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Understanding and Predicting the Stress Relaxation Behavior of Short-Fiber Composites By: Numaira Obaid A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Department of Chemical Engineering and Applied Chemistry University of Toronto © Copyright by Numaira Obaid 2018
Transcript of Understanding and Predicting the Stress Relaxation ... · Understanding and Predicting the Stress...
Understanding and Predicting the
Stress Relaxation Behavior of
Short-Fiber Composites
By:
Numaira Obaid
A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy
Department of Chemical Engineering and Applied Chemistry
University of Toronto
© Copyright by Numaira Obaid 2018
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Understanding and Predicting the Stress Relaxation Behavior
of Short-Fiber Composites
Numaira Obaid
Doctor of Philosophy
Department of Chemical Engineering and Applied Chemistry
University of Toronto
2018
Abstract
The viscoelastic properties of short-fiber composites are complex and not well-
understood. Previous experimental work has shown that the viscoelastic properties of short-fiber
composites are affected by both elastic fibers and the matrix, which is baffling since elastic fibers
do not exhibit any time-dependence of their own. The goal of this study was to understand why
and how elastic fibers can alter time-dependent behavior when contained in a composite.
In this thesis, conventional shear-lag theory was adapted to include a time-dependent
matrix and a novel analytical model was used to predict the tensile relaxation modulus of short-
fiber composites. The model highlighted the importance of incorporating both the time-
dependent tensile modulus of the matrix as well as its time-dependent shear modulus.
Investigations using the model showed that since stress transfer in a short-fiber composite occurs
through interfacial shearing, the time-dependent shear modulus of the matrix results in time-
varying stress transfer the fiber. Since the stress in the fiber is time-dependent, it exhibits an
apparent stress relaxation stemming from the relaxing shear modulus of the matrix. The model
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predictions were validated using finite-element simulations and experimental data. Comparison
to real data confirmed the hypothesis that the time-dependency observed in elastic fibers
stemmed from the indirect time-dependency imposed by the time-varying stress transfer from the
matrix.
The model was also used to determine the effect of various parameters including fiber
aspect ratio and fiber volume fraction. For the first time, a critical aspect ratio for viscoelasticity
was introduced. This was defined as the aspect ratio at which the contribution to composite stress
relaxation by the fiber is maximized. The effect of fiber orientation was also examined, and an
analytical model was developed to predict the stress relaxation of composites containing
randomly-oriented fibers. It was found that random orientation in the plane would shrink the
effect of fibers by one-third of what would be observed in oriented composites.
In the last part of the thesis, we investigated the strain rate-dependence of short fiber-
reinforced foams. The study highlighted a potential area where knowledge of the stress
relaxation behavior of the short-fiber composites could prove useful.
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Acknowledgements
First and foremost, I would like to thank my supervisors, Professor Mark Kortschot and
Professor Mohini Sain. As a teacher and mentor, Professor Kortschot has taught me more than I
could ever express on paper. He has shown me the importance of being a commendable person
before being a successful academic. Similarly, I would like to thank Professor Sain for his
compassion and understanding, and for his encouragement when I was faced with both personal
and technical challenges.
I would like to thank the members of my reading committee, Professor Don Kirk and
Professor Ning Yan, who have provided me with useful feedback and guidance throughout the
course of my studies. Additionally, I am grateful to my colleagues, Dr. Sadakat Hussain, Billy
Cheng, Dr. Omar Faruk, Dr. Mahi Fahimian, Shiang Law, and several friends – all of whom
have not only provided me with guidance, but also with emotional support throughout my
studies.
I would like to thank Allah for granting me opportunities and the ability to make the most
of them. My doctoral studies would not have been possible without the love and support of my
parents who have always encouraged me to realize my dreams. I hope to always do you proud. I
would like to thank my grandmother for teaching me, by example, the importance of an
empowered and confident woman. I would like to thank my three lovely sisters who have been
the unfortunate recipient of the worst parts of my graduate school: the late-night pickups, and
hours of venting, anxiety, and frustration. Thank you for your patience, thank you for everything.
I would like to extend my gratitude to my wonderful brothers who have never failed to tell me
how proud I made them.
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Special thanks are owed to my in-laws for their help in supervising my newborn while I
completed the various deliverables required for my thesis. Thank you for your love, support, and
encouragement.
There are no words to describe the gratitude I feel towards my husband, Bilal Arshad,
who has been my greatest cheerleader in this journey. There is a rare sincerity in your
encouragement that led to my hubris that I might actually be able to achieve anything that I set
my heart on. Thank you for being the calming voice in my moments of stress (and there were
many) and for being the most patient and perfect partner. I, truly, could not have done it without
you.
And lastly but most importantly, to my beloved son, Rayyaan, who gave me everything
that I never knew I needed. Thank you for teaching me that although this degree was important
for my career, my life was incomplete until I held you.
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Table of Contents
Acknowledgements ........................................................................................................................ iv
Table of Contents ........................................................................................................................... vi
List of Tables ................................................................................................................................. xi
List of Figures ............................................................................................................................... xii
Chapter 1. Introduction ................................................................................................................... 1
1.1 Introduction to Short-Fiber Composites ................................................................................ 1
1.2 Stress Relaxation of Composites ........................................................................................... 3
1.3 Previous Models .................................................................................................................... 5
1.4 Thesis Objectives .................................................................................................................. 7
1.5 Overview of Contents............................................................................................................ 8
Chapter References ................................................................................................................... 10
Chapter 2. Background Information ............................................................................................. 14
2.1 Micromechanical Modeling of Composites ........................................................................ 14
2.1.1 Rule of Mixtures ........................................................................................................... 15
2.1.2 Cox Shear Lag Theory .................................................................................................. 16
2.1.3 Nairn’s Correction to Shear-Lag Theory ...................................................................... 19
2.2 Viscoelasticity of Polymers ................................................................................................. 20
2.2.1 Spring and Dashpot Models ......................................................................................... 20
2.2.2 Stress Relaxation .......................................................................................................... 22
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2.2.3 Degree and Rate of Stress Relaxation .......................................................................... 24
2.3 Previous Models of Viscoelastic Behavior of Short-Fiber Composites .............................. 24
2.3.1 Phenomenological Models ........................................................................................... 25
2.3.2 Tensor Analysis ............................................................................................................ 28
2.3.3 Finite Element Models.................................................................................................. 30
2.3.4 Shear Lag in Broken Continuous-Fiber Composites .................................................... 31
2.3.5 Other Models ................................................................................................................ 31
2.3.6 Conclusions .................................................................................................................. 32
Chapter References: .................................................................................................................. 33
Chapter 3. Understanding the Stress Relaxation Behavior of Polymers Reinforced with Short
Elastic Fibers ................................................................................................................................. 38
3.1 Introduction ......................................................................................................................... 38
3.1.1 Polymer Viscoelasticity ................................................................................................ 43
3.1.2 Micromechanics of Short-Fiber Composites ................................................................ 44
3.1.3 Modelling Approach ..................................................................................................... 46
3.2 Proposed Model................................................................................................................... 47
3.3 Parametric Study ................................................................................................................. 48
3.3.1 Properties of the Matrix and Fiber ................................................................................ 48
3.3.2 Effect of Fiber Content ................................................................................................. 49
3.3.3 Effect of Fiber Aspect Ratio ......................................................................................... 50
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3.4 Finite-Element Analysis ...................................................................................................... 55
3.4.1 Modelling Approach ..................................................................................................... 55
3.4.2 FEA Results .................................................................................................................. 57
3.5 Conclusions ......................................................................................................................... 60
Chapter References ................................................................................................................... 61
Chapter 4. Predicting the Stress Relaxation Behavior of Glass-Fiber Reinforced Polypropylene
Composites .................................................................................................................................... 66
4.1 Introduction ......................................................................................................................... 66
4.2 Experimental ....................................................................................................................... 68
4.2.1 Sample Preparation ....................................................................................................... 68
4.2.2 Experimental Design .................................................................................................... 69
4.2.3 Fiber Characterization .................................................................................................. 70
4.2.4 Stress Relaxation Tests ................................................................................................. 70
4.3 Analytical Model ................................................................................................................. 71
4.3.1 Elastic Properties .......................................................................................................... 71
4.3.3 Fiber Orientation and Aspect Ratio .............................................................................. 75
4.3.4 Effect of Fiber Content on Stress Relaxation Behavior of PP (without MAPP) .......... 76
4.3.5 Effect of MAPP Addition ............................................................................................. 79
4.3.6 Effect of Fiber Content on Stress Relaxation Behavior of PP (with MAPP) ............... 81
4.4 Conclusions ......................................................................................................................... 83
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Chapter References ................................................................................................................... 84
Chapter 5. Modeling and Predicting the Stress Relaxation of Composites with Short and
Randomly Oriented Fibers ............................................................................................................ 88
5.1 Introduction ......................................................................................................................... 88
5.2 Analytical Model ................................................................................................................. 91
5.3 Finite Element Simulations ................................................................................................. 97
5.4 Results ................................................................................................................................. 99
5.5 Conclusions ....................................................................................................................... 104
Chapter References ................................................................................................................. 106
Chapter 6. Investigating the Mechanical Response of Soy-Based Polyurethane Foams with Glass
Fibers under Compression at Various Rates ............................................................................... 107
6.1 Introduction ....................................................................................................................... 107
6.2 Literature Review .............................................................................................................. 108
6.2.1 Foam Compression ..................................................................................................... 108
6.2.2 Fiber Reinforcement in Foams ................................................................................... 111
6.2.3 Strain Rate Dependence in Foams .............................................................................. 112
6.2.4 Strain-Rate Dependence and Stress Relaxation Behavior .......................................... 113
6.5 Experimental Procedure .................................................................................................... 116
6.5.1 Materials ..................................................................................................................... 116
6.5.2 Sample Preparation ..................................................................................................... 117
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6.5.3 Foam Properties .......................................................................................................... 117
6.5.4 Visual Characterization .............................................................................................. 118
6.5.5 Compression Testing .................................................................................................. 118
6.6 Results and Discussion ...................................................................................................... 119
6.6.1 Foam Morphology ...................................................................................................... 119
6.6.2 Mechanical Properties ................................................................................................ 121
6.7 Conclusions ....................................................................................................................... 126
Chapter References ................................................................................................................. 126
Chapter 7. Conclusions ............................................................................................................... 133
Chapter 8. Recommendations ..................................................................................................... 137
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List of Tables
Table 1. Details of the chemical components used in the study ................................................... 69
Table 2. Experimental design for this study ................................................................................. 69
Table 3. Chemical Formulation for PU Foams ........................................................................... 116
Table 4. Effect of fiber on foam density and cell size ................................................................ 119
Table 5. Empirical relationships relating elastic modulus and plateau strength to foam density.
These empirical relationships were used to calculate the properties of the equivalent
neat foam. ..................................................................................................................... 123
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List of Figures
Figure 1. Schematic representing the two phases in a long-fiber composite ................................ 15
Figure 2. The shear-lag model assumes that in short-fiber composites, stress transfer from the
matrix to the fiber occurs through interfacial shearing. ............................................... 17
Figure 3. Stress in the fiber varies with the fiber length, where the tensile forces are maximized
at the center of the fiber. ............................................................................................... 17
Figure 4. Phenomenological modelling of viscoelastic materials is based on a “spring” to
represent the elastic character of the material and a “dashpot” to represent its viscous
character. ...................................................................................................................... 21
Figure 5. The standard linear solid model is commonly used to describe the behavior of
viscoelastic solids ......................................................................................................... 22
Figure 6. The stress in viscoelastic materials relaxes under constant strain ................................. 23
Figure 7. The five-component model used in the study by Somashekar et al. (2012).................. 26
Figure 8. The ten-component model used in the study by Kim et al. (1991) where 𝝈𝒊 is the initial
stress applied onto the sample while 𝝈(0) is the initial reactive stresses from each of
the spring components in the phenomenological model .............................................. 27
Figure 9. A comparison between (a) continuous fiber composites and (b) discontinuous fiber
composites. ................................................................................................................... 45
Figure 10. In a short-fiber composite, the matrix adjacent to the fiber is at a different stress state
than the bulk matrix, resulting in a shear force along the interface. These interfacial
shear stresses are responsible for stress transfer to the fibers in the composite. (a)
Unstressed State; (b) Displacement under uniaxial tension. ........................................ 45
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Figure 11. The normalized elastic modulus of the polyurethane-glass composites under stress-
relaxation. The fiber content ranges from 0% to 50%. The initial elastic modulus
depends on the fiber content, but the data here have been normalized by the modulus
of the unreinforced polymer so that the stress relaxation is highlighted. ..................... 49
Figure 12. Higher fiber contents resulted in an increase in the relaxation time constant indicating
that the rate of relaxation had slowed. This showed that increasing the fiber fraction
slowed the relaxation of the composite. ....................................................................... 50
Figure 13. This graph depicts the change in normalized elastic modulus with fiber content at
various fiber aspect ratios including (a) aspect ratio of 10; (b) aspect ratio of 50; (c)
aspect ratio of 100 and (d) aspect ratio of 100,000. It can be observed that as the fiber
aspect ratio is increased, the long-term modulus increases because the longer fibers are
more efficient reinforcements. ..................................................................................... 52
Figure 14. As the aspect ratio of the fiber increases, more load is transferred from the matrix to
the fiber. This increased shear force on the fiber results in a higher relaxation time. .. 53
Figure 15. As the aspect ratio is continually increased, a larger fraction of the fiber is under
tensile loading and the influence of the shear loading zone decreases. At very high
aspect ratios, the composite begins to approach the properties of a continuous fiber
composite, with no change in the relaxation time. ....................................................... 53
Figure 16. The tensile stresses in the fiber are dependent on its aspect ratio. The aspect ratio (B)
is the value at which the maximum stress transfer begins to occur in the fiber. If the
aspect ratio of the fibers is too low (A), there is inadequate stress transfer between the
fiber and matrix. If the aspect ratio is too high (C), the properties of the composite
approach that of a long-fiber composite. ...................................................................... 55
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Figure 17. A comparison of the overall stress relaxation profile of short-fiber composites shows
excellent agreement between the predictions of the analytical model (-) and the results
obtained from the finite-element simulations (▪). The error bars represent the standard
deviation resulting from five runs of the FEA model material with differing random
fiber placements. .......................................................................................................... 57
Figure 18. Comparison of the analytical model predictions to the finite-element simulation
results shows good agreement between the two at low volume fraction; however, at
volume fractions equal to 20% and greater, the finite-element results deviate from the
predictions of the analytical model. ............................................................................. 58
Figure 19. Good agreement is obtained between the instantaneous (a) and long-term (b) modulus
values obtained from the analytical model (-) and the finite-element simulations (●). 59
Figure 20. Good agreement is obtained between the relaxation time constant obtained from the
analytical model (-) and the finite-element simulations (●). ........................................ 60
Figure 21. Shear stresses in a short-fiber composite ..................................................................... 72
Figure 22. In a stress relaxation test, stresses in the fiber decrease over time due to the decay in
the matrix modulus during stress relaxation................................................................. 73
Figure 23. The glass fibers were well-oriented within the matrix in the direction of loading
(𝑽𝒇 = 𝟓%) ................................................................................................................... 75
Figure 24. Post-processing fiber aspect ratios were measured via matrix burnout; approximately
100 fibers were measured at each fiber content. .......................................................... 76
Figure 25. Stress relaxation behavior of PP/GF composites reinforced with various fiber volume
fractions. ....................................................................................................................... 77
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Figure 26. Comparison of experimental stress relaxation (○) to the analytical model (--) shows
good agreement between the two at all fiber fractions; the error bars are based on a
90% confidence interval. .............................................................................................. 79
Figure 27. The effect of MAPP addition on the stress relaxation behavior of the base polymer
was evaluated by comparing the behavior of polypropylene without MAPP [S1 (○)] to
polypropylene with MAPP [S5 (●)]. ............................................................................ 80
Figure 28. Stress relaxation behavior of composites reinforced with 5%, 10%, and 15% fiber
volume fractions. This experimental data is for samples containing MAPP (●), and has
been compared to an analytical model based on S5 as the matrix (-) to understand the
effect of covalent bonding only. ................................................................................... 82
Figure 29. The load carried by a fiber in the loading axis can be calculated through a cross-line
perpendicular to the loading direction. ......................................................................... 94
Figure 30. A comparison of the overall stress relaxation profile of short-fiber composites shows
excellent agreement between the predictions of the analytical model (-) and the results
obtained from the finite element experiments (■). ..................................................... 100
Figure 31. Comparison of the analytical model predictions to the finite element (FE) simulation
results shows good agreement between the two at low volume fraction; however, at
volume fractions equal to 30% and greater, the finite element results deviate from the
predictions of the analytical model. ........................................................................... 101
Figure 32. Good agreement is obtained between the instantaneous (a) and long-term (b) moduli
values obtained from the analytical model (-) and the finite element simulations (●).
.................................................................................................................................... 102
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Figure 33. Good agreement is obtained between the relaxation rate constant obtained from the
analytical model (-) and the finite element simulations (●). ...................................... 103
Figure 34. Effect of fiber orientation on the properties of the composite as obtained from finite
element experiments. .................................................................................................. 104
Figure 35. Typical stress-strain behaviour of foam under compressive loading ........................ 109
Figure 36. The micromechanical deformation of a cellular material based on the cubic lattice
model .......................................................................................................................... 110
Figure 37. The application of a constant strain can be approximated as several unit step functions
.................................................................................................................................... 114
Figure 38. The effect of stress relaxation behavior of a material on its modulus at the same strain
rate .............................................................................................................................. 115
Figure 39. The strain rate dependence of viscoelastic materials stems from its stress relaxation
behavior ...................................................................................................................... 116
Figure 40. Cellular structure of foams with various fiber contents, as observed under a scanning
electron microscope .................................................................................................... 121
Figure 41. Increasing the strain rate results in an increase in the modulus and plateau stress of
neat polyurethane foam .............................................................................................. 122
Figure 42. The modulus-based reinforcement factor of composite polyurethane foams varied
with both fiber content and strain rate. ....................................................................... 123
Figure 43. The plateau stress-based reinforcement factor of composite polyurethane foams varied
with both fiber content and strain rate. ....................................................................... 125
1
Chapter 1. Introduction
The widespread use of polymers stems from their low cost and the ease of processing;
however, their low strength-to-weight ratio is a critical limitation in several applications. This
can be overcome by the addition of high-modulus fibers to the polymer. The mechanical
properties of the resulting two-phase composite can be tailored to suit various applications by
modifying the type of polymer or the type, size, and volume fraction of the fibers that are added
into the polymer matrix.
The addition of fibers increases the modulus and strength of fibers without a substantial
increase in weight. Additionally, composites can be manufactured using a variety of relatively
low-temperature and inexpensive techniques compared to metal manufacturing processes where
high temperatures are often required. Additionally, components manufactured using composite
are corrosion-resistant, and thus, have lower maintenance costs compared to their metal-based
counterparts. This is important in both the automotive and aerospace industries, which
continually seek innovative methods to reduce weight without compromising performance.
These distinct advantages have allowed composites to become a thriving billion-dollar market,
which continues to attract both industry professionals and researchers alike.
1.1 Introduction to Short-Fiber Composites
Polymer matrix composites are classified as either continuous or discontinuous based on
the length of the fiber. Continuous fiber composites contain long fibers that span the entire length
of the composite while discontinuous composites contain finite length fibers dispersed in a
polymer matrix. Although long-fiber composites have a higher load carrying capacity, these
composites can only be processed using limited manufacturing techniques to ensure that the
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fibers do not break during processing. One such processing method is pultrusion, where fibers or
woven fiber mats from a roll are pulled through a bath of polymer resin and then put through a
heated die where the polymerization occurs. Long-fiber composites can also be processed using
pre-peg tapes, which contain fibers coated with partially-polymerized resin. The tape is typically
laid up by hand and placed in a heated mold to polymerize the resin. Such processes are both
labor-intensive and time-intensive, making them expensive for high-output manufacturing. In
addition, the possibility of fiber breakage makes it difficult to produce long-fiber composites
with complex shapes.
Short-fiber composites have a comparatively lower modulus and strength than long-fiber
composites, but they can be mixed with molten thermoplastic polymers that is easy to process
using a variety of techniques such as injection molding or extrusion. This makes short-fiber
composites suitable for applications that require complex geometries and provides a cost-
effective alternative for components that do not require a high load-carrying capacity.
The static properties of short-fiber composites are relatively simple to understand: the
addition of a stiff, elastic reinforcing phase into a softer polymer matrix typically increases both
stiffness and strength. However, the time dependent properties of the matrix make the polymer-
based composites prone to creep and stress relaxation, which is a challenge when considering
composites for long-term applications. There is a substantial body of theory to predict the static
properties of polymer matrix composites based on the size, shape, and orientation of the
reinforcing phase [1,2]; however, the viscoelastic properties are significantly more complex. A
better understanding of composite viscoelasticity is needed in order to provide guidance for
optimizing composite structure, and for predicting long-term properties.
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1.2 Stress Relaxation of Composites
Stress relaxation experiments, in which a specimen is strained to a fixed level and the
slow decay of stress is monitored, present a simple method of investigating the time-dependent
modulus of reinforced polymers. Since all viscoelastic properties stem from the same basic
mechanisms, an understanding of composite stress relaxation would also provide insight into
composite creep and dynamic mechanical behaviour.
Stress relaxation is an important property that can have adverse effects on components
such as bolted joints and springs. Bolting is a conventional joining technique where a large
amount of stress is applied when the nut is tightened. Over time, that stress dissipates via stress
relaxation, causing a reduction in the clamping force with which the joints are held together.
Similarly, springs are subject to stress relaxation, which causes a change in their spring constant
and deformation during their lifetime. An understanding of stress relaxation and the ability to
control it can prevent critical failures occurring from the eventual loosening of bolted joints and
springs.
In practice, stress relaxation also influences the dissipation of residual stress and warpage
of molded short-fiber composite parts. This makes it critical to not only understand the stress
relaxation of composites, but also to have an analytical model that can be used as a tool to
predict the rate at which various short-fiber composites dissipate stress. Such models can be used
to select appropriate composite formulations for various applications, for example, a composite
with slower stress relaxation would also be better suited in applications where creep is an
important consideration; otherwise, faster stress relaxation could result in quicker and anisotropic
shrinkage.
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Stress relaxation is also a critical property in biological materials. For example, the
human skin is often held under constant strain through stretching or during weight fluctuations.
This makes it important to consider stress relaxation in the design of artificial skin grafts [3-5].
Similarly, it is because of the stress relaxation of skeletal muscles that we are able to stretch our
arms and legs for prolonged periods of time. During such movements, the human skeletal muscle
is stretched under tension at a fixed length. If the muscles did not dissipate the stress resulting
from such stretches, movement would become extremely painful. Thus, stress relaxation is an
important property that must be considered in the development of artificial skeletal muscle
tissues [6,7].
