Embed Size (px)
Citation preview
Understanding and Improving the Elastic Compressive
Modulus of Fibre Reinforced Soy-Based Polyurethane
Foams
By:
Sadakat Hussain
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 Sadakat Hussain 2014
ii
Understanding and Improving the Elastic Compressive Modulus
of Fibre Reinforced Soy-Based Polyurethane Foams
Sadakat Hussain
Doctor of Philosophy
Department of Chemical Engineering and Applied Chemistry
University of Toronto
2014
Abstract
Soy-based polyurethane foams (PUFs) were reinforced with fibres of different aspect
ratios to improve the compressive modulus. Each of the three fibre types reinforced PUF
differently. Shorter micro-crystalline cellulose fibres were found embedded inside the cell struts
of PUF and reinforced them. The reinforcement was attributed to be stress transfer from the
matrix to the fibre by comparing the experimental results to those predicted by micro-mechanical
models for short fibre reinforced composites. The reinforced cell struts increased the overall
compressive modulus of the foam. Longer glass fibres (470 microns, length) provided the best
reinforcement. These fibres were found to be larger than the cell diameters. The micro-
mechanical models could not predict the reinforcement provided by the longer glass fibres. The
models predicted negligible reinforcement because the very low modulus PUF should not
transfer load to the higher modulus fibres. However, using a finite element model, it was
determined that the fibres were providing reinforcement through direct fibre interaction with
each other. Intermediate length glass fibres (260 microns, length) were found to poorly reinforce
iii
the PUF and should be avoided. These fibres were too short to interact with each other and were
on average too large to embed and reinforce cell struts.
In order to produce natural fibre reinforced PUFs in the future, a novel device was
invented. The purpose of the device is to deliver natural fibres at a constant mass flow rate. The
device was found to consistently meter individual loose natural fibre tufts at a mass flow rate of 2
grams per second. However, the device is not robust and requires further development to deliver
a fine stream of natural fibre that can mix and interact with the curing polymeric components of
PUF. A design plan was proposed to address the remaining issues with the device.
iv
Acknowledgments
I would like to acknowledge my supervisor, Professor Mark Kortschot for his guidance and
support. As well, my reading committee members, Professor Glen Hibbard, Professor Chandra
Veer Singh and Professor Ning Yan for their feedback and suggestions for improvement. My lab
mates for their support. Particularly, Ricky Chang who developed the foam samples used in this
study. Brandon Haw for generating the images used in the compression study inside the X-ray
Tomography system. My other lab mates for their useful advice throughout the research process,
Billy Cheng, Adrew Barquin, Zach Fishman and Numaira Obaid. Mr. Shiang Law for his
assistance in training me in using the various equipment in our laboratory. Finally, my wife and
family for their love and support.
v
Table of Contents
Acknowledgments ..................................................................................................................... iv
Table of Contents ....................................................................................................................... v
List of Tables ............................................................................................................................ ix
List of Figures ............................................................................................................................ x
List of Appendices ................................................................................................................ xviii
Nomenclature .......................................................................................................................... xix
Chapter 1 Introduction ................................................................................................................ 1
1.1 Overview of thesis .......................................................................................................... 1
1.2 Thesis Objectives ............................................................................................................ 3
1.3 Limitations of Work ........................................................................................................ 4
Chapter 2 Literature review ........................................................................................................ 6
2.1 Polyurethane Foams (PUFs) ............................................................................................ 6
2.2 Structural insulated panels (SIPs) .................................................................................... 9
2.2.1 Expected failure modes for SIPs ........................................................................ 11
2.2.2 Global and Local Buckling in Sandwich Panels ................................................. 12
2.2.3 Rationale for improving Compressive and Shear Modulus of the foam core ...... 16
2.3 Natural PUFs ................................................................................................................ 16
2.4 Fibre reinforced foams .................................................................................................. 17
2.5 Introduction to micromechanical models to predict the compressive modulus of fibre
reinforced foams ........................................................................................................... 19
2.5.1 Rule of mixtures (ROM) equation ..................................................................... 21
2.5.2 Inverse rule of mixtures (IROM) equation ......................................................... 22
2.5.3 Cox’s shear-lag theory ....................................................................................... 24
2.5.4 Nairn’s shear-lag theory .................................................................................... 27
vi
2.5.5 Three-dimensional orientation factor ................................................................. 27
2.5.6 Eshelby’s equivalent inclusion and Mori-Tanaka theory .................................... 30
2.5.7 Percolation theory ............................................................................................. 35
2.6 Conclusions .................................................................................................................. 37
Chapter 3 Sample preparation and testing ................................................................................. 38
3.1 Polyurethane foam component materials ....................................................................... 38
3.2 Sample preparation ....................................................................................................... 39
3.3 Compression testing ...................................................................................................... 40
3.4 X-ray Tomography analysis .......................................................................................... 41
3.5 Fibre and foam properties ............................................................................................. 45
3.6 Conclusions .................................................................................................................. 47
Chapter 4 The effects of cell architecture and fibre aspect ratio on the compressive modulus
of PUFs ............................................................................................................................... 48
4.1 Gibson-Ashby theory for foam mechanics .................................................................... 48
4.2 Modified Gibson-Ashby equation using volume fraction of solids in foam (VFS) .......... 52
4.3 The effect of fibres on the volume fraction of solids (VFS) ............................................ 53
4.4 Assumptions on foam reinforcement and fibre placement based on fibre aspect ratio
and cell architecture ...................................................................................................... 54
4.4.1 Short-fibre reinforced foams .............................................................................. 55
4.4.2 Long-fibre reinforced foams .............................................................................. 56
4.4.3 Intermediate-fibre reinforced foams ................................................................... 60
4.5 Conclusions .................................................................................................................. 63
Chapter 5 Analyzing the compressive properties of micro crystalline cellulose (MCC) fibre
reinforced PUFs ................................................................................................................... 64
5.1 Introduction .................................................................................................................. 64
5.2 Micro-structure of MCC fibre reinforced PUFs ............................................................. 64
5.3 Changes to the MCC fibre-foam composite compressive modulus with increases in
MCC fibre weight fraction ............................................................................................ 68
vii
5.4 Conclusions .................................................................................................................. 73
Chapter 6 Analyzing the compressive properties of long-fibre reinforced PUFs ........................ 74
6.1 Introduction .................................................................................................................. 74
6.2 Long glass fibre reinforced PUFs .................................................................................. 74
6.2.1 Micro-structure of long glass fibre reinforced PUFs .......................................... 74
6.2.2 Changes to long glass fibre-foam composite compressive modulus with
increases in long glass fibre weight fraction ....................................................... 78
6.2.3 Geometric percolation of long glass fibre .......................................................... 83
6.3 Predicting the compressive modulus of long glass fibre reinforced PUFs using finite
element modelling (FEM) ............................................................................................. 90
6.3.1 Model overview and parameters ........................................................................ 90
6.3.2 Mesh Verification .............................................................................................. 93
6.3.3 Comparing the strain energy of an isolated fibre and a networked fibre using
FEM analysis .................................................................................................... 97
6.3.4 Compressive modulus prediction from FEM analysis ....................................... 101
6.4 Hemp fibre reinforced PUFs ........................................................................................ 103
6.5 Conclusions ................................................................................................................. 106
Chapter 7 Analyzing the compressive properties of intermediate length glass fibre reinforced
PUFs ................................................................................................................................... 108
7.1 Introduction ................................................................................................................. 108
7.2 Micro-structure of intermediate length glass fibre reinforced PUFs .............................. 108
7.3 Changes to intermediate glass fibre foam composite compressive modulus with
increases in intermediate glass fibre weight fraction ..................................................... 113
7.4 FEM analysis theorizing fibre reinforcement due to fibre-cell bridging ........................ 119
7.5 Conclusions ................................................................................................................. 122
Chapter 8 Novel method to meter and deliver natural fibre for reinforcement of PUFs ............ 124
8.1 Introduction ................................................................................................................. 124
8.2 Existing art .................................................................................................................. 126
viii
8.3 Metering natural fibre .................................................................................................. 128
8.3.1 Concept description .......................................................................................... 129
8.3.2 Mathematical relationships ............................................................................... 131
8.4 Theoretical modelling through simulations .................................................................. 135
8.5 Results for prototype fibre metering device .................................................................. 141
8.6 Limitations and ideas for complete system ................................................................... 143
8.7 Conclusions ................................................................................................................. 145
Chapter 9 Conclusions and Recommendations ......................................................................... 147
9.1 Conclusions ................................................................................................................. 147
9.2 Recommendations ........................................................................................................ 149
Bibliography ........................................................................................................................... 151
Appendices .............................................................................................................................. 158
ix
List of Tables
Table 1: Three categories for PUFs based on common identifiers. Adapted from [5] and [6]. ..... 7
Table 2: Polyurethane foam chemical formulation used. ........................................................... 38
Table 3: Material properties required for physical and analytical modelling .............................. 45
Table 4: Dimensions of different fibre types used and cell diameter of fibre-foam composites. N
= 100. ....................................................................................................................................... 46
Table 5: Sum of squared error normalized to compare micro-mechanical model predictions to the
actual experimental results ....................................................................................................... 72
Table 6: Sum of squared error normalized to compare micro-mechanical model predictions and
FEM simulation results to the actual experimental results for long glass fibre-foam composites
............................................................................................................................................... 102
Table 7: Sum of squared error normalized to compare micro-mechanical model predictions to the
actual experimental results for intermediate glass fibre-foam composites where the fibres are
assumed to reinforce cell struts. .............................................................................................. 117
Table 8: Sum of squared error normalized to compare micro-mechanical model predictions to the
actual experimental results for intermediate glass fibre-foam composites where the fibres are
assumed to reinforce a homogenous PUF matrix .................................................................... 119
Table 9: Results for the simulation to determine the nip gap (D). ............................................ 137
Table 10: Results from the second part of the simulation for fibre distribution. ....................... 139
Table 11: Results of mass flow rate from the developed prototype. ......................................... 142
x
List of Figures
Figure 1: PUF cellular structure. Image taken with X-ray Tomography. 2.0 micron resolution
used. ........................................................................................................................................... 1
Figure 2: PUF used as insulation foam and seat cushions in the automotive industry [1] [2]. ...... 2
Figure 3: Foams used to form the core in rigid structural panels for home building [3]. .............. 3
Figure 4: Structural Insulated Panel with an insulating core bonded to two face panels ............. 10
Figure 5: Components of a sandwich panel where c = core thickness and f = face sheet thickness
and b = width. Depth is represented by the thickness c and f. .................................................... 13
Figure 6: General global buckling of a sandwich panel ............................................................. 14
Figure 7: Local buckling of the skins in a sandwich panel ........................................................ 15
Figure 8: Schematic outlining the structural scenario for ROM. Fibre spans entire composite and
aligned longitudinally with the applied load. Arrows indicate applied load. .............................. 21
Figure 9: Schematic outlining the structural scenario for IROM. Fibre spans entire composite and
aligned perpendicular to the applied load. Arrows indicate applied load. .................................. 23
Figure 10: Shear-stress lines shown transferring load into fibre. Fibre is debonded at the ends. 25
Figure 11: Stress conditions in fibre. Tensile stress is a maximum at fibre centre and shear stress
is a maximum at fibre ends. ...................................................................................................... 26
Figure 12: Fibre of length Lf and angle θ that crosses the scan line and is not aligned to the
loading direction. ...................................................................................................................... 28
Figure 13: Homogenous inclusion undergoes transformation strain as a separate body and is then
embedded back into matrix as a bonded entity. ......................................................................... 31
xi
Figure 14: A- Low volume fraction, no percolation. B - Some fibre interaction. C - High volume
fraction, percolated network of fibres formed D – Low fibre aspect ratio with high volume
fraction, no percolation. [69] .................................................................................................... 36
Figure 15: Diagram of foam sample preparation. ...................................................................... 40
Figure 16: Two-dimensional shadow image of a fibre-foam composite. .................................... 42
Figure 17: A two-dimensional projection of the three-dimensional structure of a fibre-foam
composite. The thicker dark cylinders represent fibres. Thin struts represent the polyurethane
foam structure. ......................................................................................................................... 43
Figure 18: SkyScan Material Testing Stage (MTS-50N) used to compress samples for imaging.
................................................................................................................................................. 44
Figure 19: Three regions in PUF compressive deformation ....................................................... 48
Figure 20: Simple cubic structure denotes foam cell structure. Figure displays before and after
load is applied to the cell structure scenarios. ........................................................................... 49
Figure 21: The solid density of the pure foam is the density of polyurethane. However, the solid
density of the composite foam is unknown. .............................................................................. 52
Figure 22: Short fibres embedded within cell struts .................................................................. 55
Figure 23: Hemp fibres that are much larger than cell diameters spanning multiple cells .......... 57
Figure 24: 470 micron glass fibres that are larger than a single cell diameter are not embedded
within the cell struts ................................................................................................................. 57
Figure 25: For the long fibre case, cell structure will be ignored and treated as a homogenous
matrix with the same elastic modulus ....................................................................................... 58
Figure 26: Long fibres are embedded in homogenous matrix for physical modelling
considerations .......................................................................................................................... 59
Figure 27: Fibres bridging individual cells in different orientations .......................................... 61
xii
Figure 28: Fibres not bridging individual cells in different orientations..................................... 62
Figure 29: 5 wt % (0.08 volume %) MCC fibres are mostly embedded in cell struts. Fibres do not
span cells. Note: depth of field of image is 1.0 mm ................................................................... 65
Figure 30: 20 wt % (0.36 volume %) MCC fibres. MCC fibres are not seen. ............................ 66
Figure 31: 20 wt % (0.36 volume %) MCC fibres. MCC fibres are not seen. ............................ 66
Figure 32: 20 wt % (0.36 volume %) MCC fibres. MCC fibres are not seen. ............................ 67
Figure 33: 20 wt % (0.36 volume %) MCC fibres. SEM image showing individual cell strut that
has MCC fibres embedded in it. Fibres are considered to be more aligned than randomly
oriented. ................................................................................................................................... 67
Figure 34: Volume fraction of solids decreases as MCC fibre weight percentage increases. ...... 69
Figure 35: No-fibre base foam modulus expected through Gibson-Ashby theory (based on
volume fraction of solids at given fibre wt %) compared to experimental results. ..................... 69
Figure 36 a and b: Compressive modulus of MCC fibre-foam composite samples in comparison
to micro-mechanical models. .................................................................................................... 71
Figure 37: 20 wt % (0.3 volume %) 470 micron glass fibre embedded in foam. Intact Fibres span
multiple cells. Note: depth of field of image is 2.0 mm ............................................................. 76
Figure 38: 5 wt % (0.07 volume %) 470 micron glass fibre embedded in foam. Intact Fibres span
multiple cells. Note: depth of field of image is 2.0 mm. Arrows indicate Fibre bundles. ........... 76
Figure 39: 10 wt % (0.12 volume %) 470 micron glass fibre embedded in foam. Intact Fibres
span multiple cells. Note: depth of field of image is 2.0 mm. Arrows indicate Fibre bundles. ... 77
Figure 40: 15 wt % (0.19 volume %) 470 micron glass fibre embedded in foam. Intact Fibres
span multiple cells. Note: depth of field of image is 2.0 mm. Arrows indicate Fibre bundles. ... 77
Figure 41: Volume fraction of solids decreases as long glass fibre weight percentage increases 79
xiii
Figure 42: No-fibre base foam modulus expected through Gibson-Ashby theory compared to
experimental results. Significant fibre effect observed. ............................................................. 79
Figure 43 a and b: Compressive modulus of MCC fibre-foam composite samples in comparison
to micro-mechanical models. .................................................................................................... 81
Figure 44: 20 wt % (0.3 volume %) 470 micron glass fibre embedded in foam. Densely packed
fibres are not isolated. Note: depth of field of image is 2.0 mm. ................................................ 85
Figure 45: 20 wt % (0.3 volume %) 470 micron glass fibre embedded in foam. Densely packed
fibres demonstrate are not isolated. Note: depth of field of image is 2.0 mm. ............................ 85
Figure 46: 5 wt % (0.07 volume %) 470 micron glass fibre embedded in foam. Fibres are not
densely packed. Some fibre interaction. Note: depth of field of image is 1.8 mm. ..................... 86
Figure 47: 20 wt % (0.3 volume %) 470 micron glass fibre embedded in foam. Fibre that is
marked by arrow bends during compression due to interaction with fibre. Mechanical
percolation. Note: depth of field of image is 1.6 mm. Only fibres shown as PUF were removed
with threshold imaging techniques. ........................................................................................... 87
Figure 48: 20 wt % (0.3 volume %) 0.470 mm glass fibre embedded in foam. Fibre that is
marked by arrow bends during compression due to interaction with fibre. Mechanical
percolation. Note: depth of field of image is 1.6 mm. Only fibres shown as PUF were removed
with threshold imaging techniques. ........................................................................................... 88
Figure 49: 20 wt % (0.3 volume %) 0.470 mm glass fibre embedded in foam. Uncompressed
foam. Depth of field of image is 1.6 mm. Only fibres shown. ................................................... 89
Figure 50: 20 wt % (0.3 volume %) 0.470 mm glass fibre embedded in foam. Note: depth of field
of image is 1.6 mm. Only fibres shown as PUF were removed with threshold imaging
techniques. Increasing compression from top left (10%) to bottom right (60%). ....................... 89
Figure 51: Compression of fibre-foam composite samples in FEM simulation. Rigid Bodies are
solid and do not deform. For rendering purposes it is shown as hollow. .................................... 91
xiv
Figure 52: Nodal regions with embedded objects. Node A of the embedded object shares node a
of the host object. Image taken from the ABAQUS 6.1.2 Theory Manual. ................................ 93
Figure 53: Mesh 1 and Mesh 3 provided the same results. Mesh 1 is coarser and requires less
processing time......................................................................................................................... 95
Figure 54: Mesh 2. The mesh size is too coarse. Fibres are smaller than the hexagonal mesh
blocks. The mesh blocks distort during compression................................................................. 95
Figure 55: Mesh 1. The mesh size is suitable. Fibres are larger than the hexagonal mesh blocks.
The mesh blocks do not distort during compression. ................................................................. 96
Figure 56: Mesh 3. The mesh size is very fine. Fibres are larger than the hexagonal mesh blocks.
The mesh blocks do not distort during compression. However, the processing time is much
longer than Mesh 1 but this does not improve upon the results. ................................................. 96
Figure 57: ABAQUS set-up showing fibre scenarios. Fibre 1 is a close-contact fibre where three
other fibres are in close contact to it. Fibre 2 is an isolated fibre. .............................................. 98
Figure 58: Fibre 1 – close contact fibre has significantly higher strain energy than Fibre 2 –
isolated fibre after the simulation completes. FEM Simulation. ................................................ 99
Figure 59: Total strain energy and total fibre strain energy at 5 % compressive deformation. FEM
Simulation. ............................................................................................................................. 100
Figure 60: Fibre strain energy percentage contribution to total system strain energy at 5 %
compressive deformation. FEM Simulation. ........................................................................... 101
Figure 61: Compressive modulus of long glass fibre-foam composite samples in comparison to
micro-mechanical models and FEM simulations. .................................................................... 102
Figure 62: Hemp fibre-foam composite sample highlighting the long hemp fibres and large pores
present. ................................................................................................................................... 104
Figure 63: 5 wt % hemp fibre-foam composite. Hemp fibre not shown. Note: depth of field of
image is 2.0 mm. .................................................................................................................... 105
xv
Figure 64:5 wt % hemp fibre-foam composite. Hemp fibre not shown. Note: depth of field of
image is 1.0 mm. Image thresholding techniques used to isolate PU struts. ............................. 106
Figure 65: 5 wt % (0.07 volume %) 260 micron glass fibres embedded in foam. Fibres not
confined to cell struts. Darker and thicker cylindrical objects are fibres. The image contrast levels
have been adjusted to show the glass fibres more clearly. Note: depth of field of image is 0.8
mm. ........................................................................................................................................ 109
Figure 66: 5 wt % (0.07 volume %) 260 um Glass fibres embedded in foam. Fibres not confined
to cell struts. Darker and thicker cylindrical objects are fibres. Thinner and lighter objects are the
cell structure. The image contrast levels have been adjusted to show the glass fibres more clearly
and to display how they are much larger than the cell struts but not cell diameters. Note: depth of
field of image is 0.8 mm. ........................................................................................................ 110
Figure 67: 20 wt % (0.3 volume %) 260 um Glass fibres embedded in foam. Fibres not confined
to cell struts. Darker and thicker cylindrical objects are fibres. Thinner and lighter objects are the
cell structure. The image contrast levels have been adjusted to show the glass fibres more clearly
and to display how they are much larger than the cell struts but not cell diameters. Note: depth of
field of image is 0.8 mm. ........................................................................................................ 111
Figure 68: 5 wt % (0.07 volume %) 0.260 mm glass fibre embedded in foam. Fibres are not
densely packed. Very limited fibre interaction. Depth of field of image is 0.8 mm. ................. 112
Figure 69: 20 wt % (0.3 volume %) 0.260 mm glass fibre embedded in foam. Fibres are not as
densely packed as 0.470 mm glass fibre (long fibre) case at 20 wt % (0.3 volume %). Limited
fibre interaction. Depth of field of image is 0.8 mm. ............................................................... 112
Figure 70: Volume fraction of solids decreases as intermediate glass fibre weight percentage
increases. ................................................................................................................................ 114
Figure 71: No-fibre base foam modulus expected through Gibson-Ashby theory to experimental
results. .................................................................................................................................... 114
xvi
Figure 72 a and b: Compressive modulus of intermediate glass fibre-foam composite samples in
comparison to micro-mechanical models where glass fibres are assumed to reinforce cell struts.
............................................................................................................................................... 116
Figure 73 a and b: Compressive modulus of intermediate glass fibre-foam composite samples
where glass fibres are assumed to reinforce a homogenous PUF matrix. ................................. 118
Figure 74: Seven fibre bridging cases investigated with FEM analysis. Vertical loading. ........ 120
Figure 75: Dimensions of cell and fibre used in FEM analysis ................................................ 120
Figure 76: Deformation of intermediate cases with vertical loading. Images taken directly from
ABAQUS. .............................................................................................................................. 121
Figure 77: Force-Displacement curves for each of the seven cases. ......................................... 122
Figure 78: Maple fibre is clearly concentrated to end of the PUF sample. Region with more fibre
was stiffer. Scale bar is 2 cm. ................................................................................................. 125
Figure 79: Fibre being placed into cup of foam mixture for hand mixing. ............................... 125
Figure 80: After hand mixing in viscous foam mixture, fibre is not evenly distributed. ........... 126
Figure 81: Typical chopper gun where stream of metered chopper glass is released to interact in
air with polymeric streams [86]. Image used directly from source. .......................................... 127
Figure 82: Loose natural fibre. Added manufacturing steps required to make into yarn for
chopping. ............................................................................................................................... 128
Figure 83: Natural fibre approaching rolling wheels. Driver wheel is powered to rotate counter-
clockwise and forces driven wheel to rotate clockwise. Driven wheel can move only horizontally.
Applied force on driven wheel forces it to contact the fixed driver wheel. Frame of apparatus not
shown. Direction rotations are based on a front view .............................................................. 129
Figure 84: Residual natural fibre separating rolling wheels. Applied force on driven wheel
compresses residual natural fibre. Direction rotations are based on a front view. .................... 130
xvii
Figure 85: Applied Pressure (kPa) and Bulk Density (kg/m3) of hemp fibre at a mass of 2 grams.
Note: curve is the median curve of the five samples tested. ..................................................... 132
Figure 86: Important dimensions to consider when fibre is compressed between roller wheels.
Frame of apparatus not shown. Front view shown. T is the width of the roller and is
perpendicular to the front view. .............................................................................................. 133
Figure 87: Sections of fibre blocks between the wheels. Block 2 is compressed by the wheels.
