specimen gage
length and the distance between the two bar ends. In this thesis, a series of specimens
were tested using strain gages glued directly on the specimen to determine the effective
gage length and as a result ensure the accuracy of dynamic strains.
The effective gage length of the specimen is obtained using these instrumented tensile
tests. It is worth noting that the effective gage length is greater than the actual gage length
(10 mm). This increase in the effective gage length is due to the fact that the radius of
curvature on the geometry of the specimen is very smooth, causing the occurrence of
some deformation outside the actual gage length. A comparison between the dynamic
strains of the Hopkinson bar and the instrumented strain gage results help obtain this
effective gage length. Figure 3.18 schematically shows a typical effective gage length and
the actual gage length on a dog-bone specimen. The procedure and calculations for
obtaining the effective gage length will be explained in chapter 4.
60
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
10 mm
Figure 3.18 Schematic comparison of actual versus effective gage length
For the purpose of comparison, the stress strain relationship of the three results, one from
the strain gage signal and the other two from the Hopkinson bar results using effective
gage length and actual gage length, are plotted in Figure 3.19. It is interesting to note that
accurate dynamic behaviour of the material can be calculated using the Hopkinson bar
results when using the appropriate effective gage length.
1200
strain gage measurements
HB- actual gage length
HB-effective gage length
Stress [MPa]
1000
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
Strain
Figure 3.19 Typical behaviour using different gage lengths
61
EXPERIMENTAL SETUP
The size of the strain gage used for this purpose is of great importance. If the strain gage
is too small it might be bonded with only the fibre part or the matrix part of the composite
and hence result in low strains. The strain gages have to be large enough to cover a good
representative surface of the specimen as shown in Figure 3.20. The strain gages used
were 350 Ω (EA-06-125BZ-LE-350 -Measurement Group Inc.) having an overall
dimension of 3.3 mm by 7.4 mm. It is very important to prepare the surface of the
specimen according to the instructions to maximize the bond between the strain gage and
the specimen surface.
Figure 3.20 Woven composite specimen with mounted strain gage
3.4
Static experiments
Static tensile tests were conducted at strain rates of approximately 10-3/sec. These low
strain rate loading tests were carried out using a servo-hydraulic MTS testing machine.
Experiments under displacement control conditions yield constant static strain-rate tests.
The force is measured by using a 5 kN load cell. It is worth noting that for consistency,
the same fixture as the one used in dynamic tensile tests is used here. Also, the
specimen’s geometry is kept the same so that no unknown inaccuracy might be
introduced due to size effects. Figure 3.21 shows the hydraulic MTS testing machine used
for the static tensile experiments. The specimen can be seen mounted on the machine with
extensometer attached to it.
62
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Figure 3.21 Servo-hydraulic MTS testing machine, with the specimen fixed in place
3.4.1 Uniaxial tension tests
The specimens cut in (0°) and (90°) directions are used to obtain the tensile behaviour in
the normal directions. The strains in the specimen can be obtained in three ways: an
extensometer, gluing strain gages directly to the specimen, and calculating the strains
from the test machine actuator displacement. The strains obtained using machine actuator
displacements are usually over estimated, as the measured displacements are not
exclusive to the gage length of the specimen.
The extensometer used has a 10mm gage length with maximum strain of 15%. As can be
seen in Figure 3.21, the space available to attach the extensometer is limited; therefore the
arms of the extensometer (MTS 632.13F-20) had to be replaced by longer ones. It is
obvious that the increase in the arms length would alter the output strains, which is
adjusted by applying the following correction.
63
EXPERIMENTAL SETUP
Consider the schematic diagram shown in Figure 3.22, where “c” is the length difference
of the original and elongated extensometer arms. In a tensile test, the strains are obtained
from the displacement of the lower arm of the extensometer. Let “ΔL1” be the
displacement when using the original arms and “ΔL2” that of the elongated arms. The
maximum values of these displacements, “ΔL1max and ΔL2max, can be measured by
subtracting the gage length of 10 mm from the maximum opening at the tip of the
extensometer arms. Using the similar triangle rule, one can write
ΔL1max a
=
ΔL2 max b
(3.17)
Knowing the values of c = 15 mm , ΔL1max = 2 mm and ΔL2 max = 3.35 mm , the values of
a = 22.22 mm and b = 37.22 mm can be calculated.
b
a
c
ΔL1
ΔL2
Figure 3.22 Correction for the extensometer arms length
The extensometer strain and the actual strain can, respectively, be written as
ε ext =
ΔL1 ΔL1
=
GL 10
⇒
ΔL1 = 10ε ext
(3.18)
ε act =
ΔL2 ΔL2
=
10
GL
⇒
ΔL2 = 10ε act
(3.19)
Using the similar triangle rule, one can write
64
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
ΔL1 a
=
ΔL2 b
⇒
ΔL2 = ΔL1
b
b
= 10ε ext
a
a
(3.20)
On substituting equation (3.19) in equation (3.20), the actual strain is obtained by
b
a
ε act = ε ext = 1.675ε ext
(3.21)
In the third method which is using instrumented specimens, similar to the dynamic tests,
the strain gages used have to be large enough to cover a good representative surface of the
specimen. The strain gages used were 350 Ω (EA-06-125BZ-LE-350-Measurement
Group Inc.) having an overall dimension of 3.3 mm by 7.4 mm. A half Wheatstone bridge
and the Nicolet Oscilloscope were used to obtain the results of the instrumented static
tensile tests. In these tests the extensometer was also attached simultaneously to compare
the results. A typical comparison of these two methods which are presented in Figure 3.23
show little variation in the results. Therefore, for the static tests, strains are obtained using
the extensometer.
strain gage
extensometer
1.2%
700
Stress [MPa]
1.0%
0.8%
Strain
strain gage
extensometer
800
0.6%
600
500
400
300
0.4%
200
0.2%
100
0.0%
0
20
40
Time [Sec]
60
80
0
0.0%
0.4%
0.8%
1.2%
Strain
Figure 3.23 Strain comparison using extensometer and strain gage
65
EXPERIMENTAL SETUP
It is also important to obtain the behaviour of the material in the direction normal to the
loading direction. In a tensile test, while the strain in the loading direction increases, there
exists a compressive strain in the normal direction to loading. To measure this transverse
strain, the specimens need to be instrumented with strain gages mounted in the normal
direction to loading as shown in Figure 3.24. The same type of strain gages could be used,
as their gage length was small enough to cover the width of the dog-bone specimen.
Figure 3.24 Woven composite specimen with perpendicular strain gage
The Poisson’s ratio of a material is defined as the ratio of the transverse strain ( ε yy = ε 22 )
to the axial strain ( ε xx = ε11 ). In uniaxial tensile tests in which the loading (x,y,z) and the
fibres (1,2,3) coordinate systems coincide, the ratio of the extensometer strain versus
strain-gage strain represents the Poisson’s ratio of the material, as illustrated in Figure
3.25. The slope of this line is the Poisson’s ratio.
In an orthotropic material, where the Poisson's ratio is different in each direction (1, 2 and
3 axis) the relation between Young's modulus and Poisson's ratio is described as follows:
ν12
E11
=
ν 21
E22
,
ν13
E11
=
ν 31
E33
,
ν 32
E33
=
ν 23
E22
(3.22)
In the woven composite materials, if the weaves of the fabric are balanced, the Young's
modulus and the Poisson’s ratio in the two fibre directions would be similar.
66
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Gage-Strain-ε 22
0.00%
-0.05%
-0.10%
-0.15%
0.0%
0.5%
1.0%
1.5%
Extensometer Strain-ε11
Figure 3.25 Typical result to obtain the Poisson’s ratio
3.4.2 Bias extension shear tests
In order to characterize the shear response of woven composite materials, two test
methods in general have received wide application, namely the picture frame test and the
bias extension test. In a picture frame test, a tensile force is applied across diagonally
opposite corners of the picture frame rig to move from an initially square configuration
into a rhomboid. Consequently the sample clamped within the frame undergoes pure
shear. Whereas, a bias extension test involves clamping a rectangular piece of woven
material in such a way that the warp and weft directions of the tows are oriented at 45
degree to the direction of tensile loading (Lebrun et al. 2003)
Some researchers like Leburn et al.(2003) and Harrison et al. (2004) have evaluated both
the methods and have found comparable results in the two tests. In this research, it is
needed to obtain the shear response in both static and dynamic loading. However due to
the nature of the Hopkinson bar setup, only the bias extension shear test can be used for
dynamic loading. Therefore, for consistency, the bias extension shear test is used for both
static and dynamic tests to determine the nonlinear shear stiffness of woven composites
67
EXPERIMENTAL SETUP
in pure shear (Lee et al. 2008). The specimens are cut in a way that the yarns ran at ±45°
with respect to the loading direction. Samples of each of the materials are tested in
tension using the same setup and fixture as the axial tensile tests.
It is worth mentioning that the state of stress in the 45 degree cut specimen is not pure
shear, and tensile normal stresses, “σ1” and “σ2”, in addition to the desired shear stress
“σ12” is present in each lamina of the specimen. However, the bias extension test method
is considered as a reliable method of obtaining the shear strength and modulus of the
material for the following reasons.
Firstly, the shear response of many types of woven composite materials is non-linear, and
exhibit strain softening characteristics. As a result, even though the biaxial state of stress
present in the specimen likely causes the value of shear strength to be lower than the true
value, the reduction may be small due to the nonlinear softening response (Carlsson et al.
2002). Secondly, as the magnitudes of the normal stresses present are significantly lower
than the ultimate stress in the fibre direction, the loading can be approximated to be one
of pure shear.
The dog-bone specimens are prepared and tested in tension to ultimate failure.
Determination of the shear properties from tension test results uses a stress analysis of the
45 degree cut specimen. Using a transformation matrix, the results obtained in the loading
direction can be transformed into the stresses and strains in the fibre direction. It is
important to note that the x-direction is considered to be the loading direction throughout
the thesis. Let (x,y) and (1,2) denote the two in-plane coordinate systems of loading and
fibres directions, respectively, with a common origin as shown in Figure 3.26. The
cosines of the angles between the coordinate axes (x,y) and the coordinate axes (1,2) are
listed in Table 3.3. These direction cosines are all equal to either 1/√2 or -1/√2 simply
because the angle between the two coordinates is 45 degrees.
68
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
y
2
1
σ xx
x
σ xx
Figure 3.26 Loading direction and fibre direction coordinate systems
Table 3.3 Direction cosines
x
y
1
l1
m1
2
l2
m2
The stress components of the (1,2) coordinated system is given by (Boresi et al. 1993)
σ11 = l12σ xx + m12σ yy + 2l1m1σ xy
σ 22 = l22σ xx + m22σ yy + 2l2 m2σ xy
(3.23)
σ12 = l1l2σ xx + m1m2σ yy + (l1m2 + l2 m1 )σ xy
In the bias extension test, the axial stress in the loading direction is the only nonzero
stress component; hence the stresses in the fibre direction can be obtained as
σ11 = l12σ xx
σ 22 = l22σ xx
σ12 = l1l2σ xx = σ xx / 2
(3.24)
Also, the transformation of strain tensor under a rotation from axes (x,y) to axes (1,2) is
given by (Boresi et al. 1993)
69
EXPERIMENTAL SETUP
ε11 = l12ε xx + m12ε yy + 2l1m1ε xy
ε 22 = l22ε xx + m22ε yy + 2l2 m2ε xy
(3.25)
ε12 = 12 γ 12 = l1l2ε xx + m1m2ε yy + (l1m2 + l2 m1 )ε xy
As it was shown by Adams et al. (2003) the shear strain is given by
ε12 = (ε xx − ε yy ) / 2
(3.26)
where the axial strain, ε xx , is measured using the extensometer, however, due to the fact
that not all the tests are instrumented with strain gage in the transverse direction, the
Poisson’s ratio obtained from the instrumented tests is used to calculate the transverse
strains as ε yy = −ν xyε xx in the non-instrumented tests and hence the shear strain is
calculated as
ε12 = ε xx (1 +ν xy ) / 2
(3.27)
It is worth mentioning that due to the small size of the specimens, small strain gages were
used in these experiments. It was noted that for some of the materials tested in the 45degree direction, the transverse strain is higher than 6% at the maximum stress, which is
greater than the gage capacity. It is interesting to note that the transverse strain measured
before the gage failure, showed linear behaviour with respect to time. This linearity was
used as an approximation to obtain the transverse strain at the onset of failure in the
specimen, which is the time of drop in stress. Consequently, the Poisson’s ratio is
obtained from the ratio of the transverse strain to axial strain at the time of stress drop. A
typical graph showing the relationship of axial versus transverse strain up to the point of
stress drop is illustrated in Figure 3.27 in which the slope of the curve gives the Poisson’s
ratio in the 45 degree cut specimen.
70
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
0%
Gage- Strain-εyy
-1%
-2%
-3%
-4%
-5%
-6%
-7%
0%
5%
10%
15%
Extensometer Strain-εxx
Figure 3.27 Typical result for bias extension specimen to obtain Poisson’s ratio
The in-plane undamaged shear modulus “G12” of the composite materials can be
determined by plotting σ12 versus γ 12 and establishing the slope of the initial portion of
the curve. It is believed that deviation from the linearity behaviour is the onset of damage.
An alternate method of determining the undamaged shearing modulus is to measure the
initial axial stiffness (Ex) and the Poisson’s ratio (νxy) of the 45-degree specimens, and
then calculate the undamaged shear modulus according to the following relationship.
G12 =
Ex
2(1 +ν xy )
(3.28)
It is worth mentioning that some researchers have assumed a nonlinear elastic behaviour
for these materials (Hahn and Tsai 1973; Ogihara and Reifsnider 2002; Shokrieh and
Lessard 2000). However, the nonlinear behaviour explained in these papers contribute to
the very beginning of the curve, a little after deviation from linearity, and does not
consider the results until failure. On applying a tensile load to the 45 degree cut specimen,
the matrix crack due to presence of a scissoring effect. When unloading the sample, the
stress-strain curve does not unload over the initial loading curve; hence the deviation
from the linearity behaviour is attributed to the onset of damage. And researchers like
71
EXPERIMENTAL SETUP
Johnson et al. 2001 (2001), Matzenmiller et al. (1995), and Marguet et al. (2007) have
reported the nonlinear shear behaviour and included the shear damage parameter in their
models.
3.5
Experiments summary
Several types of experiments were discussed in this chapter. Static uniaxial tensile tests
were conducted in order to obtain the static behaviour of the woven composite materials
tested. Some of the tests were instrumented with strain gages in the axial direction to
obtain the most accurate tensile strains. While some other tests were conducted with
strain gages attached perpendicular to the loading direction in order to obtain the
transverse strains, which is required to obtain the Poisson’s ratio of the materials. In
addition, bias extension shear tests were conducted on 45-degree cut specimens to obtain
the shear stress-strain curves. Some of these tests were also instrumented with strain
gages in the perpendicular to loading direction in order to extract both the axial and
transverse strains from the tests. The ratio of the two strains gives the Poisson’s ratio in
the 45-degree cut specimens, which is required in order to convert the axial behaviour in
the loading direction, to the shear behaviour in the fibre direction.
Furthermore, dynamic tensile tests as well as dynamic bias extension tests were needed to
capture the behaviour of these woven composite materials at high strain rates. These tests
were performed using the tensile Hopkinson bar setup. Some of these tests were
performed with strain gages attached to the specimen in order to obtain their effective
gage length. It was confirmed that these materials are strain rate dependent.
Moreover, these experiments are used as a tool to develop a constitutive damage model
and its associated damage parameters which can predict the behaviour of woven
composite materials. The constitutive damage model is implemented into a two
dimensional VUMAT subroutine to be used for explicit dynamic finite element
simulations.
72
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Finally, as a general summary, Tables 3.4 and 3.5 list all the experiments conducted,
along with the number of repeats each experiment was performed. Due to the more
repeatable nature of static tests, fewer numbers of repeat was needed compared to the
dynamic tests. Based on the results, it was decided to have a minimum of three repeats for
the static tests, and a minimum of four repeats for the dynamic tests, and at least one test
was performed in the cases where instrumented results were required. The discussions of
the test results are presented in chapter 4 of this thesis.