Several experimental studies have been conducted to understand the stress relaxation
behavior of composites. The addition of short, elastic fibers in a composite has been repeatedly
observed to slow the rate of stress relaxation of a composite [8-12]. This has been a subject of
interest because elastic fibers do not exhibit time-dependent behavior, but nevertheless appear to
do so when embedded in a viscoelastic matrix.
Two mechanisms have been proposed in the short fibre composites literature to explain
the change in the rate of stress relaxation with the addition of elastic fibers. The first explanation
is that since stress relaxation occurs by rearrangement of the secondary bonds in a polymer, the
physical presence of fibers impedes molecular rearrangement of the polymer near the
fiber/matrix interface, resulting in slower relaxation of the matrix. However, considering the
scale at which the molecular rearrangements of secondary bonds occur (angstroms), it seems
unlikely that the volume of polymer close enough to a fiber surface to be affected could result in
significant changes to the bulk properties of the composite.
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The second theory proposed in literature focuses on chemical bonding at the fiber/matrix
interface [13-17]. These studies suggest that the presence of a fiber creates additional covalent
bonds at the fiber/matrix interface, and that breaking these additional bonds is a prerequisite to
polymer mobility and relaxation. It was thus proposed that the rate of stress relaxation in
composites was related to how quickly the bonds can be broken, and therefore, how quickly the
polymer could become mobile again.
Both previously proposed mechanisms rely on speculation regarding the molecular
interactions at the fiber/matrix interface. These explanations rely primarily on conjecture and do
not obviate the need for an analytical model that can be used to make numerical predictions.
Although previous experimental studies agree that the presence of elastic fibers can slow the
stress relaxation rate of polymers in which they are embedded, the explanation for this
observation remains elusive.
1.3 Previous Models
Although a number of qualitative explanations for the observed stress relaxation
phenomenon have been proposed, as discussed previously, there have been relatively few
attempts to derive a predictive model. Some studies have used conventional spring/dashpot
models to characterize viscoelasticity, but although these phenomenological models can be used
to characterize the behaviour of specific composites, they do not provide any guidance for
optimizing material structure [18,19]. Similarly, others have used finite element models
incorporating the viscoelastic matrices and short elastic fibers [20-22]. These models highlight
the importance of matrix viscoelasticity, but do not obviate the need for a simple analytical
model.
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Short-fiber composite viscoelasticity has also been extensively modeled using a tensor
elasticity approach to compute the stress field around elastic inclusions embedded in viscoelastic
matrices [23,24]. Unfortunately, the mathematical complexity of these formulations makes it
very challenging to make predictions of viscoelasticity based on measurable material properties.
While some authors have compared their results to experimental data [17], this is usually very
limited.
Other researchers have been hindered by the inversion of highly complex Laplace
transforms generated through the derivations. As a result, such studies have been unable to
produce a useable analytical model [25,26], and often, a numerical solution was used to make
predictions of stress along the length of the short fiber for only one set of parameters. No study
of this type has been able to yield a simple, closed-form solution for stress relaxation, and hence
they did not provide a way of investigating critical issues such as the effect of fiber/matrix
modulus ratio and fiber aspect ratio on the stress relaxation behavior.
Even with the extensive modelling of composite stress relaxation and viscoelasticity,
none of the studies reduce the need for a simple and concise analytical model that can be applied
to composites with various matrices and fibers. Previous phenomenological and finite-element
models only empirically represent specific cases and do not contribute to understanding the
effect of fibers on the stress relaxation of composites. Similarly, although presumably accurate
tensor analyses have been conducted in several studies, the complexity of these models limits the
use of these tools in developing a thorough understanding of generalized composite stress
relaxation behavior.
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1.4 Thesis Objectives
The overarching goals of this thesis are to understand why elastic fibers change the stress
relaxation behavior of composites, and to develop an analytical model to predict the stress
relaxation behavior of composites with various fiber aspect ratios and fiber volume fractions.
Through this study, it will be demonstrated that in polymer-matrix composites, although it is
well-established that the polymer matrix relaxes under tension, another equally important factor
is the time-dependent shear stress transfer at the fiber-matrix interface. We will show that the
time-dependent shear stress transfer causes an apparent stress relaxation in the fibers, which
plays a critical role in the stress relaxation of the overall composite. To the best of our
knowledge, there have been no previous studies that examine the contribution of the time-
dependent shear stress transfer on the stress relaxation of short-fiber composites. In addition, this
hypothesis has not been used in any previous studies to derive an adequate analytical model that
can be used to accurately predict the stress relaxation behavior of various short-fiber composites.
The study is based on three main objectives, which will help understand and predict the
stress relaxation behavior of composites.
1. In the first part of this thesis, an analytical model for short-fiber composites with
viscoelastic matrices was developed. The model accounts for the time-dependence of
the matrix shear modulus, which in turn results in time-dependent stress transfer from
the matrix to the fiber. This time-dependent stress transfer alters the time-dependence
of the overall composite upon the addition of short fibers. The analytical model
provides a tool based entirely on micromechanics and without any chemical bonding
assumptions, which can be used to predict stress relaxation behavior.
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2. In the second part of the study, finite-element simulations were performed and
experimental stress relaxation data was collected. These results were then compared
to predictions from the analytical model to determine the role of micromechanics on
stress relaxation, and to establish the accuracy of the analytical model. Good
agreement of the simulations and experimental data to the analytical model indicates
that micromechanics play a significant role on the effect of fibers on stress relaxation
behavior of composites.
3. The last part of the study examined the effect of making chemical changes at the
interface by the addition of an interfacial coupling agent, and the effect of these
changes on stress relaxation behavior of the composite.
Once the validity of the analytical model was established by both finite-element analysis
and experimental work, the model was used to predict the effect of various parameters including
fiber content and fiber aspect ratio.
In another part of this thesis, studies of the effect of fiber orientation on the stress
relaxation behavior of short-fiber composites are reported. Establishing an accurate analytical
model for misoriented fibers is particularly important due to the increased use of randomly
oriented short-fiber composites in industrial applications.
The last part of this thesis focuses on an application of composite stress relaxation
behavior to investigate the strain-rate dependence of short fiber-reinforced foams.
1.5 Overview of Contents
A total of four journal papers have been prepared from this thesis, which have all been
accepted and published in various journals, and the overall structure of this thesis is based on
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those papers. Chapters 3, 4, and 5 contain the manuscripts in the same condition that they were
published by their respective journals and no changes have been made to their overall content.
Some modifications have been made to Chapter 6, since its publication, to better explain the
relationship between strain-rate dependence and stress relaxation behavior; however, there have
been no changes made to the presented figures or conclusions. In each case, a brief introductory
paragraph is used to link the papers to the overall thesis.
A more detailed literature review on polymer viscoelasticity and common composite
micromechanical models is presented in Chapter 2. This chapter covers a brief overview of the
literature, and additional literature pertaining to each chapter is presented at the beginning of that
chapter.
The development of the analytical model to predict the stress relaxation behavior of
composites containing oriented, elastic, short-fiber composites is presented in Chapter 3. Based
on the analytical model, a parametric study was conducted to investigate the effect of fiber
content and fiber aspect ratio. Additionally, the model predictions were compared to results
obtained from finite-element simulations. Both of these results are also presented in Chapter 3.
The development of the analytical model to predict the stress relaxation behavior of
randomly-oriented, elastic, short-fiber composites is presented in Chapter 4. This chapter also
contains a comparison of the analytical model predictions to results obtained from finite-element
simulations as well as a discussion on the effect of fiber orientation on stress relaxation behavior.
Chapter 5 contains experimental stress relaxation tests performed on polypropylene-
based composites containing various contents of oriented glass fibers. The experimental data was
compared to the analytical model developed in Chapter 3.
10
At the end of the thesis, Chapter 6 presents an example of how the theory presented in
this thesis can be applied to an industrial application. This chapter investigates the effect of
deformation rate on the mechanical properties of fiber-reinforced foams.
The thesis conclusions and recommendations are presented in Chapter 7.
Chapter References
1. Piggott, M. R.; Taplin, D. M. R. Load Bearing Fiber Composites; Pergamon Press: New
York, NY, USA, 1980.
2. Jones, R. M. Mechanics of Composite Materials; CRC Process: Washington D.C, USA,
1975.
3. Reihsner, R.; Menzel, E. J. Two-dimensional stress-relaxation behavior of human skin as
influenced by non-enzymatic glycation and the inhibitory agent aminoguanidine. J. Biomech.
1998, 31(11), 985-993.
4. Cooper, M. L.; Hansbrough, J. F. Use of a composite skin graft composed of cultured human
keratinocytes and fibroblasts and a collagen-GAG matrix to cover full-thickness wounds on
athymic mice. Surgery. 1991, 109(2), 198-207.
5. Gurunluoglu, R.; Shafighi, M.; Gardetto, A.; Piza-Katzer, H. Composite skin grafts for basal
cell carcinoma defects of the nose. Aesthetic Plast. Surg. 2003, 27(4), 286-292.
6. McKeon-Fischer, K. D.; Rossmeisl, J. H.; Whittington, A. R.; Freeman, J. W. In Vivo
Skeletal Muscle Biocompatibility of Composite, Coaxial Electrospun, and Microfibrous
Scaffolds. Tissue Eng. Part A. 2014, 20(13-14), 1961-1970.
7. McHugh, M. P.; Magnusson, S. P.; Gleim, G. W.; Nicholas, J. A. Viscoelastic stress
relaxation in human skeletal muscle. Med Sci. Sports Exerc. 1992, 24(12), 1375-1382.
11
8. Kutty, S. K.; Nando, G. B. Short Kevlar fiber-thermoplastic polyurethane composite. J. Appl.
Polym. Sci. 1991, 43, 1913–1923.
9. Sreekala, M. S.; Kumaran, M. G.; Joseph, R.; Thomas, S. Stress relaxation behavior in
composites based on short oil-palm fibres and phenol formaldehyde resins. Compos. Sci.
Technol. 2001, 61, 1175–1188.
10. Suhara, F.; Kutty, S. K.; Nando, G. B. Stress relaxation of polyester fiber-polyurethane
elastomer composite with different interfacial bonding agents. J. Elastom. Plast. 1998, 30,
103–117.
11. Saeed, U.; Hussain, K.; Rizvi, G. HDPE reinforced with glass fibers: rheology, tensile
properties, stress relaxation, and orientation of fibers. Polym. Compos. 2014, 35, 2159–2169.
12. Stan, F.; Fetecau, C. Study of stress relaxation in polytetraflyoroethylene composites by
cylindrical macroindentation. Compos. Part B. Eng. 2013, 47, 298–307.
13. George, J.; Sreekala, M. S.; Thomas, S.; Bhagawan, S. S.; Neelakantan, N. R. Stress
relaxation behavior of short pineapple fiber reinforced polyethylene composites. J. Reinf.
Plast. Compos. 1998, 17, 651–672.
14. Geethamma, V. G.; Pothan, L. A.; Rhao, B.; Neelakantan, N. R.; Thomas, S. Tensile stress
relaxation of short-coir-fiber reinforced natural rubber composites. J. Appl. Polym. Sci. 2004,
94, 96–104.
15. Mirzaei, B.; Tajvidi, M.; Falk, R. H.; Felton, C. Stress relaxation behavior of lignocellulosic-
high density polyethylene composites. J. Reinf. Plast. Compos. 2011, 30, 875–881.
16. Pothan, L. A.; Neelakantan, N. R.; Rao, B.; Thomas, S. Stress relaxation behavior of banana
fiber-reinforced polyester composites. J. Reinf. Plast. Compos. 2004, 23, 153–165.
12
17. Boukettaya, S.; Almaskari, F.; Abdala, A.; Alawar, A.; Daly, H. B.; Hammami, A. Water
absorption and stress relaxation behavior of PP/date palm fiber composite materials, in:
Chouchance, M.; Fakhfakh, T.; Daly, H.; Aifaoui, N.; Chaari, F. (Eds.), Design and
Modeling of Mechanical Systems -II; Springer: Hammamet, Tunisia, 2015, 437–445.
18. Somashekar, A. A.; Bickerton, S.; Battacharyya, D. Modelling the viscoelastic stress
relaxation of glass fibre reinforcement under constant compaction strain during composites
manufacturing. Compos. Part A. 2012, 43, 1044–1052.
19. Safraoui, L.; Haddout, A.; Benhadou, M.; Rhrich, F.; Villoutreix, G. Experimental study and
modeling of the relaxation behavior of the injected polypropylene composites reinforced with
short glass fibers. Int. J. Emerg. Technol. Adv. Eng. 2014, 4, 81–87.
20. Naik, A.; Abolfathi, N.; Karami, G.; Ziejewski, M. Micromechanical viscoelastic
characterization of Fibrous Composites. J. Compos. Mater. 2008, 42, 1179–1204.
21. Brinson, L. C.; Lin, W. S. Comparison of micromechanics methods for effective properties
of multiphase viscoelastic composites. Compos. Struct. 2013, 41, 353–367.
22. Fisher, F. T.; Brinson, L. C. Viscoelastic interphases in polymer-matrix composites:
Theoretical models and finite-element analysis. Compos. Sci. Technol. 2003, 61, 731–748.
23. Sevostianov, I.; Levin, V.; Radi, E. Effective viscoelastic properties of short-fiber reinforced
composites. Int. J. Eng. Sci. 2016, 100, 61–73.
24. Smith, N.; Medvedev, G. A.; Pipes, R. B. Viscoelastic shear lag analysis of the discontinuous
fiber composite in Proceedings of the 19th International Conference on Composite
Materials, Montreal, QC, Canada, 28 July–2 August 2013.
25. Yancey, R. N.; Pindera, M. J. Micromechanical analysis of the creep response of
unidirectional composites. J. Eng. Mater. Techol. 1990, 112, 157–163.
13
26. Barbero, E. J.; Luciano, R. Micromechanical formulas for the relaxation tensor of linear
viscoelastic composites with transversely isotropic fibers. Int. J. Solids Struct. 1995, 32,
1859–1872.
14
Chapter 2. Background Information
2.1 Micromechanical Modeling of Composites
The use of polymers in high-performance applications is limited by their low mechanical
properties such as modulus and strength. The addition of stiff fibers to a soft polymer matrix
increases both its modulus and strength. Different combinations of fibers and matrices, and their
relative volume fractions, can be used to produce polymer-based composites with different
properties. The static mechanical properties of these composites are simple to understand and
have been investigated both experimentally and via analytical and numerical models.
Micromechanical models are commonly used to predict the properties of composites. In
such models, a representative volume element is selected to represent the composite on a micro-
scale, and the bulk composite is considered as a homogeneous continuum of these elements. The
interactions between the phases in the element can be extrapolated to predict the interactions
occurring within the entire composite, and thus, they can be used to predict the properties of the
bulk composite. The models are based on the mechanical interactions between the phases and
specific molecular interactions are typically ignored.
Micromechanical models are particularly useful because they can predict the properties
of a composite based only on easily-measurable properties of the individual phases comprising
the material, such as the individual stiffness, Poisson’s ratio, fiber aspect ratio, and volume
fraction of the phases.
However, as with any model, micromechanical models are only a starting point that can
be used in the initial stages of material selection and are not a replacement for experimentation
15
because the exact microstructure of a composite is not known. Local deformities and
inconsistencies can decrease the actual properties of the material when compared to theoretical
predictions.
This thesis requires an understanding of the principles of shear lag, which is the most
common model used to predict the mechanical behavior of short-fiber composites. Since this
model is an extension of the Rule-of-Mixtures model used to predict the behavior of continuous-
fiber composites, both micromechanical models will be reviewed briefly in this section.
2.1.1 Rule of Mixtures
The Rule of Mixtures is the simplest micromechanical model and is used to predict the
mechanical properties of continuous-fiber composites [1]. In this model, since the fibers extend
to the edge of the matrix, it is assumed that equal strain is experienced by the fiber and the matrix
(see Figure 1).
Figure 1. Schematic representing the two phases in a long-fiber composite
16
The modulus of the composite in the loading axis is calculated as an average of the
individual moduli of the matrix (𝑬𝒎) and the fiber (𝑬𝒇), weighted by their respective volume
fractions (𝑽𝒎 and 𝑽𝒇 respectively). The final equation is shown in Equation (1).
𝐸𝑐 = 𝐸𝑚𝑉𝑚 + 𝐸𝑓𝑉𝑓 Equation 1
This model estimates the maximum possible modulus of a composite consisting of a
specific fiber and matrix combination at a pre-selected volume fraction. The actual properties of
the composite may be lower than the properties that are predicted by the model. This model
forms the basis for many other micromechanical models.
2.1.2 Cox Shear Lag Theory
The stress transfer in short-fiber composites is more complex than that in continuous-
fiber composites because the short fibers are embedded within the matrix, and thus, any external
strain is applied to the matrix and the fibers are not directly stretched. There are several empirical
models for predicting the properties of short-fiber composites; however, empirical models have
significant limitations. Firstly, they do not provide any understanding of the physics behind a
material’s behavior and properties. Secondly, they often utilize curve-fitting parameters, which
must be determined through experimentation.
Cox’s shear lag theory is the most common analytical model used to predict the behavior
of short-fiber composites [2]. The theory examines a unit cell of a cylinder of matrix containing a
cylindrical fiber at its center (see Figure 2). At the outer edge of the cylinder, the matrix
experiences a strain that is equal to the strain applied to the composite. However, due to the
presence of the fiber, the strain in the matrix is non-uniform. At the fiber-matrix interface, the
matrix is constricted by the fiber and experiences a strain much lower than the global strain.
17
Figure 2. The shear-lag model assumes that in short-fiber composites, stress transfer from the matrix to the
fiber occurs through interfacial shearing.
Cox relates the interfacial shear stress to the matrix shear modulus and the difference
between the global displacement (at a remote distance, R, which is unaffected by the fiber) and
the displacement of the fiber. This difference is written as 𝒖𝑹 − 𝒖𝒇. Since the model assumes
that the fiber tensile stress is equal to the shear stress at the interface, the fiber tensile stress can
then be calculated.
The shear lag model assumes that the ends of the fiber do not carry any load, and thus,
the tensile stresses are at the fiber ends must be zero. When a strain is applied to a short-fiber
composite, shear stresses are generated at the interface and load is gradually transferred from the
matrix to the fiber. As a result, the tensile stresses vary along the fiber length and are maximized
at the center of the fiber (see Figure 3).
Figure 3. Stress in the fiber varies with the fiber length, where the tensile forces are maximized at the center
of the fiber.
18
The equations generated by Cox’s shear lag theory are summarized in Equations (2) and
(3). This is analogous to the Rule-of-Mixtures, where the contribution of the fiber and the matrix
are weighted by their respective volume fractions (𝑽𝒇 and 𝑽𝒎); however, the stresses in the fiber
are scaled down by an effectiveness factor 𝜼𝒄𝒐𝒙 which varies between 0 and 1. This adjustment
accounts for the variation in the strain of the fiber along its length. If the entire length of the fiber
is fully strained, the effectiveness factor approaches 1 and the equation approaches the rule-of-
mixture.
𝐸𝑐 = 𝐸𝑚𝑉𝑚 + (1 −tanh (
𝜂𝑐𝑜𝑥𝐿𝑓
2)
𝜂𝑐𝑜𝑥𝐿𝑓
2
)𝐸𝑓𝑉𝑓 Equation 2
𝜂𝑐𝑜𝑥 =1
𝑟[ 2𝐸𝑚
𝐸𝑓(1 + 𝑣𝑚) ln (𝑃𝑓
𝑉𝑓)]
12
Equation 3
Here, the modulus of the composite (𝑬𝒄) is related to the moduli of the fiber (𝑬𝒇) and the
matrix (𝑬𝒎), their respective volume fractions (𝑽𝒇 and 𝑽𝒎), the fiber length and radius (𝑳𝒇 and
𝒓), and the shear modulus of the matrix (𝑮𝒎). The model also requires a packing factor (𝑷𝒇),
which is indicative of the geometrical arrangement of the fibers and is often simplified to be
approximately equal to 1.
Like any analytical model, the shear-lag theory is based on several key idealized
assumptions. Firstly, it is assumed that the interfacial bonding between the fiber and matrix is
continuous and perfect, and that no slippage occurs at the interface. It is also assumed that both
the fiber and the matrix are homogeneous elastic and isotropic materials.
19
2.1.3 Nairn’s Correction to Shear-Lag Theory
Since its initial introduction by Cox, the shear-lag theory has been corrected by Nairn
(1997) [3] and others including Nayfeh (1977) [4] and McCartney (1992) [5]. The main dispute
discussed by Nairn is that the original theory neglects radial displacements in its calculation of
the shear strain. In other words, the shear stress generated within the matrix is said to be
proportional only to gradients in the axial displacements within the matrix, whereas by an exact
elasticity analysis, the shear stress should be proportional to the sum of the derivatives of both
the axial and radial displacements. The derivation by Nairn results in a corrected effectiveness
factor (𝜼𝒏𝒆𝒘) as shown in Equation (4) below.
𝜂𝑛𝑒𝑤 = [2
𝑟2𝐸𝑓𝐸𝑚
(𝐸𝑓𝑉𝑓 + 𝐸𝑚𝑉𝑚
𝑉𝑚4𝐺𝑓
+1
2𝐺𝑚(
1𝑉𝑚
ln (1𝑉𝑓
) − 1 −𝑉𝑚2
))]
12
Equation 4
The classic shear lag model, as presented by Cox, can be thought of as an idealized
upper-bound for the properties of short-fiber composites, since the incorporation of radial
displacements would decrease the shear strain at the interface [6].
As mentioned in previous work, shear lag analysis is based on an assumption of perfect
bonding and no slip or plastic deformation at the interface, i.e. only within the elastic region [6].
In the case of stress relaxation at very low strains, this condition is satisfied since deformation is
confined to the elastic region, and thus, the analysis can be used here. Although the model by
Nairn provides a more accurate solution, many studies have found that the original shear-lag
equation is sufficient when attempting to model the behavior of composites with a high fiber-to-
20
matrix modulus ratio (𝑬𝒇
𝑬𝒎≥ 𝟏𝟎𝟎) [7,8]. Since the system used in this thesis has a modulus ratio
within this range, the original shear-lag equation will be used.
It is important to note that since this thesis will incorporate the time-dependent shear
modulus and investigate its contribution on the stress relaxation behavior of a composite, the use
of either method would not change the conclusions drawn from this work.
2.2 Viscoelasticity of Polymers
Viscoelastic materials are a subclass of materials that display both elastic and viscous
behavior. Compared to an elastic material, which exhibits strain-rate-independent responses to
external forces, a viscoelastic material exhibits responses that vary with time. When an elastic
material is subjected to a cycle of loading and unloading, the material deforms proportional to
the strain during loading and then instantaneously reverts to its original dimensions upon
unloading. However, a viscoelastic material exhibits a time-lag and reverts to its original shape
slowly after unloading. There is a loss of energy during a loading-unloading cycle, which is not
true of an elastic material. The viscoelasticity of polymers gives rise to several measurable
properties such as creep, stress relaxation, and strain rate-dependence of the loading response.
2.2.1 Spring and Dashpot Models
The most common method to model viscoelastic properties is through phenomenological
models known as spring-and-dashpot models.
