COMPRESSIVE BEHAVIOUR OF CARBON-FIBRE POLYMER COMPOSITE
MATERIALS
by
Stergios Goutianos
An Abstract
Of a thesis submitted for the degree of
Doctor of Philosophy at the
University of London
December 2003
Thesis supervisor: Dr Ton Peijs, Prof. Costas Galiotis
1
ABSTRACT
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Date
COMPRESSIVE BEHAVIOUR OF CARBON-FIBRE POLYMER COMPOSITE
MATERIALS
by
Stergios Goutianos
A thesis submitted for the degree of
Doctor of Philosophy at the
University of London
December 2003
Thesis supervisor: Dr Ton Peijs, Prof. Costas Galiotis
Copyright by
STERGIOS GOUTIANOS
1998
All Rights Reserved
Department of Materials
Queen Mary, University of London
Mile End Road, London, E1 4NS
CERTIFICATE OF APPROVAL
PH.D. THESIS
This is to certify that the Ph.D. thesis of
Stergios Goutianos
has been approved by the Examining Committee
for the thesis requirement for the Doctor of
Philosophy degree in Materials at the December 2003
graduation.
Thesis committee:
Thesis supervisor
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Member
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to end with a period
ii
ACKNOWLEDGMENTS
noindent I would like to thank Prof. Costas Galiotis, my supervisor, for his
many suggestions and constant support during this research from the Institute of
Chemical Engineering and High Temperature Chemical Processes, Patras, Greece.
Thanks is extended to Prof. Ton Peijs for his help from the Materials Department
at Queen Mary University of London, UK.
I would like to express my thanks to Dr. A. Karantzalis (Institute of Chemical
Engineering and High Temperature Chemical Processes), without his help the thesis
would have looked totally different.
I should also mention that my graduate studies were supported by the Institute of
Chemical Engineering and High Temperature Chemical Processes, Foundation of
Research and Technology - Hellas, Patras, Greece.
Of course, I am grateful to my parents for their patience and love. Without them
this work would never have come into existence (literally).
Finally, I wish to thank the following:
Dr. T. Fragos, Dr. G. C. Psarras, Dr.
J. Parthenios and C. Koimtzoglou from the Institute of Chemical Engineering and
High Temperature Chemical Processes (for their help with respect to Laser Raman
Spectroscopy); Dr. S. Sirivedin from the Kinkg’s College, University of London (for
his help with the Finite Element Method).
iii
ABSTRACT
This is the abstract that I want bound with my thesis. It’s optional.
iv
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
CHAPTER
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 Fracture Mechanisms in Compression . . . . . . . . . . .
2.1 Full Composite Specimens . . . . . . . . . . . . . . .
2.1.1 Kink band formation and propagation . . . .
2.1.2 Failure mechanisms upon compressive loading
2.2 Model composite specimens . . . . . . . . . . . . . .
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3
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3 Modeling of the Compressive Behaviour
3.1 Introduction . . . . . . . . . . . .
3.1.1 Elastic microbuckling . . .
3.1.2 Fibre crushing . . . . . . .
3.1.3 Matrix failure . . . . . . .
3.1.4 Plastic microbuckling . . .
3.2 Fibre Microbuckling Models . . . .
3.3 Kink Band Formation Models . . .
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4 Experimental . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . .
4.2 Materials and Specimen Preparation . . . . . . . .
4.3 Specimen Testing and Raman Spectra Acquisition
4.4 Analysis of the Experimental Data . . . . . . . . .
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5 Modeling of the Stress Transfer from the Matrix to the Fibre . . . .
5.1 Shear-Lag Analyses . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Axisymmetric Analyses . . . . . . . . . . . . . . . . . . . . . .
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6 Experimental Results . . . . . .
6.1 Compressive behaviour . .
6.1.1 Elastic case . . . . .
6.1.2 Compressive failure
6.2 Tensile behaviour . . . . .
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7 Finite Element Analysis . . . . . . . . . . . .
7.1 Finite element details . . . . . . . . . .
7.1.1 Boundary conditions and applied
7.1.2 Material properties . . . . . . .
7.1.3 FE Model’s dimensions . . . . .
7.2 Results . . . . . . . . . . . . . . . . . .
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loads
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8 Validation of the Analytical Stress Transfer Models . . . . . . . . . .
8.1 Shear-Lag Analyses . . . . . . . . . . . . . . . . . . . . . . . .
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9 Discussion of the Experimental Results . . . . . . . . . . . . . . . .
9.1 Compressive failure . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Comparison of Compressive versus Tensile Behaviour . . . . .
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10 Numerical analysis of the stress transfer in compression in the case of
a broken fibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 FE model for fibre fracture in compression . . . . . . . . . . .
10.2.1 Finite element details . . . . . . . . . . . . . . . . . . .
10.3 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . .
10.3.1 Effect of fibre geometrical discontinuity on the stress field
10.3.2 Effect of friction between the fibre fragments . . . . . .
10.3.3 Stress field in the vicinity of a compressive fibre break .
10.3.4 Effect of matrix modulus . . . . . . . . . . . . . . . . .
10.3.5 Effect of matrix yield stress . . . . . . . . . . . . . . . .
10.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
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11 Conclusions
132
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12 Future Work . . . . . . . . . . . . . . . .
12.1 Introduction . . . . . . . . . . . . .
12.2 Fibre strain distributions . . . . . .
12.3 Mapping of strains in adjacent fibres
12.4 Failure mechanisms . . . . . . . . .
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134
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A Raman spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . .
147
APPENDIX
A.1 Raman spectroscopy . . . . . . . . . . . . . .
A.1.1 Introduction . . . . . . . . . . . . . .
A.1.2 The Raman effect . . . . . . . . . . .
A.1.3 Quantum theory of Raman scattering
A.1.4 Classical theory of Raman scattering .
vi
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A.2 Noise in Raman measurements . . . . . . . . . . . . . . . . . .
A.3 Cubic spline fitting of Raman measurements . . . . . . . . . .
151
153
B Matrix creep effects on the fibre stress profiles . . . . . . . . . . . .
158
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
159
159
C Fibre strain distribution in carbon fibre-epoxy composites under compressive stress field . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163
C.1 Fibre strain histograms . . . . . . . . . . . . . . . . . . . . . . 164
C.2 Strain distribution in adjacent fibres . . . . . . . . . . . . . . . 164
D Nomeclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
170
LIST OF TABLES
Table
4.1
Page
Mechanical properties . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
44
LIST OF FIGURES
Figure
2.1
2.2
2.3
2.4
2.5
2.6
4.7
4.8
4.9
4.10
4.11
4.12
6.13
Page
A schematic illustration of a kink band showing several parameters.
φ + φ is the kink band orientation angle, β is kink band angle, and
wk is the kink band width. . . . . . . . . . . . . . . . . . . . . . . .
Kink band in unidirectional PAN graphite-reinforced PEEK, Lankford Lankford (1995). . . . . . . . . . . . . . . . . . . . . . . . . . .
Photomicrograph of a kink band formed in a AS4/PEEK unidirectional composite, Vogler et al Vogler et al. (2000). . . . . . . . . . .
Overall load-end shortening behavior of unidirectional fibre composite compressed in the fibre direction when kinking occurs. The load
levels at incipient kinking, propagation across the specimen’s width
and transient and steady-state band width broadening are indicated.
Moran et al Moran et al. (1995). . . . . . . . . . . . . . . . . . . .
Schematic failure sequence: (a) microcrack initiation, (b, c) shear
failure, (d) crushing damage, (e) shear sliding, (f) longitudinal splitting, (g) bifurcation of shear failure and (h) multiple fracture. (Boll
et al Boll et al. (1990)) . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic illustration of (a) fibre bent, (b) ejected wedge of fibre
and locally debonded region and (c) fibre sliding in the debonded
zone without the ejection of a fibre wedge. Narayanan and Schadler
Narayanan and Schadler (1999a) . . . . . . . . . . . . . . . . . . . .
