COMPUTATIONAL MODELLING OF PROGRESSIVE DAMAGE
IN VISCOELASTIC-VISCOPLASTIC COMPOSITE MATERIALS
by
Thomas John Berton
A thesis submitted in conformity with the requirements
for the degree of
Doctor of Philosophy
Department of Materials Science and Engineering
University of Toronto
© Copyright by Thomas John Berton (2019)
ii
Computational modelling of progressive damage in viscoelastic-viscoplastic composite materials
Thomas John Berton
Doctor of Philosophy
Department of Materials Science and Engineering
University of Toronto
2019
ABSTRACT
Current governmental requirements for greater fuel efficiency are pushing automotive
manufacturers to find innovative ways of reducing car weight, while still maintaining mechanical
performance to ensure passenger safety. Composite materials would allow for significant weight
reduction, while still maintaining a safe environment for the passengers. However, composite
materials are known to exhibit significant time-dependent behaviour, as well as progressive damage
which affect structural performance. Bio-composites in particular are known to undergo complex
damage evolution during their lifetime. In this thesis, a multi-scale computational approach has been
adopted to take these effects into account.
A new modelling approach based on Synergistic Damage Mechanics to understand damage
evolution in composite materials undergoing time-dependent deformation is developed. The model is
applied to matrix micro-cracking in laminates. Computational micro-damage mechanics is combined
with a continuum level description of stiffness degradation to predict the evolution of micro-cracking
while requiring minimal experimental calibration. The time-dependent behaviour is modelled using
Schapery’s theory of viscoelasticity and viscoplasticity, and the evolution of matrix micro-cracking
during creep is predicted. The predictions of the model for different stacking sequences, ply
thicknesses and working temperatures show that viscoelasticity and viscoplasticity have a significant
effect on the long-term response of laminates undergoing matrix micro-cracking.
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The multi-scale modelling approach is extended to short fiber bio-composites by constructing a
micro-damage model of fiber-matrix debonding using a Cohesive Zone Model. The effect of strain
rate on fiber-matrix debonding is analyzed by implementing a non-linear viscoelastic model for the
matrix, and it is shown that viscoelastic behaviour leads to a competition between void nucleation and
debonding. The effects of fiber stiffness, dimensions and interfacial shear strength are evaluated.
Following the micro-damage analyses, the thesis focuses on damage evolution at the structural
scale by implementing the damage model in Finite Element Analysis software to analyze the dynamic
response of a car bumper. The rate-dependent evolution of damage is predicted for different stacking
sequences and material systems.
iv
To my parents, and brothers
v
ACKNOWELDGEMENTS
Many people have helped me in different ways through this challenging research journey. First and
foremost, I wish to thank my parents and my brothers for their unwavering emotional support during the
past five years.
I wish to thank my supervisor, Prof. Chandra Veer Singh, for his insightful guidance throughout my
entire research work. His vision for the potential of computational modelling of damage in composite
materials has allowed me to learn and make contributions to this field that I never thought would be
possible at the start of my PhD. I also would like to acknowledge my other PhD committee members,
Prof. Hani Naguib and Prof. Mohini Sain, who provided me with valuable support and advice throughout
my studies. Their encouraging attitude and willingness to review my work in detail on a regular basis have
allowed me to finish my thesis in the best conditions possible. I also am grateful for the support of my
external reviewer, Prof. Xinran Xiao, as well as Prof. Ben Hatton and Prof. Yu Zou who provided excellent
feedback to improve my final thesis.
I wish to thank all the members of the Computational Materials Engineering lab at the University of
Toronto, as well as the members of the Centre for Bio-composites and Bio-materials Processing. I have
felt at home in this research environment, and much of my ability to conduct research for the past five
years was made possible by the support of these two research groups.
I received technical help from several people. In particular, I want to thank Prof. John Montesano,
Yangjie, Dr. Sandip Haldar and Farzin for their technical support. I have learned a tremendous amount
about finite element modelling and solid mechanics from them, and they made my PhD research possible.
Finally, I want to thank my friends: Sami, Jean-Nicolas, Mark, Matt, Mireille, Thomas, Mohit,
Shwetank, Kulbir, Nav, Sean, Ashok, Gurjot, Meysam, and Hao. We supported each other through the
difficulties all of us had to face at some point, and cheered each other on through the many successes and
victories. They made Toronto my home.
vi
TABLE OF CONTENTS
ABSTRACT ......................................................................................................................................ii
Dedication ..................................................................................................................................... iv
ACKNOWELDGEMENTS ......................................................................................................................... v
Chapter 1. Introduction .................................................................................................................. 1
1.1 Lightweighting in the transportation sector ................................................................................. 1
1.2 Modelling the mechanical performance of composites ............................................................... 5
1.3 Thesis objectives .......................................................................................................................... 6
Chapter 2. Literature review .......................................................................................................... 9
2.1 Composite materials .................................................................................................................... 9
2.2 Damage mechanisms in composites .......................................................................................... 10
2.2.1 Matrix micro-cracking in laminates ....................................................................................... 11
2.2.2 Damage mechanisms in short fiber composites ..................................................................... 13
2.2.3 Interaction between viscoelasticity, viscoplasticity and damage evolution in composites .... 14
2.3 Models to predict matrix micro-cracking in laminates .............................................................. 16
2.4 Models to predict the combined effects of damage, viscoelasticity and viscoplasticity in
composites ........................................................................................................................................... 20
2.5 Structural analysis of composite structures under low-velocity impact..................................... 22
2.6 Micro-mechanical models to predict damage evolution in short fiber composites.................... 24
2.7 Gaps in the literature .................................................................................................................. 26
2.8 Thesis methodology ................................................................................................................... 26
Chapter 3. Methodology .............................................................................................................. 28
3.1 Prediction of damage in viscoelastic-viscoplastic laminates under multi-axial creep loading .. 28
3.1.1 Synergistic Damage Mechanics (SDM) model ...................................................................... 29
3.1.2 FE micro-damage model ........................................................................................................ 30
3.1.3 Energy-based crack multiplication model .............................................................................. 31
3.1.4 Viscoelasticity and viscoplasticity models ............................................................................. 33
3.1.5 Combined viscoelastic-viscoplastic-damage model ............................................................... 34
3.1.6 Numerical implementation ..................................................................................................... 35
3.1.7 Detailed implementation of the model into MATLAB .......................................................... 36
3.1.8 Development of an RVE to evaluate the effect of damage on laminate stiffness .................. 44
3.1.9 Assumptions of the model and range of applicability ............................................................ 47
3.2 Micro-damage model for a short fibre composite ...................................................................... 48
3.2.1 Geometry ................................................................................................................................ 48
3.2.2 Material models ...................................................................................................................... 48
3.2.3 Validation of the implementation ........................................................................................... 52
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3.3 Structural analysis of a composite car bumper .......................................................................... 54
Chapter 4. Development of an SDM-based viscoelastic creep damage model ............................ 57
4.1 Multidirectional laminates undergoing cracking in multiple plies. ........................................... 57
4.2 Cross-ply laminates undergoing cracking in transverse plies. ................................................... 61
4.3 Predictions for [±60/902]s and [±75/902]s CFRP laminates. ....................................................... 64
Chapter 5. Time-dependent bi-axial stiffness degradation envelopes for structural composites . 70
5.1 Model validation ........................................................................................................................ 70
5.1.1 Validation of the viscoelastic-viscoplastic implementation ..................................................... 71
5.1.2 Validation of the crack multiplication model ........................................................................... 73
5.1.3 Validation of the SDM model .................................................................................................. 73
5.2 Model predictions ...................................................................................................................... 74
5.2.1 GFRP ........................................................................................................................................ 74
5.2.2 CFRP ........................................................................................................................................ 79
5.2.3 Effect of temperature on the stiffness degradation envelopes .................................................. 81
5.3 Discussion ................................................................................................................................... 83
Chapter 6. Strain rate-dependent damage model for a short fiber composite .............................. 87
6.1 Model validation ......................................................................................................................... 87
6.2 Effect of strain rate on damage evolution ................................................................................... 87
Chapter 7. Structural analysis using a rate-dependent SDM model ........................................... 102
7.1 Implementation validation ........................................................................................................ 102
7.2 Model validation ....................................................................................................................... 103
7.3 Parametric studies ..................................................................................................................... 104
Chapter 8. Conclusions and future work .................................................................................... 115
8.1. Conclusions .............................................................................................................................. 115
8.2 Future work ............................................................................................................................... 117
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LIST OF TABLES
Table 3-1. Elastic properties of the constituents of the RVE. .................................................................... 51
Table 3-2. Parameters of the Cohesive Zone Model used to describe the fiber-matrix interface. ............. 51
Table 4-1. Elastic ply properties for the [±45/902]s CFRP laminate studied by Asadi [26] ....................... 58
Table 4-2. Parameters for the non-linear creep model described by Eq. (3.14), which were obtained using
the experimental data in Asadi [26]. .......................................................................................................... 58
Table 4-3. Parameters used for the creep compliance model, Eq. (3.14), as obtained by fitting the
experimental creep curve for 90° unidirectional plies, at a stress level of 510 MPa [121] ........................ 63
Table 5-1. Parameters for the viscoelastic part of the model [187] ........................................................... 72
Table 5-2. Elastic properties of the GFRP plies, obtained from [187]. ...................................................... 72
Table 5-3. Damage model parameters for the glass fiber/epoxy composite .............................................. 73
Table 5-4. Parameters for the viscoelastic part of the model [24] ............................................................. 80
Table 5-5. Elastic properties of the plies, measured at 93 °C, obtained from Tuttle et al [24] .................. 80
Table 5-6. Parameters for the viscoplastic part of the model [24] ............................................................. 80
ix
Table 5-7. Damage model parameters for IM7/5260 ................................................................................. 80
Table 6-1. Fiber dimensions studied in this thesis ..................................................................................... 95
Table 7-1. Material properties of CFRP and GFRP unidirectional lamina .............................................. 106
Table 7-2. Maximum degradation of material properties of laminates with different stacking sequences due
to low-velocity impact. ............................................................................................................................. 110
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LIST OF FIGURES
Figure 1-1. U.S. greenhouse gas emissions breakdown by economic sector [10] ---------------------- 2
Figure 1-2. Proportion of materials, by weight, making up the structure of the modern car[19] ------------ 3
Figure 1-3. CO2 emissions of natural (NMT) and glass fibre (GMT) composites over their lifetime [16].
From “Biocomposites reinforced with natural fibers: 2000-2010” by Faruk O. et al., 2010 Prog. Polym.
Sci., Vol. 37, p. 1570. Copyright (2012) by Elsevier. Reprinted with permission. ----------------------------- 4
Figure 1-4. Market growth for natural fibre composites [21] ------------------------------------------------------- 5
Figure 1-5. Improvement of NF-DLFT hybrid bio-composite impact strength with glass fiber hybridization
[40]. From “Injection-Molded Short Hemp Fiber / Glass Fiber- Reinforced Polypropylene Hybrid
Composites — Mechanical , Water Absorption and Thermal Properties”, by Panthapulakkal S. and Sain
M., 2006 J. Appl. Polym. Sci., Vol. 103, p. 2438. Copyright (2006) by Wiley Periodical, Inc. Reprinted
with permission. ------------------------------------------------------------------------------------------------------------- 6
Figure 1-6. Creep behavior of NF-DLFT for different amounts of glass and hemp fibres[39]. -------------- 8
Figure 2-1. Schematic explaining the structure of a multi-directional laminate [44]. ------------------------- 10
Figure 2-2. Schematic explaining the difference between a short fiber composite and a laminate [43]. -- 10
Figure 2-3. Schematic showing matrix micro-cracking in a cross-ply laminate [47]. ------------------------ 11
xi
Figure 2-4. Crack density evolution versus applied stress in the 90° ply of a glass/epoxy cross-ply. From
“On matrix crack growth in quasi-isotropic laminates - I. Experimental investigation”, by Tong J. et al,
1997 Compos. Sci. Technol., Vol. 57, p. 1529. Copyright (1997) by Elsevier. Reprinted with permission.
--------------------------------------------------------------------------------------------------------------------------------- 13
Figure 2-5. Damage mechanisms in a short fiber composite. (a) Fiber-matrix debonding mechanism [66].
(b) Fiber pull-out [67]. (c) Fiber fracture [68]. (d) Matrix micro-cracking initiating from fiber tips [61].
Last figure from “Microfailure behaviour of randomly dispersed short fiber reinforced thermoplastic
composites obtained by direct SEM observation.”, by Sato N. et al, 1991 J. Mater. Sci., Vol. 26, p. 3892.
Copyright (1991) by Chapman and Hall Ltd. Reprinted with permission. -------------------------------------- 14
Figure 2-6. Effect of multiple creep-recovery tests on the time-dependent strain response of a bio-
composite material. From “Characterization and modeling of performance of Polymer Composites
Reinforced with Highly Non-Linear Cellulosic Fibers” by Rozite L. et al, 2012 IOP Conf. Ser. Mater. Sci.
Eng. , Vol. 31, p. 4. Copyright (2012) by IOP Publishing Ltd. Reprinted with permission. ---------------- 15
Figure 2-7. Evolution of damage quantified at the microscopic level (a) and at the macroscopic level (b),
for a sheet molding compound versus applied strain for different strain rates. From “Multi-scales
modelling of dynamic behaviour for discontinuous fibre SMC composites” by Jendli et al., 2009 Compos.
Sci. Technol., Vol. 69, p. 101. Copyright (2008) by Elsevier. Reprinted with permission.------------------ 16
Figure 3-1. Schematic showing the different modes of ply micro-cracking in a multi-directional laminate.
VE refers to the viscoelastic properties of the plies; VP refers to the viscoplastic behaviour; QS is for quasi-
static loading. The top right graph shows the loading scheme used in the current thesis. -------------------- 29
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Figure 3-2. Flowchart explaining the MATLAB program used to performed the progressive damage
simulations under viscoelastic-viscoplastic creep. ------------------------------------------------------------------- 36
Figure 3-3. (a) SDM micro-damage FE model for a [0/90/∓45]s quasi-isotropic laminate, with micro-
cracks in the 0, 90 and -45 plies and periodic in-plane boundaries. (b) Stress contour (𝜎𝑦𝑦) in the RVE
under uniaxial strain (휀𝑥𝑥 = 0.5%, 휀𝑦𝑦 = 0, 𝛾𝑥𝑦 = 0) with a single crack in the 90° ply. ----------------- 45
Figure 3-4 . Model development for the short fiber RVE. (a) Short fiber RVE used in this thesis to study
micro-damage mechanisms in short fiber composites. (b) Top face of the RVE. (c) Side face of the RVE.
Image from “Effect of the matrix behavior on the damage of ethylene-propylene glass fiber reinforced
composite subjected to high strain rate tension”, by Fitoussi J. et al, 2013 Composites Part B, Vol. 45(1),
p. 1183. Copyright (2012) by Elsevier. Reprinted with permission. --------------------------------------------- 50
Figure 3-5. FE model of the short fiber composite. ------------------------------------------------------------------ 50
Figure 3-6. (a) Validation of the implementation of the non-linear viscoelasticity model for HDPE into
Abaqus in tabular form. (b) Yield stress versus applied strain rate. --------------------------------------------- 52
Figure 3-7. Effect of interface stiffness on composite RVE modulus -------------------------------------------- 53
Figure 3-8. (a) Effect of mesh size on stress-strain response of composite RVE with linear elastic
components. The legend refers to the element size in the central region of the RVE around the fiber tips ;
when the mesh size was constant throughout the RVE, the term ‘uniform’ is used. (b) Effect of mesh size
on stress-strain response of composite RVE with a non-linear viscoelastic matrix under a strain rate of 10-5
s-1. ------------------------------------------------------------------------------------------------------------------------------ 54
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Figure 3-9. Synergistic Damage Mechanics (SDM) RVE, showing the main features of the model,
including uniform ply crack spacing, cracks parallel to the fibers in each layer, cracking in multiple off-
axis orientations and multi-axial loading scenarios. ----------------------------------------------------------------- 56
Figure 3-10. General framework for the numerical implementation of the hierarchical multi-scale
methodology. ---------------------------------------------------------------------------------------------------------------- 56
Figure 4-1. SDM model results and comparison to experimental data [26] for a quasi-static test performed
on a [±45/902]s carbon-fibre/epoxy laminate: a) crack density evolution in different plies; b) stiffness
degradation; c) stress-strain curve. The SDM model used here does not include viscoelastic effects. ---- 59
Figure 4-2. Results of the extended SDM model for creep strain (a) and damage evolution in the 90° ply
(b) for a [±45/902]s laminate subjected to a constant creep stress of 45 MPa, and comparison to
experimental data [32]. Note that the initial experimental creep strain in a) at time t = 0 s was adjusted so
that the predicted results and the experimental data coincided at the onset of creep. ------------------------- 60
Figure 4-3. Predicted crack density evolution for a [03/903]s cross-ply laminate for different levels of strain
rate (low, medium and high) using the SDM methodology, along with experimental data from Nguyen and
Gamby [89]. ----------------------------------------------------------------------------------------------------------------- 62
Figure 4-4. Predicted and experimental crack density evolution during quasi-static loading for a [0/903]s
cross-ply laminate; the prediction assumes no viscoelastic effects. ---------------------------------------------- 62
Figure 4-5. a) Predicted and experimental [121] 90° ply crack density evolution for a [0/903]s cross-ply
laminate subjected to a constant stress of 530 MPa and b) corresponding laminate creep strain. ---------- 64
xiv
Figure 4-6. Predicted creep strain evolution for [±60/902]s and [±75/902]s CFRP laminates, with and
without the effects of damage, subjected to a constant creep stress of 19 MPa. ------------------------------- 65
Figure 4-7. The predicted evolution of compliance is shown here for the [±75/902]s laminate at a creep
stress of 19 MPa: a) different components of compliance (creep and elastic), b) stiffness degradation with
crack density in the 90° ply, and c) evolution of 90° ply crack density for creep and quasi-static loading.
--------------------------------------------------------------------------------------------------------------------------------- 66
Figure 4-8. Predicted crack density evolution in a [±75/902]s laminate in the angle plies, under creep at 19
MPa and quasi-static loading. The figure shows that cracks evolve more slowly during creep than during
quasi-static loading at low strain levels. ------------------------------------------------------------------------------- 67
Figure 4-9. Predicted COD evolution for a 90° ply in a [±75/902]s laminate undergoing creep at 19 MPa,
showing a clear increase linked to creep strain evolution. The increased in COD affects crack density
evolution through its effect on the energy release rate WI. --------------------------------------------------------- 68
Figure 4-10. a) Plot showing the predicted evolution of the elastic and time-dependent (creep) components
of the [±75/902]s laminate compliance, at two different stress levels, demonstrating the impact of damage
on compliance increase. b) Plot showing the predicted evolution of the different components (creep and
elastic) of the compliance for the two different laminates, at a stress of 19 MPa. ----------------------------- 68
Figure 5-1. Schematic showing the different modes of ply micro-cracking in a multi-directional laminate.
VE refers to the viscoelastic properties of the plies; VP refers to the viscoplastic behaviour; QS is for quasi-
static loading. The top right graph shows the loading scheme used in the current chapter. ----------------- 71
xv
Figure 5-2. Flowchart explaining the MATLAB program used to perform the progressive damage
simulations under viscoelastic-viscoplastic creep. ------------------------------------------------------------------- 72
Figure 5-3. a) Viscoelastic-viscoplastic creep strain prediction for a [±45]s GFRP laminate under a creep
stress of 50 MPa, and comparison to a previous model from the literature [187]. b) Evolution of crack
density under quasi-static loading for a cross-ply GFRP composite, and comparison to experimental data
[188] c) Time-dependent crack density evolution in each layer of a [±45/902]s CFRP laminate predicted by
the model, and comparison to experimental data under a creep stress of 45 MPa [26]. (d) Predictions of
stiffness degradation with respect to crack density in a [0/90/∓45]s GFPR laminate using the SDM model
(Eqs. 3.1-3.3) and independent FEA simulations in APDL -------------------------------------------------------- 75
Figure 5-4. Crack density evolutions versus simulation time for a [0/90/∓45]s GFRP laminate loaded to an
axial load of 210 MPa. a) 90° ply crack density. b) 45° ply crack density. ------------------------------------- 76
Figure 5-5. Crack density versus axial stress with different levels of transverse stress in the 90° and 45°
plies of the quasi-isotropic GFRP [0/90/∓45]s CFRP laminate before (t=0) and after (t= tend) creep
deformation. ----------------------------------------------------------------------------------------------------------------- 76
Figure 5-6. Evolution of crack density and normalized stiffness for the GFRP composite under different
levels of transverse loading: (a) Quasi-isotropic [0/90/∓45]s laminate under an axial load of 150 MPa, and
(b) cross-ply laminate under an axial load of 150 MPa ------------------------------------------------------------- 77
Figure 5-7. Axial modulus degradation versus applied axial stress for (a) [0/90/∓45]s and (b) [0/90]s GFRP
laminates, at the start of the creep tests, and at the end, for different biaxial loads. --------------------------- 78
xvi
Figure 5-8. The lines in the stiffness degradation contour map correspond to constant levels of stiffness
loss after time-dependent deformation under biaxial loading of (a) [0/90/∓45]s and (b) [0/90]s GFRP
laminate. ---------------------------------------------------------------------------------------------------------------------- 79
Figure 5-9. Stiffness degradation contour lines. The lines in the stiffness degradation contour map
correspond to constant levels of stiffness loss after viscoplastic deformation under biaxial loading of (a)
[0/90/∓45]s and (b) [0/90]s CFRP laminate. -------------------------------------------------------------------------- 82
Figure 5-10. Stiffness degradation contour lines for different temperatures. a) Stiffness degradation contour
lines at the end of the creep simulations (T = tend) at two different temperatures for (a) [0/90/∓45]s and (b)
[0/90]s IM7/5260 laminates. This material has a glass transition temperature of 260 °C. ------------------- 83
Figure 5-11. (a) Stress-strain curves for a [0/90/∓45]s GFRP laminate under uniaxial loading using
different material models. (b) Stress-strain curves for a [0/90/∓45]s CFRP laminate. ----------------------- 84
Figure 5-12. (a) Time-dependent evolution of transverse ply stress for the [0/90/∓45]s under uniaxial
constant load of 300 MPa, during ramp-up and subsequent creep deformation. (b) Effect of including
viscoelastic-viscoplastic stress relaxation on crack multiplication predictions for a [0/90/∓45]s GFRP
laminate. ---------------------------------------------------------------------------------------------------------------------- 85
Figure 6-1. (a)Von Mises stress contour for the short fiber composite RVE undergoing longitudinal
deformation with inset showing the stress in the neighborhood of the fiber tips. Stresses are expressed in
MPa. (b) Longitudinal displacement contour plot of the RVE under an applied strain of 0.1 %.
Displacements are expressed in μm. (c) Local damage variable for a fiber Df = 40 μm and Lf = 200 μm.
(d) Global damage variable versus applied strain and secant modulus loss curve ----------------------------- 89
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Figure 6-2. (a) Axisymmetric short glass fiber/epoxy composite RVE used to validate the model. (b)
Displacement response under 1% strain. (c) Debonded area fraction versus applied strain for a short glass
fiber/epoxy composite, and comparison to the work of Pan and Pelegri [200]. -------------------------------- 90
Figure 6-3. Von Mises stress distribution in the short fiber RVE with Df = 40 µm and Lf = 200 µm under
different strain levels with a linear elastic HDPE matrix. ---------------------------------------------------------- 91
Figure 6-4. Von Mises stress distribution in the short fiber RVE with Df = 40 µm and Lf = 200 µm under
different strain levels for a viscoelastic matrix under a strain rate of 10-5 s-1. ---------------------------------- 92
Figure 6-5. (a) Evolution of the damage variable (fraction of debonded area) with respect to applied strain
for different strain rates in a short glass fiber composite with fiber diameter Df = 40 μm. (b) Stress-strain
curves. ------------------------------------------------------------------------------------------------------------------------ 94
Figure 6-6. (a) Energy dissipation in the RVE of a short glass fiber composite (Df =40 μm) for a viscoelastic
matrix deformed at 10-1 s-1. (b) Inelastic strain evolution under different strain rates. ------------------------ 94
Figure 6-7. Evolution of COD under different strain rates for an interfacial crack at the fiber tips in a short
glass fiber composite with Df = 40 μm. -------------------------------------------------------------------------------- 94
Figure 6-8. (a) Shear stress distribution along the fiber matrix interface at two different applied strains, for
a linearly elastic matrix, and a viscoelastic matrix under a strain rate of 10-5 s-1. (b) Axial stress distribution.
--------------------------------------------------------------------------------------------------------------------------------- 96
Figure 6-9. (a) Damage variable versus applied strain for 3 fiber dimensions with a linearly elastic matrix.
(b) Maximum damage reached for 3 fiber dimensions as a function of strain rate. --------------------------- 96
xviii
Figure 6-10. Effect of IFSS on strain energy density at 4 % applied strain: (a) 𝜏𝐼𝐹𝑆𝑆 = 11 𝑀𝑃𝑎, (b) 𝜏𝐼𝐹𝑆𝑆 =
20 𝑀𝑃𝑎, (c) 𝜏𝐼𝐹𝑆𝑆 = 30 𝑀𝑃𝑎 for a matrix loaded under a strain rate of 10-1 s-1. The glass fiber diameter is
20 μm. ------------------------------------------------------------------------------------------------------------------------- 98
Figure 6-11. Evolution of maximum strain energy density versus applied strain for three different values
of the interfacial shear strength. The glass fiber diameter is 20 μm. --------------------------------------------- 98
Figure 6-12. (a) Damage evolution for different types of fibers with different mechanical properties with
a linear elastic matrix. (b) Damage variable at 4% strain as a function of strain rate for different fiber types.
In all cases, the fiber diameter is Df = 20 µm. ------------------------------------------------------------------------ 99
Figure 7-1. Validation of the implementation of the SDM model into Abaqus FEA software using a
VUMAT subroutine, as compared to the MATLAB predictions: (a) Modulus degradation. (b) Crack
multiplication in the 90º ply of the [0/90/∓45]s GFRP composite. --------------------------------------------- 103
Figure 7-2. Effect of strain rate on stiffness degradation as predicted by the rate-dependent SDM model
implemented into MATLAB for the [0/90/∓45]s GFRP composite. ------------------------------------------- 103
Figure 7-3. Comparisons of time histories of contact force for a [0/90]6s composite laminate between
simulation and experiments. -------------------------------------------------------------------------------------------- 104
Figure 7-4. Schematic of an automotive bumper subject to low-velocity impact. (a) an example of the mesh
pattern, (b) geometrical dimensions of the bumper and impactor. ---------------------------------------------- 106
Figure 7-5. Effect of the impactor’s initial velocity on time histories of the contact force for a [0/90/∓45]s
GFRP laminate. ----------------------------------------------------------------------------------------------------------- 107
xix
Figure 7-6. Time history of the evolution of crack density for the different plies of a [0/90/∓45]s GFRP
laminate. Initial velocity of the impactor is equal to : (a) 4.6 m/s, (b) 8.4 m/s, (c) 13.1 m/s. -------------- 107
Figure 7-7. Contour of 0° ply crack density for [0/90/∓45]s GFRP laminate for different initial velocities
of the impactor: (a) v = 4.6 m/s, (b) v = 8.4 m/s, (c) v =13.1 m/s. ---------------------------------------------- 108
Figure 7-8. Effect of laminate stacking sequence on (a) displacement versus contact force and, (b) time
histories of contact force for GFRP laminates under an impactor velocity of 4.6 m/s. --------------------- 108
Figure 7-9. Effect of laminate stacking sequence on (a) displacement versus contact force, and (b) time
histories of contact force for CFRP laminates under an impactor velocity of 4.6 m/s ---------------------- 109
Figure 7-10. Effect of rate-dependency on (a) displacement versus contact force, (b) time histories of
contact force for a CFRP [0/90/∓45]s laminate. -------------------------------------------------------------------- 110
Figure 7-11. Effect of the cross section profile of the bumper on 0º crack density contours for a [0/90/∓60]s
CFRP laminate under an impactor velocity of 8.4 m/s. ----------------------------------------------------------- 111
Figure 7-12. Effect of bumper cross-section shapes on (a) displacement versus contact force and, (b) time
histories of contact force for CFRP laminates for the [0/90/∓60]s laminate. --------------------------------- 111
Figure 7-13. Crack density in the 0° ply of a GFRP [0/90/∓45]s laminate under low-velocity impact at 13.1
m/s : (a) neglecting rate-dependent behavior and (b) including rate-dependent behavior. ----------------- 112
xx
Figure 7-14. Effect of including rate-dependency on crack density evolution in the 0° ply of the [0/90/∓45]s
GFRP bumper under low velocity impact. -------------------------------------------------------------------------- 112
1
Chapter 1. Introduction
In this chapter, the main motivations for the research conducted in this PhD thesis are provided. The
overall economic and societal challenges the automotive industry is facing are explained, and the gaps in
scientific knowledge preventing the incorporation of composite materials into the transportation system
are reviewed.
1.1 Lightweighting in the transportation sector
The transportation sector is responsible for an estimated 27 % of total anthropogenic green house gas
emissions (see Figure 1-1) and is therefore a major contributor to global warming. In order to reduce
mankind’s reliance on fossil fuels, while still maintaining the strong economic performance that the
transportation systems affords our societies, the efficiency of vehicles needs to be improved so that
performance is maintained, while fuel consumption is reduced [1]. Given the initial reluctance of car
manufacturers to implement effective strategies for improving the efficiency of their vehicles,
governmental regulations [2] have been enacted in North America and the European Union for CO2
emissions reductions. Countries in the EU are requiring a reduction in CO2 emissions from 140 grams of
CO2 emitted per car per kilometer driven in 2010, to 75 grams of CO2/km in 2025: if a car manufactured
in 2025 does not meet the CO2 emission target set out by the EU, a penalty of 12 350 EUR for every vehicle
produced will be imposed.
The efficiency of the transportation infrastructure can be improved by improving the energy efficiency
of the drivetrain, and reducing air drag, vehicle weight and rolling resistance. It has been estimated that by
2030 fuel economy can be increased between 50 and 100% by using these methods [1]. More
technologically advanced methods would involve the full electrification of the car and truck fleet [2]. An
easier method of improving efficiency is the reduction of the weight of cars, trucks and buses. It has been
estimated that 0.08 g CO2 /km can be saved per kilogram of car weight reduction[2]. A recent MIT study
showed that a weight reduction of 35 % of the entire transportation fleet could be achieved by 2035 [3],
reducing fuel consumption by 12 to 20 %, which would lead to significantly lower GHG emissions. Two
recent Life Cycle Analyses [4], [5] have also shown that lightweighting through material substitution
would lead to significant reductions in CO2 emissions over the entire lifecycle over the vehicles.
Lightweighting is achieved by replacing the materials in the vehicle by lighter materials that can
sustain equal mechanical constraints. By improving the mechanical properties of the materials used in a
vehicle, the weight can be reduced while still maintaining a safe environment for the passengers and
satisfactory structural performance, thereby increasing gas mileage. As shown in Figure 1-2, current cars
2
are mainly made with plastics, aluminum and steel [2]. The production of these materials is highly energy
intensive; they could be replaced by lighter materials, such as High Strength Steels (HSS), aluminum alloys
and carbon fiber composites [2]. Polymer-based composite materials are a candidate of choice to replace
the traditional metallic materials, due to their very high stiffness-to-weight and strength-to-weight ratios.
Composite materials being considered for automotive applications combine a stiff and strong fibrous
material such as carbon or glass fiber, with a weaker polymer which protects the fibers from the
environment, and transfers the loads applied to the composite onto the fibers [6]. Composites are generally
as strong as metals but are less dense and can therefore provide significant weight savings.
Several automotive companies are trying to optimize the material design process to enhance the
properties of the metals in their vehicles. Ford Company, in collaboration with Magna, are reducing car
weight by designing lighter and stronger metals through improved processing [7]. General Motors have
publicized similar plans of weight reduction through optimization of material performance [8]. High-
strength steel, aluminum, magnesium, as well as carbon fiber composites are being considered for reducing
car weight [2]. Among these, carbon fiber composites are the lightest alternative, but also the most
expensive. Market forecasts suggest that the price of carbon fiber will decrease significantly within the
coming years, which together with its much higher performance would make it competitive against
metallic alloys. Moreover, although the production of a car containing a significant fraction of carbon fiber
materials might only be feasible for high-end luxury and sports cars, incorporating them in at least some
strategic components in an average vehicle would also be a useful strategy for lightweighting. Formula 1
racing car chassis are already made of carbon fiber [9] due to its very low density and very high rigidity.