Phenomenological models are not derived from mechanistic theory but are fit well to the
observed time-dependent behavior. In a spring-and-dashpot model, the behavior of a viscoelastic
material is divided into its elastic character, which is described using a spring, and its viscous
character, which exhibits behavior analogous to a dashpot (see Figure 4).
21
𝜎 = 𝐸𝜀 𝜎 = 𝜂𝑑𝜀
𝑑𝑡
Figure 4. Phenomenological modelling of viscoelastic materials is based on a “spring” to represent the elastic
character of the material and a “dashpot” to represent its viscous character.
The behavior of the spring is described using a spring constant, 𝑬, which reduces the
equation to Hooke’s Law. The dashpot is assumed to contain a fluid with a viscosity of η, so that
stress depends on the rate of change of strain, rather than the strain itself. Both, E and η are
parameters that are assigned to the stiffness of the elastic component and viscosity of the viscous
component, respectively, of a viscoelastic material. An additional time-based parameter, τ, is
defined as the ratio of the viscosity to the stiffness. It is indicative of the response time of the
viscous component of the material.
The behavior of various viscoelastic materials can be modelled by altering the number of
springs and dashpots and their arrangement. The most common phenomenological model used to
describe viscoelastic solids is the Standard Linear Solid Model (see Figure 5) [9]. The model
consists of a single spring arranged in parallel with a spring and dashpot. The equation
describing the behavior of this model is provided in Equation (5).
22
Figure 5. The standard linear solid model is commonly used to describe the behavior of viscoelastic solids
𝑑𝜀
𝑑𝑡=
𝑑𝜎𝑑𝑡
+𝐸2
𝜂(𝜎 − 𝐸1𝜀)
𝐸1 + 𝐸2
Equation 5
There are several other phenomenological models, each representing a different
configuration of springs and dashpots. These other models will not be discussed in detail here.
2.2.2 Stress Relaxation
As mentioned earlier, viscoelastic materials exhibit complex time-dependent behavior. If
a viscoelastic material is subjected to an instantaneous deformation, which is then held constant
with time, it exhibits a relaxation of stress with time. Comparatively, elastic materials remain at
the same stressed state for as long as the strain is being applied (see Figure 6).
E2
E1
𝜂
23
Figure 6. The stress in viscoelastic materials relaxes under constant strain
The standard linear solid model described in Equation 5 above can be solved for the case
of stress relaxation, where there is no change in strain (𝒅𝜺
𝒅𝒕= 𝟎). The relaxed modulus of the
material is then described by Equation (6):
𝐸(𝑡) = 𝐸1 + 𝐸2𝑒𝑥𝑝 (−𝑡/𝜏) Equation 6
However, it is more common to rewrite this equation in terms of the instantaneous (𝑬𝟎)
and long-term moduli (𝑬∞) of the material. Since the strain in the spring (𝑬𝟏) is equal to the
strain on the spring-and-dashpot side, the stress is shared between the two. On the spring-and-
dashpot side, equal stress is carried by the spring (𝑬𝟐) and the dashpot. Over an infinite amount
of time, the dashpot relaxes until it carries zero stress and accordingly, the stress in the spring
(𝑬𝟐) is also zero. At this point, all of the load is carried by the single spring (𝑬𝟏), and thus, 𝑬∞=
𝑬𝟏. At the initial instantaneous application of the strain, the dashpot displacement is zero, the
load is shared between both sides, the load in each side is represented by the springs, as 𝝈𝟏 =
𝑬𝟏𝜺 and 𝝈𝟐 = 𝑬𝟐𝜺, respectively, and therefore, 𝑬𝟎 = 𝑬𝟏 + 𝑬𝟐. Equation (6) can be rewritten in
terms of these new parameters as shown in Equation (7), and the parameters can be easily
determined via a stress relaxation test.
24
𝐸(𝑡) = 𝐸∞ + (𝐸0 − 𝐸∞)𝑒𝑥𝑝(−𝑡/𝜏) Equation 7
2.2.3 Degree and Rate of Stress Relaxation
Two metrics are important when describing the stress relaxation behavior of a material:
the rate and the amount of relaxation. Under constant strain, the stress relaxes with time;
however, the stress in a viscoelastic material such as a polymer does not decrease to zero and a
fraction of the stress remains in the material. The remaining stress represents the time-
independent character of the material and represents the “elastic” component of the material. The
amount of relaxation refers to the fraction that the stress decreased by, representing the “viscous”
character of the material. Mathematically, this is represented by the difference between the initial
and long term moduli.
The rate of stress relaxation refers to the time it takes for the material to reach its long-
term state. If a material attains its final state quickly, it is said to relax faster. The stress
relaxation rate constant, τ, is an empirical metric that is commonly used to evaluate the stress
relaxation rate. For the standard linear solid model, τ is defined as the time it takes for the stress
(or modulus) to decrease from its initial value by 1/e of its total relaxation (i.e. 𝑬𝟎−𝑬∞
𝒆).
Thermoplastic and thermoset polymers differ in their stress relaxation mechanisms.
Unlike for thermoplastics, the crosslinks in thermosets prevent the polymers from fully relaxing.
The different mechanisms are important but this property is reflected in the model parameters.
2.3 Previous Models of Viscoelastic Behavior of Short-Fiber Composites
The properties of short-fiber composites become significantly more complex when
incorporating a viscoelastic matrix. There have been several =attempts to investigate the time-
25
dependent response of viscoelastic composites both experimentally and via models. A thorough
review of previous experimental studies is incorporated in Chapters 3, 4, 5, and 6 and will not be
presented here. Each chapter also includes a review of past modelling attempts; however, the
previous models are discussed in greater detail in the following sections.
2.3.1 Phenomenological Models
Polymer-based composites retain the viscoelastic character of their polymeric matrix and
can also be modelled using phenomenological models. Several studies have modelled the stress
relaxation of polymer-based composites using Maxwell elements in various forms such as the
Maxwell-Wiechert model and the Generalized Maxwell model. Somashekar et al. (2012) [10]
conducted a study investigating the relaxation of compaction stress in composites reinforced with
various glass-fiber mats. The experimental behavior of the composites was modelled using a
five-component phenomenological model. The arrangement of the elements in the model is
shown in Figure 7.
26
Figure 7. The five-component model used in the study by Somashekar et al. (2012)
For each composite, the fiber volume fraction was also varied, and the experimental data
were curve-fit to the phenomenological model to determine the model parameters. The effect of
fiber volume fraction was assessed based on changes in the values of the model parameters. It
was concluded that increasing the fiber volume fraction resulted in an increase in the value of the
final stress and an increase in the relaxation time, i.e. increasing the fiber content resulted in a
decrease in the amount of stress relaxed by the composite and also slowed the relaxation process.
The same phenomenological model was used in another study conducted by Kelly et al. (2006)
[11].
Similarly, a study conducted by Kim et al. (1991) [12] also evaluated the relaxation of
compaction stress from composites containing various types of fiber mats and rovings, and found
that the experimental behavior could be best fit to a Maxwell-Wiechert model. The arrangement
27
of the model elements and their corresponding equation are shown in Figure 8 and Equation (8)
and (9) below.
Figure 8. The ten-component model used in the study by Kim et al. (1991) where 𝝈𝒊 is the initial stress applied
onto the sample while 𝝈(0) is the initial reactive stresses from each of the spring components in the
phenomenological model
𝜎(𝑡)
𝜎𝑖
=𝜎(0)1
𝜎𝑖
𝑒−
𝑡𝜏1 +
𝜎(0)2
𝜎𝑖
𝑒−
𝑡𝜏2 +
𝜎(0)3
𝜎𝑖
𝑒−
𝑡𝜏3 +
𝜎(0)4
𝜎𝑖
𝑒−
𝑡𝜏4 +
𝜎(0)5
𝜎𝑖
𝑒−
𝑡𝜏5 Equation 8
𝜎(𝑡)
𝜎𝑖
= 𝐵1𝑒−
𝑡𝜏1 + 𝐵2𝑒
−𝑡𝜏2 + 𝐵3𝑒
−𝑡𝜏3 + 𝐵4𝑒
−𝑡𝜏4 + 𝐵5 Equation 9
Once again, the experimental data was fit to the model to determine the parameters, and
these parameters were used to investigate the effect of orientation. It was concluded that an
increase in fiber misorientation resulted in lower relaxation times, and thus, unidirectional
alignment of the fibers resulted in slower stress relaxation.
E1 E2 E3 E4 E5
η1 η2 η3 η4 η5
28
Another similar study has been conducted by Penneru et al. (2006) [13], who modelled
experimental data on stress relaxation using a standard linear solid model.
Although phenomenological models can be used to make important predictions of the
behavior of the specific composites and used to calibrate them, they do not provide guidance for
optimizing material structure. Since such models are based on at least five curve-fitting
parameters, it is likely that the models will fit the experimental data. Additionally, since the
models are purely empirical and the parameters are case-specific, they cannot be used to predict
the relaxation behavior of composites with different matrix/fiber formulations or different
volume fractions - they are simply unsuitable for a general application.
2.3.2 Tensor Analysis
A second body of work uses an exact elasticity analysis to understand the stress transfer
in viscoelastic composites. Smith et al. (2013) derived shear-lag transfer equations from first
principles in a discontinuous fiber composite with a viscoelastic polymer matrix [14]. Analytical
models were formulated for both the stress relaxation and creep behavior of short-fiber
composites as shown in Equations (10) and (11).
𝜎1 = 𝜈𝑓 [𝐸𝑓𝜀1(𝑠) +𝐸𝑓
𝑠𝜀1(+0)] {1 −
tanh (𝜂∗𝐿𝑟0
)
𝜂∗𝐿𝑟0
} + (1 − 𝑣𝑓)𝜎𝑚̅̅ ̅̅ Equation 10
𝜀1(𝑠) =𝜎1(𝑠) +
1𝑠𝜎1(+0)
𝑣𝑓𝐸𝑓 [1 −tanh (
𝜂𝑐𝑟𝑒𝑒𝑝𝐿𝑟0
)
𝜂𝑐𝑟𝑒𝑒𝑝𝐿𝑟0
] +(1 − 𝑣𝑓)𝑠
𝐷𝑚
Equation 11
Because of the complexity of the Laplace transform arising from the derivation, it could
not be inverted to produce a usable analytical model, which is a common issue [15, 16]. In this
study, a numerical solution was used to make predictions of stress along the length of the short
fiber for only one set of parameters. Although the study was used to conduct some basic analysis,
29
no experiments or simulations were conducted to validate the numerical solution of the analytical
model.
The study also assumes that the behavior of the matrix follows a simple exponential
decay (𝑬(𝒕) = 𝑬𝟎𝒆𝒙𝒑 (−𝒕
𝝉)). The study does not indicate any reasoning for the selection of this
equation; however, this equation is typically used to model the behavior of viscous liquids and
not viscoelastic solids. This is because viscoelastic solids typically exhibit a relaxed stress that is
greater than zero, while the equation (derived from a simple Maxwell model) assumes a final
stress state of zero.
There have been numerous other attempts to model the stress relaxation behavior of
short-fiber composites from first-principles. Merodio et al. (2006) used tensor analysis to derive
18 invariants associated with viscoelastic composite deformation [17]. Drozdov et al. (2003)
approached the viscoelasticity of composites by using an energy balance approach [18]. Despite
these attempts, no study has yielded a simple, closed-form solution for stress relaxation, and
hence, studies of this type do not provide a way to investigate critical issues such as the effect of
fiber/matrix modulus ratio and fiber aspect ratio on the stress relaxation behavior. The lack of a
closed-form solution in these cases requires numerical approximation to obtain a solution, which
is too complex, time-intensive and computationally expensive. This issue was also faced by
Yancey et al. (1990) [19] and Barbero et al. (1995) [20].
A study by Mondali and Abedian (2013) [21] used first principles to successfully derive
an analytical model for the second stage rate of creep in short-fiber composites, which showed
good agreement with finite-element simulations. The model is quite simple; however, since it
provides a method to approximate only the second stage creep rate and not the entire creep
behavior, it can not be readily converted to model the stress relaxation behavior of a short-fiber
30
composite. The same issue is faced for other studies that have modelled the creep behavior of
short-fiber composites [22-25].
2.3.3 Finite Element Models
A few finite-element models have also been used to study the behavior of short-fiber
composites. A study by Malekmohammadi et al. (2014) modelled the stress relaxation behavior
of wood strand-based composites [26]. No specific conclusions were made in this study since
most of the investigation modelled the elastic properties of the composites.
An important study conducted by Black et al. (2012) used finite-element simulations to
investigate the load transfer under creep between hydroxyapatite crystals contained in a collagen
matrix in cortical bones [27]. It was found that with increasing creep time, there was an increase
in the elastic strain within the hydroxyapatite crystals when contained in the collagen matrix.
This change would not be expected in pure hydroxyapatite and was assumed to occur due to load
transfer from the creeping viscoelastic matrix. A similar observation has been made in another
study by Daymond et al. (1999) which had evaluated the strain distribution in particulate-
reinforced metal-based composites under high-temperature creep [28].
Finite-element studies investigating the basic creep and stress relaxation behavior of
short-fiber composites are limited. Even if a myriad of such studies had been conducted, finite-
element investigations are both time- and computationally-intensive. The investigations are often
conducted for one specific case and can be used to make some observations but are not sufficient
to understand the mechanisms in greater detail and cannot be used to predict the properties of
composites that were not investigated in the simulation.
31
2.3.4 Shear Lag in Broken Continuous-Fiber Composites
The idea of using shear-lag to investigate time-dependent behavior is not new and is
found in the literature of continuous fiber composites. Subjecting continuous fiber composites to
high stresses can result in fiber fracture, decreasing the ability of the broken fiber to carry load.
In such cases, the stress is transferred through interfacial shearing at the fiber-matrix interface.
Since the matrix is viscoelastic, the stresses in the matrix decrease with time, resulting in
decreased stress transfer to the broken fiber and a higher dependency on the intact fibers. This
results in increased stress on the intact fibers, causing more fibers to fail, leading to a
propagation of fiber fractures which eventually causes the composite to fail.
Several attempts have been made to model the time-dependent shear stress transfer to the
broken fibers, both analytically and through finite-element modelling [29-31].
There are two things to note; however. Firstly, although adequate and perhaps accurate
analytical models have been developed in these studies, the problem is too complex to address
with one all-inclusive analytical model [32,33]. Often constants are preselected, and the complex
analytical model is solved for those cases. Secondly, in several of these studies, fiber breakage is
a critical aspect of the model, and thus, the modelling often utilizes the work conducted by
Weibull on fiber fracture or some parameter to describe the fiber fracture [34-37].
2.3.5 Other Models
In the area of polymers reinforced with nanofibers, studies have concluded the presence
of a third interfacial phase where the stress transfer in the resultant three-layer structure has been
modelled. Examples of these studies include Zhang and He [38], Fatemifar et al. (2014) [39], and
Papanicolaou et al. (2011) [40]. While the presence of a third interfacial phase may be important
32
for nanofiber-reinforced composites, there is limited evidence to suggest that this is also the case
for a short-fiber composite.
2.3.6 Conclusions
The viscoelasticity of short-fiber composites continues to be an important property that
has been probed both experimentally and via analytical and numerical models. To date, however,
there is no method to accurately and easily model the viscoelastic behavior of short-fiber
composites. Previous attempts have attempted to use tensor analysis, which requires complex
numerical approximation, making the models too difficult and time-intensive for widespread use.
Even phenomenological models are generally not adequate because they require numerous
experiments to determine all of the parameters for curve-fitting. Even after the curve has been fit,
the model can only be used to predict the properties of one matrix/fiber system, and the model
has to be re-evaluated for any changes are made to the system. Since the models are so specific,
they are not useful in understanding the general viscoelastic behavior of short-fiber composite
systems. The same problem exists with finite-element simulations, which are often
computationally and time intensive, making it unfeasible to develop a detailed understanding of
the viscoelastic behavior of a general short-fiber composite system. In addition, even after
several simulations, an equation-based model can not be derived.
This literature search highlights the need for a simple analytical model that can be used to
predict the viscoelastic behavior of short-fiber composites. Once validated, a simple analytical
model can be used to isolate the stress-transfer mechanisms that are involved in the viscoelastic
behavior of a short-fiber composite and can be used to assess the effect of various parameters
including aspect ratio and fiber content. This form of parametric analysis is difficult to conduct
using complex analytical models, finite-element simulations, or phenomenological models.
33
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5. McCartney, L. N. Stress transfer for multiple perfectly bonded concentric cylinder models of
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Art of Predictive Damage Modelling; Woodhead Publishing Limited: Cambridge, England,
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7. Tucker III, C. L.; Liang, E. Stiffness predictions for unidirectional short-fiber composites:
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8. Xia, Z.; Okabe, T.; Curtin, W. A. Shear-lag versus finite element models for stress transfer in
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10. Somashekar, A. A.; Bickerton, S.; Battacharyya, D. Modelling the viscoelastic stress
relaxation of glass fibre reinforcement under constant compaction strain during composites
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13. Penneru, A. P.; Jayaraman, K.; Bhattacharyya, D. Viscoelastic behaviour of solid wood
under compressive loading. Holzforschung. 2006, 60, 294-298.
14. Smith, N.; Medvedev, G. A.; Pipes, R. B. Viscoelastic shear lag analysis of the discontinuous
fiber composite in Proceedings of the 19th International Conference on Composite Materials,
Montreal, QC, Canada, 28 July–2 August 2013.
15. Flink, P.; Stenberg, B. An indirect method which ranks the adhesion in natural rubber filled
with different types of cellulose fibres by plots of E(t)/E(t = 0) versus logt. Br. Polym. J.
1990, 22, 193–199.
16. Kutty, S. K.; Nando, G. B. Short Kevlar fiber-thermoplastic polyurethane composite. J. Appl.
Polym. Sci. 1991, 43, 1913–1923.
17. Merodio, J. On constitutive equations for fiber-reinforced nonlinear viscoelastic solids.
Mech. Res. Commun. 2006, 33, 764–770.
18. Drozdov, A. D.; Al-Mulla, A.; Gupta, R. K. The viscoelastic and viscoplastic behavior of
polymer composites: polycarbonate reinforced with short glass fibers. Comput. Mater. Sci.
2003, 28, 16–30.
35
19. Yancey, R. N.; Pindera, M. J. Micromechanical analysis of the creep response of
unidirectional composites. J. Eng. Mater. Techol. 1990, 112, 157–163.
20. Barbero, E. J.; Luciano, R. Micromechanical formulas for the relaxation tensor of linear
viscoelastic composites with transversely isotropic fibers. Int. J. Solids Struct. 1995, 32,
1859–1872.
21. Mondali, M.; Abedian, A. An analytical model for stress analysis of short fibers in power law
creep matrix. Int. J. Non-Linear Mech. 2013, 57, 39-49.
22. Kelly, A.; Tyson, W. R. Tensile properties of fibre-reinforced metals: copper/ tungsten and
copper/molybdenum. J. Mech. Phys. Solids. 1965, 13, 329–338.
23. Kelly, A.; Tyson, W. R. Tensile properties of fibre reinforced metals—I. Creep of silver-
tungsten. J. Mech. Phys. Solids. 1966, 14, 177–184.
24. Doruk, M.; Yue, A. S. Creep behavior of fiber reinforced metal matrix composites. Metall.
Mater. Trans. 1976, A7, 1465–1468.
25. Nieh, T. G. Creep rupture of a silicon-carbide reinforced aluminum composite. Metall.
Mater. Trans. 1984, A15, 139–146.
26. Malekmohammadi, S.; Tressou, B.; Nadot-Matrin, C.; Ellyin, F.; Vaziri, R. Analytical
micromechanics equations for elastic and viscoelastic properties of strand-based composites.
J. Compos. Mater. 2014, 48(15), 1857-1874.
27. Deymier-Black, A. C.; Yuan, F.; Singhai, A.; Almer, J. D.; Brinson, L. C.; Dunand, D. C.
Evolution of load transfer between hydroxyapatite and collagen during creep deformation of
bone. Acta Biomater. 2012, 8, 253-261.
36
28. Daymond, M. R.; Lund, C.; Bourke, M. A. M.; Dunand, D. C. Elastic phase-strain
distribution in a particulate-reinforced metal-matrix composite deforming by slip or creep.
Metall. Mater. Transac. A. 1999. 30(11), 2989-2997.
29. Okabe, T.; Nishikawa, M. GLS strength prediction of glass-fiber-reinforced polypropylene.
J. Mater. Sci. 2009, 44, 331–334.
30. Okabe, T.; Nishikawa, M.; Takeda, N. Micromechanics on the rate-dependent fracture of
discontinuous fiber-reinforced plastics. Int. J. Damage Mechanics. 2010, 19, 339–360.
31. Hashimoto, M.; Okabe, T.; Sasayama, T.; Matsutani, H.; Nishikawa, M. Prediction of tensile
strength of discontinuous carbon fiber/polypropylene composite with fiber orientation
distribution. Comp. Part A. 2012, 43, 1791–1799.
32. Beyerlein, J.; Pheonix, S. L.; Raj, R. Time evolution of stress redistribution around multiple
fiber breaks in a composite with viscous and viscoelastic matrices. Int. J. Solids Struct. 1998,
35, 3177–3211.
33. Iyengar, N.; Curtin, W. A. Time-dependent failure in fiber-reinforced composites by matrix
and interface shear creep. Acta Mater. 1997, 45, 3419–3429.
34. Du, Z. Z.; McMeeking, R. M. Creep models for metal matrix composites with long brittle
fibers. J. Mech. Phys. Solids. 1995, 43(5), 701–706.
35. Weber, C. H.; Du, Z. Z.; Zok, F. W. High temperature deformation and fracture of a fiber
reinforced titanium matrix composite. Acta Mater. 1996, 44, 683–695.
36. Fabeny, B.; Curtin, W. A. Damage-enhanced creep and rupture in fiber-reinforced
composites. Acta Mater. 1996, 44, 3439–3451.
37. Lagoudas, D. C.; Hui, C. Y.; Pheonix, S. L. Time evolution of overstress profiles near broken
fibers in a composite with a viscoelastic matrix. Int. J. Solids Struct. 1989, 25 (1), 45–66.
37
38. Zhang, J.; He, C. A three-phase cylindrical shear-lag model for carbon nanotube composites.
Acta. Mech. 2008, 196, 33–54.
39. Fatemifar, F.; Salehi, M.; Adibipoor, R. Three-phase modeling of viscoelastic nanofiber-
reinforced matrix. J. Mech. Sci. Tech. 2014, 28(3), 1039-1044.
40. Papanicolaou, G. C.; Xepapadaki, A. G.; Drakopoulos, E. D.; Papaefthymiou, K. P.; Portan,
D. V. Interphasial viscoelastic behavior of CNT reinforced nanocomposites studied by means
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1588.