Geometry of single-fibre specimens: (a) prism geometry for compression tests and (b) dogbone geometry for tension tests (all dimensions
in mm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Micrograph of a typical fiber end. . . . . . . . . . . . . . . . . . . .
Schematic illustration of the experimental set-up of the remote Raman microprobe. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Raman spectra of the M40 carbon fiber embedded in the epoxy matrix showing the shift of the 2760 cm−1 Raman band with applied
strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Raman spectrum of the M40 carbon fiber embedded in the epoxy
matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Raman frequency shift versus strain for the M40-40B carbon fiber. .
(a) Axial fiber stress profile of an embedded fiber in an epoxy matrix loaded at strains of 0% and -0.1% respectively. The open circles/squares correspond to experimental data whereas the solid lines
represent the corresponding cubic spline fits, (b) Corresponding interfacial shear stresses. . . . . . . . . . . . . . . . . . . . . . . . . .
ix
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6.14 (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix loaded at strains of -0.2% and -0.3% respectively. The open
circles/squares correspond to experimental data whereas the solid
lines represent the corresponding cubic spline fits, (b) Corresponding
interfacial shear stresses. . . . . . . . . . . . . . . . . . . . . . . . .
6.15 (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix loaded at strains of 0% and -0.34% respectively. The open circles/squares correspond to experimental data whereas the solid lines
represent the corresponding cubic spline fits, (b) Corresponding interfacial shear stresses. . . . . . . . . . . . . . . . . . . . . . . . . .
6.16 (a)Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at a strain of -0.56%. The open squares correspond to experimental data whereas the solid line represents the corresponding
cubic spline fit, (b) Corresponding interfacial shear stresses. . . . .
6.17 (a)Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at a strain of -0.86%. The open squares correspond to experimental data whereas the solid line represents the corresponding
cubic spline fit, (b) Corresponding interfacial shear stresses. . . . .
6.18 Micrograph of a typical shear compressive fiber break. . . . . . . . .
6.19 Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at a strain of -0.86%. . . . . . . . . . . . . . . . . . . . . . .
6.20 (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix loaded at strains of 0%, 0.2% and 0.4% respectively. The open
diamonds/circles/squares correspond to experimental data whereas
the solid lines represent the corresponding cubic spline fits, (b) Corresponding interfacial shear stresses. . . . . . . . . . . . . . . . . . .
6.21 (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix loaded at strains of 0.6% and 0.8% respectively. The open circles/squares correspond to experimental data whereas the solid lines
represent the corresponding cubic spline fits, (b) Corresponding interfacial shear stresses. . . . . . . . . . . . . . . . . . . . . . . . . .
6.22 (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix loaded at strains of 1.1% and 1.25% respectively. The open
circles/squares correspond to experimental data whereas the solid
lines represent the corresponding cubic spline fits, (b) Corresponding
interfacial shear stresses. . . . . . . . . . . . . . . . . . . . . . . . .
7.23 Schematic representation of the axisymmetric finite element model.
rf and Rm are the fibre and matrix radii respectively, whereas lim is
the length of the imaginary fibres. . . . . . . . . . . . . . . . . . . .
7.24 Finite element mesh . . . . . . . . . . . . . . . . . . . . . . . . . .
7.25 The independence (a) of the axial fibre stress at the fibre end and
(b) of the far-field matrix stress on the matrix width. . . . . . . . .
7.26 The independence (a) of the axial fibre stress at the fibre end and
(b) of the far-field matrix stress on the imaginary fibre length. . . .
x
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7.27 Axial fibre stress at the fibre end versus the applied strain. Comparison of the FEA predictions with the predictions of the shear-lag
models of Hsueh Hsueh (1995) and Nair and Kim Nair and Kim
(1992) models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.28 Schematic illustration of the shear-lag model for stress transfer from
the matrix to the embedded fibre. Imaginary fibres are used in the
case of fibre ends bonded to the matrix . . . . . . . . . . . . . . . .
8.29 Comparison of the Shear-lag predictions using two different β (βcox
and βnairn ). (a) Axial fibre stress profiles for a Rcox /rf ratio equal
to 3, 10, 25, 315, ∞ (uf → 0), respectively. (b) Corresponding
interfacial shear stresses (The ISS in the case of βnairn and Rcox /rf =3
is not plotted, the maximum values are higher than 185MPa in that
case). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.30 Comparison of Mendels’s and Cox’s Shear-lag predictions. (a) Axial
fibre stress profiles for a Rcox /rf ratio equal to 3, 10, 25, 315, ∞
(uf → 0), respectively. (b) Corresponding interfacial shear stresses.
8.31 Fibre stress at the fibre end upon tensile loading. Parameters used
in the Shear-lag models of Hsueh and Nair & Kim: Rcox /rf =315, l′
(length of imaginary fibres)=100µm. . . . . . . . . . . . . . . . . .
8.32 Fibre stress at the fibre end upon compressive loading. Parameters
used in the Shear-lag models of Hsueh and Nair & Kim: Rcox /rf =315,
l′ (length of imaginary fibres)=100µm. . . . . . . . . . . . . . . . .
8.33 Comparison of the cubic spline fit with Shear-lag models in the case
of interfacial failure (ǫ=1.25%). Heuvel’s analysis is used for two
cases: (i) yielding of the interphase and (ii) debonding followed by
yielding of the interphase. The parameters used to fit the results are:
Rcox =85µm, σym =35MPa, τym =17.5MPa (Tresca), case (i): yielding
zone = 250µm, case (ii): debond length = 100µm and yielding zone
= 180µm. (a) Axial fibre stress profiles and (b) Corresponding ISS
profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.34 Axial far-field fiber strain (ǫf iber = ǫmeasured − ǫresidual ) versus applied
strain. The dotted line represents the 1:1 relation between the applied
strain and the far-field fiber strain, whereas the solid line represents
represents a 3rd degree polynomial fit of the experimental data (◦).
max
9.35 Maximum ISS values (ISSmax = ISSmax
measured - ISSǫ=0% ) versus applied
strain. The different symbols in the left part (compression) of the
graph correspond to different experiments, whereas the solid squares
in the tensile section of graph represent measurements at a distance
from the fiber ends. The solid line represents the linear fir of the
experimental data for -0.3%< ǫ <0.3%. . . . . . . . . . . . . . . . .
10.36(a) Micrograph of a typical compression induced shear fibre break,
(b) micrograph of a fracture site showing fibres failed in shear at a
well defined plane, (c) SEM picture showing again co-operative fibre
shear failure in compression. . . . . . . . . . . . . . . . . . . . . . .
10.37Schematic representation of the FE model used. . . . . . . . . . . .
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113
10.38Stress-strain curve for the epoxy matrix. . . . . . . . . . . . . . . .
10.39Axial fibre stress of an embedded fibre (Lf = 2 mm) in an epoxy
matrix, loaded at strain of 0.3%, -0.4%, and 0.5%, respectively. . . .
10.40Corresponding interfacial shear stresses of Fig. 4. The applied strains
are 0.3%, -0.4%, and 0.5%, respectively . . . . . . . . . . . . . . . .
10.41Shear stress field around the compressive fibre break. a) applied
strain = -0.465% (no fibre break is introduced), b) applied strain
= -0.55% (introduction of the fibre break), and c) applied strain =
-0.7%. (all values in MPa). The coefficient of friction between the
fibre fragments is 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . .
10.42Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=0µm). The applied strain is 0.55%, and the coefficient of friction
between the fibre fragments is 0.4. . . . . . . . . . . . . . . . . . . .
10.43Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=0µm). The applied strain is 0.55%, and the coefficient of friction
between the fibre fragments is 0.4 and 0.6, respectively. . . . . . . .