Figure 1-1. U.S. greenhouse gas emissions breakdown by economic sector [10] .
The automotive industry is already using composite materials in some applications. According to
Lucintel [11], in 2018 the total demand for composite materials in the automotive industry reached 4 billion
lbs. These composites would include glass fiber composites and bio-composites, as well as carbon fiber
components. Demand for carbon fiber composites is expected to reach 90 million lbs by 2025 [12]. The
3
carbon fiber composites can be used in highly stressed regions of the vehicles, such as the chassis and side
panels. Less strong composites such as bio-composites are already being used as seat backs and door
panels.
Significant car weight reduction could be achieved by incorporation of carbon fiber composites. Other
types of composites could also be used to improve the overall sustainability of the transportation sector.
For example, bio-composites are a particular type of composite materials in which at least one of the
constituents is derived from the natural environment [5], [13]. Bio-composites present several advantages
over plastics used traditionally. According to [14], some bio-composites can offer the same mechanical
performance but for a lower weight (the stiffness of bio-based fibres can be comparable to that of glass
fibres [15]), which will reduce vehicle weight. They are made using renewable resources, such as wood,
hemp and natural materials, which can be grown easily, reducing mankind’s reliance on limited fossil fuel
supplies. The manufacturing process is also sustainable as it requires 80 % less energy than glass fiber-
based composites (see Figure 1-3[16]), currently being used as a replacement to metals. They are easy to
recycle. Some bio-composites can also be bio-degradable, and could be buried in the ground without
impacting the environment [14]. Currently, the equivalent of 25 % of the weight of a car cannot be recycled.
Replacing thermosets with bio-composites and biodegradable and/or recyclable plastics will help alleviate
this issue [17]. They can also be incinerated without releasing as much GHGs as synthetic materials, or
they can be composted. Finally, their cost is low compared to synthetic materials [16] such as glass fiber
composites. Automakers also wish to project a good public image to customers, and at a time when
environmental problems are of increasing concern to our societies, investing in more sustainable products
is considered by some as a judicious policy [18] to improve brand reputation. Due to these factors, the
market for natural fiber composites is expected to continue growing in the coming years (see Figure 1-4).
Figure 1-2. Proportion of materials, by weight, making up the structure of the modern car[19]
4
Figure 1-3. CO2 emissions of natural (NMT) and glass fibre (GMT) composites over their lifetime [16]. From “Biocomposites
reinforced with natural fibers: 2000-2010” by Faruk O. et al., 2010 Prog. Polym. Sci., Vol. 37, p. 1570. Copyright (2012) by
Elsevier. Reprinted with permission.
A significant impediment to the large-scale implementation of bio-composites as car components is
their lower impact strength and their highly non-linear behaviour. The improvement of impact behaviour
is critical for automotive components. Because the natural fibres are short, the energy dissipation
mechanisms occurring upon impact are not sufficient to absorb the collision energy. Natural fibre enhanced
direct-long fibre thermoplastics (NF-DLFT) manufactured by the Center for Bio-composites and Bio-
materials Processing (CBBP) at the University of Toronto are a possible compromise between the
mechanical performance of synthetic glass fibre-based composites and the environmental sustainability of
bio-composites. NF-DLFTs are hybrid bio-composites reinforced with both natural hemp fibres and glass
fibres, embedded in a polypropylene (thermoplastic) matrix. The presence of glass fibres increases the
strength of the material, making them suitable for structural applications, and improves the impact strength
by providing more energy dissipation mechanisms (see Figure 1-5). A new manufacturing process has
recently been developed to further increase the length of the glass fibers and thereby improve impact
performance [20].
Overall, both continuous fiber composites and bio-composites (such as NF-DLFT) can, if properly
implemented into the transportation infrastructure, play a major role in the lightweighting targets of the
automotive industry. Continuous fiber composites can be used in the structurally demanding components
of the vehicle, and bio-composites can serve as minor load-bearing components while weighing less than
traditional materials and increase the overall sustainability of the car. However, in order to incorporate
composite materials in the automotive industry, significant technical challenges still need to be overcome.
5
Figure 1-4. Market growth for natural fibre composites [21]
1.2 Modelling the mechanical performance of composites
Reducing the weight of vehicles will help reduce greenhouse gas emissions from vehicles. However,
in order to design vehicle with composite components that demonstrate equivalent performance to the
metals used currently, their mechanical behaviour needs to be predictable. Current mathematical models
are not accurate enough to predict the constitutive response of multi-directional composites loaded under
arbitrary multi-axial scenarios and undergoing progressive damage in multiple plies at the same time.
Moreover, time-dependent effects induced by the viscoelastic and viscoplastic properties of the polymer
matrix are still not completely understood. According to the National Research Council [22], the lack of
predictive tools for the evolution of damage in composite materials has led manufacturers to overdesign
their products, preventing the full utilization of the mechanical capabilities of composite materials. This
is an especially important issue in the context of lightweighting.
Composite materials are known to exhibit time-dependent behavior, due to the viscoelasticity and
viscoplasticity of the matrix [23], [24]. Bio-composites are especially prone to time-dependent
deformation, even under relatively low loads [25] due to the shortness of the fibers and the mechanical
properties of the matrix. In addition to causing supplementary deformation which leads to lower composite
stiffness over time, these non-linear properties are known to affect the evolution of damage in laminates,
as well as short fiber composites [26], [27]. Studies on laminates have shown that viscoelasticity and
viscoplasticity can cause the evolution of damage during creep loading [26]. Studies on short fiber
composites have also shown that the rate of loading affects damage processes at the microscopic scale
[28], [29], most likely because of the time-dependent behavior of the matrix (see Figure 1-6 for the creep
behavior of NF-DLFT).
While there have been many studies investigating damage mechanisms in composites [30]–[36], either
from a modelling perspective or an experimental approach, most models currently available are only
approximately accurate, or are restricted to very specific types of composites, such as cross-plies. In order
6
to improve the efficiency of current design processes, a multi-scale computational approach is needed,
which can provide much faster assessment of the performance of a composite than an experimental
approach. In particular, a quantitative understanding of the process of matrix micro-cracking and fiber-
matrix debonding at the microscopic scale will accelerate material selection and the overall design phase
of new composite materials. Moreover, as mentioned above, the time-dependent properties of the matrix,
typically modelled using the theories of viscoelasticity and viscoplasticity, can affect damage progression
in the composite [37]–[39]. To effectively design composite materials which do not fail prematurely due
to the time-dependent properties of the matrix, an understanding of the interaction between viscoelasticity,
viscoplasticity and damage is required.
Figure 1-5. Improvement of NF-DLFT hybrid bio-composite impact strength with glass fiber hybridization [40]. From “Injection-
Molded Short Hemp Fiber / Glass Fiber- Reinforced Polypropylene Hybrid Composites — Mechanical , Water Absorption and
Thermal Properties”, by Panthapulakkal S. and Sain M., 2006 J. Appl. Polym. Sci., Vol. 103, p. 2438. Copyright (2006) by Wiley
Periodical, Inc. Reprinted with permission.
The subject of this thesis is concerned with predicting damage evolution in both continuous fiber
composites and short fiber bio-composites, while taking into account the viscoelastic-viscoplastic
properties of the constituents. Laminates and short fiber composites have different mechanical properties
and undergo damage in fundamentally different ways. However, as will be shown in this thesis, the issues
facing damage modelling in these two types of composites are the same and can be addressed using a
similar methodology. It will be shown the tools that are developed for continuous fiber composites are
applicable to short fiber composites as well.
1.3 Thesis objectives
A proper prediction of the progression of damage and the effects of viscoelasticity and viscoplasticity
is necessary to optimize the performance of composites by tailoring the manufacturing process and the
microstructure (fibre geometry, fibre volume fraction and distribution, thermoplastic mechanical
properties etc.). The motivations behind this thesis are therefore to assist in the design of composites that
permit greater environmental sustainability, and that can perform sufficiently well mechanically to be used
as structural car components. The goal of this work in particular is to provide a comprehensive predictive
7
methodology capable of assessing the long-term mechanical performance of composite structures, using a
multi-scale modelling methodology to predict damage evolution in viscoelastic-viscoplastic composites.
The multi-scale model developed in this thesis includes the microscopic scale of the fiber-matrix
interface, the mesoscopic level of the homogenized plies, and the structural level of an automotive
component. To achieve the thesis goal, the following objectives have been decided upon:
1. Predict damage evolution in viscoelastic/viscoplastic laminates undergoing creep deformation.
2. Implement the damage model into FEA software and examine the effect of quasi-static loading
and impact on a composite structure. This will provide a predictive tool for the performance of
composites, whose results can be compared to experiments and used to design new structures.
3. Determine the effect of viscoelastic and viscoplastic matrix behavior on the evolution of damage
and the competition between damage mechanisms under varying strain rates in short fiber
composites.
In this thesis, a multi-scale approach based on micro-damage FE modelling of matrix micro-cracking
is combined with Schapery’s equations for calculating creep strain arising from viscoelasticity and
viscoplasticity in multi-directional laminates. The model is applied to different composite structures
(different stacking sequences and ply thicknesses) under arbitrary in-plane loading scenarios. The results
of this analysis are reported in Chapters 4 and 5 of this thesis and provide detailed insights into the
interactions between damage evolution and creep deformation in structural laminates.
In Chapter 6, the multi-scale methodology used for laminates is extended by developing a Cohesive
Zone Model to predict fiber/matrix debonding in a polyethylene-based short fiber composite material, for
potential use in automotive applications. The effect of strain rate is studied, which illustrates and quantifies
the effect of matrix viscoelasticity on competing deformation processes in short fiber composites.
Lastly, in Chapter 7, the multi-scale framework is extended to the structural scale by implementing
the viscoelastic-viscoplastic damage model for laminates as an Abaqus VUMAT. A structural analysis of
a car bumper under low-velocity impact is conducted, and the effects of stacking sequence, material system
and impactor velocity are studied. Rate-dependency is included in the custom material model to represent
the effects of viscoelastic and viscoplastic material properties of the matrix.
8
Figure 1-6. Creep behavior of NF-DLFT for different amounts of glass and hemp fibres[39].
9
Chapter 2. Literature review
In this chapter, both the nature and the quantitative evolution of damage in laminates and short fiber
composites which have been observed in experiments are summarized. The different models published in
the literature which have been used to predict damage in viscoelastic-viscoplastic laminates and short fiber
composites are then presented and evaluated. Based on this literature review, the main gaps in the current
understanding of the interaction between damage and time-dependent properties are identified to motivate
the thesis objectives presented in Chapter 1.
2.1 Composite materials
A composite is a material which consists of at least two heterogeneous phases [6] with significantly
different mechanical properties. For the composites studied in this thesis, one phase is fibrous and is the
primary load-bearing component of the composite, while the matrix phase is more compliant and weaker,
and surrounds the fibers. The advantages of composites with a polymer matrix and stiff synthetic fibers
such as glass or carbon relative to ceramics and metallic materials are improved control of the mechanical
properties, and better specific strength and stiffness [41]. Also, their fatigue life is often better than for
aluminum [6] and their thermal stability can also be tailored more easily, which makes them a prime
candidate for aerospace applications.
There are different classes of composite materials, which are distinguished in this thesis by the
configuration and distribution of the fibers. In the case of laminates, shown schematically in Figure 2-1,
the fibers are continuous throughout the length of the component, and are arranged parallel to each other
in a ply; the laminate consists of multiple plies, oriented in different directions, and stacked on top of one
another. A laminate’s properties are in part governed by its stacking sequence, which corresponds to the
orientations of the fibers in the plies with respect to the primary loading axis. In this thesis, the stacking
sequences will all be symmetrical with respect to the mid-plane of the laminate, which allows for the study
of in-plane mechanical behaviour. Three types of laminates will be studied in this work: symmetric cross-
plies ([0n/90m]s) which consist of plies oriented along the loading axis and perpendicular to it, laminates
with [±θ/902]s stacking sequence, and symmetric multi-directional laminates [0/90/∓θ]s, where θ is an
angle between 45 and 75°. Depending on the stacking sequence, the elastic properties, the time-dependent
properties, as well as the damage evolution of the composite, will be affected. The choice of stacking
sequence is therefore an effective design tool for engineers which allows them to tailor the properties of
the composite for specific thermo-mechanical loading scenarios.
10
In the case of short fiber composites, the fibers are much shorter (on the order of less than a millimeter
to a few centimeters), and are dispersed throughout the matrix [42]. The fibers can be randomly oriented
or aligned, depending on the manufacturing process. The length of the fibers, their orientation, and their
distribution affect the macroscopic behaviour of the short fiber composite. Figure 2-2 [43] shows the
differences between a short fiber composite, and a laminate made with unidirectional fibers.
Figure 2-1. Schematic explaining the structure of a multi-directional laminate [44].
Figure 2-2. Schematic explaining the difference between a short fiber composite and a laminate [43].
2.2 Damage mechanisms in composites
The evolution of damage in composite materials is governed by the structure of the composite [41]
and the mechanical properties of the constituents. The patterns of damage evolution in laminates on the
one hand, and short fiber composites on the other hand, are known to be significantly different. First
damage mechanisms in laminates will be reviewed, concentrating specifically on matrix micro-cracking,
which is the focus of this thesis. Damage mechanisms in short fiber composites will be reviewed next.
Viscoelastic and viscoplastic behavior will be explained, and their interaction with damage evolution
observed experimentally will be discussed.
11
2.2.1 Matrix micro-cracking in laminates
Damage in laminates in the form of matrix micro-cracking initiates at the ply level [41], usually in the
plies that are oriented at more than 50° with respect to the applied load [45]. Under the transverse tensile
stress exerted on the ply, a microscopic crack is nucleated within the soft matrix, which quickly grows
parallel to the fibers and extends throughout the thickness and width of the ply (see Figure 2-3). The process
of crack nucleation in plies occurs through a combination of fiber-matrix debonding, matrix plastic damage
and failure, micro-crack nucleation and subsequent micro-crack growth throughout the thickness of the ply
and the width of the laminate [41]. Due to the constraining effect of the adjacent layers of the laminate,
the matrix micro-crack that has been nucleated will not cause immediate failure of the laminate. Instead,
as the applied load is further increased, additional cracks will nucleate, forming a series of evenly spaced
parallel cracks spanning the width and thickness of the ply. Eventually, at sufficiently high loads, the
micro-cracks can lead to other critical damage modes, such as delamination and fiber fracture [46] due to
the stress concentration induced at the crack tips.
Figure 2-3. Schematic showing matrix micro-cracking in a cross-ply laminate [47].
Many experimental studies have investigated in detail the process of matrix micro-crack initiation in
laminates [35], [48]–[51]. Ply thickness [31] has been shown to play a critical role in the crack nucleation
strain, with thinner plies leading to a larger crack initiation strain. At very small ply thicknesses, ply micro-
cracking can be suppressed entirely. Stacking sequence has also been shown to affect the initiation of
matrix micro-cracking [51]. In the case of laminates with stacking sequence [±30/90n]s and [±60/90n]s, the
orientation of the plies adjacent to the 90° ply was found to affect the ply crack initiation strain in the 90°
ply. Also, the order of the stacking sequence has been found to play a role in crack initiation [52]: when
the laminate is composed of 90° plies on the outer surface of the laminate, the crack initiation strain is
lower than for a 90° ply in the middle of the laminate. The kinetics of crack nucleation and growth depend
on the ply thickness, as well as the orientation of the cracked ply [31], [48].
12
In experiments, the first micro-cracks typically nucleate at defects such as resin-rich regions, voids,
and fiber tips which induce a stress concentration that enables the opening of a micro-crack [53]. When
the loading is increased furhter, more cracks will nucleate. Because the initial distribution of defects within
the ply is random, the first few micro-cracks will be randomly distributed throughout the length of the
laminate. However, because it is more energetically favorable for a new crack to nucleate in the central
region between two previous cracks, with increasing crack density the distribution becomes more uniform.
The effect of manufacturing defects on the first stages of crack initiation and multiplication has been
studied by several authors [54], [55].
Experimental studies of damage evolution in composites traditionally focus on microscopic defects,
which lead to the occurrence of microscopic cracks which do not cause the fracture of the composite.
Instead, the effect of these micro-cracks can be modelled by defining residual stiffness matrices. On the
other hand, if the crack that is nucleated becomes sufficiently large, a model based on Linear Elastic
Fracture Mechanics, instead of Continuum Damage Mechanics, might be necessary to predict its growth
and the fracture of the part. In this thesis, the focus will be on micro-damage mechanisms, and not on
macroscopic cracks.
Following the nucleation and growth of the first matrix micro-crack, the elastic properties of the
laminate are not significantly affected due to the small size of the crack. However, upon further loading,
supplementary cracks are nucleated [56]. In order to understand matrix micro-cracking quantitatively,
crack density as a function of globally applied stress is typically plotted [57]. As seen in Figure 2-4,
following the formation of the first matrix micro-crack, the crack density increases; after a certain amount
of loading, the curve flattens. It has been found [31] that microstructural defect at the ply level can affect
the shape of the curve at the onset of crack multiplication. The thickness of the ply has been found to also
affect on crack multiplication, with lower ply thickness leading to a larger saturation crack density. The
properties of adjacent plies also affect the rate of damage evolution, and the saturation crack density [51].
Matrix micro-cracking is typically the first mode of damage in structural laminates and dominates at
low stresses. As crack density increases, however, the matrix micro-cracks can lead to delamination [48]
at ply interfaces. This delamination induced by the micro-cracks occurs due to the strong stress
concentration at the tips of the micro-cracks [58]. The cracks are usually straight through the thickness of
the ply, however, at large crack densities, interactions between cracks can cause newly nucleated cracks to
adopt curved shapes, which can affect the delamination process [59]. Other damage modes can also occur
at higher loading, such as fiber breakage [60]. Different modes of damage occur under compressive
loading, but will not be considered in this thesis.
13
Figure 2-4. Crack density evolution versus applied stress in the 90° ply of a glass/epoxy cross-ply. From “On matrix crack growth
in quasi-isotropic laminates - I. Experimental investigation”, by Tong J. et al, 1997 Compos. Sci. Technol., Vol. 57, p. 1529.
Copyright (1997) by Elsevier. Reprinted with permission.
2.2.2 Damage mechanisms in short fiber composites
Damage mechanisms in short fiber composites include fiber-matrix debonding, matrix micro-cracking,
fiber pull-out and fiber breakage [40], [61]–[65]. The evolution of these mechanisms at the microscopic
scale are very complex to model due to the complex distribution of fibers throughout the part, and possibly
also their random orientation. In the case of polymer composites with a thermoplastic matrix, which can
demonstrate a high level of ductility and deformation prior to cracking, the sequence of damage
mechanisms under uniaxial loading leading to final failure of the material usually consists of the following
features [61]:
- Significant inelastic deformation initiates ahead of the fiber tips due to the mismatch in elastic
properties between the fiber and matrix.
- A void nucleates at the fiber tip following plastic deformation of the matrix and separation between
the fiber tip and the matrix.
- The micro-crack nucleated at the fiber tip propagates along the sides of the fiber leading to full
debonding between the fiber and matrix.
- Significant plastic deformation initiating from the debonded fibers occurs in the matrix in the
regions between the fiber tips.
- Eventually, extensive plastic deformation leads to matrix micro-cracking.
- The micro-cracks coalesce, leading to the growth of matrix macroscopic crack that can propagate
throughout the material.
14
Figure 2-5. Damage mechanisms in a short fiber composite. (a) Fiber-matrix debonding mechanism [66]. (b) Fiber pull-out [67].
(c) Fiber fracture [68]. (d) Matrix micro-cracking initiating from fiber tips [61]. Last figure from “Microfailure behaviour of
randomly dispersed short fiber reinforced thermoplastic composites obtained by direct SEM observation.”, by Sato N. et al, 1991
J. Mater. Sci., Vol. 26, p. 3892. Copyright (1991) by Chapman and Hall Ltd. Reprinted with permission.
Images of the different mechanisms obtained through experimental observation are shown in Figure
2-5. The damage mechanisms occurring in short fiber composites that will be studied in depth in Chapter
6 of this thesis are fiber-matrix debonding and matrix inelastic deformation.
It has been shown experimentally [69], [70] that the local environment around short fibers can affect
the evolution of damage. Depending on the orientation and size of adjacent fibers as well as their relative
proximity, the local evolution of damage will be affected. It has also been found that the mechanical
properties of the matrix can affect the initiation of matrix micro-cracking and debonding [71]: depending
on the strength of the matrix, the extent of debonding and matrix micro-cracking following fiber breakage
can be different.
2.2.3 Interaction between viscoelasticity, viscoplasticity and damage evolution in composites
The effects of viscoelasticity and viscoplasticity on mechanical behavior are evident during a creep
test, where time-dependent deformation will occur under a constant load (see Figure 2-6). Due to the
(a) (b)
(c) (d)
15
viscoelastic and viscoplastic properties of the material, when a load is applied, the material will keep
deforming with respect to time. When the load is removed, an immediate elastic response will be recovered.
The strain will then keep decreasing with respect to time due to viscoelasticity. A residual strain will not
be recovered with increasing time due to the viscoplastic material behaviour. A large amount of research
has been conducted on the viscoelasticity and viscoplasticity of composites [24], [72]–[79], and especially
bio-composites, due to the non-linear properties of the fibers and matrix[15], [27], [80]–[82]. While these
studies provide insights into the time-dependent behavior of composites, they do not consider the effects
of damage evolution on the mechanical response of the composites.
Figure 2-6. Effect of multiple creep-recovery tests on the time-dependent strain response of a bio-composite material. From
“Characterization and modeling of performance of Polymer Composites Reinforced with Highly Non-Linear Cellulosic Fibers”
by Rozite L. et al, 2012 IOP Conf. Ser. Mater. Sci. Eng. , Vol. 31, p. 4. Copyright (2012) by IOP Publishing Ltd. Reprinted with
permission.
The viscoelastic and viscoplastic properties of the composite can have an effect on its long-term
durability and its susceptibility to undergo progressive damage. According to a study by Guedes [83], creep
rupture can occur in laminates at loads lower than the quasi-static strength. This was attributed to the time-
dependent deformation of the composites, in which the matrix undergoes a change in structure with respect
to time, causing fracture to occur at relatively lower loads. Asadi and Raghavan [32] showed that damage
can evolve during creep loading of a CFRP laminate. Similarly, Darabi et al [84] demonstrated that rate-
dependent material properties can affect damage evolution in polymer-based composites. Practically, this
study demonstrated that for a given applied load, the damage evolution during a creep test would evolve
with respect to time. Early studies by Dillard and his collaborators [23], [85], [86] showed that under creep
loading, time-dependent cracking can occur, due to the viscoelastic properties of the laminates. Birur and
his co-workers [87], [88] similarly observed damage evolution in multi-directional laminates subject to
creep loading. Nguyen and Gamby [89] found that the matrix micro-crack multiplication in cross-ply
CFRP laminates loaded at 120 °C was affected by the loading rate, with lower loading rates leading to
higher crack density for the same amount of strain. Fitoussi et al [90] found that the rate-dependent
16
behavior of the matrix affected damage evolution in Sheet Moulding Compounds (see Figure 2-7). These
studies show that damage evolution in composites is affected by the viscoelastic and viscoplastic properties
of the matrix.
Figure 2-7. Evolution of damage quantified at the microscopic level (a) and at the macroscopic level (b), for a sheet molding
compound versus applied strain for different strain rates. From “Multi-scales modelling of dynamic behaviour for discontinuous
fibre SMC composites” by Jendli et al., 2009 Compos. Sci. Technol., Vol. 69, p. 101. Copyright (2008) by Elsevier. Reprinted
with permission.
2.3 Models to predict matrix micro-cracking in laminates
Many models have been developed to attempt to predict the progression of matrix micro-cracking in
laminates [45], [91]–[96], however, to date, no model can predict progression of damage in viscoelastic-
viscoplastic multi-directional laminates under multi-axial loading. The earlier models [91], [92] are
analytical, and use the properties of the laminate (ply thicknesses and orientations, elastic properties) to
develop closed-form solutions for stiffness loss as a function of crack density. These models can be applied
to any material with the same stacking sequence, however their accuracy is limited because of the
assumptions that are made in the stress field description within the laminate. Moreover, their extension to
stacking sequences other than cross-plies and to cracking in multiple plies at the same time is not trivial,
which severely limits their predictive abilities [46]. In the case of shear-lag models [91], the effect of a
crack on the distribution of stresses within the cracked ply is calculated by solving the differential equation
for stress equilibrium, and by assuming a functional form of the displacement profile in the different layers
of the cracked laminate. From the stress profile occurring within the layers of the laminate at different
crack spacings, the stiffness can be calculated. In the case of variational methods [92], equations describing
the relationship between applied stress and internal stress distribution are developed by assuming a
particular functional description of the different components of the stress tensor in each layer of the
laminate, from which the stored elastic energy can be calculated; the principle of minimum complementary
energy is then invoked, from which an approximate stress field can be derived using variational calculus.
17
Because this type of model is based on an approximate representation of the stress field, it demonstrates
limited accuracy.
In order to overcome the complexities involved when attempting to predict the stress field within a
multi-directional laminate containing cracks in multiple plies, the more recent models for predicting
progressive damage in laminates are based on Continuum Damage Mechanics (CDM), which take
advantage of the improving computational capabilities available to researchers. In CDM, the stiffness can
be predicted without first deriving the stress distribution within the cracked laminate. Instead, the geometry
of the cracks is used for predicting stiffness loss. In one of the first such models, Talreja [97] predicted
stiffness loss in different laminates as a function of crack density by defining a damage tensor in terms of
the size of the cracks (width and length) as well as their volume fraction. Following this work, Talreja [98]
developed a Continuum Damage Model which expressed the stiffness of a damaged composite material in
terms of the geometry of the cracks (width, thickness and volume fraction).
In the most common CDM approaches [36], [45], [94], [99], a Finite Element RVE is created in
commercial software, where each layer of a laminate is modelled as homogeneous, with transversely
isotropic elastic properties of individual plies obtained from experimental data. The orientation of the
layers is accounted for in the software by defining local coordinate systems for each layer. Once the meso-
scale model of the laminate has been created, a micro-crack is defined in the layer of interest by
disconnecting nodes along a given surface across the thickness and width of the ply. Using this RVE of a
cracked laminate, the effect of micro-cracking on stiffness can be obtained, without needing to derive
analytical expressions for stress distributions [30], [91]. However, one of the main weaknesses of such
CDM approaches is their inability to make predictions for materials with different elastic properties, with
different crack densities, and different ply thicknesses. Each time a parameter is changed, a new simulation
must be conducted.
Gudmundson and his collaborators [93], [99], [100] proposed a model in which the stiffness of a
laminate, as well as its bending stiffness, was related to the Crack Surface Displacement (Crack Opening
Displacement (COD), and Crack Sliding Displacement (CSD)) of the crack. They developed exact
expressions for stiffness reduction in terms of this parameter. This model was effective at predicting
stiffness degradations at low crack densities, when the stress fields induced by the cracks do not
significantly overlap. However, the method used to obtain the Crack Surface Displacement did not fully
account for the stacking sequence of the laminate. In a realistic application, the stiffness of adjacent layers
as well as their thickness, and the thickness of the cracked layer, will affect the possible opening of the
crack. Therefore, building on the work of Gudmundson and his co-workers, Lundmark and Varna [36],
[101] developed analytical expressions relating the COD of a matrix crack to the properties of adjacent
layers. Varna and his collaborators [102], [103] subsequently developed an expression for the evolution of
18
COD with increasing crack density, by accounting for the interaction between the cracks in a given layer.
Joffe et al [104] developed an approach to relate the COD of cracks in the 90° layer of a [±θ/904]s laminate
to the elastic properties of the constraining plies. However, the effect of cracking in multiple plies at the
same time was not predicted.
Singh and Talreja [41], [45], [105] conducted a computational implementation of Synergistic Damage
Mechanics (SDM), which combines the strengths of micro-mechanical models of stiffness degradation,
and CDM. Using experimental stiffness degradation data for a cross-ply laminate, they made predictions
of stiffness degradation for laminates with different stacking sequences. Their approach relied on the
calculation of the COD in the different layers of the laminate. By comparing their results to experimental
data, they showed that the COD is a key parameter related to stiffness degradation. They also showed that
stiffness degradation consisted of material-dependent effects, and stacking sequence-dependent effects.
The latter was accounted for through the COD. However, in these works, the interaction between adjacent
micro-cracks, and the effect of this interaction on stiffness degradation was not modelled. Lastly, the effect
of multi-axial in-plane loading scenarios was not investigated.
Montesano and Singh [94], [95], [106] developed FE micro-damage models to predict the effect of
multiple micro-cracking in multi-directional laminates. They took into account the effect of intra-ply crack
interactions, which causes the normalized COD to decrease as the crack density increases. They accounted
for multi-axial loading by defining the COD in terms of an effective strain and by using Periodic Boundary
Conditions in their model. They also showed that cracking in adjacent plies could affect the COD due to
inter-ply interactions. They proved that the results of the SDM model predictions, based on COD evolution,
were in agreement with experimental data, and independent FE calculations. Singh extended the SDM
model [107] to include non-linear relationships between the damage tensor and stiffness degradation.
However, non-linear material properties, in particular viscoelasticity and viscoplasticity were not
accounted for.
The models detailed above were concerned with the prediction of the effect of matrix micro-cracking
on the elastic properties of the laminate. However, in order to predict the overall behaviour of a laminate
under increasing loading, it is also necessary to predict the multiplication of matrix micro-cracking as a
function of applied laminate load. From this information, the full mechanical response of a laminate
undergoing micro-cracking can be calculated.
Two types of models have traditionally been used to predict crack multiplication. In the case of
strength-based theories [31], cracking is assumed to initiate when the transverse ply stress reaches a certain
threshold level. While this type of model is simple, it fails to explain the decrease in crack initiation strain
with increasing ply thickness that has been observed experimentally. Energy-based models, on the other
hand, determine crack initiation and multiplication by calculating the strain energy released upon crack
19
multiplication. The stress distribution in the cracked ply with increasing crack density is required, from
which the change in strain energy upon matrix micro-cracking can be calculated based on a critical energy
determined experimentally. In these models, cracks are assumed to nucleate and grow throughout the width
and thickness of the ply instantaneously. The concern is to predict crack multiplication, rather than
propagation [108]; the application of energy-based methods to multiple matrix micro-cracking therefore
assumes that cracks form instantaneously following nucleation. In order to make accurate predictions,
however, energy-based models do require a critical strain energy release rate value for each ply, which
depends on the properties of the matrix, the ply and also on the stacking sequence [106]. It is determined
from experimental data.
Nairn et al [56] found that an energy-based model for crack multiplication was better able to predict
crack density evolution in laminates with different stacking sequences than strength-based models. They
used a variational approach to calculate the stress field in the cracked laminates, from which the strain
energy could be calculated for different crack spacings. They defined the critical energy release rate as a
parameter to fit the experimental data of crack density versus applied laminate load. They also measured
the fracture toughness of an individual ply, and found that it was close to the critical energy release rate
for crack multiplication, but not identical. However, this model was limited to cracking in a single 90° ply,
under uniaxial loading. Moreover, the prediction of micro-crack multiplication was based on an
approximate description of the stress field; a FE-based approach would have been more accurate in
calculating this stress field.
Joffe et al [104] defined the thermo-elastic constants of a laminate containing micro-cracks in terms
of the COD. Using these expressions, they were able to develop a model which predicted crack density
while accounting for stochasticity in matrix micro-cracking. Their stochastic model was aimed at
accounting for the experimental observation that the first micro-cracks in a ply nucleate at locations where
pre-existing defects are present, which leads to non-uniform crack spacing at low crack densities. One of
the key contributions of this work was the stochastic description of micro-crack nucleation; however the
model was restricted to uniaxial loading and cracking in a single layer at a time.
Singh and Talreja [109] developed a COD-based model to obtain crack density evolution in laminates
with off-axis plies. They used FEA to develop expressions for COD evolution as a function of crack
density, from which the energy release rate for crack multiplication could be calculated. They obtained the
critical energy release rate by fitting their model to experimental data. The model was restricted to uniaxial
loading. Moreover, a fitting procedure was required to obtain the critical energy release rate for crack
multiplication.