38
Chapter 3. Understanding the Stress Relaxation Behavior of
Polymers Reinforced with Short Elastic Fibers
In this chapter, we propose an analytical model that explicitly accounts for the influence
of polymer viscoelasticity on shear stress transfer to the fibers. This model adequately explains
the effect of fiber addition on the relaxation behavior without the need to postulate structural
changes at the fiber-matrix interface. The model predictions were compared to those from
randomly generated finite-element simulations, and good agreement between the two was
observed.
This chapter has been published as: “Obaid, N.; Kortschot, M. T.; Sain, M.
Understanding the stress relaxation behavior of polymers reinforced with short elastic fibers.
Materials. 2017, 10(5), 472.”
3.1 Introduction
The interaction between the fiber and the matrix in a short-fiber composite is quite
complex. Although the effect of fibers on static properties such as modulus and strength is well
understood, it has been a challenge to understand the effect of fibers on the viscoelastic
properties of short-fiber composites. These properties are extremely important in load bearing
applications where there is the potential for creep or stress relaxation, or where the composites
are exposed to any sort of dynamic loading, and hence it is important to be able to predict the
influence of fiber reinforcement on the viscoelasticity. Composite viscoelasticity can also
influence fatigue behavior [1], and the temperature dependence of various mechanical properties,
including creep resistance [2].
39
Stress relaxation experiments, in which a specimen is strained to a fixed level and the
slow decay of stress is monitored, present a simple method of investigating the time-dependent
modulus of reinforced polymers. In practice, stress relaxation influences the residual stress and
warpage of molded short-fiber composite parts and is critical in many applications including
fasteners and gaskets. During the stress relaxation of polymer composites, the modulus of the
material typically decays from an initial value 𝑬𝟎, to a final stable value 𝑬∞. The speed of this
process, which has practical implications, is characterized in terms of a relaxation time constant
τ. The time constant is usually defined as the time needed for the modulus to decrease to 1/e of
the interval between 𝑬𝟎 and 𝑬∞.
In continuous fiber composites, such as laminated carbon fiber composites, the values of
𝑬𝟎 and 𝑬∞ depend on fiber loading, but the value of τ should not. However, it has been widely
observed that short elastic fibers (which do not themselves relax with time) alter the stress
relaxation behavior of the composite, and in particular, change the value of τ. Early research into
this phenomenon showed that short-fibers expedited the relaxation response, and many
researchers proposed mechanistic explanations, in which the fibers affect the structure of the
polymer matrix near the interface and hence modify its stress relaxation behavior. For example,
Blackley and Pike proposed that the relaxation of composites was affected by the additional
covalent bonds between the fibers and the matrix and that the rupture of these bonds during
stress relaxation caused an accelerated response, changing the relaxation time constant [3]. In
this and other early investigations, both the reinforcing fibers and the matrix were viscoelastic
materials, and although these early studies showed that the addition of short-fibers increased the
relaxation rate, almost all the more recent studies have shown the opposite effect [4,5].
40
Kutty and Nando investigated the effect of short Kevlar fibers on polyurethanes and
found that increases in fiber content slowed the stress relaxation rate [6]. Many other studies
support this observation. For example, Suhara, Kutty, and Nando also showed that increasing the
loading of short polyester fibers in polyurethane resulted in slower stress relaxation [7]. Pothan
et al. (2004) found that increased loading of banana fibers reduced the stress relaxation rate of
polyester composites [8]. Bhattacharyya et al. (2006) also showed that increasing the content of
wood fibers reduces the relaxation of polypropylene composites [9]. Saeed et al. (2014)
suggested that the presence of glass fibers resulted in decreased chain mobility in high-density
polyethylene [10]. Boukettaya et al. (2015) evaluated the stress relaxation behavior of
polypropylene composites reinforced with date palm fibers [11]. It was observed that increasing
the fiber content resulted in a decrease in the relaxation rate. Wang et al. (2012) showed that the
addition of wood flour reduced the stress relaxation rate of propylene [12].
Several studies have also investigated the stress relaxation of hybrid composites
containing more than one type of fiber. Sreekala et al. (2001) found that increasing the content of
short oil-palm fibers in a phenol formaldehyde matrix resulted in slower stress relaxation [13].
The rate of decay could be further decreased upon hybridization with glass fibers. Stan and
Fetecau (2013) investigated the stress relaxation in polytetrafluoroethylene composites [14].
Unfilled polytetrafluoroethylene (PTFE) was compared to one that was reinforced with 15%
graphite particles and a hybrid containing 32% carbon and 3% graphite. It was found that
unfilled PTFE had the fastest relaxation rate and that the addition of fillers slowed the relaxation
process.
In summary, the literature shows that the addition of fibers to a viscoelastic polymer
generally slows the relaxation process, increasing the time constant. Two main explanations have
41
been put forward to explain this phenomenon. The first explanation is that the presence of fibers
hinders molecular flow in the polymer near the interface, resulting in slower relaxation of the
matrix [15]. Geethamma et al. (2004) found that short coir fibers reduced the stress relaxation
rate of rubber and this was attributed to fibers constraining the polymeric chains thereby
preventing relaxation [16]. Mirzaei et al. (2011) investigated the effect of adding various types of
natural fibers in high-density polyethylene and drew similar conclusions [17].
An alternative explanation suggested in a number of studies centers on the potential for
chemical bonding at the fiber/matrix interface. These studies propose that breaking the additional
covalent bonds at the fiber-matrix interface is a prerequisite to polymer mobility and relaxation.
To test this idea, a number of researchers have examined the effect of various coupling agents
and their effect on stress relaxation behavior. George et al. (1998) observed that chemical
modifications via coupling agents resulted in lower rates of relaxation and hypothesized that the
surface treatment produces additional chemical bonds that hinder the movement of the polymer
[15]. Pothan et al. (2004) also showed that the stress relaxation rate is reduced with the use of a
coupling agent [7]. Boukettaya et al. (2015) proposed that from a chemical bonding perspective,
the polymeric chains are initially constrained by the fiber; however, over time, the damage of the
intermolecular linking causes the chains to once again become mobile [11]. Thus, it was
proposed that the rate of stress relaxation in a composite was related to how quickly the bonds
can be broken and, therefore, how quickly the polymer could become mobile again.
Experimental studies have confirmed that the viscoelasticity of short fiber composites is a
significant and complex phenomenon. Although a number of qualitative explanations for the
observed phenomenon have been proposed, as discussed previously, there have been relatively
few attempts to derive a predictive model. Somashekar et al. (2012) and Safraoui et al. (2014)
42
used conventional spring/dashpot models to characterize viscoelasticity, but although these
phenomenological models can be used to characterize the behaviour of particular composites,
they do not provide any guidance for optimizing material structure [18,19]. Drozdov et al. (2003)
approached the viscoelasticity of composites by using an energy balance approach [20]. Several
other groups including Naik et al. (2008), Brinson et al. (2013), and Fisher et al. (2003) have
used finite element models incorporating viscoelastic matrices and short elastic fibers [21–23].
These models highlight the importance of matrix viscoelasticity, but do not obviate the need for a
simple analytical model.
The application of shear-lag models to describe composite viscoelasticity is very limited.
Zhang and He (2008) examined the effect of nanofibers on the viscoelasticity of polymer-based
composites; however, their work was focused on the assumption that the presence of nanofibers
results in the creation of a third interfacial phase, and the shear-lag stress transfer in the resultant
three-layer structure was modelled [24].
Recently, Smith et al. (2013) derived shear-lag stress transfer equations from first
principles in a discontinuous fiber composite with a viscoelastic polymer matrix [25]. Because of
the complexity of the Laplace transform arising from the derivation, it could not be inverted to
produce a useable analytical model, which is a common issue [26,27]. A numerical solution was
used to make predictions of stress along the length of the short fiber for only one set of
parameters. Merodio (2006) used tensor analysis to derive 18 invariants associated with
viscoelastic composite deformation [28]. Neither study yielded a simple, closed-form solution
for stress relaxation, and hence they did not provide a way of investigating critical issues such as
the effect of the fiber/matrix modulus ratio and fiber aspect ratio on the stress relaxation
behavior.
43
The present study consists of two parts. We will first develop an analytical model by
explicitly considering the stress relaxation of the matrix in both tension, and critically, in the
shear stress transfer region. Through this approach, we will show that it is not necessary to infer
structural changes at the interface to explain polymer composite stress relaxation. The success of
this analytical model does not preclude the possibility that chemical or physical structural
changes at or near the interface have an effect, but it does mean that these changes might not be
important. The analytical model generated in this paper can be used to parametrically study the
response of short-fiber composites with various fiber volume fractions and aspect ratios without
reliance on numerical integration or finite element analysis. In the second part of this paper, we
will compare the predictions from the analytical model to the results obtained from the finite-
element simulations.
3.1.1 Polymer Viscoelasticity
In order to develop a simple model, the basic principles of polymer viscoelasticity and
short-fiber reinforcement must be reviewed briefly. A stress relaxation test is a simple means of
investigating the viscoelasticity of a polymer. To perform this test, a fixed tensile or compressive
strain is applied to a sample, and the stress, which decays over time, is monitored. The decrease
in stress at a constant strain corresponds to a decrease in the apparent modulus of the polymer.
The modulus of a viscoelastic material during a stress relaxation test is often modeled using
Equation (12) below:
𝐸(𝑡) = 𝐸∞ + (𝐸0 − 𝐸∞) exp (−
𝑡
𝜏) Equation 12
where 𝑬𝟎 and 𝑬∞ are the instantaneous and long-term elastic modulus of the material
respectively, t is time, and τ is the relaxation time constant. As discussed previously, the
44
uncertainty in the literature concerns the origin of changes in the time constant commonly
observed when fibers are added to a polymer.
In an isotropic solid, the shear modulus (G) and elastic modulus (E) are related by
Poisson’s ratio (ν) as shown in Equation (13a). For an isotropic, viscoelastic material, at each
point in time, the same relationship should hold, as shown in Equation (13b). Poisson’s ratio is
usually considered to be constant in this treatment.
𝐸 = 2𝐺(1 + 𝑣) Equation 13a
𝐸(𝑡) = 2𝐺(𝑡)(1 + 𝑣) Equation 13b
Thus, Equations (12) and (13) can be used to obtain the time-dependence of the shear
modulus of a viscoelastic material, as shown in Equation (14).
2𝐺(𝑡)(1 + 𝑣) = 2𝐺∞(1 + 𝑣) + 2(1 + 𝑣)(𝐺0 − 𝐺∞) exp (−
𝑡
𝜏) Equation 14a
𝐺(𝑡) = 𝐺∞ + (𝐺0 − 𝐺∞) exp (−
𝑡
𝜏) Equation 14b
3.1.2 Micromechanics of Short-Fiber Composites
The mechanism of fiber reinforcement in a composite depends on the aspect ratio of the
fibers. Figure 9 compares the micromechanical structure of a composite reinforced with
continuous and discontinuous fibers. When a continuous-fiber composite is stressed in tension,
both the fiber and the matrix are equally strained. In a short-fiber composite, however, the stress
required to strain the fibers is transferred through interfacial shearing (Figure 10), with the ends
of the fiber being entirely unloaded. Cox developed a widely cited analytical model for the
modulus of an elastic composite based on this assumption [29].
45
(a) (b)
Figure 9. A comparison between (a) continuous fiber composites and (b) discontinuous fiber composites.
(a)
(b)
Figure 10. In a short-fiber composite, the matrix adjacent to the fiber is at a different stress state than the
bulk matrix, resulting in a shear force along the interface. These interfacial shear stresses are responsible for
stress transfer to the fibers in the composite. (a) Unstressed State; (b) Displacement under uniaxial tension.
Cox’s model, which is commonly referred to as a “shear-lag” model, proposes that the
effectiveness of load transfer in a short-fiber composite is related to the modulus of both the fiber
and the matrix, as shown in Equations (15) and (16).
𝐸𝑐 = 𝑉𝑓𝐸𝑓 (1 −
tanh(𝜂𝑠)
𝜂𝑠) + 𝑉𝑚𝐸𝑚 Equation 15
𝜂 =
[ 4
𝐸𝑓 ln (𝑃𝑓
𝑉𝑓)]
12
[𝐺𝑚]12 Equation 16
This model relates the elastic modulus of the composite (𝑬𝒄) to the elastic modulus of the
fiber (𝑬𝒇) and matrix (𝑬𝒎). The contribution of each component in the composite is based on its
volume fraction (𝑽𝒇 and 𝑽𝒎). In this equation, the contribution of the fiber is scaled by a
46
multiplication factor (𝜼) which represents the effectiveness of the load transfer to the fibers with
a specified aspect ratio (s) and packing (𝑷𝒇). The load transfer depends on the ratio of the tensile
modulus of the fiber to the shear modulus (𝑮𝒎) of the matrix, as well as the fiber aspect ratio.
Nairn identified some limitations of Cox’s shear-lag model and derived an alternate
effectiveness factor [30]. However, when the ratio of the matrix to fiber modulus is sufficiently
high that the shear deformation within the fiber is not significant, the predictions of Cox’s shear-
lag model have been found to be reasonably accurate [31,32] and have shown good agreement
with the widely used Halpin-Tsai model [33–35]. In this study, the matrix to fiber modulus ratio
was very high, and thus Cox’s shear-lag model was deemed to be adequate for the present
purposes.
3.1.3 Modelling Approach
The shear lag model makes it clear that the shear modulus of the matrix is a critical factor
in determining the effectiveness of fiber reinforcement and hence the modulus of a short-fiber
composite. In a viscoelastic polymer, it is well known that the effective tensile modulus decays
with time in stress relaxation, and it is obvious that the effective shear modulus must also decay.
It is extremely surprising therefore that virtually all previous studies of the stress relaxation of
short-fiber composites have overlooked the time-dependence of the shear modulus.
In this paper, we show that the stress-relaxation behaviour in a short-fiber composite is
significantly affected by the relaxation of the shear modulus of the matrix. In addition to a decay
in matrix modulus over time, relaxation of the shear modulus means that the load transfer to the
fibers is decreased over time, decreasing their contribution, even when the fibers are purely
elastic. However, this typically happens more slowly than the relaxation of the polymer tensile
47
modulus, leading to a predicted increase in the effective time constant for the composite without
any need to hypothesize chemical or physical structural changes at the interface.
3.2 Proposed Model
The modulus of a perfectly bonded short-fiber composite (with a fiber aspect ratio and
Ef/Em ratio sufficiently high that shear deformation within the fiber can be ignored) can be
calculated using Equations (15) and (16).
The time-dependence of the elastic and shear moduli can be calculating using Equations
(12) and (14b) assuming that the matrix is an isotropic, viscoelastic material.
Combining Equations (12) and (15), the time-dependent elastic modulus of the composite
can be written as Equation (17b).
𝐸𝑐(𝑡) = 𝑉𝑓𝐸𝑓 (1 −
tanh(𝜂𝑠)
𝜂𝑠) + 𝑉𝑚𝐸𝑚(𝑡) Equation 17a
𝐸𝑐(𝑡) = 𝑉𝑓𝐸𝑓 (1 −
tanh(𝜂𝑠)
𝜂𝑠) + 𝑉𝑚 [𝐸∞ + (𝐸0 − 𝐸∞) exp (−
𝑡
𝜏)]
Equation 17b
However, the stress-transfer to the fiber is also time-dependent due to the time-
dependence of the matrix shear modulus. Thus, the stress-transfer coefficient in Equation (16)
can be combined with Equation (14b) as shown in Equation (18).
𝜂 =
[ 4
𝐸𝑓 ln (𝑃𝑓
𝑉𝑓)]
12
[𝐺𝑚(𝑡)]12 Equation 18a
𝜂 =
[ 4
𝐸𝑓 ln (𝑃𝑓
𝑉𝑓)]
12
[𝐺∞ + (𝐺0 − 𝐺∞) exp (−𝑡
𝜏)]
12
Equation 18b
48
The time-dependent elastic modulus of a short-fiber composite is therefore fully
characterized by Equation (19).
𝐸𝑐(𝑡) = 𝑉𝑓𝐸𝑓 (1 −tanh(𝜂(𝑡)𝑠)
𝜂(𝑡)𝑠) + 𝑉𝑚 [𝐸∞ + (𝐸0 − 𝐸∞) exp (−
𝑡
𝜏)]
Equation 19a
𝜂(𝑡) =
[ 4
𝐸𝑓 ln (𝑃𝑓
𝑉𝑓)]
12
[𝐺∞ + (𝐺0 − 𝐺∞) exp (−𝑡
𝜏)]
12 Equation 19b
Equation 19 predicts the composite modulus decay, 𝑬𝒄(𝒕), as a function of the time
dependent tensile and shear moduli of the matrix.
3.3 Parametric Study
3.3.1 Properties of the Matrix and Fiber
Equation 8 was used to make modulus predictions for a glass-fiber reinforced
polyurethane composite. The glass fiber was assigned an elastic modulus of 80 GPa, typical of
that reported in the literature. For simplicity, the fibers were assumed to have a packing factor of
1 (hexagonal packing). The Poisson’s ratio of the matrix was assumed to be 0.3. The fiber
modulus was two orders of magnitude above that of the PU matrix, and hence the Cox model
was sufficient for the shear lag computations needed in this study.
49
3.3.2 Effect of Fiber Content
Figure 11. The normalized elastic modulus of the polyurethane-glass composites under stress-relaxation. The
fiber content ranges from 0% to 50%. The initial elastic modulus depends on the fiber content, but the data
here have been normalized by the modulus of the unreinforced polymer so that the stress relaxation is
highlighted.
The modulus of the composite at various fiber fractions was calculated using Equation
(19) and is shown in Figure 11. In order to further examine the rate of relaxation, the predicted
values were fitted to Equation (12) to determine the relaxation rate constant (τ) and the fractional
deterioration in the modulus of the composite (𝑬𝟎 − 𝑬∞)/ 𝑬𝟎0, as shown in Figure 12. For this
part of the study, the matrix was assumed to have an instantaneous modulus of 450 MPa, a long-
term elastic modulus of 100 MPa, and a relaxation time constant of 150 seconds. The fiber was
assumed to have an aspect ratio of 10.
Figure 11 shows that the model predicts a change in the shape of the stress-relaxation
curve upon the addition of purely elastic fibers to a viscoelastic matrix, affecting the rate at
which the modulus deteriorates. If the elastic fibers did not introduce any additional viscoelastic
50
effects, their presence would result in equal reinforcement at each instant in time causing no
change in the relaxation time constant (τ) [36]. However, the model shows that increasing the
fiber content resulted in a longer relaxation time constant, indicating that the presence of fibers
slowed the rate of relaxation. It is precisely this change that elicited the various mechanistic
explanations previously reported in the literature, but Figure 12 demonstrates that the change is a
simple consequence of time dependent shear stress transfer to short-fibers. Of course, these
results do not prove that there are no structural changes at the fiber interface, only that they are
not necessary to produce the observed behaviour.
Figure 12. Higher fiber contents resulted in an increase in the relaxation time constant indicating that the
rate of relaxation had slowed. This showed that increasing the fiber fraction slowed the relaxation of the
composite.
3.3.3 Effect of Fiber Aspect Ratio
We also investigated the effect of the fiber aspect ratio on stress relaxation of the
polyurethane-glass fiber composites. For this part of the study, the matrix was assumed to have
an instantaneous modulus of 450 MPa, a long-term elastic modulus of 100 MPa, and a relaxation
51
time constant of 150 seconds. The normalized relaxation modulus with various fiber contents and
fiber aspect ratios is shown in Figure 13.
It was observed that increasing the fiber aspect ratio resulted in an increase in the long-
term relaxation modulus of the composite (in comparison to the modulus of the unreinforced
polymer, all the data is normalized). This indicated that higher aspect ratio fibers more
effectively reduce the stress relaxation. As the aspect ratio of the fibers increases, the behavior of
the composite approaches that of a continuous fiber composite.
The effect of the fiber aspect ratio on the relaxation time constant was extracted from the
data in Figure 13 and is shown in Figure 14 and Figure 15.
52
Figure 13. This graph depicts the change in normalized elastic modulus with fiber content at various fiber
aspect ratios including (a) aspect ratio of 10; (b) aspect ratio of 50; (c) aspect ratio of 100 and (d) aspect ratio
of 100,000. It can be observed that as the fiber aspect ratio is increased, the long-term modulus increases
because the longer fibers are more efficient reinforcements.
53
Figure 14. As the aspect ratio of the fiber increases, more load is transferred from the matrix to the fiber.
This increased shear force on the fiber results in a higher relaxation time.
Figure 15. As the aspect ratio is continually increased, a larger fraction of the fiber is under tensile loading
and the influence of the shear loading zone decreases. At very high aspect ratios, the composite begins to
approach the properties of a continuous fiber composite, with no change in the relaxation time.
It appears that changing the fiber aspect ratio results in two regimes: for a fiber aspect
ratio below a critical value, increasing the aspect ratio increases the relaxation time. However,
the model also predicts that when the aspect ratio is above a critical value, the opposite effect
will occur. For the modulus ratio (𝑬𝒇/𝑬𝒎) used here, the transition occurs at an aspect ratio of
~100. The critical aspect ratio for viscoelasticity may now be defined as the aspect ratio
corresponding to the maximum relaxation time. Expressed another way, this is the aspect ratio
for which the shear stress transfer into the fibers is the most critical to stress relaxation of the
54
composite. This result is surprising but is a simple outcome of shear lag analysis incorporating a
time dependent shear modulus.
For very low aspect fibers, there is little stress transfer to the fiber. Consequently, the
matrix dominates the behaviour, and the addition of fibers has little effect on the composite
viscoelasticity. The numerical value of a “low” aspect ratio depends entirely on the modulus
ratio 𝑬𝒇/𝑬𝒎, but simply means that the fiber is not effective because it is not long enough to be
strained effectively (see Figure 16).
As the fiber aspect ratio increases, the fiber takes a larger fraction of the composite load,
and the time-dependent shear stress transfer between the matrix and fiber becomes important, so
there is a significant change in the time constant as we add more fibers to the mix.
As the aspect ratio increases further, the composite begins to resemble a long fiber
composite, and the shear stress transfer is no longer important since it is confined to end sections
that are a trivial fraction of the overall fiber length [29]. Once the shear modulus of the matrix is
not important, the model developed here is not needed, and as expected, the relaxation time
constant for a continuous fiber composite (aspect ratio of 100,000 in Figure 15) is not affected by
fiber content.
55
Figure 16. The tensile stresses in the fiber are dependent on its aspect ratio. The aspect ratio (B) is the value
at which the maximum stress transfer begins to occur in the fiber. If the aspect ratio of the fibers is too low
(A), there is inadequate stress transfer between the fiber and matrix. If the aspect ratio is too high (C), the
properties of the composite approach that of a long-fiber composite.
None of this behaviour has anything to do with chemical bonding or chain mobility
hindrance at the fiber matrix interface; these phenomena are not affected by the fiber aspect ratio,
and of course, are not part of the model.