Table 3.4 List of dynamic experiments performed
Experiment type
Dynamic uniaxial tension
Material
6
90-degree
6
0-degree
5
90-degree
5
0-degree
5
90-degree
5
material 1
45-degree
6
material 2
45-degree
5
material 3
45-degree
5
0-degree
2
90-degree
1
0-degree
1
90-degree
1
0-degree
1
90-degree
1
material 3
Dynamic uniaxial tension
with axial strain gage
Number of
test repeats
0-degree
material 1
material 2
Dynamic bias extension
Orientation of
cut specimen
material 1
material 2
material 3
Dynamic bias extension
material 1
45-degree
2
with axial strain gage
material 2
45-degree
1
material 3
45-degree
1
73
EXPERIMENTAL SETUP
Table 3.5 List of static experiments performed
Experiment type
Static uniaxial tension
Material
4
90-degree
3
0-degree
4
90-degree
3
0-degree
4
90-degree
4
material 1
45-degree
4
material 2
45-degree
3
material 3
45-degree
4
0-degree
1
90-degree
1
0-degree
1
90-degree
1
0-degree
1
90-degree
1
material 3
Static uniaxial tension
transverse strain gage
Number of
test repeats
0-degree
material 1
material 2
Static bias extension
Orientation of
cut specimen
material 1
material 2
material 3
Static bias extension
material 1
45-degree
1
transverse strain gage
material 2
45-degree
1
material 3
45-degree
1
0-degree
1
90-degree
1
Static uniaxial tension
axial strain gage in
74
material 1
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
4
Equation Section (Next)
EXPERIMENTAL RESULTS
4.1
Static Experiments
In a static tension test, the tensile force is recorded as a function of the displacement of
the machine head. The deformation measured by the machine belongs to the whole
system attached between the two heads of the machine, that is, the specimen as well as its
gripping system. Therefore, force versus displacement would be of little value to
accurately describe the material behaviour. Accordingly, normalization with respect to
specimen dimensions would lead to more accurate representation of the material
response. With that in mind, the engineering stress, σ eng , is given as
σ eng =
Fi
A0
(4.1)
75
EXPERIMENTAL RESULTS
where Fi is the instantaneous tensile load measured by the load cell during testing, and
A0 is the initial cross sectional area in the gage section. The engineering strain, ε eng , can
be obtained from the change in the gage section length, such that
ε eng =
li − l0
l0
where l0
(4.2)
and li
are the initial and instantaneous gage lengths, respectively. The
engineering strain is measured directly during the test by attaching an extensometer to the
gage section as explained in section 3.4 of this thesis. As a result, a record of
instantaneous force and strain is obtained while loading.
It is worth mentioning that the stress is calculated as a function of initial area. The true
stress in a material is defined as a function of the instantaneous area, which is changing
during loading, and the formulation used for the true stress and strain can be written as
σ = σ eng (1 + ε eng )
⎛ li ⎞
⎟ = ln 1 + ε eng
⎝ l0 ⎠
ε = ln ⎜
(
)
(4.3)
However, for the woven composite materials being tested in this research, the strain-tofailure is small contributing to very little difference between the engineering and true
values. Therefore, in this study, the engineering strain on the test specimen is measured
using an extensometer with 10mm gage length as the load is applied, and the engineering
stress is calculated according to equation (4.1).
4.1.1 Tensile modulus and tension test results
Static tensile tests are performed on the dog-bone specimens cut from both the fibre
directions of all the three materials namely carbon/epoxy prepreg with 2×2 twill weave,
carbon/BMI prepreg with 8 harness satin weave and carbon/epoxy prepreg with plain
76
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
weave. These results are presented in Figure 4.1 to Figure 4.3. And a summary of the
static tensile results are presented in Table 4.1.
1000
Material - 1
M11S1
Static 11- direction
M11S2
M11S3
Stress [MPa]
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
1000
Material - 1
M12S1
Static 22- direction
M12S2
M12S3
Stress [MPa]
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
Figure 4.1 Stress-strain curves for static tensile tests of material one (woven
carbon/epoxy prepreg [0º]6) in the two fibre directions
77
EXPERIMENTAL RESULTS
1000
M21S1
M21S2
M21S3
Material - 2
Static 11- direction
Stress [MPa]
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
1000
M22S1
M22S2
M22S3
Material - 2
Static 22- direction
Stress [MPa]
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
Figure 4.2 Stress-strain curves for static tensile tests of material two (woven
carbon/BMI prepreg [0º]6) in the two fibre directions
78
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
M31S1
M31S2
M31S3
Material - 3
1000
Static 11- direction
Stress [MPa]
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
M32S1
M32S2
M32S3
Material - 3
1000
Static 22- direction
Stress [MPa]
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
Figure 4.3 Stress-strain curves for static tensile tests of material three (woven
carbon/epoxy prepreg [0º]8) in two fibre directions
79
EXPERIMENTAL RESULTS
Table 4.1 Summary of static tensile tests
Material
Specimen
Type
Static
Test Name
M11S1
Max. Stress
MPa
Strain
at max. stress
Modulus
(GPa)
623
1.03%
60.36
M11S2
674
1.08%
60.91
M11S3
712
1.17%
60.83
Woven
Average
670
1.09%
60.70
Carbon/
M12S1
653
1.12%
58.18
M12S2
681
1.18%
57.77
M12S3
689
1.21%
56.74
Average
674
1.17%
57.56
M21S1
611
1.01%
64.51
M21S2
726
1.18%
69.06
M21S3
812
1.30%
64.20
8-HS Woven
Average
716
1.17%
65.92
Carbon/BMI
M22S1
729
1.19%
62.14
M22S2
657
0.94%
68.58
M22S3
648
1.02%
61.27
Average
678
1.05%
63.99
M31S1
925
1.35%
68.75
M31S2
899
1.40%
64.03
M31S3
810
1.22%
66.29
Average
878
1.32%
66.36
M32S1
883
1.36%
64.77
M32S2
786
1.21%
65.10
M32S3
901
1.41%
63.87
Average
857
1.33%
64.58
Material-1:
2×2 Twill
Epoxy
Prepreg
Material-2:
Prepreg
Material-3:
[0°]6
[90°]6
[0°]6
[90°]6
[0°]8
Plain woven
Carbon/
Epoxy
Prepreg
80
[90°]8
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
It is observed that material three is the strongest material which has an average tensile
strength of 867 MPa. This is approximately 180 MPa higher than the other two materials.
In addition, material three has the maximum average strain to failure of about 1.33%
whereas material two has the minimum average strain to failure of 1.05%. Looking at the
variation of tensile strength between different repeats of the tests, one can observe that
material one has the least variability and material two has the maximum. However,
consistency is observed in the average tensile strength of the two fibre directions in each
material. The variability in the tensile strength of the three materials is graphically
illustrated in Figure 4.4 for both fibre directions.
It is worth noting that variability has been observed by other researchers as well. For
example Kim et al. (2004) have shown approximately 75 MPa variability in their tensile
strength results for the plain weaves and about 35 MPa variability in both the 2×2 twill
weave and 8 harness satin weaves.
Average tensile strength
Tensile strength [MPa]
1000
800
600
11-direction
400
22-direction
200
0
0
1
2
3
Material Number
Figure 4.4 Average static tensile strengths and their variability for the three
materials in the two fibre directions
81
EXPERIMENTAL RESULTS
As presented in the graphs, all three materials show a brittle behaviour when loaded in the
fibre directions. Hence, the static tensile modulus is calculated from the slope of the
stress-strain curve as the tensile modulus is defined by
E =σ /ε
(4.4)
It is worth mentioning that average tensile modulus of all the three materials has little
variability in the two fibre directions (Figure 4.5). Furthermore, the variability is small
even between the materials tested with different weaves, such that an average value of
63 ± 3 GPa is a good overall approximation. Kim et al. (2004) have shown approximately
18 GPa variation in the tensile modulus results of carbon-epoxy composites with all
three weave patterns, namely the 2×2 twill weave and 8 harness satin weaves, and the
plain weaves.
Average tensile modulus
Tensile modulus [GPa]
80
70
60
50
40
11-direction
30
22-direction
20
10
0
0
1
2
3
Material Number
Figure 4.5 Average static tensile modulus and their variability for the three
materials in the two fibre directions
To calculate the Poisson’s ratio of the three materials, their behaviour in the loading
direction as well as in the direction normal to the loading direction is required, as was
explained in section 3.4.1. The two in-plane coordinate systems in this thesis, are (x,y) for
82
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
the loading direction and (1,2) for the fibre directions. The strains in the loading direction
(ε xx = ε11 ) is measured by the extensometer and for the measurement of the transverse
strain, (ε yy = ε 22 ) , the specimens are instrumented with strain gages mounted in the
normal direction to loading, as was mentioned previously. The Poisson’s ratio of a
material is the ratio of these two strains (ε 22 / ε11 ) . Figure 4.6 illustrates the axial versus
the transverse strains measured in a tensile test for the three materials. Accordingly, the
Poisson’s ratio of each material is calculated from the slope of a line fit to the data, and is
tabulated in Table 4.2.
Poisson's ratio
Material-1
(Fibre direction)
0.00%
Material-2
Material-3
Strain-yy
-0.02%
-0.04%
ε yy = −0.06 ε xx
-0.06%
-0.08%
ε yy = −0.08 εxx
-0.10%
εyy = −0.087 εxx
-0.12%
-0.14%
0.0%
0.5%
Strain-xx
1.0%
1.5%
Figure 4.6 Axial versus transverse strains for all the materials in the fibre direction
Table 4.2 Poisson’s ratio measured in the fibre direction
Poisson’s Ratio
Material-1
0.087
Material-2
0.060
Material-3
0.080
83
EXPERIMENTAL RESULTS
It is interesting to note that these materials possess a relatively low Poisson’s ratio. Due to
the presence of little variability in the tensile modulus, as well as the low Poisson’s ratio
in the fibre direction of these materials, it was decided that one test result would be
sufficient in calculating this value. Small Poisson’s ratio was observed by many
researchers as well. For example Kumangai et al. (2003), who tested a 5 harness satin
weave carbon/epoxy material, measured a comparable value of 0.074 for their Poisson’s
ratio. Table 4.3 summarizes the values of Poisson’s ratio obtained by other researches.
Table 4.3 Summary of Poisson’s ratio and tensile modulus data from literature
Poisson’s Ratio
Tensile Modulus
(GPa)
2 layer
0.041
66.9
4 layer
0.039
67.7
6 layer
0.035
68.2
ߥଵଶ =0.055
‫ܧ‬ଵଵ = 73.5
ߥଶଵ =0.036
‫ܧ‬ଶଶ = 63
ߥଵଶ =0.066
‫ܧ‬ଵଵ = 70.04
ߥଶଵ =0.05
‫ܧ‬ଶଶ = 66
ߥଵଶ =0.038
‫ܧ‬ଵଵ = -
ߥଶଵ =0.077
‫ܧ‬ଶଶ = 69.6
x- direction
0.0346 ± 0.01
40.97 ± 2
y- direction
0.0666 ± 0.0203
47.30 ± 4.02
5HS weave carbon/epoxy
0.074
46.9
5HS weave carbon/epoxy
0.07
76
Paper
Material
Gao et al.
(1999)
8HS
weave
CFRP
ߝሶ = 1.2E-4 sec-1
Hou and Ruiz
(2000)
woven
CFRP
T300/914
ߝሶ = 4 sec-1
ߝሶ = 600 sec-1
Tan et al.
(2000)
Kumagai et al.
(2003)
Abot et al.
(2004)
84
3D woven
CFRP
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
4.1.2 Shear modulus and bias extension test results
To determine the shear characteristics of the three woven composite materials, dog-bone
specimens, cut from the plates with an off-axis angle of ±45°, are tested in tension using
the same setup and fixture as the axial tensile tests. As was explained in section 3.4.2 the
experimental values of the tensile stresses and tensile strains should be converted into the
shear stress and strains using the transformation of stress and strain tensors.
In order to perform this conversion, the strain experienced by the material in the
transverse direction to loading is also required. Therefore some bias extension shear tests
are performed using strain gages attached normal to the direction of loading.
Consequently, the Poisson’s ratio, ν xy , of the materials in the 45 degree cut specimens are
calculated from the slope of the transverse strain, ε yy , versus axial strain ε xx , as
presented in Figure 4.7 to Figure 4.9, and the values are given in Table 4.4.
Material - 1
(45° direction)
0.00%
Strain-yy
-0.40%
εyy = -0.38 εxx
-0.80%
-1.20%
-1.60%
-2.00%
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
Strain-xx
Figure 4.7 Poisson’s ratio of material one in 45º specimens
85
EXPERIMENTAL RESULTS
Material - 2
(45° direction)
0.00%
Strain-yy
-0.20%
εyy = -0.47 εxx
-0.40%
-0.60%
-0.80%
-1.00%
0.0%
0.5%
1.0%
1.5%
2.0%
Strain-xx
Figure 4.8 Poisson’s ratio of material two in 45º specimens
Material - 3
(45° direction)
0.00%
Strain-yy
-1.00%
-2.00%
εyy = -0.52 εxx
-3.00%
-4.00%
-5.00%
-6.00%
-7.00%
0.0%
5.0%
10.0%
15.0%
Strain-xx
Figure 4.9 Poisson’s ratio of material three in 45º specimens
86
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Table 4.4 Poisson’s ratio measured in the 45º specimens
Poisson’s Ratio
Material-1
0.38
Material-2
0.47
Material-3
0.52
It is interesting to note that other researchers have found similar values of Poisson’s ratio
for 45 degree cut specimens of different composite materials. Gilesche et al. (2005) tested
different structures of woven carbon/epoxy composites and reported a Poisson’s ratio of
0.70-0.75, Ogihara and Reifsnider (2002) found that the Poisson’s ratio in the 45 degree
cut specimens of four-harness satin woven glass composites was approximately 0.543,
and Kashaba (2004) tested cross-ply of GFRE with [45/-45]2S stacking sequence and
reported a Poisson’s ratio of 0.67.
Using the equations given in section 3.4.2 the shear stress and strains are calculated.
Figure 4.10 to Figure 4.12 shows the shear results for the three materials each with three
repeats of the test. From the curves it can be seen that all the materials show an initial
linear elastic region followed by nonlinear deformation behaviour. This kind of nonlinear
behaviour was observed by other researchers such as Gliesche et al. (2005), Abot et al.
(2004), and Daniel et al. (2008).
87
EXPERIMENTAL RESULTS
120
Material - 1
M13S1
Static 12- direction
M13S2
M13S3
Shear stress [MPa]
100
80
60
40
20
0
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
Shear strain
Figure 4.10 Shear stress -strain curves for static bias extension tests of material one
(woven carbon/epoxy prepreg [0º]6) in the 45º specimen
120
M23S1
Material - 2
M23S2
Static 12- direction
M23S3
Shear stress [MPa]
100
80
60
40
20
0
0.0%
2.0%
4.0%
6.0%
Shear strain
Figure 4.11 Shear stress -strain curves for static bias extension tests of material two
(woven carbon/BMI prepreg [0º]6) in the 45º specimen
88
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
M33S1
Material - 3
120
M33S2
Static 12- direction
M33S3
Shear stress [MPa]
100
80
60
40
20
0
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
Shear strain
Figure 4.12 Shear stress-strain curves for static bias extension tests of material three
(woven carbon/epoxy prepreg [0º]8) in the 45º specimen
As expected, material one with the 2×2 twill weaves is stiffer than material two with the
8HS weave (Kim et al. 2004). Material three shows the maximum strain to failure of
about 12%, which is believed to be due to the fact that the weaves in this material are not
as compact as the other two. For the purpose of comparison, one shear result from each
material is presented in Figure 4.13. The average shear strength is between 81 to 101 MPa
with material three having the maximum value.
The initial undamaged shear modulus can be determined by plotting shear stress
(σ12 = σ xx / 2) versus the engineering shear strain (γ 12 = 2ε12 ) and establishing the initial
slope of the curve. An alternate method which would yield a similar undamaged shear
modulus is using equation 3.28 together with the axial stiffness, Exx , and Poisson’s ratio,
ν xy , of the 45 degree cut specimen. From the data, the undamaged shear modulus is
between 5-6 GPa for the three materials. The summary of the shear results is presented in
Table 4.5.
89
EXPERIMENTAL RESULTS
Material-1
Static 12- direction
120
Material-2
Material-3
Shear stress [MPa]
100
80
60
40
20
0
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
Shear strain
Figure 4.13 Comparison of shear stress-strain curves of the three materials
Table 4.5 Summary of static bias extension tests
Material
Material-1:
Material-2:
Material-3:
90
Specimen
Type
[45°]6
[45°]6
[45°]8
Static
Test Name
Max. Shear
Stress (MPa)
Shear Strain
at max. stress
Modulus
Initial (GPa)
M13S1
83
2.73%
5.29
M13S2
88
2.40%
5.31
M13S3
91
2.25%
4.22
Average
87
2.46%
4.94
M23S1
79
1.50%
5.08
M23S2
81
1.33%
5.91
M23S3
84
1.42%
6.90
Average
81
1.42%
5.97
M33S1
103
7.48%
4.86
M33S2
102
7.36%
4.70
M33S3
98
6.68%
4.87
Average
101
7.17%
4.81
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
4.2
Dynamic Experiments
The dynamic tests are performed on a tensile version of the Hopkinson bar. As discussed
in section 3.31, a tensile wave is generated by the impact of the striker, and the
propagation of this wave is analysed assuming the one dimensional theory, and measuring
the elastic deformations produced in different positions on the bar system with strain
gages. The stress, strain and strain rate in the specimen are determined with the traditional
procedure based on the hypothesis of equilibrium and uniform strain along the specimen.
It is worth mentioning that the same types of specimens are used in both static and
dynamic experiments in order to eliminate any potential size effects.
4.2.1 Effective gage length
Additional strain measurements are obtained for each material in the three directions
following the procedure explained in section 3.3.3, that is, strain gages are glued directly
on the dog-bone specimen in order to measure the exact axial strains. A comparison
between the dynamic strains of the Hopkinson bar and the instrumented strain gage
results can be used to calculate the effective gage length. As mentioned before, upon
loading, the specimen undergoes some deformation outside its actual gage length. This
could be due to the presence of a large and smooth radius of curvature on the dog-bone
specimens as also observed by many other researchers such as Verleysen and Degrieck
(2004) or due to the fact that the length between the grips is larger than the gage length
and so the deformation between the grips is the sum of the deformation of the entire open
length of the material under load. Therefore, a correction factor is used in order to
calculate an effective gage length which would yield correct strains in the gage area.