10.44Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=1000µm) at different applied strains (ǫ11 ), -0.55%, -0.60%, and
0.70%, respectively. The coefficient of friction between the fibre fragments is 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.45Interfacial shear stress (σ12 ) in the vicinity of a compressive fibre
break (x=1000µm) at different applied strains (σ11 ) -0.55%, -0.60%,
and 0.70%, respectively. The coefficient of friction between the fibre
fragments is 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.46Transverse stress (σ12 ) at the fibre matrix interface at different applied strains (σ11 ), -0.55%, -0.60%, and 0.70% respectively. The coefficient of friction between the fibre fragments is 0.4. a) Transverse
stresses at the left side of the fibre break, and b) Transverse stresses
at the right side of the fibre break. . . . . . . . . . . . . . . . . . .
10.47Transverse stress contours around the compressive fibre break at an
applied strain of -0.55% (introduction of the fibre break). The coefficient of friction between the fibre fragments is 0.4. (All values in
MPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.48Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=1000µm) for three different matrix modulus, 1.5GPa,2 GPa, and
2.5GPa respectively. The coefficient of friction between the fibre
fragments is 0.4 and the applied strain is 0.55%. . . . . . . . . . . .
10.49Interfacial shear stress (σ12 ) in the vicinity of a compressive fibre
break (x=1000µm) for three different matrix modulus, 1.5GPa, 2GPa,
and 2.5GPa respectively. The coefficient of friction between the fibre
fragments is 0.4 and the applied strain is 0.55%. . . . . . . . . . . .
10.50Transverse stress (σ22 ) in the vicinity of a compressive fibre break
(x=1000µm) for three different matrix modulus, 1.5GPa, 2GPa, and
2.5GPa respectively. The coefficient of friction between the fibre
fragments is 0.4 and the applied strain is 0.55%. . . . . . . . . . . .
xii
114
115
116
117
118
119
120
121
122
123
124
125
126
10.51Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=1000µm) for three different matrix yield stresses, 30MPa, 45MPa,
and 60MPa, respectively. The coefficient of friction between the fibre
fragments is 0.4 and the applied strain is 0.55%, and the matrix
modulus is 2.0GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.52Interfacial shear stress (σ12 ) in the vicinity of a compressive fibre
break (x=1000µm) for three different matrix yield stresses, 30MPa,
45MPa, and60 MPa, respectively. The coefficient of friction between
the fibre fragments is 0.4 and the applied strain is 0.55%, and the
matrix modulus is 2.0GPa. . . . . . . . . . . . . . . . . . . . . . . . 128
10.53Transverse stress (σ22 ) in the vicinity of a compressive fibre break
(x=1000µm) for three different matrix yield stresses, 30MPa, 45MPa,
and 60MPa, respectively. The coefficient of friction between the fibre
fragments is 0.4 and the applied strain is 0.55%, and the matrix
modulus is 2.0GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10.54Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=1000µm) for a) an elastic matrix, and b) an elastic-perfectly plastic matrix (σym =45 MPa) . The coefficient of friction between the
fibre fragments is 0.4 and the applied strain is 0.55%, and the matrix
modulus is 1.5GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
10.55Interfacial shear stress (σ12 ) in the vicinity of a compressive fibre
break (x=1000µm) for a) an elastic matrix, and b) an elastic-perfectly
plastic matrix (σym =45 MPa) . The coefficient of friction between the
fibre fragments is 0.4 and the applied strain is 0.55%, and the matrix
modulus is 1.5GPa. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
12.56Geometry of multi-fibre specimens (all dimensions in mm). . . . . . 139
12.57Fibre strain histogram. Residual fibre strains (200 measurements). . 140
12.58Fibre strain histogram. Externally applied strain -0.6% (200 measurements). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
12.59Gaussian strain distributions. (200 measurements). . . . . . . . . . 141
12.60Mean value of fibres strain (laser Raman measurements) as a function
of applied strain (strain gauge measurements). . . . . . . . . . . . . 141
12.61Width of the Gaussian distributions as a function of applied strain
(strain gauge measurements). . . . . . . . . . . . . . . . . . . . . . 142
12.62Fibre strain profiles. The spacing between the fibres is 3µm and the
applied strain -0.9%. . . . . . . . . . . . . . . . . . . . . . . . . . . 142
12.63Optical micrograph of fibre fractures . . . . . . . . . . . . . . . . . 143
12.64Optical micrograph of fibre fractures . . . . . . . . . . . . . . . . . 144
12.65Optical micrograph of fibre fractures . . . . . . . . . . . . . . . . . 145
12.66Optical micrograph of fibre fractures . . . . . . . . . . . . . . . . . 146
A.1 Raman frequency shift distribution. . . . . . . . . . . . . . . . . . . 155
A.2 FFT spectra of Raman measurements (applied tensile strain, ǫ=1.1%).156
A.3 Filtered Raman measurements (applied tensile strain, ǫ=1.1%). . . 156
A.4 Cubic spline fitting (applied compressive strain, ǫ=-0.3%). . . . . . 157
A.5 ISS distribution obtained from Figure A.4. . . . . . . . . . . . . . . 157
xiii
B.1
B.2
B.3
C.1
C.2
C.3
C.4
C.5
C.6
C.7
C.8
C.9
Matrix relaxation at a low applied strain (load). . . . . . . . . . . .
Fiber stress evolution with time and Global matrix creep strain. . .
Fiber stress evolution with time and Global matrix creep strain. . .
Fibre strain histogram. Externally applied strain -0.1% (200 measurements). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fibre strain histogram. Externally applied strain -0.2% (200 measurements). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fibre strain histogram. Externally applied strain -0.3% (200 measurements). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fibre strain histogram. Externally applied strain -0.4% (200 measurements). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fibre strain histogram. Externally applied strain -0.5% (200 measurements). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fibre strain profiles. The spacings between the fibres are: 8µm (fibre
1-fibre 2) and 4µm (fibre 2-fibre 3). The applied strain -0.6%. . . .
Fibre strain profiles. The spacings between the fibres are: 4.5µm
(fibre 1-fibre 2), 3µm (fibre 2-fibre 3), and 3µm (fibre 3-fibre 4). The
applied strain -0.78%. . . . . . . . . . . . . . . . . . . . . . . . . . .
Fibre strain profiles. The spacings between the fibres are: 4µm (fibre
1-fibre 2) and 4µm (fibre 2-fibre 3). The applied strain -0.9%. . . .
Fibre strain profiles. The spacings between the fibres are: 4µm (fibre
1-fibre 2) and 4µm (fibre 2-fibre 3). The applied strain -0.9%. . . .
xiv
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162
162
165
166
166
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169
169
CHAPTER 1
INTRODUCTION
Laminated fibre reinforced polymer matrix composites (PMC’s) are finding increased use in a broad range of industrial applications, particularly in the Aerospace
and Automotive sectors. These materials are attractive for a number of reasons,
noteworthy amongst these being their mechanical properties. Composite materials
can be constructed to have a range of mechanical properties (e.g. stiffness and
strength) by selecting the constituent materials appropriately, by engineering the
interfaces between constituents, and by devising the geometrical placement of the reinforcing constituent in the matrix. Issues such as damage tolerance and durability,
performance degradation due to aging, fatigue under multiaxial loads and response
at elevated temperature are currently being investigated with a view to enhance the
confidence levels associated with PMC applications. Cost effective, novel manufacturing techniques, having a minimum impact on the performance indices, are also
being sought.