3.4 Finite-Element Analysis
3.4.1 Modelling Approach
To confirm the analytical results, finite element analysis using multiple trials with
randomly deposited short fibers was conducted in Abaqus CAE. For this work, the matrix was
defined as a viscoelastic material with an instantaneous elastic modulus of 1000 MPa, a long-
term modulus of 500 MPa, a relaxation-time constant of 100 seconds, and a Poisson’s ratio of
0.5. The fibers were defined as E-glass fibers having an elastic modulus of 80 GPa and a
Poisson’s ratio of 0.2; these values were obtained from the literature [37]. The fibers selected for
this study had a diameter of 16 microns and a length of 260 microns.
The model consisted of a three-component system including a matrix, fibers, and a rigid
body used to apply fixed displacement to the upper surface of the specimens, corresponding to a
stress relaxation experiment. The matrix was defined as a 3D deformable object while the fibers
56
were defined as beam elements with circular cross-sections for computational efficiency. An
embedded constraint was applied between the matrix and fibers representing a perfect bond. The
total number of fibers dispersed in the system was adjusted to represent various fiber volume
fractions. The matrix mesh consisted of 2211 3D standard quadratic (C3D20R) elements while
the fiber mesh consisted of 40 standard quadratic beam elements (B32) per fiber.
For each finite-element run, the positions of the fiber centres were randomly generated
using a Python script. All fibers were aligned in the load direction. Five replicate simulations
with differing but random fiber locations were conducted for each volume fraction.
The finite-element analysis stress-relaxation test consisted of two analysis steps: the
instantaneous application of strain, followed by 400 seconds of stress decay monitoring, while
the mesh was held at fixed deformation. The modulus of the composite was determined in the
conventional way using the cross-sectional area, and the applied force and displacement of the
rigid body. This approach was validated by comparing the input modulus to the calculated
modulus from the simulation outputs for an isotropic one-phase system. The data was obtained at
10-second intervals with a minimum increment time step of 0.004 seconds.
57
3.4.2 FEA Results
Figure 17. A comparison of the overall stress relaxation profile of short-fiber composites shows excellent
agreement between the predictions of the analytical model (-) and the results obtained from the finite-element
simulations (▪). The error bars represent the standard deviation resulting from five runs of the FEA model
material with differing random fiber placements.
The analytical model was re-evaluated using a set of material properties identical to that
used in the finite-element simulations and a comparison of the results is shown in Figure 17.
Excellent agreement is observed between the finite-element simulations and the predictions
made using the analytical model for fiber contents up to 15% (by volume). This agreement
supports the validity of the analytical model, and a key finding is that both the FEA and the
analytical model yielded changes in the relaxation time constant with fiber loading.
In Figure 17, it is apparent that the analytical model overpredicts the modulus at all times
for fiber loadings greater than 15%. The Cox shear lag model, upon which the current model is
58
based, assumes that each fiber is sitting in an isolated pocket of resin, and that everywhere within
the perimeter of the pocket is experiencing the remote strain. At higher volume fractions, the
situation is clearly more complex than this, as fibers approach and even touch each other, and the
model is not expected to be accurate. In fact, studies that have compared the elastic modulus for
composites have shown that the shear-lag model often overpredicts the actual modulus [38].
The fit was also examined by comparing the analytical model predictions to the finite
element simulation results at the each point in time. For a good fit between the two, the two
values should be almost equal, resulting in a slope close to one. Thus, closer proximity to a 𝒚 =
𝒙 line can represent a better fit between the two approaches (see Figure 18).
Figure 18. Comparison of the analytical model predictions to the finite-element simulation results shows good
agreement between the two at low volume fraction; however, at volume fractions equal to 20% and greater,
the finite-element results deviate from the predictions of the analytical model.
Figure 18 shows a strong agreement between the analytical model and the finite-element
simulations at low fiber volume fractions as observed by their proximity to the 𝒚 = 𝒙 line.
59
Since it has been determined that the shear-lag model is not applicable for fiber volume fractions
greater than 20%, composites with fiber volume fractions above this threshold have not been
used in further analysis.
(a) (b) Figure 19. Good agreement is obtained between the instantaneous (a) and long-term (b) modulus values
obtained from the analytical model (-) and the finite-element simulations (●).
The results from both the analytical model and finite-element results were fit to a simple
Prony Series, and three key parameters were obtained: the instantaneous modulus, the long-term
modulus, and the stress relaxation constant. The addition of elastic fibers results in an increase in
both moduli as expected and there was excellent agreement between the analytical and finite
element models (see Figure 19).
In Figure 20, the clear dependency of the relaxation rate constant on fiber content is
illustrated. Both the FEA and the analytical model predicted this trend. It is important to note
that the analytical model indicated that the change in relaxation rate constant stemmed from the
time-dependent shear modulus of the matrix, which resulted in time-dependent shear stress
transfer to the fibers, causing the stress within the fiber to be time-dependent, and thus, having an
indirect effect on the stress relaxation of a short-fiber composite.
60
Figure 20. Good agreement is obtained between the relaxation time constant obtained from the analytical
model (-) and the finite-element simulations (●).
3.5 Conclusions
Although it has been experimentally shown that the presence of short-fibers slows the
relaxation process in composites, the underlying phenomenon is complex and was not well
understood. Previous studies have postulated either microstructural or chemical interactions
between the fiber and matrix on a molecular scale in order to explain the observed changes in
relaxation, but in this study, we have shown that the effect of fibers on the stress relaxation
behaviour of a composite can be explained by simply considering the fundamentals of shear
stress-transfer at the fiber-matrix interface in short-fiber composites. The fiber-matrix interface is
simply considered to be an infinitely thin, perfectly bonded zone.
This study shows that the stress relaxation of a composite is influenced by two
phenomena: firstly, the elastic modulus of the matrix is time-dependent, and secondly, the shear
modulus of the matrix is also time-dependent and causes a time-dependent stress-transfer
between the fiber and the matrix. As the fiber content increases, the relative importance of the
61
shear stress transfer zone increases, causing an increase in the time constant for relaxation. This
effect is largest for intermediate aspect ratios where the fibers are long enough to carry a
significant fraction of the load, but short enough to be affected by the shear stress transfer from
the matrix over a significant portion of their length.
The concept of a critical fiber length (or aspect ratio) is widely used with respect to the
strength, modulus, and toughness of short fiber composites. We have identified a critical fiber
aspect ratio for viscoelasticity as the aspect ratio for which the shear lag stress transfer zone is
most influential in determining the overall load carrying ability of the composite, and hence most
critical in determining the effect of fiber loading on the time constant for stress relaxation.
In summary, an explicit accounting of the relaxation of shear modulus and the effect of
this on the reinforcement efficiency factor can adequately explain the effect of short fibers on
stress relaxation in polymer composites without any inference of structural changes at the
interface. Since viscoelastic behaviour of short fiber composites is extremely important in many
applications, this model should find wide applications.
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Chapter 4. Predicting the Stress Relaxation Behavior of Glass-Fiber
Reinforced Polypropylene Composites
In this chapter, the stress relaxation behavior of glass fiber-reinforced polypropylene
composites was experimentally measured and compared to the predictions from the analytical
model developed in Chapter 3. Further, the effect of additional covalent bonding at the fiber-
matrix interface was studied experimentally by introducing an interfacial coupling agent.
This chapter has been published as: “Obaid, N.; Kortschot, M. T.; Sain, M. Predicting the
stress relaxation behavior of glass-fiber reinforced polypropylene composites. Compos. Sci. &
Tech..2018, 161, 85-91.”
4.1 Introduction
Short, elastic fibers are routinely incorporated in polymers to improve mechanical
properties such as modulus and strength. The static properties of polymers and their composites
are relatively simple to understand: the addition of a stiff, elastic reinforcing phase into a softer
polymer matrix typically increases both stiffness and strength. There is a substantial body of
theory to predict these mechanical properties based on the size, shape, and orientation of the
reinforcing phase [1,2].
The viscoelastic properties of composites are significantly more complex. The time-
dependent properties of the matrix makes polymer-based composites prone to creep and stress
relaxation, which is a challenge when considering composites for long-term applications. A
better understanding of composite viscoelasticity is needed in order to provide guidance for
optimizing composite structure, and for predicting long-term properties.
67
Stress relaxation is a straightforward way of characterising polymer viscoelasticity. Since
all viscoelastic properties stem from the same basic mechanisms, a model for composite stress
relaxation would also provide insight into composite creep and dynamic mechanical behaviour.
The addition of short fibers in a composite has been repeatedly observed to slow the rate
of stress relaxation of a composite [3-7]. This has been a subject of interest because elastic fibers
do not exhibit time-dependent behaviour, but nevertheless appear to do so when embedded in a
viscoelastic matrix. The explanation for this observation remains elusive and previous studies in
literature have focused on attributing the effect of fibers on stress relaxation to chemical bonding
at the fiber/matrix interface [8-12]. These studies have proposed that covalent interfacial bonds
between the two phases inhibit polymer mobility and thus slow stress relaxation.
In a previous paper, we proposed a novel explanation for the effect of elastic fibers on the
stress relaxation of polymer matrix composites [13]. We proposed that the time-dependent shear
stress transfer at the fiber-matrix interface, and not increased covalent bonding at the interface,
was primarily responsible for altering the viscoelasticity of the composite. In that study, a
quantitative model was developed based on composite micromechanics, and was used to predict
the stress relaxation of composites without postulating changes in structure near the fiber
interface.
There have been other attempts to model the behaviour of viscoelastic and viscoplastic
matrices reinforced with elastic fibers. Several studies have used a tensor approach to model the
full stress field around elastic inclusions in viscoelastic matrices [14-16]. This approach, though
accurate, does not provide a simple analytical model useful for predicting stress relaxation in a
short fiber reinforced polymer as a function of fiber loading and aspect ratio.
68
There is also an extensive body of literature for high volume fraction continuous fiber
composites, where at high stress, fiber breakage results in concentrated matrix shear stresses near
the end of the broken fibre, and the decay of these stresses with time has been modelled [17-19].
Solutions to such problems tend to be quite specific for the particulars of the geometry chosen
[20, 21], although the conceptual underpinning is the same. Fibre fracture can be modelled with a
Weibull approach [22-25], and understanding the causes and evolution of fibre fractures is often
the focus of these studies.
The purpose of the present study was to compare the experimental stress relaxation
behavior of glass fiber-reinforced polypropylene composites to predictions from a simplified
analytical model. Additionally, the hypothesis that there is additional covalent bonding at the
fiber-matrix interface on the stress relaxation behavior of composites was examined
experimentally.
4.2 Experimental
4.2.1 Sample Preparation
Two batches of glass fiber reinforced polypropylene composites were prepared: one
without any compatibilizer or coupling agent and another containing maleic anhydride-grafted
polypropylene (MAPP, 5% by weight) to improve interfacial bonding. Polypropylene (with and
without MAPP) was first melted in a C.W. Brabender Compounder (Type R.E.E.6) at 185 °C
and 20 RPM for 10 minutes. Glass fibers were added to the mixer and the mixing speed was
increased to 60 RPM, and the compound was mixed for five additional minutes. The details of all
the chemical components used in this study are summarized in Table 1.
69
The compounded mixture was granulated using a C.W. Brabender Granulator (Model S9-
10). The samples were prepared via an Engel ES-28 injection molder with an injection
temperature of 210 °C, an injection time of 8 seconds, a cooling time of 38 seconds, and a mold
opening time of 2 seconds. The injection mold produced ASTM tensile, flexural and impact test
specimens.
Table 1. Details of the chemical components used in the study
Component Company Product Name
Polypropylene Total Petrochemicals Inc. PP3622
E-Glass Fibers Johns Manville StarStrain EC14 738
Maleic anhydride grafted polypropylene
(MAPP)
Eastman G-3003 Polymer
4.2.2 Experimental Design
The composites examined in this study consisted of 10%, 20%, and 30% mass fraction of
fibers (5%, 10%, and 15% by volume). Table 1 summarizes all the samples that were prepared
for this study and the number of replicates conducted for each sample. The stress-relaxation
behavior of the composites without MAPP was compared to that of neat polypropylene without
MAPP (S1). Similarly, the stress relaxation behavior of composites with MAPP was compared to
neat polypropylene containing MAPP (S5). A comparison between samples S1 and S5 (see Table
2) was also conducted to determine whether or not the presence of MAPP resulted in any
changes to the properties of the base matrix itself.
Table 2. Experimental design for this study
Sample PP GF MAPP Replicates
Set 1: S1 100% - - 5
70
Samples
without
MAPP
S2 90% 10% - 5
S3 80% 20% - 3
S4 70% 30% - 4
Set 2:
Samples
with
MAPP
S5 95% - 5% 5
S6 85% 10% 5% 5
S7 75% 20% 5% 5
S8 65% 30% 5% 5
4.2.3 Fiber Characterization
The fiber orientation in the samples was examined via x-ray tomography. The post-
processing aspect ratio of the fibers was measured by examining 100 fibers for each fiber content
after a matrix burnout at 600 °C for 6 hours. The post-processing fiber aspect ratio was not
observed to vary with fiber content significantly.
4.2.4 Stress Relaxation Tests
The stress relaxation samples were milled from a region near the edge of the injection
molded flexural modulus samples where the fibers were well oriented. The final samples had an
approximate dimension of 3.5 mm (W), 1.3 mm (T) and 26 mm (L). The stress relaxation
behavior was evaluated using a TA Q800 Dynamic Mechanical Analyser (DMA) with a tensile
clamp. The tests were conducted at 30 °C with a soaking time of 150 minutes to ensure that the
entire thickness of the sample had reached the target temperature, followed by the application of
a constant strain of 0.05% for 90 minutes.
71
4.3 Analytical Model
The analytical model proposed previously was based on the shear-lag theory, as defined
by Cox [26]. Although the theory has been extensively used to understand and predict the elastic
properties of composites, it had not been used to predict their stress relaxation behaviour before
our work [13]. The previously developed analytical model can be best understood by first
examining the elastic case presented in Cox’s shear-lag theory.
4.3.1 Elastic Properties
In short-fiber composites, the fiber end faces are assumed to carry no load, and the stress-
transfer between the fiber and matrix occurs through shear stresses stemming from the
significant mismatch in the modulus of the matrix and fiber. Upon tensile loading the matrix
remote from the fiber is assumed to experience the global strain applied to the composite.
However, the matrix adjacent to the fiber is assumed to be bonded to the fiber and thus
constrained, creating a gradient of stress within the matrix, and shear stresses at the interface
(Figure 21). Based on the ratio of the matrix and fiber moduli, a fraction of the fiber will
experience maximum tensile strain while a fraction near the ends will experience both shear
strain and tensile strain; the tensile strain increases inwards from the fiber ends. Accordingly,
only a fraction of the fiber length will experience maximum tensile stresses, while a fraction near
the ends of the fiber will experience tensile stresses less than the maximum. The total
contribution of a fiber to the composite modulus depends on the average tensile stress over its
entire length.
72
Figure 21. Shear stresses in a short-fiber composite 4.3.2 Stress Relaxation Behavior
In the case of a viscoelastic matrix, where the Young’s modulus of the matrix decreases
with time, there is an increase in the mismatch between the fiber and the matrix moduli. At the
same time, the matrix shear modulus must also decrease with time, decreasing the shear stresses
at the interface, causing the tensile strain and stresses in the fiber to decrease with time as
illustrated in Figure 22.
73
Before Deformation
Figure 22. In a stress relaxation test, stresses in the fiber decrease over time due to the decay in the matrix
modulus during stress relaxation.
In a short-fiber composite, the elastic fibers can, in this way, exhibit a decrease in their
effective reinforcement factor with time when embedded in a viscoelastic matrix, although this
pseudo time-dependence would not be observed if the elastic fibers were tested directly. Our
model predicts that these underlying changes in the matrix shear modulus are responsible for the
changes in stress relaxation rate induced by the addition of elastic fibers in a composite. An
analytical model based on these ideas was derived by incorporating the time-dependent shear
modulus of the matrix into Cox’s shear-lag equation (summarized in Equations (20) and (21))
[13].
74
𝐸𝑐(𝑡) = 𝑉𝑚𝐸𝑚(𝑡) + 𝑉𝑓𝐸𝑓 (1 −𝑡𝑎𝑛ℎ(𝑛(𝑡)𝑠)
𝑛(𝑡)𝑠) Equation 20
𝑛(𝑡) =
[ 4
𝐸𝑓 ln (𝑃𝑓
𝑉𝑓)]
12
[𝐺𝑚(𝑡)]12 Equation 21
Here, the relaxed modulus of the composite at any point in time is related to the matrix
and fiber moduli (𝑬𝒎(𝒕) and 𝑬𝒇) scaled by their respective volume fractions (𝑽𝒎 and 𝑽𝒇). The
time-dependence of the elastic modulus of the matrix has significant influence on the time-
dependence of the composite. However, our model highlights the importance of incorporating a
time-dependent reinforcement effectiveness factor (𝒏(𝒕)), which defines the rate at which the
stress is transferred between the matrix and fiber. The time dependence of this factor stems from
the time-dependence of the matrix shear modulus (𝑮𝒎(𝒕)) (see Equation (21)).
As mentioned previously, there is a significant body of literature presenting a full tensor
analysis of viscoelastic materials containing elastic inclusions, which are often ellipsoidal bodies
for computational reasons. While these analyses do, in principle, explicitly account for changes
in shear stress transfer, they do not, unfortunately, provide much physical insight into the
mechanics of the process, nor do they lend themselves to experimental verification. The
advantage of the simplified shear lag model used in the current work is that it clearly identifies
the time dependent stress transfer efficiency factor as the underlying cause of the changes in
stress relaxation behaviour with fiber loading. If the model is validated by experiments, then it
can be used to make very practical predictions, such as the dependence of stress relaxation on
particle aspect ratio.
75
4.3.3 Fiber Orientation and Aspect Ratio
Since the analytical model assumes the fibers to be completely oriented in the loading
direction, it was important to validate this assumption for accurate comparison between the
experimental results and the model. X-ray tomography of our samples indicated the fibers to be
well-oriented in the loading direction (see Figure 23). It was also observed that the fibers were
well-dispersed within the polypropylene matrix.
Figure 23. The glass fibers were well-oriented within the matrix in the direction of loading (𝑽𝒇 = 𝟓%)
High shear-forces during compounding and injection molding are expected to cause fiber
degradation. Since the fiber aspect ratio has significant influence on the expected properties of
the composite, comparison with the analytical model requires accurate measurement of the
average post-processing fiber aspect ratio (shown in Figure 24).
76
5% 10% 15%
Average Aspect Ratio:
11.3 ± 1.0
Average Aspect Ratio:
10.6 ± 0.8
Average Aspect Ratio:
9.2 ± 0.8
Figure 24. Post-processing fiber aspect ratios were measured via matrix burnout; approximately 100 fibers
were measured at each fiber content.
4.3.4 Effect of Fiber Content on Stress Relaxation Behavior of PP (without MAPP)
The overall stress relaxation of the composites is shown in Figure 25. The addition of
elastic fibers increases the absolute modulus of the composite at all time periods, which is
expected due to the high modulus of the fibers. It is also important to observe from Figure 25
that the fibers not only improve the absolute values of modulus but also delay stress relaxation,
which agrees with past experimental studies. This is not predicted by the Rule of Mixtures
equation (Equation (20)) if only the Em is a function of t but the reinforcement efficiency factor
is treated as a constant, but it is predicted if the reinforcement efficiency is also treated as a
function of time.
77
Figure 25. Stress relaxation behavior of PP/GF composites reinforced with various fiber volume fractions.
The experimental data were compared to predictions from the analytical model derived in
Equations (20) and (21), where Equation (21) incorporates the effect of changing Gm on the
reinforcement efficiency factor (Figure 6). The model predictions were calculated using the post-
processing aspect ratios measured via matrix burnout. The relaxing shear modulus of the matrix
(𝑮𝒎(𝒕)) was calculated using the experimentally-measured relaxing elastic modulus (𝑬𝒎(𝒕)) of
the neat matrix and by assuming an isotropic material with a Poisson’s ratio of 0.3.
0
500
1000
1500
2000
2500
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3500
4000
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Pa)
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78
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2500
0 20 40 60 80
Re
laxa
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3500
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Re
laxa
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n M
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5%
10%
79
Figure 26. Comparison of experimental stress relaxation (○) to the analytical model (--) shows good
agreement between the two at all fiber fractions; the error bars are based on a 90% confidence interval.
Figure 26 shows that the experimental data aligns quite well with the model predictions.
It is important to note that the analytical model is based on easily measurable properties of the
matrix and fiber parameters without any curve-fitting parameters. Based on only these properties,
it is remarkable that such a good fit between the experimental data and the analytical model was
obtained. The strong fit between the experimental data and the model predictions at various fiber
contents provides evidence that the underlying assumptions of the analytical model must be
correct, and it is apparent that the influence of fibers on the stress relaxation behavior of
composites can be largely explained by the incorporation of a time-dependent shear-stress
transfer coefficient.
4.3.5 Effect of MAPP Addition
Previous studies have neglected the time dependence of Gm on the reinforcement
efficiency and have instead attributed the effect of fibers on composite viscoelasticity to the
formation of additional covalent bonds at the fiber-matrix interface. The addition of MAPP can
result in two effects: firstly, it can homogeneously mix with the polymer matrix and thus alter the
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 20 40 60 80
Re
laxa
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n M
od
ulu
s (M
Pa)
Time (min)
15%
80
stress relaxation behavior of this material, which would, of course, alter the stress relaxation
behavior of the composite. Secondly, the MAPP can promote additional bonding at the fiber-
matrix interface and restrict polymer chain mobility in the surrounding polymer, as suggested in
the literature.
A comparison of the stress relaxation behavior of the neat polypropylene films with and
without MAPP (S5 and S1, respectively) was used to understand the effect of 5% MAPP
addition on the base matrix itself (see Figure 27).
Figure 27. The effect of MAPP addition on the stress relaxation behavior of the base polymer was evaluated
by comparing the behavior of polypropylene without MAPP [S1 (○)] to polypropylene with MAPP [S5 (●)].
It is evident that the addition of the low-molecular weight MAPP restricted stress
relaxation of the pure polypropylene matrix. Investigating the reasons for this effect was beyond
the scope of this study but could be related to possible changes in the crystallinity of the base
polymer [27, 28]. Thus, when using an analytical model to predict the stress relaxation behavior
of the glass-reinforced polypropylene samples containing MAPP, it is more appropriate to use
0
200
400
600
800
1000
1200
1400
1600
0 20 40 60 80 100
Re
laxa
tio
n M
od
ulu
s (M
Pa)
Time (min)
81
the stress relaxation behavior of polypropylene with MAPP (S5) as the matrix rather than
comparing composite behaviour to polypropylene without MAPP (S1).
4.3.6 Effect of Fiber Content on Stress Relaxation Behavior of PP (with MAPP)
The experimental data for the stress relaxation in fiber-reinforced samples containing
MAPP was compared to the predictions from the analytical model (see Figure 28).