Blocks are not drawn to scale. ................................................................................................ 136
Figure 88: Mock SIP divided into equal sized boxes. The simulation was used to deposit blocks
of fibre into each deposition zone. .......................................................................................... 138
Figure 89: Prototype used for fibre mass metering including an LVDT, pneumatic air cylinder
and roller wheels. ................................................................................................................... 141
Figure 90: Design Apparatus of the complete set-up to collect, loosen, meter and deliver natural
fibre as a stream with a constant mass flow rate. Note: The image is not to scale and some
components have been enlarged to show functionality. Ideally, the components will be compact
and placed into a box for portability. ...................................................................................... 144
xviii
List of Appendices
Appendix 1 ............................................................................................................................. 158
Appendix 2 ............................................................................................................................. 159
xix
Nomenclature
A cross sectional area of fibre in roll nip
Ac composite cross-sectional area
AE Eshelby’s stress concentration factor
Af fibre cross-sectional area
AH Hill’s stress concentration factor
Am matrix cross-sectional area
AMT Mori-Tanaka’s stress concentration factor
AR aspect ratio
b width of beam in moment of inertia equation
c core thickness of sandwich panel
CF elastic modulus of fibre/inclusion
CM elastic modulus of matrix
CPP orientation correction factor
d depth of beam in moment of inertia equation
D nip gap distance
Dc total deformation in composite
xx
Df deformation contribution by fibre
Dm deformation contribution by matrix
E elastic modulus of a given material
E* effective elastic modulus of foam
Ec composite elastic modulus
Ef fibre elastic modulus
Em matrix elastic modulus
EPU elastic modulus of polyurethane
ERS elastic modulus of reinforced cell strut
f face sheet thickness of sandwich panel
F applied load
Fc total force in composite
Ff force contribution by fibre
Fm force contribution by matrix
g(ϴ) probability density function
Gf shear modulus of fibre
Gm shear modulus of matrix
xxi
H height of fibre-wheel interaction
I moment of inertia
J identity matrix
K constant in Euler equation
Kf bulk modulus of fibre
Km bulk modulus of matrix
L length of object in Euler equation
Lc total length in composite
Lf fibre length
Lm matrix length
LS strut length in Gibson-Ashby equation
M mass of fibre in roll nip
Mfoam mass of foam
MFR desired mass flow rate of fibre in grams per second
MPU mass of polyurethane in foam
n correction factor in shear-lag equation
P applied pressure on fibres in roll nip
xxii
Pe Euler load for buckling
Pf fibre packing factor
r fibre radius
RS rotational speed of driver wheel in revolutions per minute
S Eshelby’s tensor
t strut thickness in Gibson-Ashby equation
T width of roller
V volume of fibre in roll nip
Vf volume fraction of fibre in composite
Vfoam volume of foam
VFS volume fraction of solids in foam
Vm volume fraction of matrix in composite
VPU volume of polyurethane in foam
Z constant of proportionality in Gibson-Ashby equation
α largest angle fibre makes with respect to scan line
δ deflection of a beam
ε global strain
xxiii
εA uniform applied strain
εAVG average strain in composite
εc composite strain
εC perturbed strain in inclusion
εC(x) perturbed strain in matrix
εf fibre strain
εFAVG average strain in fibre/inclusion
εm matrix strain
εMAVG average strain in matrix
εT transformation strain in inclusion
ϴ angle fibre makes with respect to scan line
λ proportionally ratio of the circumference of the roller wheels to the (H) height of fibre
being compressed (i.e. the dimension of the compressed fibre that contacts the wheel surface).
νm Poisson’s ratio of matrix
νs Poisson’s ratio of matrix in Jayaraman and Kortschot equation
ρ bulk density
ρ* apparently density of foam
xxiv
ρpu density of solid polyurethane
ρs density of solid material in Gibson-Ashby equation
σAVG average stress in composite
σc stress in composite
σel elastic stress (the stress which occurs at the onset of buckling)
σf stress in fibre
σFAVG average stress in fibre/inclusion
σI inclusion stress
σM matrix stress
σm stress in matrix
σMAVG average stress in matrix
1
Chapter 1
Introduction
1.1 Overview of thesis
In recent years, there has been a push to replace petroleum-based products, and the
market for renewable polymers is rapidly growing. Polyurethane foams (PUFs) are made by
curing petroleum-sourced polyol with excess isocyanate through addition polymerization. A
foaming agent such as distilled water is used to react with the surplus isocyanate to produce
carbon dioxide gas that creates a cellular structure in the polyurethane. Figure 1 shows a typical
PUF cellular structure.
Figure 1: PUF cellular structure. Image taken with X-ray Tomography. 2.0 micron
resolution used.
In this study, the polyol component for PUF is sourced from the oil found in soya beans and will
be referred to as soyol. Soya beans are naturally occurring, renewable and provide an alternative
2
to synthetic polyols that are produced from crude oil. With the depletion of oil reserves and the
fluctuating cost of petroleum; having an alternative natural source for polyols for use in PUF
production is necessary.
The cellular structure of PUF allows for it to be used in a wide range of applications.
Some of the applications for PUF include being used as shock absorbers and cushioning for seats
and for structural applications. Figure 2 highlights some of the uses for PUF.
Figure 2: PUF used as insulation foam and seat cushions in the automotive industry [1] [2].
A new use for PUF is in forming the core material of structural insulated panels, which are
panels that can be used as walls. These panels offer excellent insulating properties due to the
PUF core. As well, the PUF core provides rigidity for the overall panel. Figure 3 shows foam
being used to form the core in a rigid panel for use in home building.
3
Figure 3: Foams used to form the core in rigid structural panels for home building [3].
Soyol based PUFs would be desirable to be used in structural insulated panels due to the
environmentally friendly, lightweight and cost-effective properties associated with them.
However, these structural insulated panels experience excessive compression and bending loads
and therefore must resist buckling and localized skin wrinkling effects. In particular, the PUF
core must have excellent stiffness under compression and shear loading.
1.2 Thesis Objectives
The main purpose of this project was to reinforce PUFs with fibres (natural and glass) to
increase the stiffness of the foam under compression. Fibres have been widely used to reinforce
soft thermoplastics, increasing the rigidity with increased fibre content. These composite
materials (fibre reinforced thermoplastics) are commonly found in high strength low weight
applications which include panels in airplanes and cars. Traditionally, glass, carbon and Kevlar
fibre have been used to reinforce thermoplastics. These synthetic fibres have high rigidity and
can be readily produced on a large scale to accommodate the production volumes in the
automotive and aerospace sectors. Recently, natural fibres have emerged as a viable reinforcing
4
filler for thermoplastics. Natural fibres such as maple, hemp and aspen are abundant, cost-
effective and have lower densities than comparable synthetic fibres [4]. These attributes are
important because as the fibres improve the rigidity of composites, the weight and cost does not
increase significantly.
Many physical and analytical models have been developed to understand and explain
how fibres reinforce a weaker solid matrix. Typically, the weaker matrix will transfer loading
into the fibre resulting in fibre deformation. Deformation of the higher modulus fibre results in
an increase in overall composite material rigidity. However, no significant work has been done
on developing physical models to explain how fibres reinforce foams. In this study, various fibre
types (natural and glass) at different fibre lengths and fibre content levels were embedded inside
soyol based PUF. Compression testing, finite element modelling and microstructure analysis
were completed in order to develop a physical model to explain fibre reinforcement of very low
density and modulus cellular materials.
Initially, it was hypothesized where fibres of different aspect ratios are placed inside the
foam cell structure and how that fibre replacement affects foam reinforcement. Using
microscopic tools (X-ray Tomography and Scanning Electron Microscopy), the microstructure is
visualized to verify the fibre placement hypotheses. Finally, after compression testing, micro-
mechanical models and finite element modelling were used to understand how fibres of distinct
aspect ratios and fibre placement inside the cell structure reinforce PUF.
1.3 Limitations of Work
In this study, the foams synthesized were open cell PUFs. These foams are not ideal for
insulation and therefore the focus of the analysis was solely on the mechanical properties
5
(modulus) of the composite foams developed. Furthermore, the failure of these reinforced foams
was not studied. As well, glass fibre was used as a model fibre and for its ability to be
distinguished from polyurethane in X-ray analysis. Composite samples beyond 20 wt % fibre
could not be synthesized due to high viscosity during fibre mixing, limiting the reinforcement
capability. Finally, environmental degradation effects on foam samples were not studied and
should be completed in the future.
6
Chapter 2
Literature review
2.1 Polyurethane Foams (PUFs)
Polyurethanes are produced by reacting excess isocyanate with polyol in an addition
reaction. The chemical formulas for isocyanate and polyol and the polyurethane linkage that
forms from the reaction are shown below.
Isocyanate 𝑶 = 𝑪 = 𝑵 − 𝑹
The isocyanate group (O = C = N) and R group – Typically an aromatic ring
Polyol 𝑶𝑯 − 𝑪𝑯𝟐 − 𝑹𝟏
The polyol group and R1 group – repeated unit
Polyurethane 𝑹 − 𝑵𝑯 − 𝑪𝑶𝑶 − 𝑪𝑯𝟐 − 𝑹𝟏
Polyurethane chemical structure with urethane linkage
In order to create a foamed structure, distilled water is introduced to react with isocyanate to
form urea linkages and carbon dioxide (CO2). Carbon dioxide gas creates bubbles within the
solid polyurethane polymer to create the foamed structure. PUFs have both urethane linkages
and urea linkages. PUF formation is shown below.
7
Gas formation 𝑶 = 𝑪 = 𝑵 − 𝑹 + 𝑯𝟐𝑶 → 𝑹 − 𝑵𝑯𝟐 + 𝑪𝑶𝟐 (𝒈𝒂𝒔)
The formation of carbon dioxide gas.
Urea formation 𝑶 = 𝑪 = 𝑵 − 𝑹 + 𝑹 − 𝑵𝑯𝟐 → 𝑹 − 𝑵𝑯 − 𝑪𝑶 − 𝑵𝑯 − 𝑹
The formation of urea from isocyanate and an amine group.
PUFs are divided into three categories based on rigidity: rigid, semi-rigid and flexible
foams [5][6][7]. Rigid PUFs are typically found in structural applications such as sandwich
panels in walls, skis and snowboards. Semi-rigid PUFs can be used as trim in automobile
interiors and seating. Flexible PUFs can be used in vibration dampening and shock absorption.
Table 1 highlights the average elastic modulus associated with each type of PUF. The rigidity of
PUF is based on the type and content of isocyanate (functionality and index) and polyol (OH
No.) present in the chemical formulation.
Table 1: Three categories for PUFs based on common identifiers. Adapted from [5] and [6].
Rigid Semi-Rigid Flexible
OH No. 350 – 560 100 – 200 5.6 – 70
Isocyanate
Functionality
3.0 – 8.0 3.0 – 3.5 2.0 – 3.1
Elastic Modulus
(MPa)
>700 700 – 70 < 70
8
The chemical formulation of isocyanate has a significant impact on the rigidity of PUF.
The R group in isocyanate is typically a highly reactive aromatic ring. Aliphatic rings are also
possible but are less commonly used due to lower reactivity. Based on the chemical formulation
of the aromatic ring it will either form the molecule methylenediphenyl diisocyanate (MDI) or
toluene diisocyanate (TDI) when it is combined with the isocyanate group (N=C=O). In this
study, MDI will be used since TDI is chemically volatile when used in enclosed spaces which
will be required for wall insulation [7]. As well, MDI is more cost-effective. The aromatic rings
have multiple isocyanate groups attached where the isocyanate functionality (number of reactive
isocyanate group molecules bonded to the aromatic ring, moles of reactive isocyanate groups
divided by moles of MDI) is typically at an average of 2.7 for flexible foams and in the range of
2.0 to 3.1 as shown in table 1 [7]. Another common indicator for isocyanate content is the
isocyanate index, which is the percentage of the actual amount (in moles) of isocyanate used
divided by the theoretical amount (in moles) of isocyanate required for a 1:1 ratio with the polyol
content. Higher isocyanate index values indicates more isocyanate groups available for reactions.
The isocyanate index for PUFs is typically in the range of 105 to 115 [6]. Having multiple
reactive isocyanate groups attached increases cross-linking potential with polyol components.
The cross-linking increases the rigidity and toughness of the polyurethane. Therefore, increasing
the isocyanate content is one way to increase the rigidity and elastic modulus of PUFs.
Polyols are long flexible segments in urethane linkages and give soft elastic properties to
PUF. Polyols are typically sourced from petroleum. Increasing the polyol content and lowering
the isocyanate content and the degree of cross-linking will decrease the rigidity and elastic
modulus of PUF.. The hydroxyl number (OH No.) is a measure of the amount of repeated OH
groups found in the polyol. It is measured as the milligrams of potassium hydroxide (KOH)
equivalent to the OH content in one gram of polyol. It is measured through titration of KOH into
9
a solution of polyol that has a known excess amount of acetic acid. Some of the acetyl groups
will bond to the polyol OH groups and the remaining acetic acid can be neutralized by the
titration of KOH. This will then allow for the calculation of the hydroxyl number. The higher
the hydroxyl number, the more likelihood of cross-linking between the polyols and isocyanates
since there are more reactive OH groups available for bonding. Therefore, PUFs with higher
hydroxyl numbers have higher rigidity due to an increase in cross-linking. Table 1 shows the
typical OH numbers for the three categories of PUFs.
Surfactants such as polysiloxane are used to promote the nucleation and stabilization of
gas bubbles. They lower the surface tension of carbon dioxide gas in polyurethane to encourage
foaming and bubble formation. They help regulate cell size and provide colloidal stability.
Furthermore, they act as emulsifiers to aid in the mixing of the polyol with isocyanate and
catalysts.
2.2 Structural insulated panels (SIPs)
Structural insulated panels (SIPs) are sandwich panels that are composed of two facings
such as oriented strand board, separated by a foamed plastic insulation core. They provide both
insulation and load-bearing capabilities. A typical SIP is shown in Figure 4.
10
Figure 4: Structural Insulated Panel with an insulating core bonded to two face panels
The foamed core is usually either expanded polystyrene or PUF. The faces can also be made
from metal sheets or fibre composites giving the panels more strength and stiffness.
SIPs are one of the fastest growing products for use in home building in the United
States, where their use doubled between 1997 and 2006. In 2006 alone, 43 million square feet of
SIPs were used in home building [8]. The energy savings and the simplicity during installation
associated with using SIPs are the main reasons for this growth. For example, in comparison to
the traditional fibreglass batting home insulation products, SIPs outperform in trapping air inside
the home. The difference in thermal resistance capability can be attributed to the envelope that
SIPs create as opposed to the pocket structure of fibreglass bats. Furthermore, it is also easier to
install SIPs as they come pre-assembled. SIPs avoid the time-consuming tasks of traditional
home building that includes framing and sheathing stick framed walls.
11
SIPs also provide rigidity and strength to the structure of homes. They must resist
bending and buckling under loading from the roof and top floors of the house. As well, they must
be able to withstand high-winds. For these reasons it is imperative to improve the rigidity and
buckling resistance of SIPs. By enhancing these properties, opportunities for SIPs to be used in
new applications that require higher rigidity and good insulation would emerge while preventing
failure in current applications. The following sub-section discusses the mechanics involved for
SIPs.
2.2.1 Expected failure modes for SIPs
Many studies have addressed the mechanics of SIPs and sandwich panels. Mousa and
Uddin [9] developed SIPs (8 feet by 4 feet by 1 foot) with expanded polystyrene as the core and
glass-polypropylene face sheets and compressed them vertically to mimic the loading that SIPs
experience when they are used as walls in homes. They found that the mode of failure was face
sheet wrinkling and local buckling. In studies by Daniel et al [10] and Gdoutos et al [11], they
determined that sandwich panel failures are likely to occur from either core failure or face
wrinkling and are based on the span length of the beam. Shorter beams experienced core failures
due to shear and compressive loadings, which occur when the panels bend or buckle globally.
Longer beams experienced face sheet wrinkling effects due to localized buckling. With localized
buckling, the sheets can either buckle outwards or inwards. In this study the focus will be on
when the sheets buckle inwards. When the sheets buckle inwards, they crush the foam core with
shear and compressive deformation. The following section provides further details about the
failure modes and the desired properties of the foam core to resist wrinkling failures.
12
2.2.2 Global and Local Buckling in Sandwich Panels
SIPs are sandwich panels. Other products that are made up of sandwich panels include
skis, snowboards, and corrugated cardboard. Each of these sandwich panels has two faces that
are stiff and are separated by a lightweight core. The advantage of adding a lightweight core is
the improvement to the second moment of inertia, which improves the buckling resistance of
sandwich panels without significantly increasing the overall weight of the structure. The
following series of equations highlight the improvement of the second moment of inertia and
buckling resistance. Equation 1 shows the moment of inertia for a solid beam.
,
Equation 1: 𝑰 = 𝒃𝒅𝟑
𝟏𝟐
Euler’s Formula for the second moment of inertia in a rectangular member. I = moment of
inertia, d = depth and b = width [12]
As can be noted from Equation 1, an increase in the depth or width of the beam will increase its
moment of inertia. Figure 5 shows a sandwich panel with the dimensions of the skins and the
core.
13
Figure 5: Components of a sandwich panel where c = core thickness and f = face sheet
thickness and b = width. Depth is represented by the thickness c and f.
The moment of inertia for a sandwich panel is based on equation 1 and the composite area
theorem. It is assumed that the inner core of the sandwich panel has zero modulus in comparison
to the modulus of the face sheets. Therefore, the sandwich panel can be treated as a beam with a
missing core in the middle. Equation 2 shows the manipulation to determine the moment of
inertia for a sandwich panel.
Equation 2: 𝑰 = (𝒃𝒅𝟑
𝟏𝟐) 𝒘𝒉𝒐𝒍𝒆 − (
𝒃𝒅𝟑
𝟏𝟐) 𝒊𝒏𝒏𝒆𝒓 𝒄𝒐𝒓𝒆
Using Figure 5, the depth outside (the whole panel) is equal to 2f + c and the depth of the inner
core (inside) is equal to c. Furthermore, it can be assumed that f is negligible in size in
comparison to c, leading to the result in Equation 3.
Equation 3: 𝑰 = (𝒃(𝟐𝒇+𝒄)𝟑
𝟏𝟐) − (
𝒃(𝒄)𝟑
𝟏𝟐) =
𝒃
𝟏𝟐(𝟖𝒇𝟑 + 𝟏𝟐𝒇𝟐𝒄 + 𝟔𝒄𝟐𝒇) ≈ 𝒃𝒇𝟐𝒄 +
𝒃𝒄𝟐𝒇
𝟐≈
𝒃𝒄𝟐𝒇
𝟐
14
Therefore, Equation 3 suggests that for sandwich panels that have skins of fixed thickness, as the
core thickness increases the second moment of inertia of the panel increases by a power of 2.
Buckling resistance is directly proportional to the second moment of inertia. SIPs must
resist buckling under loading from the top floors and the roof of a home. Buckling instability is a
phenomenon that occurs when a straight and slender object that is fixed at its ends experiences
excessive compressive loads and is allowed to bend. The object bends to a point where it
“buckles” in the transverse direction. Figure 6 shows general global buckling of a sandwich
panel that is under axial compressive loading.
Figure 6: General global buckling of a sandwich panel
The critical load that causes the object to buckle is called the Euler load and is defined in
Equation 4.
15
Equation 4: 𝑷𝒆 = 𝝅𝟐𝑬𝑰
𝑲𝑳𝟐
The Euler load (Pe) for buckling where E = elastic modulus/stiffness of object, I = Second
moment of inertia, K = constant and L = length of object.
Equation 4 suggests that by increasing the second moment of inertia the resistance to global
buckling is improved. Equation 3 suggests that introducing a foam core into a sandwich panel
increases the second moment of inertia and thus improve the buckling resistance. Furthermore,
increasing the stiffness of the foam core, the buckling resistance will also improve.
Another possible form of buckling can occur in the faces of SIPs. This is known as local
wrinkling/buckling effects where the skins of sandwich panels warp in shape under compressive
loadings. Figure 7 highlights local buckling of sandwich panels/SIPs under axial compressive
loading.
Figure 7: Local buckling of the skins in a sandwich panel
16
The skins, when very long and slender can easily buckle under compressive loading. In this
project, the concern is on the inward buckling that places compressive and shear loadings on the
foam core. The core of SIPs inhibits wrinkling by the skins by resisting the bending of the skins
in the transverse direction.
2.2.3 Rationale for improving Compressive and Shear Modulus of the
foam core
It is expected that if global buckling occurs, the sandwich panel will bend and the foam
core could fail due to excessive shear and compressive deformation. As well, in the scenario for
local buckling, once a critical load is reached the skins can buckle inwards and crush the interior
core in compression and shear [10] . This leads to failure of the sandwich panel. In order to
resist the deformation, high shear and compressive modulus for the PUF are desired. By having a
higher modulus, greater loads will be required to deform the foam during bending. Specifically,
Yeh [8] developed a standard for SIPs where it is recommended that the foam core of SIPs must
have high compressive and shear modulus.
In this project, fibres will be introduced into PUF to increase its compressive modulus.
This will hinder global and localized skin buckling by increasing the stiffness of the PUF in the
transverse direction and thus increase the load needed for buckling.
2.3 Natural PUFs
PUFs are made by reacting isocyanate with polyol in an addition polymerization reaction.
The cellular structure is created by adding a foaming agent such as distilled water to generate gas
bubbles. Traditionally, the polyol is sourced as a by-product from petroleum production.
17
However, with rising energy costs and depleting petroleum reserves, many alternatives to
petroleum sourced polyols have been studied [13]–[24]. These alternatives are renewable and
include polyols derived from castor and canola based oils. In this study the polyol is sourced
from soybeans and is known as soyol. Soyol derived PUFs have been synthetized successfully
[25]–[29]. However, the compressive stress at 10 % compressive strain for the soybean based
PUF formulation identical to the one used in this study is lower than that required by the SIP
standard [8], [25]. The SIP standard suggests a stress minimum value of 110 kPa, where as the
value in the literature reported is 40 kPa for neat PUF. As discussed earlier, one way to improve
the rigidity of PUFs is to increase the isocyanate group content (index and functionality).
However, isocyanate is a non-renewable component and therefore should be limited in a foam
derived from renewable polyol sources. Another method of increasing the compressive stiffness
of PUF, while maintaining low isocyanate content, is to add fibre reinforcement. The following
section discusses the reinforcement and increase in rigidity of foams due to the addition of fibres.
2.4 Fibre reinforced foams
PUFs are lightweight and cost-effective making them highly attractive to be used in SIPs
as the core material. However, as discussed in the previous section, the soy based PUF used in
this study lacks acceptable rigidity for use in SIPs. Therefore, the main goal is to reinforce the
soy based PUFs with fibres to increase its stiffness so that it can be used in the future in SIPs. In
this project, glass fibres, micro-crystalline cellulose and natural fibres (hemp fibre) were used to
reinforce soy based PUF. Glass fibres have been used widely to reinforce solid polymers
successfully and are commonly found in automotive and aerospace products. Micro-crystalline
cellulose fibres are shorter fibres and were analyzed to determine if they reinforce PUFs
differently in comparison to the longer glass fibres. Natural and wood fibres are materials created
18
in the wood and plant processing business, which includes lumber development, textiles and pulp
and paper processing. More importantly, these fibres are an excellent reinforcing component for
polymers. They contain cellulose, lignin, and hemicellulose and have a high modulus and
excellent strength properties and form bonds readily with common polymer functional groups.
These attributes are important because the fibres will improve the rigidity and buckling
resistance of SIPs, but the weight and cost will not increase. Furthermore, in PUF, these natural
fibres can act as fillers and increase the overall "green" content.
Numerous attempts have been made to reinforce polymeric foams with fibres [25], [30]–
[44]. Gu et al [25] (using a PUF composition identical to that used here) discovered that the
compressive strength (defined as compressive loading at 10 % compressive strain) of PUF can
increase by a factor of two when wood fibres are introduced. Doroudiani and Kortschot [30]
determined that increasing wood fibre content enhanced the tensile modulus and impact strength
of microcellular polystyrene foam composites. They observed an increase in impact strength by a
factor of three when 20 weight percent of cherry hardwood fibre was added. Many other studies
reported increases in mechanical properties when adding fibres to PUF. Bledzki et al [43],
observed increases in shear modulus and impact strength with an increase in natural fibre
content. Cotgreave and Shortall [32] observed an increase in tensile modulus and strength with
the increase of glass fibre content. Finally, Siegmann et al [33], found increases in tensile
modulus, and compressive modulus with the introduction of glass fibre and powder to PUF.
Based on the results from the literature, introducing fibres into PUFs should increase the stiffness
in compression. However, the previous studies do not include comprehensive physical models to
explain how the fibres improve the modulus of foams. In this study, the focus is on
understanding the physical reinforcement of low density soy based PUF by fibres.
19
2.5 Introduction to micromechanical models to predict the
compressive modulus of fibre reinforced foams
Micromechanical models are mathematical models that can predict the elastic modulus
and strength properties of composite materials. In this project, the micromechanical models
pertaining to elastic modulus will be explored. These analytical models are physical models, as
opposed to empirical models, since they explain the physical mechanisms involved in the
reinforcement of a softer matrix by stiff fibres. Having an analytical model that can predict the
elastic modulus reduces the need for expensive experiments.