Experience with applications of PMC’s for rotor blades, pressure vessels, and other
situations that demand for tensile stiffness and strength have shown that polymer
matrix composites have superior tensile properties relative to compressive stiffness
and strength. Experiments on carbon fibre PMC’s have indicated tensile strengths
that are about twice the reported compressive strength values (e.g. Refs Ahn and
Waas (1999); Berbinau et al. (1999); Budiansky and Fleck (1993); Fleck and Liu
1
2
(2001)). This low compressive strength is a limiting design parameter which decreases the exploitation of these materials. As a result, the compressive response of
the composite materials with long continuous fibres has been a subject of intense
research over the past thirty years, and today a substantial body of experimental
and theoretical results has enriched our knowledge on this topic. Some fundamental
issues, however, still remain open and will be addressed in the current work.
The contents of this thesis are organised as follows. In Chapter 2, a short literature
review on the experimental observations is given. Chapter 3 deals with the existing
analytical models to predict the composite compressive strength. Chapter 4 contains
the experimental details (i.e experimental technique, specimen’s geometries etc).
In Chapter 5, a literature review on the analytical stress transfer models is given.
The experimental results are presented in Chapter 6, whereas some preliminary
numerical (FEA) results are given in Chapter 7. In Chapter 8, a validation of the
analytical stress transfer problems is made based on the experimental results. A
discussion on the experimental findings of this project is made in Chapter 9, followed
by the conclusions in Chapter 10. Chapter 11 contains the future work with some
preliminary results.
CHAPTER 2
FRACTURE MECHANISMS IN COMPRESSION
In this chapter a short literature review on the experimental observations is
given.
2.1
Full Composite Specimens
Kink band formation and propagation has been recognized as the primary
failure compressive mechanism in limiting compressive strength of unidirectional
fibre-reinforced composites Jelf and Fleck (1992); Soutis and Fleck (1990); Budiansky (1983). A typical kink band is shown in Figure 2.1. φ + φ is the kink band
orientation angle, β is kink band angle, and wk is the kink band width, whereas
in Figures 2.2 and 2.3 photomicrographs of kink bands are shown. There is experimental evidence to suggest that the initial stages of this mode of failure in
carbon fibre epoxy composites are controlled by fibre misalignment and shear yield
strength of the matrix Jelf and Fleck (1992); Budiansky (1983); Argon (1972). This
Section is dealing with the initiation and propagation of kink bands observed in
fibre composites, during compression tests.
2.1.1 Kink band formation and propagation
Moran et al Moran et al. (1995) carried out compression tests in unidirectional
fibre composites consisted of IM7 carbon fibres (60% volume fraction) and a relative
ductile PEEK matrix. Figure 2.4 depicts a typical load curve and the associated
3
4
stages of kinking observed in their work. The specimens had notches to facilitate
kinking formation. The initial deformation is linear elastic. As it can be seen in
Figure 2.4, the load-end shortening curve begins to show non-linearity when the
matrix in a small region ahead of the notch begins to flow plastically. Moran et
al Moran et al. (1995) call this stress, at the beginning of plastic flow, as incipient
kinking stress. The peak stress is shortly reached after non-linearity is spotted.
The kink band angle at this early stage is found to be very small, and lies between
100 -150 from the normal to the fibres plane. The band is narrow, about ten fibre
diameters wide. The fibres within the band rotate slowly until they lie 150-200
from the vertical. At this point the rotation becomes unstable, the fibres rotate
rapidly to an orientation of 400 -450 from the vertical, where they lock-up. This fibre
lock-up, forces the kink band to spread into the unkinked (unaffected) material by
band width broadening. The band width is increased from approximately 50µm to
270µm, while the fibres, during this process, remain roughly fixed in orientation,
close to their initial lock-up direction. Moran et al Moran et al. (1995) believe
that the essence of band broadening is the outward propagation of the localized
bends in the fibres. Behind the bends, the fibres are straight but are rotated away
from the original fibre direction. The matrix within the band is sheared by an
extense comparable to the fibre rotation angle. As they state, the process of band
broadening is analogous to neck propagation, in certain polymers, often called ”cold
drawing”.
Kyriakides et al Kyriakides et al. (1995) studied the compression failure mechanisms in AS4/PEEK composites. They reported that failure was sudden and the
load after reached its maximum value dropped about 30% in a dynamic way. The
5
failure is local, of the form of kinking, initiated at one end of the specimen, and arrested due to the radial confinement employed. They proposed the following failure
mechanism: Initially, the loaded specimen has sufficient energy for the formation
of more than one kink plane each of which is initiated from the weakened region
formed at the termination of the previous kink plane. In other words, the termination of a kink plane provides the imperfection for the initiation of the next kink
plane. This energy, depends on the unloading of the surrounding material during
the incipient load drop. Hence. the existence of confinement limits the extent to
which the kink bands can deform to and, as a result, the energy that can be dissipated. Each load peak on the plateau value corresponds to the initiation of one
or more additional kink planes. All kink planes are oriented in such a way so that
their normals are approximately co-planar. The orientation of the kinked fibres, in
adjacent kink planes, is opposite. Once failure is initiated, more kink planes can
be developed at well-defined stress, which is lower compared to that required for
failure initiation. This phenomenon is quite similar to a propagating instability
known to affect several materials Kyriakides et al. (1995). Such instability consists
of three steps. The first is the initiation phase where the structure becomes unstable for the first time. This global instability is usually associated with a maximum
load (limit-load). The limit-load is usually followed by localization of deformation,
which is the second phase. Localized deformation is then arrested due to reasons
that are unique for each of the problems exhibiting this behavior. Finally the instability starts propagating into the intact domain of the structure. The lowest
load at which the instability will spread, corresponds to steady-state quasi-static
propagation and is called ”propagation load”. Usually the propagation load is significantly lower than the load required to initiate instability in the intact structure.
6
As Kyriakides et al Kyriakides et al. (1995) state the issue of compression has the
three phases described above, with kinking being the final stage of localization. In
other problems, which exhibit propagating instabilities, the propagation stress is
characteristic of the material or structure Kyriakides et al. (1995). In the case of
kink band propagation, the associated stress is a characteristic stress of the bulk
composite material. Vogler and Kyriakides Vogler and Kyriakides (1997) have outlined the requirements should pre-exist during experiments, in order this stress to
be a true material property.
Kyriakides et al Kyriakides et al. (1995) studied in detail the characteristics of
the kink bands and kink planes (kink bands on the surface and in the interior of
the specimens, respectively) by means of optical and scanning electron microscopy.
They found that kink bands are bands of broken fibres, in contrast with Moran et
al Moran et al. (1995) who observed that local fibre bending does not cause fibre
breaking in their systems. The kink band angle orientation was found to be 120 -160 ,
and the widths of these kink bands were found to vary from 76µm to 255µm (from
11 to 36 fibre diameters). The kink bands in the interior of the specimens found
to be quite straight, and their width varies less than those on the surface of the
specimens. The kink band angles vary between 120 -16.50 , which are identical to
the kink bands angles on the surface of the specimens. The kink band widths vary
between 150µm - 220µm (approximately 20-39 fibre diameters). In all the cases, the
rotations of the kinked fibres (φ) were in the range of 130 -240, that is significantly
less than the expected lock-up rotation of 2β. But as Kyriakides et al Kyriakides
et al. (1995) suggest this a direct result of the radial constraint. Kyriakides et al
Kyriakides et al. (1995) observed that as the confining increased, the rotation of
broken fibres in the kink bands was reduced.
7
The confinement, used in reference Kyriakides et al. (1995) to preserve the kink
bands, limited the post-failure deformation and possible prevented in this way, the
attainment of the natural post-failure equilibrium state in this way. For this reason,
Vogler and Kyriakides Vogler and Kyriakides (1997) employed a different experimental set-up to measure the post-failure strength of composites in compression (for
more details the reader should refer in Ref. Vogler and Kyriakides (1997)). In order
to study the nature of kink initiation, they concluded experiments where the specimens were immediately unloaded, after the first failure occurred. The kink band
orientation angle was found to be about 150 , which is very close to the average value
of the inclinations of the kink bands in the confined experiments Kyriakides et al.