Montesano and Singh [106] extended this previous model by accounting for multi-axial loading, from
which they were able to plot stiffness degradation curves describing the stiffness loss in terms of the applied
20
bi-axial loads. Moreover, they developed a computational methodology for calculating the critical energy
release rate for crack multiplication, based on the crack initiation strain for each ply of a laminate. Lastly,
by using a periodic RVE to calculate the COD in each layer of the laminate, they were able to predict the
effect of multi-axial loading on cracking in multiple plies at the same time. This type of model will be used
in this thesis due to its high accuracy and large range of applicability
2.4 Models to predict the combined effects of damage, viscoelasticity and viscoplasticity in composites
Viscoelasticity and viscoplasticity in composites were initially comprehensively modelled by
Schapery and his collaborators [77], [110] [24], [27], [32], [111], [112]. Their model requires creep-
recovery data under different loading levels for parameter calibration. Schapery’s model expresses the
time-dependent strain of a composite using the following equation:
휀(𝑡) = 𝑔0𝐴𝜎(𝑡) + 𝑔1∫ 𝛥𝐴(𝜓 − 𝜓′) 𝑑(𝑔2𝜎)
𝑑𝜏𝑑𝜏
𝑡
0− (2.1)
Where 휀(𝑡) is the time-dependent strain, 𝐴 is the instantaneous elastic compliance, 𝜎(𝑡) is the time-
dependent applied stress, and 𝛥𝐴 is a function which represents the increase in compliance with respect to
time. The parameters 𝑔[0−2]are functions of stress and temperature, and account for the change in
compliance with respect to stress. Lastly, the variable 𝜓 is related to the variable representing time, and is
scaled to take non-linear behaviour into account.
Eq. (2.1) is applicable for uniaxial loading scenarios. In the case of unidirectional plies, the transverse
compliance, and the shear compliance will be accurately modelled by an expression such as Eq. (2.1) as
these components are dominated by the mechanical behaviour of the matrix. The parameters will be
different for these two different components of the compliance matrix. In order to account for multi-axial
loading scenarios to be expected in laminates, the approach used by Tuttle et al [24] can be utilized. In
their approach, a non-linear time-dependent compliance model is defined for both the transverse
compliance, and the shear compliance. When a ply undergoes multi-axial stresses, the non-linear
parameters appearing in Eq. (2.1) are defined in terms of the octahedral shear stress.
Guedes et al [113] extended these works by developing an algorithm capable of predicting the
viscoelastic-viscoplastic response of arbitrary laminates under various loading conditions. Zaoutsos et al
[114] used Schapery’s theory to model the viscoelastic response of a carbon fiber/epoxy composite. Al-
Haik et al [115] showed that artificial neural networks could also be used to predict the viscoelastic
behaviour of composites at high temperatures. Korontzis et al [116] developed a procedure to obtain the
linear viscoelastic properties of a laminate from those of a ply, which were obtained experimentally.
The model of Tuttle et al [24] has been used in this thesis to predict the creep response of laminates
because it is an accurate model with a large range of applicability. However, the model has been improved
21
upon by combining it with a novel multi-scale modelling methodology to couple viscoelasticity and
viscoplasticity with damage evolution.
As explained in section 2.2.3, the viscoelastic and viscoplastic properties of composites have been
shown to affect the long-term strength of laminates. Under a constant load, the viscoelastic and viscoplastic
behaviour of the matrix will cause changes in the structure of the composite by inducing the nucleation of
micro-cracks, which will eventually lead to critical damage modes, such as delamination [26], [83], [88].
In order to predict the long-term mechanical response of laminates, the progression of damage during creep
loading needs to be modelled. A robust tool for designing high-performance composites that considers
structural durability under demanding thermo-mechanical loading can then be developed.
Towards that goal, Park and Schapery [74] developed one of the earliest theoretical models to predict
progressive damage in a linear viscoelastic particulate composite. They derived an expression for
predicting how micro-damage affects the viscoelastic response of the composite. However, the model
required extensive experiments for calibration. Al-Rub et al [84] and Darabi et al [117] combined several
models to account for non-linear viscoelasticity, viscoplasticity and damage in asphaltic materials and
polymer-based composites. A relatively simple procedure was used to predict the effects of damage on the
mechanical performance. Good agreement with experimental data was observed, however, due to the
complex properties being modelled, extensive experimental calibration was required for the different
components of the constitutive model. Marklund et al [27] showed that damage, viscoelasticity and
viscoplasticity could be combined in a relatively simple constitutive law to predict the creep response of a
bio-composite material with a polymer matrix, using a simpler testing sequence where these properties
were individually tested. Although the constitutive model combining damage with viscoelasticity and
viscoplasticity was accurate, the multiplication of damage was not in fact predicted, but instead was
measured experimentally, limiting the usefulness of the model.
Varna et al [103] and Kumar and Talreja [118] developed a CDM model for linearly viscoelastic cross-
plies under static loading. They developed a FE micro-mechanical model combined with a transversely
isotropic linear viscoelastic model for each ply to predict the evolution of the COD under constant strain
and relate it to the stress relaxation induced by matrix micro-cracking. These models provided new insights
into the interaction between damage and viscoelasticity in composites, however, crack multiplication was
not predicted, the analysis was restricted to cross-plies with cracks in a single layer under uniaxial loading,
and the material model was restricted to linear viscoelasticity, which is usually not accurate at high stresses.
Ahci and Talreja [119] proposed a CDM model accounting for non-linear viscoelasticity and damage to
predict the creep response of a cross-ply laminate undergoing matrix micro-cracking, using experimental
data for model calibration. Crack multiplication was not predicted, and the analysis was restricted to cross-
22
plies. Giannadakis and Varna [120] proposed a model for predicting the effect of shear damage on the
viscoelastic-viscoplastic creep response of a [±45]s laminate, which used experimental data to determine
the combined effects of damage and time-dependent behaviour on the creep response of the laminate.
Although the constitutive relation describing the combined effects of damage, viscoelasticity and
viscoplasticity on total strain was accurate, the model was highly empirical, and could not be easily
extended to other systems. Nguyen and Gamby [89] developed a shear-lag model to investigate the effects
of loading rate on cracking in non-linearly viscoelastic CFRP laminates. They found that the multiplication
of micro-cracks under varying strain rates could be related to the rate-dependent fracture behavior of the
transverse plies. However, the model was not applicable to different stacking sequences with more
complex damage evolution scenarios. The analysis focused on crack multiplication, instead of creep strain
prediction. The experimental study of Asadi and Raghavan [32] showed that under creep loading
conditions, a laminate can undergo time-dependent micro-cracking due to the viscoelastic properties of the
matrix. They developed a CDM model to explain the effect of matrix micro-cracking on the creep response
of the laminate, which showed excellent accuracy but was highly empirical. A previous study by Birur et
al [87] showed similar experimental results. Similarly, Ogi and his collaborators [121]–[123] studied the
effect of matrix micro-cracking in a cross-ply on the stress relaxation response. They were able to predict
the effect of micro-cracks on stress relaxation by using a time-dependent shear-lag model. However, crack
multiplication was not predicted and the analysis was restricted to cross-plies with cracks in the 90º layer.
Asadi [26] and Asadi and Raghavan [124] extended these previous works; they studied the evolution of
crack densities in a [±45/902]s CFRP laminate in all layers under different creep loads, and temperatures.
Asadi [26] developed a micro-mechanical variational model to predict the time-dependent increase in
micro-crack density, as well as the performance degradation due to the defects. However, the analysis was
restricted to a specific stacking sequence and required complex numerical modelling to solve the
differential equations describing the distribution of stresses in a the cracked laminate.
From this literature review, it is clear that although there are some models capable of predicting the
effects of viscoelasticity and viscoplasticity on damage evolution and the ensuing creep response of
laminates, to date, there is no definitive physics-based model that can predict both matrix micro-crack
density evolution and creep strain evolution in multi-directional laminates with cracks in multiple layers
under multi-axial loading scenarios.
2.5 Structural analysis of composite structures under low-velocity impact
Composite materials are being increasingly used in structural applications due to their high mechanical
performance and lower density as compared to traditional materials. In the case of automotive structures,
impact performance is an important property to control to guarantee the safety of passengers, as well as
23
pedestrians. Novel composite material systems have been successfully manufactured with improved
impact response [125]–[130], which shows that impact performance can be improved for composites by
optimizing the structure. Under low-velocity impact, the evolution of micro-damage mechanisms might
not be noticeable, however their effect on mechanical properties is very significant, which motivates the
need for an accurate model to predict progressive damage. A damage model for impact damage should be
able to relate the laminate material properties, including ply thickness, stacking sequence and material
system to the evolution of damage throughout the structure. Such a model would allow engineers to
improve the low-velocity impact response of automotive components, guarantee the safety of pedestrians,
and help avoid costly vehicle repairs.
To understand the impact response of advanced laminates, experimental approaches have been used
in the past to investigate the effect of different parameters on micro-damage evolution [126]–[134].
Through a proper control of the material properties and geometrical configurations of the laminate,
significant gains in impact performance have been achieved. For example, Sarasini et al [127] found that
damage evolution in flax/carbon fiber cross-plies depended on the presence of flax plies on the outside or
in the middle of the laminate. Ravandi et al [128] studied the effect of inter-ply stitching on micro-crack
propagation. However, while such studies have been extremely useful in improving the understanding of
impact damage, these experimental works are limited in their ability to generalize their findings to different
material systems. Moreover, these studies are expensive, time-consuming and difficult to conduct. The
results are also sensitive to environmental conditions such as humidity and temperature [135], [136].
On the other hand, numerical studies of micro-damage evolution under impact loading have also been
conducted [137]–[142]. A damage model is typically implemented for each ply, which predicts the loss in
stiffness in terms of applied load. A delamination model can also be used to predict structural failure. Most
of these models are based on Continuum Damage Mechanics (CDM) and make use of empirical parameters
to describe the loss in load-carrying capacity with increasing damage. Some of these models have been
applied to car bumpers [143], [144], allowing for improvements in structural design. However, in all cases,
model parameters need to be obtained prior to simulations, which severely limits the range of applicability
of these models, because parameters need to be experimentally determined for each new laminate structure.
CDM models can predict progressive damage under mechanical loading based on experimentally-
obtained damage parameters. However, the Synergistic Damage Mechanics (SDM) model which has been
developed can help by-pass the need for complicated and costly experiments [59], [105], [107], [145].
SDM is based on micromechanical FEA calculations, which calibrate the parameters of a CDM model,
by-passing costly and complicated experiments. A stiffness degradation matrix is defined for symmetric
laminates, and the evolution of Crack Opening Displacement (COD) with respect to crack density is
calculated using an RVE of the laminate with cracks introduced in the plies. From these FE calculations,
24
the evolution of stiffness with respect to crack density can be calculated without the use of any
experimental measurements. Based on this modelling approach, crack multiplication can also be calculated
using the crack initiation strain in each ply which can be obtained experimentally. The latest developments
of SDM have incorporated multi-axial tensile loading effects [94], [106] by using periodic RVEs of a
cracked laminate. The advantages of this model include its minimal reliance on experimental data for
predictions, accurate modelling of complex multi-axial loading scenarios, and its relative simplicity, which
allows for its implementation in FEA software. Two previous papers have implemented the SDM model
into commercial FEA software to conduct structural analyses which included the effects of progressive
damage [146], [147]. The model was applied to calculate fatigue damage in composite wind turbine blades,
as well as the delamination response. The model has been highly successful in predicting damage evolution
patterns in complex structures, and the progressive failure of the wind turbine leading to full delamination
was predicted accurately.
From this literature review, it is clear that there is a need for an accurate model of rate-dependent
matrix micro-crack evolution at the structural level. Therefore, the model developed in this thesis should
be implemented in FE software as an advanced damage analysis tool that relies on minimal experimental
data, is highly accurate, can be generalized to many materials, and accounts for time-dependent properties.
2.6 Micro-mechanical models to predict damage evolution in short fiber composites
The failure of short fiber polymer-based composites is governed by the evolution of subcritical micro-
damage. The sequence of damage mechanisms, which include fiber-matrix debonding, matrix micro-
cracking and fiber fracture [61], eventually leads to the failure of the material. Due to the mismatch in
elastic properties between the stiff fiber and more compliant matrix, a shear stress concentration occurs at
the fiber tip, which promotes the nucleation of an interfacial crack. This crack may grow along the fiber-
matrix interface, or propagate into the matrix [71], depending on the properties of the interface and matrix.
Experimental studies have shown [61] that once initial damage initiates from the fiber tips, extensive non-
linear deformation occurs in the matrix, which eventually leads to the nucleation of matrix cracks which
propagate throughout the structure. The evolution of these mechanisms is inherently linked to the
microstructure, and to the mechanical properties of the constituents [69].
The process of fiber-matrix debonding has been studied in depth over the past decades both
experimentally and theoretically [148]–[160], due to its fundamental importance in the onset of critical
damage modes in short fiber composites. Computational modelling of fiber-matrix debonding can provide
more detailed insights than experiments into the effects of material properties and reinforcement
configuration on damage evolution in short fiber composites. Initial studies by Tvergaard and his
collaborators [161]–[163] focused on the behavior of metal matrix composites reinforced with ceramic
25
whiskers. They were able to predict the effects of material properties, fiber dimensions and loading
scenarios on the evolution of damage in the form of debonding and fiber fracture. A review of the current
literature shows many significant works on the modelling of micro-damage mechanisms in short fiber
composites since those seminal papers [29], [156], [164]–[170]. Some of these investigations have
focussed on modelling micro-damage mechanisms explicitly through the development of complex RVEs
which take into account heterogeneous microstructures [165], [166], [171] and use non-linear material
models for the matrix, fiber, and interface. Other models have been developed which do not explicitly
model a composite microstructure, but instead predict the behavior of macroscopic composite structures
using analytical relationships relating damage density to the local stress field throughout the structure. For
example, Notta-Cuvier et al [156] used analytical relationships relating debonded length to the strain
applied to a short fiber in order to predict the behavior of a composite consisting of many fibers. These
models rely on accurate analytical relationships in order to make accurate predictions of composite
behavior. The parameters used in these models need to be obtained from experimental data, which can be
complicated to conduct. Lastly, many computational studies have analyzed fiber-matrix debonding in
unidirectional composites under transverse tensile loading [148], [158], [172]–[175]. However, although
these studies are very useful for unidirectional composites, the evolution of micro-damage in short fiber
composites is inherently different due to the size of the fibers which induces stress concentration at the
fiber tips.
While much work has been done on modelling short fiber composites using computational methods,
the effect of strain rate on the behavior of thermoplastic short fiber composites is still not fully understood.
Some of the previous FE simulations which explicitly modelled the complex microstructure [165], [166],
[171], [176] assumed in-plane deformation, and did not look at rate-dependent effects in detail. Moreover,
accurate relationships between debond length, microcrack density and local strain levels still need to be
developed in order for higher level analytical models to make accurate predictions of structural behavior.
Strain rate behavior is especially important to account for in the case of automotive applications, for which
crashworthiness needs to be understood. Recent developments have also shown increased interest from the
automotive industry in thermoplastic short fiber composites, which are known to exhibit significant rate-
dependent behavior [38], [177]. Several experimental studies [29], [90], [178] have also shown that the
rate-dependent properties of the matrix can affect the kinetics of damage in glass fiber composites, as well
as the nature of the damage mechanisms themselves. However, to date, no definitive computational
description of the effect of matrix viscoelasticity on damage processes at the fiber scale has been
undertaken. While experimental measurements can provide an averaged quantification of damage
processes, for example through acoustic emission analysis [179], computational modelling is required in
order to obtain detailed insights into the progression of damage.
26
2.7 Gaps in the literature
The literature review of the previous sections has shown that previous research has been conducted on
the evolution of damage in viscoelastic-viscoplastic composite materials. However, some fundamental
gaps in the literature remain, which can be summarized as follows:
No current model can accurately and efficiently predict the time-dependent multiplication of crack
density under constant multi-axial loading in multi-directional viscoelastic-viscoplastic laminates.
Some models have relied on experimental testing for calibration, and their applicability to different
stacking sequences is limited. Other models have relied on complex micro-mechanical modelling
of ply stress evolution during micro-cracking, but their mathematical formulation restricts them to
particular stacking sequences, thereby preventing their applicability to other composites.
The effect of damage on time-dependent creep strain in laminates is not yet clear. Some previous
models have been developed to quantify this effect, but they are empirical and require extensive
testing for parameter calibration. There is a need for a fundamental understanding of the interaction
between time-dependent properties and damage evolution in laminates.
As shown in the previous sections, studies on the evolution of damage mechanisms in short fiber
composites have not accounted for the effect of the viscoelastic and viscoplastic properties of the
matrix, even though it has been established experimentally that rate-dependent loading affects the
nature of damage mechanisms, as well as the evolution of damage. These effects need to be
accounted for in the development of accurate models to understand the mechanical behavior of
automotive composite materials.
Current structural analyses of composite materials rely on empirical relationships to predict the
evolution of damage and its effect on mechanical performance. These models use simplified
expressions of damage evolution which do not fully account for the physics of crack multiplication
under multi-axial loading, and the non-linear relationship between damage and crack density.
Moreover, rate-dependent behavior of the composites arising from the viscoelastic-viscoplastic
properties of the matrix is usually not modelled. These limitations affect the accuracy of structural
analyses. Therefore, there is a need for the implementation of an accurate damage model into FE
software for the analysis of realistic composite structures under dynamic loading.
2.8 Thesis methodology
The SDM model developed by Singh and Talreja [45] and extended by Montesano and Singh [94] has
proven to be a highly versatile and efficient framework for predicting damage evolution in multi-
directional laminates under multi-axial loading scenarios. It has been combined with an energy-based crack
27
multiplication model [109], [180] to predict crack density evolution with respect to stress. This multi-scale
methodology can easily be applied to many stacking sequences and requires minimal experimental input:
the elastic properties of the plies, and the ply crack initiation strain. Based on these inputs, micro-damage
FE models are generated to calculate the parameters of a Continuum Damage Mechanics model. These
equations can be used to predict stiffness degradation under complex multi-axial loading conditions. In
this thesis, the strengths of the SDM model are further improved by including time-dependent properties
to predict progressive damage evolution under creep deformation. This improvement would provide
further understanding into the nature of the effect of viscoelastic-viscoplastic matrix behavior on damage
evolution in laminates.
The SDM methodology is however restricted to laminates, in which micro-cracks are assumed equally-
spaced, parallel to the continuous fibers, and to span the thickness of the ply and the width of the laminate.
Damage evolution in short fiber composites has been shown experimentally to be fundamentally different
and less regular. Although models are available for these more complex micro-structures, they do not
account for the viscoelastic and viscoplastic properties of the matrix. Considering the strengths of the SDM
methodology, a highly effective approach for understanding the evolution of micro-damage mechanisms
is the development of a micro-mechanical FE model, where damage mechanisms are explicitly taken into
account. Such a methodology would by-pass the need for complicated experimental calibration, and could
efficiently assess the effects of fiber, matrix and interfacial properties on the progression of damage.
Moreover, the interaction between the viscoelastic-viscoplastic properties of the matrix, and the evolution
of damage could be modelled, and provide insights into experimental observations of damage evolution in
polymer-based short fiber composites. This would constitute a key contribution to the field of damage
mechanics of short fiber composites.
Lastly, the SDM model can be implemented into FE software through user-defined subroutines, which
could include rate-dependent behavior. This would extend the applicability of SDM to the structural scale
and would also provide improved insights into the effects of viscoelasticity and viscoplasticity on damage
evolution at the component level under dynamic loading. The application of interest in this thesis would
be low-velocity impact of an automotive component.
In summary, the overall goal of this thesis is the development of a multi-scale framework to predict
damage evolution in viscoelastic-viscoplastic composites, both with continuous fiber and short fibers. This
multi-scale model should span from the microscopic scale of the fiber-matrix interface, to the structural
level of an automotive component, and include the mesoscopic level of the homogenized plies and the
continuum level of a laminate.
28
Chapter 3. Methodology
In this chapter, the methodology used to address the objectives of the thesis detailed in the previous
chapters are provided. The main equations and concepts are explained, and the numerical implementation
in FE and numerical software is given.
3.1 Prediction of damage in viscoelastic-viscoplastic laminates under multi-axial creep loading
The Synergistic Damage Mechanics (SDM) model is a CDM-based damage model that can quantify
the stiffness degradation due to matrix micro-cracks in the plies of a symmetric laminate. For each ply
orientation, the micro-cracks in a given layer will be parallel to the fibers. A mode of damage corresponds
to a specific crack orientation. Each mode of damage in the different plies of the laminate is described in
terms of a damage tensor. The reduced stiffness due to the matrix micro-cracks can then be calculated with
evolving crack densities in the different plies. In SDM, it is assumed that cracks are parallel to each other,
equally spaced, span the whole thickness of the ply and the whole width of the laminate (Figure 3-1).
Although a given layer may have micro-cracks, it can still carry load because each micro-crack is
constrained by the adjoining plies of the laminate that are oriented at different angles. Moreover, although
it has been observed that micro-cracks might not open completely throughout the width of the laminate,
for the purposes of this thesis, it is assumed that once micro-cracks nucleate, they span the width of the
laminate and the thickness of the ply.
29
Figure 3-1. Schematic showing the different modes of ply micro-cracking in a multi-directional laminate. VE refers
to the viscoelastic properties of the plies; VP refers to the viscoplastic behaviour; QS is for quasi-static loading. The
top right graph shows the loading scheme used in the current thesis.
3.1.1 Synergistic Damage Mechanics (SDM) model
Each mode of damage (𝛼) corresponds to a different ply crack orientation dictated by the ply
orientation, and the damage in the ply is represented by a damage tensor of the following form:
𝐷𝑖𝑗(𝛼) =
𝜅𝛼𝑡𝛼2
𝑠𝛼𝑡𝑛𝑖𝑛𝑗 = 𝐷𝛼𝑛𝑖𝑛𝑗 (3.1)
where, 𝑡𝛼 is the thickness of the cracked ply, 𝑠𝛼 is the spacing between cracks in the ply, 𝑡 is the thickness
of the laminate, 𝑛𝑖 represents the components of the vector normal to the crack surface relative to the
loading direction, and 𝜅𝛼 accounts for the effect of adjacent cracks on the COD of the ply crack. Due to
the interactions of the stress fields between cracks in different layers, and within the same layer, 𝜅𝛼 will
change depending on the crack density of each ply. The stiffness of a laminate that has undergone
progressive ply cracking can be defined by:
𝐶𝑖𝑗 =
[
𝐸𝑥0
1 − 𝜈𝑥𝑦0 𝜈𝑦𝑥
0
𝜈𝑥𝑦0 𝐸𝑦
0
1 − 𝜈𝑥𝑦0 𝜈𝑦𝑥
0 0
𝜈𝑥𝑦0 𝐸𝑦
0
1 − 𝜈𝑥𝑦0 𝜈𝑦𝑥
0
𝐸𝑦0
1 − 𝜈𝑥𝑦0 𝜈𝑦𝑥
0 0
0 0 𝐺𝑥𝑦0]
−∑𝑏𝛼𝐷𝛼 [
2𝑎1(𝛼) 𝑎4
(𝛼) 0
𝑎4(𝛼) 2𝑎2
(𝛼) 0
0 0 2𝑎3(𝛼)
]
𝛼
(3.2)
30
𝐶𝑖𝑗 is the 3x3 stiffness tensor of a symmetric laminate written in Voigt notation, under plane stress. 𝐸𝑥0,
𝐸𝑦0 are the Young’s moduli in axial and transverse directions, 𝜈𝑥𝑦
0 , 𝜈𝑦𝑥0 are the major and minor Poisson’s
ratios, and 𝐺𝑥𝑦0 is the shear modulus of the undamaged laminate. The first term is the stiffness of the
undamaged laminate and can be calculated using Classical Laminate Theory (CLT) from the ply properties.
The second term represents the reduction in stiffness of the laminate due to matrix micro-cracks. It depends
on the damage tensor and a set of stiffness degradation parameters 𝑎[1−4]𝛼 , where there are 4 parameters
for each mode of damage 𝛼. The parameter 𝑏𝛼 depends on the stacking sequence of the laminate, and is
equal to 1 for the ply adjacent to the mid-plane of the laminate and 2 for all other plies. The parameters
𝑎[1−4]𝛼 are obtained by computing the decrease in stiffness of the laminate using FE micro-damage model
for a given crack density; they do not change with increasing crack density.
3.1.2 FE micro-damage model
FE micro-damage modelling of a laminate with matrix micro-cracks is used to evaluate the effect of
matrix micro-cracking on the stiffness of the laminate. The individual plies of the laminate are modelled
in FE using a representative volume element (RVE) with transversely isotropic elastic ply properties. The
required input parameters are elastic constants of the plies, their thickness and the stacking sequence of the
laminate. The cracks are represented in the FE mesh by disconnecting nodes across pre-defined crack
surfaces. Using periodic boundary conditions to the RVE, the crack density can be changed by changing
the size of the model. A certain amount of strain is then applied to the laminate, and the Crack Opening
Displacements (COD) of the cracks in the different layers of the laminate are then calculated. For each
mode of damage, the evolution of the COD of each layer with increase in crack density is recorded. As the
crack density increases in a given layer, the COD for cracks in this same layer decreases due to the
interaction between the stress fields of the cracks. Using the results of these simulations, the evolution of
the COD in terms of the crack density in each layer is approximated by a function consisting of three fitting
parameters, in the following form:
𝜅𝛼 = (∆𝑢2̅̅ ̅̅ ̅)𝛼휀𝑒𝑓𝑓𝑡𝛼
=𝑐1
1 + (𝑐2𝜌𝛼)𝑐3
(3.3)
where (∆𝑢2̅̅ ̅̅ ̅)𝛼 is the average crack opening displacement (COD), 𝜅𝛼 is the normalized COD, and 휀𝑒𝑓𝑓 is
the effective strain causing the cracks to open given by 휀𝑒𝑓𝑓 = 휀22 + 𝜈12휀11 +1
8𝜈21𝛾12. 𝑡𝛼 is the thickness
of the cracked ply, 𝜌𝛼 is the crack density corresponding to the mode of damage 𝛼, and 𝑐1, 𝑐2, and 𝑐3 are
31
fitting parameters. Once the fitting parameters are obtained, a relationship between crack density and COD
can be established, and the COD can be predicted for a known crack density using the above relation. The
construction of the FE micro-damage model has been discussed in further detail by Montesano and Singh
[94]. Lastly, the average COD appearing in Eq. (3.3) is given by the following equation: (∆𝑢2̅̅ ̅̅ ̅)𝛼 =
1
𝑡𝛼∫ 𝛥𝑢𝑦(𝑧)𝑑𝑧𝑡𝛼2
−𝑡𝛼2
, where 𝛥𝑢𝑦(𝑧) is the distance between nodes on each side of the crack.
3.1.3 Energy-based crack multiplication model
The SDM model described in the previous section can predict the stiffness degradation for a given
state of damage or density of matrix micro-cracks. In order to predict the constitutive response of a laminate
undergoing progressive damage, the evolution of the density of matrix microcracks is required. The
evolution of crack density in each layer of the laminate can be evaluated by using an energy-based model
[106]. This model is inspired by Irwin’s theory of fracture mechanics [109], which states that the strain
energy released upon crack propagation is equal to the amount of work required to close the cracks
following their nucleation. This theory has been modified to predict the multiplication of cracks in the
different layers of the laminate [106], [109].
Based on the evolution of COD with the density of matrix micro-cracks obtained using the FE micro-
damage model, the energy density release rate for crack multiplication can be obtained using the following
equation:
𝑊𝐼 = (𝜎2𝛼)2𝑡𝛼𝐸2
[2�̃�𝑛𝛼 (𝑠𝛼2) − �̃�𝑛
𝛼(𝑠𝛼)] (3.4)
Here, �̃�𝑛𝛼(𝑠𝛼) is the normalized COD for a crack spacing 𝑠𝛼 (equal to 𝜅𝛼), 𝐸2 is the modulus of the ply
in the transverse direction, and 𝜎2𝛼 is the local ply transverse stress (perpendicular to the crack surfaces).
𝜎2𝛼 is calculated based on the local ply strain values, and the linear elastic properties of the plies. Using the
strain energy density release rate calculated with Eq. (3.4), a numerical procedure implemented in
MATLAB is used to predict crack density evolution versus applied global strain on the laminate.
In order to predict crack multiplication, the critical strain energy density release rate 𝐺𝐼𝑐 of the ply is
required. 𝐺𝐼𝑐 is itself calculated based on experimental data on ply cracking, combined with a numerical
procedure described in [106]. After the crack density in each layer has been evaluated, the stiffness of the
laminate can be determined by using Eq. (3.1) - (3.3) with the parameters obtained from the FE micro-
damage model. To incorporate different material models into the framework of SDM model, the global
strain in the material should be determined using the material model. Next, the local stress (𝜎2𝛼) can be
computed from the local strain, which is obtained by transforming the global strain to ply coordinate
32
system. From there, the energy release rate can be calculated using Eq. (3.4), and the crack density can be
updated.
The crack multiplication model is implemented into MATLAB. At the beginning of the numerical
simulations, each ply is divided into a certain number of segments. This number of segments is given by
two parameters, as follows: 𝑁𝑣 =𝐿𝑜
𝑡𝑝𝑙𝑦. 𝐿𝑜is the length of the laminate, which can be set equal to the length
of the laminates that have been studied experimentally in the literature; 𝑡𝑝𝑙𝑦 is the thickness of the ply; 𝑁𝑣
is the maximum number of cracks in the ply. Once 𝑁𝑣 has been computed, a for loop is initiated; at each
iteration of the for loop, the energy release rate for crack multiplication is calculated using Eq. (3.4); this
energy effectively corresponds to the amount of strain energy that would be released if the segment under
consideration underwent cracking. If 𝑊𝐼 > 𝐺𝐼𝑐, the segment is assumed to have cracked; the number of
cracks in the ply is incremented and, conversely, the number of segments available for cracking is
decremented by 1. The for loop continues to the next segments and the number of cracks is incremented
until the crack multiplication criterion is not satisfied anymore. At that point, the for loop is terminated
and the next loading step is applied. If the crack multiplication criterion is always satisfied, the loop stops
after 𝑁𝑣 iterations, at which point the number of cracks has reached its maximum.
As per the work of Montesano and Singh [106], stochastic behavior is taken into account in this damage
model. At each iteration of the for loop described in the previous paragraph, the value of 𝐺𝐼𝑐is in fact
computed using the following equation: 𝐺𝐼𝑐 = 𝐺𝑜 [𝑙𝑛 (1
1−𝐹)]
1
𝑚, where 𝐺𝑜and 𝑚 are Weibull parameters,
and 𝐹 is a randomly generated number between 0 and 1. Therefore, at each iteration of the loop, there will
be slight variations in the values of 𝐺𝐼𝑐, which has been found to better match experimental data [181].
This can be explained based on the random location of defects in the plies such as inhomogeneous fiber
distributions or voids between the fibers. The parameters 𝐺𝑜and 𝑚 can be adjusted to better represent the
distribution of defects in each ply.
It has been observed experimentally [182] that 𝐺𝐼𝑐 increases with increasing crack density. This effect
has been taken into account in this thesis using the following relationship 𝐺𝐼𝑐 = 𝐺𝐼𝑐𝑜 + 𝐺𝐼𝑐𝑟(1 − 𝑒𝑥𝑝(𝑟𝜌𝛼)
where 𝐺𝐼𝑐𝑟 = 0.8 describes the magnitude of the increase in critical energy release rate for crack
multiplication, and 𝑟 = 15 is a material parameter. 𝐺𝐼𝑐𝑜 is the stochastic value of critical energy release
rate obtained using the Weibull distribution.
33
3.1.4 Viscoelasticity and viscoplasticity models
The creep strain of a viscoelastic-viscoplastic ply can be calculated as a function of time for different
creep loading levels using Schapery’s thermodynamics-based constitutive model [110]. The creep strain
tensor can be defined using the following equation:
휀𝑖(𝑡) = [𝑔0𝑡𝐴0 + 𝑔1
𝑡𝑔2𝑡∑𝐴𝑟[1 − 𝑒
−𝜆𝑟𝑡
𝑎𝜎𝑇𝑡]
𝑘
𝑟 =1
] 𝜎𝑖 + {𝐶𝑖𝜎𝑖𝑁𝑡}
𝑛 (𝑖 = 2,6)
(3.5)
Where 𝑖 = 2 for the transverse strain component, and 𝑖 = 6 for the engineering shear strain component.
The first term in square brackets represents the viscoelastic response of the material. The 𝑔[0−2]𝑡 and 𝑎𝜎𝑇
𝑡
parameters are stress- and temperature-dependent parameters describing the effect of non-linearity on the
compliance. The summation term corresponds to a Prony series consisting of 𝑘 terms, where 𝐴𝑟 is the
increase in time-dependent compliance corresponding to the 𝑟𝑡ℎterm of the Prony series, and 𝜆𝑟is the
inverse of the time-constant describing the rate of increase in compliance with time. The viscoelastic
model, as well as the non-linearizing parameters, are consistent with Schapery’s formulation of non-linear
viscoelasticity [110]. The second term represents viscoplastic strain expressed in the form of a Zapas-
Crissman [183] functional, which is commonly used to describe the viscoplastic behavior of composite
materials. The constants 𝐶, 𝑁 and 𝑛 define the viscoplastic response of the material. The parameters to
describe the response of the composites investigated here were obtained through fits to experimental data,
as explained by Tuttle et al [24].