The significant physical properties used in micromechanical models include the elastic
modulus (E), the volume fractions of constituents (V), the fibre aspect ratio (AR) and the
Poisson’s ratio (ν).
The following micromechanical models will be used to predict the compressive modulus
of fibre reinforced foams:
1. Rule of mixtures (ROM)
2. Inverse rule of mixtures (IROM)
3. Cox’s shear-lag theory [45][46]
4. Nairn’s shear-lag model [47]
5. Mori-Tanaka model [48]
6. Mechanical percolation theory [49], [50]
20
The micromechanical models (1-5) have been used previously to predict the tensile modulus of
short fibre reinforced thermoplastics [51]–[55] with some success. However, they have not been
used to predict the tensile or compressive modulus of fibre reinforced foams. These models all
assume that the fibres are well bonded to the matrix, aligned and are well dispersed. The models
assume that there is elastic stress transfer between the fibre and the matrix and that the matrix
and fibre do not yield and there is no slip. The models assume that the fibres are not interacting
with each other and are isolated in pockets of matrix. These models help in understanding
physically how fibres reinforce PUFs. The ROM models the ideal reinforcement scenario and
provides the largest increase to the compressive modulus of fibre reinforced PUF. The IROM
provides the worst case reinforcement scenario for any well bonded fibre-matrix composite. The
samples generated in this project produce compressive modulus results that are below the results
predicted by the ROM but higher than the results predicted by the IROM. Shear-lag theory and
Mori-Tanaka theory provides models that can predict how short-fibres (fibres that do not span
the length of the specimen) reinforce a weaker matrix. In this project, short fibres are used.
Mechanical percolation theory has been used to predict the shear modulus of fibre
reinforced low stiffness matrix composites before [49], [50]. However, it has not been applied to
predict reinforcement in cellular materials. Mechanical percolation theory assumes that high
aspect-ratio (AR) fibres at low volume fractions interact with close contact. As the fibre content
increases, fibres form networks of close contact throughout the matrix. During compression of
the composite matrix, these networked fibres will bend each other. This bending of fibres will
provide reinforcement to the matrix since more energy is required to deform the very stiff fibres
in comparison to the weak matrix.
21
2.5.1 Rule of mixtures (ROM) equation
The rule of mixtures (ROM) was developed by Voigt in 1887 and is the simplest
micromechanical model for explaining fibre reinforcement in a composite. It is based on an ideal
fibre reinforcement scenario, where fibres span the entire length of the composite and are
oriented in the loading direction as shown in Figure 8. The composite modulus predicted by
ROM is the highest possible elastic modulus for a composite material.
Figure 8: Schematic outlining the structural scenario for ROM. Fibre spans entire
composite and aligned longitudinally with the applied load. Arrows indicate applied load.
The ROM assumes that the strain exhibited by the composite globally would equal the
strain experienced in the fibres and the matrix as shown in Equation 5.
Equation 5: 𝜺𝒎 = 𝜺𝒇 = 𝜺𝒄
The total force (FC) being applied to the composite in Figure 8 is the summation of the force on
the fibre and the matrix as shown in Equation 6.
22
Equation 6: 𝑭𝒄 = 𝑭𝒇 + 𝑭𝒎
Equation 6 can be re-written to include stress and area terms.
Equation 7: 𝝈𝒄𝑨𝒄 = 𝝈𝒇𝑨𝒇 + 𝝈𝒎𝑨𝒎
The total area is divided out to gain the volume fraction terms. It is assumed that the through
thickness of the composite in Figure 8 is homogenous and can be ignored to change area
fractions to volume fractions.
Equation 8: 𝝈𝒄 = 𝑽𝒎𝝈𝒎 + 𝑽𝒇𝝈𝒇
By substituting Hooke's law = 𝜎 into Equation 8, and the assumption from Equation 5, the
ROM equation can be developed as shown in Equation 9.
Equation 9: 𝑬𝒄 = 𝑽𝒎𝑬𝒎 + 𝑽𝒇𝑬𝒇
Equation 9 suggests that the ROM predicts the modulus of the composite to be based on the
moduli of the fibre and the matrix and their respective volume fractions. This model represents
an ideal upper bound case where any well-bonded fibre reinforced composite modulus predicted
cannot exceed the values predicted by the ROM.
2.5.2 Inverse rule of mixtures (IROM) equation
The inverse rule of mixtures (IROM) was developed by Reuss and is another
micromechanical model for explaining fibre reinforcement in a composite. It is the worst case
fibre reinforcement scenario for any well-bonded fibre matrix composite. The fibres are assumed
23
to span the entire length of the composite but are oriented perpendicular to the loading direction
as shown in Figure 9. The composite modulus predicted by IROM will always be the lowest
possible modulus for a composite material with well-bonded fibres.
Figure 9: Schematic outlining the structural scenario for IROM. Fibre spans entire
composite and aligned perpendicular to the applied load. Arrows indicate applied load.
The IROM assumes that the stress exhibited by the composite globally would equal the
stress experienced in the fibres and the matrix as shown in Equation 10.
Equation 10: 𝝈𝒎 = 𝝈𝒇 = 𝝈𝒄
The total deformation (DC) of the composite in Figure 9 is the summation of the deformation on
the fibre and the matrix when loading is applied as shown in Equation 11.
Equation 11: 𝑫𝒄 = 𝑫𝒇 + 𝑫𝒎
Equation 11 can be re-written to include strain and length terms.
24
Equation 12: 𝜺𝒄𝑳𝒄 = 𝜺𝒇𝑳𝒇 + 𝜺𝒎𝑳𝒎
The total length is divided out to gain the volume fraction terms. It is assumed the composite in
Figure 9 is homogenous and the other dimensions can be ignored to alter length fractions to
volume fractions. The total strain on the composite is the summation of the strain on the matrix
and the fibre proportioned to their volume fractions as shown in Equation 13.
Equation 13: 𝜺𝒄 = 𝑽𝒎𝜺𝒎 + 𝑽𝒇𝜺𝒇
By substituting Hooke's law = 𝜎 into Equation 13, and the assumption from Equation 10, the
IROM equation can be developed as shown in Equation 14.
Equation 14: 𝑬𝒄 = 𝑬𝒇𝑬𝒎
𝑽𝒎𝑬𝒇+𝑽𝒇𝑬𝒎
Equation 14 suggests that the IROM predicts the modulus of the composite to be related to the
moduli of the fibre and the matrix and their respective volume fractions. This model represents a
lower bound case where any well-bonded fibre reinforced composite modulus predicted cannot
be lower than the values predicted by the ROM.
2.5.3 Cox’s shear-lag theory
Shear-lag theory is a commonly accepted micro-mechanical model to describe the fibre
reinforcement of short fibre reinforced composites. The theory was developed by Cox [46] in
1954 and was one of the first analytical models for short fibre composites. It was modified for
fibre packing considerations later by Piggott [45]. Cox’s work is based on understanding the
elasticity of paper and other fibrous materials. The theory assumes that the matrix transfers the
25
applied load to the fibre through interfacial shear stresses as shown in Figure 10. The fibre is
debonded at its ends and is isolated and not interacting with neighbouring fibres. The fibre is
stretched with shear stresses from the matrix with a maximum shear stress load at the ends of the
fibre and a maximum tensile stress at the centre of the fibre as shown in Figure 11. The transfer
of applied load from the matrix to the fibre is assumed to occur with no slippage and no yielding
of either the matrix or the fibre. Equation 15 below shows the classic shear-lag model as defined
in Piggott’s work [45].
Figure 10: Shear-stress lines shown transferring load into fibre. Fibre is debonded at the
ends.
26
Figure 11: Stress conditions in fibre. Tensile stress is a maximum at fibre centre and shear
stress is a maximum at fibre ends.
Equation 15: 𝑬𝒄 = 𝑬𝒇 (𝟏 −𝒕𝒂𝒏𝒉(
𝒏𝑳𝒇
𝟐)
𝒏𝑳𝒇
𝟐
) 𝑽𝒇 + 𝑬𝒎𝑽𝒎
Where Lf is fibre length and n is a correction factor.
The correction factor accounts for lower reinforcement provided by short fibres that do not span
the length of the composite. The correction factor is based on an assumption that the fibre is a
cylinder embedded in a cylinder of matrix whose outer surface experiences the global strain of
the composite. These fibre cylinders are assumed to be well dispered and interact with the matrix
and not other fibre cylinders. Equation 16 highights the correction factor developed by Cox.
Equation 16: 𝒏 =𝟏
𝒓[
𝟐𝑬𝒎
𝑬𝒇(𝟏+𝒗𝒎)𝑳𝒏(𝑷𝒇
𝑽𝒇)]
𝟏/𝟐
27
Where 𝑣𝑚 – poisson’s ratio of matrix, and Pf – packing factor assumed to be (2π/√3) for this
study.
2.5.4 Nairn’s shear-lag theory
The assumptions made by Cox and the accuracy of his shear-lag theory were analyzed in
a study by Nairn [47]. Nairn determined a new correction factor based on the exact elasticity
equations for axisymmetric stress states in axial and radial directions for transversely isotropic
materials. The study highlights that Cox ignored the radial displacements in his fundamental
shear-lag assumption. The new correction factor was found to predict within 20 % of average
axial stress and verified with finite element models. Equation 17 highlights the correction factor
developed by Nairn.
Equation 17: 𝒏 = [𝟐
𝒓𝟐𝑬𝒇𝑬𝒎[
𝑬𝒇𝑽𝒇+𝑬𝒎𝑽𝒎
𝑽𝒎𝟒𝑮𝒇
+𝟏
𝟐𝑮𝒎[
𝟏
𝑽𝒎𝒍𝒏(
𝟏
𝑽𝒇)−𝟏−
𝑽𝒎𝟐
]
]]
𝟏/𝟐
In this study, the shear-lag theory correction factors of both Cox and Nairn will be used.
2.5.5 Three-dimensional orientation factor
The micro-mechanical models (ROM, IROM and Shear-lag theory) discussed assume
that the embedded fibres are perfectly aligned. However, in this study the fibre-foam composites
will have fibres that are randomly oriented in three-dimensions after the foam samples cure.
When fibres are not unidirectional (and aligned with the loading direction) the reinforcement
ability is decreased. To account for this decrease in composite modulus, an orientation correction
28
factor (CPP) is multiplied by the fibre modulus (Ef) to account for the decrease in fibre
contribution to the composite modulus.
Jayaraman and Kortschot [56] completed a study where they analyzed the orientation
correction factor for fibres randomly oriented in two-dimensions. They used the paper physics
approach developed by Cox where fibres cross an arbitrary scan line in a two-dimensional
composite and experiences loading. The scan line is shown in Figure 12 with a misaligned fibre
crosses it.
Figure 12: Fibre of length Lf and angle θ that crosses the scan line and is not aligned to the
loading direction.
The paper physics approach initially involves determining the number of fibres of length L and
orientation θ that cross the scan line and calculating the axial loading in each fibre. The next step
involves finding the directional components of the axial loading, particularly the component in
29
the direction of loading (x-direction). The next step involves multiplying the total number of
fibres that cross the scan line with their respective x-direction loading components. The final step
involves integrating the multiple of fibre and x-direction component over fibre length and fibre
orientation to determine the total load experienced by the fibres that cross the scan line. The
orientation and loading integrals can be separated as they are not dependent on each other to find
the equation for the orientation factor. A full derivation of the analysis can be found in
Jayaraman and Kortschot’s work [56]. The integral result for the orientation factor is shown in
Equation 18.
Equation 18: 𝑪𝑷𝑷 = ∫ 𝒈(𝜽)𝝅
𝟐⁄
𝟎(𝒄𝒐𝒔𝟒𝜽 − 𝝂𝒔𝒔𝒊𝒏𝟐𝜽𝒄𝒐𝒔𝟐𝜽)𝒅𝜽
Where g(θ) is a probability density function for orientation in two-dimensions and νs is the
Poisson’s ratio of the matrix being deformed. The probability density function g(θ) is equal to
1/α when 0 < θ < α and equals 0 when θ > α.
Jayaraman and Kortschot found that the orientation correction factor (CPP) is equal to 1/3 when
fibres are randomly oriented in two-dimensions (i.e. when α equals 90 degrees or π/2 radians).
For three-dimensionally randomly oriented fibres, Piggott’s manuscript on short-fibre reinforced
composites [57] suggests an orientation correction factor of 1/5. Kobari and Kortschot [58]
experimentally determined that the orientation factor is 1/5 for short wood fibres embedded in a
polypropylene matrix that has been compression moulded. Therefore, based on these results of
short fibres randomly oriented in a three-dimensional matrix, the CPP was assumed to be 1/5 in
this study.
30
2.5.6 Eshelby’s equivalent inclusion and Mori-Tanaka theory
Mori-Tanaka theory is another commonly accepted micro-mechanical model to predict
the improvements in elastic modulus of composite materials [48]. The work of Mori-Tanaka has
been analyzed and simplified for specific cases in many studies [53], [59]–[64]. Mori-Tanaka
theory is based on Eshelby’s equivalent inclusion theory [65]. Eshelby’s work involved solving
for the elastic stress field around an ellipsoidal particle that is perfectly bonded to an infinite
matrix. The ellipsoidal particle is the inclusion or reinforcing fibre in the matrix. This varies
from shear-lag theory which assumes that embedded fibres are cylinder and de-bonded from the
matrix at the ends. Based on the ellipsoidal particle assumption and ignoring fibre packing
considerations, Eshelby’s theory is limited in its accuracy at higher volume fractions [53]. Mori-
Tanaka theory improves on Eshelby’s theory and will be used as a micro-mechanical model for
analysis in this study.
The first part of Eshelby’s work involves a homogenous inclusion (same properties as the
matrix) undergoing a phase transformation as a separate body and experiencing a uniform strain
εT (transformation strain). Figure 13 highlights the scenario.
31
Figure 13: Homogenous inclusion undergoes transformation strain as a separate body and
is then embedded back into matrix as a bonded entity.
Once the transformed inclusion is placed inside the matrix, the previously unstrained matrix
undergoes some deformation since the inclusion and the matrix are now bounded. This
complicated perturbed strain εC(x) is relative to the original shape of the matrix and dependent on
location. The stress σM experienced by matrix can be determined through Hooke’s law as shown
in Equation 19.
Equation 19: 𝝈𝑴(𝒙) = 𝑪𝑴𝜺𝑪(𝒙)
Where CM is the matrix elastic modulus.
Similarly, the stress in the inclusion σI can be determined, however, the transformation strain is
subtracted from the perturbed strain since it does not place additional strain into the inclusion.
Equation 20 highlights the stress experienced in the inclusion.
Equation 20: 𝝈𝑰 = 𝑪𝑴(𝜺𝑪 − 𝜺𝑻)
Eshelby deduced from these relationships that the complicated perturbed strain must be related to
the transformation strain of the inclusion through a tensor known as Eshelby’s tensor S.
32
Equation 21: 𝜺𝑪 = 𝑺𝜺𝑻
Eshelby’s tensor is dependent on the inclusion’s aspect ratio and the matrix elastic properties.
The second portion of Eshelby’s work involved studying the scenario when the inclusion
is inhomogeneous (having different mechanical properties than the matrix) and finding a
relationship for rigidity of the composite. Eshelby compared the inhomogeneous inclusion case
to the homogenous inclusion case. The comparison study involved when a uniform strain εA is
applied to both cases, the stress on the respective inclusions will be different as shown in
Equation 22 (homogenous case) and Equation 23 (inhomogeneous case). The inhomogeneous
inclusion does not undergo a transformation, has a different elastic modulus and still forces the
matrix to experience a perturbed strain.
Equation 22: 𝝈𝑰 = 𝑪𝑴(𝜺𝑨 + 𝜺𝑪 − 𝜺𝑻)
Equation 23: 𝝈𝑰 = 𝑪𝑭(𝜺𝑨 + 𝜺𝑪)
By equating the two inclusion cases, the relationship between them is shown in Equation 24.
Equation 24: 𝑪𝑭(𝜺𝑨 + 𝜺𝑪) = 𝑪𝑴(𝜺𝑨 + 𝜺𝑪 − 𝜺𝑻)
The next portion of Eshelby’s work involved relating the stiffness properties of the matrix and
the inclusion to average applied strain values that could be related back to the rule of averages
for composite materials. Equation 25 shows the average applied strain for the entire composite
and Equation 26 addresses the average strain in the inclusion/fibre that is uniform.
Equation 25: 𝜺𝑨𝑽𝑮 = 𝜺𝑨
Equation 26: 𝜺𝑨𝑽𝑮𝑭 = 𝜺𝑨 + 𝜺𝑪
33
By combining Equations 21, 24, 25 and 26, Equation 27 can be developed.
Equation 27: 𝜺𝑨𝑽𝑮 = (𝑱 + 𝑺𝟏
𝑪𝑴(𝑪𝑭 − 𝑪𝑴)) 𝜺𝑨𝑽𝑮
𝑭
Eshelby then used the stress concentration factor established by Hill [66] to simplify the
relationship in Equation 27. Hill’s work outlined that a 4th order tensor (AH) must relate the
average strain in a composite to the average strain in an inclusion as shown in Equation 28.
Equation 28: 𝜺𝑨𝑽𝑮𝑭 = 𝜺𝑨𝑽𝑮𝑨𝑯
Therefore, using Hill’s concept, Eshelby determined an alternate stress-concentration factor.
Equation 29 shows Eshelby’s stress-concentration tensor (AE).
Equation 29: 𝑨𝑯 = (𝑱 + 𝑺𝟏
𝑪𝑴(𝑪𝑭 − 𝑪𝑴))
−𝟏
= 𝑨𝑬
The final component of Eshelby’s work involved connecting the stress-concentration factor to
the rule of averages for stress and strain in composite materials. By doing so, a relationship for
the composite stiffness C can be established. Equation 30 and Equation 31 show the rule of
averages for composite materials.
Equation 30: 𝝈𝑨𝑽𝑮 = 𝑽𝒎𝝈𝑨𝑽𝑮𝑴 + 𝑽𝒇𝝈𝑨𝑽𝑮
𝑭
Equation 31: 𝜺𝑨𝑽𝑮 = 𝑽𝒎𝜺𝑨𝑽𝑮𝑴 + 𝑽𝒇𝜺𝑨𝑽𝑮
𝑭
By using Equations 28, 29, 30, 31 and Hooke’s law, a relationship for the elastic modulus of a
composite material that includes inhomogeneous inclusions is established. Equation 32
highlights Eshelby’s relationship for composite stiffness.
34
Equation 32: 𝑪 = 𝑪𝑴 + 𝑽𝒇(𝑪𝑭 − 𝑪𝑴) (𝑱 + 𝑺𝟏
𝑪𝑴(𝑪𝑭 − 𝑪𝑴))
−𝟏
Mori-Tanaka further developed the work of Eshelby to include the scenario of high
inclusion volume fraction. Their theory assumes that a matrix is embedded with a concentrated
amount of inclusions. Each inclusion interacts with a far field strain that is equivalent to the
average strain experienced by the matrix. The theory differs from Eshelby’s work by assuming
that the matrix average strain is related to the fibre average strain through the Eshelby stress-
concentration factor as shown in Equation 33. Through manipulation, the Mori-Tanaka stress
concentration tensor can be developed as shown in Equation 34.
Equation 33: 𝜺𝑨𝑽𝑮𝑭 = 𝜺𝑨𝑽𝑮
𝑴 𝑨𝑬
Equation 34: 𝑨𝑴𝑻 = 𝑨𝑬 ((𝟏 − 𝑽𝒇)𝑱 + 𝑽𝒇𝑨𝑬)−𝟏
The Mori-Tanaka stress-concentration tensor can be substituted into equation 32 to develop a
relationship for the rigidity of the composite material.
The stress-concentration tensors are 4th order tensors for a generalized case of inclusions
embedding an infinite matrix. Tandon and Weng [67], [68] established simplifications for the
tensors for cases with unidirectional short fibre-like cylinders and three-dimensional randomly
oriented fibre-like cylinders. Equation 35 shows the equation for the composite elastic modulus
for when a matrix is embedded with unidirectional fibres.
Equation 35: 𝑬𝒄 =𝑬𝒎
𝟏+𝑽𝒇(𝑨𝟏+𝟐𝝂𝒎𝑨𝟐)/𝑨
The equations for A1, A2 and A can be found in Appendix 1.
35
For three-dimensional randomly oriented fibres, Tandon and Weng established a relationship for
the composite bulk modulus. The composite bulk modulus can be converted to the longitudinal
modulus by knowing the Poisson’s ratio. Equation 36 highlights the composite bulk modulus for
the fibres randomly oriented in three dimensions.
Equation 36: 𝑲𝒄
𝑲𝒎=
𝟏
𝟏+𝑽𝒇𝒑
The equation for p can be found in Appendix 2.
2.5.7 Percolation theory
The micro-mechanicals models discussed (ROM, IROM, Shear-Lag and Mori-Tanaka)
involve stress transfer from the matrix to the fibre during composite deformation. The following
micro-mechanical model involves reinforcement through close fibre contact that leads to fibre
bending. Geometric percolation theory predicts that fibres will form a continuous network at a
critical volume fraction known as the percolation threshold. The percolation threshold is
dependent on the fibre AR, where higher AR fibres will have a lower percolation threshold.
Figure 14 highlights the formation of a percolating network of fibre as the volume fraction of
fibre increases.
36
Figure 14: A- Low volume fraction, no percolation. B - Some fibre interaction. C - High
volume fraction, percolated network of fibres formed D – Low fibre aspect ratio with high
volume fraction, no percolation. [69]
Initially, percolation theory was used to justify the increase in electrical conductivity when low
volume fractions of highly conductive metal fibres were embedded in electrically insulating
polymers [69], [70]. When the fibres are not touching as shown in Figure 14a, the conductivity
is virtually identical to the matrix conductivity. However, as a network forms at a critical volume
fraction or “percolation threshold” as shown in Figure 14c, the conductivity increases by orders
of magnitude. Since percolation depends on fibre AR, Figure 14d, highlights how a low AR fibre
particle cannot achieve percolation even at high fibre content.
Recently, studies [49], [50] have suggested that geometrically percolated fibres can also
increase the rigidity of a composite material. The concept of fibres mechanically percolating is
used to explain the increase in stiffness of a soft matrix when low volume fractions of a high
37
aspect ratio rigid fibre is added. Mechanical percolation assumes that fibres in close contact will
interact directly with each other and this results in an increase of stiffness in an otherwise soft
matrix. The deformation in the fibres is not due to stress transfer from the matrix as discussed in
the previous models. The deformation is dependent on fibre interaction and fibre bending during
percolation. Favier et al. observed an increase in shear modulus by two orders when 1 volume %
of cellulose whisker was used to reinforce a latex matrix [49], [50]. The increase in shear
modulus was attributed to mechanical percolation, since the geometric percolation threshold for
the cellulose whiskers they used is approximately 1 volume % cellulose whisker content. The
increase in shear modulus was only observed above the glass transition temperature of latex
where the shear modulus of the latex dropped from 1 GPa to 0.1 MPa. This is relevant for this
study, as the stiffness of soyol based PUFs is comparable to that of latex above the glass
transition temperature.
2.6 Conclusions
Fibre reinforced thermoplastics have been widely studied. However, research into the
areas of natural PUFs, fibre reinforced PUFs and physically modelling the reinforcement of
fibres in foams is limited. The focus of this study is to increase the compressive modulus of
natural PUFs without increasing the isocyanate content and by optimizing the foam
reinforcement in terms of fibre type, length and content. Finally, the micro-mechanical models
will aid in understanding the physical mechanisms responsible for foam reinforcement due to
fibre presence.
38
Chapter 3
Sample preparation and testing
3.1 Polyurethane foam component materials
Multiple components, including surfactants, catalysts and ablowing agent were added to
the soybean polyol and isocyanate to generate the PUFs used in this project. Table 2 highlights
the individual components, the suppliers and the amounts used of each component.
Table 2: Polyurethane foam chemical formulation used.