Figure 4.14 shows typical strains measurements from the strain gage results (the dotted
line) as well as from the Hopkinson bar equations using the actual gage length (the
dashed line). It can be seen that by increasing the gage length to an effective gage length
(the solid line) the strain gage results coincide with the Hopkinson bar results. The
summary of the calculated effective gage lengths for all the materials are presented in
Table 4.6.
91
EXPERIMENTAL RESULTS
2.00%
Strain
strain gage measurements
HB- actual gage length
HB-effective gage length
1.00%
0.00%
0
10
20
30
Time [μ sec]
40
50
60
Figure 4.14 Typical comparison of different strain measurements
Table 4.6 Summary of effective gage lengths
Material
Material-1:
Material-2:
Material-3:
92
Specimen Type
Effective gage length
(mm)
[0°]6
25
[90°]6
25
[45°]6
20
[0°]6
18
[90°]6
18
[45°]6
14
[0°]8
17
[90°]8
17
[45°]8
12
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
4.2.2 Tension test results
High strain rate tensile tests are performed on the dog-bone specimens cut from both the
fibre directions of all the three materials using the Hopkinson bar setup. The strain rates
in these experiments vary from approximately 200 /sec up to 550 /sec. These results are
presented in Figure 4.15 to Figure 4.17. From the curves it can be seen that all the
materials show an initial linear elastic region followed by nonlinear deformation until the
ultimate stress which precedes a nonlinear strain softening behaviour. This kind of
nonlinear behaviour was observed by other researchers such as Kawata et al. (1981),
Welsh and Harding (1985), Harding et al. (1989), Rodriguez et al. (1996), and Sham et
al. (2000) who studied the tensile behavior of different woven composites materials. A
summary of the dynamic tensile results are presented in Table 4.7.
93
EXPERIMENTAL RESULTS
1200
Material - 1
M11D1
Dynamic 11- direction
M11D2
M11D3
1000
Stress [MPa]
M11D4
800
600
400
200
0
0.0%
1.0%
2.0%
3.0%
Strain
1200
Material - 1
M12D1
Dynamic 22- direction
M12D2
M12D3
1000
Stress [MPa]
M12D4
800
600
400
200
0
0.0%
1.0%
2.0%
3.0%
Strain
Figure 4.15 Stress-strain curves for dynamic tensile tests of material one in the two
fibre directions
94
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
1200
Material - 2
M21D1
Dynamic 11- direction
M21D2
M21D3
1000
Stress [MPa]
M21D4
800
600
400
200
0
0.0%
1.0%
2.0%
3.0%
Strain
1200
Material - 2
M22D1
Dynamic 22- direction
M22D2
M22D3
1000
Stress [MPa]
M22D4
800
600
400
200
0
0.0%
1.0%
2.0%
3.0%
Strain
Figure 4.16 Stress-strain curves for dynamic tensile tests of material two in the two
fibre directions
95
EXPERIMENTAL RESULTS
1200
Material - 3
M31D1
Dynamic 11- direction
M31D2
M31D3
1000
Stress [MPa]
M31D4
800
600
400
200
0
0.0%
1.0%
2.0%
3.0%
Strain
1200
Material - 3
M32D1
Dynamic 22- direction
M32D2
M32D3
1000
Stress [MPa]
M32D4
800
600
400
200
0
0.0%
1.0%
2.0%
3.0%
Strain
Figure 4.17 Stress-strain curves for dynamic tensile tests of material three in the two
fibre directions
96
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Table 4.7 Summary of dynamic tensile tests
Material
Specimen
Type
[0°]6
Material-1
[90°]6
[0°]6
Material-2
[90°]6
[0°]8
Material-3
[90°]8
(MPa)
Strain at
max. stress
Strain Rate
at max. stress
M11D1
764
0.63%
274
M11D2
M11D3
760
742
0.57%
0.59%
200
251
M11D4
726
0.68%
314
Average
748
0.62%
260
M12D1
M12D2
868
814
0.52%
0.50%
215
200
M12D3
690
0.51%
189
M12D4
657
0.46%
209
Average
757
0.50%
203
M21D1
1043
1.22%
457
M21D2
930
1.15%
424
M21D3
924
0.87%
347
M21D4
913
0.92%
311
Average
953
1.04%
385
M22D1
1074
1.32%
565
M22D2
1039
1.15%
460
M22D3
923
0.87%
336
M22D4
921
1.16%
465
Average
989
1.13%
456
M31D1
1074
0.83%
308
M31D2
1022
0.90%
331
M31D3
963
0.76%
284
M31D4
Average
819
0.88%
346
969
0.84%
317
M32D1
951
0.80%
300
M32D2
877
0.81%
319
M32D3
M32D4
872
854
0.88%
0.79%
340
294
Average
889
0.82%
313
Test Name
Max. Stress
97
EXPERIMENTAL RESULTS
It is observed that material two has the highest average tensile strength of 979 MPa
followed by materials one and three with 929 MPa and 752 MPa average tensile strength,
respectively. Looking at the variation of dynamic tensile strength between different
repeats of the tests, it is observed that material two has the least variability and material
three has the maximum. However, consistency is observed in the average tensile strength
of the two fibre directions in each material. The variability in the tensile strength of the
three materials is graphically illustrated in Figure 4.18 for both fibre directions.
Average dynamic tensile strength
Tensile strength [MPa]
1200
1000
800
600
11-direction
22-direction
400
200
0
0
1
2
3
Material Number
Figure 4.18 Average dynamic tensile strengths and their variability for the three
materials in the two fibre directions
Furthermore, the highest average strain to failure of about 2.45% is observed for material
two and the lowest (1.1%) is detected for material one. Looking at the variation of
average strain at ultimate stress between different repeats of the tests, it is observed that
material two has the maximum variability, whereas the other two materials have minimal
variability. Moreover, variability in the average strain at ultimate stress is also observed
between the two fibre directions in each material. It is important to point out that the
strain rate, at which each test is performed, cannot be precisely controlled. As a result
98
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
these test repeats are each performed at a somewhat different strain rate, which does
contribute to these inconsistencies. Also, the values presented in Table 4.7 shows that the
tests with the higher maximum stress, are mostly performed at higher strain rate. Figure
4.19 illustrates graphically the average dynamic strains variability in the three materials
for both fibre directions. It is interesting to see that dynamic results of the two normal
directions are comparable for each material. This fact was also observed in the static
results, confirming that these materials are balanced, which justifies the use of the biased
extension shear test to obtain the in-plane shear properties.
Average strain
Strain at max. stress
1.5%
1.0%
11-direction
22-direction
0.5%
0.0%
0
1
2
3
Material Number
Figure 4.19 Average dynamic strains at maximum stress and their variability for the
three materials in the two fibre directions
4.2.3 Bias extension test results
To determine the high strain rate shear characteristics of the three woven composite
materials, dog-bone specimens, cut from the plates with an off-axis angle of ±45°, are
tested using the same Hopkinson bar setup and fixture as the axial tensile tests. The strain
rates in these experiments vary from about 170 /sec up to 580 /sec. As was explained in
section 3.4.2 the experimental values of the tensile stresses and tensile strains are
99
EXPERIMENTAL RESULTS
converted into the shear stress and strains using the transformation of stress and strain
tensors.
The Poisson’s ratio, ν xy , of the materials in the 45 degree cut specimens, obtained in
section 4.1.2, along with the equations given in section 3.4.3 are used to calculate the
dynamic shear stress and strains. Figure 4.20 to Figure 4.22 represents the dynamic shear
behaviour of the three materials with four test repeats each. The behaviour of all the three
materials is similar, that is, an initial linear elastic region followed by nonlinear
deformation until the ultimate stress which precedes a nonlinear strain softening
behaviour. Table 4.8 summarizes the dynamic bias extension test results. In the bias
extension test results, as expected, the material can withstand less strength as compared to
the normal directions; however they fail at much higher strains. Material three has the
highest average strength of 171 MPa, followed by material one with 164 MPa, and
material two with 132 MPa.
200
Material - 1
M13D1
Dynamic 12- direction
M13D2
Shear stress [MPa]
M13D3
150
M13D4
100
50
0
0.0%
1.0%
2.0%
3.0%
4.0%
Shear strain
Figure 4.20 Stress-strain curves for dynamic bias extension tests of material one in
the 45º specimen
100
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
200
Material - 2
M23D1
Dynamic 12- direction
M23D2
Shear stress [MPa]
M23D3
150
M23D4
100
50
0
0.0%
1.0%
2.0%
3.0%
4.0%
Shear strain
Figure 4.21 Stress-strain curves for dynamic bias extension tests of material two in
the 45º specimen
200
Material - 3
M33D1
Dynamic 12- direction
M33D2
Shear stress [MPa]
M33D3
150
M33D4
100
50
0
0.0%
1.0%
2.0%
3.0%
4.0%
Shear strain
Figure 4.22 Stress-strain curves for dynamic bias extension tests of material three in
the 45º specimen
101
EXPERIMENTAL RESULTS
Table 4.8 Summary of dynamic bias extension tests
Material
Material-1
Material-2
Material-3
Specimen
Type
[45°]6
[45°]6
[45°]8
(MPa)
Strain at
max. stress
Strain Rate
at max. stress
M13D1
178
0.51%
198
M13D2
175
0.51%
190
M13D3
160
0.47%
186
M13D4
144
0.46%
171
Average
164
0.49%
186
M23D1
136
0.61%
279
M23D2
134
0.44%
263
M23D3
134
0.53%
269
M23D4
125
0.36%
243
Average
132
0.49%
263
M33D1
181
1.22%
516
M33D2
171
1.39%
589
M33D3
166
1.07%
423
M33D4
165
1.15%
446
Average
171
1.21%
493
Test Name
Max. Stress
Furthermore, material three has the highest average strain to failure of about 3.60%,
followed by material two and one with 1.43% and 1.08%, respectively. From the test
results, it can be deducted that material one has the maximum variability in the shear
strength, whereas material two has the maximum variability in the shear strain (Figure
4.23).
It is worth mentioning, even though not much data is available on the comparison of inplane static and dynamic shear response of woven composite materials, it is well
understood in literature that the behaviour in shear is strongly nonlinear and irreversible
as was mentioned in many papers for example by Marguet et al. (2007) and Johnson
(2001). This could be due to the fact that in shear experiments, the nonlinear properties of
the resin constituent of the composite materials is no longer insignificant, and while the
102
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
fibre in a [45°] specimen are trying to align themselves in the loading direction,
nonlinearity is evident in both the static and dynamic results.
2.0%
150
1.5%
100
1.0%
50
0.5%
Stress
Strain
0
Shear strain at max. stress
Shear strength [MPa]
Average dynamic shear result
200
0.0%
0
1
2
3
Material Number
Figure 4.23 Average dynamic shear strength and strains at maximum stress
including their variability for the three materials in the 45º direction
4.3
Comparison of Static and Dynamic Results
The experimental results presented in the previous sections clearly show that strain rate
has an effect on both the stress and the strain. Figure 4.24 to Figure 4.26 graphically
illustrate the comparison between typical static and dynamic test results for the three
materials in the specimens cut in both the fibre as well as the 45 degree directions. This
strain rate sensitivity confirms the need for a reliable constitutive rate dependent damage
model capable of modelling the response of these woven composite materials under high
velocity loadings.
103
EXPERIMENTAL RESULTS
1200
Material - 1
M1FD
Fibre direction
M1FS
Stress [MPa]
1000
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
(a)
Shear stress [MPa]
200
Material - 1
M13S
12- direction
M13D
150
100
50
0
0.0%
1.0%
2.0%
3.0%
4.0%
Shear strain
(b)
Figure 4.24 Comparison of static and dynamic behaviour in material one (a) 0º or
90º specimens, (b) 45º specimens
104
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
1200
Material - 2
M2FD
Fibre direction
M2FS
Stress [MPa]
1000
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
Strain
(a)
Shear stress [MPa]
200
Material - 2
M23D
12- direction
M23S
150
100
50
0
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
6.0%
7.0%
Shear strain
(b)
Figure 4.25 Comparison of static and dynamic behaviour in material two (a) 0º or
90º specimens, (b) 45º specimens
105
EXPERIMENTAL RESULTS
1200
Material - 3
M3FS
Fibre direction
M3FD
Stress [MPa]
1000
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
2.0%
Strain
(a)
Shear stress [MPa]
200
Material - 3
M33D
12- direction
M33S
150
100
50
0
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
Shear strain
(b)
Figure 4.26 Comparison of static and dynamic behaviour in material three (a) 0º or
90º specimens, (b) 45º specimens
106
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
A look at the initial portion of the stress-strain curves reveal that the dynamic curves are
typically steeper than the static ones. This confirms the increase in the initial undamaged
elastic modulus with increasing strain rate.
In general, on comparing the static and dynamic results, a higher stress is observed for
dynamic tests, whereas the strains at the maximum stress are higher for the static tests.
Looking at the dynamic results in the normal direction, it is evident that material two
experiences the maximum increase (about 38%) in the tensile strength and the minimum
decrease in the strain at maximum stress relative to quasi-static results. On the other hand,
the dynamic versus static shear behaviour of the three materials reveal a large increase
(between 63% to 89%) in the shear strength and a large decrease (between 66% to 83%)
in the strains obtained at the maximum stresses. These are presented in Table 4.9.
Table 4.9 Comparison of the average of static and dynamic results
Average stress [MPa]
Description
Material 1
Material 2
Material 3
Static
Dynamic Increase
Average strain at max. stress
Static
Dynamic
Decrease
Normal
672
752.5
12%
1.13%
0.56%
51%
Shear
87
164
89%
2.46%
0.49%
80%
Normal
704
971
38%
1.12%
1.08%
3%
Shear
81
132
63%
1.42%
0.49%
66%
Normal
868
929
7%
1.33%
0.83%
37%
Shear
101
171
69%
7.17%
1.21%
83%
107
EXPERIMENTAL RESULTS
108
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
5
Equation Section (Next)
DEVELOPMENT OF A RATE
DEPENDENT CONTINUUM DAMAGE
MODEL
5.1
Introduction
This chapter focuses on the development of a stress rate continuum damage mechanics
(CDM) based model for composite materials. A physical treatment of growth of damage
based on the extensive experimental results is combined with the frame work of
continuum damage mechanics models to form the foundation of the model for materials
whose response is governed by elastic deformation coupled with damage.
109
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
The basis for the stress-strain relationship in many CDM approaches is the concept of
stress (or strain) equivalence (Lemaitre (1996) among others). The effective stress
concept states that any deformation behaviour, whether uniaxial, multi-axial, elastic,
plastic or viscoplastic, of a damaged material is presented by the constitutive laws of the
virgin material in which the applied stress σ is replaced by the effective stress σ by the
following equation;
σ = M (d ) σ
(5.1)
where M(d ) is a fourth order damage effect tensor developed for this purpose which
provides a linear mapping of the stress tensor to the effective stress tensor, resulting in a
redistribution of stresses over the reduced remaining area, and thus higher local stresses.
A hypothesis of strain equivalence for isotropic damage was proposed by Lemaitre in
which the Cauchy stress was replaced by the effective stress in the constitutive equation.
For solving the damage problems, one must establish that the strain tensor is the same in
the effective and damaged state, that is
ε =ε
(5.2)
The notation used for a second order tensor is bold letter, and the one used for a fourth
order tensor is italic bold letter. The relationship between the effective stress and strain in
linear elasticity has the simple form of
σ = C 0ε
(5.3)
0
where C is the original undamaged stiffness tensor. The stress-strain relationship in a
damaged material can be obtained by rearranging the terms of equations (5.1) and (5.3) as
follows:
σ = M ( d ) −1 C 0 ε
110
(5.4)
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
The damaged elastic stiffness tensor C (d ) can be defined as the combination of the
damage effect tensor with the undamaged stiffness tensor
C (d ) = M (d ) −1 C 0
(5.5)
Therefore, a general constitutive relationship for a CDM model is obtained as
σ = C (d ) ε
(5.6)
On performing the time derivative on both sides of equation (5.6), this elastic CDM
model can be expanded to consider the rate effects, ε , for the materials that are strain
rate dependant (Armero and Oller 2000; Chaboche 1988a; 1988b; Dechaene et al. 2002;
Nemes and Speciel 1996), such that;
σ = C (d ) ε + C (d , d ) ε
(5.7)
From the formulation it can be clearly seen that the change in stress results not only from
the increments of strain, but also from the rate of damage development, which is
responsible for the stress softening that occurs. The main role of a CDM model is to
provide a mathematical description of the dependence of elastic coefficients on the
damage state as well as the damage evolution. The key to the success of all CDM models
is to maintain a reasonable link with the physical and experimental observations of
damage growth and material response.
As a preliminary step in the development of a new CDM model for woven composite
materials, an existing composite damage model by Dechaene et al. (2002) was
considered. The selection of this model over other models presented in the literature is
due to existence of rate dependency in the material behaviour. The Dechaene et al. model
is developed for woven glass fibre/ epoxy composite materials in which the fabric warp or
weft layer is treated as an equivalent unidirectional layer, that is, the fabric layer is treated
111
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
as a 0/90 cross-ply composite, even though, the behaviour of 0/90 cross-ply composites
can be different from that of the woven fabric composite.