For orthotropic viscoelastic-viscoplastic plies, the transverse and shear strains are time-dependent,
while the longitudinal strain can be assumed to be linearly elastic due to the large stiffness of the fibers
[24]. In order to evaluate the viscoelastic-viscoplastic behaviour of the laminate, an approach utilizing the
framework of CLT developed by Dillard [85], and Tuttle and Brinson [184] is adopted in this work. At
each creep time-step in the simulation, the local stress in the ply is used to obtain the non-linear parameters
and the local ply compliance using Eq. (3.5), from which the overall laminate non-linear time-dependent
compliance is obtained using CLT. The unconstrained strain in each ply which would occur if the ply were
not constrained by the overall laminate is then obtained, resulting in an equivalent laminate load, which is
used with the laminate compliance to calculate the overall creep strain of the laminate. Local ply stresses
are then adjusted to account for the difference between desired unconstrained ply strain, and actual
laminate strain. The details of the algorithm implemented in MATLAB are provided in the section 3.1.7.
It should be noted that the critical energy density release rate for viscoelastic materials is dependent
on loading rate [89]. The stresses in the plies as well vary slightly over time due to the different viscoelastic
compliance of each layer. Therefore, the critical energy density release rate can be expressed as a function
of time. In order to obtain the time-dependent form of the critical energy release rate, Asadi and Raghavan
34
[185] performed experiments on individual plies for the same strain rate and at different temperatures, and
recorded the failure strain and stress. Using a model based on thermal activation theory, they obtained the
time-dependent critical energy release rates from their experiments. In order to predict crack density
evolution during creep loading, when the time-dependent properties of the plies will dominate, in this work
the form of 𝐺𝐼𝑐(𝑡) was chosen to best fit available experimental data of crack density evolution versus
time. The values of 𝐺𝐼𝑐(0) for the different plies were obtained using the numerical procedure of
Montesano and Singh [94]. The precise forms of 𝐺𝐼𝑐(𝑡) for each ply for the laminates considered in this
study are given in chapters 4 and 5.
3.1.5 Combined viscoelastic-viscoplastic-damage model
The present thesis is concerned with the progression of damage during creep loading, and the resulting
creep strain. During creep loading, the crack density in the different plies of the laminate will keep
increasing due to the increase in strain. The model calculates the effect of damage on the time-dependent
behavior through the following strain formulation:
휀𝑖𝐿(𝑡) = 𝑆𝑖𝑗
𝐿 (𝐷)𝜎𝐶𝑅𝐸𝐸𝑃,𝑗𝐿 + 휀𝐶𝑅𝐸𝐸𝑃,𝑖
𝐿 (𝑡) (3.6)
In Eq. (3.6), 휀𝑖𝐿(𝑡) is the time-dependent strain tensor that includes the effects of damage (with 3
components as per the plane-stress approximation, and in Voigt notation). The first term is the elastic part
of the strain under a creep load 𝜎𝐶𝑅𝐸𝐸𝑃,𝑗𝐿 and depends on the damage variable 𝐷 that dictates the increase
in laminate compliance 𝑆𝑖𝑗𝐿 due to damage. The increase in laminate compliance is obtained using the SDM
model using Eq. (3.1)-(3.3) [186]. The second term is the viscoelastic-viscoplastic strain as per Eq. (3.5).
During creep loading, the increase in local ply strain will lead to an increase in the crack density and COD.
The increase in crack density during constant loading will increase the state of damage expressed in terms
of the damage variable 𝐷, thereby enhancing the creep response through its effect on the elastic component
of strain as per Eq. (3.6). Therefore, time-dependent properties will affect damage evolution and the total
creep strain in two ways. First, the time-dependent strain will cause an increase in the local ply strain
during loading, which will increase the time-dependent crack density. Second, the time-dependent strain
will directly increase the total creep strain. In this way, therefore, even though damage is assumed not to
affect the purely time-dependent response of the laminate, the time-dependent response of the plies affect
damage evolution.
35
3.1.6 Numerical implementation
The SDM model combined with a viscoelastic-viscoplastic material model has been implemented in
MATLAB and the computation steps have been demonstrated in the flowchart in Figure 3-2. The
relationship between crack density and COD has been determined from FE micro-damage model
independently. The viscoelastic-viscoplastic material model parameters were determined from the
literature [24], [26]. The simulation was performed in two steps, (a) quasi-static loading and (b) creep
deformation. At each simulation step, the local energy density release rate for crack multiplication was
calculated using Eq. (3.4). The crack density was then updated when the energy release rate (𝑊𝐼) was
found to be higher than the critical energy release rate (𝐺𝐼𝑐) using the methodology described by
Montesano and Singh [106]. Using the SDM model, the stiffness of the laminate can then be updated, and
the stress-strain response predicted. The axial and transverse loads are increased to their respective creep
load levels (𝜎𝑥0 and 𝜎𝑦
0). Following the quasi-static loading, the loads were maintained constant. A similar
procedure was followed for time stepping during the viscoelastic-viscoplastic deformation except that the
time dependent critical energy release rate (𝐺𝐼𝑐(𝑡)) was used to determine the crack multiplication. Based
on the time-dependent crack density, the total creep strain of the laminated can be computed using Eq.
(3.6).
36
Figure 3-2. Flowchart explaining the MATLAB program used to performed the progressive damage simulations
under viscoelastic-viscoplastic creep.
3.1.7 Detailed implementation of the model into MATLAB
The constitutive model presented in this chapter has been implemented into MATLAB. Here, the steps
implemented into the code are explained. The algorithm is partly based on the method used to calculate
residual thermal strains in a composite undergoing thermal stresses [6]. In this section, the non-linear
viscoelastic-viscoplastic model parameters obtained from the literature are also provided.
3.1.7.1 Quasi-static loading
The elastic properties of a laminate are dependent on the elastic properties of the individual plies, the
thickness of the plies, and the stacking sequence. In the remainder of this thesis, the mechanical response
Apply stress increment dσ to laminate
Get local crack driving stress 𝜎2𝛼using CLT.
Calculate WI (Eq. 3-4) using 𝑢𝑛𝛼 computed from Eq. 3-3. If WI>GIc
increment crack density using the crack multiplication model
Update laminate stiffness matrix (Eq. 3-2). Obtain laminate strain from new
stiffness and stress values.
Apply time increment dt
Get laminate creep strain 휀𝐶𝑅𝐸𝐸𝑃,𝑖𝐿 (𝑡) using material model (Eq. 3-5)
Calculate WI (Eq. 3-4). If WI> GIc (t), increment crack density using the crack
multiplication model
Update elastic part of compliance (Eq. 3-2). Get elastic strain
𝑆𝑖𝑗𝐿𝜎𝐶 𝑁𝑆𝑇.𝑗𝐿 𝑡 and total strain 휀𝑖
𝐿(𝑡)(Eq. 3-6)
No
Yes
Input ply properties and damage model parameters 𝑎 1−4𝛼 , 𝑐1−3, 𝐺𝐼𝑐
Initialize time t=0 and set 휀𝐶𝑅𝐸𝐸𝑃,𝑖𝐿 (0) =0
Yes
Output variables of
interest (e.g. crack
density, strain, stiffness)
No
𝜎𝑥 = 𝜎𝐶 𝑁𝑆𝑇,𝑥 𝜎𝑦 = 𝜎𝐶 𝑁𝑆𝑇,𝑦
T = tend
37
of a laminate undergoing damage will need to be calculated. The equations used to obtain the elastic
response of a laminate will be given here, and are explained in detail in the textbook by Herakovich [6].
The thickness of each ply in the laminate is given by 𝑡𝑘. Their orientation is denoted 𝜃𝑘. The direction
along the unidirectional fibers is denoted by “1”, while the local transverse direction is denoted by “2”.
The out-of-plane thickness direction is denoted by “3”. In the case of the coordinate system of the laminate,
the loading axis, which is parallel to direction “1” in the ply with orientation 𝜃𝑘 = 0°, is denoted as “x”.
The transverse laminate direction, which is parallel to the direction “1” in the ply with orientation 𝜃𝑘 =
90° is denoted by “y”. The out-of-plane direction, which in this thesis is always parallel to the ply direction
“3”, is denoted by “z”.
To obtain the relationship between stress and strain in a ply, plane stress conditions are assumed, such
that the only non-zero components of the Cauchy stress tensor are the in-plane components. For the
purposes of this thesis, the analysis is restricted to in-plane loading conditions. Therefore, the ply strain
tensor and the ply stress tensor can be expressed as single columns.
In order to calculate the stiffness matrix of a multi-directional laminate, the stiffness tensor of an
individual ply is needed. In the composites studied in this thesis, it is assumed that the ply is transversely
isotropic. The Young’s modulus of the ply in the “1” direction (parallel to fibers) is denoted 𝐸1, while the
Young’s modulus in the “2” direction (perpendicular to fibers) is denoted 𝐸2. The major Poisson’s ratio,
which relates the transverse strain to the applied axial strain, is denoted 𝜈12. The shear modulus is denoted
𝐺12. The relationship between stress and strain for an individual ply is then given by:
[
𝜎1𝜎2𝜏12] =
[
𝐸11 − 𝜈12 𝜈21
𝜈12𝐸21 − 𝜈12 𝜈21
0
𝜈12𝐸21 − 𝜈12 𝜈21
𝐸21 − 𝜈12 𝜈21
0
0 0 𝐺12]
[
휀1휀2𝛾12] (3.7)
Where 𝛾12is the engineering in-plane shear strain, and 𝜏12is the in-plane shear stress. These columns
are expressed in the local ply coordinate system (1-2-3), and the transverse symmetry of the ply prevents
coupling between normal stresses and shear deformation. We also have 𝜈21 = 𝜈12𝐸2
𝐸1 as the minor Poisson’s
ratio. In the equation above, it is assumed that the coordinate system in which the stress and strain tensors
are expressed is aligned with the fibers. The stress values appearing in the stress tensor, and the strain
values appearing in the strain tensor depend on the coordinate system in which these tensors are expressed.
When the coordinate system is rotated relative to the longitudinal axis, the values appearing in these two
vectors will change.
When a ply is oriented at an angle 𝜃𝑘 different from 0° relative to the loading axis, the relationship
between the global stress tensor, [𝜎]𝐺𝐿 𝐵and the global strain tensor, [휀]𝐺𝐿 𝐵 expressed in the coordinate
38
system of the laminate depends on the angle 𝜃𝑘. In order to obtain the relationship between the global
stress tensor and the global strain tensor, the stiffness matrix of the ply oriented at an angle relative to the
loading direction “x” needs to be modified. To do so, two coordinate transformation matrices are defined
as follows:
𝑇1 = [𝑚2 𝑛2 2𝑚𝑛𝑛2 𝑚2 −2𝑚𝑛−𝑚𝑛 𝑚𝑛 𝑚2 − 𝑛2
]
𝑇2 = [𝑚2 𝑛2 𝑚𝑛𝑛2 𝑚2 −𝑚𝑛−2𝑚𝑛 2𝑚𝑛 𝑚2 − 𝑛2
]
(3.8)
Where 𝑚 = 𝑐𝑜𝑠 (𝜃𝑘) and 𝑛 = 𝑠𝑖𝑛(𝜃𝑘). These matrices relate the global stress tensor and the global strain
tensor to the local stress tensor and the local strain tensor, according to the following relationships:
[
𝜎1𝜎2𝜏12] = 𝑇1 [
𝜎𝑥𝜎𝑦𝜏𝑥𝑦]
[
휀1휀2𝛾12] = 𝑇2 [
휀𝑥휀𝑦𝛾𝑥𝑦]
(3.9)
In order to obtain the stiffness tensor for a ply oriented at angle 𝜃𝑘with respect to the loading axis, the
following equation is used:
Q̅ = 𝑇1−1Q𝑇2 (3.10)
Where Q is the stiffness tensor of the ply in the ply’s coordinate system (see Eq. 3.7), and Q̅ is the ply’s
stiffness expressed in the global coordinate system of the laminate.
Once the global stiffness tensor of each ply has been obtained, the total stiffness of the laminate
sequence can be obtained using the following expression
𝐶𝑝𝑞 =1
𝑡𝑡𝑜𝑡∑ 𝑡𝑘
𝑛𝑝𝑙𝑦
𝑘=1
Qpq̅̅ ̅̅ ̅ (3.11)
Where 𝐶𝑝𝑞is the overall stiffness tensor of the laminate, and 𝑡𝑡𝑜𝑡is the thickness of the laminate. From the
stiffness of the laminate, the elastic response under any in-plane loading scenario can be calculated,
provided the laminate has not undergone any form of damage which would degrade the elastic properties
of the composite.
The first part of the simulations corresponds to quasi-static loading up to the creep load level. During
this portion of the simulations, the viscoelastic and viscoplastic properties of the plies are ignored. This is
39
reasonable based on the extensive time scales over which time-dependent deformation start becoming
significant. The damage evolution is calculated using the crack multiplication model, and the stiffness
degradation occurring during the quasi-static loading is obtained using the SDM model.
At each loading step, the value of applied stress on the laminate is incremented. Based on the stiffness
of the laminate at that step, the laminate overall strain can be calculated using the following equation:
휀𝑖 = 𝑆𝑖𝑗(𝐷)𝜎𝑗 (3.12)
Where D describes the level of damage in the laminate, and 𝑆 = 𝐶−1 is the compliance matrix of the
laminate. The transverse stress in each ply can then be updated using the elastic properties of the individual
ply:
𝜎2 =𝐸2
1 − 𝜈12𝜈21𝜖2 −
𝜈12𝐸21 − 𝜈12𝜈21
𝜖1 (3.13)
Based on the updated transverse stress of each ply, the energy release rate for cracking in each ply can
be calculated, using Eq. (3.4). The normalized COD values appearing in Eq. (3.4) are obtained from Eq.
(3.3), where the 𝑐[1−3] parameters were derived from the FE micro-damage model.
The crack multiplication model accounts for the stochasticity in ply micro-cracking that has been
reported experimentally [30], and which Montesano and Singh implemented numerically [106]. Moreover,
the increase in critical energy release rate (𝐺𝐼𝑐) with higher crack density has also been accounted for (see
section 3.1.3). At the start of the quasi-static simulations, each ply is divided into a certain number of
segments. At each loading step, a for loop is run through all the intact segments in each ply; the energy
release rate calculated using Eq. (3.4) is compared to the critical energy release rate at each segment. If
𝑊𝐼 > 𝐺𝐼𝑐, the number of cracks is incremented by 1. As soon as this crack multiplication condition is not
satisfied anymore, the crack density in the ply is calculated, and the loop is terminated. This is performed
for each ply.
Once the crack density in each ply has been obtained by using the crack multiplication model, the
stiffness of the damaged laminate is calculated using Eq. (3.1) - (3.3) The laminate strain is then updated
as per Eq. (3.12).
3.1.7.2 Viscoelastic and viscoplastic material properties
The approach used to calculate the creep response of viscoelastic-viscoplastic laminates was taken
from the literature on time-dependent behavior in composites [26], [32]. The first step in the algorithm is
to provide the model parameters, and the viscoelastic-viscoplastic material model, as per Eq. (3.5). The
stress-dependent and temperature-dependent parameters are obtained for a carbon-fiber/bismaleimide
IM7/5260 composite studied by Tuttle et al [24].
40
Different material systems are characterized by different non-linear viscoelastic behavior. In the results
reported in the next chapter, the exact form of Eq. (3.5) is changed by replacing the Prony series description
of the time-dependent compliance by a Kohlrausch-Williams-Watts (KWW) stretched exponential, so that
the compliance of a ply is given by the following equation:
𝑆𝑖𝑗(𝑡) = 𝑔0𝑆𝑖𝑗0 + 𝑔1𝑔2𝛥𝑆𝑖𝑗 (1 − 𝑒𝑥𝑝 (−[(
𝑡
𝑎𝜎)𝑐
])) (𝑖 = 2,6) (3.14)
The 𝑖 and 𝑗 subscripts denote the component of the strain tensor. In the current work, the Voigt notation is
used, and it is assumed that plane stress conditions apply in the laminate. Therefore, a value of 𝑖 equal to
2 will denote the transverse component of the strain tensor, while a value of 6 denotes the engineering
shear strain component. The other terms in the compliance matrix of the ply are time-independent.
The 𝑔[0−2]parameters are temperature- and stress-dependent non-linearizing parameters which depend
on the material system. For the carbon-fiber/epoxy material studied by Asadi and Raghavan [26], [32],
[185], these non-linearizing parameters are given in [32].
In the case of the GFRP material system studied by Megnis and Varna [187], the viscoelastic-
viscoplastic model is different. The creep strain of a viscoelastic-viscoplastic ply can be calculated as a
function of time for different creep loading levels using Schapery’s thermodynamics-based constitutive
model [110]. The creep strain is defined for GFRP using the following equation [187]:
where 𝑖 = 2 for the transverse strain component, and 𝑖 = 3 for the shear strain component. The first term in
square brackets represents the viscoelastic response of the material. In the material system studied here,
Megnis and Varna [187] showed that the viscoelastic deformation could be assumed to be linear. The
summation term corresponds to a Prony series consisting of 𝑘 terms, where 𝐴𝑟 is the increase in time-
dependent compliance corresponding to the 𝑟𝑡ℎterm of the Prony series, and 𝜏𝑟 is the time-constant
describing the rate of increase in compliance with time. The second term in Eq. (3.15) represents
viscoplastic strain expressed in the form of the product of a stress-dependent function, and a time-
dependent function. The exact expressions for these functions are given by: 휁66 = 5.974 ∙ 10−9𝜏12
2 +
3.419 ∙ 10−8𝜏12 + 3.621 ∙ 10−7 where 𝜏12 is the shear stress in MPa, and 𝜉(𝑡) = 0.003𝑡0.43. We should
note that according to the experimental study, viscoplastic deformation only occurred in the shear direction.
Following the implementation of Eq. (3.15) for each ply, a numerical procedure developed by Dillard et
휀𝑖(𝑡) = [𝐴0 +∑ 𝐴𝑟[1 − 𝑒−𝑡𝜏𝑟]
𝑘
𝑟 =1
] 𝜎𝑖 + 휁𝑘𝑖(𝜎)𝜉(𝑡)𝜎𝑖 (𝑖 = 2,3) (3.15)
41
al. [85] was used to obtain the creep behavior of the laminate from the creep behavior of the individual
plies.
In order to obtain the creep deformation of the laminate, the following steps are implemented into
MATLAB following the quasi-static simulations, in a series of functions:
1. Increase the value of the time parameter by the time-step value, dt
2. For each ply, calculate the time-dependent components of the compliance matrix at time t, 𝑆22 and
𝑆66. This is done by using the expressions for the non-linearizing parameters, and Eq. (3.5)/Eq.
(3.14)/Eq.(3.15), depending on the material system of interest. The local ply stress tensor [
𝜎1𝜎2𝜏12] is
used to calculate the non-linearizing parameters. The time-independent components of the
compliance matrix are obtained using the linear elastic properties of each ply. The resulting matrix
is denoted by 𝑆𝑖𝑗𝑡
3. For each ply, repeat the procedure described in point 2, but evaluate the compliance matrix at time
t + dt, where dt is the timestep used in the simulations. It is assumed that the local ply stress tensor
is constant over the time step. The resulting matrix is denoted by 𝑆𝑖𝑗𝑡+𝑑𝑡.
4. Calculate the increase in strain that each ply would undergo if it were not constrained by the other
neighboring plies by using the following equation, where 𝛥휀𝑖 is the increase in the local ply strain
tensor:
𝛥휀𝑖 = (𝑆𝑖𝑗𝑡+𝑑𝑡 − 𝑆𝑖𝑗
𝑡 )𝜎𝑗
5. Using the compliance matrix 𝑆𝑖𝑗𝑡 calculated for each ply at time t in step 2, obtain the compliance
matrix of the laminate at time t using the following equations, based on Classical Laminate Plate
Theory [6]:
Qply = St−1
QLAM̅̅ ̅̅ ̅̅ ̅ =1
𝑡𝑡𝑜𝑡∑ 𝑇1
−1Qply𝑇2𝑡𝑘𝑛𝑝𝑙𝑦
𝑆̅𝐿𝐴𝑀 = QLAM̅̅ ̅̅ ̅̅ ̅−1
In the second of these equations, the matrices 𝑇1and 𝑇2 are used to obtain the stiffness of each ply
in the global coordinate system of the laminate. They are provided above.
6. Using the stiffness matrix of each ply obtained in step 5, as well as the unconstrained strain for
each ply obtained in step 4, calculate the equivalent load per unit width necessary to cause such
deformation, using the following equation, where 𝑡𝑝𝑙𝑦 is the thickness of the ply:
42
𝛥𝑁𝑖 = 𝑄𝑖𝑗𝑝𝑙𝑦̅̅ ̅̅ ̅̅𝛥휀𝑗𝑡𝑝𝑙𝑦
7. Obtain the total load per unit width that would have to be applied to the laminate to cause the
unconstrained deformation of each ply, using the following equation
𝛥𝑁𝐿𝐴𝑀 = ∑ 𝛥𝑁𝑖𝑛𝑝𝑙𝑦
8. Using the laminate compliance obtained in step 5, and the equivalent load increment obtained in
step 7, obtain the increment in strain of the whole laminate, using the following equation:
𝛥휀𝑖𝐿𝐴𝑀̅̅ ̅̅ ̅̅ ̅̅ =
1
𝑡𝑡𝑜𝑡𝑆�̅�𝑗𝐿𝐴𝑀𝛥𝑁𝑗
𝐿𝐴𝑀
9. In order to account for the difference between the strain of the laminate, and the unconstrained
strain of each ply, the local ply stresses must be adjusted, using the following equation, where the
strains are expressed in the local ply coordinate system:
𝜎𝑖𝑡+𝑑𝑡 = 𝑄𝑖𝑗
𝑝𝑙𝑦(휀𝑗𝐿𝐴𝑀 − 𝛥휀𝑗
𝑝𝑙𝑦)
3.1.7.3 Including the effects of damage during creep loading
In order to predict damage evolution during creep loading, the time-dependent crack density and the
ensuing creep strain response must be calculated. To do so, at each increment of time, after the creep strain
of the laminate has been calculated using the approach described in section 3.1.7.2, the crack density in
each ply must be calculated. This is done using the same approach as in section 3.1.7.1 for quasi-static
loading conditions, except that the critical energy release rate for crack multiplication is a decaying
function of time. The precise form of 𝐺𝐼𝑐(𝑡) is given in the next section for each material system studied
herein. Once the crack density for each ply has been calculated, the stiffness matrix of the laminate is
calculated, and the laminate strain adjusted, using Eq. (3.2) and Eq. (3.6).
We should note, however, that the local ply stress appearing in Eq. (3.4) to calculate the energy release
rate for crack multiplication is not equal to the local ply stress calculated in step 9 of the algorithm
described in section 3.1.7.2. Instead, the local ply stress used in Eq. (3.4) is obtained using the following
equation:
𝜎2 = −𝜈12𝐸2
1 − 𝜈12𝜈21휀1 +
𝐸21 − 𝜈12𝜈21
휀2
Where all the constants correspond to the linear elastic properties of each ply. This form of the local
stress allows to better take into account the time-dependent fracture mechanics of the composite plies.
43
3.1.7.4 Time-dependent critical energy release rate for crack multiplication.
In order to correctly predict crack density evolution during a creep test, the crack multiplication model
must take into account the rate-dependent fracture behavior of the polymeric matrix. According to the work
of Asadi and Raghavan [185], during a creep test, this behavior can be accounted for by using a time-
decaying function to represent the critical energy density release rate for crack multiplication. They
developed an experimental procedure to identify the exact form of this function. This approach is necessary
in order to account for the microscopic structural changes happening in the polymer during loading.
In the present work, a time-decaying function has been used to represent the reduction in critical energy
release rate for crack multiplication with respect to time. In the case of the material system modelled with
Eq. (3.14) for a CFRP material (Toho G30-500/F263-7), the exact form of this function was identified by
comparing the predictions of the crack multiplication model explained in section 3.1.3 to the experimental
data of time-dependent crack multiplication provided by Asadi [26] and choosing the form that provided
the best agreement. It was an approach of trial-and-error. An exponentially decaying formulation for
critical energy release rate is used herein for the 90º ply, i.e. 𝐺𝐼𝑐(𝑡) = 𝐺𝐼𝑐(0) 𝑒𝑥𝑝(−(𝑡 ∙ 10−10)0.105).
In the case of the material system studied by Tuttle et al [24], namely CFRP IM7/5620, no time-
dependent crack multiplication data was available. Therefore, a different approach was used. Nguyen and
Gamby [89] studied the rate-dependent crack multiplication in a carbon fiber/epoxy cross-ply. They
developed a rate-dependent model for the critical energy release rate for crack multiplication. The time-
dependent compliance function that they developed in their paper for this particular material system was
plotted, and found it was found that this function was very close to the behavior of IM7/5620. Therefore,
it was assumed that the time-dependent critical energy release rate for the two materials would be similar.
The rate-dependent critical energy release rate was converted to a time-dependent function by plotting the
compliance as a function of time and taking the derivative of the compliance to obtain the rate of
deformation; from these two curves, a time-dependent critical energy release rate could be obtained.
In order to take the effects of temperature into account, a different form of the time-dependent decay
in critical energy release rate had to be chosen. In the non-linear viscoelasticity model developed by
Schapery, temperature affects the rate of increase in compliance through a temperature-shift factor, 𝑎𝜎𝑇.
Physically, it was assumed in this thesis that the effect of temperature on the time-dependent compliance
would be similar to its effect on the rate of decrease in critical energy release rate. Therefore, the
temperature-shift factor originally developed for the description of the change in compliance with respect
to time was applied to the time-decaying critical energy release rate equation by multiplying it with the
time constant.
44
In the case of the GFRP material system studied by Varna and Megnis [187], 𝐺𝐼𝑐(0) was determined
by using experimental data published in the literature [188]. Shokrieh et al [189] obtained a rate-dependent
fracture energy for glass fiber composites. The creep strain predicted by the current model was used to
calculate the time-dependent creep strain rate. This time-dependent equation was then combined with the
strain-rate dependent fracture model of Shokrieh et al to obtain the decay in 𝐺𝐼𝑐(𝑡). The final form is given
by:
𝐺𝐼𝑐(𝑡) =𝐺𝐼𝑐0
2+𝐺𝐼𝑐0
2𝑒𝑥𝑝 (−(
𝑡
104))
0.807
(3.16)
where 𝐺𝐼𝑐0 is the critical energy for crack multiplication at the start of creep and 𝑡 is time since the start
of the creep simulation.
It should be noted that in this part of the model, the time-dependent properties of the matrix are
accounted for through the time-dependent GIc(t), and the time-dependent COD evolution. The ply
properties and the local stresses are assumed to be linearly elastic. This is deemed reasonable given that
the stress relaxation caused by the time-dependent behavior of the matrix are very small.
3.1.8 Development of an RVE to evaluate the effect of damage on laminate stiffness
The SDM model relies on a computational micro-damage FE model to evaluate the constants appearing
in the equations relating crack density in each ply of the laminate to its overall stiffness degradation. In
this section, the construction of the RVE used to evaluate the parameters of the SDM model is reviewed.
The FEA simulations performed to obtain the parameters are then described.
The micro-damage RVE can be built in any commercial FEA software. In the present work, ANSYS
mechanical APDL [190] software has been used. An example is shown in Figure 3-3. Each layer of the
laminate is modelled as a transversely isotropic linearly elastic material, with material properties obtained
from experimental data. Each ply is homogeneous and, therefore, the fibers are not explicitly modelled.
The mesh applied to the RVE is constructed such that the separate plies are perfectly bonded (i.e. no
delamination is modelled, and the only mode of damage considered is matrix micro-cracking). The in-
plane boundaries of the RVE are periodic. This is achieved using the approach described by Montesano
and Singh [94]: the mesh on each side of the RVE is set to match the mesh on the opposite side. The
displacement at each node on each boundary is constrained using equation-type constraints to enforce
Periodic Boundary Conditions. The bottom boundary of the RVE is symmetrical, in order to account for
the symmetry of the laminate with respect to the mid-plane.
The orientation of each ply is taken into account by defining a local coordinate system (CS) orientation
relative to the global coordinate system. This local CS is defined by rotating the X and Y axes around the
Z-axis (laminate thickness direction) so that the local X-axis coincides with the longitudinal direction of
45
the fibers and the local Y-axis is perpendicular to the fibers. The effect of ply orientation is then accounted
for through the anisotropic properties of the plies. The cracks are created by defining crack surfaces before
meshing the RVE. Each crack face is defined for each ply to be parallel to the fibers. The crack surfaces
are defined at the start of model creation for each ply. In the simulations considered in this thesis, cracking
was modelled in one ply at a time. For each simulation performed, cracking was defined in one ply, and
all other ply cracks were closed by merging nodes across the crack surfaces. In the work of Montesano and
Singh [94], cracking was modelled in several plies at once to obtain the interaction between cracks in
separate layers of the laminate. This was not done in this thesis. Lastly, it should be noted that in SDM
and, more broadly, CDM of composites, the RVE is assumed to represent the mechanical behavior of a
laminate. Due to the periodicity of the boundary conditions, inhomogeneity in crack spacing is not
considered.
An example of a deformed RVE and the resulting stress distribution in shown in Figure 3-3(b). The
90° layer contains a single crack which spans the thickness of the ply, and the width of the RVE. The stress
distribution around the crack tip is non-uniform, as expected. In fact, the highly stressed region around the
crack-tips is susceptible to delamination crack initiation, as shown analytically by Carraro et al [191].
However, this mechanism of damage has not been studied in this thesis.
(a) (b)
Figure 3-3. (a) SDM micro-damage FE model for a [0/90/∓45]s quasi-isotropic laminate, with micro-cracks in the 0, 90 and -45
plies and periodic in-plane boundaries. (b) Stress contour (𝜎𝑦𝑦) in the RVE under uniaxial strain (휀𝑥𝑥 = 0.5%, 휀𝑦𝑦 = 0, 𝛾𝑥𝑦 = 0)
with a single crack in the 90° ply.
The RVE is used to obtain the 𝑎[1−4]𝛼 parameters, as well as the COD model parameters, 𝑐1−3. The
𝑎[1−4]𝛼 parameters describe the loss in stiffness, while the 𝑐1−3 parameters describe the interactions between
adjacent cracks, the constraining effects of adjacent plies, as well as between cracks in different layers.
The 𝑎[1−4]𝛼 parameters are obtained by modelling cracking in each ply, and running three separate
simulations, one for each in-plane strain state (휀𝑥𝑥, 휀𝑦𝑦, 𝛾𝑥𝑦). The value of applied strain can be arbitrary,
46
however in this thesis a value of 0.5% was chosen for all SDM micro-mechanical simulations. From the
stress distribution in the structure for a given amount of applied strain, the average stress and strain can be
calculated using the following equations [94]:
𝜎𝑖𝑗̅̅̅̅ =1
𝑉∫𝜎𝑖𝑗𝑑𝑉𝑣
=1
𝑉∑𝜎𝑖𝑗
𝑛𝑉𝑛
𝑛
휀𝑖𝑗̅̅ ̅ = 1
𝑉∫휀𝑖𝑗𝑑𝑉𝑣
=1
𝑉∑휀𝑖𝑗
𝑛𝑉𝑛
𝑛
Where 𝑛 is the total number of elements, and 𝑉𝑛 is the volume of the element. These summations were
calculated using Ansys ADPL, by defining a script that iterated through each element in the RVE to obtain
the average stress and strain tensors. From the average stress and strain tensor, the plane-stress stiffness
matrix for the damaged RVE can be obtained as follows, using the results from the three separate
simulations corresponding to the three separate values of the applied in-plane strain:
𝑄11𝐿𝐴𝑀 =
𝜎𝑥𝑥휀𝑥𝑥
𝜈21𝐿𝐴𝑀 =
𝜎𝑦𝑦
휀𝑥𝑥
𝑄22𝐿𝐴𝑀 =
𝜎𝑦𝑦
휀𝑦𝑦
𝑄66𝐿𝐴𝑀 =
𝜏𝑥𝑦
𝛾𝑥𝑦
In order to obtain the parameters appearing in Eq. (3.2), the COD corresponding to the crack density
chosen in these simulations needs to be obtained. Based on this value of the COD parameter, the thickness
of the ply, and the crack spacing, the 𝑎[1−4]𝛼 parameters can be chosen such that the values of the stiffness
tensor predicted by Eq. (3.2) correspond to the values calculated from the FEA model.