0
500
1000
1500
2000
2500
3000
0 20 40 60 80
Re
laxa
tio
n M
od
ulu
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Pa)
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4500
0 20 40 60 80
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Pa)
Time (min)
10%
5%
82
Figure 28. Stress relaxation behavior of composites reinforced with 5%, 10%, and 15% fiber volume
fractions. This experimental data is for samples containing MAPP (●), and has been compared to an
analytical model based on S5 as the matrix (-) to understand the effect of covalent bonding only.
Figure 28 shows that the experimental data aligns well with the analytical model;
however, it is apparent from the composites containing 10% and 15% fiber volume fractions that
the addition of MAPP does hinder stress relaxation somewhat, although the effect is quite small.
The effect of MAPP is more pronounced at shorter relaxation times, evident by decreased
agreement between the model predictions and the experimental data. It is possible that the
addition of MAPP results in increased inhomogeneity in the matrix, causing the matrix closer to
the fiber to crystallize differently than that in the bulk to create an “interphase”, which slows
stress relaxation. However, since the analytical model is based only on the time-dependent shear
stress transfer between the two phases and does a reasonably good job of predicting the
experimental data, it is still safe to conclude that most of the stress relaxation behavior of
composites can be explained by incorporating time-dependent shear stress transfer without
inferring matrix changes near the interface.
0
1000
2000
3000
4000
5000
6000
0 20 40 60 80
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laxa
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n M
od
ulu
s (M
Pa)
Time (min)
15%
83
4.4 Conclusions
In the first part of this study, stress relaxation experiments were conducted on glass fiber-
reinforced polypropylene. The experimental results were compared to a previously published
analytical model, which can be used to predict the stress relaxation behavior of short-fiber
composites using a viscoelastic shear-lag approach. The detailed development of the analytical
model is described in our previous work [13].
A good fit was observed between the analytical model and the experimental results,
indicating that the analytical model was an adequate and accurate tool to predict the stress
relaxation behavior of short, elastic fiber-reinforced composites with various fiber fractions. The
agreement with the model indicates that the stress relaxation behavior of a short-fiber composite
can be explained by incorporating a time-dependent shear stress transfer at the fiber-matrix
interface.
Previous studies investigating the effect of short fibers on the stress relaxation behavior
of a composite have attributed changes in the relaxation time constant to increased covalent
bonding at the fiber-matrix interface. The second part of the study evaluated this hypothesis and
investigated the role of interfacial covalent bonds. This was conducted by examining the stress
relaxation behavior of composites containing MAPP as a coupling agent, and then comparing the
experimental findings to predictions made using the analytical model. The addition of MAPP
was found to make changes to the properties of the bulk matrix, and thus, the analytical model
predictions were calculated using the new matrix properties.
It was found that even when MAPP was added to the system to alter the fiber/matrix
interface, the experimental data remained well-aligned with the analytical model predictions
84
which do not depend on modelling interfacial changes. In fact, since the analytical model
predictions align quite closely with the experimental data, it was concluded that most of the
stress relaxation behavior of a composite can predicted using a simple model incorporating the
time-dependent matrix modulus and the time dependent shear stress transfer efficiency. The
fibers are gradually unloaded during a stress relaxation experiment, even though they themselves
are perfectly elastic.
The analytical model was able to predict the experimental stress relaxation behavior of
composites, both with and without MAPP, with a high level of accuracy. This highlights its value
as an accurate tool to predict the stress relaxation behavior of short-fiber composites.
Chapter References
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5. Saeed, U.; Hussain, K.; Rizvi, G. HDPE reinforced with glass fibers: rheology, tensile
properties, stress relaxation, and orientation of fibers. Polym. Compos. 2014, 35, 2159–2169.
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6. Sreekala, M. S.; Kumaran, M. G.; Joseph, R.; Thomas, S. Stress relaxation behavior in
composites based on short oil-palm fibers and phenol formaldehyde resins. Compos. Sci.
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7. Stan, F.; Fetecau, C. Study of stress relaxation in polytetraflyoroethylene composites by
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relaxation behavior of short pineapple fiber reinforced polyethylene composites. J. Reinf.
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9. Geethamma, V. G.; Pothan, L. A.; Rhao, B.; Neelakantan, N. R.; Thomas, S. Tensile stress
relaxation of short-coir-fiber reinforced natural rubber composites. J. Appl. Polym. Sci. 2004,
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10. Mirzaei, B.; Tajvidi, M.; Falk, R. H.; Felton, C. Stress relaxation behavior of lignocellulosic-
high density polyethylene composites. J. Reinf. Plast. Compos. 2011, 30, 875–881.
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absorption and stress relaxation behavior of PP/date palm fiber composite materials, in:
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Modeling of Mechanical Systems -II; Springer: Hammamet, Tunisia, 2015, 437–445.
13. Obaid, N.; Kortschot, M. T.; Sain, M. Understanding the stress relaxation behavior of
polymers reinforced with short elastic fibers. Materials, 2017, 10, 472.
14. Sevostianov, I.; Levin, V.; Radi, E. Effective viscoelastic properties of short-fiber reinforced
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15. Smith, N.; Medvedev, G. A.; Pipes, R. B. Viscoelastic shear lag analysis of the discontinuous
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16. Luciano, R.; Barbero, E. J. Analytical expressions for the relaxation moduli of linear
viscoelastic composites with periodic microstructure. J. Appl. Mech. 1995, 62, 786–793.
17. Okabe, T.; Nishikawa, M. GLS strength prediction of glass-fiber-reinforced polypropylene.
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18. Okabe, T.; Nishikawa, M.; Takeda, N. Micromechanics on the rate-dependent fracture of
discontinuous fiber-reinforced plastics. Int. J. Damage Mechanics. 2010, 19, 339–360.
19. Hashimoto, M.; Okabe, T.; Sasayama, T.; Matsutani, H.; Nishikawa, M. Prediction of tensile
strength of discontinuous carbon fiber/polypropylene composite with fiber orientation
distribution. Comp. Part A. 2012, 43, 1791–1799.
20. Beyerlein, J.; Pheonix, S. L.; Raj, R. Time evolution of stress redistribution around multiple
fiber breaks in a composite with viscous and viscoelastic matrices. Int. J. Solids Struct. 1998,
35, 3177–3211.
21. Iyengar, N.; Curtin, W. A. Time-dependent failure in fiber-reinforced composites by matrix
and interface shear creep. Acta Mater. 1997, 45, 3419–3429.
22. Du, Z. Z.; McMeeking, R. M. Creep models for metal matrix composites with long brittle
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25. Lagoudas, D. C.; Hui, C. Y.; Pheonix, S. L. Time evolution of overstress profiles near broken
fibers in a composite with a viscoelastic matrix. Int. J. Solids Struct. 1989, 25 (1), 45–66.
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88
Chapter 5. Modeling and Predicting the Stress Relaxation of
Composites with Short and Randomly Oriented Fibers
In the previous chapter, we developed an analytical model for the effect of fully aligned
short fibers, and the model predictions were successfully compared to finite element simulations.
However, in most industrial applications of short-fiber composites, fibers are not aligned, and
hence it is necessary to examine the time dependence of viscoelastic polymers containing
randomly oriented short fibers. In this chapter, we propose an analytical model to predict the
stress relaxation behavior of short-fiber composites where the fibers are randomly oriented. The
model predictions were compared to results obtained from finite element simulations.
This chapter has been published as: “Obaid, N.; Kortschot, M. T.; Sain, M. Modeling and
predicting the stress relaxation of composites with short and randomly oriented fibers.
Materials.2017, 10(10), 1207.”
5.1 Introduction
The high modulus of short-fiber-reinforced polymers has made them useful in demanding
load-bearing applications such as high-performance sporting equipment. However, because of
the inherent viscoelasticity of the matrix phase, polymer composites are prone to creep [1] and
stress relaxation, making it a challenge when considering composites for long-term applications.
A better understanding of composite viscoelasticity is needed so that long-term behavior can be
better predicted.
A stress relaxation test, where a constant strain is applied to a specimen and the decay of
stress is monitored, is a straightforward way of characterizing polymer viscoelasticity. Since
89
viscoelastic properties all result from the same molecular mechanisms, a model for stress
relaxation in composites would also shed light on creep or dynamic mechanical behavior.
The interaction between the fiber and matrix in a short-fiber composite is quite complex,
and it has been a challenge to understand the effect of fibers on the viscoelastic properties of
these materials. In particular, the effect of fibers on stress relaxation behavior is still not well-
understood. Purely elastic fibers should, of course, increase the elastic component (i.e., increase
the absolute stiffness), but would not be expected to alter the time-dependent behavior of the
material, which is traditionally attributed to the viscoelastic matrix. However, in many studies,
researchers have found that the addition of short fibers to a polymer slows the stress-relaxation
process. Kutty and Nando [2] investigated the effect of short Kevlar fibers on polyurethanes and
found that increasing the fiber content resulted in slower stress relaxation rates. Suhara, Kutty,
and Nando [3] showed that increasing the content of short polyester fibers decreased the stress
relaxation rate in polyurethanes. The same effects were observed by Saeed et al. [4] and Sreekala
et al. [5] for the effect of glass fibers embedded in different matrices.
Two main mechanisms have been proposed in the literature to explain the change in the
time constant for stress relaxation with the addition of elastic fibers. The first explanation is that
since stress relaxation occurs by rearrangement of the secondary bonds in a polymer, the
physical presence of fibers impedes molecular rearrangement of the polymer near the interface,
resulting in a slower relaxation of the matrix. Considering the scale at which the molecular
rearrangements of secondary bonds occur (angstroms), it seems unlikely that the volume of
polymer close enough to a fiber surface to be affected could result in significant changes to the
bulk properties of the composite.
90
The second theory proposed in the literature focuses on chemical bonding at the
fiber/matrix interface. These studies propose that when a fiber is added into a matrix, strong
covalent bonds are formed at the interface. During stress relaxation, these additional covalent
bonds must be overcome to allow polymer mobility and stress relaxation. Due to these additional
bonds, the stress relaxation rates are expected to be slower. George et al. [6], and later
Geethamma et al. [7] and Mirzaei et al. [8], have all suggested this mechanism.
Experimental studies agree that the presence of elastic fibers can slow the stress
relaxation rate of polymers in which they are embedded. Both of the previously proposed
mechanisms rely on speculation regarding the molecular interactions at the fiber-matrix
interface. Furthermore, they do not provide any quantitative predictions of the changes in stress
relaxation due to fiber addition.
Short-fiber composite viscoelasticity has also been extensively modeled using a tensor elasticity
approach to compute the stress field around elastic inclusions embedded in viscoelastic matrices
[9,10]. Unfortunately, the mathematical complexity of these formulations makes it very
challenging to make predictions of viscoelasticity based on measurable material properties.
While some authors have compared their results to experimental data [11], this is usually very
limited.
In a previous paper, we proposed a novel explanation for the effect of elastic fibers on the
time-dependency of a polymer matrix composite [12]. We developed a quantitative model based
on simple composite micromechanics, by explicitly accounting for the time-dependent shear
stress transfer at the fiber-matrix interface. In contrast to previous theories, which were primarily
focused on attributing the effect of fibers to changes in the structure near the fiber interface, our
model can predict stress relaxation without postulating such changes.
91
In our previous study, we specifically examined the effect of short fibers fully aligned
with the load direction on composite stress relaxation. In that study, an analytical model was
developed, studied parametrically, and then validated by finite element experiments. The
analytical model showed that the time-dependence of polymer composites was influenced by two
key factors. Firstly, the tensile modulus of the matrix is, of course, time-dependent. However,
equally important is its time-dependent shear modulus, which results in a time-dependent shear
stress transfer to the fiber. This causes the stress carried by the fiber to also decay with time,
causing the overall time-dependence of the composite to alter upon the addition of elastic fibers.
In this study, we extend the model to account for random fiber orientation and validate it
using a finite element simulation. In these simulations, the results were obtained and conclusions
were made based on repeated random sampling. In this study, five replicates were conducted at
each fiber content. For each replicate, fibers with random positions and orientations were
generated to ensure that the results accurately represented a truly random system. These
individual replicates were averaged to understand the effect of fiber content. Since these
simulations incorporate the inherent randomness of fiber positions and orientations, by adopting
this approach these simulations can be considered equivalent to lab-scale experiments.
We also use the model to understand critical differences in the expected viscoelastic
behavior of aligned versus misaligned fiber composites.
5.2 Analytical Model
The analytical model proposed previously was based on a simple shear-lag model,
originally developed by Cox [13]. While this model has been widely used to predict the elastic
properties of short-fiber composites, it had not previously been used to model stress relaxation.
92
For polymer composites containing short fibers aligned in the load direction, the stress relaxation
modulus was derived in Reference [12] and the result of this derivation is expressed in Equations
(22) and (23).
𝐸𝑐(𝑡) = 𝑉𝑚 [𝐸∞ + (𝐸0 − 𝐸∞) exp (−
𝑡
𝜏)] + 𝑉𝑓𝐸𝑓 (1 −
𝑡𝑎𝑛ℎ(𝑛(𝑡)𝑠)
𝑛(𝑡)𝑠) Equation 22
𝑛(𝑡) =
[ 4
𝐸𝑓 ln (𝑃𝑓
𝑉𝑓)]
12
[𝐺∞ + (𝐺0 − 𝐺∞) exp (−𝑡
𝜏)]
12 Equation 23
Equation (22) is a modified rule of mixtures equation for the modulus of composites
containing short fibers with an aspect ratio (the length-to-diameter ratio of the fiber) of s. The
first term represents the matrix contribution scaled by its volume fraction (𝑽𝒎). The matrix
modulus is allowed to decay with time (t), as it must in a polymer matrix composite. The
relaxation of the matrix modulus is modeled using a one-component Prony Series consisting of
the instantaneous modulus (𝑬𝟎), long-term modulus (𝑬∞), and the relaxation time constant (τ).
The second term accounts for the contribution of the fibers, where 𝑽𝒇𝑬𝒇 (the product of the fiber
volume fraction and fiber modulus) is reduced by the so-called shear-lag factor to account for
the fact that the short fibers are not fully loaded along their length. The shear-lag factor (𝜼(𝒕)) is
determined by Equation (23), in keeping with the original derivation by Cox which consisted of
fiber characteristics such as their packing factor (𝑷𝒇), fiber volume fraction, and fiber modulus,
as well as the shear modulus of the matrix. However, in this analytical model, the shear modulus
of the matrix is allowed to decay with time. The critical effect of this decay in shear modulus
was overlooked in previous studies of polymer composite viscoelasticity. The equation assumed
the polymer to be an isotropic material, and thus, the elastic and shear moduli were related via
93
Poisson’s ratio and the decay in the matrix shear modulus also followed a one-component Prony
Series analogous to that of the elastic modulus.
Equations (22) and (23) assume that the stress relaxation modulus decays following a
Prony Series, and that the rate of relaxation is characterized by a relaxation rate constant (τ). τ
represents the time required for the modulus to drop to a fraction of 𝟏/𝒆 of its initial value
during a stress relaxation test. Previous studies have shown that increasing the fiber content
resulted in an increase in the relaxation rate constant, representing a change in the effective time-
dependency of the material. Our previous paper showed that the change in relaxation rate
constant can be predicted quite well by incorporating time-dependent shear stress transfer at the
fiber-matrix interface.
In many practical applications, especially those that utilize short fibers, fibers are not
highly aligned in the load direction. The misorientation of fibers changes the overall stress
distribution in a composite, and it is therefore important to develop an accurate analytical model
to predict the stress relaxation behavior of composites with misoriented fibers.
The model for the elastic modulus of a short-fiber composite with misoriented fibers was
developed in the original paper by Cox [14], but here we will use the derivation presented by
Jayaraman and Kortschot for fibers that are randomly oriented in the plane of the specimen [15].
The model considers a fiber of length L and cross-sectional area 𝑨𝒇 = 𝝅𝒓𝒇𝟐 embedded within a
matrix and misoriented at an angle θ from the direction of loading, as shown in Figure 29. An
imaginary cross-section, perpendicular to the loading direction, is used to assess the total
contribution of the fiber to the stress in the composite normal to the cross-section.
94
Figure 29. The load carried by a fiber in the loading axis can be calculated through a cross-line perpendicular
to the loading direction.
It is assumed that the fibers have a distribution in orientation with probability density
functions, as defined in Equation (24), respectively.
∫ 𝑔(𝜃)𝑑
𝜋/2
0
𝜃 = 1 Equation 24
Because the fiber is not oriented in the loading direction, the projected length of the fiber
in the loading direction is then:
𝐿𝑥 = 𝐿𝑐𝑜𝑠𝜃 Equation 25
The volume fraction of the fibers, taking into account the total number of fibers (N) and
their individual fiber volumes (area of the fiber (𝑨𝒇) multiplied by their length), is calculated as:
𝑉𝑓 =𝑇𝑜𝑡𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝐹𝑖𝑏𝑒𝑟𝑠
𝑇𝑜𝑡𝑎𝑙 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑆𝑝𝑒𝑐𝑖𝑚𝑒𝑛=
𝑁𝐴𝑓𝐿
𝑎𝑏𝑐 Equation 26
This equation can be rewritten to calculate the total number of fibers in the specimen as
follows:
𝑁 =
𝑉𝑓𝑎𝑏𝑐
𝐴𝑓𝐿 Equation 27
95
The number of fibers with an orientation between (𝜽 + 𝒅𝜽) is calculated using its
probability density function:
𝑁𝜃 = [
𝑉𝑓𝑎𝑏𝑐
𝐴𝑓𝐿] (𝑔(𝜃)𝑑𝜃) Equation 28
The total length of fibers oriented in the θ direction is calculated by considering the
projected length of these fibers along the loading direction (from Equation (25)):
𝐿𝑇 = 𝑁𝜃𝐿𝑥 Equation 29
𝐿𝑇 = [
𝑉𝑓𝑎𝑏𝑐
𝐴𝑓𝐿] 𝑔(𝜃)𝑑𝜃(𝐿𝑐𝑜𝑠𝜃) Equation 30
Now, we can reconsider the concept of the imaginary cross-section to assess the total
load carried by the fibers. The number of fibers intersecting a cross-section can be simplified as
the total projected length of the fibers in the loading axis divided by the total length of the
specimen in the loading direction.
𝑁𝑆𝐶𝐴𝑁 =
𝐿𝑇
𝑐 Equation 31
The total load carried by all fibers in the cross-section is:
𝐹𝑇 = ∑𝑁𝑆𝐶𝐴𝑁𝐹𝑥
𝜃
Equation 32
𝑭𝒙 has been developed previously as [16]:
𝐹𝑥 = 𝐹�̅�𝑐𝑜𝑠𝜃 = ∅𝐴𝑓𝐸𝑓𝜀0(cos2 𝜃 − 𝑣𝑠 sin2 𝜃 𝑐𝑜𝑠𝜃) Equation 33
where 𝒗𝒔 is the Poisson’s ratio of the material, 𝜺𝟎 is the strain in the material, and ∅ is the
shear-lag factor defined by Cox [10]. Thus,
𝐹𝑇 = ∫ ∅𝐴𝑓𝐸𝑓𝜀0(cos2 𝜃 − 𝑣𝑠 sin2 𝜃 𝑐𝑜𝑠𝜃)
𝜋/2
0
[[𝑉𝑓𝑎𝑏
𝐴𝑓𝐿] 𝑔(𝜃)𝑑𝜃] Equation 34
The integral can be solved as shown below with an orientation-based factor (CPP):
96
𝐹𝑇 = 𝑉𝑓𝐸𝑓𝑎𝑏𝜀0[𝐶𝑃𝑃] Equation 35
𝐶𝑃𝑃 = ∫ 𝑔(𝜃)
𝜋/2
0
(cos4 𝜃 − 𝑣𝑠 sin2 𝜃 cos2 𝜃)𝑑𝜃 Equation 36
If the model is evaluated considering a truly random orientation in the fibers, then there is
equal probability that the fiber has a misorientation angle (θ) between 0 and π/2:
𝑔(𝜃) =1
𝜋/2 if 0 < 𝜃 ≤ 𝜋/2 Equation 37
𝑔(𝜃) = 0 if 𝜃 > 𝜋/2 Equation 38
If g(θ)=2/π, the orientation-based factor further simplifies to:
𝐶𝑃𝑃 =
2
𝜋 (
3
8) (
𝜋
2) −
2
𝜋 𝑣𝑠 (
𝜋
16) Equation 39
The Poisson’s ratio of a typical polymeric material is 𝒗𝒔= 1/3; however, during stress
relaxation, the polymer relaxes and its Poisson’s ratio approaches 𝒗𝒔= 1/2 [17]. Using 𝒗𝒔= 1/2,
the orientation-based factor simplifies to:
𝐶𝑃𝑃 =
5
16 Equation 40
Equation (35) can be simplified to:
𝐹𝑇 = 𝑉𝑓𝐸𝑓𝑎𝑏𝜀0 (
5
16) Equation 41
The contribution by the fibers is therefore reduced by a factor of 5/16 when the fibers are
randomly oriented in a plane. The elastic modulus of such a composite must be similarly
reduced, and thus, Cox’s shear lag prediction can be modified to incorporate this shrinkage:
𝐸𝑐 =
5𝑉𝑓𝐸𝑓
16(1 −
𝑡𝑎𝑛ℎ(𝑛𝑠)
𝑛𝑠) + 𝑉𝑚𝐸𝑚 Equation 42
The following model is derived for the stress relaxation behavior of composites
consisting of a viscoelastic matrix embedded with short, elastic fibers oriented randomly in the
plane. The analytical model takes into account the time-dependent shear modulus of the matrix
in addition to its time-dependent elastic modulus.
97
𝐸𝑐 =
5𝑉𝑓𝐸𝑓
16(1 −
𝑡𝑎𝑛ℎ(𝑛(𝑡)𝑠)
𝑛(𝑡)𝑠) + 𝑉𝑚𝐸𝑚(𝑡) Equation 43
𝑛(𝑡) =
[ 4
𝐸𝑓 ln (𝑃𝑓
𝑉𝑓)]
12
[𝐺𝑚(𝑡)]12 Equation 44
Note that 𝑬𝒎 and 𝑮𝒎 are both functions of t, but that 𝑬𝒇 is a fixed quantity for glass
fibers. The model can be used to predict that, in the case of randomly oriented fibers, the time-
dependent effectiveness factor is scaled by a factor of 5/16, reducing the effect of fibers on the
overall stress relaxation behavior when compared to their oriented counterparts. Thus, the model
shows that, unlike oriented composites, where time-dependence is influenced by both the shear
and tensile moduli of the matrix, the time-dependence of misoriented composites is primarily
influenced by the tensile modulus of the matrix and the effect of the time-dependent shear
modulus is comparatively lower. We would therefore expect a more subtle relationship between
fiber volume fraction and the relaxation time constant in randomly oriented fibers.
5.3 Finite Element Simulations
Finite Element (FE) experiments were conducted in Abaqus/CAE (version 6.16, Simulia,
Dassault Systemes, Paris, France) and were compared to the predictions from the analytical
model to determine its accuracy. In these FE experiments, the matrix was defined as a
viscoelastic material having an instantaneous modulus of 1 GPa, a long-term modulus of 0.5
GPa, a relaxation-time constant of 100 s, and a Poisson’s ratio of 0.5. The fibers were defined as
having an elastic modulus of 80 GPa, a Poisson’s ratio of 0.2, a length of 260 μm, and an aspect
ratio of 16.