Their model is a stress rate formulation, similar to equation (5.7), consisting of two parts.
The first term is defined as the elastic change in stress with reduced elastic stiffness due
to damage, and the second term is the drop in stress due to growth of damage resulting
from matrix cracks as well as fibre failure (Iannucci et al. (2001). The latter term is a
function of damage and rate of damage.
σ = C (d ) ε + σ d (σ ,d , d )
(5.8)
Dechaene et al. (2002) introduce six damage parameters to track the damage growth. The
damage state of the lamina is divided into three categories, matrix cracks between fibre in
the warp (tracked by d1 in the x-direction and d4 in the y-direction), fibre breakage and
pull out (tracked by d2 in the x-direction and d5 in the y-direction), and shear matrix
cracks representing the horizontal component of matrix crack, or small scale delamination
(tracked by d3 in the xz-plane and d6 in the yz-plane). Figure 5.1 illustrates all these
damage variables.
112
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Z
d3
Y
X
d2
d6
d5
d1
d4
Figure 5.1 Schematic view of the six damage variables in a composite element
(adapted from Dechaene et al. 2002)
In their model, the evolution of damage is expressed as an increasing function of stress
and damage which includes two stages, nucleation and growth. Nucleation refers to
initiation of cracks and it is assumed that the rate of nucleation depends only on the level
of stress. The rate of evolution of the damage, which is the growth of the existing cracks,
depends on the amount of cracks already present, or in other words, on the existing level
of damage. In addition, they have introduced a threshold stress value, σ 0 , below which
there is no nucleation nor growth of damage. For growth of damage, the equation
suggested by Dechaene et al. (2002) has the form,
d1 = ( A0 + A1d1 )
σ xx
σ 0 (1 − c1d1 )(1 − c2 d 2 )
2
−1
(5.9)
113
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
They mention that similar expressions apply for the other damage rates. On the other
hand, in an earlier paper by Iannucci et al. (2001), a different form for damage evolution
is suggested. It is worth mentioning that the purpose here is not to judge their model but
rather point out the existence of various empirical forms to represent the growth of
damage and hence obtain the best form that can characterize the behaviour of the woven
composite materials based on the accomplished experimental results. While in those
papers the authors mention that their proposed damage model can predict the damage
modes observed in the laboratory experiments, a number of issues were raised during this
investigation which should be addressed by a new damage model. These include:
1. Effect of strain rate on the undamaged elastic stiffness
2. Presence of rate-dependent shear damage in the xy-plane
3. Effect of strain rate on the undamaged shear modulus
4. Material characterization and quantitative comparisons to experimental results
5. A methodology to obtain the material constants in the new CDM model
The first three items were not only observed by many researchers (Gilat et al. 2002;
Harding et al. 1989; Johnson et al. 2001; Marguet et al. 2007), but also were confirmed by
the experimental results presented in chapter 4. Later in this chapter all these items will be
dealt with in detail. Nonetheless, it is important to point out that the new CDM model
concentrates on the in-plane response of the woven carbon composite materials. It is
believed that the concept of delamination can be used to consider the effects of damage
on the out-of plane or through-thickness direction. Nevertheless, this concept is deemed
to be beyond the scope of the present work. This is not to say, however, that its
importance is being downplayed. Rather, a reasonable structure for the model must first
be developed before through-thickness damage can be adequately addressed.
5.2
Rate dependent Foroutan-Nemes damage Model
The main approach in developing a complete 3D rate dependant composite constitutive
damage model is to divide the model into two sections; one concentrating on the in-plane
response , which is the subject of this thesis, and the other on the out-of plane or through114
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
thickness response, which can be considered as the inter-laminar response. In the
following sections the formulation for the Foroutan-Nemes stress rate CDM model for
woven composite materials is presented.
5.2.1 Model assumptions
The proposed composite CDM model is capable, but not limited to, predicting the
development of damage due to high strain rate loading, i.e., the model is able to predict
the behaviour of the woven composite materials undergoing dynamic as well as static
loading. Details and material constants of the developed model may be different for every
type of woven composite material; however, the model is independent of the geometry
and boundary conditions of the structure for the particular composite material.
The constitutive damage model is developed for woven carbon fibre fabric with an epoxy
or BMI matrix. However, it is believed that the model can be used to predict the
behaviour of other strain rate dependant woven composite materials which have similar
response. The Foroutan-Nemes rate dependant CDM model is based on the following
assumptions:
1. Plastic strain in these materials is negligible; hence the constitutive equation is
divided into an elastic part and a damage part, therefore, eliminating the need
for a plastic part.
2. The stress and strain are considered as continuous functions within an element.
This means that the discontinuous displacements due to crack opening are
smeared out, or in other words damage is assumed to be evenly distributed
throughout the element.
3. A limited number of scalar damage variables, dij, characterize the state of
damage at a point, where i variables represent the particular direction and j
variables represent the type of damage in that particular direction. These
variables are defined by their effect on the composite mechanical properties and
are always ascending, as the damage process is irreversible and no healing
115
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
occurs during a test. Clearly this principle can only apply if the damage is made
up of a large number of minute cracks evenly distributed throughout the
element.
4. The nucleation and growth of damage should be considered as two distinct
stages of damage development. Nucleation means that damage starts from
nowhere and it is assumed that the rate of nucleation depends on the level of
stress as well as rate of loading. In addition, the growth of damage depends on
the existing amount of damage already present. This dependence on loading rate
contributes to the difference between the behaviour under the static and under
the dynamic loading. Besides, the rate of damage growth creates a time lag
between damage and stress resulting in strain rate sensitivity.
5. Start of damage is apparent when the stress level exceeds certain threshold
values. The threshold for nucleation and growth of damage are assumed to be
equivalent.
6. The onset of damage at the threshold stress level results in the reduction of the
elastic stiffness.
There is no absolute need for damage variables to have a geometric meaning, such as size
or number of cracks of a particular type. Only clear relations between the damage
variables and the components of stress and strains are required. The value of the damage
variables can never decrease, as there is no healing of the material. For the ease of
understanding and calculations, it is possible to restrict these variables within the interval
[0,1], zero meaning no damage and unity meaning that the damage is complete as far as
that particular type is concerned.
5.2.2 Internal state variables
The internal state variables (damage variables), besides being capable of representing the
complex micro defect features, should be simple enough to make the model applicable to
engineering applications. Traditionally, damage parameters are associated with cracking
(e.g. reduction in the cross sectional area of the load bearing material) in the principal
116
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
lamina directions, parallel and perpendicular to the fibre direction, corresponding to
matrix and fibre dominated damage modes.
In the Foroutan-Nemes damage model three zones are assumed in the stress-strain
response. Figure 5.2 illustrates a typical stress-strain response with the three zones
indicated. The first zone is the elastic zone where no damage is present; hence the stress
in this zone is below the threshold stress. Zone II starts when the stress exceeds the
particular threshold stress; by definition, this is the nucleation stress level below which no
damage is present in the material. The damage di1, which is the damage due to matrix
cracks and some fibre failure, is present in this zone. In the third zone, the damage di2 is
in effect. This damage can be related to the fibre pullout and fibre breakage which causes
the softening of the material beyond the ultimate tensile stress. Therefore, six damage
variables can be introduced in the developed 3D stress-rate formulation with in-plane
damage, which are defined as following:
d11 – damage due to matrix cracks, due to x- direction loading
d12 – damage due to fibre breakage and pullout, due to x- direction loading
d 21 – damage due to matrix cracks, due to y- direction loading
d 22 – damage due to fibre breakage and pullout, due to y- direction loading
d 41 – damage due to shear matrix cracks, due to shear loading
d 42 – damage due to fibre breakage and pullout, due to shear loading
117
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
Ultimate stress
Stress
Zone II,
di1 = 0 to di1max
di 2 = 0
Zone III,
di1 = di1max
di 2 = 0 to di 2 max
Threshold stress
Zone I, di1 = di 2 = 0
0
0.0%
Strain
Figure 5.2 Typical stress-strain response showing zones of damage
In a general three dimensional case for a linear elastic material, each of the stress
components is a linear function of the strain tensor; resulting in 36 elastic coefficients,
Cij . As explained in Boresi et al. (1993) these coefficients are constants that are
characteristics of the material and it is mathematically shown, with double differentiation
from the strain energy density function, that the elastic coefficients are symmetrical, i.e.
Cij = C ji reducing the elastic coefficients to 21. However, woven composite materials are
orthotropic in nature, that is, they possess three orthogonal planes of material symmetry
and three corresponding orthogonal axes. It is shown in Boresi et al. that the most general
orthotropic material contains nine elastic coefficients relative to the orthotropic axes. In
an in-plane two dimensional case, the number is hence reduced to 4 elastic coefficients.
In the Foroutan-Nemes damage model the notation Eij is used instead of Cij to represent
the elastic coefficients. In the first zone where no damage is present, the elastic
coefficients considered are the initial undamaged coefficients, which are characteristics of
the material.
118
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Since damage reduces the elastic stiffness of the specimen, a set of equations has to be
developed which describes this reduction in the elastic moduli due to the types of damage
considered. The developed set of equations would serve as the definition of the damage
variables. A survey of approaches taken to this problem in other CDM models shows a
somewhat varied treatment (Chow and Wang 1987; Dechaene et al. 2002; Matzenmiller
et al. 1995; Yazdchi et al. 1996). However, the modulus of elasticity formulation adopted
here considering the six damage parameters associated with modulus reduction can be
written as
0
E11 = E11
0
E22 = E22
0
E44 = E44
[(1 − α11 d11 )(1 − α d12 )][(1 − α 21 d21 )][(1 − α 41d 41 )(1 − α d42 ) ]
[(1 − α 21 d21 )(1 − α d 22 )][(1 − α11 d11 )][(1 − α 41 d 41 )(1 − α d42 )]
[(1 − α 41 d41 )(1 − α d 42 )][(1 − α11 d11 )(1 − α d12 )][(1 − α 21 d21 )(1 − α d 22 )]
(5.10)
0
0
where E11
and E22
are the initial undamaged Young’s modulus in the x- and y-directions,
0
E44
is the initial undamaged shear modulus in the xy-plane, αi1 is a constant that controls
the maximum value of damage di1 at the end of zone II, and α is a constant usually set to
0.9999 prevent the value of (1 − α 2 di 2 ) going to zero when di2 are equal to unity.
It is important to point out that the value of initial undamaged modulus is dependent on
the rate of loading; therefore the following expression is used to obtain the undamaged
rate dependant modulus:
0
E11
= C 2 (1 + C1 ε11 )
C3
0
E22
= C2 (1 + C1 ε22 )
C3
0
E44
= C4 (1 + C1 ε44 )
C5
(5.11)
where C 1 to C5 are the material constants which are determined using the modulus at
quasi static and high rates of strain.
119
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
5.2.3 Evolution of Damage
In a rate dependant damage model, the presumption of the evolution of damage equations
is perhaps the most arbitrary part of the model development, and various forms can be
used for these equations. Nevertheless, an attempt is made to use a logical and
convincible method in developing a set of successful evolution equations.
The rate of damage is expressed as a function of stress, damage and strain rate. The
empirical form for the rate of growth of damages in the x-direction for the damages d11
and d12 - damage due to matrix cracks and damage due to fibre fracture respectively - are
given as
⎛
⎞
σ 11
0
⎜ σ (1 − α d )(1 − α d ) ⎟⎟
11 11
12 ⎠
⎝ 11
0.5
d11 = Z11 [ A11 + A12 d11 ] ⎜
⎛
⎞
σ11
d12 = Z12 [ A13d11 + A14 d12 ] ⎜ 0
⎜ σ (1 − α d )(1 − α d ) ⎟⎟
11 11
12 ⎠
⎝ 11
(5.12)
0.5
(5.13)
where A11 to A14 are the material constants, σ 110 is the loading rate independent, threshold
stress below which there is no damage, and Z11 and Z12 are functions of the strain rates
with the following equations
Z11 = ⎡ (W11 ε11 ) W12 ⎤
⎣
Z12 = ⎡ Z 11
⎣⎢
⎦
W13
⎤
⎦⎥
(5.14)
where W11 to W13 are the material constants. It is important to mention that the constant
A11 in equation (5.12) contributes to the nucleation of damage, as it allows the rate of
damage to start from a state of no damage. Also, equation (5.13) is chosen such that
evolution of damage depends on the existing damage present in zone II of the stress-strain
response, with A13 being the responsible constant.
120
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Similar expressions apply for d21 and d22 -damage rates related to damages in the ydirection. However, the rate of damage in the shear xy-direction is expressed as
⎛
⎞
σ 44
d41 = Z 4 [ A 41 + A 42 d 41 ] ⎜ 0
⎜ σ (1 − α d )(1 − α d ) ⎟⎟
41 41
42 ⎠
⎝ 44
0.5
⎛
⎞
σ 44
d42 = Z 4 [ A 43d 41 + A 44 d 42 ] ⎜ 0
⎜ σ (1 − α d )(1 − α d ) ⎟⎟
41 41
42 ⎠
⎝ 44
(5.15)
0.5
(5.16)
where A41 to A44 are the material constants, Z 4 is a function of strain rate with the
following expression:
Z 4 = ⎡ (W 41 ε44 ) W42 ⎤
⎣
⎦
(5.17)
0
where W41 and W42 are the material constants, and σ 44
is the rate dependent threshold
shear stress.
5.2.4 Rate sensitive constitutive CDM model
Having defined the damage parameters as well as the evolution of damage equations, it is
possible to derive the Foroutan-Nemes rate dependant CDM model. For ease of
understanding the features of the model, the resulting stress-strain response for a one
dimensional case is considered initially. Figure 5.3 shows the basic idea for the evolution
of stress and strain. If there was no growth of damage, the stress increment corresponding
to the stain increment Δε which takes place in the time interval Δt , would be Δσ = E1Δε ,
shown as AD in the figure. However, during the same time interval, the presence of
damage causes a drop in stress by an increment value of Δσ d as shown by BD, and this
depends on damage, rate of damage, as well as stress. Hence, the general form of the
stress rate formulation in one-dimensional form is
σ = E ε − σ d (σ , d , d )
(5.18)
121
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
Δε d
D
C
Stress
σ2
A
σ1
Δσ d
B
Δσ
Δε
E2
E1
0
0.0%
ε1
ε2
Strain
Figure 5.3 Dynamic stress-strain curve showing increment of stress due to damage
In equation (5.18) the first term is the elastic change in stress with reduced elastic
modulus due to damage, and the second term is the drop in stress due to growth of
damage resulting from matrix cracks as well as fibre failure. Due to presence of two types
of damage, it is wise to divide the drop in stress increment due to damage into two
components, one mainly dealing with the damage due to matrix cracks, di1, and the other
dealing with the damage due to fibre breakage and fibre pullout, di2. As a result, the
following equation is used for the stress rate formulation in the 11- direction.
σ11 = E11ε11 −
d11 ( B11 + B12σ 11 ) d12 ( B13 + B14σ11 )
(1 − α11d11 )
−
(1 − α d12 )
(5.19)
where B11 to B14 are the material damage constants, related to the 11-direction, to be
obtained from experiments. In the equation, the damaged rate of stress is dependent on
the stress, the rate of damage, as well as the damage itself.
122
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Having defined the one dimensional formulation, it is now possible to return to the
general constitutive equation and present the final form of the stress rate constitutive
equation. As explained previously, the general CDM model has the form of
σ = C (d ) ε + C (d , d ) ε
(5.20)
which can be written as
σ = C (d ) ε + σ d (σ ,d , d )
(5.21)
where the coefficients of the constitutive secant stiffness tensor, C(d), are functions of the
0
0
0
damage state, d ij , and the undamaged elastic constants, E11
, E22
, and E44
(as presented
in equation (5.10)). Therefore, the full 3D stress rate constitutive equation is written as
σ11 = C11ε11 −
d11 ( B11 + B12 σ 11 )
−
d12 ( B13 + B14 σ 11 )
+ C ε + C ε
12 22
13 33
(1 − α d12 )
(1 − α11d11 )
d ( B σ + B22 ) d22 ( B23 + B24σ 22 )
−
σ 22 = C22ε22 − 21 21 22
+ C21ε22 + C23ε33
(1 − α 21d 21 )
(1 − α d 22 )
σ 33 = C13ε11 + C23ε22 + C33ε33
σ 44 = C44ε44 −
d41 ( B41 + B42σ 44 )
(1 − α 41d 41 )
(5.22)
−
d42 ( B43 + B44σ 44 )
(1 − α d 42 )
σ 55 = C55ε55
σ 66 = C66ε66
where Bij are the material damage constants, d ij and dij are the damage and rate of
damage evolution, respectively and Cij are damaged elastic stiffness tensors presented by
the following for an orthotropic material:
123
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
⎡ C11 C12
⎢C
⎢ 12 C22
⎢C13 C23
⎢
0
⎢ 0
⎢ 0
0
⎢
0
⎢⎣ 0
C13
0
C23 0
C33 0
0 C44
0
0
0
0
0
0
0
0
C55
0
0 ⎤
0 ⎥⎥
0 ⎥
⎥
0 ⎥
0 ⎥
⎥
C66 ⎥⎦
(5.23)
In the current 3D formulation, the through thickness stresses and stiffness are not affected
by the damage propagation. Models dealing with the inter-laminar failure known as
delamination can be combined with this model to include damage that might be present
through the thickness. Hence this model is reduced to an in-plane stress rate formulation.