In order to account for the effect of crack density multiplication on stiffness, the evolution of the COD
needs to be calculated. In this thesis, cracking is modelled in one ply at a time. In the case of the 𝑐1−3
parameters, a set of simulations is run with increasing crack density, with one set of simulations per ply.
Since the RVE is periodic, the crack density can be increased by decreasing the size of the RVE. For each
simulation, the average COD is calculated through post-processing of the ANSYS output data. The
relationship between COD and crack density can then be approximated by the relationship of Eq. (3.3). In
the present thesis, it is assumed that cracks in separate plies do not interact, so that the COD evolution in
each ply is independent of the evolution in adjacent plies. In reality, cracks in separate plies interact, which
휀𝑥𝑥 = 0.005
휀𝑦𝑦 = 0.005
𝛾𝑥𝑦 = 0.005
47
can be taken into account by defining a COD that depends on cracks in multiple layers, as shown by
Montesano and Singh [94].
In order to account for multi-axial effects, the role of out-of-plane contraction under transverse load
needs to be assessed. In the case of previous papers on the SDM model [45], [105], simulations were
restricted to uniaxial loading. In order to account for the increase in COD with increasing strain, the COD
was normalized (Eq. 3.3) with respect to an effective strain, and the cracked ply thickness. The effective
strain is given by 휀𝑒𝑓𝑓 = 휀22 + 𝜈12휀11 +1
8𝜈21𝛾12 where 𝜈12 and 𝜈21 are the major and minor Poisson’s
ratios of the ply, respectively, and 𝛾12 is the engineering shear strain. To account for multi-axial effects,
simulations are performed with a transverse strain applied to the laminate, in addition to the axial strain.
The evolution of the COD with respect to crack density is recorded. Following this set of simulations, the
COD is normalized with respect to the effective strain, instead of the transverse strain 휀22. The evolution
of normalized COD with respect to crack density is then approximated using Eq. (3.3) by selecting
appropriate values for the 𝑐[1−3] parameters.
In order to improve the efficiency of the SDM model, Montesano et al [192] developed analytical
expressions for the 𝑐[1−3] parameters appearing in Eq. (3.3). The parameters for each ply are based on the
ply’s thickness, as well as the stiffness and thickness of adjacent plies. In this thesis, these expressions
were implemented into MATLAB.
3.1.9 Assumptions of the model and range of applicability
As explained at the beginning of this chapter, the RVE used in the SDM modelling framework makes
certain assumptions about the distribution of damage throughout the laminate:
The cracks are equally spaced throughout the laminate
They are parallel to the fibers
The growth of the micro-crack is not modelled and once it nucleates, it spans the width of
the laminate and thickness of the ply
The presence of defects within plies leads to a distribution of matrix micro-cracks that is initially not
uniform throughout the laminate. However, with increased loading, the average spacing between matrix
micro-cracks becomes approximately uniform. Stochastic effects are considered in this study by defining
the critical energy release rate for crack multiplication (𝐺𝐼𝑐) in terms of a random variable (see section
3.1.3)
The analysis conducted in this thesis has solely focused on symmetric laminates under in-plane
loading. Laminates that are not symmetric will undergo bending and torsion when loaded in tension or
48
compression. Further analysis would be required to account for these effects, and they are not the focus of
this thesis.
Lastly, the formalism of SDM is applicable to thin laminates, and there is no out-of-plane stress across
the top and bottom boundaries of the laminate. This assumption leads to the stiffness degradation equations
(see Eq. (3.2)). Further improvements on the model would be required to extend it to thick structures.
3.2 Micro-damage model for a short fibre composite
The fourth goal of this thesis is to understand damage evolution mechanisms in short fiber composites
with a ductile thermoplastic matrix. Specifically, the role of strain rate on the process of fiber-matrix
debonding is to be studied. The modelling approach used in this section is based on FE micro-damage
mechanics, in which a Representative Volume Element (RVE) of a short fiber composite is generated. A
Cohesive Zone Model has been used to predict debonding at the fiber-matrix interface. A constitutive
model for a nonlinear viscoelastic matrix has also been implemented.
3.2.1 Geometry
A schematic of the RVE is shown in Figure 3-4. A rectangular box of height L0 and width W0 was
created. Two half fibers were inserted into the RVE, both with diameter Df and length Lf. The fiber ends
were separated by a small distance (1/20 Lf) to model the microstructure of short fiber composites and the
effect of stress concentration at the fiber tips on micro-damage processes. The RVE was meshed with solid
quadrilateral element C3D8R in Abaqus. The bottom faced of the RVE was pinned by preventing
displacement in all directions. A uniform displacement was applied to the top face in the longitudinal
direction. The sides of the box were maintained plane through multi-point constraints, as recommended by
Liang and Tucker [193], in order to appropriately model the mechanical response of the RVE.
3.2.2 Material models
Cohesive zone models define the stress distribution at an interface in terms of separation [162], [163],
[194], [195]. Needleman [194] used CZM to describe debonding in short fiber composites. Ahead of a
crack-tip, a material will undergo changes in its microstructure due to the applied stress. Due to the
evolution of these damage mechanisms, the separation at the interface increases, until the crack grows
along the interface. This behavior is defined through the CZM with a traction-separation law, which relates
the separation at the interface to the stress at the interface. It is a phenomenological model, which does not
account for the micro-damage mechanisms explicitly, but instead accounts for them in an averaged way
through the traction-separation curve. The CZM for a fiber-matrix interface can be obtained by monitoring
the local separation at the interface, and relating it to the applied normal stress. Once the traction-separation
49
law has been obtained, the behavior of the interface under different loading conditions, and different
interface shapes can be calculated.
The fiber-matrix interface was modelled in Abaqus using a surface-based Cohesive Zone Model, with
parameters of Li et al [172] fitted experimentally to debonding of a glass fiber/epoxy unidirectional
composite. The CZM is based on a linear representation of the interfacial traction-separation behavior. The
initial elastic portion of the CZM represents the traction-separation behavior when the interface has not
undergone damage. The traction across the interface 𝑻 = {
𝑇𝑛𝑇𝑡1𝑇𝑡2
} is related to the separation 𝜹 = {
𝛿𝑛𝛿𝑡1𝛿𝑡2
}
(where 𝑛 denotes the normal direction, and 𝑡1 and 𝑡2 denote the two shear directions) through the following
equation:
It is therefore assumed that the normal and shear interfacial elastic responses are uncoupled. When the
traction across the interface reaches a critical level, damage initiates, which causes a degradation in the
interfacial stiffness matrix. The critical stress levels in the shear and normal directions are provided in
Table 3-2
In this thesis, the maximum stress criterion for damage initiation is used, which states that damage
initiates when any of the traction forces is larger than its critical value (see Table 3-2)
Following damage initiation, the traction at the interface is related to the separation through the
following expressions: 𝑻 = (1 − 𝐷)𝑻, where 𝑻 is the traction calculated in the absence of damage. The
damage variable 𝐷 describing the loss in load transfer capability at the interface is given by the following
expression:
𝐷 =𝛿𝑚𝑓 (𝛿𝑚 𝑚𝑎𝑥 − 𝛿𝑚
0 )
𝛿𝑚𝑚𝑎𝑥(𝛿𝑚
𝑓− 𝛿𝑚
0 ) (3.18)
where 𝛿𝑚𝑚𝑎𝑥 is the maximum separation that has occurred at the interface; 𝛿𝑚
0 is the interfacial separation
at which damage initiates; and 𝛿𝑚𝑓
is the interfacial separation at which full debonding occurs. In this thesis,
the shear and normal directions have a different value for 𝛿𝑚0 . In the case of mixed-mode behavior, where
both normal and shear separations arise at the fiber-matrix interface, the evolution of the damage variable
is described in terms of the effective separation 𝛿𝑒𝑓𝑓 = √(𝛿𝑛2 + 𝛿𝑡1
2 + 𝛿𝑡22 ) .
𝑻 = [
𝐾𝑛𝑛 0 00 𝐾𝑡1𝑡1 00 0 𝐾𝑡2𝑡2
] 𝜹. (3.17)
50
Figure 3-4 . Model development for the short fiber RVE. (a) Short fiber RVE used in this thesis to study micro-damage
mechanisms in short fiber composites. (b) Top face of the RVE. (c) Side face of the RVE. Image from “Effect of the matrix
behavior on the damage of ethylene-propylene glass fiber reinforced composite subjected to high strain rate tension”, by Fitoussi
J. et al, 2013 Composites Part B, Vol. 45(1), p. 1183. Copyright (2012) by Elsevier. Reprinted with permission.
Figure 3-5. FE model of the short fiber composite.
The two fibers were modelled using the linear elastic properties measured experimentally for a glass
fiber by Li et al [172]. Other types of fibers were considered as well: carbon and hemp. The matrix was
Top fiber
HDPE matrix
Bottom fiber
(a) (b)
(c)
51
modelled using different mechanical models in different simulation cases to understand the effect of matrix
properties on damage evolution in the RVE. The first material model assumed perfect linear elasticity of
the matrix. In this case the only damage mechanism accounted for was fiber-matrix debonding. In order to
understand the effects of strain rate on the evolution of damage in the short fiber RVE, the viscoelastic
model developed by Popelar et al [196] for HDPE was then implemented into Abaqus to obtain the stress-
strain response of the RVE under different loading rates. Popelar et al [196] used a non-linear viscoelastic
model for HDPE based on a Prony-series description of time-dependent decay in modulus of elasticity.
The constitutive behavior is given by the following expression:
𝜎(𝑡) = ∫ 𝐸𝑅(𝑡 − 𝜏)𝑑[ℎ(휀)휀]
𝑑𝑡𝑑𝑡
𝑡
0
(3.19)
where 𝐸𝑅(𝑡) represents the time-decay in elastic modulus, 휀 is the strain, and ℎ(휀) is a function of strain
representing the non-linear viscoelastic part of the model, given by ℎ(휀) = (1 + 50휀)−1. The time-
dependent modulus is given by:
𝐸𝑅(𝑡) = 𝐸0 +∑𝐸𝑖(1 − 𝑒−𝑡𝜏𝑖)
10
𝑖=1
(3.20)
where the Prony series parameters are given by Popelar et al [196]. Using Eq. (3.19) and Eq. (3.20), an
analytical relationship was derived for the viscoelastic response under constant strain rate, which was then
implemented into Abaqus.
Table 3-1. Elastic properties of the constituents of the RVE.
Fiber and matrix
Glass Fiber Hemp fiber Carbon fiber HDPE Matrix
Young’s modulus (GPa) 64 [172] 8.5[197] 200 [198] 1.86 [196]
Poisson’s ratio 0.2 0.3 0.3 0.42
Table 3-2. Parameters of the Cohesive Zone Model used to describe the fiber-matrix interface.
Direction Critical stress
(MPa)
Critical
separation (μm)
Initial stiffness
(N/mm3)
Normal 7.1 1 107
Tangential 11.0 1 107
A non-linear static analysis was conducted in Abaqus to predict the deformational response of the
RVE. Dynamic effects were ignored, and the rate of deformation of the RVE was assumed to be uniform.
Output from the simulations was written to the database files at regular intervals to obtain sufficiently
detailed information about the deformation and damage processes.
52
3.2.3 Validation of the implementation
The stress-strain response of the ductile HDPE matrix was implemented into Abaqus in tabular form
by approximating it as a plastic material with isotropic hardening behavior. The implementation was
validated by comparing the results of the model of Popelar et al [196] to the output data provided by
Abaqus. The results are shown in Figure 3-6 (a). Clearly, the model has been correctly implemented.
Subplot (b) shows the 0.2% offset yield strength for HDPE, as a function of strain rate. While inelastic
behavior in polymers is linked to different mechanisms than in metals, this approximation was deemed
reasonable a low strain levels.
(a) (b)
Figure 3-6. (a) Validation of the implementation of the non-linear viscoelasticity model for HDPE into Abaqus in tabular form.
(b) Yield stress versus applied strain rate.
This particular implementation has several limitations. First of all, it is known that the compressive
behavior of HDPE is different from the tensile behavior [199]. This can be taken into account through a
pressure-dependent plasticity model. However, as suggested by Pan and Pelegri [200], the compressive
behavior of the matrix under tensile loading of a short fiber RVE can be neglected with reasonable
accuracy. Secondly, the unloading response of HDPE cannot be accurately predicted with the current
model, since it assumes all non-linear behavior is due to plasticity, which is not necessarily the case for a
viscoelastic material. Since the current thesis is focused on monotonic loading, this approximation is
deemed reasonable and sufficient for the purposes of this work. Lastly, the behavior at large strains is not
accurately taken into account in the current implementation. Following initial non-linear softening, glassy
polymers such as HDPE are known to undergo significant strain hardening [201] at high strains. However,
the current analysis focuses on the low range of polymer deformation, and therefore the accuracy of the
constitutive model is deemed sufficient.
The CZM is used to represent the fiber-matrix debonding process. In order to accurately model this
process, the interfacial stiffness before the initiation of debonding should be large enough that the load is
carried by the composite constituents instead of being artificially carried by the interface. On the other
53
hand, the interface stiffness should be low enough that the stresses around the interface can be accurately
obtained without spurious oscillations of the stress field in the RVE, and the debonding progression
correctly modelled. The reader is referred to the work of Turon et al [202] for further details on the issues
surrounding the choice of an appropriate value for interface stiffness. In this thesis, the effect of interface
stiffness was evaluated in order to identify reasonable values to accurately model the process of fiber-
matrix debonding. The Young’s modulus of the RVE has been plotted in Figure 3-7 for different values of
the interfacial stiffness. The modulus of the RVE reaches a constant value of 18 GPa for an interfacial
stiffness of 107 N/mm3 or more. At lower values of interfacial stiffness, the modulus of the RVE is under-
predicted. Inspection of the RVE showed large separation at the fiber tip, even without any interface
damage, suggesting that low values of interfacial stiffness do not accurately represent the mechanical
behavior of the short fiber composite. A supplementary simulation was performed where the fiber and
matrix meshes were merged, and a modulus of 18.5 GPa was obtained, thus showing that the modulus
obtained using CZM with high values of interfacial stiffness is correct. An interfacial stiffness of 107
N/mm3 was used in this thesis to accurately model the interface, while still avoiding spurious stress
oscillations in the RVE. Moreover, the mesh was refined around the fiber tips to better account for the
large stress gradients occurring in these regions.
In order to further verify the accuracy of the model, a mesh size sensitivity analysis was conducted.
The results are shown in Figure 3-8 (a) in the form of stress-strain curves under different strain rates.
Clearly, there is little mesh size sensitivity for an element length less than 4 µm. This was therefore chosen
as the element size in the central region of the RVE. A larger element size was used away from the fiber
tips, where the stress gradients are lower. The effect of element size far from the fiber tips was also
evaluated in the case of a non-linear viscoelastic matrix (see subplot (b)). The element size is clearly not
critical on the stress-strain behavior of the RVE with a viscoelastic matrix, and therefore an element size
of 8 µm (away from the fiber tips) was used in subsequent simulations.
Figure 3-7. Effect of interface stiffness on composite RVE modulus
54
(a) (b)
Figure 3-8. (a) Effect of mesh size on stress-strain response of composite RVE with linear elastic components. The legend refers
to the element size in the central region of the RVE around the fiber tips ; when the mesh size was constant throughout the RVE,
the term ‘uniform’ is used. (b) Effect of mesh size on stress-strain response of composite RVE with a non-linear viscoelastic
matrix under a strain rate of 10-5 s-1.
3.3 Structural analysis of a composite car bumper
The second objective of this thesis is the implementation of the SDM model described in section 3.1
into FE software to conduct a rate-dependent damage analysis of an automotive component. In this section,
low-velocity impact on a bumper is simulated using explicit dynamics FEA [203].
Figure 3-10 shows the general steps of the numerical implementation of the multi-scale damage model.
First, the SDM model parameters are obtained using the approach provided in section 3.1. Next, an FE
model of a composite car bumper is created in Abaqus, and the low velocity impact response of the
structure is calculated using explicit dynamics. The evolution of damage is predicted, and the effect of
strain-rate dependent behavior is evaluated.
Figure 3-10 shows an automotive bumper subjected to low-velocity external impactor. The impactor
is modelled as a rigid hemispherical shell with diameter 198 mm and mass 10.4 kg and is meshed with a
4-node 3-D bilinear quadrilateral (R3D4) element. As the impactor is rigid, a reference point is considered
to represent the geometry of the impactor. The reference point is constrained in 5 degrees of freedom (X
and Y translations and 3 rotations) and it is only free in the impact direction (Z translation). An initial
velocity is applied to the reference point to perform the impact simulation. Two materials, IM7/5260
Carbon Fiber/Bismaleimide (CFRP) and VICOTEX NVE 913/28%/192/EC9756 (GFRP) are considered
for this study; the material properties are listed in Chapter 5.
In order to take rate-dependency into account, the critical energy release rate for crack multiplication,
𝐺𝐼𝑐, is defined in terms of the loading rate, as per the work of Nguyen and Gamby[89]. The following
equation is used:
55
Where 𝜎�̇� = 1.3216 𝑀𝑃𝑎/𝑚𝑖𝑛 is a reference loading rate, and 𝑚3 = 0.2 describes the rate-dependency of
𝐺𝐼𝑐. 𝐺𝐼𝑐𝑜 is the critical energy for crack multiplication at the beginning of the cracking process and is
provided for each system in Chapters 4 and 5. 𝐺𝐼𝑐𝑟 = 0.8 𝐺𝐼𝑐𝑜and the shape factor 𝑟 = 1.5 describe the
increase in critical energy release rate with respect to increasing crack density (see [180])
The geometrical dimensions of the bumper are given in Figure 3-10. It is meshed with a 4-node doubly
curved shell element with reduced integration formulation (S4R). The transverse shear stiffness values
were equal to 19.8 MN/m for GFRP and 23.24 MN/m for CFRP. These values were obtained using CLT
and the stiffness properties of the plies. The average element aspect ratio is about 2.0 to maintain an
acceptable accuracy in the results. Moreover, as the subroutine is called at every integration point during
the simulation procedure, the element size is considered larger than the micromechanical RVE in order to
ensure the applicability of the constitutive equations. The contact between the impactor and the bumper is
defined via a general explicit contact algorithm in which the contact force is generated by using the penalty
enforcement contact method. Moreover, a tangential interaction with Coulomb friction coefficient is
considered to account for the shear component of the surface traction τ , which is related to the normal
contact pressure by 𝜏 = 𝜇𝑝. The friction coefficient, 𝜇, depends on the surfaces of the contact materials
and has been calculated for various materials. Here 𝜇 is considered to be 0.2 between the rigid surface and
the composite laminate [137]. To implement the constitutive equations and incorporate the damage model,
a vectored user-defined material (VUMAT) subroutine is developed. At each time increment and at every
integration point in the elements, Abaqus calls the VUMAT subroutine to update the state of the material
and the material mechanical response (i.e. stress and energy) based on the strain increment applied at the
integration point. During this process, if the criterion of damage initiation is satisfied (see Eq. (3.4)), the
crack density is updated and the stiffness degradation is calculated to determine the stress increment at that
integration point. This process continues until the end of the simulation. The flowchart of the VUMAT
subroutine is presented in Figure 3-10. Lastly, it should be noted that delamination is not taken into account
in the current model. The focus is on matrix micro-cracking, which will eventually lead to delamination at
high densities.
𝐺𝐼𝑐(�̇�) = 𝐺𝐼𝑐𝑜 + 𝐺𝐼𝑐𝑟(1 − 𝑒𝑥𝑝(−𝑟 𝜌𝛼)) (�̇�
𝜎�̇�)𝑚3
(3.21)
56
Figure 3-9. Synergistic Damage Mechanics (SDM) RVE, showing the main features of the model, including uniform ply crack
spacing, cracks parallel to the fibers in each layer, cracking in multiple off-axis orientations and multi-axial loading scenarios.
Figure 3-10. General framework for the numerical implementation of the hierarchical multi-scale methodology.
57
Chapter 4. Development of an SDM-based
viscoelastic creep damage model
In the present chapter, the results of the predictions of the viscoelastic damage developed in the
previous section are provided, and interpreted. First, the model is validated by comparing its predictions
to experimental data obtained from the literature. Second, the predictions of the model for different
stacking sequences and different loading scenarios are explained and discussed. Note that this chapter only
considers uniaxial loading and viscoelasticity; viscoplasticity and multi-axial loading will be considered
in Chapter 5.
4.1 Multidirectional laminates undergoing cracking in multiple plies.1
Before investigating the effects of creep on damage evolution and performance degradation, the quasi-
static response of the [±45/902]s CFRP laminate is predicted and compared to the experimental test data
reported by Asadi [26]. Eq. (3.2) constitutes the basic constitutive model for the damaged laminate, where
the evolution of ply cracking is predicted using the energy-based approach, see Eq. (3.4). The elastic
properties for the carbon/epoxy plies are shown in Table 4-1. The predicted and experimental crack density
evolution data during quasi-static loading are shown in Figure 4-1(a). The ply cracks initiate in the 90°
plies first, as expected, at about 40MPa applied stress, consistent with experimental observations.
Qualitatively, the overall trends in the evolution of crack densities in different plies agree well with the
test data. Quantitatively, it is clear that the model predictions are accurate for the 90° ply cracks, but less
accurate for the ±45° plies at higher applied stresses. In the study by Asadi [26], experimental tests revealed
that delamination was observed at applied stresses of 75MPa. Since this mechanism of damage is not
considered by the prediction model, which only takes into account sub-critical matrix micro-cracking, this
deviation at higher stresses is expected. Nonetheless, the model is accurate at lower applied stresses, and
since the creep simulations will be performed at approximately 50 MPa prior to the onset of delamination,
this is deemed acceptable. The corresponding quasi-static axial stiffness degradation results are shown in
Figure 4-1 (b). Again, the predictions correlate well with the available experimental data, which are well
within the error bars shown in the plot. The sudden changes in stiffness due to the onset of 90° ply cracks
and 45° ply cracks are also clear. Figure 4-1 (c) shows the predicted stress–strain response, which also
1Reprinted from Int. J. Damage Mech., vol. 25, no. 7, T. Berton, J. Montesano and C.V. Singh, Development of a
synergistic damage mechanics model to predict evolution of ply cracking and stiffness changes in multidirectional
composite laminates under creep, pp. 1060-1078, Copyright (2015), with permission from SAGE Publications. DOI:
10.1177/1056789515605569
58
shows good agreement with the experimental data. Figure 4-1 thus validates the developed model for
multidirectional laminates during quasi-static loading.
Table 4-1. Elastic ply properties for the [±45/902]s CFRP laminate studied by Asadi [26]
Material property Value
E1 (GPa) 113
E2 (GPa) 9.68 G12 (GPa) 4.83
ν12 0.3
Table 4-2. Parameters for the non-linear creep model described by Eq. (3.14), which were obtained using the experimental data
in Asadi [26].
S22 (transverse) S66 (shear)
S0 (MPa-1) 1.033 x 10-4 S0 (MPa-1) 2.07 x 10-4
ΔS (MPa-1) 0.002675 ΔS (MPa-1) 0.002113
τ (s) 3.49 x 1014 τ (s) 1.52 x 109
c 0.2863 c 0.3638
Next, the deformational response of the [±45/902]s CFRP laminate under creep loading is predicted
and compared to the experimental test data. The model parameters for ply creep compliance (see Eq.
(3.14)) are provided in Table 4-2. It is notable that according to previous experimental studies, only the
transverse and shear components of the ply compliance tensor were found to be time-dependent. The non-
linear stress-dependent ply parameters are also obtained from the experimental data reported in Asadi [26]
and are defined as functions of temperature, T, by the expressions provided in the PhD thesis by Asadi.
59
Figure 4-1. SDM model results and comparison to experimental data [26] for a quasi-static test performed on a [±45/902]s carbon-
fibre/epoxy laminate: a) crack density evolution in different plies; b) stiffness degradation; c) stress-strain curve. The SDM model
used here does not include viscoelastic effects.
0 0.2 0.4 0.6 0.80
20
40
60
80
100
120
140
Strain (percent)
Str
ess
(MP
a)
Exp.
SDM model
0 20 40 60 80 1005
10
15
20
25
Stress (MPa)
Lo
ng
itu
din
al
Mo
du
lus
(GP
a)
Exp.
SDM Model
0 20 40 60 800
0.5
1
1.5
2
Stress (MPa)
Cra
ck d
ensi
ty (
/mm
)
SDM Model (+45° ply)
SDM Model (-45° ply)
SDM Model (90° ply)
Exp. (90° ply)
Exp. (-45° ply)
Exp. (+45° ply)
+45° ply crack
initiation strain
90° ply crack
initiation strain
(b)
(c)
(a)
60
Figure 4-2. Results of the extended SDM model for creep strain (a) and damage evolution in the 90° ply (b) for a [±45/902]s
laminate subjected to a constant creep stress of 45 MPa, and comparison to experimental data [32]. Note that the initial
experimental creep strain in a) at time t = 0 s was adjusted so that the predicted results and the experimental data coincided at the
onset of creep.
Two simulations were conducted to predict the laminate creep strain evolution, the first without the
effects of ply cracking (i.e. damage) and the second whereby the evolution of cracking was considered.
The results for the laminate creep strain evolution under a constant creep stress of 45 MPa are shown in
Figure 4-2(a) along with the experimental data. It can be seen that the inclusion of damage in the developed
model significantly improves the predicted creep strain evolution, showing that damage is an important
phenomenon affecting creep for these laminates. The corresponding predicted 90° ply crack density
evolution is shown in Figure 4-2(b), along with the experimental data. The initial crack density at a stress
of 45 MPa is accurately predicted at approximately 0.1 cracks/mm. The subsequent crack evolution
prediction also correlates reasonably well with the experimental data and evolves somewhat similarly to
the creep strain due to an apparent proportionality between creep strain and CODs and thus the crack
density evolution. The crack density at longer times is determined to approach a saturation value of 0.5
cracks/mm, which is slightly lower than the experimental data.
0 5 10 15
x 105
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (s)
Cra
ck D
ensi
ty (
/mm
)
Exp.
Extended SDM model
0 2 4 6
x 105
0.22
0.225
0.23
0.235
0.24
0.245
0.25
Time (s)
Str
ain
(%
)
Exp.
Extended SDM model
Creep model without damage
Increase in
creep strain due
to damage (a)
(b)
Lower creep strain
without damage
61
As depicted above, using the ply elastic and viscoelastic properties along with the critical energy
release rate, the developed model can predict strain and damage evolutions reasonably well. Minor
discrepancies between predictions and test data can be attributed to inherently different nature of damage
processes under quasi-static and viscoelastic conditions, which the current model may not capture fully.
For instance, the crack density is well estimated up until 6x105 s, whereas the crack density is slightly
under-predicted beyond this time range. Nonetheless, the lack of experimental creep data beyond 106 s
precludes a direct comparison with the model predictions. It should be noted, however, that viscoplasticity
may cause supplementary strains at long times, which are not currently accounted for by the prediction
model. Moreover, the model assumes damage only affects the elastic properties of the laminate and not its
viscoelastic time-dependent compliance. As shown by Varna et al., [103] matrix cracks will cause
degradation in time-dependent stiffness, so that accounting for this effect may aid in improving the results.
Future work should consider all these effects to improve the accuracy of the model.
4.2 Cross-ply laminates undergoing cracking in transverse plies.
In order to showcase the versatility of the model, the quasi-static response of a [03/903]s cross-ply
laminate is also evaluated because it has different thickness ratios of cracking and non-cracking plies, and
the test data are from a different source [89]. Although the experimental data revealed that the laminate
exhibited little time-dependent response, it was stated in the study that the crack density evolution under
quasi-static loading was dependent on the applied strain rate. In order to account for the variation in crack
density evolution at different applied strain rates for our predictions, the critical energy release rate GIc was
defined to be a function of the applied strain rate based on data provided in the cited study. The predicted
and experimental results are shown in Figure 4-3, which demonstrates that the current model accurately
predicts crack initiation at lower stresses for lower applied loading rates, which is expected. It can also be
seen that at high loading rates, when time-dependent effects are minimal, the prediction model is very
accurate. As the loading rate is decreased by a factor of 100, the model predictions are less accurate at
higher applied stresses as shown in Figure 4-3. This is most likely due to the fact that time-dependent
effects were neglected for the quasi-static simulations.
62
Figure 4-3. Predicted crack density evolution for a [03/903]s cross-ply laminate for different levels of strain rate (low, medium
and high) using the SDM methodology, along with experimental data from Nguyen and Gamby [89].
It is expected that at very low applied strain rates, the laminate will exhibit some creep behaviour due
to the increased loading time, which will drive crack multiplication slightly higher. In order to provide
further validation for the current model, our predictions will be compared to the available experimental
results of Ogi and Takao [121] for a [0/903]s CFRP cross-ply laminate. A quasi-static simulation was first
performed using the elastic ply properties presented in Table 4-3, where the predicted and experimental
crack density evolution results are shown in Figure 4-4. A good agreement between the model predictions
and the test data is shown. The accuracy of the results suggests that time-dependent effects were not
important during loading due to the relatively high applied strain rate.
Figure 4-4. Predicted and experimental crack density evolution during quasi-static loading for a [0/903]s cross-ply laminate; the
prediction assumes no viscoelastic effects.
The creep model presented in the previous section is now applied for the [0/903]s CFRP laminate,
where the ply transverse creep properties were determined by fitting the unidirectional ply creep curves
provided in Ogi and Takao [121] with Eq. (3.14) at an applied stress of 510MPa. It was assumed that all
nonlinear parameters were equal to 1 in this instance because there were not sufficient data in the cited
source to determine them precisely. This implies that the nonlinear parameters are stress independent and
thus, the current analysis is only valid for stresses close to 510MPa. Since the magnitude of creep stress
0 500 1000 15000
0.2
0.4
0.6
0.8
1
Stress (MPa)
Cra
ck d
ensi
ty (
/mm
)
High
Low
Medium
Low
Medium
High
SDM
Model
Exp.
0 200 400 600 8000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Stress (MPa)
Cra
ck d
ensi
ty (
/mm
)
SDM Model
Exp.
63
we consider is 530 MPa, this is deemed acceptable. The creep parameters obtained through data fitting are
presented in Table 4-3, where a time constant of 319 seconds was used to evaluate GIc(t). It should be noted
that the magnitude of the ΔS22 value presented in Table 4-3 is notably lower compared with the values
shown in Table 4-2, which is a result of forcing the nonlinear parameters to be stress independent. The
predicted and experimental crack density evolution and laminate creep strain results are shown in Figure
4-5(a) and (b), respectively, for a constant applied creep stress of 530MPa. From Figure 4-5(a), it is clear
that the crack density evolution in the 90° ply correlates well with the experimental data when t>2x105 s,
but is slightly under-predicted during the early stages of creep. From Figure 4-5(b), it can be seen that
including the effect of damage evolution on the elastic properties greatly enhances the accuracy of the
predicted laminate creep strain. The laminate creep strain at longer times is slightly underestimated by the
prediction model and is possibly due to neglected viscoplastic strains.
Table 4-3. Parameters used for the creep compliance model, Eq. (3.14), as obtained by fitting the experimental creep
curve for 90° unidirectional plies, at a stress level of 510 MPa [121]
Non-linear Creep
Parameter Value
dS22 (Pa-1) 9.80 X 10-13
τ22 (s) 319
64
Figure 4-5. a) Predicted and experimental [121] 90° ply crack density evolution for a [0/903]s cross-ply laminate subjected to a
constant stress of 530 MPa and b) corresponding laminate creep strain.
4.3 Predictions for [±60/902]s and [±75/902]s CFRP laminates.
In order to showcase the predictive capabilities of the developed model for general multidirectional
laminates, the results for two additional CFRP laminates, namely [±60/902]s and [±75/902]s, are analyzed
here. It should be noted that no supplementary data were required to make these predictions. The elastic
and time-dependent ply properties were given in Table 4-1 and Table 4-2 and by Eq. (3.14). Two creep
simulations were performed for two different constant stresses of 16 MPa and 19 MPa, respectively, using
the developed MATLAB algorithm (see Chapter 3).