The model consisted of a two-phase system including a matrix and fibers. A rigid body
was used to apply boundary conditions that corresponded to a stress relaxation experiment. A tie
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constraint was applied between the rigid body and the matrix surfaces. The matrix was defined as
a three-dimensional deformable object while the fibers were defined as one-dimensional beam
elements for computational efficiency. The interaction between the two phases was defined by
defining the fibers as being embedded in the matrix with perfect bonding. High mesh densities
were applied to both the matrix and the fiber: the matrix mesh consisted of a total of 2211
quadratic elements of type C3D20R while the fiber mesh consisted of 40 quadratic elements of
type B32 per fiber.
The experiments were conducted at various fiber contents and the number of fibers was
altered in accordance with the desired fiber volume fraction. Five replicate simulations were
conducted for each fiber content. For each replicate, a Python script was used to assign a random
position and random orientation to each fiber. This ensured that each new experiment contained
fiber placements that were random and different from previous replicates. The mesh was
autogenerated on the edges of the fiber.
The finite element experiment consisted of two analysis steps: An instantaneous strain
was applied, followed by 400 s during which stress decay was observed. Default solver settings
were used in both analysis steps. The modulus of the composite was determined using the cross-
sectional area, the applied force, and displacement of the rigid body. The data was collected at 10
s intervals with a minimum increment time step of 0.004 s.
99
5.4 Results
The accuracy of the closed-form analytical model was evaluated through comparison to
the finite element experiments conducted in Abaqus CAE. The fit between the two was
compared by two methods. First,
Figure 30 compares the overall stress relaxation behavior as predicted by the analytical
model against the data obtained through the finite element experiments. The accuracy of the
model was also determined by plotting the analytical model predictions against the finite element
experimental results (at the same point in time), and comparing the slope of this line to that of a
𝒚 = 𝒙 line. If the model is a good fit, both values should be equal and result in a slope close to
one. This comparison is shown in Figure 31.
100
Figure 30. A comparison of the overall stress relaxation profile of short-fiber composites shows excellent
agreement between the predictions of the analytical model (-) and the results obtained from the finite element
experiments (■).
101
Figure 31. Comparison of the analytical model predictions to the finite element (FE) simulation results shows
good agreement between the two at low volume fraction; however, at volume fractions equal to 30% and
greater, the finite element results deviate from the predictions of the analytical model.
Figure 30 and Figure 31 show excellent agreement between the analytical model and the
finite element model at fiber volume fractions below 30%; however, at fiber content higher than
30%, the analytical model is no longer able to accurately predict the stress relaxation behavior of
the composite. This is expected because the analytical model is based on the simplified shear-lag
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model, which is known to be accurate only at low fiber volume fractions. Shear-lag models
assume that fibers are located in an isolated pocket of resin and that the outer surface of this
pocket experiences a uniform strain equal to the global strain. However, at higher volume
fractions, as the fibers approach each other, the situation becomes more complex and the model
is not expected to be accurate. Studies that have compared the elastic modulus of composites in a
static case have shown that the shear-lag model often overpredicts the actual modulus [18].
Figure 32 compares the analytical predictions and the results from the finite element
simulation for the instantaneous and long-term moduli of the composites. An increase in the fiber
content resulted in an increase in both moduli, which is a trivial effect of adding stiff glass fibers
in a polymeric matrix. The results from the simulations were well-predicted by the analytical
model for misoriented fibers, validating the model.
Figure 32. Good agreement is obtained between the instantaneous (a) and long-term (b) moduli values
obtained from the analytical model (-) and the finite element simulations (●).
Figure 33 shows the relaxation rate constant as a function of volume fraction of fibers. It
is important to note that the analytical model, which is based entirely on micromechanics and
makes no assertions about chemical interactions between the fiber and matrix, is able to predict a
103
relationship between the rate constant and fiber volume fraction. The finite element results are
well-predicted by the analytical model for composites containing misoriented fibers. As with any
experimental study, some deviations will exist between the relaxation rate constant predicted via
the analytical model and those that were obtained through the finite element experiments. In
these simulations, the finite element analysis consisted of randomized specimens, and additional
replicates could be used to further improve the match between the two datasets. The strong
agreement between the results validates the ability of the model to predict the entire stress
relaxation behavior for misoriented-fiber composites at low fiber volume fractions.
Figure 33. Good agreement is obtained between the relaxation rate constant obtained from the
analytical model (-) and the finite element simulations (●).
In a previous study, we conducted finite element experiments on composites with
oriented fibers instead of misoriented fibers; the settings used in that study were the same as
those described here [12]. Figure 34 compares the effect of orientation on the results obtained
from both finite element simulations.
As mentioned earlier, the misorientation of the fibers results in a reduction in their ability
to carry stress in the loading direction. Thus, for misoriented composites, the influence of the
104
time-dependent shear lag effectiveness factor shrinks, reducing the effect of fibers on the
relaxation rate constant of the composite.
Figure 34. Effect of fiber orientation on the properties of the composite as obtained from finite element
experiments.
The analytical model predicts that there should be a greater difference between the
relaxation time constant for oriented and misoriented samples compared to the results shown in
Figure 34. However, it is important to note that the data shown in this figure was obtained
through the finite element experiments and is thus subject to error due to the averaging of five
replicates. Also, as mentioned earlier, the concept of a relaxation time constant depends heavily
on the fit of the experimental data to a one-component Prony Series. As a result, the relaxation
time constant is very sensitive to small errors in curve-fitting and variance between replicates.
Although the relaxation time constant is a useful means to predict the stress relaxation rate and
for comparison of overall trends, its absolute value is highly sensitive to small errors.
5.5 Conclusions
The goal of this study was to investigate the role of short fibers on stress relaxation
behavior by examining the micromechanics at the fiber-matrix interface in composites with
105
randomly oriented fibers. This novel perspective differs from previous investigations, which
have focused on attributing the effect of fibers to chemical or structural changes at the interface
between the fiber and the matrix. The study aimed to not only understand the role of
micromechanics, but also to develop an analytical model that could be used to make predictions
regarding the stress relaxation of composites with varying fiber content, orientations, and aspect
ratios.
Good agreement was observed between the results obtained via finite element
experiments and the predictions from the analytical model. Thus, the analytical model developed
in this study provides an adequate and accurate tool to predict the stress relaxation behavior of
misoriented short-fiber composites with varying fiber content. It was found that the rate of stress
relaxation is influenced by both the time-dependent elastic modulus of the matrix and the time-
dependent shear stress transfer at the fiber-matrix interface, which stems from the time-
dependent shear modulus of the matrix. However, the misorientation of fibers shrinks the
contribution of the time-dependent shear stress transfer by a factor of one-third compared to the
contribution of oriented fibers.
The analytical model was validated using finite element experiments conducted in
Abaqus CAE. Excellent agreement was observed between the analytical model and experiments
at fiber volume fractions below 30%; the deviation from the analytical model beyond 30% was
attributed to the inaccuracy of the shear-lag model at higher fiber volume fractions. Since the
experiments were conducted for random fiber positions and orientations, its agreement with the
analytical model further signifies the importance of incorporating time-dependent shear stress
transfer at the fiber-matrix interface.
106
Chapter References
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fiber reinforced thermoplastics. Compos. Part B Eng. 2016, 97, 68–83.
2. Kutty, S. K.; Nando, G. B. Short Kevlar fiber-thermoplastic polyurethane composite. J. Appl.
Polym. Sci. 1991, 43, 1913–1923.
3. Suhara, F.; Kutty, S. K.; Nando, G. B. Stress relaxation of polyester fiber-polyurethane
elastomer composite with different interfacial bonding agents. J. Elastom. Plast. 1998, 30,
103–117.
4. Saeed, U.; Hussain, K.; Rizvi, G. HDPE reinforced with glass fibers: Rheology, tensile
properties, stress relaxation, and orientation of fibers. Polym. Compos. 2014, 35, 2159–2169.
5. Sreekala, M. S.; Kumaran, M. G.; Joseph, R.; Thomas, S. Stress relaxation behavior in
composites based on short oil-palm fibers and phenol formaldehyde resins. Compos. Sci.
Technol. 2001, 61, 1175–1188.
6. George, J.; Sreekala, M. S.; Thomas, S.; Bhagawan, S. S.; Neelakantan, N. R. Stress
relaxation behavior of short pineapple fiber reinforced polyethylene composites. J. Reinf.
Plast. Compos. 1998, 17, 651–672.
7. Geethamma, V. G.; Pothen, L. A.; Rhao, B.; Neelakantan, N. R.; Thomas, S. Tensile stress
relaxation of short-coir-fiber reinforced natural rubber composites. J. Appl. Polym.
Sci. 2004, 94, 96–104.
8. Mirzaei, B.; Tajvidi, M.; Falk, R. H.; Felton, C. Stress relaxation behavior of lignocellulosic-
high density polyethylene composites. J. Reinf. Plast. Compos. 2011, 30, 875–881.
9. Sevostianov, I.; Levin, V.; Radi, E. Effective viscoelastic properties of short-fiber reinforced
composites. Int. J. Eng. Sci. 2016, 100, 61–73.
107
10. Smith, N.; Medvedev, G. A.; Pipes, R. B. Viscoelastic shear lag analysis of the discontinuous
fiber composite in Proceedings of the 19th International Conference on Composite
Materials, Montreal, QC, Canada, 28 July–2 August 2013.
11. Luciano, R.; Barbero, E. J. Analytical expressions for the relaxation moduli of linear
viscoelastic composites with periodic microstructure. J. Appl. Mech. 1995, 62, 786–793.
12. Obaid, N.; Kortschot, M. T.; Sain, M. Understanding the stress relaxation behavior of
polymers reinforced with short elastic fibers. Materials. 2017, 10, 472.
13. Piggott, M. R.; Taplin, D. M. R. Load Bearing Fiber Composites; Pergamon Press: New
York, NY, USA, 1980.
14. Cox, H. L. The elasticity and strength of paper and other fibrous materials. Br. J. Appl.
Phys. 1952, 3, 72–79.
15. Jayaraman, K.; Kortschot, M. T. Correction to the Fukuda-Kawata Young’s modulus theory
and the Fukuda-Chou strength theory for short fiber-reinforced composite materials. J.
Mater. Sci. 1996, 31, 2059–2064.
16. Fukuda, H.; Kawata, K. On Young’s modulus of short-fiber composites. Fiber Sci.
Technol. 1974, 7, 207–222.
17. Lakes, R. S.; Wineman, A. On Poisson’s ratio in linearly viscoelastic solids. J. Elast.
2006, 85, 45–63.
18. Facca, A. G.; Kortschot, M. T.; Yan, N. Predicting the elastic modulus of natural fiber
reinforced thermoplastics. Compos. Part A. 2006, 37, 1660–1671.
107
Chapter 6. Investigating the Mechanical Response of Soy-Based
Polyurethane Foams with Glass Fibers under Compression at
Various Rates
The previous chapters in this thesis have focussed on investigating the stress relaxation
behavior of composites; however, viscoelasticity also affects the strain-rate dependence of the
material. This chapter presents a scenario in which an understanding of composite stress
relaxation can be useful in predicting the mechanical behavior of reinforced polyurethane foams
at various strain rates.
This chapter has been published as: “Obaid, N.; Kortschot, M. T.; Sain, M. Investigating
the mechanical response of soy-based polyurethane foams with glass fibers under compression at
various rates. Cellular Polymers. 2015, 34(6), 281-298.” Some modifications have been made to
the chapter since its original publication to better explain the relationship between strain-rate
dependence and stress relaxation behavior; however, there have been no changes made to the
presented results, figures, findings, and conclusions.
6.1 Introduction
Polyurethane foams account for the largest proportion of total polyurethane production in
the world. Their versatile properties make them suitable for various applications including
packaging, cushioning, insulation, and as the core material of sandwich panels. Despite their
desirable properties, polyurethanes and their foams have adverse effects on the environment due
to their lack of degradability.
108
Several groups have investigated the use of soy-oil based polyols, known as soyol, to
replace the otherwise petroleum-based polyol used to form polyurethanes. The production of
polyurethane foams from soyol also results in better biodegradability when compared to their
petroleum-sourced counterparts [1]. Although the use of soyol presents a relatively ‘greener’
approach to producing polyurethane foams, its mechanical properties need to be investigated and
improved.
Short fiber reinforcement has often been used to improve the compressive modulus and
plateau stress of these foams. However, since polymeric foams are derived from viscoelastic
materials, the time-dependent properties of the solid material are retained in the foamed form as
well. The addition of short fibers can also impact the viscoelastic behavior of the foams, and to
the best of our knowledge, there are limited studies that have investigated this effect. In this
study, elastic glass fibers were used to reinforce viscoelastic polyurethane foams and the
compressive behaviour of the resulting foam was observed at various strain rates by measuring
the modulus and yield strength. The strain-rate dependence of the composite foams was used to
make indirect observations of their viscoelastic behavior.
6.2 Literature Review
6.2.1 Foam Compression
Industrially-available foams are often subjected to compressive loading conditions. The
properties of the cellular structures are dependent on the base material from which the foam was
derived as well as the foam architecture such as its density. The properties of a foam are often
modelled using the reduced density (∅𝑹), which is a ratio of the density of the foam to the
density of the solid from which it was derived as shown in Equation (45).
109
∅𝑅 =
𝜌𝐹𝑂𝐴𝑀
𝜌𝑆𝑂𝐿𝐼𝐷
=𝑉𝑠𝑜𝑙𝑖𝑑
𝑉𝑓𝑜𝑎𝑚
Equation 45
The typical engineering stress-strain curve of foams is shown in Figure 35. A simplified
micromechanical model was developed by Gibson to explain the observed foam behaviour [2].
This model assumes the foam to be made of repeating units of a cubic lattice. The fundamental
equations developed through this model are reasonable in predicting the compressive modulus
and plateau stress of the foam.
Figure 35. Typical stress-strain behaviour of foam under compressive loading
The typical deformation of foam can be divided into three main phases. The cubic lattice
model shows the micromechanical deformation of the unit cell in each phase, as summarized in
Figure 36.
0
0.5
1
1.5
2
2.5
3
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Str
ess
Strain
Stage II:
Plateau Region
Stage III:
Densification
Stage I:
Linear Elastic
110
Before Deformation Stage I: Linear Elastic Stage II: Plateau Region Stage III: Densification
Figure 36. The micromechanical deformation of a cellular material based on the cubic lattice model
The first part of the curve is a linear region where the stress and strain are related by
Hooke’s law. The cubic lattice model assumes that this region of the curve is primarily
dominated by the elastic bending of the cell struts. Equation (46) below shows that the elastic
modulus of foams is related to reduced density and the modulus of the solid material from which
the foam was derived.
𝐸𝑓𝑜𝑎𝑚 = 𝐸𝑠𝑜𝑙𝑖𝑑(∅2) Equation 46
This model does not account for strain-rate effects and the relationship can be assumed to
exist for various rates of deformation.
The second region is a long plateau at an almost constant stress, governed by the buckling
of the cell struts. Equation (47) shows that the plateau stress of foams is related to reduced
density and the yield strength of the solid material from which the foam was derived.
𝜎𝑝𝑙𝑓𝑜𝑎𝑚= 0.3 𝜎𝑌𝑆(∅
1.5) Equation 47
The plateau region continues until all the cells in the structure have collapsed, leading to
the third region. The last region of the curve is known as the densification regime. Here, the cells
are fully collapsed and opposite sides of the unit cell begin to interact. Any advantage gained by
111
the foam architecture is lost in this stage as the material begins to approach the modulus of the
solid material from which it was derived.
6.2.2 Fiber Reinforcement in Foams
The size of the fiber relative to the cell size influences the type of reinforcement it
provides. For foams, long fibers are defined as those that span multiple cells while short fibers
are those that are confined to the cell struts. In the case of short fibers, each strut can then be
treated as a short-fiber composite, increasing their bending stiffness and thus, improving the
overall properties.
Based on composite micromechanics, fibers reinforce the matrix if the load is transferred
between the matrix and the fiber as a result of interfacial shearing [3]. Based on the shear lag
model, the effectiveness of load transfer is related to the modulus and aspect ratio of the fibers
and the moduli of the matrix as shown in Equations (48) and (49).
𝐸𝑐 = 𝐸𝑚𝑉𝑚 + (1 −tanh (
𝜂𝑐𝑜𝑥𝐿𝑓
2)
𝜂𝑐𝑜𝑥𝐿𝑓
2
)𝐸𝑓𝑉𝑓 Equation 48
𝜂𝑐𝑜𝑥 =1
𝑟[ 2𝐸𝑚
𝐸𝑓(1 + 𝑣𝑚) ln (𝑃𝑓
𝑉𝑓)]
12
Equation 49
Several studies have found that short fiber reinforcement can improve the mechanical
properties of foams under compression. Hussain et al. showed that the addition of glass fibers
resulted in an increase in the compression modulus of foams. This study used different
mechanical models to predict the elastic modulus of the composite foams [4]. Another study
showed that the addition of a small concentration of nanocellulose resulted in an increase in the
112
compressive modulus of the foam [5]. This was attributed to the increased bending stiffness of
the cell struts themselves. Similar observations were also made on the use of titanium oxide and
carbon nanopowder as reinforcing agents [6].
Most of the work conducted on compression of composite foams has been limited to low
strain rates where the behaviour of the material is in a quasi-static compression regime. To the
best of our knowledge, the experimental and theoretical work in the area of high strain rate
compression of fiber-reinforced foams is quite limited, and yet many foams are used in energy
absorption applications such as helmets, where high strain rate deformation is critical.
6.2.3 Strain Rate Dependence in Foams
The compressive properties of polymer foams depend on the rate at which they are
deformed. There are two primary explanations for this phenomenon. Compression of foams
induces intercellular transport of the entrapped air and for rapid compression, the time delay is
not sufficient to allow for dissipation of the air, causing it to act as a stagnant medium with
possible load bearing capacity. In addition, the properties of foam stem from the polymer from
which it was derived, and since polymers are inherently viscoelastic materials, this results in a
strain-rate effect.
Several studies have investigated the strain rate effect in foams. One study observed that
an increase in the strain rate resulted in an increase in the elastic modulus and yield strength of
the material; however, the maximum strain to fracture decreased. It was also observed that the
rate sensitivity increased at higher foam densities [7]. This has been observed in other studies as
well [8-10].
113
A few studies have found composite foams to exhibit strain rate dependent behaviour as
well [11-13]; however, there is insufficient literature that examines the effect of fibers on the
viscoelastic behavior of the foam, and their corresponding effect on the mechanical behaviour
observed at various strain rates.
6.2.4 Strain-Rate Dependence and Stress Relaxation Behavior
As mentioned previously, polymeric foams exhibit strain-rate dependence. One of the
explanations for this is that the properties of the solid material from which a foam is derived
influences its properties. Since polymers are viscoelastic materials, polymeric foams would also
be expected to display viscoelastic behavior such as strain-rate dependence.
Strain-rate dependence refers to the dependence of a viscoelastic material’s properties
(such as modulus and strength) on the rate at which it is deformed. The dependence of a
material’s mechanical properties on strain rate stem from the its stress relaxation behavior. When
a material is loaded at a specific rate, the continuous deformation rate can be modelled as a
discrete function consisting of an incremental application of a strain applied at an increment in
time. The deformation rate can then be approximated as step functions where an instantaneous
strain is applied at each increment in time and then held constant until the next increment (see
Figure 37). This is then similar to several stress relaxation tests being conducted one after the
other.
114
Figure 37. The application of a constant strain rate can be approximated as several unit step functions
For a viscoelastic material, each unit-step increment in the strain is followed by the
relaxation of stress. At the initial application of strain, the material would be stressed, and then
continue to relax. The next increment of strain would result in a stress added to its current
stressed state. This can be further understood by examining Figure 38 where we consider two
materials (Material 1 and Material 2), where Material 2 relaxes slower than Material 1.
For Material 1, the initial application of strain causes the material to almost fully relax
before the next increment in strain is applied. The next application of strain is then applied when
the stress state in the material is very low. This continues until the total application of a strain of
𝜀 would result in a total stressed state of 𝝈𝟏. Comparatively, Material 2 is not fully relaxed
before the next incremental strain is applied, and thus, the total stress in the material (𝝈𝟐) is
higher after an application of the same total strain. Since Hooke’s Law states that the modulus of
the material is a ratio of the stress to the applied strain (𝑬 = 𝝈/𝜺), Material 2 would have a
higher modulus than Material 1 at the same strain rate.
115
Figure 38. The effect of stress relaxation behavior of a material on its modulus at the same strain rate
Similarly, the effect of stress relaxation behavior on strain rate can be better understood
by examining Figure 39 below. If the same Material 1 described previously was subjected to two
strain rates, 𝜺�̇� and 𝜺�̇�, where 𝜺�̇� > 𝜺�̇�. At a higher strain rate, the increments of strain are
applied quicker than at a lower strain rate, and thus, the material does not have sufficient time to
relax completely. As a result, the stress in the material (and its modulus) is higher at a higher
strain rate. These concepts have been expressed in literature and form the basis of the Boltzmann
Superposition Principle.
116
Figure 39. The strain rate dependence of viscoelastic materials stems from its stress relaxation behavior
6.5 Experimental Procedure
6.5.1 Materials
Rigid polyurethane foams were prepared based on a formulation of commercially
available soybean oil-derived polyol, isocyanate, a silicone-based surfactant, and catalysts for
both the blowing and gelling reactions. Distilled water was used as the blowing agent for
foaming of the polyurethanes. Some of the foams produced for this study consisted of varying
concentrations of glass fibers to be evaluated as reinforcement. Table 3 below summarizes the
suppliers and approximate contents (by weight) of each of the constituents in the formulation.
Table 3. Chemical Formulation for PU Foams
Chemical Commercial Name & Supplier Content
Polyol BiOH-X005 by Cargill 100.0
Isocyanate Rubinate by Huntsman 85.0
117
DI water - 1.3
Catalyst Polycat 9 0.3
Catalyst 33LV 2.0
Surfactant DC 5357 1.5
Glass Fibers
9907D by Fibertech (E-glass fibers
with a diameter of 16 microns and a
length of 260 microns)
Variable
6.5.2 Sample Preparation
To produce neat (unreinforced) foam, soy polyol was mixed with the remaining
chemicals excluding isocyanate at ambient conditions for 20 minutes; the isocyanate was then
added in and mixed for an additional minute or until a colour change was observed. The mixture
was then poured into an aluminum mold and allowed to expand via a free rising method. The
sample was cured at room temperature for approximately 24 hours prior to cutting to create
individual specimens.
For the samples prepared with reinforcement, the glass fibers were first added to the
polyol and mixed for 20 minutes prior to the addition of the remaining chemicals. Beyond this,
the method remained the same as for the neat foam. The fiber-filled foams had a longer curing
time, ranging between 7 to 10 days prior to cutting.