Material Type Specific Type Supplier/Details
Weight (g) /
Type
Surfactant
Pyrrolidine,
Polysiloxane
Air Products
1.2, 2.0
Catalyst
Tertiary amine,
Ethanolamine
Sigma Aldrich
and Air Products
1.8
1.2
Blowing agent Distilled Water
Available in
Laboratory
4.0
Iso-cyanate
Polymeric
diphenylmethane
diisocyanate (PMDI)
Specific gravity of
1.23 g/mL at
25°C, NCO
NCO index: 100
39
content of 31.2%,
and functionality
of 2.7 was
supplied by
Huntsman
Soybean based polyol Soyol® 2102
Hydroxyl number
of 63 mg KOH/g
was received from
Urethane Soy
Systems (Volga,
South Dakota,
USA)
60
3.2 Sample preparation
The PUFs used in this study were produced by reacting soyol with isocyanate and
distilled water. Fibre was introduced into the procedure to create fibre-reinforced PUF. The types
of fibres used were glass, MCC and hemp as discussed earlier. Figure 15 highlights the
procedure used to develop the foam samples.
40
Figure 15: Diagram of foam sample preparation.
Figure 15 outlines the free-rising method. Fibres were added into the soy based polyol and hand
stirred for 20 minutes to disperse the fibres in the mixture. The fibre amounts are dependent on
the weight fraction desired. After fibre mixing, the catalysts, surfactants and a blowing agent
were added and mixed for an additional 5 minutes to ensure complete dispersion. Isocyanate was
the last component added and the mixture was stirred briefly for 20 seconds. The mixture was
then poured into an open mould where the foam was allowed to rise freely. The foam was kept in
room temperature for 7 days to properly cure.
3.3 Compression testing
Compression tests were performed on five samples at various weight fractions (0, 5, 10,
15 and 20 wt %) for the different types of fibre-foam composites generated (glass, MCC and
hemp, see section 3.5). The compressive testing was performed according to ASTM - D 1621-10
to determine the compressive modulus [71]. Foams samples were cut into 5 cm x 5 cm x 3 cm
41
specimens (cross-sectional area). The foams were compressed in a Sintech 20 using a 453 kg
load cell. The foams were compressed to 15 % deformation at a speed of 2.5 mm/sec.
3.4 X-ray Tomography analysis
In order to visualize the three-dimensional structure of the fibre reinforced foams, X-ray
tomography images of the fibre-reinforced samples with various weight percentage fibre were
taken. Electron and optical microscopy only provide images of two-dimensional sections of the
fibre-foam composites. The X-ray scanning was performed using a SkyScan 1172 with a 10
Megapixel camera at a beam accelerating voltage of 45 Kilovolts and a current density of 171.
The size of the samples analyzed was approximately 15 mm x 4 mm x 2 mm (height x length x
width). The maximum magnification was used, giving an image resolution of approximately 2.0
microns.
An X-ray tomography system takes a series of two-dimensional shadow images of a
specimen that is being examined by an X-ray beam. The X-ray beam hits the specimen where
some of the X-rays are absorbed. The unabsorbed X-rays transmit through the specimen and are
collected. The intensity of the collected X-rays and their location as it penetrates the specimen
provides the necessary information to create the shadow images. The specimen is rotated 180
degrees and a shadow image is taken at 0.3 degree intervals. Figure 16 shows a two-dimensional
shadow image taken of a fibre-foam composite.
42
Figure 16: Two-dimensional shadow image of a fibre-foam composite.
Figure 16 is a grayscale image that is made up of an array of pixels (picture elements) that are
2.0 microns in size. Each pixel has a grayscale value between 0 and 255 to represent the
grayscale intensity. Darker pixels have higher grayscale values in Figure 16. Higher grayscale
intensity values (darker pixels) indicate more X-ray absorption. The pixels with the minimal
grayscale intensity value (0, lightly coloured pixels) contain only air. The grayscale value of each
pixel is dependent on the material present in the pixel. The chemical elements that make up the
material and the density of the material determines the X-ray absorption of the material. For
example, heavier elements such as Lead absorb X-rays readily and therefore typically have
higher grayscale intensities. In this study, the cell pores (air), the polyurethane struts and the
fibres absorb X-rays uniquely and should be easily distinguished during examination.
4 mm
43
In order to develop a three-dimensional image of the fibre-foam composites, a
reconstruction software (NRecon – supplied by SkyScan) was used. The software uses
deconvolution to determine the only possible three dimensional image from the series of two-
dimensional shadow images generated. Figure 17 highlights an example of a three dimensional
image generated from X-ray tomography analysis.
Figure 17: A two-dimensional projection of the three-dimensional structure of a fibre-foam
composite. The thicker dark cylinders represent fibres. Thin struts represent the
polyurethane foam structure.
The second portion of X-ray Tomography analysis involved imaging compressed fibre
reinforced PUF samples using the SkyScan Material Testing Stage (MTS-50N) shown in Figure
44
18. These experiments were performed to visualize the microstructure of compressed fibre
reinforced PUFs. The testing stage was placed directly inside the SkyScan apparatus and
replaced the original sample holder. The samples were placed inside the testing stage for analysis
and compressed to the desired strain. The samples were investigated individually. The X-ray
beam is focused on the sample inside the testing stage (sample was located in position 4 of
Figure 18) and is imaged as discussed earlier. The samples analyzed were 10 mm x 5 mm x 5
mm (height x length x width) in size. The sample length was decreased and the sample width
was increased in comparison to the first batch of uncompressed samples. This size was selected
to ensure buckling would not occur during compression
Figure 18: SkyScan Material Testing Stage (MTS-50N) used to compress samples for
imaging.
The samples were secured to the lower platform (3). Platforms (3) and (5) compressed
(move towards each other with the aid of the mechanisms in (1) and (2)) the sample inside the
chamber to pre-defined levels through user input commands in the MTS-50N control software.
The samples were compressed to 10, 20, 30, 40, 50 and 60 percent strain and then imaged. All
compression was completed prior to commencing of imaging. The platforms did not move
during imaging to ensure stability of images being taken.
45
3.5 Fibre and foam properties
For physical and analytical modelling, some physical and mechanical properties of the
fibre types and foam used in this study must be known. These properties include the elastic
modulus, density, Poisson’s ratio of PU and PUF, fibre AR and average cell diameter for each
fibre-foam composite. Table 3 highlights the values for the material properties used in this study.
Table 3: Material properties required for physical and analytical modelling
Material Type Elastic Modulus
(GPa)
Poisson’s Ratio Density (g/cm3) Supplier
Polyurethane 0.65 0.5 [72] 1.2 [73] -
PUF 0.0000629 0 [74] - Measured for
each sample
-
E - Glass Fibre ~ 80 [75] - 2.5 [75] Fibertec Inc.
MCC Fibre ~ 25 [76] 0.3 [77] ~ 1.5 [78] Sigma-Aldrich
Hemp Fibre ~ 69 - 1.5 Hempline Inc.
Table 3 shows that the elastic moduli of the fibres is significantly larger than the elastic modulus
of the PUF and should provide reinforcement. As well, a pure PU matrix is much stiffer than a
PUF matrix, indicating the rigidity loss when a polymer is foamed.
46
Table 4 highlights the dimensions used for the fibre types and the average cell diameters
for each fibre-foam composite. The average dimension values were established from a sample
size of 100.
Table 4: Dimensions of different fibre types used and cell diameter of fibre-foam
composites. N = 100.
Fibre Type Average Fibre Length
(µm)
Diameter Average Cell
Diameter of Fibre-
Foam Composite
260 µm Glass Fibre 210a 16* 313
470 µm Glass Fibre 401a 16* 312
MCC Fibre 50** 10 [79] 314
Hemp Fibre 2025b - 722
No-Fibre Neat Foam - - 242
*Value provided by supplier Fibretec Inc. **Value provided by supplier Sigma-Aldrich. a fibre
length measured by X-ray tomography when embedded in PUF. b fibre length measured by
optical microscope
Table 4 shows that there was some fibre length degradation of the glass fibre after mixing. As
well, Table 4 shows that the average cell diameter of the PUF varies when fibre is introduced
into the matrix. As well, each fibre type has a unique size relationship with their respective
47
average cell diameter. In this study, this size difference will be explored in terms of how the fibre
reinforces the PUF. Chapter 4 offers hypotheses for how each fibre type will reinforce PUF.
3.6 Conclusions
In this chapter, the formulation used to develop the soyol based PUF was highlighted. As
well, a detailed description on how to use this formulation to produce the fibre reinforced PUF
was also provided. The characterization technique used in this study to analyze the three-
dimensional structure of the fibre-foam composites was also described and some sample images
were provided. Finally, the material and physical properties of the fibre types used and the PUF
were given.
48
Chapter 4
The effects of cell architecture and fibre aspect ratio on the
compressive modulus of PUFs
4.1 Gibson-Ashby theory for foam mechanics
The theories developed by Gibson and Ashby [73], [80] are the most widely accepted
analytical and structural models for foam compression deformation. For PUF compression
deformation, their model separates the compression stress-strain relationship of PUF into three
regions as shown in figure 19. Region 1 is an elastic-linear region where the cells begin to
deform. In region 2, the foam still behaves elastically but the deformation is non-linear. Region 3
is a non-linear and permanent deformation region of densification. In this thesis, the compressive
modulus is the critical parameter, and hence the focus will be on the elastic-linear regime.
Figure 19: Three regions in PUF compressive deformation
49
Foams are made up of a series of cells. The cells of the PUF studied were primarily open
cell. Gibson and Ashby developed their model for open cell foams by assuming that the foam
had a unit cell with a simple cubic structure as shown in figure 20.
Figure 20: Simple cubic structure denotes foam cell structure. Figure displays before and
after load is applied to the cell structure scenarios.
50
When the foam is initially compressed at small applied loads, the struts at the top and bottom of
the cell bend leading to deflection. This initial cell deformation due to bending has a linear
stress-strain relationship that is completely elastic. Gibson and Ashby assumed that the cell struts
bend according to the standard beam theory in order to relate the applied load to the deflection of
the cell struts as shown in equation 37.
Equation 37: 𝜹 ∝ 𝑭𝑳𝒔
𝟑
𝑬𝑰
Where 𝛿 = deflection, F = applied load, LS = length of strut, E = modulus of material and I =
second moment of inertia.
The applied load F can be related to the global stress, 𝐹 ∝ 𝜎𝐿𝑆2 and the deflection to the global
strain, 휀 = 𝛿
𝐿𝑆 allowing equation 37 to be redefined to the following:
Equation 38: 𝜺 ∝ 𝝈𝑳𝑺
𝟒
𝑬𝑰
The effective modulus of the foam can be related to the global stress and strain, 𝐸∗ = 𝜎 and 𝐼 ∝
𝑡4 , where t is defined as the thickness of the struts. Therefore, equation 38 can be rewritten to
equation 39 (where Z is a constant of proportionality).
Equation 39: 𝑬∗
𝑬=
𝒁𝒕𝟒
𝑳𝑺𝟒
The constant of proportionality is approximately 1 according to analysis performed by Gibson
and Ashby. Therefore, the effective modulus of the foam can be approximated and is related to
the ratio between the strut length LS and thickness t of the cell struts that compose the foam.
Since it is difficult to measure the individual length and thickness of each cell strut, Gibson-
51
Ashby further simplified equation 39 to a ratio of density as shown in the following equation,
which is the main relationship of the Gibson-Ashby Theory.
Equation 40: 𝑬∗
𝑬=
𝒁𝒕𝟒
𝑳𝑺𝟒 = (
𝒕𝟐𝑳𝑺
𝑳𝑺𝟑 )
𝟐
≈ (𝑴𝒇𝒐𝒂𝒎
𝑽𝒇𝒐𝒂𝒎⁄
𝑴𝑷𝒖𝑽𝑷𝒖⁄
)
𝟐
≈ (𝝆∗
𝝆𝒔)
𝟐
Here, 𝝆∗ is the apparent density of the foam and 𝝆𝒔 is the density of the solid material that
composes the foam (polyurethane in this thesis). Mfoam and Vfoam represent the mass and volume
of the entire foam sample including solid polyurethane and air. MPU and VPU represent the mass
and volume of only the polyurethane in the foam. The mass of the foam will equal the mass of
the solid polyurethane since it is the only material that composes the cell structure. Equation 40
shows that the compressive modulus of foams is independent of cell size and dependent only on
the apparent density (porosity) of the foam. Therefore, various foam samples may have different
cell sizes but would share the same compressive modulus if they had the same apparent foam
density.
Region 2 begins at the onset of the non-linear elastic deformation portion of the curve in
figure 19. This region results in a sharp increase in deformation with minimal additional applied
loading. Gibson and Ashby attribute this phenomenon to be the result of the onset of buckling for
the vertically aligned struts in figure 20. Based on a similar derivation used in region 1, the
following equation is defined for region 2.
Equation 41: 𝝈𝒆𝒍
𝑬= 𝒁 (
𝝆∗
𝝆𝒔)
𝟐
𝜎𝑒𝑙 , is defined as the elastic stress (the stress which occurs at the onset of buckling). Therefore,
similar to region 1, the thickness and length of the cell struts and the respective densities
contribute to the onset of buckling and non-linear elastic deformation.
52
The deformation the foam would undergo in region 3 is related to the densification of the
solid material. The cells compress to a point where the materials in the struts contact leading to
increased loads with very little further remote strain. Air voids in the foam matrix have been
eliminated and the foam deformation stress-strain curve matches a solid block being compressed.
This explains the sharp increase in loading required to deform the foam in region 3. Both region
2 and 3 are not studied in this thesis because the facings of a SIP would have already buckled
and failed in order to deform the interior foam core to these levels. Therefore, optimizing regions
2 and 3 are not a part of the motivation for this project.
4.2 Modified Gibson-Ashby equation using volume fraction of
solids in foam (VFS)
When fibres are added to the foam matrix to create a composite material, fibres could be
embedded into the cell structure as shown in figure 21.
Figure 21: The solid density of the pure foam is the density of polyurethane. However, the
solid density of the composite foam is unknown.
53
This structural scenario would result in the density of the solid material term in equation 40
being difficult to determine. The density of the solid material in the pure foam is simply the
density of polyurethane, however, in the composite foam this would be a density that is a
combination of both fibres and solid polyurethane. Therefore, an alternative physical relationship
must be used to predict the effective modulus of the foam from the cell structure.
The volume fraction of solids of the foam (VFS) is a representation of the porosity of the
foam and can be calculated more easily than the density of the solid material present in the foam
when fibres are present. The volume fraction of solids in the foam can be related to the solid
density of the foam and the effective modulus of the foam as shown in Equation 42.
Equation 42: 𝑬∗
𝑬= (
𝝆∗
𝝆𝒔)
𝟐
= (𝑴𝒇𝒐𝒂𝒎
𝑽𝒇𝒐𝒂𝒎⁄
𝑴𝑷𝒖𝑽𝑷𝒖⁄
)
𝟐
= (𝑽𝑷𝒖
𝑽𝒇𝒐𝒂𝒎)
𝟐
= (𝑽𝑭𝑺)𝟐
The volume of polyurethane present in the foam VPU and the volume of the foam Vfoam can easily
be calculated by knowing the weight fractions of fibre and polyurethane in the composite foam
during manufacturing. This outcome makes equation 42 easier to use to predict the effective
foam modulus when fibres are embedded. Equation 42 further illustrates that the porosity of the
foam is the critical parameter in predicting the effective modulus of the foam according to
Gibson-Ashby theory and not cell size.
4.3 The effect of fibres on the volume fraction of solids (VFS)
The volume fraction of solids and porosity is altered when fibres are introduced into the
foam system. The fibres act as nucleating agents for gas bubble formation. Therefore, with
increased fibre content there is an increase in gas bubble formation. With increased gas content
in a given foam system, the porosity must increase and the volume fraction of solids must
54
decrease. With a decrease in volume fraction of solids, a decrease in effective foam modulus is
also expected.
Fibres such as wood and natural fibres and silane treated glass fibres form chemical
bonds to the hydroxyl groups along the polyol chains during mixing and foaming. This increase
in chemical bonding in the system will increase the viscosity of the fibre and polymeric mixture.
With an increase in the mixture viscosity, expansion during the foaming stage is limited. By
limiting the foam expansion in volume, the porosity is also decreased and the volume fraction of
solids is increased. This outcome will result in an increased effective modulus of the composite
foam according to equation 42. Therefore, there are two competing factors that alter the effective
modulus of the foam when fibres are introduced.
Since there are two competing factors that affect the effective foam modulus when fibres
are introduced, it is critical to determine the volume fraction of solids for each composite foam
sample generated in this thesis. By knowing the volume fraction of solids, the theoretical
effective modulus of the foam sample without fibre reinforcement can be determined. As well, it
would be possible to determine whether gas bubble nucleation or increased mixture viscosity has
a larger impact on the volume fraction of solids and effective foam modulus for the composites
generated in this project.
4.4 Assumptions on foam reinforcement and fibre placement
based on fibre aspect ratio and cell architecture
Fibre length and fibre placement within the cell architecture have a significant effect on
how fibres reinforce individual cells and the entire foam matrix. The three regimes that are
investigated in this thesis include short fibres that are found within the cell struts, intermediate
55
fibres that have a similar length to the cell diameter and long fibres that span more than one cell.
Each regime will offer a different physical reinforcing mechanism to the cells and foam matrix.
4.4.1 Short-fibre reinforced foams
Short fibres that have fibre lengths that are much shorter than the cell diameters in a foam
matrix will be embedded within the cell struts. In this project, MCC fibres are predicted to
behave like short fibres based on the data outlined in chapter 3. The MCC fibres have an average
fibre length that is much shorter than the average cell diameter of the foam matrix that they
embed. Figure 22 is a schematic of short fibres embedded within the cell struts.
Figure 22: Short fibres embedded within cell struts
These short fibres are initially placed inside in a liquid polyol mixture that contains no voids. As
discussed in section 4.3, cells nucleate along the fibre surface once the foaming process begins.
Many cells will form along the fibres and neighboring cells will coalesce into a larger cell with a
smaller surface area to limit the surface energy caused by the introduction of gas bubbles.
56
Furthermore, three previous studies [32], [81], [82] found that short fibres were embedded within
the cell struts of PUF.
The cells embedded within the cell struts are expected to reinforce them. The
reinforcement is predicted to follow a physical mechanism based on a micro-mechanical model
such as ROM, IROM, Shear-Lag Theory or Mori-Tanaka (Cylindrical modified version). The
fibres are expected to be parallel to the cell struts and therefore the unidirectional versions of the
models can be used. With the fibre reinforcement, the cell struts will be stiffer, which will in turn
increase the stiffness of the entire foam based on Gibson-Ashby theory. Equation 43 highlights
that ERS (fibre reinforced strut modulus predicted by micro-mechanical models) replaces E in the
Gibson-Ashby model for short fibres. As well, since ERS is larger than EPU (in this study, the
material that makes up the foam is polyurethane), the effective modulus of the composite foam Ec
must also be larger than the neat foam modulus E* provided VFS does not decrease.
Equation 43: 𝑬𝒄
𝑬𝑹𝑺 = (𝑽𝑭𝑺)𝟐
4.4.2 Long-fibre reinforced foams
Long fibres that have fibre lengths that are larger than the cell diameters in a foam matrix
will span cells and will not be embedded within the cell struts. The fibres would be too large to
fit within the cell struts and therefore will span cells instead. The 470 micron glass fibre and the
hemp fibre used in this study are expected to behave as long fibres. Chapter 3 highlighted that
the average fibre length of both fibre types was larger than the average cell diameters of the
foams that they embedded. Figure 23 and 24 show how these long fibres would be placed inside
a series of cells in a foam matrix at a two-dimensional level.
57
Figure 23: Hemp fibres that are much larger than cell diameters spanning multiple cells
Figure 24: 470 micron glass fibres that are larger than a single cell diameter are not
embedded within the cell struts
58
Since the long fibres are larger than the cell diameters, it is hypothesized for physical
modelling purposes that the cell structure can be ignored and the foam matrix can be treated as a
homogenous matrix. This homogenous matrix will have well-defined physical properties,
including an elastic modulus that matches the elastic modulus of the cell based foam matrix
predicted by Gibson-Ashby Theory. Figures 25 and 26 highlight these assumptions.
Figure 25: For the long fibre case, cell structure will be ignored and treated as a
homogenous matrix with the same elastic modulus
59
Figure 26: Long fibres are embedded in homogenous matrix for physical modelling
considerations
In order to understand how the physical scenario depicted in figure 26 will result in foam
reinforcement, two hypotheses will be investigated. One hypothesis for foam reinforcement by
long fibres assumes that the fibres would be embedded within the homogenous matrix and
reinforce it according to traditional micro-mechanical models. The matrix would transfer stress
into the fibres and traditional theories such as ROM, IROM, Shear-Lag and Mori-Tanaka would
apply. The fibres are expected to be randomly oriented and therefore the models must account
for this with an orientation correction factor. An orientation correction factor will be used for
ROM, IROM, and Shear-Lag where the fibre modulus is multiplied by 1/5 as discussed in
section 2.5.5. The work of Tandon and Weng [68] discussed in 2.5.6 will be used to correct the
model for Mori-Tanaka theory. The significant parameters required to input into each of the
analytical models would include the elastic and shear modulus of the homogenous matrix, the
60
elastic modulus of the long fibre and the volume fractions of both fibre and homogenous matrix.
Each of these parameters are known or can be theorized by Gibson-Ashby Theory.
The second hypothesis assumes that mechanical percolation is the physical mechanism
behind foam reinforcement due to fibres. The long fibres have high aspect ratios which may
cause them to geometrically percolate or have significant fibre interaction. During compression
these fibres may mechanically percolate leading to an overall increase in foam modulus. In order
to evaluate the possibility of mechanical percolation, the composite foam samples must be
investigated with X-ray Tomography to confirm the possibility of geometric percolation or fibre
interaction. Furthermore, a finite element model was developed to verify if mechanical
percolation is occurring for long fibre reinforced PUFs.
4.4.3 Intermediate-fibre reinforced foams
Intermediate length fibres are shorter or similar to the average cell diameter but are too
large to be embedded within the cell struts. Therefore, these fibres are assumed to have a direct
interaction and relationship with the cell as a whole, which is something not predicted for the
other two regimes. Based on fibre length and cell diameter analysis in chapter 3, the 260 micron
fibre length composite samples studied in this project are expected to behave as intermediate
length fibres.
There are two hypotheses that will be investigated to predict how intermediate length
fibres will affect foam reinforcement. One hypothesis assumes that since intermediate fibre
length is similar to the cell diameter, the fibres can form “bridges” between cell struts. Figure 27
highlights four examples of fibre bridging.
61
Figure 27: Fibres bridging individual cells in different orientations
The four fibre bridging cases in figure 27 will reinforce differently. In the first case (top left
corner), the fibre will reinforce the cell by restricting the vertical cell struts from bending during
compression which is expected from figure 19. In order for the vertical struts to bend, the fibre
must be elongated. This will require an increase in compressive energy to provide the strain
energy to elongate the fibre. The increase in energy requirements means that the overall
composite system is stiffer with a higher compressive modulus. For the second case (top right
corner), the fibre would be purely compressed which will increase the overall stiffness of the
cell. The third and fourth case is hypothesized to share similar reinforcement mechanisms as
seen in both the first two cases. In order to verify how these cases will reinforce individual cells
a finite element model was developed.
The second hypothesis for how intermediate fibres would affect foams involves a no
reinforcement scenario. This scenario would occur when the fibres do not bridge between the
62
cell struts. Figure 28 highlights how intermediate fibres would not bridge and not reinforce the
individual cell.
Figure 28: Fibres not bridging individual cells in different orientations
In the four cases in figure 28, the cells would be able to deform normally without requiring the
fibres to be deformed. Furthermore, the fibres could be too short to adequately reinforce the
foam matrix through traditional micro-mechanical models or mechanical percolation. As well,
they are too large to reinforce the cell struts. Therefore, the fibres may not increase the
compressive modulus of the foam matrix and could have a detrimental effect on the foam
modulus.
63
4.5 Conclusions
Three different fibre reinforcement regimes are hypothesized. Short fibres are expected to
be embedded within cell struts and reinforce these struts according to traditional micro-
mechanical models. The reinforced struts would increase the overall foam modulus according to
Gibson-Ashby theory. Long fibres are expected to span multiple cells. The cell structure of the
foam will be ignored and treated as a homogenous matrix. This homogenous matrix is expected
to be reinforced by the long fibres through traditional micro-mechanical models or mechanical
percolation. Intermediate length fibres are expected to either bridge cell struts to provide cell
reinforcement or to provide no reinforcement to the foam matrix.