The creep strain results for the two laminates studied are shown in Figure 4-6. Since the [±75/902]s is
more compliant in the loading direction, the initial strain at time t = 0 s for a constant creep stress of 19
MPa is notably higher compared to the [±60/902]s laminate, which is expected. The effect of damage on
the predicted creep strain is also more noticeable for the [±75/902]s laminate – the [±60/902]s laminate did
not exhibit any significant damage evolution (see Figure 4-6). The strain increases by approximately 0.10%
for the [±75/902]s laminate when damage is not considered, whereas the increase is approximately 0.38%
when the effects of damage are considered. This is due to two main factors. First, the ply crack driving
stresses, 𝜎2𝛼 , from Eq. (3.4) are higher for the more compliant [±75/902]s laminate, which undergoes
greater laminate strains in the loading direction. Thus, as the laminate creep strain continues to increase,
0 2 4 6 8 10 12
x 104
1.3
1.4
1.5
1.6
1.7
Time (s)
Str
ain
(%
)
Creep model without damage
Exp.
Extended SDM model
0 1 2 3 4
x 105
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
Cra
ck d
ensi
ty (
/mm
)
Exp.
Extended SDM model
(a)
(b)
65
the energy release rate WI increases by a greater magnitude, which favours crack multiplication in both the
90° and 75° plies. For the [±60/ 902]s laminate, the laminate creep strain increases by a lower magnitude,
and thus, its effects on crack multiplication in the 90° and 60° plies is minimal. This same effect can also
be seen in Figure 4-2 for the [±45/902]s laminate since the crack density evolution in the 90° ply has a
similar shape to the creep strain curve; it should be noted that the applied loading for the [±45/902]s
laminate was greater at 45MPa. Second, since the critical energy release rate GIc is an exponentially
decreasing function of time, for the [±75/902]s laminate, it directly enhanced the crack multiplication
process due to the adequate laminate strain magnitudes. For the [±60/902]s laminate, the laminate creep
strain magnitudes were not sufficiently high to promote damage even though GIc decreased with time.
Figure 4-6. Predicted creep strain evolution for [±60/902]s and [±75/902]s CFRP laminates, with and without the effects of
damage, subjected to a constant creep stress of 19 MPa.
Figure 4-7 (a), (b) and (c), respectively, show for the [±75/902]s laminate the evolution of compliance
with laminate strain, the evolution of longitudinal stiffness with crack density, and the evolution of crack
density in the 90° ply. From Figure 4-7(a), we can see that there is no notable change in the compliance
before approximately 0.20% strain, which corresponds to the start of the creep portion of the simulation.
The evolution of crack density, and thus elastic compliance, is minimal during the quasi-static portion,
which is due to the relatively small applied stress of 19 MPa. After the onset of creep, the elastic
compliance increases by 0.075 GPa-1 during the duration of creep loading (up to 0.4% strain), which is a
result of notable crack evolution in the 90° ply. The time-dependent creep compliance increases by 0.02
GPa-1 during the same creep interval, which is approximately a quarter of the increase in the elastic
compliance. This shows that the influence of ply crack evolution during creep loading in the studied
laminates is significant and is a result of the specific layup considered. From Figure 4-7(b), we can see that
laminate stiffness, which is affected by the crack density evolution in all plies, decreases linearly during
the initial stages of creep. It should be noted that during creep loading, the ±75° ply crack density is less
0 2 4 6 8 10
x 105
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time (s)
Str
ain
(%
)
[±75/902]s without damage
[±75/902]s with damage
[±60/902]s with damage
[±60/902]s without damage
Clearer effect of
damage due to
higher compliance
of the laminate
66
advanced when compared to a quasi-static loading case, as is shown in Figure 4-8. Therefore, stiffness
degradation is greater for the quasi-static case when compared to creep loading (see Figure 4-7(b)).
Figure 4-7. The predicted evolution of compliance is shown here for the [±75/902]s laminate at a creep stress of 19
MPa: a) different components of compliance (creep and elastic), b) stiffness degradation with crack density in the
90° ply, and c) evolution of 90° ply crack density for creep and quasi-static loading.
From Figure 4-7(c), we can see that crack density evolution in the 90° ply during the load ramp-up is
minor, and that it increases significantly during creep loading. The crack density under creep at a specific
strain magnitude is greater when compared to the quasi-static loading case (see Figure 4-7(c)), which is a
result of time-dependent effects in the thicker 90° ply. This is manifested through the critical energy release
rate and ply CODs, which enhance ply crack multiplication. An increase in crack density tends to decrease
0 0.2 0.4 0.64
5
6
7
8
9
10
Crack density (/mm)
La
min
ate
sti
ffn
ess
(GP
a)
Quasi-static loading
Creep loading
0 0.1 0.2 0.3 0.40
0.05
0.1
0.15
0.2
0.25
Total laminate strain (%)
Co
mp
lia
nce
(/G
Pa
)
Total compliance
Elastic compliance
Creep compliance
Onset of
creep
Significant
cracking
during creep (a)
(b)
Onset of
creep
0 0.1 0.2 0.3 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Total laminate strain (%)
Cra
ck d
ensi
ty (
/mm
)
Quasi-static loading
Creep loading
Onset of
creep
(c)
Ramp-up for
creep
67
COD for elastic laminates [94]; however, in the 90° ply the COD increases (see Figure 4-9), which is a
result of increasing creep strain. This demonstrates that the time-dependent effect on the 90° ply COD is
notable for the laminates studied, which is an important result in this study. Furthermore, it can be seen
from Figure 4-8 that the crack density begins to plateau at higher strains; however, the characteristic
damage state (CDS) is not reached before the end of the creep test. This plateau is due to the fact that the
creep strain also reaches a plateau after a notable time increment, which limits the crack driving stresses
during the latter stages of creep loading.
Figure 4-8. Predicted crack density evolution in a [±75/902]s laminate in the angle plies, under creep at 19 MPa and quasi-static
loading. The figure shows that cracks evolve more slowly during creep than during quasi-static loading at low strain levels.
Figure 4-10(a) shows the evolution of laminate axial compliance for two different stress levels during
creep loading of a [±75/902]s laminate. It can be seen that there is a small increase in the creep compliance
component when the stress is increased from 16 MPa to 19 MPa. This is due to the nonlinearity in the
KWW creep model (see Eq. (3.14)), which accounts for a higher compliance at higher stresses through the
nonlinear parameters defined in Eq. (3.14). On the other hand, the elastic compliance at the 16 MPa applied
stress increases from 0.1 GPa-1 to 0.2 GPa-1 during creep, whereas it increases from 0.1GPa-1 to 0.3 GPa-1
at the 19 MPa applied stress. This is due to enhanced crack evolution at the 19 MPa applied stress level,
leading to a greater effect on stiffness degradation. Once again it is demonstrated that ply crack evolution
directly influences the long-term laminate response. It can be inferred from Figure 4-10 that ply cracking
leads to a significant increase in total laminate compliance, even over narrow stress ranges. Figure 4-10(b)
shows the different components of the compliance (creep and elastic) for the [±60/902]s and [±75/902]s
laminates at the same stress level of 19 MPa. The creep compliance of the [±60/902]s laminate is
approximately half that of the [±75/902]s laminate. This is due to the difference in the stacking sequence
and points to the importance of an accurate creep model that considers deformation in both the shear and
transverse directions. The elastic compliance is also much lower for the [±60/902]s and evolves very little
due to the decreased amount of ply cracking, which is a result of lower energy release rates caused by
lower laminate creep strains.
0 0.1 0.2 0.3 0.40
0.5
1
1.5
2
Strain (%)
Cra
ck d
ensi
ty (
/mm
)
Quasi-static loading, +75° ply
Quasi-static loading, -75° ply
Creep loading, -75° ply
Creep loading, 75° ply
68
Figure 4-9. Predicted COD evolution for a 90° ply in a [±75/902]s laminate undergoing creep at 19 MPa, showing a clear increase
linked to creep strain evolution. The increased in COD affects crack density evolution through its effect on the energy release rate
WI.
Figure 4-10. a) Plot showing the predicted evolution of the elastic and time-dependent (creep) components of the
[±75/902]s laminate compliance, at two different stress levels, demonstrating the impact of damage on compliance
increase. b) Plot showing the predicted evolution of the different components (creep and elastic) of the compliance
for the two different laminates, at a stress of 19 MPa.
0 2 4 6 8 10
x 105
0.8
0.9
1
1.1
1.2x 10
-6
Time (s)
CO
D (
mu
m)
0 2 4 6 8 10
x 105
0
0.1
0.2
0.3
0.4
0.5
0.6
Time (s)
Com
pli
an
ce (
/GP
a)
Creep
Elastic
Total ([±60/902]s)
Total ([±75/902]s)
Elastic
Creep
0 2 4 6 8 10
x 105
0
0.1
0.2
0.3
0.4
Time (s)
Com
pli
an
ce (
/GP
a)
Total (19 MPa)
Creep
Elastic
Total (16 MPa)
Elastic
Creep
Change in creep
compliance
Significant
increase in elastic
compliance
(a)
(b)
69
Summary for Chapter 4
This chapter presented a physics-based multi-scale model to predict the simultaneous ply crack
evolution and time-dependent deformational response of general multidirectional laminates subjected to
quasi-static and creep loads. The model follows the framework of SDM and utilizes computational
micromechanics to define the corresponding damaged laminate stiffness degradation parameters, requiring
only ply-level elastic and viscoelastic empirical data as input. The critical energy release rates associated
with ply crack multiplication are also evaluated computationally through a crack closure concept. A
nonlinear viscoelastic Schapery-type model is used to represent the ply behaviour, which is cast into the
framework of CLT for predicting the total laminate time-dependent response. The effects of time-
dependent behaviour and damage evolution are superimposed to evaluate the laminate compliance and the
corresponding creep strains for various cross-ply and multidirectional laminates. Model predictions for the
studied laminates subjected to quasi-static and creep loading correlated well with available experimental
data, with respect to the evolution of ply crack density and laminate creep strain. The predictive capabilities
of the model were further shown through a parametric evaluation of [±θ/902]s laminates subjected to creep
loading at various applied stresses. The developed model is the first reported model that considers the
effects of viscoelasticity, ply crack evolution and stiffness degradation for general multidirectional
laminates simultaneously and will provide an effective means to assess the long-term durability of
structures made from these laminates.
The model presented in this chapter is applicable to uniaxial loading scenarios and the ensuing
evolution in micro-crack density in the different plies of multi-directional laminates. The model is has been
developed for symmetric laminates specifically, with a focus on in-plane mechanical properties. Constant
loading was the focus of this chapter; future works should consider the effects of different loading
scenarios. In the next chapter, the effects of viscoplasticity, as well as bi-axial loading scenarios, are
considered.
70
Chapter 5. Time-dependent bi-axial stiffness
degradation envelopes for structural
composites2
In this chapter, the SDM model is extended to include viscoplastic behavior of the plies. The effect of
bi-axial loading scenarios on damage evolution in symmetric laminates is studied by using COD-based
descriptions of progressive damage. Carbon fiber and glass fiber composites are studied. The chapter
concludes with the development of stiffness degradation envelopes which describe loss in performance
with respect to time for different biaxial loading scenarios.
5.1 Model validation
In order to demonstrate the accuracy of the current model, and its ability to predict the time-dependent
behavior of viscoelastic-viscoplastic composites, the model has been validated with respect to
experimental data, as well as previous models. Due to the current unavailability of experimental data for
time-dependent matrix micro-crack multiplication in viscoelastic-viscoplastic laminates, the different
components of the model have been validated separately. The three parts of the model were presented in
Chapter 3, and consist of a viscoelastic-viscoplastic material model for laminates (not including the effects
of damage), a time-dependent micro-crack multiplication model, and a stiffness degradation model based
on Synergistic Damage Mechanics. While the combined effects of viscoelasticity, viscoplasticity and
damage have not been studied experimentally in laminates in a single procedure, the validation of the
separate parts of the model is sufficient for the purposes of this thesis given the significant quality of
research that has been conducted in the past on the separate issues of matrix micro-cracking and
viscoelastic-viscoplastic behavior in laminates.
2Reprinted from Compos. Struct., vol. 203, March, T. Berton, S. Haldar, J. Montesano and C.V. Singh, Time-
dependent damage analysis for viscoelastic-viscoplastic structural laminates under biaxial loading, pp. 60-70,
Copyright (2018), with permission from Elsevier. DOI: 10.1016/j.compstruct.2018.06.117
71
Figure 5-1. Schematic showing the different modes of ply micro-cracking in a multi-directional laminate. VE refers to the
viscoelastic properties of the plies; VP refers to the viscoplastic behaviour; QS is for quasi-static loading. The top right graph
shows the loading scheme used in the current chapter.
5.1.1 Validation of the viscoelastic-viscoplastic implementation
We first validate the nonlinear viscoelastic-viscoplastic model, without any damage, by comparing the
predictions from our implementation to the theoretical creep simulation results for a GFRP laminate under
uniaxial loading previously published by Megnis and Varna [187]. The parameters to describe the
viscoelastic-viscoplastic behavior of the individual plies of the GFRP laminate as defined in Eq. (3.15) are
given in Table 5-1. Figure 5-3(a) shows the prediction of creep strain following a quasi-static loading to a
stress of 50 MPa in the [±45]2s GFRP laminate and comparison with the material model previously
developed by Megnis and Varna for viscoelastic-viscoplastic plies [187]. It can be observed from the figure
that the predictions of the current model match very well with the previous work, thus validating the creep
model.
72
Figure 5-2. Flowchart explaining the MATLAB program used to perform the progressive damage simulations under
viscoelastic-viscoplastic creep.
Table 5-1. Parameters for the viscoelastic part of the model [187]
A1 (Pa-1) A2 (Pa-1) A3 (Pa-1) A4 (Pa-1)
Transverse
Shear
𝜏𝑟(s)
8.12 x 10-12
3.24 x 10-11
2400
2.81 x 10-12
-3.09 x 10-11
14000
6.14 x 10-12
4.64 x 10-11
25000
8.25 x 10-12
2.29 x 10-11
550000
Table 5-2. Elastic properties of the GFRP plies, obtained from [187].
Elastic property Value
E1(GPa)
E2(GPa)
𝜈12 G12 (GPa)
45
14.6
0.32
4.95
Apply stress increment dσ to laminate
Get local crack driving stress 𝜎2𝛼using CLT.
Calculate WI (Eq. 3-4) using 𝑢𝑛𝛼 computed from Eq. 3-3. If WI>GIc
increment crack density using the crack multiplication model
Update laminate stiffness matrix (Eq. 3-2). Obtain laminate strain from new
stiffness and stress values.
Apply time increment dt
Get laminate creep strain 휀𝐶𝑅𝐸𝐸𝑃,𝑖𝐿 (𝑡) using material model (Eq. 3-5)
Calculate WI (Eq. 3-4). If WI> GIc (t), increment crack density using the crack
multiplication model
Update elastic part of compliance (Eq. 3-2). Get elastic strain
𝑆𝑖𝑗𝐿𝜎𝐶 𝑁𝑆𝑇.𝑗𝐿 𝑡 and total strain 휀𝑖
𝐿(𝑡)(Eq. 3-6)
No
Yes
Input ply properties and damage model parameters 𝑎 1−4𝛼 , 𝑐1−3, 𝐺𝐼𝑐
Initialize time t=0 and set 휀𝐶𝑅𝐸𝐸𝑃,𝑖𝐿 (0) =0
Yes
Output variables of
interest (e.g. crack
density, strain, stiffness)
No
𝜎𝑥 = 𝜎𝐶 𝑁𝑆𝑇,𝑥 𝜎𝑦 = 𝜎𝐶 𝑁𝑆𝑇,𝑦
T = tend
73
Table 5-3. Damage model parameters for the glass fiber/epoxy composite
Laminate [0/90/∓45]s [0/90]s
Ply orientation 0° 90° -45° 45° 0° 90°
Stiffness
degradation
parameter
(GPa)
a1 0.77 8.08 5 5 0.77 8.14
a2 7.45 0.84 4.92 5.01 7.55 0.82
a3 1.74 1.54 0.90 0.84 1.30 1.61
a4 4.78 5.37 3.48 2.56 6.64 3.07
Critical
energy
release rate 𝐺𝐼𝑐0 300 300 300 300 300 300
COD
evolution
parameters
c1 3.15 1.28 1.28 1.26 3.15 1.26
c2 0.95 0.35 0.35 0.7 0.95 0.7
c3 1.69 1.63 1.63 1.61 1.69 1.61
5.1.2 Validation of the crack multiplication model
The crack multiplication was validated by comparing its predictions to experimental results provided
in the literature for the glass fiber/epoxy material system studied in this chapter [188]. The predictions of
the crack multiplication model are compared to the experimental data in Figure 5-3(b). The critical energy
release rates for crack multiplication in each ply were obtained through trial-and-error, until agreement
with experiments was satisfactory. The values 𝐺𝐼𝑐0 for each laminate and each ply are given in Table 5-3.
In the case of the simulations performed in this chapter, it was necessary to predict the evolution of
cracking during constant creep loading. As explained in Chapter 3, this was accomplished by implementing
a time-dependent critical crack multiplication energy, 𝐺𝐼𝑐(𝑡). Using this time-dependent critical energy,
as well as the creep model for viscoelastic-viscoplastic composites, crack multiplication was predicted for
a [±45/902]s CFRP laminate for which time-dependent micro-cracking has been studied experimentally by
Asadi [26]. The results of the model have been compared to the experimental measurements in Figure
5-3(c), showing almost perfect agreement.
5.1.3 Validation of the SDM model
The stiffness degradation parameters used for the SDM model, namely 𝑎1−4𝛼 , were obtained for the
GFRP laminates using FEA micro-mechanical simulations and are provided in Table 5-3.The parameters
for the evolution of COD with respect to crack density are provided in Table 5-3 as well. In order to validate
the SDM model for the GFRP, independent FEA simulations were performed for different crack densities
from which stiffness was obtained. The predictions of the independent FEA simulations were compared to
the predictions of the SDM model, as shown in Figure 5-3(d). Clearly, the predictions of the SDM model
74
are extremely close to the independent simulation results, thus validating the analytical COD formulation,
and the 𝑎[1−4 ]𝛼 parameters.
5.2 Model predictions
5.2.1 GFRP
Two laminates with different stacking sequences and the same ply material properties were considered in
this work to investigate the effect of ply lay-up. A detailed parametric study has been performed to predict
the crack density and stiffness degradation in the [0/90/∓45]s and [0/90]s glass fiber/epoxy laminates with
viscoelastic-viscoplastic properties described by Eq. (3.15) and material properties obtained from Megnis
and Varna [187]. The parameters for the energy-based crack multiplication model and the SDM model are
given in Table 5-3. The ply thickness was set to 0.5 mm.
Figure 5-4(a) shows the evolution of crack density in the 90° ply of a [0/90/∓45]s GFRP quasi-isotropic
viscoelastic-viscoplastic laminate during quasi-static and under creep deformation at a constant axial load
of 210 MPa. The different curves correspond to different transverse loads. The initial portion of the curves
to the left of the red vertical line corresponds to the quasi-static loading, while the portion to the right of
the red line corresponds to the creep loading. It can be observed from the figure that cracking initiates
during the quasi-static loading, and the crack density continues to increase during the creep test. The crack
density evolution shown in Figure 5-4(a) can also be qualitatively compared to experimental measurements
of crack density evolution during creep loading (see [32]). From Figure 5-4(a), it can be seen that an
increasing applied transverse stress 𝜎𝑦 does not lead to a large difference in crack density evolution in the
90° layer. The crack density in the 90° ply reaches approximately 1.5 cr/mm at long times under all
transverse loads. The evolution of crack density in the +45° ply is shown in Figure 5-4(b). The overall
trends of the evolution of crack density are similar to that in the 90° ply. However, transverse loading has
a large effect on the crack density in the off-axis ply. This effect is due to the orientation of the plies for
which a transverse laminate stress causes a larger local transverse stress driving crack multiplication. The
crack density is equal to 0.6 cr/mm under uniaxial loading in the 45° ply, while it reaches 1.4 cr/mm under
a transverse load of 180 MPa.
To quantify the increase in crack density during creep under biaxial loading, crack densities before
(t=0) and after (t = tend) the viscoelastic-viscoplastic deformation are shown in Figure 5-5 for the 90° and -
45° plies of the [0/90/∓45]s quasi-isotropic laminate. In the case of the 90° ply, the crack density increases
with the applied axial load 𝜎𝑥, with crack initiation occurring at around 140 MPa, as can be seen in subplot
(a). The effect of creep strain is to lower the stress necessary to reach a given crack density, shifting the
crack initiation stress to around 80 MPa, as can be seen in subplot (a). The transverse load has very little
75
effect on the crack density evolution, both before and after creep loading. This can be explained from the
stacking sequence of this laminate. In the case of the 90° ply of the quasi-isotropic laminate, the transverse
load causes a slight contraction of the laminate in the axial direction, preventing cracking. However, the
out-of-plane contraction enhances crack multiplication. In this particular stacking sequence, these effects
cancel out, such that the transverse load has almost no effect on crack density evolution in the 90° ply. In
the case of the -45° ply, shown in subplot (b), the crack density also increases with increasing axial stress.
The transverse stress has a very large effect on the crack initiation stress, however, reducing it from more
than 240 MPa to about 80 MPa at time t=0, and from 150 MPa to less than 0 MPa at time t = tend as seen
in subplot (b). This can be explained from the fact that the transverse stress causes an increase in the local
ply COD and local transverse stress by increasing the local ply transverse strain. After creep loading has
occurred, the crack density in the -45° ply has significantly increased.
(a) (b)
(c) (d)
Figure 5-3. a) Viscoelastic-viscoplastic creep strain prediction for a [±45]s GFRP laminate under a creep stress of 50
MPa, and comparison to a previous model from the literature [187]. b) Evolution of crack density under quasi-static
loading for a cross-ply GFRP composite, and comparison to experimental data [188] c) Time-dependent crack density
evolution in each layer of a [±45/902]s CFRP laminate predicted by the model, and comparison to experimental data
under a creep stress of 45 MPa [26]. (d) Predictions of stiffness degradation with respect to crack density in a
[0/90/∓45]s GFPR laminate using the SDM model (Eqs. 3.1-3.3) and independent FEA simulations in APDL
76
(a) (b)
Figure 5-4. Crack density evolutions versus simulation time for a [0/90/∓45]s GFRP laminate loaded to an axial
load of 210 MPa. a) 90° ply crack density. b) 45° ply crack density.
(a) (b)
Figure 5-5. Crack density versus axial stress with different levels of transverse stress in the 90° and 45° plies of the
quasi-isotropic GFRP [0/90/∓45]s CFRP laminate before (t=0) and after (t= tend) creep deformation.
To show the effect of biaxial load ratio on crack density evolution and stiffness, the evolution of crack
density in the different plies of the [0/90/∓45]s laminate and its axial and transverse stiffness with respect
to the biaxial load ratio are shown in Figure 5-6(a). The results correspond to an axial load 𝜎𝑥 level of 150
MPa and a varying transverse load 𝜎𝑦 of 0 – 300 MPa. It can be seen for the quasi-isotropic laminate that
the crack density in 45o and -45o plies are most affected by the transverse load. While the crack density in
90o ply increases from 0.9 to 1.15 cr/mm, the crack density in -45o ply increases from around 0 cr/mm to
more than 1.7 cr/mm. As expected, the increase in transverse load significantly affects the transverse
stiffness as can be seen in the figure. It also affects the axial stiffness significantly, leading to a decrease
in modulus from 0.91 to 0.7 of the initial value. In the case of the cross-ply laminate shown in subplot (b),
crack density in both layers increases due to transverse load, leading to more severe stiffness degradation
in both laminate directions. The predictions of crack density evolution up to around 0.8 cr/mm and
77
associated stiffness degradation by 20 % in these laminates are in good agreement with experimental
observations [54]. We should note that the results of Figure 5-6 are taken at the end of the creep
simulations.
The axial stiffness due to damage progression under quasi-static loading (i.e. before creep) and that
under creep is depicted in Figure 5-7 for the two GFRP laminates. The solid lines represent the stiffness
of the laminates after quasi-static load, i.e. before creep, and the dashed lines represent the stiffness of the
laminates after creep. As can be seen in Figure 5-7(a), the initial axial stiffness of the undamaged
[0/90/∓45]s CFRP starts degrading beyond 140 MPa of uniaxial quasi-static loading and keeps decreasing
with increasing applied load. The normalized stiffness reduces to a value of 85 % of the undamaged value
under a uniaxial load of 300 MPa. Low transverse loads do not have a noticeable effect on damage
initiation; however, when the transverse load reaches 180 MPa, damage initiates at an axial load of 80
MPa. Under transverse loading, cracking is enhanced in all layers of the laminate, causing a larger loss in
stiffness with axial loading. Under a transverse load of 180 MPa, the stiffness has reduced to 77 % of the
undamaged value. Creep deformation enhances cracking, and causes further stiffness degradation. Figure
5-7 (b) shows the axial stiffness of the [0/90]s CFRP laminate under quasi-static loading (before creep) and
after creep. Under uniaxial loading, the stiffness starts degrading under a load of 150 MPa and keeps
decreasing with higher loading levels, due to increased cracking. At the end of 300 MPa of quasi-static
loading the axial stiffness reduces to a value of 85 % of the initial value. The presence of transverse stress
enhances the damage progression leading to a larger reduction in stiffness as can be seen by comparing the
solid lines of different colors. Similarly, creep deformation causes further stiffness degradation, with the
stiffness reaching a minimum value of 79%, at which point crack density is saturated.
(a) (b)
Figure 5-6. Evolution of crack density and normalized stiffness for the GFRP composite under different levels of
transverse loading: (a) Quasi-isotropic [0/90/∓45]s laminate under an axial load of 150 MPa, and (b) cross-ply
laminate under an axial load of 150 MPa
78
(a) (b)
Figure 5-7. Axial modulus degradation versus applied axial stress for (a) [0/90/∓45]s and (b) [0/90]s GFRP laminates,
at the start of the creep tests, and at the end, for different biaxial loads.
In order to better understand the effects of biaxial loading on stiffness degradation, time-dependent
stiffness degradation envelopes have been developed to determine the amount of stiffness loss for different
values of biaxial loading in both laminates studied in this chapter. In Figure 5-8 (a), we show the stiffness
degradation contour lines in terms of the axial and transverse loads for the [0/90/∓45]s GFRP laminate,
before the creep test, and at the end. In each case, the stiffness degradation contour lines correspond to a
loss of 5 %, 10% and 15% in the initial axial modulus, 𝐸𝑥0, or in the initial transverse modulus, 𝐸𝑦
0. Looking
first at the 5% stiffness degradation contour, at time t = 0 (red), we can see that for low transverse loads
the line is nearly vertical, showing that the transverse load does not affect crack density evolution. This
can be confirmed by inspecting Figure 5-5, where it is seen that the crack density in the 90° ply is barely
affected by the transverse load. Although the 45° and -45° layer are affected by the transverse load, their
effect on stiffness loss at low loads is much lower. Similarly, the 5% contour line is horizontal when the
transverse loads are much higher, at which point the stiffness loss is dominated by cracking in the 0° ply.
Looking at the trends for the two other stiffness contour lines, corresponding to a loss in stiffness of 10%
and 15%, the lines are more slanted. This is due to the effect that transverse loads have on crack
multiplication at high loading levels, in the +45° and -45° layer. From inspection of the stiffness
degradation contour lines at the end of the creep simulations, it is clear that the lines have been shifted to
much lower values of stress. This suggests that the viscoelastic-viscoplastic properties of the matrix affect
the patterns of stiffness degradation under uniaxial loading. For example, the 5% stiffness degradation
contour line starts at 125 MPa of axial load, instead of 180 MPa. Under transverse loading, the 5% stiffness
degradation contour line crosses the vertical axis at 80 MPa when viscoelastic-viscoplastic deformation
has been taken into account, while it only crosses the axis at 120 MPa when creep viscoelastic-viscoplastic
deformation is ignored. These marked effects are due to two factors which affect the multiplication of
micro-cracks in the different plies of the laminate. First, viscoelastic-viscoplastic deformation increases
79
the driving force for crack multiplication by increasing the amount of strain for a given applied load.
Second, under time-dependent deformation, the critical energy release rate (see Eq. (3.16)) degrades due
to the behavior of the matrix, which enhances crack multiplication under constant loading.
In Figure 5-8(b), we have shown the stiffness degradation contour lines for the [0/90]s GFRP cross-
ply, with the same ply properties as the [0/90/∓45]s laminate. The contour lines correspond to 5%, 10%
and 15% stiffness loss in either the axial, 𝐸𝑥0, or transverse, 𝐸𝑦
0, modulus. Looking at the 5% stiffness loss
line at time t = 0, we can see that the 5% stiffness degradation occurs at 175 MPa in the axial, and 100
MPa in the transverse direction. When viscoelastic deformation is taken into account (there is no
viscoplasticity in the cross-ply because of the absence of shear stress), 5% stiffness degradation occurs at
125 MPa of axial load, and 75 MPa of transverse load. All the contour lines are slanted, due to the effect
of transverse loading on crack multiplication. This can be explained by inspection of Figure 5-5(b), where
it can be seen that a transverse stress causes crack density in the 90° ply to increase, for a given axial load.
In the case of the 0° ply, large transverse stresses cause an enhancement in crack density, leading to further
stiffness degradation. As for the [0/90/∓45]s laminate, the creep strains cause stiffness degradation contour
lines to be shifted to much lower stresses; the shapes are also changed because of the creep strain
deformation that occurs in the laminate.
(a) (b)
Figure 5-8. The lines in the stiffness degradation contour map correspond to constant levels of stiffness loss after
time-dependent deformation under biaxial loading of (a) [0/90/∓45]s and (b) [0/90]s GFRP laminate.
5.2.2 CFRP
A detailed parametric study has been performed to predict the crack density and stiffness degradation
in the [0/90/∓45]s and [0/90]s carbon fiber-bismaleimide (IM7/5260) laminates with viscoelastic-
viscoplastic properties described by Eq. (3.5) and material properties obtained from Tuttle et al [16]. The
ply thickness was set to 0.14 mm. The parameters for the energy-based crack multiplication model and the
SDM model are given in Table 5-4. The effect of both axial (𝜎𝑥) and transverse (𝜎𝑦) tensile loads on the
evolution of crack density in the plies of the two CFRP laminates has been predicted, as well as their effect
80
on the laminate stiffness in the axial (X) and transverse (Y) directions. When a transverse load (𝜎𝑦) is
applied to the laminate along with an axial load, the crack density in each layer for a given axial stress (𝜎𝑥)
may or may not be significantly affected depending on the orientation of the ply and the stacking sequence
of the laminate.
Table 5-4. Parameters for the viscoelastic part of the model [24]
A1 (Pa-1) A2 (Pa-1) A3 (Pa-1) A4 (Pa-1)
Transverse
Shear
𝜆𝑟(min-1)
1.200 x 10-12
5.723 x 10-12
0.001
1.795 x 10-12
3.942 x 10-12
0.01
8.823 x 10-13
3.874 x 10-12
0.1
1.954 x 10-13
1.001 x 10-12
1
Table 5-5. Elastic properties of the plies, measured at 93 °C, obtained from Tuttle et al [24]
Elastic property Value
E1(GPa)
E2(GPa)
𝜈12 G12 (GPa)
157
8.3
0.3
5.81
Table 5-6. Parameters for the viscoplastic part of the model [24]
n N
Transverse component
Shear component 0.16
0.33
6.25
10.11
Table 5-7. Damage model parameters for IM7/5260
Laminate [0/90/∓45]s [0/90]s
Ply angle 0° 90° -45° 45° 0° 90°
Stiffness
degradation
parameter
(GPa)
a1 0.41 4.17 3.92 4.01 0.41 4.28
a2 4 0.44 3.53 4.03 4.03 0.44
a3 1.66 1.84 0.48 0.5 2.08 2.27
a4 2.56 2.68 -0.036 -0.5 2.57 2.73
Critical
energy
release rate
GIc0
80 80 80 80 100 100
COD
evolution
parameters
c1 3.1159 1.1957 1.1957 1.1773 3.1159 1.1773
c2 0.266 0.098 0.098 0.196 0.266 0.196
c3 1.5484 1.5422 1.5422 1.5248 1.5484 1.5248
Time-dependent stiffness degradation envelopes have been developed to determine the amount of
stiffness loss for different values of biaxial loading in both laminates studied in this thesis. In Figure 5-9(a),
we show the stiffness degradation contour lines in terms of the axial and transverse loads for the
[0/90/∓45]s CFRP laminate, before the creep test, and at the end. In each case, the stiffness degradation
contour lines correspond to a loss of 1 %, 2% and 3% in the initial axial modulus, 𝐸𝑥0, or in the initial
81
transverse modulus, 𝐸𝑦0. Looking first at the 1% stiffness degradation contour, at time t = 0 (red), we can
see that for low transverse loads the line is vertical, showing that the transverse load does not affect crack
density evolution. Similarly, the 1% contour line is horizontal when the transverse loads are much higher,
at which point the stiffness loss is dominated by cracking in the 0° ply. Looking at the trends for the two
other stiffness contour lines, corresponding to a loss in stiffness of 2% and 3%, the lines are much more
slanted. Looking next to the stiffness degradation contour lines at the end of the creep simulations, it is
clear that the shapes are very similar. However, the contour lines are shifted to lower loads. The 1%
stiffness degradation contour line starts at 500 MPa of axial load, instead of 750 MPa.