6.5.3 Foam Properties
The average neat foam density was measured using the mass and volume of five samples
with approximately constant dimensions. For the fiber-reinforced foams, the density
measurement was a bit more complex.
118
The properties of composite foam are dependent on both foam density and fiber
reinforcement. However, the presence of fibers also changes the foam density due to increased
heterogeneous nucleation and increased viscosity. Unfortunately, due to the complicated
interrelationship between the fiber content and foam density, it is not possible to produce foams
with the same density. This makes it difficult to isolate the effect of fiber reinforcement from the
effect of fiber on foam density. Fortunately, the effect of foam density on the properties of a neat
foam has been quantified in previous models, as shown in Equation (46) and Equation (47).
Thus, once a composite foam is produced, an equivalent neat foam density must be calculated to
account for the structural changes to the base foam induced by the addition of fibers. The ENF
density is calculated as shown in Equation (50) below.
𝜌𝐸𝑁𝐹 =𝑚𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 − 𝑚𝑓𝑖𝑏𝑒𝑟𝑠
𝑉𝑡𝑜𝑡𝑎𝑙
Equation 50
The effect of fibers on foam properties can then be represented by a ratio of the
composite foam properties to those of an equivalent neat foam (e.g. 𝑬𝒄/𝑬𝑬𝑵𝑭); this is called the
reinforcement factor.
6.5.4 Visual Characterization
The samples were characterized by using SEM analysis on a Hitachi SU3500 scanning
electron microscope. The samples were prepared by using a sharp blade to cut the foams into the
desired dimensions and were gold sputtered prior to imaging.
6.5.5 Compression Testing
Uniaxial compression testing was performed in accordance with ASTM D1621-10 using
a Sintech 20 using a 20,000 lb load cell. A displacement was applied onto the plates at a fixed
rate, and the force on the sample was measured. The rates of displacement were 30 mm per
119
minute and 500 mm per minute, producing strains of 1.7E-2 s-1 and 2.8E-1 s-1, respectively. The
dimensions of the samples were measured and used to convert the force-displacement data to
stress-strain curves. The samples were prepared by cutting using a band saw and finished using
fine grit sand paper. The tested samples were rectangular prisms of 5 cm in width, 5 cm in
length, and 3 cm in height.
The dynamic compression testing of the samples was conducted using an Instron
Dynatup tester, inducing a strain rate of approximately 1.1E+5 s-1. These samples were prepared
similar to those used for low speed compression testing.
6.6 Results and Discussion
6.6.1 Foam Morphology
Adding fibers to an uncured polymer mixture can have competing effects. The fiber
surface may act as an area with reduced surface energy available for heterogeneous cell
nucleation. This additional nucleation may either increase cell size or it could also cause adjacent
cells to interact, or coalesce, increasing the apparent cell size of the foam. On the other hand, the
fibers can also increase the viscosity and rigidity of the polymer prior to curing. This can restrict
the expansion of cells during the growth phase. These two competing effects are responsible for
the structure of composite foams. Table 4 summarizes the foam density and the cell sizes
obtained from samples with various fiber contents.
Table 4. Effect of fiber on foam density and cell size
Fiber Weight Fraction ENF Density (g/cm3) Cell Size (micron)
0% 0.107 ± 0.004 492 ± 149
3% 0.102 ± 0.001 1,152 ± 399
5% 0.062 ± 0.001 567 ± 212
120
8% 0.108 ± 0.001 576 ± 222
11% 0.109 ± 0.006 518 ± 234
It was observed that the fibers generally resulted in a negligible change in the density of
the base foam. However, the foams prepared with 5% fibers resulted in a slight decrease in the
base foam density. The effect of the fibers on cell size was also negligible; however, an addition
of 3% fibers resulted in an increase in the cell size. Further investigation is required to better
understand the effect of fibers on cell structure. Although measurement of the cell size was
required to characterize the foam structure, cell size has negligible effect on mechanical
properties [5].
The foams were also visually inspected using scanning electron microscopy, as shown in
Figure 40 below. The cells were isotropic in shape and contained thin membranes in all foams
regardless of the presence of fibers, indicating the fibers to be well-incorporated into the foam
without significant disruption in cell structure.
(a) Neat polyurethane foam
(b) Polyurethane foam containing 3% glass fibers
121
(c) Polyurethane foam containing 5% glass fibres
(d) Polyurethane foam containing 8% glass fibres
(e) Polyurethane foam containing 11% glass fibres
Figure 40. Cellular structure of foams with various fiber contents, as observed under a scanning electron
microscope
6.6.2 Mechanical Properties
For each test, the prepared foams were compressed to a maximum strain beyond the start
of the densification regime. This produced a complete stress-strain curve for each sample, which
was used to determine the elastic modulus in the linear region and the plateau stress.
Neat Foam Properties
Because polyurethane foam is viscoelastic, its properties are expected to vary with strain
rate. Figure 41 displays the effect of strain rate on the properties of neat polyurethane foam. As is
expected for a viscoelastic material, the modulus and plateau stress of polyurethane foam
increased upon increasing strain rate.
122
Figure 41. Increasing the strain rate results in an increase in the modulus and plateau stress of neat
polyurethane foam
As mentioned previously, fibers have a two-fold effect on the properties of foams: the
fibers act as reinforcing agents but also alter the base density of the foam. The effect of fiber
reinforcement can be isolated by calculating the properties of a hypothetical neat foam which has
the same density as the composite foam but does not contain the fibers. The properties of the
equivalent neat foam (ENF) serve as an adjusted baseline to determine the effect of fiber addition
only. As specified by Ashby and Gibson, the density of the foam is related to its modulus and
plateau stress via Equations (46) and (47), where the constants in the equations were determined
via these neat foam experiments. The empirical relationships between foam density and its
modulus and plateau stress are specified in Table 5 below.
10.00
10.50
11.00
11.50
12.00
12.50
1.7E-02 2.8E-01 1.1E+05E
last
ic M
od
ulu
s (M
Pa)
Strain Rate (1/s)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.7E-02 2.8E-01 1.1E+05
Pla
teau
Str
ess
(MP
a)
Strain Rate (1/s)
123
Table 5. Empirical relationships relating elastic modulus and plateau strength to foam density. These
empirical relationships were used to calculate the properties of the equivalent neat foam.
Elastic Modulus Plateau Strength
𝐸(𝜌) = 939 𝜌𝑓2 𝑎𝑡 𝜀̇ = 1.7𝐸 − 2 𝑠−1
𝐸(𝜌) = 942 𝜌𝑓2 𝑎𝑡 𝜀̇ = 2.8𝐸 − 1 𝑠−1
𝐸(𝜌) = 1057 𝜌𝑓2 𝑎𝑡 𝜀̇ = 1.1𝐸 + 5 𝑠−1
𝜎𝑃𝐿(𝜌) = 22 𝜌𝑓1.5 𝑎𝑡 𝜀̇ = 1.7𝐸 − 2 𝑠−1
𝜎𝑃𝐿(𝜌) = 29 𝜌𝑓1.5 𝑎𝑡 𝜀̇ = 2.8𝐸 − 1 𝑠−1
𝜎𝑃𝐿(𝜌) = 30 𝜌𝑓1.5 𝑎𝑡 𝜀̇ = 1.1𝐸 + 5 𝑠−1
Elastic Modulus
Fiber reinforcement in foams can be better understood by calculating a reinforcement
factor, which is a ratio of the properties of a composite foam to those of the equivalent neat foam
(e.g. 𝑬𝒄/𝑬𝑬𝑵𝑭). The properties of the equivalent neat foam were calculated using their densities
and the models developed in Table 5. Figure 42 shows the modulus-based reinforcement factor
of composites containing various fiber contents at different rates of deformation.
Figure 42. The modulus-based reinforcement factor of composite polyurethane foams varied with both fiber
content and strain rate.
0
1
2
3
4
5
6
7
8
9
0% 3% 5% 8% 11%
Rei
nfo
rcem
ent
Fac
tor
Weight fraction of fibers
1.7E-02 (1/s) 2.8E-1 (1/s) 1.1E5 (1/s)
124
Figure 42 shows that regardless of strain rate, foams with a fiber content of 5% had the
highest reinforcement factor. This was expected because the presence of the stiffer fibers
increases the stiffness of the composite foam. It was also observed that a further increase in the
fiber content beyond 5% resulted in a deterioration of the modulus. This could stem from higher
fiber contents resulting in poorer microstructure of the cell struts and increasing their brittleness,
causing the fibers to act as poor reinforcement.
The results in Figure 42 also showed that at all fiber contents, the reinforcement
efficiency increased at higher strain rates. This is non-trivial because although it is expected that
the modulus of a foam would increase with strain rate due to polymeric viscoelasticity, an
increase in the reinforcement efficiency indicates that not only does the modulus increase but the
fibers act as better reinforcing agents at higher fiber contents. Furthermore, the increase in
reinforcement efficiency varies with fiber content, indicating that the fiber content alters the
strain-rate dependence of the foams. The fibers used in this study; however, are perfectly elastic
and are not expected to exhibit any time-dependence independently. One possible explanation
for this observation is that the fibers somehow alter the viscoelastic behavior of the foams.
Particularly, as shown in this thesis, increasing the fiber content slows the stress relaxation
behavior of the foams. As a result, foams with a higher fiber content will exhibit a slower
relaxation of stress and thus, these foams will exhibit a higher modulus at higher strain rates than
a foam deformed at the same strain rate but having a lower fiber content.
In this case; however, increasing the fiber content beyond 5% results in decreased
reinforcement efficiency. As mentioned previously, the structure of foams is complex,
particularly those reinforced with composites, and although the stress relaxation behavior of the
cell wall material may impact the foam’s strain rate dependence, it is important to consider other
125
competing effects of the fibers such as cell rupture and increased friability. Experimental
investigation of the stress relaxation behavior of the solid material contained in the cell struts can
help develop a better understanding of the influence of the strain-rate dependence of composite
foams.
Plateau Stress
The next stage of foam deformation is a long plateau region where the cell struts buckle
in bands until the entire foam has collapsed; the reinforcement factors for plateau stress are
shown in Figure 43. It was observed that compared to the elastic modulus, the plateau stress of
reinforced foams was less dependent on strain rate.
Figure 43. The plateau stress-based reinforcement factor of composite polyurethane foams varied with both
fiber content and strain rate.
Similar to the trends observed for elastic modulus, there appeared to be an optimal fiber
content for the plateau stress as well. In this region, at lower fiber fractions, the reinforcement
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0% 3% 5% 8% 11%
Rei
nfo
rcem
ent
Fac
tor
Weight fraction of fibers
1.7E-2 (1/s) 2.8E-1 (1/s) 1.1E5 (1/s)
126
factors continually increased with increasing fiber content; however, foams with 8% and 11%
fibers had equal or poorer reinforcement compared to the neat foams.
6.7 Conclusions
The modulus and plateau stress of polyurethane foam were found to be strain-rate
dependent, as was expected due to the viscoelastic nature of the solid polyurethane contained in
the cell struts. It was also observed that the plateau stress was less dependent on strain-rate when
compared to the modulus. As well, the reinforcement factors of the foam modulus were much
higher than those of plateau stress, indicating that the fibers had a larger influence in the initial
stage of deformation.
At all fiber contents, the reinforced foams had higher moduli than the unreinforced
foams. The reinforcement factor was also observed to increase with increasing strain rate,
indicating that the fibers acted as better reinforcing agents at higher strain rates. It was also
observed that the increase in the reinforcement factor varied with fiber content as well. This
indicated that changing the fiber content altered the viscoelastic behavior of the foams – an
expected observation because the fibers used in the study were elastic and incapable of
displaying viscoelastic properties on their own. This observation agreed with the hypothesis in
this thesis that foams with higher fiber content exhibit a slower relaxation of stress, and thus,
these foams will exhibit a higher modulus at higher strain rates than those deformed at the same
strain rate but containing a lower fiber content.
Chapter References
1. Chen, Q.; Li, R.; Sun, K.; Li, J.; Liu, C. Preparation of Bio-Degradable Polyurethane Foams
from Liquefied Wheat Straw. Adv. Mater. Res. 2011, 217-218, 1239-1244.
127
2. Gibson, L. J.; Ashby, M. F., Cellular Solids: Structures and Properties; Cambridge University
Press: New York, USA, 1997.
3. Piggott, M. R.; Taplin, D. M. R. Load Bearing Fiber Composites; Pergamon Press: New
York, NY, USA, 1980.
4. Hussain, S; Kortschot, M. T. Polyurethane foam mechanical reinforcement by low-aspect
ratio micro-crystalline cellulose and glass fibres. J. Cell. Plast. 2015, 51(1), 59-73.
5. Li, Y.; Ren, H.; Ragauskas, A. J. Rigid polyurethane foam reinforced with cellulose
whiskers: Synthesis and characterization. Nano-micro Lett. 2010, 2(2), 89-94.
6. Uddin, M. F.; Mahfuz, H.; Zainuddin, S.; Jeelani, S. Infusion of spherical and acicular
nanoparticles into polyurethane foam and their influences on dynamic performances in
Proceedings of the International Symposium on MEMS and Nanotechnology. 2005, 6, 147-
153.
7. Subhash, G.; Liu, Q.; Gao, X. Quasistatic and high strain rate uniaxial compressive response
of polymeric structural foams. Int. J. Impact. Eng. 2006, 32(7), 1113-1126.
8. Chen, W.; Lu, F.; Winfree, N. High-strain-rate compressive behavior of a rigid polyurethane
foam with various densities. Exp. Mech. 2002, 42(1), 65-73.
9. Luong, D. D.; Gupta, N. Compressive properties of closed-cell polyvinyl chloride foams at
low and high strain rates: Experimental investigation and critical review of state of the art.
Comp. Part B. 2013, 44(1), 403-416.
10. Linuel, E.; Marsavina, L.; Voiconi, T.; Sadowski, T. Study of factors influencing the
mechanical properties of polyurethane foams under dynamic compression. J. Phys.:
Conference Series. 2013, 451, 1-4.
128
11. Luo, H.; Zhang, Y.; Wang, B.; Lu, H. Compressive Behavior of Glass Fiber Reinforced
Polyurethane Foam. SEM Annual Conference on Experimental Mechanics, Albuquerque,
NM, 2009.
12. Luo, H.; Zhang, Y.; Wang, B.; Lu, H. J. Characterization of the compressive behavior of
glass fiber reinforced polyurethane foam at different strain rates. Offshore Mech. Arctic Eng.
2010, 132(2), 021301-1-12.
13. Poveda, R. L.; Gupta, N. Carbon-Nanofiber-Reinforced Syntactic Foams: Compressive
Properties and Strain Rate Sensitivity. The J. Miner. Met. & Mater. Soc. 2013, 66(1), 66-77.
133
Chapter 7. Conclusions
The goal of this thesis was to examine the role of fiber reinforcement on the stress
relaxation behavior of short-fiber composites. It has been shown in several previous studies that
the stress relaxation behavior of polymers can be altered by the addition of elastic fibers. This is
counterintuitive because elastic fibers do not exhibit any time-dependence on their own and thus,
they should not alter the time-dependence of a polymer-based composite either. The goal of this
thesis was to understand why elastic fibers changed the stress relaxation behavior of composites.
In this thesis, we investigated the role of short fibers on stress relaxation behavior by
examining the micromechanics at the fiber-matrix interface in composites with short, elastic
fibers. This novel perspective differs from that of previous investigations, which have focused on
attributing the effect of fibers to chemical or structural changes at the interface between the fiber
and the matrix. We aimed to not only understand the role of micromechanics, but also to develop
a simple analytical model that could be used to make predictions regarding the stress relaxation
of composites with varying fiber content, orientations, and aspect ratios.
In the first part of this thesis, we developed an analytical model to predict the stress
relaxation behavior of short-fiber composites with various fiber aspect ratios and fiber volume
fractions. A parametric study based on the model showed that in polymer-matrix composites,
although it is well-established that the polymer matrix relaxes under tension, another equally
important factor is the time-dependent shear stress transfer at the fiber-matrix interface. Since
the shear modulus of the matrix is time-dependent, it results in time-dependent stress transfer
from the matrix to the fiber, causing the short fibers to influence the time-dependence of the
overall composite.
134
The study showed that the predictions from the analytical model matched finite-element
simulations as well as experimental studies conducted on glass fiber-reinforced polypropylene.
Excellent agreement of the simulations and experimental data to the analytical model indicated
that micromechanics play a significant role on the effect of fibers on stress relaxation behavior of
composites. Since the model aligns well with real experimental data, it can be used as a tool to
predict the stress relaxation behavior of various other composites as well.
The analytical model was used to conduct a parametric study of the effect of fiber aspect
ratio on composite stress relaxation. It was observed that increasing the fiber aspect ratio had a
two-fold effect on the stress relaxation behavior of the composites: increased load transfer due to
higher interfacial area and a decrease in the fraction of the fiber length experiencing interfacial
shearing. Since the effect of fibers on the stress relaxation of a short-fiber composite is due to
time-dependent shear stress transfer, there is a critical aspect ratio at which the fiber is long
enough to make a significant contribution to the properties of the composite but also sufficiently
short for the interfacial shear zones to be relevant. This is defined as the critical aspect ratio for
viscoelasticity. This was a novel contribution because although the concept of a critical fiber
length (or aspect ratio) is widely used with respect to the strength, modulus, and toughness of
short fiber composites, it had not been defined for viscoelasticity. Based on this finding, it can be
recommended that for applications requiring slower relaxation of stress, the critical aspect ratio
for viscoelasticity should be used. On the other hand, in applications that require for faster
relaxation of stress, fibers with an aspect ratio higher than the critical aspect ratio does not
provide much benefit over very low aspect ratios.
Previous studies investigating the effect of short fibers on the stress relaxation behavior
of a composite had attributed changes in the relaxation time constant to increased covalent
135
bonding at the fiber-matrix interface. This hypothesis was investigated by evaluating the stress
relaxation behavior of glass-reinforced polypropylene containing MAPP as a coupling agent, and
then comparing the experimental findings to predictions made using the analytical model. It was
found that even when MAPP was added to the system to alter the fiber/matrix interface, the
experimental data remained well-aligned with the analytical model predictions that do not
depend on modelling interfacial changes. Since the analytical model predictions align quite
closely with the experimental data, it was concluded that most of the stress relaxation behavior of
a composite can be predicted using a simple model incorporating the time-dependent matrix
modulus and the time dependent shear stress transfer efficiency.
The effect of fiber orientation on the stress relaxation behavior of short-fiber composites
was also studied. An analytical model was developed to predict the stress relaxation behavior of
composites containing randomly oriented elastic fibers. The analytical model was validated using
finite element experiments and excellent agreement was observed between the analytical model
and experiments at fiber volume fractions below 30%. The results from the simulations of these
composites were compared to those for oriented composites. It was found that the misorientation
of fibers shrinks the contribution of the time-dependent shear stress transfer by a factor of one-
third compared to the contribution of oriented fibers. It is recommended that if the fibers are
being used to prevent the relaxation of stress, it is better to take measures to improve the fiber
orientation rather than increasing the fiber content or fiber aspect ratio. Using perfect orientation
allows the fibers to impede stress relaxation three-times better than if they were randomly
oriented.
The analytical model is subject to several assumptions that can influence its accuracy.
Since the model is based on Cox’s shear lag, the assumptions of this model must be valid. These
136
include perfect interfacial bonding between the matrix and the fiber and that the fibers do not
carry any tensile stress. The composite must also have a narrow distribution of fiber length. The
model only approximates the behavior of the composite within the linear viscoelastic region.
Lastly, the model assumes that the Poisson’s ratio of the composite is constant and does not vary
with time.
The last part of this thesis focuses on an application of composite stress relaxation
behavior to investigate the strain-rate dependence of short fiber-reinforced foams. It was
assumed that the strain-rate dependence of a composite foam stem from the viscoelastic
properties of the solid material from which the foam is derived. However, there is currently no
adequate analytical model which can predict the viscoelastic properties of composites. Thus, the
analytical model presented here can be used to understand the viscoelastic properties of
composite foams as well. In this part of the thesis, the strain rate-dependent properties of glass-
fiber reinforced polyurethane foams were investigated. At all fiber contents, the reinforced foams
had higher moduli than the unreinforced foams. Both the modulus and plateau stress of
polyurethane foam were found to be strain-rate dependent; however, the plateau stress was less
dependent on strain-rate than the modulus.
It was found that the strain-rate dependence of the composites varied with fiber content,
which indicated that changing the fiber content altered the viscoelastic behavior of the foams.
This observation agreed with the hypothesis in this thesis that foams with higher fiber content
exhibit a slower relaxation of stress, and thus, these foams will exhibit a higher modulus at
higher strain rates than those deformed at the same strain rate but containing a lower fiber
content.
137
Chapter 8. Recommendations
The work in this thesis can be used to identify areas that require further investigation.
Since the analytical model predictions are in good agreement with both experimental data and
the results from finite-element simulations, this form of analysis is a suitable foundation for other
models. It would be particularly useful if this work could be expanded to account for a
viscoelastic fiber contained in a viscoelastic matrix. With increasing interest in sustainable
materials, many researchers are using plant and wood-based fibers as reinforcement in polymer-
based composites. Since natural fibers are viscoelastic, the new analytical model would have to
account for three factors: the time-dependency of the matrix, the time-dependency of the fiber,
and the time-dependent stress transfer from the matrix to the fiber.
The analytical model predictions could be compared to other fiber/matrix systems to
determine its validity for other cases such as polyurethanes and other thermosets. It would also
be beneficial to further investigate the effect of aspect ratio. This study introduced the novel idea
of a critical aspect ratio for viscoelasticity, and it would be beneficial to analyze this
experimentally. Experimental work of this type; however, is challenging since the properties are
dependent on the post-processing fiber aspect ratio, which is difficult to control.
The study could also be extended to investigate hybrid systems such as for composites
with both longer and shorter fibers. This thesis showed that longer fibers decrease the total
amount of deformation while fibers at the critical aspect ratio decrease the rate at which
deformation occurs, a combination of the two fiber types could result in composites that
exhibited minimum stress relaxation. Studies in this area could prove beneficial for applications
where stress relaxation is a significant issue. Investigating other hybrid systems such as a
combination of viscoelastic and elastic fibers could be beneficial as well.
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Stress relaxation behavior can also be used to investigate other viscoelastic properties
such as strain-rate dependence in a tensile test. Further investigations could use the model in
conjunction with the Boltzmann Superposition Principle to develop a model to predict the
modulus of composites at various strain rates. If such an analytical model was validated
experimentally or via simulations, it could prove beneficial for several applications.
Finally, an analytical model of this type can help control stress relaxation, and thus,
control the strain-rate dependence of composites using very simple parameters such as fiber
volume fraction and fiber aspect ratio, allowing for better tailoring of materials for specific
applications. In the case of composite foams, it would be particularly useful to determine the
stress relaxation behavior of the cell wall material and using that to understand and predict the
modulus of the foam and how it varies with strain rate.