64
Chapter 5
Analyzing the compressive properties of micro crystalline
cellulose (MCC) fibre reinforced PUFs
Note: Some of the findings in this chapter can be found in our manuscript: Polyurethane
foam mechanical reinforcement by low-aspect ratio micro-crystalline cellulose and glass fibres
Journal of Cellular Plastics, Released Online, 2014. DOI: 10.1177/0021955X14529137
5.1 Introduction
Micro-crystalline Cellulose (MCC) fibres were studied as potential reinforcing fibres for
PUF. MCC fibres contain a high content of cellulose which has high stiffness and are capable of
reinforcing a weaker polymer matrix. MCC fibres were selected for investigation to create the
short-fibre reinforcement scenario discussed earlier. The fibre aspect ratio of MCC fibre is
comparable to the cell strut size and therefore the fibres should be embedded inside the cell struts
of PUF.
5.2 Micro-structure of MCC fibre reinforced PUFs
Microscopic analysis through X-ray tomography and Scanning Electron Microscopy
(SEM) supports the claim that MCC fibre-foam composites are representative of the short-fibre
reinforcement scenario discussed in chapter 4. Figure 29 shows that at 5 wt %, the MCC fibres
are not spanning the cells and are not visible. Therefore, the MCC fibres must be embedded
within the cell struts where they would be covered by the polyurethane.
65
Figure 29: 5 wt % (0.08 volume %) MCC fibres are mostly embedded in cell struts. Fibres
do not span cells. Note: depth of field of image is 1.0 mm
SEM was used to examine the cells and cell struts of the MCC fibre-foam composite with
higher magnification. Figures 30, 31 and 32 show that fibres are not spanning the cells. Figure 30
shows a series of cells highlighting that the fibres do not span multiple cells and therefore the
MCC fibre-foam composite is not the long-fibre foam composite scenario. Figures 31 and 32
show individual cells where there is no evidence of fibre bridging or fibres lying within the cells.
This suggests that the MCC fibres do not behave like intermediate length fibres. Finally, Figure
33 shows MCC fibres embedded within a cell strut showing that the MCC fibre-foam composite
represents the short fibre scenario discussed in chapter 4. The fibres appear to be more aligned
than randomly oriented. Therefore, the MCC fibres should reinforce the individual cell struts
according to micro-mechanical modelling. This cell strut reinforcement will result in an increase
66
in the compressive modulus of the entire composite foam matrix according to Gibson-Ashby
theory.
Figure 30: 20 wt % (0.36 volume %) MCC fibres. MCC fibres are not seen.
Figure 31: 20 wt % (0.36 volume %) MCC fibres. MCC fibres are not seen.
67
Figure 32: 20 wt % (0.36 volume %) MCC fibres. MCC fibres are not seen.
Figure 33: 20 wt % (0.36 volume %) MCC fibres. SEM image showing individual cell strut
that has MCC fibres embedded in it. Fibres are considered to be more aligned than
randomly oriented.
68
5.3 Changes to the MCC fibre-foam composite compressive
modulus with increases in MCC fibre weight fraction
The foam matrix does not remain consistent as fibres are added as discussed in chapter 4.
MCC fibres act as nucleating sites for cells leading to a substantial increase in gas content which
in turn lowers the volume fraction of solids. According to Gibson-Ashby theory (Equation 42),
lowering the solid volume fraction in a foam will decrease the compressive modulus of the foam.
Another possibility is that the fibres increase the viscosity of the liquid polyol before the
isocyanate is added by bonding and interacting with the hydroxyl groups in the polyol. With an
increase in viscosity, the foam expands less during the foaming stage, resulting in a lower sample
volume. With a lower sample volume, the solid volume fraction would increase. These
competing factors where the fibres both nucleate cells and increase viscosity will result in the
solid volume fraction being lower when small amounts of fibre are added but eventually
stabilizing. Figures 34 and 35 highlight the decrease in solid volume fraction and the expected
decrease in the compressive modulus of the no fibre base foam (dependant on Equation 42) as
MCC fibre weight percentage is increased and volume fraction of solids is decreased. The no-
fibre base foam case is dependent on the volume fraction of solids for each sample type at 5, 10,
15, and 20 wt %.
69
Figure 34: Volume fraction of solids decreases as MCC fibre weight percentage increases.
Figure 35: No-fibre base foam modulus expected through Gibson-Ashby theory (based on
volume fraction of solids at given fibre wt %) compared to experimental results.
Fibre Effect
70
When samples with various MCC fibre weight percentages (0, 5, 10, 15, 20 wt %) were tested
for compressive modulus, the experimental results did not correlate with the calculated no fibre
base foam modulus curve in Figure 35, suggesting that there must be some reinforcement of the
foams due to the presence of MCC fibres. The fibres added must be contributing to increasing
the compressive modulus by reinforcing the cell struts. The results suggest that if the density
were held constant through formulation and processing modifications, the MCC reinforcement of
the struts would increase the foam modulus, as expected, although the low aspect ratio MCC is
not providing very effective reinforcement.
In order to understand the foam reinforcement due to the introduction of MCC fibres into
the cell struts, Figure 36 and Table 5 compares the experimental results with various micro-
mechanical predictions. The micro-mechanical models (ROM, IROM, Cox, Mori-Tanaka
(cylinder modified version), and Nairn) predict ERS in Equation 43, which in turn determines EC,
the composite foam compressive modulus which are the values plotted in Figure 36. The aligned
(unidirectional) fibre case for micro-mechanical models is used to solve the micro-mechanical
models as discussed in chapter 3 based on the assumptions made from analyzing the SEM
images. As well, during cell growth during foaming, the cell struts will be extended in length
which should align embedded MCC fibres.
71
Figure 36 a and b: Compressive modulus of MCC fibre-foam composite samples in
comparison to micro-mechanical models.
72
Table 5: Sum of squared error normalized to compare micro-mechanical model predictions
to the actual experimental results
Micro-Mechanical Models Sum of squared error
Sum of squared error
normalized
ROM 7.7 x 106 216.35
IROM 5.2 x 104 1.47
Cox 3.6 x 104 1
Nairn 2.9 x 106 80.29
Mori-Tanaka 2 x 105 5.69
The experimental results were only slightly higher than the IROM curve, which would be the
worst-case scenario for fibre reinforcement. This further supports the claim that there is some
fibre reinforcement, although it is limited. The experimental curve is well below the ROM curve
as expected, suggesting that the fibres are not being strained as much as the struts are being
strained globally. Cox’s shear-lag theory correlates well with the experimental results as shown
in Table 5 and Figure 36, but this may not indicate that the physical assumptions underlying the
theory are valid. However, it does provide support that the short-fibre reinforcement scenario
discussed in chapter 4 is acceptable.
The results are much lower than the values suggested by Nairn’s correction factor for
Shear-Lag. As well, Mori-Tanaka does not correlate well beyond 10 wt % MCC fibre content.
73
There are multiple reasons that might explain why the experimental results would be lower than
the values predicted by Nairn’s Shear-lag Theory and Mori-Tanaka. Micro-mechanical models
assume that fibres are being strained by a matrix evenly and typically in complete tension. For
fibres embedded in the cell struts, the situation differs, since the struts will actually bend and not
be completely in tension. The MCC fibres that will be on the compression side of the cell strut
bending might buckle early, leading to less load transfer into the fibres. Furthermore, the models
assume that fibres are well dispersed, which is unlikely to be the scenario within the cell struts.
As well, some of the fibres may be exposed and stick out of the cell struts. The exposed fibre
surface areas would not contribute to reinforcement. From the results in Figure 36 and Table 5, it
can be concluded that MCC fibres embedded inside PUF cell struts do reinforce the cell struts.
However, the reinforcement is quite poor and the long-fibre reinforcement scenario should
provide better results.
5.4 Conclusions
The introduction of MCC fibres into a PUF matrix resulted in a decrease of solid volume
fraction. However, the compressive modulus of the MCC fibre-foam composites did not decrease
as significantly as predicted by Gibson-Ashby theory. This suggested that the MCC fibres were
reinforcing the PUF matrix to a certain degree. Through X-ray tomography and SEM analysis it
was determined that MCC fibres were embedded within the cell struts of PUF. Therefore, the
MCC fibre-foam composite was assumed to represent the short-fibre scenario for reinforcement.
The experimental results were compared to micro-mechanical models assuming a short-fibre
reinforcement scenario to understand the physical mechanisms behind the foam reinforcement.
Mori-Tanaka, Cox’s Shear-Lag theory and IROM were found to adequately predict the
reinforcement by the MCC fibres.
74
Chapter 6
Analyzing the compressive properties of long-fibre reinforced
PUFs
Note: Some of the findings in this chapter can be found in our manuscript: Understanding the
effects of high aspect ratio fibers on the mechanical reinforcement of soybean based
polyurethane foam, Journal of Cellular Polymers, Vol 33, Issue 1, 2014. Pg 1-20.
6.1 Introduction
Glass fibres (470 microns in length) and hemp fibres were studied as potential reinforcing
fibres inside PUF. Glass fibre has been traditionally used in reinforcement of weaker plastics.
Hemp fibres contain a high content of cellulose that is stiff and capable of reinforcement of the
lower modulus polyurethane. These longer fibres are expected to span cells and replicate the
long-fibre reinforcement case discussed in chapter 4.
6.2 Long glass fibre reinforced PUFs
Glass fibres have a much higher elastic modulus than PUF and therefore should reinforce
it. Based on the analysis in chapter 3, these glass fibres are too large to be within the cell struts
and could possibly span cells.
6.2.1 Micro-structure of long glass fibre reinforced PUFs
X-ray Tomography was used to understand the structure of the long glass fibre reinforced
PUFs. Figures 37 to 40 show that long glass fibres span multiple cells. These glass fibres are
longer than the cell diameters and are not solely confined to the cell struts. Since the glass fibres
75
are much longer than the cells it is assumed that 470 micron glass fibre-foam composites
represent the long-fibre scenario discussed in chapter 4. For the purposes for modelling, the glass
fibres are assumed to be embedded in a low modulus homogenous matrix and the specifics of the
cell structure will be ignored. The compressive modulus of this homogenous polymeric matrix is
determined by Gibson-Ashby Theory as discussed earlier.
Figure 37 highlights that even at low fibre volume percentages of 0.3 volume %, the high
glass fibre aspect ratio produces a system that appears to be near geometric percolation. The
glass fibres are not isolated and are in close-contact even with some glass fibre length
degradation during mixing. This leads to the possibility of mechanical percolation as a foam
reinforcement mechanism by the fibres during compression.
Figures 38 to 40 show some fibre bundles (arrows point to fibre bundles). These bundles
may be detrimental to foam reinforcement since the fibres within the bundles could slide against
each other during compression and remain unloaded without stress transfer from the matrix.
However, since the fibres are indeed close enough together to be interacting, this leads to
possible bending of the fibres during compression. This further supports the possibility of
mechanical percolation.
76
Figure 37: 20 wt % (0.3 volume %) 470 micron glass fibre embedded in foam. Intact Fibres
span multiple cells. Note: depth of field of image is 2.0 mm
Figure 38: 5 wt % (0.07 volume %) 470 micron glass fibre embedded in foam. Intact Fibres
span multiple cells. Note: depth of field of image is 2.0 mm. Arrows indicate Fibre bundles.
77
Figure 39: 10 wt % (0.12 volume %) 470 micron glass fibre embedded in foam. Intact
Fibres span multiple cells. Note: depth of field of image is 2.0 mm. Arrows indicate Fibre
bundles.
Figure 40: 15 wt % (0.19 volume %) 470 micron glass fibre embedded in foam. Intact
Fibres span multiple cells. Note: depth of field of image is 2.0 mm. Arrows indicate Fibre
bundles.
78
6.2.2 Changes to long glass fibre-foam composite compressive modulus
with increases in long glass fibre weight fraction
Similar to MCC fibres, long glass fibres alter the porosity of the foam matrix. The long
glass fibres act as nucleating sites for cell formation, which leads to an increase in gas content
present in the fibre-foam composite. With an increase in the porosity and gas content there is a
decrease in the volume fraction of solids as shown in figure 41. Correspondingly, Gibson-Ashby
theory (equation 42) predicts that the compressive modulus of an unreinforced foam (no fibre
base foam) should decrease if the volume fraction of solid decreases. Figure 42 shows the
compressive modulus of the no fibre base foam curve decreasing as the long fibre glass content
is increased and the volume fraction of solids in the foam decreases. However, the experimental
results show that the long glass fibre-foam composites consistently have a higher compressive
modulus than those predicted by the baseline curve suggesting that the long glass fibres are
indeed reinforcing the foam matrix. Significant improvements in the compressive modulus for
the fibre-foam composites are observed at glass fibre content beyond 10 wt % suggesting a
threshold fibre content for foam reinforcement. Furthermore, the improvements in compressive
modulus for long glass fibre-foam composites are significantly better than the improvements
observed to MCC fibre-foam composites in chapter 5. For example, at 20 wt % long glass fibre,
there is an average improvement of 400 kPa between the experimental results the baseline curve.
However, there is only an average improvement of 100 kPa at 20 wt % MCC fibre which
actually represents a higher volume % of fibre due to the lower density of MCC in comparison to
glass.
79
Figure 41: Volume fraction of solids decreases as long glass fibre weight percentage
increases
Figure 42: No-fibre base foam modulus expected through Gibson-Ashby theory compared
to experimental results. Significant fibre effect observed.
Fibre Effect
80
The experimental results were compared to the accepted micro-mechanical models for
fibre reinforced polymers (ROM, IROM, Mori-Tanaka, Cox and Nairn) in order to understand
the physical mechanisms of foam reinforcement by the long glass fibres. Figure 43 shows that
the micro-mechanical models do not predict the experimental results well. Based on the
micrographs shown in section 6.2.1 it was assumed that the fibres are randomly oriented in three-
dimensional space. As discussed in chapter 3, the orientation correction factor (CPP) of 1/5 was
used for ROM, IROM, Cox and Nairn. The randomly oriented fibre model for Mori-Tanaka
Theory developed by Tandon and Weng [68] was also used.
81
Figure 43 a and b: Compressive modulus of MCC fibre-foam composite samples in
comparison to micro-mechanical models.
82
In figure 43, the ROM provides the upper possible bound for compressive modulus of
the long glass fibre-foam composite where the long glass fibres are assumed to be oriented in the
direction of loading and span the entire length of the foam. The IROM provides the lower bound
of compressive modulus considering foam reinforcement where the long glass fibres are assumed
to be perpendicular to the direction of loading. The experimental results curve in figure 43 is
much higher than the IROM curve suggesting that fibres are significantly contributing to
improving the fibre-foam compressive modulus.
However, the Cox and Nairn shear lag models and Mori-Tanaka theory fail to predict the
improvement of the compressive modulus of the fibre-foam composite samples correctly. Shear-
lag theory models the shear-transfer of load from the matrix to the ends of the fibre that have de-
bonded. Since the long glass fibres have an elastic modulus that is one hundred thousand times
larger than the compressive modulus of the foam, Cox’s shear lag theory predicts that the shear
stress transfer is essentially negligible due to the disparity in modulus. This implies that the long
glass fibres are essentially not loaded and explains why Cox’s predicted line is identical to the
IROM line where there is poor foam reinforcement due to fibres. Similarly, Mori-Tanaka theory
also predicts limited foam reinforcement by fibre deformation due to the disparity in elastic
modulus between the long glass fibres and the foam matrix.
Nairn’s shear lag model improves some of the basic assumptions in Cox’s original model
and was found to be more accurate with finite element modelling in Nairn’s study. However, it
significantly over predicts the experimental results in figure 43. One possible rationale for the
poor prediction of the experimental results is that Nairn analyzed composites where the fibre
modulus was at a maximum hundred times larger than the modulus of the matrix. Furthermore,
Nairn suggested that shear-lag theory could not work well when fibres are encased in an infinite
83
matrix volume. For long glass fibre-foam composites with 20 wt % glass fibre, the fibre volume
% (volume of fibre over entire volume of foamed composite) is only 0.3 %. This low volume of
fibre and the extremely low matrix modulus explains the discrepancy between Nairn’s shear lag
model and the experimental results.
None of the analytical micro-mechanical models were able to predict the compressive
modulus of the fibre-foam composites containing long glass fibre. Therefore, the hypothesis that
the foam reinforcement could be predicted by micro-mechanical models assuming that long glass
fibres are embedded in a homogenous matrix is incorrect. Cox’s shear lag theory and Mori-
Tanaka theory predict limited foam reinforcement improvement in composite modulus at high
fibre loads due to limited stress transfer from the matrix into the long glass fibres. However, the
fibres must be nevertheless loaded since the compressive modulus of the fibre-foam composites
increased with higher long glass fibre content. Therefore, since the fibres are loaded and the
matrix is not responsible, a direct fibre-to-fibre load transfer mechanism is likely. This suggests
that the second hypothesis discussed in chapter 4, where the long glass fibres mechanically
percolate could form the basis of an appropriate micro-mechanical model.
6.2.3 Geometric percolation of long glass fibre
Geometric percolation of fibre-foam composites occurs when a network of fibres that are
interconnected form throughout a given specimen as discussed in chapter 2. Geometric
percolation occurs when the percolation threshold (a specific volume % of fibre) is reached. The
percolation threshold is dependent on fibre aspect ratio [49], [50], [69]. When the aspect ratio of
fibre increases, the percolation threshold decreases. The fibres in a percolated network are
84
theorized to transfer load to each other through direct interaction when the fibre-foam composite
is compressed. This fibre load transfer will lead to fibre deformation and loading and will
increase the compressive modulus of the fibre-foam composite. This foam reinforcement
mechanism through direct fibre interaction is known as mechanical percolation. Mechanical
percolation may also occur before complete geometric percolation when there is significant
localized fibre interaction leading to fibre bending when the fibre-foam composite is
compressed.
Figures 44 and 45 show that at 20 wt % long glass fibre, there is long range fibre
interaction and geometric percolation (Note: Image thresholding was used to isolate fibres).
According to figures 42 and 43 there is significant foam reinforcement when 20 wt % long glass
fibre is introduced into the PUF matrix. Therefore, it is likely that mechanical percolation is
occurring. Figure 46 shows that at 5 wt % long glass fibre there is some short order fibre
interaction but the fibres are not geometrically percolating. This is expected since there is
negligible foam reinforcement when 5 wt % fibre is introduced into the PUF matrix. Therefore, it
is likely that at some fibre content level after 5 wt % and before 20 wt % long glass fibre (0.07
volume % to 0.3 volume %), a fibre interaction threshold is reached leading to mechanical
percolation. This finding is comparable to the results observed by Favier et al [49], [50] when
they used cellulose nano whiskers to reinforce latex.
85
Figure 44: 20 wt % (0.3 volume %) 470 micron glass fibre embedded in foam. Densely
packed fibres are not isolated. Note: depth of field of image is 2.0 mm.
Figure 45: 20 wt % (0.3 volume %) 470 micron glass fibre embedded in foam. Densely
packed fibres demonstrate are not isolated. Note: depth of field of image is 2.0 mm.
86
Figure 46: 5 wt % (0.07 volume %) 470 micron glass fibre embedded in foam. Fibres are
not densely packed. Some fibre interaction. Note: depth of field of image is 1.8 mm.
In order to verify if the long glass fibres were mechanically percolating and bending
during compression, fibre-foam composite samples were compressed and analyzed under X-ray
Tomography. Using the SkyScan X-ray Tomography (MTS-50N) compressive analysis system,
long glass fibre-foam composite samples were compressed to 20 % strain to visualize if the long
glass fibres were mechanically percolating. Figures 47 and 48 show two separate isolated long
glass fibres in a fibre-foam composite that bend during compression due to fibre interaction.
87
Figure 47: 20 wt % (0.3 volume %) 470 micron glass fibre embedded in foam. Fibre that is
marked by arrow bends during compression due to interaction with fibre. Mechanical
percolation. Note: depth of field of image is 1.6 mm. Only fibres shown as PUF were
removed with threshold imaging techniques.
88
Figure 48: 20 wt % (0.3 volume %) 0.470 mm glass fibre embedded in foam. Fibre that is
marked by arrow bends during compression due to interaction with fibre. Mechanical
percolation. Note: depth of field of image is 1.6 mm. Only fibres shown as PUF were
removed with threshold imaging techniques.
The fibres are straight and unloaded when the composite is uncompressed. However, when the
composite is compressed to 20 % strain the isolated fibres are clearly bending and taking on
loading. Therefore, the fibres are mechanically percolating and there is fibre load transfer due to
direct interaction between fibres in these networks. Figures 49 and 50 show long glass fibre-
foam composites that are uncompressed and compressed up to 60 % strain. Initially, when there
is no compression (figure 49), the fibres are unloaded. However, at increased compression the
fibres mechanically percolate leading to more fibre loading. The fibre-foam composite that is
compressed to 60 % strain has the most fibre bending indicating that the effect of mechanical
percolation increases when there is more compression.
89
Figure 49: 20 wt % (0.3 volume %) 0.470 mm glass fibre embedded in foam. Uncompressed
foam. Depth of field of image is 1.6 mm. Only fibres shown.
Figure 50: 20 wt % (0.3 volume %) 0.470 mm glass fibre embedded in foam. Note: depth of
field of image is 1.6 mm. Only fibres shown as PUF were removed with threshold imaging
techniques. Increasing compression from top left (10%) to bottom right (60%).
1400 Microns
90
X-ray tomography imaging analysis highlighted that long glass fibres are geometrically
percolated at 20 wt % fibre content. Compressing the long glass fibre-foam composite samples
and visualizing them under the X-ray tomography highlighted that the long glass fibres were
mechanically percolating when the fibres were in a geometrically percolated network. In order to
verify that mechanical percolation of the long glass fibres is the physical mechanism that can
account for the increase in compressive modulus, a finite element model was developed. The
model and the analysis is discussed in the following section.
6.3 Predicting the compressive modulus of long glass fibre
reinforced PUFs using finite element modelling (FEM)
A finite element model (FEM) was developed in ABAQUS 6.1.2. CAE to understand and
predict physically the deformation of long glass fibre reinforced PUF. Previous studies [83], [84]
were successful in using FEM to predict mechanical properties of composite materials. The
experimental results from figures 42 and 43 and the FEM simulation results are compared to
deduce if there were any correlations.
6.3.1 Model overview and parameters
The FEM assumes that a deformable block of long glass fibre-foam composite is placed
in between two non-deformable rigid bodies. The fibre-foam composite block is deformed to 10
percent strain in compression for the main analysis by the rigid bodies. The fibre content is
varied from 0 to 20 wt % fibre content for the main analysis. The dimensions of the long glass
fibre-foam composite block are 4700 μm x 4700 μm x 4700 μm. Figure 51 shows the assembly
set up of the FEM in the ABAQUS environment.
91
Figure 51: Compression of fibre-foam composite samples in FEM simulation. Rigid Bodies
are solid and do not deform. For rendering purposes it is shown as hollow.
The foam is assumed to be a homogenous matrix where the cell structure is ignored for
simplicity. The compressive modulus of the foam matrix is based on the no fibre baseline curve
of figure 42, where foam modulus was adjusted for any changes in volume fraction of solids
(induced by cell nucleation) at the different long glass fibre content levels. The Poisson’s ratio
of the PUF was assumed to be 0 based on the information in chapter 3. Long glass fibres are
embedded within the foam homogenous matrix.
The fibres used in the FEM were uniformly sized cylinders, 16 microns in diameter and
470 microns in length. The elastic modulus and density values for the fibres were based on those
of E-glass (Elastic Modulus = 80 GPa, Density = 2550 Kg/m3, information also used for micro-
Rigid bodies
Foam Sample
92
mechanical models and discussed in chapter 3). Fibres were represented by beam elements in the
ABAQUS model. Beam elements in ABAQUS must be slender since they are treated as one-
dimensional objects in three-dimensional space. Their cross-sectional dimensions must be small
compared to dimensions along the axial direction allowing for thickness to be ignored and for the
elements to be treated as one-dimensional. The long glass fibres have a high aspect ratio and
therefore can be treated as beams. The long glass fibres embedded inside the foam are straight
initially. They are allowed to bend, buckle and elongate during deformation. The fibres were
randomly positioned and oriented inside the foam block structure by a Python script written to
set up the model in ABAQUS. Initially, the script selects an X, Y, Z coordinate randomly for one
end of the fibre to be placed. It then selects randomly three angles respective to the three axes to
orient the fibre in three-dimensional space. The process was repeated until the desired fibre
weight fraction was obtained. The fibres were treated as embedded elements in ABAQUS as
shown in figure 52. The nodes of embedded elements were joined with the nearest node of the
host element for simulation calculations. The host element would be the foam in this situation
and the nodes are essentially bonded. This represents how a fibre would be encased by the foam
matrix in a real setting. As well, this scenario represents perfecting bonding which is expected
since the micro-mechanical models predict very little shear stress transfer from the matrix to the
embedded fibres. Therefore, it is assumed that the fibres are perfectly bonded to the matrix
throughout the deformation since there is little chance of debonding or slippage to occur.