(1995). Close to the two edges of the specimens, Vogler and Kyriakides Vogler and
Kyriakides (1997) observed a transition zone in which the inclination of the bands
was slightly higher than 150 due to the influence of the free edges. The kink band
width was about 2000µm, that is, an order of magnitude larger than the values
in the smaller confined specimens. The fibres inside the kink band were found to
have an average rotation value, corresponding approximately to 2β. Therefore, the
assumption of Kyriakides et al Kyriakides et al. (1995) that, in the confined experiments the angle φ is quite smaller than 2β because of the confinement, seems
to be realistic. In reality, however, the orientation angle of the kinked fibres inside
the kink bands is higher than 2β, because the angles β and φ were measured after
unloading the specimens. Fibres were clearly broken at the boundaries of the kink
bands, while there are broken fibres inside the bands as well. The distance between
two breaks in a fibre ranges from 35µm to 70µm.
In order to study the kink propagation, Vogler and Kyriakides Vogler and Kyriakides
(1997) performed experiments in which the compression continued after the onset
8
of first failure. The variables which determine the kink bands found to be the same.
In other words, β found to be 150 and φ equal to 290 , that is a value very close to 2β.
Once again, just before unloading, the values of the axial and lateral displacement
are higher; this fact supports their suggestion that in the presence of stress the
material inside the kink band is sheared to an angle beyond 2β. This issue will
be presented in details in the next paragraph. Contrary to the values of φ and β,
the kink band width was increased from 1400µm after first failure to 2900µm as
might be expected. Contact of the straight fibres outside the band with the rotated
fibres inside tends to bend the straight fibres. Excessive bending results in breaking
of a narrow strip of these fibres all along the boundary, which then rotate further
and conform with the deformation inside the original band, resulting in a process of
increasing width. Moreover, the edges of the broadened kink bands are not perfectly
straight, making the interface even more imperfect, and, subsequently, the bending
of the incoming fibres easier. As Vogler and Kyriakides Vogler and Kyriakides (1997)
stated this ’mask’ the ’initiation’ of the next zone of fibres entering the kink band
and results in the flat stress plateau seen experimentally. According to Budiansky
and Fleck Budiansky and Fleck (1993) (a) the transverse strains (ǫT ), (b) the shear
strains (γ) inside the kink band, and (c) the angles β, φ are related as follows:
ǫT =
ln(cos(β − φ))
cosβ)
γ=φ
(2.1)
(2.2)
Thus, for φ >2β the transverse strain becomes compressive. Once the fibres inside
the kink band have rotated beyond 2β, the band material is under transverse compression and starts to resist further shearing. This local recovery in stiffness arrests
9
the shearing and makes the broadening of the band the preferred mode of deformation; the same conclusions have been made by Moran et al Moran et al. (1995),
as mentioned above. However, they Moran et al. (1995) reached their conclusion
assuming a different mechanism for the fibre lock-up.
The literature survey reveals that the experimental data concerning the rotation
angle of the fibres inside a kink band mainly arises from unloaded specimens. Kyriakides et al Kyriakides et al. (1995,?) observed that although in the absence of load
the fibres inside a kink band were rotated to an angle approximately 2β, in the presence of load the rotations angles were significantly larger than 2β. To investigate
this phenomenon Vogler and Kyriakides Vogler and Kyriakides (1999) developed a
new experimental set-up to measure directly the deformation inside a kink band. In
their compression test (specimen geometry and experimental details are described
in Ref. Vogler and Kyriakides (1999)]) the first part of the response was, similarly,
nearly linear. Failure involved the dynamic formation of a kink band, which traversed the specimen; at this point the specimen was unloaded and the characteristic
variables of the formed kink band were measured. The kink band inclination was
approximately, the fibre average rotation was 310 , and the kink band width was approximately 2200µm (about 300 fibre diameters). As the band was reloaded, fibres
inside the kink band rotated (340 , 370 ) while the band width remained unchanged.
As the stress level reached the axial propagation stress, the band started to propagated (wk ∼
=2500µm, φ=390). During the stress plateau, φ remained unchanged
(between 400 and 410 ), the band width propagated steadily and reached a width
of approximately 3400µm, and β increased slightly to 170 . During unloading (end
of the experiment) fibre rotation remained relatively unchanged at the beginning
and subsequently decreased, while the band width remained constant. By the time
10
the specimen was fully unloaded, φ returned to 320 . Hence, Vogler and Kyriakides
concluded that the assumption, commonly made, that φ=2β, in the presence of
load, is not correct.
2.1.2 Failure mechanisms upon compressive loading
Although kinking is the dominant compression failure mechanism in unidirectional composites, usually, more competing modes of failure can be present.
Moreover, kinking process is usually associated with initial fibre misalignment effects Argon (1972); Budiansky (1983); Fleck et al. (1995). Piggott Piggott (1984);
Martinez et al. (1997), however, demonstrated that a variation of the initial fibre
misalignment angle, φ, from 0.50 to 150 in carbon and polymer fibre reinforced polymer composites did not lead to any reduction in the composite strength. Creighton
and Clyne Creighton and Clyne (2000) studied the compressive response of carbonfibre/epoxy composites produced by pultrusion, in which the fibre misalignment
was estimated to be below 10 . However, the failure mechanism was always via kink
band formation (β ∼
= 200 ). They also found that there was a substantial reduction
in strength due to the presence of pores in the matrix material. These results show
that the fibre misalignment may not be the main cause of the compressive failure
in the form of kinking. Bazhenov et al Bazhenov et al. (1989) observed kinks to be
initiated by the compressive failure of fibres themselves and consequent formation of
instability regions in polymer (PABI or PPT) fibre reinforced polymer composites.
Kozey Kozey (1993b,a) observed splitting to occur in a local mode in CFRP, and
the split crack quickly triggered the final kinking failure. Failure by splitting, was
also observed by Andrews et al Andrews et al. (1998) in four-point bending tests of
aramid fibre reinforced pultruded rods based on a vinyl ester resin matrix, whereas
11
this was not the case when an epoxy matrix was used. Similar results also reported
by Piggott Piggott and Harris (1980). This difference is, most likely, due to the
better transverse strengths measured in the rods made from epoxy resin matrix.
Furthermore, Andrews et al Andrews et al. (1998), by using the technique of Laser
Raman spectroscopy, observed that the compressive failure stress of the aramid fibres in the rods (for both types of resin used) was significantly lower than the failure
stress of individual aramid fibres. They suggested that this lower compressive failure stress may be due to compressive failure of fibres occuring by cooperative kink
band formation in the compressive rods.
Karayaka and Sehitoglou Karayaka and Sehitoglou (1996) experimentally observed
that temperature has a remarkable effect on the compressive strength and associated failure mechanisms of unidirectional AS4 carbon-fibre/epoxy composites. At
temperatures below 1000 C failure was found to be due to uniform development of
longitudinal matrix cracks followed by transverse shear failure of the composite. At
temperatures above 1000C, localised matrix failure lead to decrease in the lateral
support provided by the matrix on the fibres and formation of kink bands. The kink
band boundaries were well defined by fractured fibres, while there was no evidence
of interfacial failure away from the kink bands. Within the kink bands the fibre
remained attached to the epoxy resin.
2.1.2.1 Effect of fibre/matrix adhesion on compressive
composite mechanical properties
Madhukar and Drzal Madhukar and Drzal (1992) quantified experimentally
the relationship between fibre/matrix adhesion with the compressive properties and
failure modes of unidirectional carbon-fibre/epoxy composites. They used three
12
identical sets of composites differing only in their fibre/matrix interfacial shear
strength. The interfacial shear strength was altered by using the same carbon fibres
with different surface treatments. They found that fibre surface treatment affects
only slightly the compressive modulus. The compressive strength and maximum
compressive strain, however, according to these authors, are highly sensitive to the
fibre surface treatment. Both strength and strain increase with increasing the interfacial shear strength. Concerning the failure modes, they Madhukar and Drzal
(1992), observed that when the fibre/matrix bonding is poor, failure occurs by delamination and global delamination buckling. An increase in the interfacial strength
leads to shear failure driven by inphase microbuckling of the fibres. For the highest
values of interfacial shear strength, they observed yielded, ’stepped’, fracture surfaces, which show compressive failure of fibres in several planes perpendicular to
the fibre axis. Ha and Nairn Ha and Nairn (1992) observed similar results.