A summary of the material constants present in the model along with their units as well as
the equations is presented in Table 5.1. It can be seen that overall 29 constants are present
in the Foroutan-Nemes rate dependant CDM model.
124
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Table 5.1 Material constants present in the Foroutan-Nemes CDM model
Damaged
No. modulus
constants
Unit
1
C1
sec
2
C2
Pa
C
0
E11
= C 2 (1 + C1 ε11 ) 3
3
C3
unit less
0
E22
= C2 (1 + C1 ε22 )
C3
4
C4
Pa
0
E44
= C4 (1 + C1 ε44 )
C5
5
C5
unit less
6
α
unit less
0
E11 = E11
7
α11 = α 21
unit less
0
E22 = E22
8
α41
unit less
0
E44 = E44
Equations
[(1 − α11 d11 )(1 − α d12 )]
[(1 − α 21 d 21 )(1 − α d 22 )]
[(1 − α 41 d 41 )(1 − α d 42 )]
Damage rate
No. dependent
constants
Unit
Equations
9
W11
sec
10
W12
unit less
Z11 = ⎡ (W11 ε11 ) W12 ⎤
11
W13
unit less
12
W41
sec
13
W42
unit less
14
A11
sec-1
15
A12
sec-1
16
A13
sec-1
17
A14
sec-1
18
A41
sec-1
19
A42
sec-1
20
A43
sec-1
21
A44
sec-1
⎣
Z12 = ⎡ Z 11
⎦
⎤
⎣⎢
⎦⎥
Z 4 = ⎡ (W 41 ε11 ) W42 ⎤
⎣
⎦
W13
0.5
⎛
⎞
σ11
d11 = Z11 [ A11 + A12 d11 ] ⎜ 0
⎜ σ (1 − α d )(1 − α d ) ⎟⎟
11 11
12 ⎠
⎝ 11
0.5
⎛
⎞
σ
11
d12 = Z12 [ A13d11 + A14 d12 ] ⎜ 0
⎜ σ (1 − α d )(1 − α d ) ⎟⎟
11 11
12 ⎠
⎝ 11
0.5
⎛
⎞
σ 44
d41 = Z 4 [ A 41 + A 42 d 41 ] ⎜ 0
⎜ σ (1 − α d )(1 − α d ) ⎟⎟
41 41
42 ⎠
⎝ 44
0.5
⎛
⎞
σ
44
d42 = Z 4 [ A 43d 41 + A 44 d 42 ] ⎜ 0
⎜ σ (1 − α d )(1 − α d ) ⎟⎟
41 41
42 ⎠
⎝ 44
125
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
Table 5.1 (continues) Material constants present in the Foroutan-Nemes CDM
model
No. Stress rate
constants
Unit
22
B11=B21
Pa
23
B12=B22
unit less
24
B13=B23
Pa
25
B14=B24
unit less
26
B41
Pa
27
B42
unit less
28
B43
Pa
29
B44
unit less
5.3
Equations
σ11 = E11ε11 −
d11 ( B11 + B12 σ 11 )
−
d12 ( B13 + B14 σ 11 )
(1 − α d12 )
(1 − α11d11 )
d ( B σ + B22 ) d22 ( B23 + B24σ 22 )
σ 22 = E22ε22 − 21 21 22
−
(1 − α d 22 )
(1 − α 21d 21 )
d ( B + B42σ 44 ) d42 ( B43 + B44σ 44 )
σ 44 = 2 E44ε44 − 41 41
−
(1 − α d 42 )
(1 − α 41d 41 )
Material characterization
The determination of the material parameters required in the available CDM models has
been one of the most difficult tasks, and for most cases, has not been covered in detail in
the literature. The damage laws are frequently abstract and the means of determining the
parameters required are often not clear. In the current approach, the growth of damage as
well as the stiffness reduction functions have been chosen to be representative of the
performed experimental observations, even though they might be more complex than
many other CDM models. It is believed that this complexity is of little importance as long
as a proper procedure is explained for obtaining the required material constants.
It is possible to divide the material parameters required by the Foroutan-Nemes model
into four groups. The first group consists of the material properties such as the tensile and
shear stiffness and the threshold stresses which are required by many CDM models. The
other three groups are the specific requirements of the Foroutan-Nemes damage model.
These groups contain the parameters that define the reduction in stiffness, (equation set
126
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
(5.10)), the rate of nucleation and growth of damage, (equations (5.12) to (5.16)), and the
stress rate formulation (equation set (5.22)).
5.3.1 Undamaged Stiffness and Strength Constants
The first and foremost step required for measuring material elastic and strength constants
is to perform a series of static and dynamic tests as was explained in chapter 4 of this
thesis. It is important to recall that these woven composite materials show a high
variability in the tensile test results. Therefore a typical average result should be
considered in determining the constants.
Figure 5.4 illustrates a typical dynamic as well as static result in the fibre direction. Also
for the ease of calculations a smooth curve is considered for each of the static and
dynamic responses which show the respective approximate behaviour. The slope of the
approximate static response is considered as the static tensile modulus of the material and
the initial slope of the smooth dynamic curve, obtained from the average of Hopkinson
bar response, is defined as the undamaged tensile stiffness at high rates of strain, E110 H .
This is possible because in the initial part of the dynamic response the elastic undamaged
tensile modulus is represented by:
0H
E11
= σ 11 / ε 11
(5.24)
As explained in section 5.2.2, this corresponds to zone I of the stress-strain curves, where
no damage is present. By definition, the initiation of damage, causes a reduction in the
tensile stiffness, which is the start of zone II, and the stress level at this initiation is
0
.
defined as the threshold stress, σ 11
127
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
Dynamic Result
approx. Dynamic
Static Result
approx. Static
800
700
Stress [MPa]
600
500
400
300
200
100
0
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
Strain
Figure 5.4 Typical stress-strain response of a woven composite material
It is worth mentioning that the woven composite materials tested were all balanced,
meaning that the tensile responses in the two fibre directions are very similar, hence, one
set of constants is sufficient for the 11- and 22- directions.
In the same context, Figure 5.5 represents the shear response of the woven composite
material with both the smooth approximation curves. The undamaged static and dynamic
shear modulus, is half the initial slope of the corresponding shear stress–shear strain
curves as it is characterized by
0
E44
= σ 44 / 2ε 44 = σ 44 / γ 44
(5.25)
And the reduction in the shear modulus marks the start of zone II, with the threshold
0
. Therefore, it is worth mentioning that the undamaged tensile and shear
stress, σ 44
0
0
and σ 44
) are
modulus as well as the tensile and shear threshold stress levels ( σ 11
calculated from the initial portion of the stress-strain curves.
128
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Dynamic Result
approx. Dynamic
Static Result
approx. Static
200
Shear stress [MPa]
180
160
140
120
100
80
60
40
20
0
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
Shear strain (ε 44 =γ 44 /2)
Figure 5.5 Typical shear stress-strain response of a woven composite material
The rate dependant undamaged modulus defined in the developed CDM model is
calculated using equation (5.11) which is
0
E11
= C 2 (1 + C1 ε11 )
C3
0
E22
= C2 (1 + C1 ε22 )
C3
0
E44
= C4 (1 + C1 ε44 )
C5
(5.26)
where the value of C1 is equal to unity and the constants C2 to C5 are obtained using data
fit algorithm between the strain rate data and the modulus at quasi static and high rates of
strain.
In the material model, it is necessary to define a threshold or nucleation criterion which
can determine the nucleation of damage. Clearly, if the nucleation function is greater than
unity, growth of damage has initiated in the material.
Several failure criteria exist in the literature. A brief description of some of these failure
criteria are presented here. One of the most known criteria is the von Mises criterion,
129
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
which is used to define yield for a combined state of stress in isotropic materials (Boresi
et al. 1993). However, woven composite materials are orthotropic in nature and hence the
von Mises criterion cannot be used.
The yield criterion proposed by Hill (1983) predicts yielding of anisotropic materials and
accounts for six different material properties in the various material direction. In order to
determine the material constants, it is necessary to measure the yield stresses
corresponding to the three normal and three pure shear stresses in the principal directions
and orthogonal planes of anisotropy, respectively (Boresi et al. 1993). The Hill criterion
is the generalization of the von Mises criterion and under plane stress, the theory predicts
yielding would initiate when the magnitude of stresses reaches the following condition
2
2
2
1 ⎞
⎛ σ 11 ⎞ ⎛ σ 22 ⎞ ⎛ 1
⎛ σ 44 ⎞
⎜
⎟ +⎜
⎟ − ⎜ 2 + 2 ⎟ σ 11σ 22 + ⎜
⎟ =1
Y ⎠
⎝ X ⎠ ⎝ Y ⎠ ⎝X
⎝ S ⎠
(5.27)
where “X” and “Y” are the yield strengths under uniaxial loading in the two in plane
directions, and “S” is the in plane shearing yield strength in the xy-plane (Lessard 2002).
Azzi and Tsai (1965) extended the Hill theory to anisotropic unidirectional fibre
reinforced composites. According to Tsai-Hill criterion, failure occurs when the following
equation is equal to or greater than one.
2
2
2
⎛ σ 11 ⎞ ⎛ σ 22 ⎞ ⎛ σ11σ 22 ⎞ ⎛ σ 44 ⎞
+
⎜
⎟ +⎜
⎟ −⎜
⎟ =1
2 ⎟ ⎜
⎝ X ⎠ ⎝ Y ⎠ ⎝ X ⎠ ⎝ S ⎠
(5.28)
The Tsai-Wu failure criterion is a failure theory which uses different strengths in tension
and compression (Tsai and Wu 1971). For the case of plane stress, the Tsai-Wu failure
criterion reduces to
2
2
⎛ σ 11 σ 11 ⎞ ⎛ σ 22 σ 22 ⎞ ⎛ σ 11 ⎞ ⎛ σ 22 ⎞ ⎛ σ 11σ 22
⎜ t − c ⎟ + ⎜ t − c ⎟ + ⎜ t c ⎟ + ⎜ t c ⎟ − ⎜⎜
X ⎠ ⎝Y
Y ⎠ ⎝ X X ⎠ ⎝ Y Y ⎠ ⎝ Xt X c Yt Y c
⎝X
2 ⎞
⎞ ⎛ σ 22
⎟⎟ + ⎜ 2 ⎟ = 1 (5.29)
⎠ ⎝ S ⎠
where the superscripts t and c refer to the tensile and compression strengths, respectively.
130
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Hashin (1980) developed a three dimensional failure criteria for unidirectional lamina that
recognized two distinct and uncoupled failure modes. While the failure criterion was
based on composite stresses, it constructs a piecewise continuous failure form based on
failure modes. Hashin identified two failure modes, fibre versus matrix. He developed
separate equations based on the failure mode to determine a failure state for unidirectional
composite laminates. For the case of plane stress, the Hashin failure criterion has the
following two conditions in tension:
2
2
⎛ σ 11 ⎞ ⎛ σ 44 ⎞
⎜ t ⎟ +⎜
⎟ =1
⎝X ⎠ ⎝ S ⎠
2
Tensile fibre mode
(5.30)
2
⎛ σ 22 ⎞ ⎛ σ 44 ⎞
⎜ t ⎟ +⎜
⎟ =1
⎝ Y ⎠ ⎝ S ⎠
Tensile matrix mode
In this thesis, it is intended to use the failure criterion concept to determine the nucleation
of damage. Therefore, instead of using the corresponding strengths, the threshold stresses
are used.
2
2
⎛ σ 11 ⎞ ⎛ σ 44 ⎞
⎜ 0 ⎟ +⎜ 0 ⎟ =1
⎝ σ 11 ⎠ ⎝ σ 44 ⎠
2
Tensile fibre mode
(5.31)
2
⎛ σ 22 ⎞ ⎛ σ 44 ⎞
⎜ 0 ⎟ +⎜ 0 ⎟ =1
⎝ σ 22 ⎠ ⎝ σ 44 ⎠
Tensile matrix mode
And a nucleation function, F 0 , can be defined as
2
2
⎛ σ ⎞ ⎛ σ 22 ⎞ ⎛ σ 44 ⎞
+⎜ 0 ⎟ +⎜ 0 ⎟
F = ⎜ 11
0 ⎟
⎝ σ 11 ⎠ ⎝ σ 22 ⎠ ⎝ σ 44 ⎠
2
0
(5.32)
0
0
0
where σ 11
, σ 22
, and σ 44
are the threshold stress values that are obtained from
experiments. Therefore, if the nucleation function is greater than unity, damage initiates
in the material. In the special cases of uniaxial tension or pure shear, equation (5.32)
reduces to one of the following equations:
131
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
⎛σ ⎞
F = ⎜ 11
0 ⎟
⎝ σ 11 ⎠
2
0
⎛σ ⎞
F = ⎜ 22
0 ⎟
⎝ σ 22 ⎠
2
⎛σ ⎞
F = ⎜ 44
0 ⎟
⎝ σ 44 ⎠
2
(5.33)
0
0
Table 5.3 tabulates the nucleation threshold values for the three materials using
experimental data presented in Figure 4.15 to 4.17 and Figure 4.20 to 4.22.
Table 5.2 Calculated threshold values for the three woven composite materials
Constants
0
0
σ 11
and σ 22
[MPa]
0
σ 44
[MPa]
Material 1
Material 2
Material 3
109
124
165
25
23
15
It is worth mentioning that the combined stress nucleation function can predict the
initiation of damage in a material under multi-dimensional loading. To understand this
better, consider an example in which material one experiences a multi-dimensional
loading such that the two normal and shear stress are as following
σ11 = 70 MPa, σ 22 = 70 MPa, σ 44 = 15 MPa
(5.34)
Using equation (5.32) and Table 5.2, the nucleation function is calculated to be 1.09
which is greater than unity, implying that damage has initiated in the material. It can be
seen that even though the individual stress components are each lower than the
corresponding threshold stress, damage has initiated in the material.
Also, it is important to point out that the failure criteria that were briefly described at the
beginning of this section each have a physical justification, and not all can be used as a
132
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
threshold criterion for woven composite materials. To understand this better, consider
another case of multi-axial loading on material one that result in the following stress
components:
σ11 = 110 MPa, σ 22 = 110 MPa, σ 44 = 0
(5.35)
If the Hill criterion, equation (5.27), is used to calculate the nucleation function, the value
calculated would be zero, that is damage is not present. This cannot be justified since both
normal stress components have already exceeded the threshold limit, which means
damage has initiated in the material. On the other hand, if the combined stress criterion
defined in equation (5.32) is used, the nucleation function would equal 1.4, implying that
damage is present in the material.
5.3.2 Stiffness Reduction Functions
Considering the one dimensional tensile case, it becomes possible to simplify the stiffness
reduction equation (5.10) to the following:
0
E11 = E11
[(1 − α11 d11 )(1 − α d12 ) ]
(5.36)
However, beyond the threshold stress, the reduction in the stiffness is due to the presence
of damage d11. This corresponds to zone II (Figure 5.2) where, due to absence of fibre
failure and fibre pullout, the damage variable d12 is zero. Hence the value of damage d11
which is mainly due to matrix cracks can be calculated from
α11 d11 = 1 − ( E11 / E110 ) 2
at Zone II
(5.37)
It is known that the value of damage, d11, varies between zero at the threshold stress and
unity at the maximum tensile stress. Therefore, it is possible to obtain the constant α11
such that a value of 0.9999 is obtained at the end of zone II. This constant determines the
start of the strain softening in the stress-strain response.
133
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
In zone III, the damage variable d12, which is mainly due to fibre failure and fibre pullout,
can be calculated by rearranging the equation (5.36) in the following form
α d12 = 1 −
0 2
( E11 / E11
)
(1 − α11 d11 )
at Zone III
(5.38)
Here the damage variable d11 has a known value close to unity, and α, is a constant
usually set to 0.9999 which prevents the value of (1 − α di 2 ) going to zero when di2 are
equal to unity.
Following the same procedure for the shear response, one can obtain the constant α41 as
well as the increments of damage related to the shear response, that is, d41 and d42, with
the following equations:
0 2
α 41 d 41 = 1 − ( E44 / E44
)
α d 42 = 1 −
0 2
( E44 / E44
)
(1 − α 41 d 41 )
at Zone II
(5.39)
at Zone III
(5.40)
5.3.3 Damage Evolution Functions
The rate of damage in the one dimensional case can be calculated from the increments of
damage at the time increment, Δ t , for both the matrix failure as well as fibre breakage
damages, such that
Δd
d11 = 11
Δt
Δd
d12 = 12
Δt
(5.41)
′ and d12
′
In order to simplify the evolution of damage equations (5.12) and (5.13), let d11
be defined and calculated as the following
134
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
⎛ σ 0 (1 − α11d11 ) ⎞
′ = d11 ⎜ 11
d11
⎟⎟
⎜
σ 11
⎝
⎠
0.5
0
⎛ σ 11
(1 − α11d11 )(1 − α d12 ) ⎞
′
d12 = d12 ⎜
⎟⎟
⎜
σ 11
⎝
⎠
0.5
(5.42)
Hence, the evolution of damage equations (5.12) and (5.13) can be rewritten as
′ − Z11 ⎣⎡ A11 + A12 d11 ⎦⎤ = 0
d11
(5.43)
′ − Z12 ⎡⎣ A13d11 + A14 d12 ⎤⎦ = 0
d12
where the Z constants are related to loading rate of the specimen. At high loading rate at
which Hopkinson bar experiments are performed, the Z values are equal to unity, and
decreases according to equation (5.14) as the loading rate decreases to static loading. The
two sets of rate of damage parameters ( A11 , A12 and A13 , A14 ) are obtained from the
best-fit curve of d1 j verses d1′ j , where j denotes the type of damage.