In Figure 5-9(b), we have shown the stiffness degradation contour lines for the [0/90]s CFRP cross-
ply, with the same ply properties as the [0/90/∓45]s laminate. The contour lines correspond to 1%, 2%and
3% stiffness loss in either the axial, 𝐸𝑥0, or transverse, 𝐸𝑦
0, modulus. Looking at the 1% stiffness loss line
at time t = 0, we can see that the 1% stiffness degradation occurs at 900 MPa in the axial, and 500 MPa in
the transverse direction. All the contour lines are slightly curved, due to the effect of transverse loading on
crack multiplication. As for the [0/90/∓45]s laminate, the creep strains cause stiffness degradation contour
lines to be shifted to much lower stresses, however the overall shapes are maintained.
5.2.3 Effect of temperature on the stiffness degradation envelopes
The effect of temperature on the damage behavior and stiffness degradation has been studied using the
SDM model and temperature dependent material properties for [0/90/∓45]s and [0/90]s laminates. The
material model (Eq. 3.5) is applicable up to a temperature of about 150 °C [24]. Therefore 150 oC was
chosen as the maximum simulation temperature. In order to perform the simulations at high temperatures,
the parameters of the SDM model were calculated by accounting for the different elastic properties of the
plies at 150 °C. The critical energy release rate for crack multiplication was obtained using the trends
reported by Asadi [26]. The time-dependent part of the critical energy release rate was obtained using the
method detailed in Chapter 3.
82
(a) (b) Figure 5-9. Stiffness degradation contour lines. The lines in the stiffness degradation contour map correspond to constant levels
of stiffness loss after viscoplastic deformation under biaxial loading of (a) [0/90/∓45]s and (b) [0/90]s CFRP laminate.
Figure 5-10(a) shows the stiffness degradation contour lines for the [0/90/∓45]s laminate at two
different temperatures, 93 °C and 150 °C. The different lines correspond to a loss in stiffness of 1%, 2%
and 3%. It is clear that temperature has almost no effect on the shapes of the envelopes in this laminate,
and only shifts the envelopes towards lower values of stress. This suggests that the dominant effect of
viscoelasticity-viscoplasticity on damage evolution is through the change in critical energy release rate
with respect to time, and not the increase in strain. As for the lower temperature, at 150 °C, the 1% stiffness
degradation contour line is vertical for low transverse stresses and horizontal for low axial stresses, due to
the low effect of transverse load on cracking in the 90° ply of the laminate, and the low effect of axial load
on cracking in the 0° ply. At larger applied loads, the contour lines are more slanted. In Figure 5-10(b), we
have shown the stiffness degradation contour lines for the cross-ply [0/90]s laminate at two temperatures
of 93 °C and 150 °C. For the cross-ply laminate, the effect of temperature is lower than for the quasi-
isotropic laminate, and the shapes of the contour lines are not affected significantly. In fact, the 1% stiffness
degradation contour line at 150°C is located slightly beyond the 1 % stiffness degradation contour line at
93°C at some points due to the larger initial critical energy release rate (𝐺𝐼𝑐0 ) at higher temperature.
However, the rate of decay in critical energy release rate with respect to time causes most of the contour
lines at 150°C to be shifted towards lower values of biaxial loads.
83
(a) (b)
Figure 5-10. Stiffness degradation contour lines for different temperatures. a) Stiffness degradation contour lines at the end of the
creep simulations (T = tend) at two different temperatures for (a) [0/90/∓45]s and (b) [0/90]s IM7/5260 laminates. This material
has a glass transition temperature of 260 °C.
5.3 Discussion
The model results presented in this chapter are an improvement on the SDM model that includes the
effects of time-dependent damage due to viscoelasticity and viscoplasticity on the creep response of multi-
directional laminates under multi-axial loading. The viscoplastic model used in this chapter was based on
the Zapas-Crissman functional [183] which relates the viscoplastic strain to the loading history of the plies.
Different model parameters were used for the GFRP and CFRP composites.
In order to succinctly summarize the findings of this chapter, the stress-strain curves under uniaxial
loading were plotted for the two material systems in Figure 5-11. For each material system, four stress-
strain curves are plotted: linear elastic model without damage; linear elastic model with damage (before
creep); viscoelastic-viscoplastic model without time-dependent damage; viscoelastic-viscoplastic model
with time-dependent damage (after creep deformation). In the case of the GFRP composite (subplot (a)),
the effect of damage is large, and therefore, before creep deformation, significant non-linearity in the
stress-strain curve can already be observed. Time-dependent deformation, and then time-dependent
damage, cause more severe non-linearity in the stress-strain response of the composite. The curve which
includes the time-dependent damage evolution constitutes a worst-case scenario for the uniaxial
mechanical response of the composite, and could be used as a design tool for applications where constant
loading at higher temperatures might be important. In the case of CFRP, the degradation in stiffness is
much lower, therefore the stress-strain response is not affected much by micro-crack multiplication.
Similarly, the viscoelastic-viscoplastic deformation is much less significant due to the stiffer carbon fibers,
and the stress-strain response is not significantly affected by time-dependent damage. The plies are also
thinner, which constrains crack opening and leads to less stiffness degradation. However, micro-crack
density reaches high values at high loading levels, which will eventually lead to delamination and critical
84
failure. Therefore, even though the stress-strain response is not remarkably affected by micro-cracking,
the model developed in this thesis can provide information into the sequence of mechanisms leading to
failure of the laminate.
(a) (b) Figure 5-11. (a) Stress-strain curves for a [0/90/∓45]s GFRP laminate under uniaxial loading using different material models. (b)
Stress-strain curves for a [0/90/∓45]s CFRP laminate.
The viscoelastic-viscoplastic properties of the ply will cause the stress to vary with time during
constant loading. When cracking occurs, the local stress redistribution will cause the stiffness (elastic) of
the laminate to decrease, which is accounted for in the SDM model. The effect of damage on the evolution
of time-dependent stresses is not explicitly accounted for in our model, as explained in Chapter 3. As of
now, the micro-damage model is limited to the degradation in elastic properties, and viscoelastic-
viscoplastic stress redistribution under micro-cracking is beyond the capabilities of the micro-damage FE
model that has been developed. This approximation is deemed sufficient based on the results of Figure
5-3(c), which show that the model is accurate enough to predict creep strain in a viscoelastic laminate
undergoing time-dependent micro-cracking. Although cracking is assumed not to affect the creep response
under constant crack density, the crack multiplication model predicts crack multiplication under constant
loading, which affects the creep response. Therefore, damage and creep are coupled accurately, even
though the local time-dependent stress redistribution at the microscopic level that occurs upon cracking is
not accounted for explicitly.
Under constant loading, the creep strain of a multi-directional viscoelastic-viscoplastic laminate will
increase. However, even though the globally applied stress is constant, each ply undergoes time-dependent
deformation at a different rate due to their different orientations. This leads to time-dependent changes in
ply stresses. Overall, however, the laminate stress is constant and equal to the applied load, as per Classical
Laminate Plate Theory. The evolution of stresses during creep loading have been plotted in Figure 5-12(a)
for the GFRP quasi-isotropic laminate under a uniaxial constant load of 300 MPa. For both plies, the
85
transverse stress decreases with respect to time following ramp-up to the constant loading level. This is
due to the presence of a very stiff 0° ply, whose behavior in the axial direction is elastic and time-
independent. This layer inhibits time-deformation of the laminate, and effectively reduces the stress in the
more compliant plies.
As explained in Chapter 3 of this thesis, the energy release rate for crack multiplication assumes a
linear elastic relationship between stress and strain in the plies. In reality, as shown in Figure 5-12 (a), the
stresses in the plies are time-dependent due to the viscoelastic-viscoplastic behavior of the matrix. The
effect of including stress relaxation on the crack multiplication predictions has been investigated. The
effect of including stress relaxation is shown in Figure 5-12(b) for a GFRP laminate and compared to the
predictions which do not include stress relaxation. As expected, there is a slight effect on crack
multiplication, however, stress relaxation does not significantly affect the evolution of crack density. The
biggest effect of viscoelastic-viscoplastic properties on crack multiplication is through the decay in 𝐺𝐼𝑐
with respect to time, and the effect of stress relaxation can therefore be ignored.
(a) (b) Figure 5-12. (a) Time-dependent evolution of transverse ply stress for the [0/90/∓45]s under uniaxial constant load of 300 MPa,
during ramp-up and subsequent creep deformation. (b) Effect of including viscoelastic-viscoplastic stress relaxation on crack
multiplication predictions for a [0/90/∓45]s GFRP laminate.
Summary for Chapter 5
A continuum damage mechanics-based SDM model was developed to incorporate the effects of
viscoelastic-viscoplastic properties in GFRP laminates. Schapery’s thermodynamics-based viscoelastic-
viscoplastic constitutive model was implemented in the framework of the SDM model to investigate
progressive damage by matrix micro-cracking in laminates. After initial validation with available
experimental data, two GFRP laminates, namely [0/90/∓45]s and [0/90]s, were studied in this chapter.
The damage behavior was predicted under an initial quasi-static step followed by viscoelastic-viscoplastic
86
creep deformation under biaxial loading. The laminate was quasi-statically loaded up to the creep load
level and then it was allowed to undergo creep at that constant load.
It was observed that the presence of a transverse load can increase the crack density compared to that
under uniaxial loading. In the quasi-isotropic laminate, an increase of transverse load from 0 to 180 MPa
with a constant longitudinal load of 210 MPa led to increase in crack density from 0.6 cr/mm to 1.5 cr/mm
in the 45° ply. In the 90° ply, on the other hand, the crack density was barely affected when a transverse
load is applied. For a [0/90]s cross-ply, a transverse load for a given axial load enhanced cracking in both
plies.
The viscoelastic-viscoplastic deformation was found to reduce the micro-cracking initiation stress for
both laminate stacking sequences. In the quasi-isotropic laminate, damage initiation was found at a uniaxial
load of 140 MPa in the longitudinal direction under quasi-static deformation. However, when viscoelastic-
viscoplastic deformation was accounted for, the damage initiation in the laminate was at 80 MPa of
longitudinal uniaxial load. In the case of the cross-ply, the crack initiation stress decreased from 150 MPa
to 80 MPa when viscoelastic-viscoplastic deformation was taken into account. For both stacking
sequences, an increase in transverse stress enhanced stiffness degradation.
To comprehensively quantify the damage under biaxial loading, stiffness degradation maps were
developed under different combinations of axial and transverse loads. It was found that a given stiffness
degradation occurs at lower axial loading levels when a transverse load is applied to the laminate. When
the effect of creep was accounted for, the level of stiffness degradation for a given biaxial loading scenario
was always more severe. Depending on the effect of transverse load on the evolution of damage in the
different plies of the laminates, the slopes of the contour lines were different. Due to the extensive effects
of viscoelastic-viscoplastic deformation in these composites, the contour lines describing the damage state
after creep were shifted to lower loads, indicating more significant damage, and underwent changes in their
shape, due to the time-dependent evolution in creep strain.
Taken together, these results demonstrate the ability of the SDM model to predict time-dependent
damage progression in viscoelastic-viscoplastic laminates under multi-axial constant loading. Through its
ability to accurately take into account multi-axial loads, and to predict the time-dependent damage
progression in individual plies, the model paves the way for the design of damage-tolerant composites
structures which can withstand more demanding thermo-mechanical environments.
In this thesis, the focus has been on constant loading and creep deformation. The model can be further
extended to include different loading scenarios. In order to extend the methodology to time-dependent
loading, the micro-damage FE model should be augmented to be able to model the viscoelastic and
viscoplastic behavior of each ply.
87
Chapter 6. Strain rate-dependent damage model
for a short fiber composite
In this chapter, the effects of strain rate on the evolution of interfacial debonding in a short fiber
thermoplastic matrix composite have been studied using computational micro-damage modelling. A three-
dimensional RVE of a short glass fiber/thermoplastic matrix composite has been developed and a non-
linear viscoelastic model has been implemented for the matrix. A bilinear Cohesive Zone Model has been
used to model the behavior of the fiber/matrix interface. Following validation with respect to a previous
numerical study, an extensive study of the effects of strain rate, fiber diameter and fiber type on micro-
damage evolution has been performed.
6.1 Model validation
In order to validate the model, the methodology was applied first to a short carbon fiber/epoxy
composite, studied by Pan and Pelegri [200]. An axisymmetric model was created, and the interface
modelled using a CZM. The results of our implementation are shown in Figure 6-2, and compared to the
predictions of Pan and Pelegri. Clearly, the present model perfectly predicts debonding evolution versus
applied strain. Damage initiates at approximately 0.22% applied strain, and reaches its maximum value
(full debonding) at 0.6% strain. Interestingly, the shape of the damage evolution curve is similar to that
observed for crack multiplication in laminates reported in Chapters 4 and 5, even though the microscopic
phenomena that these curves describe are fundamentally different. Due to the complexity involved in
measuring debonded area fraction versus strain experimentally, the experimental validation of this curve
is not possible at this point in time, and therefore the results shown in Figure 6-2 are deemed satisfactory
for the purposes of this thesis.
6.2 Effect of strain rate on damage evolution
Following validation of the model with respect to experimental data, the CZM was applied to a short
glass fiber composite embedded in a HDPE matrix. Initially, the matrix and fiber were modelled as linearly
elastic, and the fiber-matrix interface was modelled using the CZM. The fiber diameter was set to Df = 40
μm and the length Lf = 200 μm (giving a total length of 400 μm when including both fibers). In Figure
6-1, the Von Mises stress profile has been plotted under an applied strain of 0.23 %. The contour plot in
the inset shows a zoomed-in part of the RVE around the two fiber tips. Note that interfacial damage has
not been accounted for in these results. Stresses provided in the legend are expressed in MPa, and distances
88
are in micrometers. The matrix stress is on average much lower than the fiber stress, because of its lower
modulus of elasticity. The stress in each fiber is lower at the tip, but increases along the length. At the fiber
tip, there is a mismatch between the elastic properties of the matrix and fiber, which leads to the occurrence
of a stress concentration. The maximum stress in the fiber reaches approximately 125 MPa, while the
matrix is under a stress of 10 MPa approximately. In Figure 6-1 (b), the displacement in the vertical
direction has also been plotted. Displacement is almost linear with respect to position along the RVE, going
from 0 μm at the bottom face, to 0.94 μm at the top face. Around the region of the fiber tips, the
displacement is not uniform along the cross-section of the RVE due to the stress concentration occurring
as a result of the mismatch in elastic properties of the fiber and matrix.
In the current chapter, the evolution of damage at the interface is taken into account in the calculations
by defining the degradation in stress transfer with respect to normal and shear displacements across the
interface. The CZM provides a local damage variable which describes the loss in stress transfer across the
interface. On the other hand, a global damage variable can be defined through a damage variable 𝐷 equal
to 0 when the total interface area is intact, and 1 when the interface has failed at all points. It is given by
𝐷 =𝐴
𝐴0, where 𝐴0 is the initial area of the fiber-matrix interface, and 𝐴 is the damaged area following the
onset of debonding from the fiber tip. In Figure 6-1(c), a contour plot of the bottom half-fiber shows the
value of the local damage variable, which corresponds to the loss in load-transferring capability of the
interface and is obtained using the cohesive zone model. A value of 1 shows the interface has failed, while
a value of 0 shows the interface is intact. The transition from an intact interface to a completely failed
interface occurs over a very small distance along the half-fiber, as shown by the small transition zone from
completely damaged interface to intact interface. Damage is also uniform around the circumference of the
fiber, which is as expected given the symmetry assumed in the model. In Figure 6-1(d), we have plotted
the damage variable based on debonded area fraction in red. We have also plotted the damage variable
based on the reduction in secant modulus (obtained from the stress-strain curve), defined as follows: 𝐷 =
1 −𝐸
𝐸0, where 𝐸0 is the initial secant modulus (equal to Young’s modulus of the RVE), and 𝐸 is the current
secant modulus. Clearly, the two quantities are related, as expected, and the secant modulus loses 90 % of
its original value due to debonding. This observation further validates the model.
89
(a) (b)
(c) (d)
Figure 6-1. (a)Von Mises stress contour for the short fiber composite RVE undergoing longitudinal deformation with inset
showing the stress in the neighborhood of the fiber tips. Stresses are expressed in MPa. (b) Longitudinal displacement contour
plot of the RVE under an applied strain of 0.1 %. Displacements are expressed in μm. (c) Local damage variable for a fiber Df =
40 μm and Lf = 200 μm. (d) Global damage variable versus applied strain and secant modulus loss curve
In Figure 6-3, the Von Mises stress distribution under increasing deformation has been plotted for one
of the RVEs along a vertical cross-section. The matrix here is modelled as linearly elastic; viscoelasticity
is accounted for in the next simulations. At low strain, no debonding has occurred, and stress is transferred
from the matrix to the fibers, leading to an increase in stress from the fiber tips to the midpoint. The matrix
region located directly between the fiber tips shows a very low Von Mises stress. However, the region
around the fiber tips shows a large stress concentration, with the Von Mises stress reaching more than 15
MPa. Under further loading, at 0.76% strain shown in subplot (b), some fiber-matrix debonding has
occurred. The stresses in the fibers are much lower in the regions that have undergone debonding. The
stresses in the matrix around these regions is also much higher. This observation suggests that stress around
the fiber tips is carried by the matrix, as the reinforcing effect of the fiber is inhibited by the debonding
process. Finally, at 4 % strain shown in subplot (c), the interface has almost completely failed, and the
stress in the fibers is very low; there is a slight increase far from the fiber tips, where the interface is still
intact. The stress in the matrix is much higher than in the fiber, as it is carrying almost the entire load
applied to the RVE.
𝐷 =𝐴
𝐴0
Global damage
variable
Local damage
variable
90
(a) (b) (c)
Figure 6-2. (a) Axisymmetric short glass fiber/epoxy composite RVE used to validate the model. (b) Displacement response under
1% strain. (c) Debonded area fraction versus applied strain for a short glass fiber/epoxy composite, and comparison to the work
of Pan and Pelegri [200].
In Figure 6-4, the Von Mises stress distribution under increasing deformation has been plotted for the
same RVE. The strain rate applied is 10-5 s-1. From the first image, corresponding to 0.088% strain, it can
be seen that the stress along the fibers increases from the tip to the midpoint due to stress transfer from the
matrix to the fiber. As the strain is increased further, to 0.76%, full normal separation occurs at the fiber
faces. Some partial debonding also occurs at the fiber tips. This leads to a lower stress at the ends of the
fibers. At 4 % strain, significant inelastic deformation of the matrix around the to fiber has occurred due
to the high stresses in that region. An interfacial crack has also opened at the fiber tips. The matrix region
away from the fiber tips is still in the linear range of deformation. Debonding has not extended along the
fiber-matrix interface, and there is still effective stress transfer between the matrix and fiber further away
from the fiber tips. Interestingly, the matrix region around the lower fiber tip has not undergone extensive
deformation.
91
Figure 6-3. Von Mises stress distribution in the short fiber RVE with Df = 40 µm and Lf = 200 µm under different strain levels
with a linear elastic HDPE matrix.
In order to understand the effect of matrix non-linearity on damage evolution at the fiber-matrix
interface, the strain rate applied to the RVE was changed by implementing the non-linear behavior into
Abaqus based on the stress-strain response calculated using the non-linear viscoelastic model for HDPE
(see Chapter 3).
In Figure 6-5 (a), we have plotted the global damage variable which describes the fraction of debonded
interface area with respect to applied strain for a fiber diameter Df = 40 μm and a fiber length Lf = 200 μm.
The different curves correspond to the different strain rates applied to the RVE. The simulations were
stopped at 4% strain because damage did not evolve significantly past this strain level. In the case of the
linear elastic matrix, damage initiates early on in the simulations, at approximately 0.1% . It evolves at a
constant rate with respect to applied axial strain until the global strain reaches a strain of 3.6 %, after which
damage reaches its maximum value. As the strain rate is decreased and the non-linear behavior of the
HDPE matrix becomes more pronounced, the evolution of damage is affected. In all cases, damage initiates
under an applied strain of 0.1 %. However, with decreasing strain rate, damage evolves more slowly. Under
a strain rate of 102 s-1, the damage variable reaches a maximum value of 0.95; when the strain rate is lower,
it only reaches a maximum value of 0.3. This behavior suggests that in the case of a relatively thick fiber
(Df = 40 μm), debonding evolution is significantly affected by the viscoelastic behavior of the matrix.
휀 = 0.08 % 휀 = 0.76 % 휀 = 4 %
92
Figure 6-4. Von Mises stress distribution in the short fiber RVE with Df = 40 µm and Lf = 200 µm under different strain levels
for a viscoelastic matrix under a strain rate of 10-5 s-1.
The stress-strain curves obtained for these RVEs using Abaqus are shown in Figure 6-5 (b), with
different applied strain rates. The initial part of the curves is always linear, because the matrix has not been
strained high enough to enter the non-linear region, and no debonding has occurred. In the case of a linearly
elastic matrix, the response becomes non-linear above an applied strain of 0.5 % due to the onset of fiber-
matrix debonding, which reduces the load-carrying capacity of the short fiber composite. As the strain rate
is decreased using the viscoelastic constitutive model for the matrix, the stress is lower. This non-linear
behavior of the RVE is affected by matrix non-linearity in two different ways: the viscoelastic behavior of
the matrix induces a lower matrix stress as the strain rate is decreased, and the viscoelastic behavior also
affects damage evolution. As the strain rate is decreased, the evolution of damage is delayed, because the
interface shear stresses are lower. The fact that stress is reduced when the non-linear behavior of the matrix
becomes more pronounced shows that debonding does not dominate the response of this RVE, and that
instead matrix inelasticity largely governs the stress-strain response. This can be illustrated by calculating
the amount of work applied to the RVE which is dissipated as matrix inelastic deformation, versus the
amount which is dissipated in the form of damage (i.e. debonding) evolution. These quantities have been
plotted in Figure 6-6 (a). Clearly, most of the applied work is dissipated into inelastic deformation, while
the remainder is stored in the form of elastic energy. The amount of energy dissipated in the form of
damage is minimal at lower strain rates.
The inelastic strain was calculated in the RVE using the following equation: 휀𝑝𝑙 =1
𝑉∑ 휀𝑖𝑛
𝑖 𝑑𝑉𝑖𝑖 where
𝑉 is the volume of the RVE, 𝑉𝑖 is the volume of element 𝑖, 휀𝑖𝑛𝑖 is the inelastic strain in the element obtained
4 %0.76 %0.088 %
Interfacial crack
openingPartial
debonding
along sides
Full debonding
at fiber face
93
from Abaqus, and the summation runs over all matrix elements in the RVE. In Figure 6-6 (b) the matrix
inelastic strain has been plotted, which shows that most of the deformational response of the matrix is
dominated by its inelastic behavior. By inspection of the RVE, it was found that inelastic strain was
concentrated at the fiber tips (see Figure 6-4(c)). Away from the fiber tips, strain was elastic. This
observation suggests that the stress concentration at the fiber tips induces the nucleation of a zone of
extensive inelastic deformation. Inspection of the RVE also showed the growth of interfacial opening at
the fiber tip, instead of debonding, when viscoelastic properties were accounted for.
The COD at the fiber tip obtained in these simulations (where strain rate was varied by changing the
constitutive behaviour of the matrix) describes the growth of the interfacial opening, and is shown in Figure
6-7. When the matrix is modelled with a linear elastic property, the COD is close to 0, and there is no
microcrack opening. However, as the strain rate is lowered and inelastic effects become more pronounced,
the COD increases much more significantly. Under a strain rate of 10-2 s-1, the COD reaches 6.5 μm; under
a strain rate of 10-5 s-1, the COD reaches 9 μm. This suggests that under low strain rate, microcrack opening
will dominate the response of a short fiber composite, most likely leading to crack growth into the matrix
as interfacial cracks from adjacent fibers coalesce. These observations are in agreement with experimental
observations by Jerabek et al [204], who found that under high strain rates, debonding in a glass particle
reinforced plastic dominated the response, while matrix shear deformation was more important at low
strain rates. It should be noted that the present model does not account explicitly for matrix microcracking.
Instead, the focus is on interfacial microcrack opening from the fiber tips.
The results reported here can be used in analytical studies of damage in short fiber composites. For
example, Notta-Cuvier et al [156] implemented an analytical model for predicting damage and failure in
randomly oriented short fiber composites. They defined an expression relating the applied strain to
debonded area fraction. They also postulated that at a certain critical void volume fraction, crack
propagation and critical failure of the material could be assumed. The crack opening shown in Figure 6-7
could be used in such an analytical model to predict strain-rate dependent failure induced by the
coalescence of interfacial cracks nucleated at fiber tips. These results improve on previous studies by
accurately accounting for the viscoelastic properties of the matrix.
94
(a) (b)
Figure 6-5. (a) Evolution of the damage variable (fraction of debonded area) with respect to applied strain for different strain rates
in a short glass fiber composite with fiber diameter Df = 40 μm. (b) Stress-strain curves.
(a) (b)
Figure 6-6. (a) Energy dissipation in the RVE of a short glass fiber composite (Df =40 μm) for a viscoelastic matrix deformed at
10-1 s-1. (b) Inelastic strain evolution under different strain rates.
Figure 6-7. Evolution of COD under different strain rates for an interfacial crack at the fiber tips in a short glass fiber composite
with Df = 40 μm.
In Figure 6-8 (a), the interfacial shear stress along the fiber-matrix interface has been plotted at 0.08%
strain (before the onset of debonding) and at 4% strain (after full debonding), for a linearly elastic matrix,
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and a viscoelastic matrix under a strain rate of 10-5 s-1. Subplot (b) shows the axial stress along the fiber
under the same loading conditions. Looking first at subplot (a), in the case of the linearly elastic matrix,
the shear stress is highest at the fiber tip due to the stress concentration effect. The maximum stress reaches
11 MPa, which is the stress at which mode II debonding initiates, according to the CZM implemented in
this thesis. As one moves along the length of the fiber, the shear stress eventually reaches 0. After
significant debonding has occurred (blue dotted curve), the shear stress along the interface is 0 until one
reaches a distance of 170 μm along the fiber. At this location along the fiber, there is a transition from a
fully debonded region, to a fully bonded region. The shear stress increases at that point due to the effective
load transfer enabled by the pristine section of the interface. Looking next to the non-linear matrix, prior
to debonding (yellow dashed curve), the shear stress is lower than in the case of the linearly elastic matrix,
which can be attributed to the lower load-carrying capability of the matrix. At 4 % strain (purple dashed
curve), some debonding has occurred, but it only extends up 60 μm along the fiber. The shear stress
concentration in that case only reaches 5 MPa, which is not high enough to induce mode II debonding.
This therefore suggests that the viscoelastic behavior of the matrix leads to a lower shear stress
concentration at the fiber tips, which inhibits mode II debonding.
Looking next at subplot (b), which shows the axial stress in the fiber, in the case of a linearly elastic
matrix prior to debonding, the stress increases along the length of the fiber, due to the effective transfer of
load from the matrix to the fiber. After significant debonding has occurred, the stress in the fiber is much
lower as the load is mainly carried by the matrix. The portion that is still bonded to the matrix still carries
some load. In the case of the viscoelastic matrix, prior to debonding, the stress in the fiber is lower than in
the linearly elastic matrix due to the lower shear stress across the interface. After debonding, however, the
stress in the fiber is much larger than in the linearly matrix at the same level of applied strain due to less
extensive debonding and more effective stress transfer across the interface. These results therefore suggest
that in a viscoelastic matrix, more load will be carried by the fiber, which might lead to the onset of fiber
breakage.
Table 6-1. Fiber dimensions studied in this thesis
Lf (μm) Df (μm)
200 10
200 20
200 40
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(a) (b)
Figure 6-8. (a) Shear stress distribution along the fiber matrix interface at two different applied strains, for a linearly elastic matrix,
and a viscoelastic matrix under a strain rate of 10-5 s-1. (b) Axial stress distribution.
(a) (b)
Figure 6-9. (a) Damage variable versus applied strain for 3 fiber dimensions with a linearly elastic matrix. (b) Maximum damage
reached for 3 fiber dimensions as a function of strain rate.
The effect of strain rate on debonding evolution was studied for 2 other fiber diameters, provided in
Table 6-1. The corresponding evolution of the global damage variable when the matrix is modelled as
linearly elastic is plotted in Figure 6-9 (a) as a function of applied strain. In the case of the thicker fiber,
damage initiates at a slightly lower value of applied strain and reaches its maximum more quickly. In the
case of the thinner fibers, damage initiates at larger applied strains, and does not evolve as quickly. These
results are due to the different distribution of interfacial stresses induced by fibers with different diameters.
The effects of viscoelasticity and strain rate were studied next, by plotting the maximum amount of
damage reached for the different fiber dimensions as a function of strain rate. This maximum value was
reached at a strain of 4%. The results are plotted in Figure 6-9(b). In the case of the thinner fibers, the
viscoelastic properties cause a significant decrease in the maximum debonded area fraction reached in the
simulations, even at large strain rates. In these cases, most of the applied work is dissipated into inelastic
97
deformation which induces the growth of the interfacial opening. In the case of the thickest fiber,
debonding evolution is more gradual with respect to strain rate, and even at relatively low strain rates
debonding is significant. Taken together, these results suggest that fiber diameter has an effect on the rate-
dependent micro-damage processes in short fiber composites. These observations can be rationalized based
on the different stress concentration enforced at the fiber tips under different fiber diameters. Large fibers
undergo debonding more readily at lower applied strain, at which point inelastic deformation of the matrix
is minimized. Thinner fibers, on the other hand, do not undergo debonding as readily, such that inelastic
deformation becomes dominant before debonding can initiate.
The mechanical behavior of short fiber composites is largely governed by the interfacial shear strength
(IFSS). This property can be partly controlled through proper processing. In order to understand the effect
of this parameter on the progression of micro-damage mechanisms, its value was varied for the fiber
diameter of 20 µm. The values chosen were: 11 MPa (used for the results shown in the previous
paragraphs), 20 MPa and 30 MPa. This range of interfacial shear strengths is representative of experimental
investigations for this material system. In order to understand the effect of IFSS on the deformation of the
matrix, the contour plot of strain energy density obtained using Abaqus has been plotted for these three
different values of interfacial shear strengths under a strain of 4 % in Figure 6-10 for a viscoelastic matrix
under 10-1 s-1 strain rate. The units are N/mm2. In the first case (lowest interfacial shear strength),
debonding has propagated a certain distance along the top fiber. At the same time, significant inelastic
deformation has occurred at the top fiber tip, accompanied with interfacial crack opening. The maximum
strain energy density reaches 40 MPa. In the second case, the debonded area has not propagated as far
along the top fiber, but the crack opening is larger, and the strain energy density for the same strain level
is also much larger. Lastly, in the third case, the interfacial crack opens at the bottom fiber tip, and the
strain energy density reaches an even higher value of 116 MPa.
In order to quantify the evolution of maximum strain energy density, its maximum value throughout
the RVE has been plotted for the three values of IFSS in Figure 6-11. As the applied strain increases, the
maximum strain energy density increases due to the significant inelastic deformation of the matrix at the
fiber tips. When the IFSS is increased, the maximum stored strain energy density increases significantly.
This suggests that in addition to the opening of the interfacial crack, the strain energy density in the matrix
adjacent to the crack is very large, which would most likely lead to the nucleation of a micro-crack into
the matrix region close to the fiber tips.
In summary, increasing the IFSS leads to less extensive debonding, larger crack opening displacement,
and more inelastic deformation in the matrix. Although the material model used in this chapter for the
matrix does not predict high strain behavior and matrix failure, it can be argued that the increase in strain
energy density with applied strain would lead to the nucleation of a micro-crack in the highly strained
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region around the fiber tips. With a larger IFSS, matrix micro-cracking becomes more likely. Moreover,
coalescence between interfacial cracks on adjacent fibers also becomes more likely as the COD at the fiber
tip increases.
(a) (b) (c)
Figure 6-10. Effect of IFSS on strain energy density at 4 % applied strain: (a) 𝜏𝐼𝐹𝑆𝑆 = 11 𝑀𝑃𝑎, (b) 𝜏𝐼𝐹𝑆𝑆 = 20 𝑀𝑃𝑎, (c) 𝜏𝐼𝐹𝑆𝑆 =30 𝑀𝑃𝑎 for a matrix loaded under a strain rate of 10-1 s-1. The glass fiber diameter is 20 μm.