2.2
Model composite specimens
Since a fully fibre-reinforced composite is far too complex system to study,
many researchers revert to model systems in order to study the micromechanisms
of failure in composite materials. The micromechanics of reinforcement under tensile loading has been widely investigated van den Heuvel et al. (1996a,b); Marston
and Galiotis (1998); Galiotis et al. (1999). It seems, however, that very little information about the micromechanical behavior in compression is available. Generally,
in such model composite systems, single layers of either tows or monofilaments are
embedded, commonly, in an epoxy matrix. Upon loading the specimen, subsequent
breaks occur in the tow or filaments and by registration of breaks positions and
monitoring the stress or strain along the fibres, indications of the load transfer
13
mechanisms can be obtained.
Until the development of the technique of laser Raman spectroscopy (LRS), experimental techniques, such as photoelasticity or polarized-light microscopy, could
provide information only in a qualitative manner about the failure events. Laser
Raman spectroscopy technique has been established as the only experimental technique to date that can provide information on fibre stress in a quantitative manner
at the microscopic level.
In this Section, a review of experimental results on model composites loaded in
compression will be discussed.
Boll et al Boll et al. (1990) studied the compression behavior of single carbon filaments embedded in a block of epoxy polymer with a polarized-light microscopy
technique. According to their qualitative observations, they suggested that carbon
fibres do not fail in the classical microbuckling mode Rosen (1965) but the basic
failure mechanism for an isolated filament, is shear failure. Most of the experimental
evidence in the literature suggests a micro-buckling mode based on the observation
of kink bands in post-failure examination of laminates. There have been, however,
some research works on fibre shear failure for CFRP composites in compression.
Ewins and Potter Ewins and Potter (1980) reported a distinct change in failure
mode from shear failure to microbuckling as the temperature approached the glass
transition temperature (Tg ), of the matrix in carbon-fibre epoxy composites. This
change in mechanism occurred because of the lower modulus of the matrix above Tg .
Hahn and Williams Hahn and Williams (1984) observed that compression induced
shear failure in high modulus carbon fibres (517GPa), whereas, in the case of lower
modulus (230GPa) fibres, kink bands were generated. In testing pitch-based carbon
14
fibres, Hawthorne and Teghtsoonian Hawthorne and Teghtsoonian (1975) generally
observed fine, compression-induced cracks except in the case of the lowest modulus
fibres. These cracks, for the highest modulus fibres, appeared simultaneously and
fairly evenly distributed along the fibres. Further loading of the specimens produced
a gradual coarsening of the cracks until eventually some of them developed a distinct
shear fracture (the interface and the surrounding resin was disturbed around these
shear fracture locations. For the low modulus fibres (ELf ∼
=130GPa), they could not
detect fine cracking prior to observe the beginning of a sheared fracture. This is
in contrast with the results of Hahn and Williams Hahn and Williams (1984) who
reported kink band formation in low modulus carbon fibres as previously mentioned.
Boll et al Boll et al. (1990) found that the fragment distribution differs from those
obtained in tension experiments (of the same fibre-matrix system) in that under
compression loading the average size is smaller (180µm), the size distribution is
more narrow and relatively symmetrical. The average fragment length for an AS4
fibre in tension embedded in the same epoxy is about 400µm. The uniformity of
the size distribution suggests that compressive fracture is not governed by a random
flaw distribution as in the case of tensile fracture. Boll et al concluded that compressive failure is determined by some microcrystalline structure that is uniformly
distributed along the fibre. Hawthorne and Teghtsoonian Hawthorne and Teghtsoonian (1975) reached a similar conclusion. They examined numerous compression
microcrack fractures and they found only few cases where the crack might possibly
have been associated with an observable flaw. Hence, they concluded that axial
compression fracture does not seem to be associated with gross fibre flaws. Axial
compressive strength of the anisotropic carbon fibres composed of wrinkled and entangled ribbons of turbostratic graphite layers mutually separated by microvoids.
15
The higher the Young modulus, the more the microfibrils are progressively aligned
along the fibre axis and the more they tend towards a more graphite nature (in
terms of crystalline size and perfection). An increase in radial ordering and decrease in transverse interfibrillar coupling usually accompanies an increase in axial
alignment of carbon fibres. Consequently, by growing the fibre anisotropy from the
isotropic material intrinsic (flaw free) strength value, the resultant filament compression failure may, firstly, increase, and then, as the modulus approaches very
high values, decreases. The more ordered the carbon fibres (higher modulus, the
more shear modulus decreases, resulting to a decrease in the fibre strength. The
change in appearance of compression fracture modes and the microfibrilar structure
of carbon fibres made Hawthorne and Teghtsoonian to suggest that axial compression failure of oriented carbon fibres is due to a buckling or kinking structural
instability. Microcracking may be initiated as a buckling or kinking of single microfibrils of well-ordered layer packets, which, because they are less interwoven and
coupled with their neighbours than those of lower modulus fibres, would be less
constrained to deform individually. This localised distortion does not need to involve dislocations but ”crystallite regions weak in shear”. Their conclusion is that
simple shear initiation of axial compression failure in glassy carbon fibres changes
to the microbuckling mode as fibre anisotropy increases. It should be noted that in
contrast to the compression, the tensile strength increases as fibre Young’s modulus
increases.
Based on SEM experiments, Boll et al Boll et al. (1990) proposed that compression
failure is initiated as a microcrack, which then propagates as a shear failure. The
subsequent post-failure damage may take a variety of forms (see Figure 2.5). For
comparison purposes, a single glass filament (S-glass) embedded in the same epoxy,
16
was subjected to compression loads; they reported distinct microbuckling as the
failure mode of the glass fibre.
A somewhat different compression failure mechanism has been reported from Narayanan
and Schadler Narayanan and Schadler (1999a) in a recent work. In agreement with
the previous researchers, they detected fibre surface features in the early stages of
compression (four-point bending tests of single tow composites). These features are
straight or wavy lines; wavy lines do not form from thin-straight lines. Using the
technique of laser Raman spectroscopy, they obtained the strain profiles for these
features. Thin lines did not always cause a decrease in the strain to zero, but they
show a definite drop in the load. Therefore, Narayanan and Schadler Narayanan
and Schadler (1999a) suggested that thin lines are defects but not breaks. The
wavy lines dropped the strain to zero, and some of them showed a debonded zone.
According to the authors, these observations are clear indications of complete fibre
failures. The wavy nature of these features suggests that the fibre has undergone a
shear deformation of its surface. The onion skin structure of the fibres could facilitate the shear deformation on the few graphene layers on the surface, which might
be the reason for the wavy lines. Outer layer cracks, also, recognized by Hawthorne
and Teghtsoonian Hawthorne and Teghtsoonian (1975). They observed multiple
cracks on the fibre surface. By polishing the fibre no trace of the cracks remained.
This is an indication of gradation in the severity of fracture from surface to core.
An increase in compression load causes fibre breaking at locations where there are
no wavy lines. This step involves three stages. The first stage is fibre bending with
the creation of debonding at the bent region. As the applied strain increases, the
bent fibre is forced against one side of the debonded region (Figure 2.6a). The final
stage is the breaking of the bent fibre which may occur with the ejection of a small
17
wedge of the fibre projecting out of the fibre axis as shown in Figure 2.6b. In some
cases the feature resembling the one shown in Figure 2.6b, without the small wedge
being ejected. In this case the fibre ends just slide past each other.