93
Figure 52: Nodal regions with embedded objects. Node A of the embedded object shares
node a of the host object. Image taken from the ABAQUS 6.1.2 Theory Manual.
The ABAQUS modelling technique chosen is an explicit dynamic simulation. Explicit
dynamic simulations are efficient in ABAQUS for large models that require quasi-static
deformation (similar to slowly deforming the sample in a compression tester) and can perform a
large number of small time increments efficiently. The simulation deforms the fibre-foam
composite block at 0.05 percent strain deformation for each step. There are a total of 200 steps,
leading to a total of 10 percent deformation (100 steps for 5 percent deformation). At each
deformation step, the force that is required to deform the foam block by the 0.05 percent strain
increment is calculated and recorded. Since the dimensions of the fibre-foam composite block
are known, a stress-strain curve for the deformation can be generated and the compressive
modulus can be determined.
6.3.2 Mesh Verification
In order to optimize the performance of the FEM simulation, three different foam blocks
with different mesh size elements were investigated. The foam blocks with the different mesh
94
size elements were compressed with rigid bodies to 10 percent strain. The load-displacement
curves produced from the compression simulation were analyzed to determine if they provided
accurate and acceptable results. Mesh sizes were selected to be finer than the fibre length.
Initially, larger mesh sizes (less accurate, less processing time) were analyzed. Smaller mesh
sizes were then selected until no improvement in accuracy for the results could be obtained.
Also, the FEM models that were compressed were visually analyzed to determine if the analysis
was accurate.
The three mesh sizes investigated were, Mesh 1: 100 microns, Mesh 2: 500 microns and
Mesh 3: 50 microns. The mesh types were all hexagonal. The analysis concluded that Mesh 1:
100 microns was the most suitable. Although Mesh 3 is a finer mesh, the results provided are
identical to the results obtained with Mesh 1 as shown in figure 53. Since Mesh 1 is a coarser
mesh, it requires less processing time and therefore is the preferred choice. Mesh 2 is not a
suitable choice because it is too coarse for the analysis and gives inaccurate readings as shown in
figures 53 and 54. Figure 54 highlights that mesh 2 results in distorted elements during
compression. Figures 55 and 56 show that the elements for mesh 1 and 3 do not distort during
compression and are therefore acceptable.
95
Figure 53: Mesh 1 and Mesh 3 provided the same results. Mesh 1 is coarser and requires
less processing time.
Figure 54: Mesh 2. The mesh size is too coarse. Fibres are smaller than the hexagonal mesh
blocks. The mesh blocks distort during compression.
96
Figure 55: Mesh 1. The mesh size is suitable. Fibres are larger than the hexagonal mesh
blocks. The mesh blocks do not distort during compression.
Figure 56: Mesh 3. The mesh size is very fine. Fibres are larger than the hexagonal mesh
blocks. The mesh blocks do not distort during compression. However, the processing time
is much longer than Mesh 1 but this does not improve upon the results.
97
6.3.3 Comparing the strain energy of an isolated fibre and a networked
fibre using FEM analysis
In order to understand how long glass fibre interaction and mechanical percolation affects
foam reinforcement, two fibre interaction scenarios were investigated using a FEM analysis. The
FEM simulation included a fibre-foam block that was compressed by rigid bodies to 10 percent
strain. One fibre investigated was placed in close-contact with neighbouring fibres and assumed
to interact with them during compression. This fibre represents the mechanical percolation
scenario. The other fibre that was investigated was placed inside the foam block and was
completely isolated and not interacting with any other fibres during compression. The traditional
micro-mechanical models predicted the foam matrix was too soft and would not transfer loading
into fibres. Therefore, the isolated fibre is not expected to deform during compression of the
fibre-foam composite block. Figure 57 shows a schematic of the two fibre scenarios.
98
Figure 57: ABAQUS set-up showing fibre scenarios. Fibre 1 is a close-contact fibre where
three other fibres are in close contact to it. Fibre 2 is an isolated fibre.
The strain energy was measured for both fibre 1 and fibre 2 to determine if the fibres
deform during compression. If the fibres deformed and acquired strain energy there would be
foam reinforcement since more energy (and force) would be required to compress the foam
system leading to a higher fibre-foam composite compressive modulus. The strain energy of a
fibre/system is equal to the integral of its stress strain curve of deformation. Therefore, strain
energy increases correlate directly with increases in modulus. In ABAQUS, it is easier to
measure the strain energy of individual fibres and therefore was selected. The FEM analysis
revealed that fibre 1 (non-isolated fibre) does deform during compression and the strain energy is
much higher than would be predicted by the traditional micro-mechanical models. In addition
Fibre 2 (isolated fibre) was essentially unstrained as predicted by the traditional micro-
99
mechanical models. Figure 58 shows that strain energy for fibre 1 is significantly higher than
fibre 2 during the compression analysis.
Figure 58: Fibre 1 – close contact fibre has significantly higher strain energy than Fibre 2 –
isolated fibre after the simulation completes. FEM Simulation.
Compression FEM analysis was completed on long glass fibre-foam blocks containing
randomly placed 5, 10, 15 and 20 wt % long glass fibre. These specific fibre-foam composite
blocks were compressed to 5 % strain. Figures 59 and 60 highlight that strain energy in the fibres
main contributor to the total strain energy of the composite after 5 percent compressive
deformation in the FEM simulation. As well, the fibre strain energy contribution increased at
higher weight fractions of glass fibre. In particular, the fibres contribution to the strain energy is
significant at fibre content of 15 wt % and beyond. This correlates with the micro-structure
analysis of the long glass fibre-foam composites. Therefore, mechanical percolation is believed
100
to account for the foam reinforcement when long glass fibres are embedded in the homogenous
foam matrix.
Figure 59: Total strain energy and total fibre strain energy at 5 % compressive
deformation. FEM Simulation.
101
Figure 60: Fibre strain energy percentage contribution to total system strain energy at 5 %
compressive deformation. FEM Simulation.
6.3.4 Compressive modulus prediction from FEM analysis
For the main FEM analysis, long glass fibre-foam blocks containing randomly placed 5,
10, 15 and 20 wt % long glass fibre were compressed to 10 percent strain in order to determine
the compressive modulus. Figure 61 and table 6 show that the predictions from the FEM
simulations are in line with experimental results. In fact, the FEM simulations correlate with the
experimental results much better than the traditional micro-mechanical models. This proves that
percolation and fibre AR are significant in PUF reinforcement.
102
Figure 61: Compressive modulus of long glass fibre-foam composite samples in comparison
to micro-mechanical models and FEM simulations.
Table 6: Sum of squared error normalized to compare micro-mechanical model predictions
and FEM simulation results to the actual experimental results for long glass fibre-foam
composites
Micro-Mechanical Models Sum of squared error
Sum of squared error
normalized
ROM 8.9 x 1010
5.6 x 104
IROM 3.5 x 105 5.69
Cox 3.3 x 105 5.32
103
Nairn 1.8 x 108 2.9 x 103
Mori-Tanaka 1.6 x 105 2.70
FEM 6.2 x 104 1
The combination of micro-structure analysis, experimental results and FEM simulations
strongly support the hypothesis that mechanical percolation plays a significant role in the
reinforcement of long glass fibre reinforced PUF. Therefore, the addition of long glass fibres to
the PUF system is beneficial for volume fractions greater than the percolation threshold.
6.4 Hemp fibre reinforced PUFs
Hemp fibres are much stiffer than the low modulus PUF matrix due to a high content of
cellulose. Fibres actually span multiple cells since the average hemp fibre length is much larger
than the average cell diameter measured. Therefore, the PUF matrix can be treated as a low
modulus homogenous mixture that is easily reinforced by the stiffer hemp fibres. In fact, since
the hemp fibres has the largest fibre length of the fibres investigated in this study, the foam
reinforcement was expected to be significant.
However, the hemp fibre and (Soyol® 2102) polyol mixture developed during the
manufacturing stage was too viscous to produce any samples. The high viscosity increase was
due to the long hemp fibre length and fibre entanglement. The hemp fibres bonded readily with
the polyol to further contribute to the viscosity increase. An alternate, less viscous soy-based
104
polyol (BiOH polyol-x0500) was used to produce hemp fibre-foam composites. The samples
generated were quite poor. Portions of the foam composite would flake off during handling. The
composites were very flexible and due to the destruction of the composites during handling
further mechanical stiffness studies under the compression tester were not pursued.
Figure 62 shows that the hemp fibres are much larger than the cells. However, large pores
did form that had diameters that are comparable to the length of the hemp fibres. X-ray
Tomography was used to analyze the cells of the hemp fibre reinforced PUFs as shown in figures
63 and 64. The cells are much larger than the cell diameters of the long glass fibre reinforced
PUFs. This could be detrimental. Although, the volume fraction of solids is the critical factor and
not cell diameter, it is hypothesized that the much larger cell diameters of the hemp fibre-foam
composites indicate increased porosity due to the poor mechanical properties exhibited during
handling. As well, the very thin cell struts exhibited by the hemp fibre-foam composites may
easily fracture during loading.
Figure 62: Hemp fibre-foam composite sample highlighting the long hemp fibres and large
pores present.
105
Figure 63: 5 wt % hemp fibre-foam composite. Hemp fibre not shown. Note: depth of field
of image is 2.0 mm.
106
Figure 64:5 wt % hemp fibre-foam composite. Hemp fibre not shown. Note: depth of field
of image is 1.0 mm. Image thresholding techniques used to isolate PU struts.
6.5 Conclusions
The introduction of 470 μm glass fibres into a PUF matrix resulted in a decrease of solid
volume fraction. However, the 470 μm glass fibres significantly reinforced the PUF matrix
resulting in an improved fibre-foam composite compressive modulus at higher 470 μm glass
fibre weight fractions. Through X-ray tomography it was determined that the 470 μm glass fibres
were not embedded within the PUF cell struts but were larger than the cell diameters. Therefore,
it was assumed that 470 μm glass fibre-foam composite represented the long-fibre reinforcement
107
scenario. The microstructures from X-ray tomography showed fibre interaction and percolation
at high long glass fibre content, suggesting that mechanical percolation of the long glass fibres
was possible.
The improvement in compressive modulus could not be predicted by the traditional
micro-mechanical models: Mori-Tanaka, Shear-lag theory and ROM. An FEM simulation was
developed to understand the foam reinforcement and was found to predict the experimental
compressive modulus improvements correctly. The FEM simulation attributed the improvement
in compressive modulus at higher long glass fibre content levels due to mechanical percolation
of the fibres. Specifically, improvements were observed when fibre content was high and
assumed to have crossed a geometric percolation threshold.
Finally, hemp fibre reinforced PUF was also produced but the hemp fibre introduction
did not reinforce the PUF matrix.
108
Chapter 7
Analyzing the compressive properties of intermediate length glass
fibre reinforced PUFs
7.1 Introduction
260 micron fibre length glass fibres were studied as potential reinforcing fibres inside
PUF. Glass fibres have very high stiffness in comparison to the low modulus PUF matrix and
have been used extensively to reinforce weaker polymers. The 260 micron glass fibres were
investigated to represent the intermediate-fibre reinforcement scenario discussed in chapter 4.
Based on the analysis in chapter 3, the 260 micron fibre length glass fibres are expected to lie
within the cell and any foam reinforcement by the fibres would be with direct interaction
between fibres and individual cells. These intermediate length glass fibres are not likely to be
found within the cell struts.
7.2 Micro-structure of intermediate length glass fibre reinforced
PUFs
X-ray Tomography was used to visualize the micro structure of the 260 micron glass
fibre reinforced PUFs. Figures 65 to 67 highlight that at various fibre content, the 260 micron
glass fibres are not embedded within the cell struts of the PUF but do not span cells. The figures
confirm that 260 micron glass fibre reinforced PUF represents the intermediate-fibre scenario.
Since the intermediate glass fibres are comparable in length to the cell diameter, it is assumed
that any reinforcement will depend on fibres interacting with individual cells. Two hypotheses
109
proposed in chapter 4 are investigated in order to understand how the intermediate glass fibres
will affect foam reinforcement. The first hypothesis assumes that the fibres can form bridges
between cell struts, since intermediate fibre length is similar to the cell diameter. These bridges
prevent the cell struts from bending and buckling leading to reinforcement of individual cells
which reinforce the foam overall. The alternative hypothesis is that no bridges are formed and
the fibres do not reinforce the cells and the overall foam. Based on qualitative observations of the
micro structures in figures 65 to 67, the intermediate glass fibres do not appear to form many
bridges and therefore could be detrimental to the compressive modulus of the fibre-foam
composite.
Figure 65: 5 wt % (0.07 volume %) 260 micron glass fibres embedded in foam. Fibres not
confined to cell struts. Darker and thicker cylindrical objects are fibres. The image
contrast levels have been adjusted to show the glass fibres more clearly. Note: depth of field
of image is 0.8 mm.
110
Figure 66: 5 wt % (0.07 volume %) 260 um Glass fibres embedded in foam. Fibres not
confined to cell struts. Darker and thicker cylindrical objects are fibres. Thinner and
lighter objects are the cell structure. The image contrast levels have been adjusted to show
the glass fibres more clearly and to display how they are much larger than the cell struts
but not cell diameters. Note: depth of field of image is 0.8 mm.
111
Figure 67: 20 wt % (0.3 volume %) 260 um Glass fibres embedded in foam. Fibres not
confined to cell struts. Darker and thicker cylindrical objects are fibres. Thinner and
lighter objects are the cell structure. The image contrast levels have been adjusted to show
the glass fibres more clearly and to display how they are much larger than the cell struts
but not cell diameters. Note: depth of field of image is 0.8 mm.
Mechanical percolation is unlikely to occur during compression of the intermediate glass
fibre-foam composites. Figure 68 show that at 5 wt % glass fibre there is no fibre interaction and
no geometric percolation. Figure 69 shows at 20 wt % glass fibre there is some localized
interaction but no long range geometric percolation or fibre interaction as observed in the micro
structures of long glass fibre reinforced PUF. Therefore, mechanical percolation of the
intermediate glass fibres is not likely to provide foam reinforcement.
112
Figure 68: 5 wt % (0.07 volume %) 0.260 mm glass fibre embedded in foam. Fibres are not
densely packed. Very limited fibre interaction. Depth of field of image is 0.8 mm.
Figure 69: 20 wt % (0.3 volume %) 0.260 mm glass fibre embedded in foam. Fibres are not
as densely packed as 0.470 mm glass fibre (long fibre) case at 20 wt % (0.3 volume %).
Limited fibre interaction. Depth of field of image is 0.8 mm.
113
7.3 Changes to intermediate glass fibre foam composite
compressive modulus with increases in intermediate glass
fibre weight fraction
Similar to the MCC fibres and long glass fibres, intermediate glass fibres encourages cell
nucleation in PUF. By increasing the gas content, the porosity of PUF also increases. This
explains why the volume fraction of solids the intermediate glass fibre-foam composite decreases
somewhat when intermediate glass fibre is introduced into PUF as shown in figure 70. However,
the decrease is not as pronounced as observed for the MCC fibre and long glass fibre composites.
Future work must be completed to understand this result. Correspondingly, the expected
compressive modulus for the intermediate glass fibre-foam composite should also decrease
according to Gibson-Ashby theory since the solid volume fraction decreased. Figure 71 shows
the expected compressive modulus no fibre base foam curve for the fibre-foam composite
assuming no reinforcement and the experimental results at 0, 5, 10, 15, and 20 wt % intermediate
glass fibre content. Since, the average experimental results are higher than the baseline curve at
all fibre content levels, there could be some reinforcement of the PUFs due the intermediate glass
fibres. However, the difference between experimental and base foam is quite small and is much
lower than the long glass fibre case suggesting minimal reinforcement. Furthermore, the
difference between experimental and baseline for this case is comparable, although smaller than
the MCC fibre case. The MCC fibres have less than ½ the modulus of the intermediate glass
fibres and therefore should be poorer at reinforcing the foam. Therefore, it is assumed that the
no reinforcement hypothesis proposed earlier is true and the intermediate glass fibres are
detrimental to the PUF compressive modulus and should be avoided.
114
Figure 70: Volume fraction of solids decreases as intermediate glass fibre weight
percentage increases.
Figure 71: No-fibre base foam modulus expected through Gibson-Ashby theory to
experimental results.
115
The compressive modulus experimental results for the intermediate glass fibre-foam
composites were compared to micro-mechanical models where it was assumed that the glass
fibres were actually embedded inside the cell struts. These glass fibres were assumed to reinforce
the cell struts as the MCC fibres did. The theoretical micro-mechanical models (ROM, Mori-
Tanaka, Shear-Lag Cox, Shear-Lag Nairn) in Figure 72 and table 7 show that if the intermediate
glass fibres were embedded inside the cell struts, the fibres would have significantly reinforced
the foams overall. Mori-Tanaka theory suggest improvements of nearly 1000 kPa and Cox’s
shear-lag theory predicts improvements of over 400 kPa at 10 wt % glass fibre, provided they
were reinforcing the struts by being embedded in them. Therefore, fibre placement within the
foam structure is critical to fibre reinforcement. The micrographs in section 7.2 highlight that
these intermediate length glass fibres were not embedded in the cell struts. These intermediate
length fibres should be avoided, since they would perform much better as very short fibres.
116
Figure 72 a and b: Compressive modulus of intermediate glass fibre-foam composite
samples in comparison to micro-mechanical models where glass fibres are assumed to
reinforce cell struts.
117
Table 7: Sum of squared error normalized to compare micro-mechanical model predictions
to the actual experimental results for intermediate glass fibre-foam composites where the
fibres are assumed to reinforce cell struts.
Micro-Mechanical Models Sum of squared error
Sum of squared error
normalized
ROM 7.0 x 107 2900
IROM 2.4 x 104 1
Cox 4.1 x 106 171
Nairn 4.8 x 107 2004
Mori-Tanaka 1.4 x 108 591
The compressive modulus experimental results for the intermediate glass fibre-foam
composites were compared to micro-mechanical models where it was assumed that the
intermediate glass fibres were embedded inside a homogenous matrix. This physical model
represents the long glass fibre case. Figure 73 and table 8 show that the micro-mechanical
models (Mori-Tanaka, Cox’s shear-lag theory and IROM) predict the experimental results well.
However, since the experimental results coincide with the IROM (worst-case reinforcement
scenario), the intermediate glass fibres poorly reinforce the very low modulus PUF matrix. The
poor reinforcement is attributed to a very low modulus matrix and that the intermediate length
glass fibres are too short for sufficient stress transfer from the matrix. As well, the aspect ratio of
118
these intermediate length glass fibres is too low for mechanical percolation to occur at these fibre
content levels, therefore, this suggests that the intermediate length glass fibres should be avoided.
Figure 73 a and b: Compressive modulus of intermediate glass fibre-foam composite
samples where glass fibres are assumed to reinforce a homogenous PUF matrix.
119
Table 8: Sum of squared error normalized to compare micro-mechanical model predictions
to the actual experimental results for intermediate glass fibre-foam composites where the
fibres are assumed to reinforce a homogenous PUF matrix
Micro-Mechanical Models Sum of squared error
Sum of squared error
normalized
ROM 9.9 x 1010
1.7 x 105
IROM 4.6 x 104 2.0
Cox 3.2 x 104 1.9
Nairn 1.3 x 106 58.6
Mori-Tanaka 10.0 x 104 1
7.4 FEM analysis theorizing fibre reinforcement due to fibre-cell
bridging
Fibre bridging was investigated with a FEM simulation where fibres span a single cell
and are pinned by the cell struts. Fibre bridging could possibly apply to both intermediate and
long fibres that span multiple cells. Different cases were studied where the fibre arrangement
within the cell was varied to understand their reinforcing capabilities for an individual cell. By
reinforcing individual cells, and increasing the force required to compress an individual cell, the
120
overall foam modulus should also increase. Figure 74 illustrates the seven fibre bridging cases
investigated with the FEM analysis.
Figure 74: Seven fibre bridging cases investigated with FEM analysis. Vertical loading.
The FEM created was a three dimensional explicit dynamic analysis in ABAQUS CAE
6.1.2. Each of the fibre-cell composite cases was compressed to 10 percent strain and the
compressive force required was recorded. Fibre surfaces were assumed to have perfect bonding
with the cell strut. Figure 75 shows the dimensions assumed for the cell and fibre in the analysis.
The mesh sizes used were 3 microns hexagonal on the fibre, and 2.5 microns hexagonal for the
cell and similar mesh verification was performed as described earlier. The models were also sub-
divided into half due to symmetry relationships.
Figure 75: Dimensions of cell and fibre used in FEM analysis
Loading
Direction
121
Figure 76 shows the deformation some of the cell cases when loading is applied
vertically. Both vertically and horizontally aligned fibres provide cell reinforcement.
Horizontally placed fibres restrict the vertical cell struts from bending and increase the overall
required load to deform the cell. The horizontally aligned fibres are elongated during
deformation. However, they do not restrict the top and bottom cell struts from bending.
Vertically aligned fibres undergo pure compression deformation initially which reinforces the
cell significantly as shown in figure 77 (case 3 and 6). However, the vertically aligned fibres
eventually buckle which prevents any further reinforcement. The diagonally aligned fibres do
not take on loading during deformation and do not restrict the cell struts. Therefore, they do not
provide any cell reinforcement.
Figure 76: Deformation of intermediate cases with vertical loading. Images taken directly
from ABAQUS.
122
Figure 77: Force-Displacement curves for each of the seven cases.
7.5 Conclusions
Through X-ray tomography it was determined that the 260 µm glass fibres were not
embedded within the PUF cell struts and did not span multiple cells. Therefore, it was assumed
that these glass fibres represented the intermediate length fibre reinforcement scenario. However,
the inclusion of 260 µm glass fibres into the PUF matrix produced almost no reinforcement.
These intermediate length glass fibres were not long enough to form a mechanically percolating
network, as confirmed by the tomography and the measured composite moduli. Furthermore, the
260 µm fibres do not embed or reinforce the cell struts as observed for the MCC fibres. They do
not have a high aspect ratio for reinforcement through shear stress transfer as predicted by the
micro-mechanical models of Mori-Tanaka and Shear-lag theory. Finally, fibre bridging was not
conclusively observed in the micrographs. Therefore, the results suggest that intermediate length
123
fibres should be avoided when attempting to reinforce polyurethane foams. This may be
problematic when using wood fibres for PUF reinforcement as they can have similar aspect
ratios and fibre length as 260 µm glass fibres.
An FEM analysis was conducted on seven different arrangements of fibre bridging. The
analysis showed that vertically aligned fibres that are pinned to the top and bottom cell struts and
are compressed axially will provide the best reinforcement of an individual cell. However, the
reinforcement is limited due to the fibres buckling. Horizontally aligned fibres that restrict the
vertically aligned cell struts from bending provide some reinforcement, however it is quite small
in comparison to the vertically aligned fibres. Diagonally aligned fibres provide no
reinforcement.
124
Chapter 8
Novel method to meter and deliver natural fibre for reinforcement
of PUFs
Note: This chapter is adapted from our published manuscript: A novel method to deliver
natural fibre for mechanical reinforcement of polyurethane foam - Journal of Cellular
Polymers (2014) – Volume 33, Issue 3, Pg 123-138
8.1 Introduction
Currently no mechanism exists to produce residual natural fibre reinforced PUFs on a
large scale. Residual natural fibre reinforced PUFs are manufactured on a laboratory scale by
introducing fibre into polyol and hand mixing or mixing with a mechanical stirrer for dispersion.
The residual natural fibre-polyol mixture is highly viscous so that even at the lab scale, fibre
dispersion is often a problem. Yuan and Shi [85] reported an increase of 100 to 160 poise at 20
% wood flour content. It is not possible to hand mix at these levels and mechanical mixing can
result in damaged fibres. Therefore, the amount of fibre used is often constrained, resulting in
composite PUFs that are not uniform in rigidity and are limited in their potential stiffness.
Figures 78 to 80 highlight the dispersion problems associated with the current mixing methods.
Figure 78 shows maple fibre clearly constrained to one end of the fibre-foam composite sample
leading to an uneven distribution of compressive modulus. Figures 79 and 80 are schematics to
show that after foam expansion the fibre bundle may not disperse. In this study, a novel
technique for producing high residual natural fibre content PUFs with uniform fibre dispersion at
a large scale is described.