Similarly, from the shear response, one can calculate two sets of rate of damage
parameters ( A 41 , A42 and A 43 , A44 ) from the best-fit curve of d 4 j verses d4j′ , using the
following equations:
′ − Z 4 ⎡⎣ A 41 + A 42 d 41 ⎤⎦ = 0
d41
(5.44)
′ − Z 4 ⎡⎣ A 43d 41 + A 44 d 42 ⎤⎦ = 0
d42
where
⎛ σ 0 (1 − α 41d 41 ) ⎞
′ = d41 ⎜ 44
d41
⎟⎟
⎜
σ 44
⎝
⎠
0.5
0
⎛ σ 44
(1 − α 41d 41 )(1 − α d 42 )
′ = d42 ⎜
d42
⎜
⎝
σ 44
⎞
⎟⎟
⎠
0.5
(5.45)
135
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
Z4 is related to loading rate of the specimen, calculated according to equation (5.17). It
has a value of unity at the experimented high strain rate which decreases with a decrease
in the strain rate.
5.3.4 Stress Rate Functions
The next step is to determine the constants in the stress rate CDM model presented in
equation (5.22). First, consider the one dimensional case in which the stress rate
formulation can be simplified as
σ11 = E11ε11 −
d11 ( B11 + B12 σ 11 )
(1 − α11d11 )
−
d12 ( B13 + B14 σ 11 )
(1 − α d12 )
(5.46)
As was explained in section 5.2.4, the first part is the elastic change in stress with a
reduced elastic modulus E11, the second term is the drop in stress due to the growth of
damage d11 which is mainly in the form of matrix cracks, and the third term is the drop in
stress due to growth of damage d12 resulting from fibre breakage and fibre pullout.
In the zone II region of the stress-strain response, due to absence of the growth of damage
d12, the third term is not considered. Hence, for zone II, equation (5.46) can be simplified
and rearranged as follows:
B11 + B12 σ11 =
( E11ε11 − σ11 )(1 − α11d11 )
d11
at Zone II
(5.47)
The stress rate constants B11 and B12 can be calculated from a linear fit between the stress
and incremental values obtained from the right hand side of equation (5.47) using a datafit algorithm. Here, the rate of stress is calculated from the stress increments Δσ11 at time
increments Δ t as:
σ11 =
136
Δσ 11
Δt
(5.48)
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
The values of d11 and d11 at different time intervals can be obtained using the equations
explained previously.
The two remaining unknown stress rate constants of equation (5.46), i.e. B13 and B14 can
be obtained from the zone III region of the stress-strain response. In this zone, the rate of
damage d11 is assumed to be zero; hence equation (5.46) can be simplified as follows:
B13 + B14 σ11 =
( E11ε11 − σ11 )(1 − α d12 )
d12
at Zone III
(5.49)
and the stress rate constants are calculated from a linear fit between the stress and
incremental values obtained from the right hand side of equation (5.49) using a data-fit
algorithm.
Similarly, it is possible to calculate the shear stress rate constants B 41 , B 42 and B 43 , B44
using a data-fit algorithm and the following equation
σ 44 = 2 E44ε44 −
d41 ( B41 + B42 σ 44 )
(1 − α 41d41 )
−
d12 ( B43 + B44 σ 44 )
(1 − α d42 )
(5.50)
Consequently, using this methodology, it is feasible to easily obtain all the constants
present in the Foroutan-Nemes constitutive damage model. Table 5.3 tabulates the
calculated values for these material constants for the three materials presented in this
thesis.
137
DEVELOPMENTS OF A RATE DEPENDANT CONTINUUM DAMAGE MODEL
Table 5.3 Calculated constants for the three woven composite materials
No. Constants
0
1
σ 11
[Pa]
Material 1
Material 2
Material 3
1.09E+08
1.24E+08
1.65E+08
2
0
σ 44
[Pa]
2.50E+07
2.30E+07
1.50E+07
3
ν 12
4
5
6
7
8
C1 [sec]
C2 [Pa]
C3
C4 [Pa]
C5
9
10
11
α
α11 = α 21
α41
12
13
14
15
16
W11 [sec]
W12
W13
W41 [sec]
W42
A11 [sec-1]
A12 [sec-1]
A13 [sec-1]
A14 [sec-1]
A41 [sec-1]
A42 [sec-1]
A43 [sec-1]
A44 [sec-1]
B11=B21 [Pa]
B12=B22
B13=B23 [Pa]
B14=B24
B41 [Pa]
B42
B43 [Pa]
B44
0.087
1
8.35E+10
0.2
8.04E+09
0.45
0.99999
0.73
0.775
0.0056
1.045
0.7
0.005
1.07
18000
-12000
21000
-21000
25000
-15500
13000
-13000
-1.00E+06
0.25
1.50E+05
0.35
-1.30E+06
0.15
2.00E+06
0.35
0.06
1
6.08+10
0.14
1.00E+10
0.3
0.99999
0.7
0.775
0.0029
0.97
0.7
0.004
1.05
15000
-10000
17000
-17000
32000
-19000
10000
-10000
4.00E+06
0.35
-5.00E+07
0.55
-1.50E+06
0.12
3.00E+06
0.7
0.08
1
8.58E+10
0.17
6.40E+09
0.3
0.99999
0.73
0.7
0.0038
1
0.85
0.0025
1.04
16000
-10000
18000
-14000
17000
-13500
7000
-7000
-7.00E+06
0.38
7.00E+07
0.5
-1.40E+06
0.50
1.40E+07
0.65
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
138
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
6
Equation Section (Next)
IMPLEMENTATION OF THE
DAMAGE MODEL INTO A FINITE
ELEMENT CODE
6.1
Introduction
The ultimate criterion for a successful model is its ability to predict results comparable
with those of experiments, regardless of how complex the developed mathematical model
is in describing the material behaviour. Currently, the finite element method is one of the
most powerful tools for analysing complex structures subjected to different loading
conditions. Therefore, in order to use this technique for any developed material model, it
must be incorporated into a finite element code. Although in many engineering
139
IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
applications, complicated structural geometry may be used, they mostly consist of some
fundamental structural units. This makes it reasonable to test models at high loading rates
dealing with damage analysis using simple structural units.
6.2
Model implementation in finite element
The Foroutan-Names damage model for woven composite materials developed and
parameterized in the previous sections is next implemented in a VUMAT Material User
Subroutine of the commercial finite element program ABAQUS/Explicit. This subroutine
is compiled and linked with the finite element solver and enables ABAQUS/Explicit to
obtain the needed information regarding the state of the material and the material
mechanical response during each time step, for each integration point of each element.
The basic difference between an explicit and an implicit analysis is that the former
requires no information regarding the state variables at the end of the increment. Rather it
relies on sufficiently small time increments such that system’s future state can be
extrapolated from the current state. The state variables are updated after the increment.
An implicit approach accounts for the change in state variables that occur over the step.
Hence, they require that the stress Jacobian be defined. Implicit solutions tend to be more
time consuming, however, much larger time increments are possible. In the current
situation, where short duration loading, such as impact, is of concern and total time of the
event is a fraction of a second, the explicit solution method is the reasonable analysis
technique to carry out. For a detailed review of the ABAQUS theories or their
implementation the reader is referred to the ABAQUS theory manual (ABAQUS 2006).
Hence, the model defined in Chapter 5 was programmed in FORTRAN language and
supplied as a VUMAT subroutine to the ABAQUS/Explicit finite element software
following the instruction provided in the ABAQUS documentation. The analysis
procedure involves three elements: the main program, input file and the subroutine.
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HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Figure 6.1 illustrates the interaction between these main three elements and the subroutine
procedure. In the flowchart, nblock is a term used in ABAQUS that refers to the number
of integration points to be processed in one call to the VUMAT. The FORTRAN code for
the VUMAT subroutine is provided in Appendix A. The essential features of the coupling
between the ABAQUS/Explicit finite-element solver and the VUMAT Material User
Subroutine at each time increment at each integration point of each element can be
summarized as follows:
(a) The corresponding previous time-increment, stresses, and material state
variables as well as the current time-step strain increments are provided by the
ABAQUS/Explicit finite element solver to the material subroutine. In the
present work, the strain components, the damage variables, the secant modulus
and a variable defining the deletion status of the element in question are used as
state variables; and
(b) The material stress state as well as values of the material state variables at the
end of the time increment are determined within the VUMAT and returned to
the ABAQUS/Explicit finite-element solver, using the information provided in
(a), the nucleation function define in Section 5.3.1 for damage initiation, and
the CDM material model presented in Section 5.2.
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IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
FE data base
Mesh, BC & loading
Main Program
ABAQUS/Explicit
User-supplied material
properties & constants
VUMAT subroutine
From k = 1 to
nblock
Calculate total strain
& nucleation function
Is
F >1?
0
NO
Calculate stress
σii
NO
Calculate
di2, Eii, σii
YES
Calculate
di1, Eii, σii
Is
di1 < 0.9999
?
YES
Is
di2 < 0.9999
?
NO
Element is
deleted
YES
NO
Is
k > nblock
YES
?
Next time step
t = Δt + t
YES
Is
t < total time
?
NO
STOP
Figure 6.1 Flowchart for implementation of the CDM model into ABAQUS
142
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
6.3
Implementation and comparison to experiments
The material model for woven composite materials developed and parameterized in the
previous chapter is justified in this section by carrying out a series of dynamic tensile
analyses on a plane stress single element and by comparing the computational results with
the experimental results. In the remainder of this section, a brief description of loadingunloading as well as compression tests and the nature of the results obtained in these
simulations are presented. This is followed by the description of the computational
procedure used to simulate the test.
6.3.1 Tensile test results
For the tension test, a two dimensional plane stress element is subjected to dynamic
tensile displacement at one end. The element used is a four node bilinear plane stress
quadrilateral with reduced integration point denoted by CPS4R in ABAQUS. In order to
consider the rate effect, the amplitude toolset is used which allows the user to specify a
time varying displacement for the loading, which is to say, the displacement history
extracted from the results of the Hopkinson bar testing can be applied to the simulation.
From the dynamic experimental results it was observed that the displacement, Δ , is a
function of time, which can be approximated by
Δ = A t2
(6.1)
Figure 6.2 compares the actual time-displacement data obtained from a Hopkinson bar
experiment with the profile obtained using the equation (6.1) for the three materials, and
it can be concluded that the profile from the equation is a representation. Also A is found
to be a function of average strain rate, that is,
A = Bε 2
(6.2)
where B is a constant obtained from a data fit algorithm using the displacement versus
time data.. Therefore, a displacement versus time profile can be applied at different
average strain rates to simulate the loading of the material.
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IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
Material One
Displacement [mm]
1
Fibre Direction
0.8
0.6
0.4
Mat-1 Test
Mat-1 eq
0.2
0
0
10
20
30
40
50
60
Time [μ sec]
(a)
Material Two
Displacement [mm]
2.5
Fibre Direction
2
1.5
1
Mat-2 Test
Mat-2 eq.
0.5
0
0
10
20
30
40
50
60
70
Time [μ sec]
(b)
Figure 6.2 Comparison between the displacement-time profiles obtained from
dynamic test results and the equation 6.2 (a) material one, (b) material two, (c)
material three
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HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Material Three
Fibre Direction
Displacement [mm]
1.5
1
Mat-3 eq
0.5
Mat-3 Test
0
0
10
20
30
40
50
60
Time [μ sec]
(c)
Figure 6.2 (continued) Comparison between the displacement-time profiles obtained
from dynamic test results and the equation 6.2 (c) material three
Due to symmetry only a quarter of specimen gage section is modeled. Figure 6.3
illustrates the tensile loading in the fibre direction where, the nodes on the left are
restrained in x-direction, and the nodes on the bottom are restrained in the y-direction.
The results for the two fibre direction will be similar as the properties in the two
directions were the same for all the three materials in this thesis.
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IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
y
u=0
u = Δx(t )
x
v=0
Figure 6.3 Two dimensional element, tensile simulation in ABAQUS
Using the VUMAT subroutine for the material model, the load is applied at different
strain rates and the stress-strain behaviour is obtained. The simulated strain rates vary
from quasi static value of 1E-3/sec to a value equivalent to the experimental average
strain rates for each material. The dynamic average strain rates obtained using the
Hopkinson bar setup are 230/sec, 420/sec and 315/sec for materials one to three,
respectively. For the purpose of comparison, the tensile simulation results at different
strain rates is plotted with the average experimental dynamic and static results for each
material as illustrated in Figure 6.4. The figure shows that the model can well predict the
dynamic experimental result. Table 6.1 summarizes the maximum dynamic tensile
strength obtained from the simulation and compares them with the ones from the test
data. It is found that the percentage difference is less than 5% for all three materials. It is
worth mentioning that due to presence of symmetry in the material, that is similar
behaviour in both fibre directions, the same results can be postulated for the perpendicular
y-direction.
146
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
1000
Material One
900
Fibre Direction
Dy-Test
St- Test
ABQ 1000
ABQ 230
Stress [MPa]
800
ABQ 100
700
ABQ 50
600
ABQ 5
500
ABQ 1
ABQ 1E-3
400
300
200
100
0
0.0%
0.5%
1.0%
1.5%
Strain
(a)
1400
Dy-Test
Material Two
St-Test
Fibre Direction
1200
ABQ 1000
ABQ 420
Stress [MPa]
ABQ 200
1000
ABQ 100
ABQ 50
800
ABQ 5
ABQ 1
ABQ 1E-3
600
400
200
0
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
3.0%
Strain
(b)
Figure 6.4 Comparison of the ABAQUS simulation with the average dynamic and
static tensile test results (a) material one, (b) material two, (c) material three
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IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
1400
St-Test
Fibre Direction
1200
Stress [MPa]
Dy-Test
Material Three
ABQ 1000
ABQ 315
ABQ 100
1000
ABQ 50
800
ABQ 5
600
ABQ 1E-3
ABQ 1
400
200
0
0.0%
0.5%
1.0%
1.5%
2.0%
Strain
(c)
Figure 6.4 (continued) Comparison of the ABAQUS simulation with the average
dynamic and static tensile test results (c) material three
As can be seen from the graphs, a much higher strain rate of 1000/sec, is also chosen to
illustrate the prediction of the simulation, and also compare the result with the available
dynamic test data. The model cannot perfectly predict the static behaviour of the material;
however an approximate response can be obtained. Since the main purpose of the
developed model is to predict the behaviour under dynamic loading, an approximate
prediction for the static response is accepted.
Table 6.1 Comparison of maximum dynamic tensile strength
Max. dynamic tensile stress
[MPa]
148
Material 1
(232 sec-1)
Material 2
(420 sec-1)
Material 3
(315 sec-1)
Simulation
755
990
1014
Test
752
971
969
% difference
0.3%
2.0%
4.5%
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
6.3.2 Bias extension shear test results
For the shear test, a two dimensional plane stress element is subjected to dynamic shear
displacement at the top left corner node, as illustrated in Figure 6.5. Due to the nature of
the bias extension shear test, in which the fibres are at 45 degree to the loading direction,
the whole square part of the specimen gage section is modeled. As shown in the Figure
6.5, the bottom left corner node is restrained in x-direction and y-direction. In a similar
manner to the tensile simulations, the amplitude toolset is used to specify various
displacement times for the loading, in order to consider different average strain rates as
illustrated in Figure 6.6.
y
v = Δ y (t )
u = Δx(t )
u=v=0
x
Figure 6.5 Two dimensional element, shear simulation in ABAQUS
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IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
Material One
Displacement [mm]
0.6
12- Direction
0.4
Mat-1 Test
0.2
Mat-1 eq
0
0
10
20
30
40
50
60
70
Time [μ sec]
(a)
Material Two
Displacement [mm]
0.8
12- Direction
0.6
0.4
Mat-2 Test
Mat-2 eq.
0.2
0
0
10
20
30
40
50
60
70
Time [μ sec]
(b)
Figure 6.6 Comparison between the shear displacement-time profiles obtained from
dynamic test results and the equation 6.2 (a) material one, (b) material two, (c)
material three
150
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Material Three
2
12- Direction
Displacement [mm]
1.8
1.6
1.4
1.2
1
0.8
Mat-3 eq
0.6
Mat-3 Test
0.4
0.2
0
0
20
40
60
80
Time [μ sec]
(c)
Figure 6.6 (continued) Comparison between the shear displacement-time profiles
obtained from dynamic test results and the equation 6.2 (c) material three
Using the VUMAT subroutine for the material model, the above mentioned load is
applied at different strain rates, varying from quasi-static value of 1E-3/sec to a value
equivalent to the experimented average strain rates for each material, and the stress-strain
behaviour is obtained. Using the Hopkinson bar setup, the dynamic average strain rates
for the bias extension shear experiments are calculated as 180/sec, 260/sec and 490/sec
for the materials one to three respectively. For the purpose of comparison, the shear
simulation results at different strain rates is plotted with the average experimental
dynamic and static bias extension shear results for all the three material as illustrated in
Figure 6.7. It can be seen that the model can well predict the dynamic behaviour, and the
percentage difference between the simulation and test data for the maximum shear
strength is approximately between 3-5% for the three materials as tabulated in Table 6.2.