18
Figure 2.1: A schematic illustration of a kink band showing several parameters.
φ + φ is the kink band orientation angle, β is kink band angle, and wk is the kink
band width.
19
Figure 2.2: Kink band in unidirectional PAN graphite-reinforced PEEK, Lankford
Lankford (1995).
20
Figure 2.3: Photomicrograph of a kink band formed in a AS4/PEEK unidirectional
composite, Vogler et al Vogler et al. (2000).
21
Figure 2.4: Overall load-end shortening behavior of unidirectional fibre composite
compressed in the fibre direction when kinking occurs. The load levels at incipient
kinking, propagation across the specimen’s width and transient and steady-state
band width broadening are indicated. Moran et al Moran et al. (1995).
22
Figure 2.5: Schematic failure sequence: (a) microcrack initiation, (b, c) shear failure,
(d) crushing damage, (e) shear sliding, (f) longitudinal splitting, (g) bifurcation of
shear failure and (h) multiple fracture. (Boll et al Boll et al. (1990))
23
Figure 2.6: Schematic illustration of (a) fibre bent, (b) ejected wedge of fibre and
locally debonded region and (c) fibre sliding in the debonded zone without the
ejection of a fibre wedge. Narayanan and Schadler Narayanan and Schadler (1999a)
CHAPTER 3
MODELING OF THE COMPRESSIVE BEHAVIOUR
In this chapter, a short literature review on the existing analytical models for the
prediction of the composite compressive strength is given. In general, the study
of this topic is divided into two branches: fibre microbuckling models and kink
band formation models. Although some researches have shown that microbuckling
and kink band type failure modes could be incorporated into one continuing theme
Chaudhuri (1991), these two methods are still treated as parallel models, and thus
will be presented here in this distinguishing way.
3.1
Introduction
Prior to present the main analytical models for the prediction of composite
compressive strength, the main failure mechanisms will be firstly addressed and the
corresponding formulas will be given.
3.1.1 Elastic microbuckling
Early investigations associated compressive failure with a fibre buckling process in an elastic matrix Rosen (1965). Rosen’s model Rosen (1965) considers a
two-dimensional array of layers of fibres bonded to an elastic matrix (elastic microbuckling requires the matrix to behave in a linear elastic way up to high strain.
The majority of commercially available composite materials, however, do not satisfy this requirement). Two possible modes of microbuckling are assumed (e.g. the
24
25
shear mode and the extensional mode, see Fig. The shear buckling mode is predicted for all practical cases. In composites of a significant fibre volume fraction,
e.g. uf >0.3, the shear mode governs the compressive strength. Higher-order models (e.g. see Ref. Zhang and Latour (1994)), however, have shown that the shear
mode is favoured at all values of fibre volume fraction and the predictions of the two
modes differ only slightly in the low fibre volume fraction range. The compressive
strength is simply described by the following equation:
σc =
Gm
LT ∼
= Ḡ
1 + uf
(3.3)
where Ḡ is the effective shear modulus, Gm
LT is the matrix shear modulus and uf is
the fibre volume fraction. The experiments that followed Rosen’s analysis consistently found compressive strength values much lower (3 to 4 times) than the model’s
predictions. Moreover, as stated by Niu and Talreja Niu and Talreja (2000), Equation 3.3 has not clearly been proven by a microbuckling analysis.
3.1.2 Fibre crushing
Fibre crushing occurs when the axial strain in the composite attains a critical
value equal to to the crushing strain of the fibres, ǫfc . Fibre failure can occur by
longitudinal splitting (silica glass), plastic yielding (metallic fibres), kinking within
the fibres (wood) or by plastic microbuckling of the microstructural units of the
fibre (kevlar). In these cases the compressive composite strength is given by the
rule-of-mixtures:
σc = uf σcf + (1 − uf )σym
where σcf is the fibre crushing strength and σym is the matrix yield strength.
(3.4)
26
3.1.3 Matrix failure
Matrix failure occurs when the axial strain in the composite attains the failure
strain of the matrix ǫm
c . The mechanism of failure is brittle crack propagation in the
matrix material. This failure criterion (rule-of-mixtures) predicts a failure stress,
σc of:
σc =
f
m uf EL
σc ( m
EL
+ (1 − uf ))
(3.5)
where σcm is the failure strength of the matrix itself. It should be noted that this
mechanism operates mainly in ceramic composite materials Jelf and Fleck (1992).
3.1.4 Plastic microbuckling
Argon Argon (1972) identified the composite shear strength (τyc ) and the initial fibre misalignment angle (φ) as the main factors controlling the compressive
strength. The composite strength is given by:
σc =
3.2
τyc
φ
(3.6)
Fibre Microbuckling Models
As mentioned previously, Rosen’s formula yields compressive strengths which
are significantly higher than those measured and as a result many extensions and
improvements (see Refs. Chung and Testa (1969); Greszczuk (1975)) of Rosen’s
model have been proposed. These refinements, however, have not significantly
27
lowered Rosen’s prediction of compressive strength. Steiff Steif (1987), for example, performed an exact two-dimensional micro-buckling analysis involving linear
or non-linear constituents within a bifurcation framework but his prediction was
similar to that of Equation 3.3. Maewal Maewal (1981) concluded that compressive
microbuckling can occur at small strains only if the ratio of the moduli of the constituents is relatively large. Moreover, he found that the composite is imperfection
insensitive so that the initial waviness of the fibres is not expected to reduce significantly the microbuckling stress. Budiansky Budiansky (1983) generalised Rosen’s
formula (Equation 3.3) to:
σc = Ḡ + E¯T tanβ 2
(3.7)
for β 6=0. Equation 3.7, however, gives β=0 as the critical angle for kinking - this
despite the fact that observed kink angles are usually bounded well away from zero
as it has been already mentioned.
Lagoudas et al Lagoudas et al. (1991) have tried to improve Rosen’s formula by
treating the composite as an inhomogeneous two-dimensional continuum with spatial variation in the axial Young’s modulus to account for fibres and matrix (the
periodicity is taken into account by expanding the axial Young’s modulus in a
Fourier series with wavelength the average spacing between fibres). When the fibres
are perfectly aligned their predictions coincide with Rosen’s results. By introducing,
however, initial imperfections the predicted compressive strength significantly decreases and they suggested that initial imperfections might be an important factor
in initial microbuckling. The compressive strength is given by:
28
σc = G
f
p
m
EL
−EL
sin(πuf )(cos(πuf ) − cos2 (πuf ) + 8)
2πE
f
f
m
m)
EL
−EL
4(EL
−EL
sin(πu
)(cos(πu
)
−
sin(πuf ))
f
f
2πE
πE
1+
1+
(3.8)
where E and G are the effective axial and shear stiffness of the composite, respectively. Tadjbakshs and Wang Tadjbakshs and Wang (1992) expanded the twodimensional analysis of Lagoudas et al Lagoudas et al. (1991) in order to study the
fibre buckling in three dimensional (cross-ply laminate) composites.
The above fibre microbuckling approaches focus on the initiation of buckling. However, according to Steiff Steif (1990a), only fibre buckling is not adequate. The
fibre needs to bend sufficiently far in order for fibre fracture to happen, and this
results in the kink band formation. The width of the kink band is determined by
the wavelength of the imperfection which causes microbuckling. The kink band
angles (β) predicted by this treatment Steif (1990b), however, were not within the
range of the experimentally observed values. β was found to approach 450 , but it
is observed experimentally that kink bands are inclined at β=200 ∼300 Schultheisz
and Waas (1996); Moran et al. (1995). Moreover, the idea of tensile fracture of the
fibres in bending (the fibre breaks when the maximum tensile strain in the fibre is
equal to the fibre failure strain) used in this analysis Steif (1990a,b) is not supported
experimentally Berbinau et al. (1999).