125
Figure 78: Maple fibre is clearly concentrated to end of the PUF sample. Region with more
fibre was stiffer. Scale bar is 2 cm.
Figure 79: Fibre being placed into cup of foam mixture for hand mixing.
2 cm
126
Figure 80: After hand mixing in viscous foam mixture, fibre is not evenly distributed.
8.2 Existing art
Glass fibre reinforced PUFs are easily produced. They are manufactured by chopper gun
spray systems. The glass fibre is supplied as a roving, which has continuous strands of glass fibre
wound onto a spool. One end of the roving is brought to the chopper gun by a roller-pulley
system. The chopper gun is hand held and has a sharp blade that slides down to bend and break
the glass fibre strand since the glass is brittle. The strand is broken into fibres that are of the same
length and cross-sectional area by the periodic motion of the blade. The fibres are released as a
stream with a constant mass flow rate. The stream is directed by the operator of the hand held
chopper gun. The length the fibres are cut and the speed that the fibre strand is supplied can be
changed allowing the mass flow rate to be adjusted as needed. The polyol and isocyanate are
127
supplied from tanks and are metered according to the desired chemical formulation. The metered
glass fibre mixes in air with the polymeric streams and the mixture lands on a crevice in a wall
panel and cures. This technique allows for quick and large scale production of glass fibre
reinforced plastics with high volume fractions of glass fibre. Figure 81 highlights a typical
chopper gun.
Figure 81: Typical chopper gun where stream of metered chopper glass is released to
interact in air with polymeric streams [86]. Image used directly from source.
Residual natural fibre PUFs would be difficult to make with the chopping gun apparatus.
The spray system is valid but metering loose residual natural fibre with the chopping mechanism
is not possible. Figure 82 shows loose residual natural fibre are in clumps which would require
further processing in order to be formed into yarn suitable for chopping. These extra processing
steps would increase the overall cost of residual natural fibre PUFs and therefore an alternate
metering method that does not require further processing to the residual natural fibre was
128
pursued. Some techniques have been invented to meter natural fibre flocks coarsely [87]–[90].
These methods involve using rollers and gates to control the flow of fibre. However, these
techniques would not be suitable for use with a hand held portable chopper gun spray system. In
this study, a novel invention is proposed and evaluated that was independently developed and
can be used in a hand held chopper gun-spray system.
Figure 82: Loose natural fibre. Added manufacturing steps required to make into yarn for
chopping.
8.3 Metering natural fibre
The chopper gun spray system requires a mechanism that can easily meter residual
natural fibres as they cannot be chopped and metered. The following concept proposed can meter
loose natural fibres at a high mass flow rate without requiring the loose fibres to be formed into
129
yarn. The concept can be connected with the spray system to develop natural fibre reinforced
PUFs.
8.3.1 Concept description
Residual natural fibres such as Aspen, Maple, Hemp and Hemlock would be difficult to
make into roving and are commonly delivered in bulk loose form. Longer fibres are often
entangled into loose tufts and are not free flowing. Therefore, there is a need for a system that
can meter and deliver a stream of residual natural fibre at high mass flow rates. In this study, a
novel system was developed to meter residual natural fibre consistently.
The device involves residual natural fibre being fed between two dimensionally identical
roller wheels as shown in figures 83 and 84.
Figure 83: Natural fibre approaching rolling wheels. Driver wheel is powered to rotate
counter-clockwise and forces driven wheel to rotate clockwise. Driven wheel can move only
horizontally. Applied force on driven wheel forces it to contact the fixed driver wheel.
Frame of apparatus not shown. Direction rotations are based on a front view
130
Figure 84: Residual natural fibre separating rolling wheels. Applied force on driven wheel
compresses residual natural fibre. Direction rotations are based on a front view.
The driver wheel is fixed in position in all directions with respect to the frame of the apparatus.
The driver wheel rotates counter clockwise (figures 83 and 84) with the aid of an electric motor.
The driven wheel can move laterally and rotate freely. An applied lateral load forces the driven
wheel to contact the driver wheel as shown in figure 83. With contact, the driven wheel rotates
clockwise due to the counter clockwise rotation of the driver wheel. As residual natural fibre
enters the contact region, the driven wheel will be forced laterally away from the driver wheel to
create a nip as shown in figure 84. The applied lateral load on the driven wheel compresses the
residual natural fibre to a volume that is dependent on the mass of the fibre present. Using a
pneumatic actuator, the force on the driven wheel is maintained at a fixed level, independent of
its lateral motion, and hence the bulk density of the residual natural fibre in the nip is
approximately constant. The roller wheel nip gap can be measured using an LVDT (Linear
Variable Displacement Transducer) and thus the mass of the fibre present can be determined.
131
When more fibre is drawn into the nip, the nip gap increases, since the pressure and force on the
driven wheel is constant. At the same time, the driver rotational speed is reduced (based on
equation 49) in an attempt to maintain a constant mass flow rate through the roller wheels. The
opposite happens as the nip gap decreases. This feedback loop of measuring the nip gap and
adjusting the rotational speed operates continuously.
8.3.2 Mathematical relationships
The nip gap correlates with the mass of natural fibre compressed. The applied lateral load
on the driven wheel remains constant providing a consistent applied pressure to fibres present
between the roller wheels. Experiments conducted on loose hemp fibre determined that a simple
relationship exists between applied pressure and the bulk density of hemp fibre (density of fibre
and air when fibre tuft is compressed). Therefore, the bulk density of hemp fibre is known if it is
subjected to a known pressure in the nip. If the bulk density of the fibre is known and the volume
of fibre is measured through the nip gap, the mass of the fibre can be calculated and thus the
fibre can be delivered at a constant mass flow rate by adjusting the speed of the roller wheels.
In order to determine the relationship between applied pressure and the bulk density of
hemp fibre, 5 samples of 2 grams of hemp fibre were compressed using a Sintech model # 20
compression tester fitted with a 454 kg load cell. The tester moved at 2.5 mm/sec to loads as high
as 80 kg. The relationship between the applied pressure (kPa) and bulk density of the fibre
(kg/m3) can be seen in figure 85.
132
Figure 85: Applied Pressure (kPa) and Bulk Density (kg/m3) of hemp fibre at a mass of 2
grams. Note: curve is the median curve of the five samples tested.
The Applied Pressure-Bulk Density relationship is shown in equation 44.
Equation 44 𝛒 = −𝟎. 𝟎𝟎𝟎𝟑𝑷𝟐 + 𝟎. 𝟔𝟗𝟎𝟏𝑷 + 𝟏𝟓𝟕. 𝟒
Where P = applied pressure [KPa], ρ = bulk density [kg/m3] and 157.4 is the bulk density at
atmospheric pressure.
Bulk density is a relationship between the mass of fibre (M) and volume of fibre (V) being
compressed and applied pressure (P) is a function of the force/applied load (F) on the cross-
sectional area (A) of fibre being compressed. Substituting M/V for ρ and F/A for P leads to
equation 45.
133
Equation 45 𝑴
𝑽= −𝟎. 𝟎𝟎𝟎𝟑 × (
𝑭
𝑨)
𝟐
+ 𝟎. 𝟔𝟗𝟎𝟏 × (𝑭
𝑨) + 𝟏𝟓𝟕. 𝟒
Figure 86 highlights the dimensions of the compressed fibre as it passes through the roller
wheels to further simplify V and A.
Figure 86: Important dimensions to consider when fibre is compressed between roller
wheels. Frame of apparatus not shown. Front view shown. T is the width of the roller and
is perpendicular to the front view.
The dimensions of V are a function of Height of fibre-wheel interaction (H), Nip Gap Distance
(D) and width of the roller (T). H and T are constant as they are dependent on the dimensions of
the wheels specifically. H can be determined through visual inspection when the wheels are
placed adjacently as shown in figure 86. A is function of H and T, which allows equation 2 to
simplify to equation 46.
134
Equation 46 𝑴
𝑯×𝑻×𝑫= −𝟎. 𝟎𝟎𝟎𝟑 × (
𝑭
𝑯×𝑻)
𝟐
+ 𝟎. 𝟔𝟗𝟎𝟏 × (𝑭
𝑯×𝑻) + 𝟏𝟓𝟕. 𝟒
By isolating for M, equation 47 highlights the linear relationship between mass of fibre being
compressed and the nip gap.
Equation 47 𝐌 = (𝐇 × 𝐓 × 𝐃) (−𝟎. 𝟎𝟎𝟎𝟑 × (𝑭
𝑯×𝑻)
𝟐
+ 𝟎. 𝟔𝟗𝟎𝟏 × (𝑭
𝑯×𝑻) + 𝟏𝟓𝟕. 𝟒)
The final desired relationship involves setting the rotational speed of the driver wheel based on
the measured nip gap (D). Equation 48 shows the relationship between the driver rotational
speed and the mass of fibre being compressed.
Equation 48 𝐑𝐒 =𝑴𝑭𝑹×𝟔𝟎(
𝒔
𝐦𝐢𝐧)
𝑴×𝟏𝟎𝟎𝟎(𝒈
𝒌𝒈)×𝝀
Where RS = Rotational speed of driver wheel in revolutions per minute (RPM), MFR = desired
mass flow rate of fibre in grams per second (g/s), and 𝜆 = proportionally ratio of the
circumference of the roller wheels to the (H) height of fibre being compressed (i.e. the dimension
of the compressed fibre that contacts the wheel surface).
Since, M is the mass of fibre being compressed, 𝜆 is used to account for the difference between
the mass of fibre being compressed and the total mass that would be present in one revolution of
the roller wheels. This allows the correct RS value to be set. Finally, the RS value adjusts with
changes in nip gap (D) according to equation 49.
135
Equation 49 𝐑𝐒 =𝑴𝑭𝑹×𝟔𝟎(
𝒔
𝐦𝐢𝐧)
(𝐇×𝐓×𝐃)(−𝟎.𝟎𝟎𝟎𝟑×(𝑭
𝑯×𝑻)
𝟐
+𝟎.𝟔𝟗𝟎𝟏×(𝑭
𝑯×𝑻)+𝟏𝟓𝟕.𝟒)×𝟏𝟎𝟎𝟎(
𝒈
𝒌𝒈)×𝝀
8.4 Theoretical modelling through simulations
A mathematical simulation was developed to test the viability of the natural fibre
metering concept. The simulation was developed using the fundamental equations in the
previous section and generated in Microsoft ® Excel. The simulation determined the expected
range in nip gap (D) when various tufts of fibre are fed through, which allowed for an
appropriate selection of an LVDT for the actual apparatus. The simulation also determined the
distribution mass flow rate of fibre over time, to validate the basic concept.
The simulation assumes that fibre feeding into roller wheels is sectioned off into multiple
blocks, where each block contains a certain amount of fibre as shown in figure 86. For the
simulation, it was assumed that the diameter and width (T) of the wheels was 50 mm and 23 mm
(these were the dimensions for wheels available for preliminary analysis). Through visual
analysis of the wheels compressing hemp fibre, it was assumed that each block in figure 87 has a
cross-section of 23 mm x 10 mm (T x H), and is compressed by the wheels at a pressure of 200
kPa. 200 kPa was chosen because equation 44 correlates well with the curve in figure 85 at 200
kPa. The length of the blocks is dependent on the nip gap (D).
136
Figure 87: Sections of fibre blocks between the wheels. Block 2 is compressed by the
wheels. Blocks are not drawn to scale.
Block 1 represents a new block of fibre that enters between the roller wheels. To simulate the
natural variability of mass that would be fed to the roll nip in a real system, the mass of Block 1
was assigned randomly in the range of 0.2 to 1 gram (range selected based on analysis of
available hemp fibre tufts – supplied by Hempline Inc.). Block 2 is the block that is compressed
by the roller wheels. At each iteration, (D) is calculated using equation 47. Block 2 is then sent to
block n’s location and released into the stream. Block 1 is sent to Block 2’s location and a new
137
fibre block enters into Block 1 and the iteration is complete. The simulation runs for 3000
iterations and table 9 displays the range of (D) calculated.
Table 9: Results for the simulation to determine the nip gap (D).
Range Value
Max Wheel Spacing 27.60 mm
Min Spacing 5.52 mm
Average Spacing 16.54 mm
The results from table 9 suggest acquiring an LVDT with a minimum range of 50 mm to
guarantee that the maximum nip gap can be detected.
When traditional chopper spray guns are used, the operator sprays glass fibre and the
polyurethane contents over the mould for a short period of time. The operator does not spray the
fibre in one location continuously but sprays the fibre throughout the mould and achieves a
uniform layer of fibre. Each layer that is sprayed must have a uniform distribution of fibre to
ensure that the mechanical properties are similar in every section of the composite foam
developed. The natural fibre metering concept must replicate this ability of glass fibre chopper
spray guns. Figure 88 represents a SIP that is divided into square deposition zones of equal size.
138
Figure 88: Mock SIP divided into equal sized boxes. The simulation was used to deposit
blocks of fibre into each deposition zone.
The simulation takes the fibre blocks and meters them according to equation 48 as they
pass through the roller wheels and places them into the 12 deposition zones of figure 88. The
simulation uses equation 48 to vary the rotational wheel speed as different amounts of fibre
blocks enter the roll nip, to attempt to produce a steady mass flow rate of 2 g/s. Each iteration
releases a block of fibre into the deposition zones starting with the zone at the left hand side of
figure 88 that is marked in black. The simulation will continue to place fibre blocks into the same
zone for 1 second. After 1 second, the simulation begins placing fibre blocks into the zone
immediately to the right of the first zone. This process continues until all the zones obtain fibre
going from left to right for 1 second each to create a layer of fibre. The 1 second parameter was
selected to represent a real glass chopper spray gun system case where the operator will place
fibre in a certain zone for 1 second. In the simulation, multiple blocks of fibre were placed in
each zone for the given time parameter of 1 second.
The simulation was used to examine a second scenario where the rotational wheel speed
was not adjusted and keept constant at 40 RPM (the required wheel speed to obtain an average
flow rate of 2 g/s for a mass of 0.6 grams in the fibre block, median mass of hemp fibre tufts).
The resulting variability of mass of fibre found in the deposition zones is shown in table 10. This
139
analysis determines the feasibility of the concept to deliver equal amounts of fibre to all regions
of a SIP.
Table 10: Results from the second part of the simulation for fibre distribution.
Trial # Wheel Speed
Max Zone
Mass
Min Zone
Mass
Avg. Zone
Mass
St. Deviation
1A
Variable (2
g/s)
2.6 1.73 2.02 0.259
1B
Constant Speed
– 40 RPM
3.47 1.33 1.96 0.576
2A
Variable (2
g/s)
2.42 1.69 2.02 0.260
2B
Constant Speed
– 40 RPM
2.81 1.43 2.09 0.512
3A
Variable (2
g/s)
2.39 1.51 1.99 0.268
3B
Constant Speed
– 40 RPM
2.93 1.05 2.07 0.574
4A
Variable (2
g/s)
2.62 1.68 2.06 0.303
140
4B
Constant Speed
– 40 RPM
2.72 1.04 1.95 0.469
5A
Variable (2
g/s)
2.44 1.66 2.00 0.256
5B
Constant Speed
– 40 RPM
2.73 1.38 1.94 0.398
6A
Variable (2
g/s)
2.34 1.67 1.99 0.207
6B
Constant Speed
– 40 RPM
2.51 1.39 1.89 0.331
From table 10, the range (maximum mass – minimum mass) and standard deviation of the mass
distribution in the deposition zones is significantly larger when constant speed is used. An
unpaired t-test was performed to compare the difference between the standard deviations of the
constant speed simulation results and the standard deviations of the variable speed simulation
results. A 95 percent confidence interval was determined for the difference in standard deviations
lying in the range of 0.307 to 0.129. This suggests that the standard deviation in the mass found
in the different zones will be higher when constant speed is used. Although, if the fibre blocks
were allowed to be deposited into each zone for an infinite period of time, Scenario 1 and 2
would provide equal total mass results, as they would average over time. However, each layer
141
would not be uniform in fibre mass distribution. Therefore, by varying the rotational wheel speed
according to equation 48, fibre will be distributed more uniformly.
8.5 Results for prototype fibre metering device
A prototype for metering natural fibre was developed to test the concept. Figure 89 shows
the prototype.
Figure 89: Prototype used for fibre mass metering including an LVDT, pneumatic air
cylinder and roller wheels.
A stainless steel pneumatic air cylinder (Mcmaster-Carr) was used to apply constant pressure of
200 kPa to the hemp fibre (Hempline, 1.27 cm length) as it passes through the roller wheels
(Mcmaster-Carr, 6.35 cm diameter, 23 mm thickness, 80 A Durometer, Neoprene). The RPM to
nip gap (D) relationship of equation 49 was programmed into the continuous servo motor system
142
(Galil Motion Control) that rotated the driver wheel for a desired fibre mass flow rate of 2 grams
per second. The nip gap (D) was measured by an LVDT (Omega, Model Number LD-610-25, +/-
25 mm stroke length).
Tests were conducted on the ability of the prototype to meter single hemp fibre tufts at a
uniform rate. The mass of the fibre tufts that would pass through the roller wheels was measured
on an electronic balance scale. The fibre tufts were then picked up manually and dropped into the
rotating roll nip. A stopwatch was used to measure the time required for the fibre tufts to be
processed through the roller wheels. The mass of each fibre tuft sample was divided by the time
it took to go through the roller wheels to obtain the mass flow rate. Table 11 shows the results
obtained from the experiments.
Table 11: Results of mass flow rate from the developed prototype.
# of Samples Average RPM
Average Mass Flow
Rate Calculated (g/s)
Mass Flow Rate
Variance of Samples
25 Variable (2 g/s) 2.19 0.08802
15 Constant – 100 1.69 0.2188
5 Constant – 600 4.18 3.0444*
5 Constant – 500 7.39 7.278*
Note: * These results may be limited due to the experimental apparatus
Table 11 shows that the prototype device can adequately meter hemp fibre at a constant mass
flow rate. The device varies the roller speed based on the roll nip gap measured as variable
143
amounts of fibre mass run through the nip at constant applied pressure. There is a noticeable
improvement in feed uniformity when varying the rotational speed versus keeping the speed at a
constant value. The variance between the mass flow rates of the samples lowers by more than a
factor of 2 when the rotational speed is varied. These results give confidence to proceed into
developing the device further and improve on existing results.
8.6 Limitations and ideas for complete system
The first prototype device developed has limitations, but has demonstrated the feasibility
of the basic concept. The as-received hemp fibre was clumped together and had to be loosened
and fed manually to the rotating wheels. A completely automated system would extract thick
fibre tufts from a bulk container and provide a narrow stream of metered residual natural fibre
that could be mixed with polyol and isocyanate in air. Figure 90 shows a diagram for a proposed
system and includes the various components that must be developed.
144
Figure 90: Design Apparatus of the complete set-up to collect, loosen, meter and deliver
natural fibre as a stream with a constant mass flow rate. Note: The image is not to scale
and some components have been enlarged to show functionality. Ideally, the components
will be compact and placed into a box for portability.
Component (1) is natural fibre in large clumps entering a hopper (2) where it is collected by the
system. The hopper (2) vibrates to aid in separating the fibre which is initially thick into smaller
tufts. As well, the vibration would remove loose fibre particles that can be collected and metered
separately. Component (3) is a picker/stripper roller that grabs bulky fibre and strips it into
smaller strands that will be easier to meter. Components (4), (5) and (6) combine to form a
vacuum system that uses suction to remove the fibre from the picker/stripper roller. The vacuum
system also drives the fibre to the metering station (8). Component (6) is a centrifugal fan that
uses open blades to prevent fibre trapping on the blades. Component (7) is a pipe that is
145
connected and placed directly above the metering device (8). This pipe (7) can be connected to a
cyclone apparatus to lower the velocity of the fibre in the air stream for a smoother delivery to
the metering station. Components (4 through 7) allow for a constant feeding of natural fibre to
the metering station (8) which can then produce metered fibre for the delivery station (9 and 10).
Preliminary tests were conducted where fibre was collected by the intake of a centrifugal fan and
the outward velocity of the fibre was lowered by a cyclone apparatus.
After the fibre (1) has been metered by component (8) it must be delivered as individual
fibres in a stream that can mix in air with the polymeric constituents of the PUF. Component (9)
shows two small rotating wheels that will rotate at high speeds (~ 3000 rpm) to strip strands of
fibre from fibre tufts that have been metered and drive them to the delivery nozzle of component
(10) which creates a stream of fibre. Once the fibre, polyol, water and isocyanate mix in air they
begin to cure. The mixture will hit the mould creating the desired shape of the reinforced PUF.
8.7 Conclusions
In order to develop natural fibre reinforced polyurethane foams on an industrial scale, a
mechanism must be developed that can deliver a continuous stream of loose packed fibre at a
desired mass flow rate. In this study, a device based on a variable speed roll nip was developed.
The concept involves fibres being compressed by the wheels to a constant density. The volume
of the fibre is measured and thus the mass of the fibre present between the roller wheels is
determined. A relationship between the rotational speed of the roller wheels and the mass of fibre
present is established to deliver a desired mass flow rate of fibres as it passes through the rotating
roll nip. Verification of the concept was completed with computer simulations and a preliminary
146
prototype that was designed. Finally, a design plan is provided for a complete automated system
that would take natural fibre and deliver it as a fine stream to interact with the polymeric
components that make up polyurethane foam.
147
Chapter 9
Conclusions and Recommendations
9.1 Conclusions
In this study, a natural PUF was developed from the oil contained in soybeans. These
soybean based foams were combined with four types of fibres (470 micron glass, 260 micron
glass, MCC and Hemp) separately to increase the compressive modulus for potential use as the
core material in SIPs. The fibres lowered the foam density and increased the porosity of the PUF,
which should have lowered the stiffness. However, the fibres also reinforced the PUF matrix as
the compressive modulus increased when fibre was added.
The 470 micron glass fibres provided the best foam reinforcement. They behaved as long
fibres and spanned through the cells in the foam matrix. The reinforcement could not be
predicted by traditional micro-mechanical models. Instead, it was proven through FEM analysis
and X-ray tomography analysis, that the fibres reinforced the PUF matrix through mechanical
percolation. During compression, these long glass fibres interacted directly and bent each other
causing deformation in the fibres. This bending and fibre deformation increases the overall
stiffness of the composite material. The high aspect ratio of the 470 micron glass fibres
encouraged fibre interaction and percolation, where a network of interacting fibres formed within
the PUF matrix. Hemp fibre reinforced PUFs were also investigated, however, no acceptable
samples for mechanical analysis could be developed.
148
The MCC fibres also provided foam reinforcement. Through SEM analysis and X-ray
Tomography analysis it was clear that the MCC fibres were embedded entirely inside the cell
struts because of their low aspect ratio and small size. They reinforced the cell struts according to
traditional micro-mechanical models. By increasing the rigidity of the cell struts, the overall
foam modulus also increased according to the Gibson-Ashby model. However, the reinforcement
provided by the MCC fibres was smaller than the reinforcement provided by the long glass
fibres.
Intermediate length fibres (fibres lengths that are similar to cell diameters) were found to
be detrimental to foam reinforcement and should be avoided when using low modulus open cell
PUF. 260 micron glass fibres (which behaved as intermediate length fibres) were found to not
improve the rigidity of PUF and at certain fibre content levels potentially decreased the rigidity
of the PUF. It was hypothesized that intermediate length fibres could provide reinforcement to
foams through fibre bridging. In fibre bridging, the fibres are pinned by the cell struts of an
individual cell thus forming a bridge between the cell struts. An FEM analysis was completed to
investigate the reinforcement provided by fibre bridges and it was determined that significant
reinforcement can be provided when bridged fibres are aligned in the direction of loading.
However, through microscopic analysis, no conclusive evidence of fibre bridging was observed
for the 260 micron length glass fibres or long glass fibres.
Finally, a novel device was developed in this study to help produce residual natural fibre
reinforced PUFs on a large scale. The device was successful in metering loose tufts of hemp
fibre at a rate of 2 grams per second. The device has two roller wheels that are brought together
with a constant applied lateral load. As fibre tufts pass through the nip region of the roller
wheels, the fibre is compressed to a constant bulk density. By measuring the volume of the
149
compressed fibre, and knowing the bulk density, the mass can be calculated. The speed of the
roller wheels is adjusted based on the mass of fibre in the nip region to establish a constant mass
flow rate of fibre.