A simulation for a much higher strain rate of 1000/sec, is also performed to illustrate the
prediction of the response, and also compare the result with the available dynamic test
data. It is worth mentioning that the model cannot perfectly predict the static shear
151
IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
behaviour of the material; however an approximate response can be obtained. Since the
main purpose of the developed model is to predict the behaviour under dynamic loading,
an approximate prediction for the static response is accepted.
250
St-Test
Material One
Dy-Test
Shear stress [MPa]
12- Direction
ABQ 1000
200
ABQ 165
ABQ-50
ABQ 5
150
ABQ 1
ABQ 1E-3
100
50
0
0.0%
1.0%
2.0%
3.0%
4.0%
Shear strain
(a)
Figure 6.7 Comparison of the ABAQUS simulation with the average dynamic and
static shear test results (a) material one, (b) material two, (c) material three
152
Shear stress [MPa]
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
180
Material Two
160
12- Direction
Dy-Test
St-Test
ABQ 1000
140
ABQ 260
ABQ-50
120
ABQ 5
100
ABQ 1
ABQ 1 E-3
80
60
40
20
0
0.0%
1.0%
2.0%
3.0%
4.0%
5.0%
Shear strain
(b)
200
Dy-Test
Material Three
St-Test
Shear stress [MPa]
12- Direction
ABQ 1000
ABQ 490
150
ABQ 100
ABQ-50
ABQ 5
100
ABQ 1
ABQ 1 E-3
50
0
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
Shear strain
(c)
Figure 6.7 (continued) Comparison of the ABAQUS simulation with the average
dynamic and static shear test results (b) material two, (c) material three
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IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
Table 6.2 Comparison of maximum dynamic shear strength
Max. dynamic shear stress
[MPa]
Material 1
(186 sec-1)
Material 2
(263 sec-1)
Material 3
(493 sec-1)
Simulation
172
136
165
Test
164
132
171
% difference
4.9%
3.0%
3.5%
6.3.3 Simulation results for loading-unloading-reloading
During a typical dynamic impact, a part of the impacted material can also go under
compressive loading after initially experiencing tensile loading. Hence, it is important to
check the behaviour of the material model when the material goes under a loadingunloading-reloading cycle. For this purpose, a constant strain rate is considered which is
equal to the average strain rate at which dynamic tests are performed, that is, 230/sec for
material one, 420/sec for material two and 315/ sec for material three. First the loadingunloading where the unloading starts at different time intervals is considered. In these
simulations the specimen is loaded to a fraction of the total time followed by an
unloading interval as illustrated in Figure 6.8. It is interesting to note that the response
returns to a point of zero stress and zero strain when the specimen is unloaded.
154
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Material One
900
Fiber Direction
800
Stress [MPa]
700
600
500
400
300
200
100
0
0.0%
0.5%
1.0%
Strain
(a)
Material Two
1200
Fiber Direction
Stress [MPa]
1000
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
Strain
(b)
Figure 6.8 Loading-unloading simulation results (a) material one, (b) material two,
(c) material three
155
IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
1200
Material Three
Fiber Direction
Stress [MPa]
1000
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
(c)
Figure 6.8 (continued) Loading-unloading simulation results (c) material three
In the loading-unloading-reloading simulation, after the specimen is loaded to a fraction
of the total time, two unloading scenarios are possible. In the first case the specimen is
unloaded to remove the entire initial tensile load on the specimen as illustrated in Figure
6.9, which is then followed by applying the load again with the same strain rate. It can be
seen that due to the presence of damage already present in the specimen, the reloading
response has lower initial tensile modulus. Also it is worth nothing that the tensile
strength is much lower than the initial one step loading tensile simulation due to damage
during the first loading cycle.
156
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Material One
900
Fiber Direction
800
Stress [MPa]
700
600
500
400
300
200
100
0
0.0%
0.5%
1.0%
Strain
(a)
Material Two
1200
Fiber Direction
Stress [MPa]
1000
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
2.0%
2.5%
Strain
(b)
Figure 6.9 Loading-unloading-reloading simulation results (a) material one, (b)
material two, (c) material three
157
IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
1200
Material Three
Fiber Direction
Stress [MPa]
1000
800
600
400
200
0
0.0%
0.5%
1.0%
1.5%
Strain
(c)
Figure 6.9 (continued) Loading-unloading-reloading simulation results (c) material
three
In the second case the specimen is unloaded for a longer duration and hence the specimen
would experience a compressive load as illustrated in Figure 6.10, which is followed by
reapplying the load with the same constant strain rate until failure. It can be seen that the
damage present in the specimen is larger than the previous case and the tensile strength is
much lower than first case simulation. It is important to note that the material experiences
damage even in the compression response, and as soon as there is load reversal from
compression, the growth of damage becomes zero. Hosur et al .(2003a) have shown the
high strain rate response of woven carbon epoxy materials under compression. From their
experimental results it can be concluded that the peak stress and the slope of the stressstrain response increases with strain rate in compression, which is similar to the tensile
response. It is worth mentioning that in the developed constitutive damage model, it is
assumed that the material behaviour is similar in both tension and compression.
158
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Material One
900
Fiber Direction
Stress [MPa]
600
300
0
-0.5%
0.0%
0.5%
1.0%
-300
-600
Strain
(a)
Material Two
1200
Fiber Direction
900
Stress [MPa]
600
300
0
-1.2%
-0.6%
0.0%
-300
0.6%
1.2%
1.8%
2.4%
-600
Strain
(b)
Figure 6.10 Loading-unloading to compression-reloading simulation results (a)
material one, (b) material two, (c) material three
159
IMPLEMENTATION OF THE DAMAGE MODEL IN TO A FINITE ELEMENT CODE
Material Three
Fiber Direction
1200
Stress [MPa]
800
400
0
-0.6%
0.0%
0.6%
1.2%
-400
-800
Strain
(c)
Figure 6.10 (continued) Loading-unloading to compression-reloading simulation
results (c) material three
6.4
Summary
The developed Foroutan-Names damage model was implemented in a VUMAT Material
User Subroutine of the finite element program ABAQUS/Explicit. Uniaxial tension and
shear simulations were performed on the three materials at different loading rates. The
dynamic response was well simulated, and the simulation for the static loading was
shown to be approximately close to the test data. Simulation results for higher than tested
strain rates were also presented to illustrate the response at these high rates. It was shown
that the tensile strength increases with the increase in the loading rate, and also a strain
softening phenomenon was observed after the maximum tensile strength. The loadingunloading, as well as compression response of the materials were also simulated to ensure
the capability of the model to capture these responses.
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HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
7
Equation Section (Next)
CONCLUSIONS
7.1
Summary and conclusions
A summary of the major findings drawn from this research along with its important
conclusions are provided in this section, which is divided into two parts. The first part
discusses the experimental work needed to develop a rate dependant constitutive damage
model capable of predicting the response of woven composite materials under high strain
rate loading and the second part discusses the developed CDM model that is implemented
into a finite element code along with the simulation results.
161
CONCLUSIONS
7.1.1 Experimental findings
In this work, four types of experiments were performed, namely static uniaxial tension,
static bias extension shear, dynamic uniaxial tension, and dynamic bias extension shear
experiments. The tests were conducted on three composite materials, namely a 2×2 twill
weave carbon/epoxy prepreg [0]6, a carbon/BMI prepreg [0]6 with 8 harness satin weave,
and a plain weave carbon/epoxy prepreg [0]8, all of which are balanced woven fabrics
with a zero direction staking sequence.
The static experiments were performed using a servo-hydraulic MTS testing machine,
where as the dynamic tests were conducted using a tensile version of Hopkinson bar
setup. For the dynamic experiments, the shape and size of the specimen was designed
such that, it is short enough to reach stress equilibrium but large enough to characterize
the mechanical properties respective of the material. Also to insure failure in the gage
section, the specimen’s cross sectional area must be reduced. This is done using a smooth
radius of curvature in the transition region to prevent stress concentration. For
consistency, the same specimen size and geometry are used for all static and dynamic
experiments.
The test specimens are cut from composite plate in the (0°), (90°) and (45°) directions.
The (0°) and the (90°) specimens are used to obtain static and dynamic tensile behaviour
in the normal directions, whereas the (45°) specimens are used in biased extension shear
tests to obtain the nonlinear in-plane shear properties. The specimens cut in (45°)
directions are tested in tension to ultimate failure. A stress analysis of the 45 degree cut
specimen uses the transformation matrix to transform the results obtained in the loading
direction into the stresses and strains in the fibre direction.
A special fixture was designed to hold the dog bone specimen in between the Hopkinson
bar setup while performing the experiments. The designed fixture has the following
specifications. Firstly, a large gripping force is needed to prevent the specimen from
slipping out of the fixture, secondly, it is made of the same material as the Hopkinson
162
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
bars to prevent disturbance in the wave propagation, and thirdly, the specimen must be
aligned with the fixtures to avoid eccentric loading.
In static experiments, the strains are measured using a reduced extensometer in the
loading direction, and the transverse strains are measured from the results of the strain
gages attached on the specimens perpendicular to the loading direction. In the Hopkinson
bar experiments, a calibration factor is introduced to acquire the specimen strains using
the equations of Hopkinson bar theory. This calibration factor is needed to calculate the
effective gage length of the dog-bone specimen. Due to the smooth radius of curvature in
the dog bone specimens, the specimen gage length is no longer well-defined. Hence, a
series of specimens are tested using strain gages glued directly on the specimen to
determine the calibration factor to calculate the effective gage length and as a result
ensure the accuracy of dynamic strains.
Static tensile tests are performed and a linear brittle behaviour is confirmed for all the
three materials. It is observed that material three is the strongest with an average tensile
strength of 867 MPa. This is approximately 180 MPa higher than the other two materials.
In addition, material three has the maximum average strain to failure of about 1.33%. The
average static tensile modulus of all the three materials has little variability in the two
fibre directions. Even the variability between the materials tested with different weaves is
very small, hence an average value of 63±3 GPa is a good approximation. Also a very
low Poisson’s ratio is obtained for all three materials.
In the static bias extension shear results for all the materials, an initial linear elastic region
followed by nonlinear deformation behaviour is observed. Material three has the
maximum strength, and the average shear strength for the three materials is 81-101 MPa.
And the material with the 2×2 twill weaves is stiffer than material two with the 8HS
weave. Material three shows the maximum strain to failure of about 12%, about twice the
other two materials. The initial undamaged shear modulus obtained from the initial slope
of the shear stress-strain curve is approximately 5-6 GPa for the three materials.
163
CONCLUSIONS
High strain rate tensile tests are performed in both fibre directions of all three materials
using the Hopkinson bar setup. The strain rates in these experiments vary from
approximately 200 /sec up to 550 /sec. All the materials show an initial linear elastic
region followed by nonlinear deformation until the ultimate stress which precedes a
nonlinear strain softening behaviour. It is observed that material two has the highest
average tensile strength of 979 MPa followed by materials one and three with 929 MPa
and 752 MPa average tensile strength, respectively. Furthermore, material two has the
highest average strain to failure of about 2.45%, and material one has the lowest of about
1.1%. Also, it was observed that the tensile results of the two fibre directions are
comparable for each material, which confirms that these materials are symmetric and
justifies the use of the biased extension shear test to obtain the in-plane shear properties.
The dynamic bias extension shear experiments are performed using the same Hopkinson
bar setup and fixture. The strain rates in these experiments vary from about 170 /sec up to
580 /sec. The behaviour of all the three materials is similar, that is, an initial linear elastic
region followed by nonlinear deformation until the ultimate shear stress which precedes a
nonlinear strain softening behaviour. It was observed in these experiments that the shear
strength is much lower than the tensile strength, as expected. However, the materials fail
at much higher strains. Material three has the highest average shear strength of 171 MPa,
followed by material one with 164 MPa, and material two with 132 MPa.
Comparisons of the static and dynamic experimental results clearly show that strain rate
has an effect on both the stress and the strain. A higher stress is observed for dynamic
tests, whereas the strains at the maximum stress are higher for the static experiments.
Also an increase in the initial undamaged elastic modulus, for both tensile and shear
response, with increasing strain rate is observed. This strain rate sensitivity confirms the
need for a reliable constitutive rate dependent damage model capable of modelling the
response of these woven composite materials under high velocity loadings.
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HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
7.1.2 Material model and simulation findings
A rate dependant constitutive damage model is developed in the framework of continuum
damage mechanics which exhibits elastic deformation coupled with damage. The
proposed Foroutan-Nemes stress rate CDM model concentrates on the in-plane response
of the woven carbon composite materials. The model is mainly developed to predict the
behaviour of woven composite materials undergoing dynamic loading. It cannot perfectly
predict the static brittle behaviour of the material; however, an approximate static
response can be obtained. Details and material constants of the developed model may be
different for every type of woven composite material; however, the model is independent
of the geometry and boundary conditions of the structure for the particular composite
material.
In the Foroutan-Nemes CDM model the reduction in the elastic moduli is attributed to
damage. And the evolution of damage is expressed as a function of stress, damage and
strain rate. The developed stress rate CDM formulation includes two parts, one is related
to the elastic change in stress with reduced elastic modulus, and the second is the drop in
stress due to growth of damage. The damaged rate of stress is dependent on the stress, the
rate of damage, as well as the damage itself. A comprehensive methodology is presented
in this thesis for obtaining all the damage parameters and material constants present in the
developed CDM model.
The developed Foroutan-Names damage model is implemented into a VUMAT Material
User Subroutine of the finite element program ABAQUS/Explicit. Uniaxial tension and
shear simulations are performed on the three materials at different loading rates. It is
shown that the model simulates well the dynamic and the static response. Simulation
results for higher than tested strain rates are also presented to illustrate the response at
these high rates. It is shown that the tensile strength increases with the increase in the
loading rate, and also a strain softening phenomenon is observed after the maximum
tensile strength. The loading-unloading, as well as compression response of the materials
are also simulated to ensure the capability of the model to capture these responses. This is
165
CONCLUSIONS
essential as during a typical dynamic impact, a part of the impacted material can go under
compressive loading after initially experiencing a tensile loading.
7.2
Originality and contribution to knowledge
In this work the following can be considered as a contribution to the literature:
1. A modified continuum damage mechanics model that includes shear damage is
developed for woven composite materials. The model simulates the non-linear,
rate dependant behaviour due to damage evolution. The model is implemented
into a material user subroutine of a finite element program, and the simulation
results are presented in this work.
2. A comprehensive simple methodology is established for determination of damage
parameters and the material constants present in the developed model. This
methodology simplifies the procedure for obtaining all constants present in the
model using tensile and bias extension shear experiments.
3. Characterization of the response of three differently woven composite materials,
all of which use carbon fabric as the constituent. The comparison between the
static and dynamic results obtained from both uniaxial tension and bias extension
shear experiments are also presented in this work. The stress-strain results clearly
show strain rate sensitivity in these materials.
4. The tensile version of Hopkinson bar setup, in conjunction with the designed
specimen and specimen fixture, is shown to be suitable for testing laminated
woven composite materials in uniaxial tension and bias extension shear at high
rates of strain.
5. With the increase in the use of the woven composite materials in the aerospace
industry, the developed model can be used with the ones dealing with the interlaminar failure known as delamination to simulate the response of the material
under impact.
6. The following publications resulted from the current work:
a- Foroutan, R., and Nemes, J.A., (2009) High Strain Rate Characterization of
Woven Carbon/Epoxy Composites. In: American Society for Composites and
166
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
The Canadian Association for Composite Structures and Materials, 24th ASC
Conference 2009, Newark, Delaware, United states.
b- Foroutan,
R.,
and
Nemes,
J.A.,
(2008)
Damage
Modeling
and
Characterization of Woven Composite Materials, May 2008, (Technical
Report-8)
c- Foroutan, R., and Nemes, J.A., (2007b) Woven Composite Damage Model,
November 2007, (Technical Report-7)
d- Foroutan, R., and Nemes, J.A., (2007a) Instrumented Tensile Tests, May,
2007, (Technical Report-6)
e- Foroutan, R., and Nemes, J.A.,(2006a) Material Modeling and Instrumented
Tensile Tests, November 2006, (Technical Report-5)
f- Foroutan, R., and Nemes, J.A., (2006b) Tensile Tests II, April 2006,
(Technical Report-4)
g- Foroutan, R., and Nemes, J.A., (2005b) Tensile Tests I, November 2005,
(Technical Report-3)
h- Foroutan, R., and Nemes, J.A., (2005a) Development of Tensile Test
Procedures, April 2005 (Technical Report-2)
i- Foroutan, R., and Nemes, J.A., (2004) A Review on Impact Modeling of
Composite Structures, October 2004, (Technical Report-1)
7.3
Recommended future work
The following can be considered as potential future tasks that can be done to enhance the
prediction of the response of high strain rate materials subjected to dynamic loading:
1. The experiments in this study were mainly conducted to determine the in-plane
behaviour of the woven composite material. In order to explore the concept of
delamination, it is suggested to perform Mode I and Mode II fracture tests, to
determine the inter-laminar behaviour of these materials.