Figure 6-11. Evolution of maximum strain energy density versus applied strain for three different values of the interfacial shear
strength. The glass fiber diameter is 20 μm.
The effect of fiber type was investigated next. Composites considered in the automotive industry are
based on hemp, glass and carbon fibers [205], [206]. Due to the different mechanical properties of these
fibers, specifically their Young’s moduli and Poisson’s ratios, it can be assumed that the evolution of
debonding will be affected. It is also likely that the effect of strain rate on damage evolution will depend
on the mechanical properties of the fibers. In the next set of results, the fiber diameter was maintained at
20 µm, and the length to 400 µm.
First, the evolution of debonded area fraction in terms of applied strain has been plotted for the different
fiber types (see Figure 6-12 (a)). The behavior of the carbon fiber is similar to the that of the glass fiber,
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with damage initiating at a low strain level (0.1 %). The behavior of hemp is different, with damage
initiation at approximately 0.75 %. Note that in all three cases, the interfacial shear strength is constant at
11 MPa. The different behavior of hemp can most likely attributed to its very low stiffness as compared to
the other two fiber types. This lower stiffness leads to a lower stress concentration at the fiber tips, thereby
inhibiting the growth in debonded area fraction.
In subplot (b), the maximum value of the damage variable (i.e. the debonded area fraction) has been
plotted for the three types of fibers with the RVE deformed under different strain rates. From this figure,
it is clear that the rate-dependency of debonding is similar for the types of fibers considered in this chapter.
Surprisingly, the rate-dependency of debonding for hemp is similar to that of carbon and glass at lower
strain rates, even though its Young’s modulus is much smaller. This observation suggests that the rate-
dependency of the debonding process occurring around the fibers tips is not significantly influenced by the
mechanical properties of the fibers.
(a) (b)
Figure 6-12. (a) Damage evolution for different types of fibers with different mechanical properties with a linear elastic matrix.
(b) Damage variable at 4% strain as a function of strain rate for different fiber types. In all cases, the fiber diameter is Df = 20 µm.
Summary for Chapter 6
The evolution of fiber/matrix debonding in short fiber composites with a thermoplastic matrix has been
studied with computational micro-damage mechanics using a bilinear Cohesive Zone Model for the
interface, and a nonlinear viscoelastic material model for the matrix. The CZM was validated with respect
to a previous model. The evolution of micro-damage was precisely monitored through quantitative post-
processing of the simulation data. It has been found that under high strain rate behavior, debonding
dominated the nonlinear response of the RVE, while it was less significant with decreasing strain rate. At
lower strain rates, the stress concentration at the tip of the short fibers induced the opening of an interfacial
crack. The rate of growth of this interfacial micro-crack was significantly affected by the strain rate level.
100
The effects of fiber dimensions on the evolution of micro-damage mechanisms were also studied.
Composites with thin fibers were found to be more prone to void opening. Larger fibers, on the other hand,
were found to be more susceptible to debonding, with interfacial crack opening dominating the response
at low strain rates. The effects of interfacial shear strength and fiber types were also studied. A stronger
interface led to less significant debonding and a higher likelihood of micro-cracking. The fiber type, on
the other hand, did not significantly affect the strain-rate dependency of damage evolution.
One limitation of the current approach is that microcrack growth into the matrix following interfacial
separation is not explicitly predicted. It is not clear at this point at which level of COD the interfacial
micro-crack would grow into the matrix and cause composite failure. While different values of the COD
have been calculated for the different fiber sizes and different strain rates, it is not clear whether the results
demonstrate that micro-crack growth is more likely in certain RVEs. Future studies should focus on micro-
crack growth into the matrix in order to develop accurate criteria for rate-dependent failure of short fiber
composites. The results of the analysis are applicable to matrices with high ductility (more than 100%) due
to the large deformation occurring at the fiber tips. Moreover, the analysis is restricted to the lower range
of interfacial shear strengths (IFSS). The fibers are modelled using a linear elastic isotropic material model.
The loading scenario used in this chapter is purely quasi-static, even when high loading rates are
applied. The deformation calculated using FEA assumes no dynamic behavior. Under impact loading, it is
likely that the time-dependent evolution of stresses will lead to different damage mechanisms, and different
damage evolution patterns. While these effects will be important to consider in future studies, the goal of
this chapter is to isolate the effect of matrix viscoelasticity on the competition between damage
mechanisms.
Temperature and humidity are known to affect both the viscoelastic-viscoplastic properties of the
matrix, the natural fiber and the interface [75], [207], [208]; the material models should be adjusted
accordingly for these effects to be properly accounted for. Temperature and humidity will cause different
time-dependent behavior in the polymer. Residual stresses might also affect the debonding process, which
needs to be understood in the case of natural fibers, which are highly susceptible to moisture. The behavior
of the fibers was assumed to be linearly elastic, which might not be the case in natural fibers, which could
require more sophisticated mechanical models for a proper description of damage evolution at the fiber
level [209].
In this chapter, damage evolution has been studied from a microscopic perspective, focusing on a
single fiber composite. In order to incorporate these results into a multi-scale modelling approach, the
effects of fiber orientation, and fiber distribution will need to be accounted for. Debonding and microcrack
evolution are highly sensitive to the local environment, including the orientation of neighboring fibers, and
their length [69]. Once these effects have been accounted for, expressions can be developed for predicting
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damage evolution in terms of the applied stress. These expressions can then be implemented in structural
analysis software for improving the strain-rate dependent damage predictions of composite structures with
significant viscoelastic behavior. The results of this study therefore provide valuable insights into the
processes of rate-dependent damage in thermoplastic composites and pave the way for the design of
recyclable composites with a thermoplastic matrix to be used in rate-sensitive structural applications.
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Chapter 7. Structural analysis using a rate-
dependent SDM model
In this chapter, the multi-axial SDM model developed in Chapter 3 is implemented into Abaqus by
using a VUMAT. Low-velocity impact on an automotive bumper is simulated using explicit dynamics
FEA. The effect of different design parameters on the results are evaluated and discussed.
7.1 Implementation validation
In Chapter 3, the SDM model was first developed in MATLAB (section 3.1); it was then programmed
into Abaqus using a custom VUMAT subroutine (section 3.3) which implements the SDM model
numerically in the FE software. In this section, the implementation of the SDM model into Abaqus is
validated by comparing the output from FE simulations to the results of the MATLAB implementation.
This was performed by applying a uniaxial in-plane load to a single shell element in the FE software, and
obtaining the stiffness degradations and crack densities in terms of the applied load. In Figure 7-1(a), the
stiffness predicted in MATLAB for a quasi-isotropic GFRP laminate has been plotted. Also plotted are the
results obtained from the implementation of the SDM model into Abaqus. Almost perfect agreement can
be observed. Similarly, the crack densities plotted in subfigure (b) are in good agreement with the
MATLAB numerical implementation of the SDM model.
Next, the VUMAT was augmented to include viscoelastic-viscoplastic properties through Eq. (3.21),
which defines the rate-dependency of the critical energy release rate for crack multiplication (𝐺𝐼𝑐). In
Figure 7-2, the evolution of stiffness in terms of strain has been plotted for different loading rates for a
single shell element in Abaqus loaded under different rates. At high loading rates (more than
3.61%/second), crack multiplication is inhibited due to the increase in critical energy release rate for crack
multiplication, which leads to lower crack densities in all plies of the laminate, and consequently lower
stiffness degradation. It can therefore be concluded that the rate-dependent SDM model has been correctly
implemented into Abaqus using the VUMAT.
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(a) (b)
Figure 7-1. Validation of the implementation of the SDM model into Abaqus FEA software using a VUMAT subroutine, as
compared to the MATLAB predictions: (a) Modulus degradation. (b) Crack multiplication in the 90º ply of the [0/90/∓45]s GFRP
composite.
Figure 7-2. Effect of strain rate on stiffness degradation as predicted by the rate-dependent SDM model implemented into
MATLAB for the [0/90/∓45]s GFRP composite.
7.2 Model validation
In order to verify the accuracy of the present damage model, a comparative study is undertaken here.
The results of the present study are compared with those obtained by Schoeppner and Abrate [210]. They
studied the impact response of laminate plates before and after delamination using experimental testing.
In the test, a 12.7 mm × 12.7 mm laminate plate with stacking sequence [0/90]6s was impacted by a
spherical nose impactor made of steel with diameter 25.4 mm, mass 3.1 kg and initial velocity 1.72 m/s
(or initial kinematic energy equal to 4.6 J). The orthotropic properties of the laminate are given in [210].
Figure 7-3 shows time histories of the contact force obtained in the present study alongside the results of
Schoeppner and Abrate. The comparison of total contact time shows a relative discrepancy of about 25%
between the results of the current study and those obtained by Schoeppner and Abrate. Three explanations
can be provided for the differences between the results. First, the impact event takes place in small time
intervals (milliseconds) and the results of Schoeppner and Abrate were obtained using experimental tests.
Therefore, due to the complexity involved in measuring the displacement response experimentally, there
104
might be small discrepancies between experimental results and computational results. Second, FE
problems with contact definitions such as the impact simulations considered here can involve considerable
numerical inaccuracies due to the complexity of the contact models. Moreover, when defining the
interaction properties in a contact problem, different contact algorithms (i.e. general contact, surface to
surface contact penalty and kinematic methods) can predict different contact forces. Third, the impactor of
present simulation is considered to be a rigid body, however, in the experimental test of [210], the impactor
is made of steel which is a deformable material. Based on these remarks, the differences between
simulation results and the experimental results shown in Figure 7-3 are reasonable.
Figure 7-3. Comparisons of time histories of contact force for a [0/90]6s composite laminate between simulation and experiments
[210].
7.3 Parametric studies
In this section, a parametric study is performed on the effects of the impactor's initial velocity, laminate
stacking sequence, bumper's cross section profile and rate-dependency. Figure 7-4 shows the schematic of
the assembled model of rigid impactor and the composite bumper considered for this study; the model
dimensions are also provided. The material properties are given in Table 7-1. To analyze the effect of the
impactor's initial velocity, three different sets of initial velocities, 4.6 m/s, 8.4 m/s, and 13.1 m/s are
considered. Figure 7-5 shows the time history of contact force for different initial velocities for the
[0/90/∓45]s GFRP composite. It can be found that the velocity has a significant effect on the maximum
contact force, however, the patterns in all results are almost identical. The contact force initially increases
upon impact due to resistance from the bumper. It then decreases as the strain energy of the bumper is
transferred back to the impactor. The oscillations in all three curves are due to the dynamic behavior of the
bumper.
Figure 7-6 shows the evolution of the crack density in each ply of the GFRP [0/90/∓45]s laminate as
a function of time. The crack densities plotted correspond to the maximum values reached throughout the
[210]
105
bumper. The three different plots correspond to different initial impactor velocities. Looking first to subplot
(a), the crack multiplication process initiates first in the 0° ply, at approximately 0.2 ms after impact. This
suggests that the local laminate stress is mainly transverse at this location. The crack density increases,
until it saturates at 750 cr/m. At 1.6 ms after impact, damage initiates in the ∓45° plies. The crack density
reaches a value 1200 cr/m in the -45° while it only reaches 590 cr/m in the 45° because of its lower
thickness. Crack multiplication initiates in the 90° ply at 2.5 ms, reaching a maximum value of 750 cr/m.
These trends for crack multiplication in the different plies suggest a complex local loading scenario. We
should note that the results of the simulations are written to the output files every 0.24 ms. Therefore, the
results shown in Figure 7-6 contain slight numerical inaccuracies due to the very low time scales at which
crack multiplication evolves, which cannot be exactly captured with this relatively large time step.
When the impactor velocity is increased to 8.4 m/s, cracking initiates sooner in the ∓45º plies. This is
due to the larger increase in local ply transverse stress with increasing velocity, which promotes crack
multiplication. The maximum crack density in all plies increases relative to the simulation with lower
impactor velocity. The same trends can also be observed when the impactor velocity is increased to 13.1
m/s. However, the crack density in the 0º ply does not increase significantly with higher impactor velocity.
This saturation effect can be explained based on the crack multiplication model, in which the interaction
between adjacent cracks prevents further increase in crack density.
The SDM methodology implemented in this chapter is based on shell elements, and therefore
delamination cannot be modelled explicitly using, for example, cohesive zone modelling. However, the
current model provides accurate insights into the evolution of sub-critical damage mechanisms which will
eventually lead to failure of the structure. Understanding the progression of damage in the separate plies
of the different laminates paves the way for failure models of laminates under dynamic impact loading. As
was shown in Figure 7-6, the crack density in the 0° ply saturates, which can accurately be taken into
account through the evolution of COD with respect to increasing crack density. Moreover, the local ply
stresses throughout the bumper induce a complex loading scenario on the different stacking sequences,
which will cause complex patterns of damage evolution. To our knowledge, the SDM model, which uses
fully periodic RVEs for calculating progressive damage evolution, is one of the only models which can
account for these complex multi-axial loading scenarios with minimal experimental input. Taken together,
we believe the observations reported in these figures demonstrate the effectiveness of the damage model.
Figure 7-7 illustrates the contours of 0° ply crack density for the quasi-isotropic GFRP laminate for
different initial velocities of the impactor. As seen in the three plots, the dominant damage occurs in the
region of impact. As the impactor's velocity increases, additional damage is induced in the regions close
to the boundaries. Under a velocity of 4.6 m/s, the maximum crack density in the 0° ply is 710 crack per
meter (cr/m). It reaches 910 cr/m when the impactor velocity is increased to 13.1 m/s. This increase in
106
crack density is due to the larger in-plane deformation that the impactor causes on the bumper as its velocity
is increased, which drives crack multiplication. We should note that the crack density contours shown in
this figure are envelope plots, which display results at the integration points having the largest crack density
within the shell elements used in this analysis.
Figure 7-4. Schematic of an automotive bumper subject to low-velocity impact. (a) an example of the mesh pattern, (b)
geometrical dimensions of the bumper and impactor.
Table 7-1. Material properties of CFRP and GFRP unidirectional lamina
Material Density Orthotropic properties
CFRP 1550 kg/m3 E1 =157 GPa, E2 = E3 = 8.3 GPa
𝜈12 = 𝜈13 = 0.32 𝜈23 = 0.32 G12=G13 = 5.81 GPa, G23 = 3.14GPa
GFRP 1600 kg/m3 E1 = 45 GPa, E2 = E3= 14.6 GPa
𝜈12 = 𝜈13 = 0.32 𝜈23 = 0.42 G12 = G13 = 4.95 GPa, G23 = 5.14GPa
107
Figure 7-5. Effect of the impactor’s initial velocity on time histories of the contact force for a [0/90/∓45]s GFRP laminate.
(a) (b)
(c)
Figure 7-6. Time history of the evolution of crack density for the different plies of a [0/90/∓45]s GFRP laminate. Initial velocity
of the impactor is equal to : (a) 4.6 m/s, (b) 8.4 m/s, (c) 13.1 m/s.
108
Figure 7-7. Contour of 0° ply crack density for [0/90/∓45]s GFRP laminate for different initial velocities of the impactor: (a) v =
4.6 m/s, (b) v = 8.4 m/s, (c) v =13.1 m/s.
(a) (b)
Figure 7-8. Effect of laminate stacking sequence on (a) displacement versus contact force and, (b) time histories of contact force
for GFRP laminates under an impactor velocity of 4.6 m/s.
Figure 7-8 shows the results of displacement versus contact force and time histories of the contact
force for a GFRP with three different laminate stacking sequences under an impactor velocity of 4.6 m/s.
The [0/90/∓45]s laminate has the highest maximum contact force compared to the others due to its higher
stiffness. On the other hand, the [0/90]s laminate has the highest maximum displacement in the contact
region, due its much lower torsional stiffness relative to the other two laminates. Moreover, the shear
stiffness of [0/90]s is significantly degraded under impact due to damage, which explains the lower
contact force. From Figure 7-8(b), it can be seen that the [0/90/∓45]s reaches a larger contact force (30
kN versus 23 kN for [0/90]sand 28 kN for [∓45]s). This is due to the larger stiffness properties of this
stacking sequence relative to the other two laminates.
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(a) (b)
Figure 7-9. Effect of laminate stacking sequence on (a) displacement versus contact force, and (b) time histories of contact force
for CFRP laminates under an impactor velocity of 4.6 m/s
Figure 7-9 shows the contact force - displacement curves and the contact force - time curves for three
CFRP stacking sequences under an impactor velocity of 4.6 m/s. The trends observed for the CFRP
material are the same here: the quasi-isotropic laminate induces a larger contact force on the impactor,
while the angle-ply laminate is the least resistant to impact. However, the contact force caused by the
GFRP composites is always lower than for CFRP (see Figure 7-8). For example, the GFRP quasi-isotropic
laminate induces a maximum contact force of 30 kN at 3.5 ms, while the CFRP quasi-isotropic laminate
leads to a maximum force of 60 kN at 3.5 ms. This is due to the much larger in-plane stiffness of CFRP
relative to GFRP.
Table 7-2 shows the maximum amount of damage in each laminate studied in this chapter. The CFRP
composites undergo less damage than the GFRP composites. This is due to the larger in-plane stiffness of
CFRP, which constrains the COD and inhibits stiffness degradation. The [∓45]s composite undergoes the
most stiffness degradation due to its off-axis orientation. The cross-ply and the quasi-isotropic laminate
undergo similar amounts of stiffness degradation, however the shear modulus of the cross-ply is degraded
more significantly due to the absence of ∓45° plies in its stacking sequence.
Figure 7-10 shows the effect of rate-dependence for the [0/90/∓45]s CFRP laminate, based on Eq.
(3.21). Rate-dependence has a slight effect on impact displacement and contact force evolution when the
impactor has an initial velocity of 13.1 m/s. Including rate-dependence causes micro-cracking to be
inhibited at higher loading rates. This leads to larger in-plane stiffness. This explains the slightly different
contact force observed when rate-dependent cracking is accounted for. However, due to the low loading
rates involved in low velocity impact, this effect is negligible, thus warranting the use of the simpler rate-
independent crack multiplication model developed in this chapter.
110
Table 7-2. Maximum degradation of material properties of laminates with different stacking sequences due to low-velocity impact.
Laminate Ex Ey Gxy 𝜈xy
CFRP, [0/90/∓45]s 2.8% 4.6% 3.4% +2.6 %
CFRP, [0/90]s 2.7% 3.7% 42.3% -54.8 %
CFRP, [0/90/∓60]s 4.5% 2.0% 6.4% -2.2%
CFRP, [∓45]s 74% 74% 2.7% +22 %
GFRP, [0/90/∓45]s 13.0 % 17.6 % 14.2 % +1.5 %
GFRP, [0/90]s 17.2 % 18.2% 40.4 % -64.1 %
GFRP, [0/90/∓60]s 19.6% 10.6% 16.7% -18.2%
GFRP, [∓45]s 72.9 % 72.9 % 16 % +40%
(a) (b)
Figure 7-10. Effect of rate-dependency on (a) displacement versus contact force, (b) time histories of contact force for a CFRP
[0/90/∓45]s laminate.
Figure 7-11 shows the crack density contours for the [0/90/∓60]s CFRP laminate with three different
bumper cross-sections when the impactor velocity is 8.4 m/s. The first cross-section is square, and the
maximum crack density in the 0° ply reaches 1000 cr/m. The second cross-section is curved, and shows a
maximum crack density of 820 cr/m, while the last cross-section contains grooves and the crack density
reaches 980 cr/m. These differences between the different bumpers can be attributed to the shapes of the
cross-section, which induce a different stress distribution upon impact, and therefore a different driving
force for crack multiplication. The first bumper is more susceptible to damage evolution, because of the
irregular stress state induced on its surface. These results therefore suggest that damage tolerance in
composite car bumpers is related to the shape of the cross-section, which can be accounted for using the
methodology presented in this chapter.
111
Figure 7-11. Effect of the cross section profile of the bumper on 0º crack density contours for a [0/90/∓60]s CFRP laminate under
an impactor velocity of 8.4 m/s.
Figure 7-12 shows the effect of cross-section on the contact force and displacement of the bumper for
the [0/90/∓60]s CFRP laminate at an impactor velocity of 8.4 m/s. The overall responses for all three cross-
sections is very similar. There are some significant differences in the mechanical responses of the different
bumpers: the bumper with a humped cross-sectional profile is much more rigid, leading to a maximum
force of 60 kN on the impactor. The other two types of bumpers are more compliant. This can be attributed
to the different distribution of damage in the bumper upon impact, as was illustrated in Figure 7-11. It can
also be attributed to the different stress distribution occurring in the different bumpers, which induces a
different load-displacement response, even when neglecting the effects of damage. This suggests that
cross-sectional shape partly governs the homogenized mechanical behavior of the bumper and that it has
an effect on the damage evolution patterns throughout the structure.
(a) (b)
Figure 7-12. Effect of bumper cross-section shapes on (a) displacement versus contact force and, (b) time histories of contact
force for CFRP laminates for the [0/90/∓60]s laminate.
As shown in Figure 7-10, the effect of including a rate-dependent crack multiplication criterion to
represent the viscoelastic-viscoplastic behavior of laminates did not lead to significant differences in the
112
load-displacement curve at the location of impact. In order to more fully understand rate-dependent effects
in this structure, the crack density in the 0° ply has been plotted in Figure 7-13, by ignoring rate-
dependency on the one hand, and by including it on the other hand. Clearly, even though the rate-dependent
behavior did not lead to significant changes in the load-displacement curves, the crack density distribution
is significantly affected: the maximum reaches 910 cr/m when rate-dependency is ignored, while it only
reaches 765 cr/m when rate-dependency is accounted for. Moreover, damage is less extensive throughout
the structure and covers a more limited area. This can be explained based on the increase in energy required
to increase crack density when rate-dependency is included, which reduces crack density. This effect is
more clearly shown in Figure 7-14, where the crack density in the 0° ply has been plotted. Clearly, rate-
dependency has a large effect on crack density evolution. This effect therefore needs to be accounted for
a proper prediction of damage evolution. However, the resulting effect on the contact force-displacement
curve is low due to the restricted area over which micro-cracking extends.
(a) (b)
Figure 7-13. Crack density in the 0° ply of a GFRP [0/90/∓45]s laminate under low-velocity impact at 13.1 m/s : (a) neglecting
rate-dependent behavior and (b) including rate-dependent behavior.
Figure 7-14. Effect of including rate-dependency on crack density evolution in the 0° ply of the [0/90/∓45]s GFRP bumper under
low velocity impact.
113
Summary for Chapter 7.
In this chapter, a robust and accurate damage model considering matrix micro-cracking in laminates
has been used to predict the low-velocity impact response of a car bumper. FE micro-damage modelling
is used to obtain the parameters of the damage model, in lieu of extensive and complicated experiments
that have traditionally been used. The model is then implemented into Abaqus using a VUMAT. The
impact response of different stacking sequences and different material systems is simulated using explicit
dynamics in Abaqus. The results show the following:
GFRP laminates undergo more extensive deformation than the CFRP laminates, due to their lower
in-plane stiffness, and more extensive damage evolution. The larger damage susceptibility of
GFRP can be explained from the lower stiffness of the plies relative to CFRP, which limits the
constraining effects of adjacent plies on the Crack Opening Displacements.
It was found that for both material systems, the quasi-isotropic laminate underwent less
deformation upon impact, and induced a larger contact force on the impactor; these observations
were attributed to the larger bending stiffness of this stacking sequence.
The effect of incorporating a rate-dependent crack multiplication model was also considered and
showed that rate-dependency only had some effect of damage evolution, but an insignificant effect
on the load-displacement response of the bumper.
Lastly, the effect of bumper cross-section shape was studied, and it was found that the damage
evolution of composite car bumpers is sensitive to geometry.
In summary, the results of the parametric studies demonstrate the ability of the composite damage
model to predict micro-cracking in the different plies of GFRP and CFRP laminates with different stacking
sequences and cross-sectional shapes, as well as the mechanical response of damaged composites under
dynamic loading.
The focus of this chapter has been on low-velocity impact of laminates, which is important to
understand in order to predict damage tolerance following collision of a thin composite with an impactor.
However, in the case of vehicles colliding at very high speeds, such as might be expected practically, the
modelling approach developed in this chapter would not be applicable, due to the onset of critical damage
modes such as delamination and fiber fracture, which are not accounted for in this model. Future works
should attempt to extend the model to these critical damage modes, and predict the interactions between
matrix micro-cracking, delamination and fiber fracture. Such a complete model would help elucidate the
progression of damage that unfolds during a crash event and help in the design of failure-tolerant
automotive structures [211]. Furthermore, the model implemented in this chapter is concerned with thin
structures, which can be modelled using shell elements; a rate-independent version of this model has been
114
used before [212], [213] for wind turbine blades. Thicker structures would require a different formulation
of the SDM model which accounts for out-of-plane behavior. Such improvements are left to future works.
In the case of short fiber composites, for example bio-composites, the low-velocity impact response
will be different due to the different evolution of damage processes. The in-plane mechanical properties of
the material will be isotropic due to the random orientation of the fibers. Damage will nucleate at several
locations in the structure due to the stress concentrations effected by the fiber tips. The diffuse damage
will grow with increasing loading, and eventually lead to the coalescence of matrix micro-cracks and the
nucleation of a macroscopic crack, resulting in part failure. The exact shape of the contact force response
might be different from that of the laminate due to the elastic isotropy, but the main difference between
the two types of composites will be the evolution of damage.
115
Chapter 8. Conclusions and future work
8.1. Conclusions
In this thesis, computational modelling tools have been developed to study the effects of the
viscoelasticity and viscoplasticity of polymer-based composites on the nature and evolution of damage
prior to failure. The investigation has spanned from microscopic damage modelling of debonding in short
fiber composites, to ply-level micro-crack modelling in continuous fiber composites, and up to the
structural level, where damage was predicted in an automotive composite structure. A wide range of
computational techniques have been used to complete the objectives of this thesis.
In Chapters 4 and 5 of this thesis, micro-damage modelling of matrix micro-cracking in symmetric
multi-directional laminates has been used to calibrate the equations of a Synergistic Damage Mechanics
model to predict stiffness degradation, as well as micro-crack multiplication in viscoelastic-viscoplastic
composites. An algorithm based on Classical Laminate Plate Theory has been implemented to predict the
bi-axial creep of laminates, and combined with the SDM model to predict the evolution of damage during
creep deformation. Excellent agreement of the predictions of the Continuum Damage Model with
experimental data has been obtained. The effect of viscoplasticity has also been considered in Chapter 5.
Parametric studies have been conducted to understand the effects of biaxial loading ratios, fiber type (glass
and carbon), stacking sequence and temperature. The strengths of the model include its straightforward
implementation for a large range of stacking sequences, its ability to accurately account for different multi-
axial in-plane loading scenarios, and its minimal reliance on experimental data. Therefore, objective 1 of
this thesis has been addressed. The key findings of the two chapters based on this multi-scale model are as
follows:
The evolution of creep strain under constant loading could be captured using the damage
model, as shown in Chapter 4 in Figure 4-2. Therefore, matrix micro-cracking can be assumed
to only affect the elastic properties of the laminate during creep deformation, which results in
a relatively simple constitutive model for the creep of multi-directional laminates.
The time-dependent behavior of the matrix mainly affects damage evolution through the
decrease in critical energy release rate for crack multiplication (𝐺𝐼𝑐) with respect to time. The
evolution of stress in the plies during the creep test does not significantly affect crack
multiplication, as shown in Figure 5-12.
Creep deformation shifts the stiffness degradation envelopes to lower values of stress, as
shown in Figure 5-8-Figure 5-10. These curves can be used to guide the design of composites
used in demanding thermo-mechanical applications
116
Temperature has a slight effect on damage evolution, which has been taken into account by
using temperature-dependent ply stiffness values and time-dependent critical energy release
rates for crack multiplication.
The multi-scale damage model has been implemented into FE software with a user-defined subroutine
in Chapter 7. The effect of low-velocity impact on the dynamic evolution of matrix micro-cracking has
been studied for different stacking sequences, material systems and impactor velocities. It has been found
that some stacking sequences are more prone to damage than others, as shown by the more significant
stiffness degradation levels: the angle-ply underwent the most stiffness degradation, reaching more than
73% stiffness loss when the GFRP was considered, and 74% for the CFRP. The cross-ply underwent
significant loss in shear modulus (more than 40%), while the quasi-isotropic laminates were the most
resistant to damage and underwent the least deformation. The evolution of matrix micro-cracking has been
quantified during the impact event while taking into account complex multi-axial loading scenarios. A
loading-rate dependent crack multiplication model has been implemented, and it has been shown that strain
rate has an effect on damage evolution, although the subsequent effect on the load-displacement response
of the bumper is not very significant. Objective 3 of this thesis has therefore also been completed.
Lastly, in Chapter 6, a new micro-damage model for short fiber composites has been developed to
predict debonding evolution under different strain rates. It models the behavior of the interface using a
bilinear Cohesive Zone Model, and the matrix non-linearity is accounted for using a viscoelastic model. A
range of strain rates has been studied, from 102 s-1 to 10-5 s-1. The effects of fiber diameter, fiber type and
interfacial strengths have been studied as well, providing new insights into the evolution of micro-damage
in short fiber composites with a ductile matrix, as follows:
The viscoelastic behavior of thermoplastic matrices leads to a transition from debonding to
interfacial crack opening with decreasing strain rate. For a 40 μm fiber, at 4% strain, the
debonded area fraction goes from 95% debonding at 102 s—1 strain rate, to less than 30 %
under a strain rate of 10-5 s-1.
Higher interfacial strength lead to a higher likelihood of matrix cracking, and a lower
likelihood of debonding: at 30 MPa interfacial shear strength, the strain energy density reached
in the matrix 110 MPa at 4% strain, while it only reached 40 MPa at 11 MPa interfacial shear
strength. Debonding was also less extensive with increasing interfacial shear strength.
Thinner fibers are less prone to debonding, and are more likely to lead to interfacial crack
opening: for a 40 μm diameter fiber, the debonded area fraction reaches 30 % at a strain rate
of 10-5 s-1, while for a 10 μm diameter fiber, it only reaches 15 %.
117
Hemp fibers are less susceptible to debonding, even at high strain rates, as compared to the
glass and carbon fibers.
The central focus of this thesis has been on the effect of viscoelastic and viscoplastic properties on
damage evolution in composites. These properties relate to the time-dependency of loading, and have been
studied in several ways: constant loading, in which these properties will lead to creep deformation; quasi-
static loading under different strain rates, in which viscoelasticity and viscoplasticity will lead to stress
relaxation and different shapes of the stress-strain curve; and dynamic loading, where the viscoelastic-
viscoplastic properties lead to different material responses, as well as different patterns of damage
evolution due to the rate-dependent failure of the matrix.
8.2 Future work
Based on the findings of this thesis, the following recommendations for future work can be made:
A fully non-linear FE model of matrix micro-cracking should be developed to more accurately
represent the effect of matrix viscoelasticity and viscoplasticity on damage evolution. In this thesis,
the SDM model was based on RVEs with elastic plies, and the effect of matrix micro-cracking on
viscoelastic and viscoplastic compliance were not modelled explicitly. Future work should
improve the accuracy of the model used in Chapters 4 and 5 of this thesis to apply to more complex
loading scenarios, such as dynamic loading, and fatigue. Moreover, the relationship between the
fracture toughness of the matrix, and the critical energy release rate for crack multiplication (𝐺𝐼𝑐)
should be understood, in order to by-pass the complicated set of experiments required to obtain
this parameter.
A more in-depth analysis of micro-damage mechanisms in short fiber composites is required. The
effect of surrounding fibers on the evolution of debonding and matrix cracking needs to be
evaluated in more depth so as to provide more detailed insight into the failure of short fiber
composites. This could be done by augmenting the RVE used in Chapter 6 to include multiple
fibers with different lengths and orientations. A more sophisticated model accounted for the
failure of the polymeric matrix should also be developed to account for the competition between
damage modes, especially for strong interfaces and brittle matrices, which are more likely to
undergo cracking.
The user-defined material subroutine should be extended to account for rate-dependent behavior
at high-temperatures. In this thesis, viscoelastic and viscoplastic behavior were accounted for by
118
using a rate-dependent critical energy release rate for crack multiplication, which is only valid at
room temperature. At high temperatures, the non-linear properties of the plies will also lead to
complex mechanical behavior. This should be accounted for in future investigations, so as to fully
extend the capabilities of the SDM model.
In conclusion, the results of this thesis demonstrate the accuracy of FE modelling of damage
mechanisms, and also provide key results and fundamental insights into damage evolution, at multiple
scales. The methodologies and results can be used for the design of composite components in a wide range
of applications.
119
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