9.2 Recommendations
Based on the compressive modulus results for the various fibre types used in this study, it
is proposed that hybrid fibre composite open-cell PUFs be investigated. Long fibres can provide
reinforcement through mechanical percolation while very short fibres can provide foam
reinforcement by reinforcing the cell struts. Intermediate fibres should be avoided unless a
processing method can be established to encourage fibre bridging. Furthermore, these physical
reinforcement mechanisms should be studied with other foam types which include closed-cell
foams and micro-cellular foams to determine if there are similarities in reinforcement
mechanisms. For example, closed-cell foams have trapped gas bubbles and cell faces and would
have unique reinforcing mechanisms in comparison to open-cell foams. As well, changes to the
thermal conductivity with the introduction of fibre should be studied for its insulating ability.
Mori-Tanaka theory and Cox’s shear lag theory provided the best results in comparison
to the other micro-mechanical models investigated. Nairn’s shear-lag theory should be avoided
when dealing with high modulus fibres and very low modulus PUFs. However, FEM analysis
provided the best results for the long glass fibre scenario and is the ideal tool for understanding
composite reinforcement with imbedded fibres.
In the FEM analysis performed, the fibres were assumed to be randomly oriented and
dispersed. However, in reality fibre clusters formed. Future FEM work in this area should
include a dispersion analysis to understand how fibre clustering plays a role in fibre
150
reinforcement. As well, the FEM analysis focussed on the scenario where the fibres are well
bonded to the matrix and that the matrix remains completely elastic during deformation. FEM
models in the future could include failure analysis for the matrix and fibre de-bonding to study
that effect.
Finally, the novel device is not robust and cannot handle a continuous stream of loose
fibre tufts and requires work for practical use. A design proposal was introduced for the next
stages involved in developing a complete system and should be completed.
151
Bibliography
[1] “Spray Foam Insulation - Insulation - CertainTeed.” [Online]. Available:
http://www.certainteed.com/images/products/thumbnails/aspraycathceiling2md.jpg.
[Accessed: 12-May-2014].
[2] “Bridgestone Corporation.” [Online]. Available:
http://www.bridgestone.com/corporate/news/img_news/2011091401_01.jpg. [Accessed: 12-
May-2014].
[3] “CU Boulder Solar Decathlon - The House - House Images.” [Online]. Available:
http://solar.colorado.edu/2002/house.materials3.html. [Accessed: 12-May-2014].
[4] Q. Guo, B. Cheng, M. Kortschot, M. Sain, R. Knudson, J. Deng, and A. Alemdar,
“Performance of long Canadian natural fibers as reinforcements in polymers,” J. Reinf. Plast.
Compos., vol. 29, no. 21, pp. 3197–3207, 2010.
[5] K. Ashida, Polyurethane and related foams : chemistry and technology. Boca Raton: Taylor
& Francis, 2006.
[6] Polymeric foams : mechanisms and materials. Boca Raton: CRC Press, 2004.
[7] G. Woods, The ICI Polyurethanes book. New York: Published jointly by ICI Polyurethanes
and Wiley, 1990.
[8] B. Yeh, T. Williamson, and E. Keith, “Development of Structural Insulated Panel
Standards,” Structures Congress 2008, Vancouver, BC, Canada, (2008), American Society of
Civil Engineers pp. 1-10
[9] M. A. Mousa, “Structural behavior and modeling of full-scale composite structural insulated
wall panels,” Eng. Struct., vol. 41, pp. 320–334, Aug. 2012.
[10] I. M. Daniel, E. E. Gdoutos, K.-A. Wang, and J. L. Abot, “Failure modes of composite
sandwich beams,” Int. J. Damage Mech., vol. 11, no. 4, pp. 309–334, 2002.
[11] E. E. Gdoutos, I. M. Daniel, and K.-A. Wang, “Compression facing wrinkling of
composite sandwich structures,” Mech. Mater., vol. 35, no. 3, pp. 511–522, 2003.
[12] M. F. Ashby, Materials selection in mechanical design, third edition. Boston: Elsevier
Butterworth-Heinemann, 2005.
[13] Z. S. Petrović and D. Fajnik, “Preparation and properties of castor oil-based
polyurethanes,” J. Appl. Polym. Sci., vol. 29, no. 4, pp. 1031–1040, 1984.
152
[14] A. Guo, D. Demydov, W. Zhang, and Z. S. Petrovic, “Polyols and polyurethanes from
hydroformylation of soybean oil,” J. Polym. Environ., vol. 10, no. 1–2, pp. 49–52, 2002.
[15] X. Kong and S. S. Narine, “Physical properties of polyurethane plastic sheets produced
from polyols from canola oil,” Biomacromolecules, vol. 8, no. 7, pp. 2203–2209, 2007.
[16] P. L. Nayak, S. Lenka, S. K. Panda, and T. Pattnaik, “Polymers from renewable
resources. I. Castor oil-based interpenetrating polymer networks: Thermal and mechanical
properties,” J. Appl. Polym. Sci., vol. 47, no. 6, pp. 1089–1096, 1993.
[17] L. Zhang, M. Zhang, L. Hu, and Y. Zhou, “Synthesis of rigid polyurethane foams with
castor oil-based flame retardant polyols,” Ind. Crops Prod., vol. 52, pp. 380–388, 2014.
[18] A. Palanisamy, “Water-blown polyurethane–clay nanocomposite foams from biopolyol—
effect of nanoclay on the properties,” Polym. Compos., vol. 34, no. 8, pp. 1306–1312, 2013.
[19] V. B. Veronese, R. K. Menger, M. M. de C. Forte, and C. L. Petzhold, “Rigid
polyurethane foam based on modified vegetable oil,” J. Appl. Polym. Sci., vol. 120, no. 1, pp.
530–537, 2011.
[20] A. Palanisamy, B. S. Rao, and S. Mehazabeen, “Diethanolamides of castor oil as polyols
for the development of water-blown polyurethane foam,” J. Polym. Environ., vol. 19, no. 3,
pp. 698–705, 2011.
[21] H. J. Wang, M. Z. Rong, M. Q. Zhang, J. Hu, H. W. Chen, and T. Czigány,
“Biodegradable foam plastics based on castor oil,” Biomacromolecules, vol. 9, no. 2, pp.
615–623, 2007.
[22] S. Tan, T. Abraham, D. Ference, and C. W. Macosko, “Rigid polyurethane foams from a
soybean oil-based Polyol,” Polymer, vol. 52, no. 13, pp. 2840–2846, 2011.
[23] H. Fan, A. Tekeei, G. J. Suppes, and F.-H. Hsieh, “Rigid polyurethane foams made from
high viscosity soy-polyols,” J. Appl. Polym. Sci., vol. 127, no. 3, pp. 1623–1629, 2013.
[24] L.-T. Yang, C.-S. Zhao, C.-L. Dai, L.-Y. Fu, and S.-Q. Lin, “Thermal and mechanical
properties of polyurethane rigid foam based on epoxidized soybean oil,” J. Polym. Environ.,
vol. 20, no. 1, pp. 230–236, 2012.
[25] R. Gu, M. Khazabi, and M. Sain, “Fiber reinforced soy-based polyurethane spray foam
insulation. Part 2: Thermal and Mechanical properties,” BioResources, vol. 6, no. 4, pp.
3775–3790, 2011.
[26] R. Gu and M. M. Sain, “Effects of wood fiber and microclay on the performance of soy
based polyurethane foams,” J. Polym. Environ., vol. 21, no. 1, pp. 30–38, 2013.
[27] R. Gu, S. Konar, and M. Sain, “Preparation and characterization of sustainable
polyurethane foams from soybean oils,” J. Am. Oil Chem. Soc., vol. 89, no. 11, pp. 2103–
2111, 2012.
153
[28] R. Gu, M. M. Sain, and S. K. Konar, “A feasibility study of polyurethane composite foam
with added hardwood pulp,” Ind. Crops Prod., vol. 42, pp. 273–279, 2013.
[29] S. Bandyopadhyay-Ghosh, S. B. Ghosh, and M. Sain, “Synthesis of soy-polyol by two
step continuous route and development of soy-based polyurethane foam,” J. Polym.
Environ., vol. 18, no. 3, pp. 437–442, 2010.
[30] S. Doroudiani and M. T. Kortschot, “Expanded wood fiber polystyrene composites:
processing–structure–mechanical properties relationships,” J. Thermoplast. Compos. Mater.,
vol. 17, no. 1, pp. 13–30, 2004.
[31] A. K. Bledzki, W. Zhang, and A. Chate, “Natural-fibre-reinforced polyurethane
microfoams,” Compos. Sci. Technol., vol. 61, no. 16, pp. 2405–2411, 2001.
[32] T. C. Cotgreave and J. B. Shortall, “The mechanism of reinforcement of polyurethane
foam by high-modulus chopped fibres,” J. Mater. Sci., vol. 12, no. 4, pp. 708–717, 1977.
[33] A. Siegmann, S. Kenig, D. Alperstein, and M. Narkis, “Mechanical behavior of
reinforced polyurethane foams,” Polym. Compos., vol. 4, no. 2, pp. 113–119, 1983.
[34] H. J. Wang, C. Y. Zhang, M. Z. Rong, M. Q. Zhang, and T. Czigány, “Interfacial effects
in short sisal fiber/maleated castor oil foam composites,” Compos. Interfaces, vol. 15, no. 2–
3, pp. 95–110, 2008.
[35] S. H. Kim, H. C. Park, H. M. Jeong, and B. K. Kim, “Glass fiber reinforced rigid
polyurethane foams,” J. Mater. Sci., vol. 45, no. 10, pp. 2675–2680, 2010.
[36] M. A. Mosiewicki, G. A. Dell’Arciprete, M. I. Aranguren, and N. E. Marcovich,
“Polyurethane foams obtained from castor oil-based polyol and filled with wood flour,” J.
Compos. Mater., vol. 43, no. 25, pp. 3057–3072, 2009.
[37] T. G. Rials, M. P. Wolcott, and J. M. Nassar, “Interfacial contributions in lignocellulosic
fiber-reinforced polyurethane composites,” J. Appl. Polym. Sci., vol. 80, no. 4, pp. 546–555,
2001.
[38] U. Cabulis, M. Kirpluks, U. Stirna, M. J. Lopez, M. del Carmen Vargas-Garcia, F.
Suárez-Estrella, and J. Moreno, “Rigid polyurethane foams obtained from tall oil and filled
with natural fibers: Application as a support for immobilization of lignin-degrading
microorganisms,” J. Cell. Plast., vol. 48, no. 6, pp. 500–515, 2012.
[39] E. S. Ali and S. Ahmad, “Bionanocomposite hybrid polyurethane foam reinforced with
empty fruit bunch and nanoclay,” Compos. Part B Eng., vol. 43, no. 7, pp. 2813–2816, 2012.
[40] M. Kuranska and A. Prociak, “Porous polyurethane composites with natural fibres,”
Compos. Sci. Technol., vol. 72, no. 2, pp. 299–304, 2012.
[41] V. Yakushin, U. Stirna, L. Bel’kova, L. Deme, and I. Sevastyanova, “Properties of rigid
polyurethane foams filled with milled carbon fibers,” Mech. Compos. Mater., vol. 46, no. 6,
pp. 679–688, 2011.
154
[42] L. Mkrtchyan and M. Maier, “Fibre reinforced polyurethane foam to increase passive
safety of vehicle occupants,” Plast. Rubber Compos., vol. 36, no. 10, pp. 445–454, 2007.
[43] A. K. Bledzki, J. Gassan, and W. Zhang, “Impact properties of natural fiber-reinforced
epoxy foams,” J. Cell. Plast., vol. 35, no. 6, pp. 550–562, 1999.
[44] I. Banik and M. M. Sain, “Nanoclay modified water-blown polyurethane foams derived
from bifunctional soybean oil-based polyol,” Polym.-Plast. Technol. Eng., vol. 49, no. 7, pp.
701–706, 2010.
[45] M. R. Piggott, Load-bearing fibre composites. Oxford: Pergamon Press, 1980.
[46] H. L. Cox, “The elasticity and strength of paper and other fibrous materials,” Br. J. Appl.
Phys., vol. 3, no. 3, pp. 72–79, Mar. 1952.
[47] J. A. Nairn, “On the use of shear-lag methods for analysis of stress transfer in
unidirectional composites,” Mech. Mater., vol. 26, no. 2, pp. 63–80, 1997.
[48] T. Mori and K. Tanaka, “Average stress in matrix and average elastic energy of materials
with misfitting inclusions,” Acta Metall., vol. 21, no. 5, pp. 571–574, 1973.
[49] V. Favier, J. Y. Cavaille, G. R. Canova, and S. C. Shrivastava, “Mechanical percolation
in cellulose whisker nanocomposites,” Polym. Eng. Sci., vol. 37, no. 10, pp. 1732–1739,
1997.
[50] V. Favier, G. R. Canova, J. Y. Cavaillé, H. Chanzy, A. Dufresne, and C. Gauthier,
“Nanocomposite materials from latex and cellulose whiskers,” Polym. Adv. Technol., vol. 6,
no. 5, pp. 351–355, 1995.
[51] A. G. Facca, M. T. Kortschot, and N. Yan, “Predicting the elastic modulus of natural
fibre reinforced thermoplastics,” Compos. Part Appl. Sci. Manuf., vol. 37, no. 10, pp. 1660–
1671, 2006.
[52] A. G. Facca, Predicting the tensile modulus and strength of single and hybrid natural
fibre reinforced thermoplastic composites. Doctoral Thesis. University of Toronto. 2007.
[53] C. L. Tucker III and E. Liang, “Stiffness predictions for unidirectional short-fiber
composites: review and evaluation,” Compos. Sci. Technol., vol. 59, no. 5, pp. 655–671,
1999.
[54] I. J. Beyerlein and C. M. Landis, “Shear-lag model for failure simulations of
unidirectional fiber composites including matrix stiffness,” Mech. Mater., vol. 31, no. 5, pp.
331–350, 1999.
[55] Y. Yao and S. Chen, “The effects of fiber’s surface roughness on the mechanical
properties of fiber-reinforced polymer composites,” J. Compos. Mater., vol. 47, no. 23, pp.
2909–2923, 2013.
155
[56] K. Jayaraman and M. T. Kortschot, “Correction to the Fukuda-Kawata Young’s modulus
theory and the Fukuda-Chou strength theory for short fibre-reinforced composite materials,”
J. Mater. Sci., vol. 31, no. 8, pp. 2059–2064, 1996.
[57] M. R. Piggott, Load bearing fibre composites. Boston: Kluwer Academic Publishers,
2002.
[58] L. Kobari, The effect of fibre orientation on the modulus of natural fibre reinforced
composites. Master’s Thesis. University of Toronto. 2008.
[59] R. M. Christensen, “A critical evaluation for a class of micro-mechanics models,” J.
Mech. Phys. Solids, vol. 38, no. 3, pp. 379–404, 1990.
[60] R. M. Christensen and K. H. Lo, “Solutions for effective shear properties in three phase
sphere and cylinder models,” J. Mech. Phys. Solids, vol. 27, no. 4, pp. 315–330, 1979.
[61] Y. Benveniste, “A new approach to the application of Mori-Tanaka’s theory in composite
materials,” Mech. Mater., vol. 6, no. 2, pp. 147–157, 1987.
[62] H. Tan, Y. Huang, C. Liu, and P. H. Geubelle, “The Mori-Tanaka method for composite
materials with non-linear interface debonding,” Int. J. Plast., vol. 21, pp. 1890–1918, 2005.
[63] B. Gommers, I. Verpoest, and P. Van Houtte, “The Mori–Tanaka method applied to
textile composite materials,” Acta Mater., vol. 46, no. 6, pp. 2223–2235, 1998.
[64] A. Dasgupta and S. M. Bhandarkar, “A generalized self-consistent Mori-Tanaka scheme
for fiber-composites with multiple interphases,” Mech. Mater., vol. 14, no. 1, pp. 67–82,
1992.
[65] J. D. Eshelby, “The determination of the elastic field of an ellipsoidal inclusion, and
related problems,” Proc. R. Soc. Lond. Ser. Math. Phys. Sci., vol. 241, no. 1226, pp. 376–
396, 1957.
[66] R. Hill, “Elastic properties of reinforced solids: some theoretical principles,” J. Mech.
Phys. Solids, vol. 11, no. 5, pp. 357–372, 1963.
[67] G. P. Tandon and G. J. Weng, “The effect of aspect ratio of inclusions on the elastic
properties of unidirectionally aligned composites,” Polym. Compos., vol. 5, no. 4, pp. 327–
333, 1984.
[68] G. P. Tandon and G. J. Weng, “Average stress in the matrix and effective moduli of
randomly oriented composites,” Compos. Sci. Technol., vol. 27, no. 2, pp. 111–132, 1986.
[69] M. T. Kortschot and R. T. Woodhams, “Computer simulation of the electrical
conductivity of polymer composites containing metallic fillers,” Polym. Compos., vol. 9, no.
1, pp. 60–71, 1988.
156
[70] R. A. MacDonald, C. M. Voge, M. Kariolis, and J. P. Stegemann, “Carbon nanotubes
increase the electrical conductivity of fibroblast-seeded collagen hydrogels,” Acta Biomater.,
vol. 4, no. 6, pp. 1583–1592, 2008.
[71] “Test Method for Compressive Properties of Rigid Cellular Plastics,” ASTM International,
2010.
[72] H. J. Qi and M. C. Boyce, “Stress–strain behavior of thermoplastic polyurethanes,”
Mech. Mater., vol. 37, no. 8, pp. 817–839, 2005.
[73] L. J. Gibson, Cellular solids : structure and properties. New York: Cambridge University
Press, 1997.
[74] N. Mills, Polymer foams handbook: engineering and biomechanics applications and
design guide. Butterworth-Heinemann, 2007.
[75] F. Wallenberger, J. Watson, and H. Li, “Glass Fibers,” in ASM Handbook, vol. 21 -
Composites, ASM International, 2001.
[76] S. J. Eichhorn and R. J. Young, “The Young’s modulus of a microcrystalline cellulose,”
Cellulose, vol. 8, no. 3, pp. 197–207, 2001.
[77] R. J. Roberts, R. C. Rowe, and P. York, “The Poisson’s ratio of microcrystalline
cellulose,” Int. J. Pharm., vol. 105, no. 2, pp. 177–180, 1994.
[78] C. C. Sun, “True density of microcrystalline cellulose,” J. Pharm. Sci., vol. 94, no. 10,
pp. 2132–2134, 2005.
[79] A. Ilindra and J. D. Dhake, “Microcrystalline cellulose from bagasse and rice straw,”
Indian J. Chem. Technol., vol. 15, pp. 497–499, 2008.
[80] L. J. Gibson, “Biomechanics of cellular solids,” J. Biomech., vol. 38, no. 3, pp. 377–399.
[81] S. K. Khanna, “Application of reinforced polymer foams for energy absorption in
lightweight structures,” presented at the 2002 SEM Annual Conference & Exposition on
Experimental and Applied Mechanics.
[82] M. Khazabi, R. Gu, and M. Sain, “ Fiber reinforced soy-based polyurethane spray foam
insulation. Part 1: Cell Morphologies,” BioResources, vol. 6, no. 4, 2011.
[83] B. Wang, R. Wang, and Y. Wu, “The Young’s moduli prediction of random distributed
short-fiber-reinforced polypropylene foams using finite element method,” Sci. China Ser. E
Technol. Sci., vol. 52, no. 1, pp. 72–78, 2009.
[84] Y. Wu, B. Wang, and F. Han, “A Novel Numerical Random Model of Short Fiber
Reinforced Foams,” in Advanced Intelligent Computing Theories and Applications. With
Aspects of Theoretical and Methodological Issues, Springer, 2008, pp. 321–328.
157
[85] J. Yuan and S. Q. Shi, “Effect of the addition of wood flours on the properties of rigid
polyurethane foam,” J. Appl. Polym. Sci., vol. 113, no. 5, pp. 2902–2909, 2009.
[86] G. L. Smith, “Adjustable chopper diverter for a spray gun,” US5850971 A, 22-Dec-1998.
[87] A. F. Aggen, “Material metering device,” US4436226 A, 13-Mar-1984.
[88] P. Brutsch, P. Staheli, R. Demuth, and J. Faas, “Metering method and metering apparatus
for dispensing predeterminable quantities of fiber flocks,” US5121523 A, 16-Jun-1992.
[89] S. Schlichter, “Apparatus for measuring the mass of fibre material passing through a
spinning preparation machine or system,” US7614121 B2, 10-Nov-2009.
[90] B. Rübenach, “Fiber material feeder having a spiked lattice,” US6360403 B1, 26-Mar-
2002.
158
Appendices
Appendix 1
Equations for Mori Tanaka theory derived by Tandon and Weng [67] for oriented fibres.
A1 = D1*(B4+B5)-2*B2
A2 = (1+D1)*B2-(B4+B5)
A = 2*(B2*B3)-B1*(B4+B5)
B1= Vf*D1+D2+(1-Vf)*(D1*S1111+2*S2211);
B2= Vf+D3+(1-Vf)*(D1*S1122+S2222+S2233)
B3= Vf+D3+(1-Vf)*(S1111+(1+D1)*S2211)
B4= Vf*D1+D2+(1-Vf)*(S1122+D1*S2222+S2233)
B5= Vf+D3+(1-Vf)*(S1122+S2222+D1*S2233)
D1=1+2*(Uf-Um)/(Lf-Lm)
D2=(Lm+2*Um)/(Lf-Lm)
D3=Lm/(Lf-Lm)
Lf = (Gf*(Ef-2*Gf))/(3*Gf-Ef)
Uf = Gf
Lm = (Gm*(Em-2*Gm))/(3*Gm-Em)
159
Um = Gm
S1111= (1/(2*(1-Vo)))*(1-2*Vo+(3*AR*AR-1)/(AR*AR-1)-(1-2*Vo+(3*AR*AR)/(AR*AR-
1))*g)
S2222= ((3*AR^(2))/(8*(1-Vo)*(AR^(2)-1)))+(1/(4*(1-Vo)))*(1-2*Vo-(9/(4*(AR^(2)-1))))*g
S2233= (1/(4*(1-Vo)))*((AR^(2)/(2*(AR^(2)-1)))-(1-2*Vo+(3/(4*(AR^(2)-1))))*g)
S2211= -(1/(2*(1-Vo)))*(AR^(2)/(AR^(2)-1))+(1/(4*(1-Vo)))*((3*AR^(2)/(AR^(2)-1))-(1-
2*Vo))*g
S1122= -(1/(2*(1-Vo)))*(1-2*Vo+(1/(AR^(2)-1)))+(1/(2*(1-Vo)))*(1-2*Vo+(3/(2*(AR^(2)-
1))))*g
g = (AR/(AR^(2)-1)^(3/2))*(AR*(AR^(2)-1)^(1/2)-arccosh(AR))
Appendix 2
Equations for Mori Tanaka theory derived by Tandon and Weng [68] for randomly oriented
fibres.
S1133= S1122
S3311= S2211
S3322= S2233
S3333= S2222
Km = Em/3
Kf = Ef/(3*(1-2*0.22))
160
A1 = 6*(Kf-Km)*(Gf-Gm)*(S2222+S2233-1)-2*(Km*Gf-Kf*Gm)+6*Kf*(Gf-Gm)
A2 = 6*(Kf-Km)*(Gf-Gm)*S1133+2*(Km*Gf-Kf*Gm)
A3 = -6*(Kf-Km)*(Gf-Gm)*S3311-2*(Km*Gf-Kf*Gm)
A4 = 6*(Kf-Km)*(Gf-Gm)*(S1111-1)+2*(Km*Gf-Kf*Gm)+6*Gf*(Kf-Km)
A5 = 1/(S3322-S3333+1-Gf/(Gf-Gm))
A = 6*(Kf-Km)*(Gf-Gm)*(2*S1133*S3311-(S1111-1)*(S3322+S3333-
1))+2*(Km*Gf+Kf*Gm)*(2*(S1133+S3311)+(S1111-S3322-S3333))-6*Kf*(Gf-Gm)*(S1111-
1)-6*Gf*(Kf-Km)*(S2222+S2233-1)-6*Kf*Gf
P1 = 1+Vf*(2*(S1122+S2222+S2233-1)*(A3+A4)+(S1111+2*S2211-1)*(A1-2*A2))/(3*A)
P2 = (A1-2*(A2-A3-A4))/(3*A)
P = P2/P1
![[Engelberg] Compressive Sensing](https://img.pdfslide.net/doc/110x75/55cf9985550346d0339dc8ee/engelberg-compressive-sensing.jpg)