2. The developed model can be combined with theory behind cohesive elements
available in finite element codes to simulate the three-dimensional response of the
167
CONCLUSIONS
material under impact. With the help of cohesive elements the inter-laminar
behaviour of the material can be included in the simulations.
3. In this study, the focus of the thesis is on the high strain rate tensile behaviour of
woven composite materials and it is assumed that the behaviour is similar in
compressive loading. It would be interesting to perform compressive high strain
rate experiments in order to compare the two responses.
4. At the moment, the Foroutan-Nemes model does not include permanent
deformation, so in case of load reversal, the damage stays constant, however the
material returns to zero stress at zero applied strain. It would be interesting to
investigate the presence of permanent strain in the response of these materials in
case of load reversal.
5. The developed model is shown to be very effective for predicting rate-dependant
response of the woven composite materials subjected to tension and bias extension
shear. The next interesting step is to validate the developed model which includes
the following two steps. First a high strain rate loading experiment like drop tower
experiment, punch experiment using Hopkinson bar theory, or ballistic impact
experiment must be performed on woven composite panels. Then, the performed
experiment must be simulated using the developed Foroutan-Nemes damage
model combined with the cohesive elements theory. A good agreement between
the simulation and the test data validates the model.
6. Finally, the validated material model should be checked for things like mesh
sensitivity and various loading combinations. It can then be used to simulate
different structural analysis like a bird impact on a woven composite wing leading
edge or the impact of a failed engine blade on a woven composite engine casing.
168
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
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182
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
APPENDIX A: SUBROUTINE FOR
ABAQUS/EXPLICIT
C USER SUBROUTINE VUMAT
subroutine vumat(
C Read only 1 nblock, ndir, nshr, nstatev, nfieldv, nprops, lanneal,
2 stepTime, totalTime, dt, cmname, coordMp, charLength,
3 props, density, strainInc, relSpinInc,
4 tempOld, stretchOld, defgradOld, fieldOld,
3 stressOld, stateOld, enerInternOld, enerInelasOld,
6 tempNew, stretchNew, defgradNew, fieldNew,
C Write only 5 stressNew, stateNew, enerInternNew, enerInelasNew )
C
include 'vaba_param.inc'
C
C All arrays dimensioned by (*) are not used in this algorithm
dimension props(nprops), density(nblock),
1 coordMp(nblock,*),
2 charLength(*), strainInc(nblock,ndir+nshr),
3 relSpinInc(*), tempOld(*),
4 stretchOld(*), defgradOld(*),
5 fieldOld(*), stressOld(nblock,ndir+nshr),
6 stateOld(nblock,nstatev), enerInternOld(nblock),
7 enerInelasOld(nblock), tempNew(*),
8 stretchNew(*), defgradNew(*), fieldNew(*),
183
ABAQUS VUMAT SUBROUTINE
9 stressNew(nblock,ndir+nshr), stateNew(nblock,nstatev),
1 enerInternNew(nblock), enerInelasNew(nblock)
C
character*80 cmname
CHARACTER*100 WRKDIR, FILE5
C
Sig0H1 = props(1)
Sig0H4 = props(2)
E0H11 = props(3)
E0H44 = props(4)
xnu = props(5)
C1
= props(6)
C2
= props(7)
C3
= props(8)
C4
= props(9)
C5
= Props(10)
C6
= Props(11)
alpha = props(12)
alpha11= props(13)
alpha41= props(14)
W11 = props(15)
W12 = props(16)
W13 = props(17)
W41 = props(18)
W42 = props(19)
A11 = props(20)
A12 = props(21)
A13 = props(22)
A14 = props(23)
A41 = props(24)
A42 = props(25)
184
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
A43 = props(26)
A44 = props(27)
B11 = props(28)
B12 = props(29)
B13 = props(30)
B14 = props(31)
B41 = props(32)
B42 = props(33)
B43 = props(34)
B44 = props(35)
C
WRKDIR='C:\abq test'
FILE5='R1.TXT'
OPEN(91,DEFAULTFILE=WRKDIR,FILE=FILE5,STATUS='UNKNOWN')
C
C Parameters
C
txnu = 1
C
IF (StepTime.EQ.0) Then
C
do I = 1,nblock
C
E11 = E0H11 *(C2* (1 + C1 * abs(strainInc(i,1)/dt) )**C3)
E22 = E0H11 *(C2* (1 + C1 * abs(strainInc(i,2)/dt) )**C3)
E44 = E0H44 *(C4* (1 + C1 * abs(strainInc(i,4)/dt) )**C5)
stressNew(i,1) = sig1 + E11*(strainInc(i,1)+ xnu*strainInc(i,2))
stressNew(i,2) = sig2 + E22*(strainInc(i,2)+ xnu*strainInc(i,1))
stressNew(i,4) = sig4 + E44* strainInc(i,4)
C
C
185
ABAQUS VUMAT SUBROUTINE
End Do
C
Else
C
do 100 i = 1,nblock
C
C
State variables
C
d11 = StateOld(i,1)
d12 = StateOld(i,2)
d21 = StateOld(i,3)
d22 = StateOld(i,4)
d41 = StateOld(i,5)
d42 = StateOld(i,6)
E11 = StateOld(i,7)
E22 = StateOld(i,8)
E44 = StateOld(i,9)
C
C
Strains
C
strain11 = StateOld(i,10) + strainInc(i,1)
strain22 = StateOld(i,11) + strainInc(i,2)
strain33 = - xnu * (strain11 + strain22 )
strain44 = StateOld(i,12) + strainInc(i,4)
C
F1 = StateOld(i,13)
F2 = StateOld(i,14)
F4 = StateOld(i,15)
C
E011 = StateOld(i,16)
E022 = StateOld(i,17)
186
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
E044 = StateOld(i,18)
C
dout = 1
C
C
Trial stress
sig1 = stressOld(i,1)
sig2 = stressOld(i,2)
sig4 = stressOld(i,4)
C
C
Delta strain Rate (DSR)
C
DSR1 = strainInc(i,1)/dt
DSR2 = strainInc(i,2)/dt
DSR4 = strainInc(i,4)/dt
DSR = max(abs(DSR1),abs(DSR2), abs(DSR4))
C
C
Damage rate constant (DRC) is for damage non-heeling property
C
If (strain11.GE.0) then
If (DSR1.GE.0) Then
DRC1 = 1
Else
DRC1 = 0
End If
Else
If (DSR1.LE.0) Then
DRC1 = 1
Else
DRC1 = 0
End If
End If
187
ABAQUS VUMAT SUBROUTINE
C
If (strain22.GE.0) then
If (DSR2.GE.0) Then
DRC2 = 1
Else
DRC2 = 0
End If
Else
If (DSR2.LE.0) Then
DRC2 = 1
Else
DRC2 = 0
End If
End If
C
If (strain44.GE.0) then
If (DSR4.GE.0) Then
DRC4 = 1
Else
DRC4 = 0
End If
Else
If (DSR4.LE.0) Then
DRC4 = 1
Else
DRC4 = 0
End If
End If
C
C2
E011
= E0H11 *(C2* (1 + C1 * DSR )**C3)
C2
E022
= E0H11 *(C2* (1 + C1 * DSR )**C3)
188
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
C2
E044
= E0H44 *(C4* (1 + C1 * DSR )**C5)
C
Sig01 = Sig0H1
Sig02 = Sig0H1
Sig04 = Sig0H4
C
Z11
= ( (W11 * abs(DSR1))**W12 )
Z12
= ( abs(Z11)**W13 )
Z4
= ( (W41 * abs(DSR4))**W42 )
C
If (abs(sig1).LT.sig01.and.F1.EQ.0) Then
C
E011 = E0H11 *(C2* (1 + C1 * abs(DSR1) )**C3)
E11
= E011
sig1dE = E11/txnu *(strainInc(i,1)+ xnu*strainInc(i,2))
sig1d = sig1dE
go to 20
End If
C
If (abs(sig1).GE.sig01.OR.F1.NE.0) Then
C
F1 = 1
C
If (d11.LT.0.9999) Then
C
d11d = max ( 0.000 , dt * Z11 * (A11 + A12 * d11 )*
1
sqrt(abs(sig1) / sig01 / (1 - alpha11 * d11)) * DRC1)
d11 = min(d11 + d11d, 0.9999)
d12d = 0
d12 = d12 + d12d
C
189
ABAQUS VUMAT SUBROUTINE
E11d = - alpha11 * E011 / 2 * d11d *
1
sqrt (1/(1- alpha11 * d11 ))
C
sig1dE = E11/txnu*(strainInc(i,1)+ xnu* strainInc(i,2))
sig1dD = - d11d* (B11 + B12 * sig1)/(1 - alpha11 * d11)
sig1d = sig1dE + sig1dD
C
If (strain11.GT.0) Then
If (DSR1.GE.0) Then
E11 = E11 + E11d
Else
E11 = (sig1 + sig1d)/strain11
End If
Else
If (DSR1.LE.0) Then
E11 = E11 + E11d
Else
E11 = (sig1 + sig1d)/strain11
End If
EndIf
C
go to 20
End If
C
If (d11.EQ.0.9999) Then
C
d11d = 0
d11 = 0.9999
d12d = max ( 0.0000, dt * Z12 * (A13 * d11 + A14 * d12)*
1
sqrt( abs(sig1) / sig01 / (1 - alpha11 * d11) /
1
(1-alpha*d12)) * DRC1)
190
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
d12 = min (d12 + d12d , 0.9999)
C
E11d = - alpha * E011 / 2 * d12d *
1
sqrt ((1-alpha11 * d11)/(1-alpha*d12))
C
sig1dE = E11/txnu *(strainInc(i,1)+ xnu* strainInc(i,2))
sig1dD = - d12d *( B13 + B14 * sig1)/(1-alpha*d12)
sig1d = sig1dE + sig1dD
C
If (strain11.GT.0) Then
If (DSR1.GE.0) Then
E11 = E11 + E11d
Else
E11 = (sig1 + sig1d)/strain11
End If
Else
If (DSR1.LE.0) Then
E11 = E11 + E11d
Else
E11 = (sig1 + sig1d)/strain11
End If
EndIf
C
go to 20
C
End If
C
End If
C
20
Continue
C
191
ABAQUS VUMAT SUBROUTINE
If (abs(sig2).LT.sig02.and.F2.EQ.0) Then
C
E022 = E0H11 *(C2* (1 + C1 * abs(DSR2) )**C3)
E22
= E022
sig2dE = E22/txnu *(strainInc(i,2)+ xnu*strainInc(i,1))
sig2d = sig2dE
go to 30
End If
C
If (abs(sig2).GE.sig02.OR.F2.NE.0) Then
C
F2 = 1
C
If (d21.LT.0.9999) Then
C
d21d = max ( 0.000 , dt * Z11 * (A11 + A12 * d21 )*
1
sqrt(abs(sig2) / sig02 / (1 - alpha11 * d21)) * DRC2)
d21 = min(d21 + d21d, 0.9999)
d22d = 0
d22 = d22 + d22d
C
E22d = - alpha11 * E022 / 2 * d21d *
1
sqrt (1/(1- alpha11 * d21 ))
C
sig2dE = E22/txnu*(strainInc(i,2)+ xnu* strainInc(i,1))
sig2dD = - d21d* (B11 + B12 * sig2)/(1 - alpha11 * d21)
sig2d = sig2dE + sig2dD
C
If (strain22.GT.0) Then
If (DSR2.GE.0) Then
E22 = E22 + E22d
192
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
Else
E22 = (sig2 + sig2d)/strain22
End If
Else
If (DSR2.LE.0) Then
E22 = E22 + E22d
Else
E22 = (sig2 + sig2d)/strain22
End If
EndIf
C
go to 30
End If
C
If (d21.EQ.0.9999) Then
C
d21d = 0
d21 = 0.9999
d22d = max ( 0.0000, dt * Z12 * (A13 * d21 + A14 * d22)*
1
sqrt( abs(sig2) / sig02 / (1 - alpha11 * d21) /
1
(1-alpha*d22)) * DRC2)
d22 = min (d22 + d22d , 0.9999)
C
E22d = - alpha * E022 / 2 * d22d *
1
sqrt ((1-alpha11 * d21)/(1-alpha*d22))
C
sig2dE = E22/txnu *(strainInc(i,2)+ xnu* strainInc(i,1))
sig2dD = - d22d *( B13 + B14 * sig2)/(1-alpha*d22)
sig2d = sig2dE + sig2dD
C
If (strain22.GT.0) Then
193
ABAQUS VUMAT SUBROUTINE
If (DSR2.GE.0) Then
E22 = E22 + E22d
Else
E22 = (sig2 + sig2d)/strain22
End If
Else
If (DSR2.LE.0) Then
E22 = E22 + E22d
Else
E22 = (sig2 + sig2d)/strain22
End If
EndIf
C
go to 30
C
End If
C
End If
C
30
Continue
C
If (abs(sig4).LT.sig04.and.F4.EQ.0) Then
C
E044 = E0H44 *(C4* (1 + C1 * abs(DSR4) )**C5)
E44
= E044
sig4dE = E44 * strainInc(i,4)
sig4d = sig4dE
go to 40
End If
C
If (abs(sig4).GE.sig04.OR.F4.NE.0) Then
194
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
C
F4 = 1
C
If (d41.LT.0.9999) Then
C
d41d = max ( 0.000 , dt * Z4 * (A41 + A42 * d41 )*
1
sqrt(abs(sig4) / sig04 / (1 - alpha41 * d41)) * DRC4)
d41 = min(d41 + d41d, 0.9999)
d42d = 0
d42 = d42 + d42d
C
E44d = - alpha41 * E044 / 2 * d41d *
1
sqrt (1/(1- alpha41 * d41 ))
C
sig4dE = E44 * strainInc(i,4)
sig4dD = - d41d* (B41 + B42 * sig4)/(1 - alpha41 * d41)
sig4d = sig4dE + sig4dD
C
If (strain44.GT.0) Then
If (DSR4.GE.0) Then
E44 = E44 + E44d
Else
E44 = (sig4 + sig4d)/strain44
End If
Else
If (DSR4.LE.0) Then
E44 = E44 + E44d
Else
E44 = (sig4 + sig4d)/strain44
End If
EndIf
195
ABAQUS VUMAT SUBROUTINE
C
go to 40
End If
C
If (d41.EQ.0.9999) Then
C
d41d = 0
d41 = 0.9999
d42d = max ( 0.0000, dt * Z4 * (A43 * d41 + A44 * d42)*
1
sqrt( abs(sig4) / sig04 / (1 - alpha41 * d41) /
1
(1-alpha*d42)) * DRC4)
d42 = min(d42 + d42d, 0.9999)
C
E44d = - alpha * E044 / 2 * d42d *
1
sqrt ((1-alpha41 * d41)/(1-alpha*d42))
C
sig4dE = E44 * strainInc(i,4)
sig4dD = - d42d *( B43 + B44 * sig4)/(1-alpha*d42)
sig4d = sig4dE + sig4dD
C
If (strain44.GT.0) Then
If (DSR4.GE.0) Then
E44 = E44 + E44d
Else
E44 = (sig4 + sig4d)/strain44
End If
Else
If (DSR4.LE.0) Then
E44 = E44 + E44d
Else
E44 = (sig4 + sig4d)/strain44
196
HIGH STRAIN RATE BEHAVIOUR OF WOVEN COMPOSITE MATERIALS
End If
EndIf
C
go to 40
C
End If
C
End If
C
40 Continue
C
If(d12.GT.0.999) go to 60
If(d22.GT.0.999) go to 60
If(d42.GT.0.999) go to 60
C
StateNew(i,19) = StateOld(i,19)
C
Go to 70
C
60 dout = 0
Strain11 = strain11 - StrainInc(i,1)
Strain22 = strain22 - StrainInc(i,2)
Strain33 = - xnu * (strain11 + strain22 )
Strain44 = strain44 - StrainInc(i,4)
C
C Update the stress
C
70 Continue
C
stressNew(i,1) = (sig1 + sig1d) * dout
stressNew(i,2) = (sig2 + sig2d) * dout
197
ABAQUS VUMAT SUBROUTINE
stressNew(i,4) = (sig4 + sig4d) * dout
C
C Update the state variables
StateNew(i,1) = d11
StateNew(i,2) = d12
StateNew(i,3) = d21
StateNew(i,4) = d22
StateNew(i,5) = d41
StateNew(i,6) = d42
StateNew(i,7) = E11
StateNew(i,8) = E22
StateNew(i,9) = E44
StateNew(i,10) = strain11
StateNew(i,11) = strain22
StateNew(i,12) = strain44
StateNew(i,13) = F1
StateNew(i,14) = F2
StateNew(i,15) = F4
StateNew(i,16) = E011
StateNew(i,17) = E022
StateNew(i,18) = E044
StateOld(i,19) = dout
C
WRITE(91,*) nblock
C
100 continue
C
End If
C
return
end
198