In the above microbuckling analyses, the fibre is assumed to be perfectly bonded
to the matrix, that is the fibre/matrix interface has no effect on the composite
compressive strength, Xu and Reifsneider Xu and Reifsnider (1993) developed a
micromechanical model to account for the effects of fibre debonding and matrix
slippage. Their final expression, which does not include factors for fibre misalignment and fibre initial curvature, for calculating the compressive strength in terms
29
of the constituent properties and micromechanical parameters, is as follows:
σc = Gm
LT (uf +
v
u
u
(2(1 + vm )t
√
π πnrf
Em
Em
L
L
ELm
ELf
(1 − uf ))
3 ELf (uf ELf + 1 − uf )(1 + uf vf + vm (1 − uf ))
(3.9)
sin(πξ)
)
2π
where ξ is the extent of matrix slippage and the parameter n represents the fi+1 − ξ −
bre/matrix bonding conditions. Their most striking conclusion was that a complete
slippage of the matrix reduces the compressive strength by over 50%, which implies that the interfacial shear strength has a considerable effect on the compressive
strength. Their predictions, however, do not always correlate with the experimental
data Niu and Talreja (2000). The impact of the interfacial strength on the compressive strength, has also been investigated by Williams and Cairns Williams and
Cairns (1994), who concluded that the interface instability mode occurs prior to the
buckling mode of the fibres.
The assumption of initial fibre waviness has also been adopted by Chung and Weitsman Chung and Weitsman (1994), who, furthermore, suggested that the spacings
between the fibres are random. These stochastic spacings, combined with a nonlinear shear response of the matrix, were found to result in highly localised internal
transverse loads on the fibres. Thus, they concluded that these transverse loads
may cause a transition from microbuckling to micro-kinking of the fibres. In this
work, however, the response of the fibre was modelled by means of Bernoulli-Euler
beam theory (which does not allow for shear deformation) and this approach rulled
out the formation of kinks. In a subsequent work, Chung and Weitsman Chung
and Weitsman (1995) refined this model by taking account the shear deformations
30
in the fibres, and they observed a band of discontinuity in the fibres’ shear strains
immediately after failure by buckling, which may result in kink band formation. A
similar kind of approach was followed by Lessard and Chung Lessard and Chung
(1991); Chung and Lessard (1991), who assumed nonuniform loading of the fibres.
By a variational method based on the minimum energy principle, they found that
the load distribution on the fibres has a great effect on the buckling behaviour of the
composite. More specifically, a nonuniform and more concentrated load distribution
could significantly affect load fibre buckling strength.
Similar to the majority of microbuckling analyses, Grandidier et al Grandidier et al.
(1992) presented a model where the fibre is modelled as en elastic beam and the
matrix as an elastic foundation. In contrast to the previous models, the strain
in the matrix is distributed through the thickness (thickness effect) and lowers
failure stresses. Drapier et al Drapier et al. (1998, 2001) extended the linear elastic
homogenised model of Grandidier et al Grandidier et al. (1992) to account for the
effect of initial fibre imperfection and matrix non-linearity (plastic microbuckling).
Most fibre buckling models, presented previously, are deterministic with respect to
fibre misalignment, in the sense that all the fibres are assumed to have a single
value of fibre misalignment. Thus, fibre misalignment is taken as an empirical parameter and it is set to a reasonable value so that the model predictions match the
experimental data. It is well known, however, that there is not a unique value for
fibre misalignment for all the fibres Yurgartis (1987), but, most likely, a Gaussian
distribution of misalignment Barbero and Tomblin (1996). The standard deviation
is a measure of the dispersion, and not of the expected value of the distribution.
From a purely statistical point of view, a single value of misalignment that in the
average represents the population, is the mean value. The mean value, however,
31
of the misalignment distribution is zero. Barbero Barbero (1998) developed a microbuckling analysis assuming that the area of buckled fibres is proportional to the
area of under the normal distribution located beyond the misalignment angle ±φ.
The compressive strength was then to be controlled by a dimensionless parameter
χ = GΩ/τym , where Ω is the standard deviation of the normal distribution of the
fibre misalignment angles.
3.3
Kink Band Formation Models
As stated previously, before Argon Argon (1972) was first to recognize that
the initial fibre misalignment angle, φ, would have a large degrading effect on the
compressive strength, and for a perfectly plastic composite the critical stress is given
by Equation 3.6, which is independent of the fibre volume fractions (uf ). Budiansky
Budiansky (1983) extended Argon’s formula to a more general expression (elasticperfectly plastic composite) as:
σc =
τyc
γyc
+φ
=
Ḡ
1+
φ
γyc
(3.10)
where the yield strain is defined as γyc = τyc /Ḡ. For γyc ≪ φ Equation 3.10 gives
Argon’s expression while for φ=0 leads to Rosen’s formula.
Budiansky and Fleck Budiansky and Fleck (1993) provided a more general expression through traction continuity of the kink band boundary:
σ0 − 2τ0 tanβ =
τ − τ0 + σtanβ
φ+φ
(3.11)
by connecting the applied stress σ0 , τ0 , the kink band angle, the fibre additional
rotation angle (φ), and the stresses σ and τ that develop in the kink band. In
32
the case of β=0 and a pure compressive stress, Equation 3.11 reduces to Equation
3.10. It should be noted that in this treatment fibre inextensibility is once again
assumed, and the fibre bending resistance is neglected. The matrix is described
by a plastic strain-hardening law. Finally, the deformation state is considered to
be homogeneous but different inside and outside of the kink band (the fibre/matrix
interface has no effect on the compressive behaviour), while the kink band is assumed
to pre-exist. As shown by Karayaka and Sehitoglou Karayaka and Sehitoglou (1996)
rigid fibre assumption is only valid at elevated temperatures where matrix failure
is governed by shear stresses. A somewhat similar approach has been followed by
Lagoudas and Saleh Lagoudas and Saleh (1993). The basic difference is that in this
work the modes of energy dissipation in the kink process zone are taken into account,
which enables the study of the effect of micro-geometry and phase materials in the
compressive strength, whereas in the work of Fleck and Budiansky Budiansky and
Fleck (1993), the dissipated energy is calculated through the work done at the kink
band boundaries by the applied tractions. The dissipated energy in the kink process
zone is consisted of two terms (a) the work done to break all fibres crossed by the
kink band as it propagates, and (b) the amount of work dissipated in plastically
deforming the matrix due to large rotation of the fibers (the dissipation energy
outside the kink band is neglected). Their predicted compressive strength is given
by:
σc =
s
2uf E m 2
(σ φwk + 2πdf gf )
πdf L y
(3.12)
where σym is the matrix yield stress and gf is the fibre critical energy release rate
corresponding to the fibres flexural toughness. In the case where the dissipated
plastic work is much larger than the work spent in breaking the fibres (i.e. for
33
carbon/thermoplastic composites) the compressive strength is given by:
σc =
s
2uf E m 2
σ φwk
πdf L y
(3.13)
It should be noted that Equation 3.12 gives conservative predictions of the compressive strength with respect to the experimental data, and this is not the case
for the predicted strengths of Fleck and Budiansky Budiansky and Fleck (1993).
Lagoudas and Saleh Lagoudas and Saleh (1993) compared the compressive strength
due to kinking as predicted by Equation 3.12 with the compressive strength due to
microbuckling according to Equation 3.8 for carbon/thermoplastic and boron/epoxy
composites. They found that kinking failure is more likely to occur for carbon.thermoplastic
composites for all fibre volume fractions, whereas for a wide range of uf (10-70%)
the microbuckling failure is more likely to occur for the boron/epoxy composites.
It was found experimentally, by Kyriakides et al Kyriakides et al. (1995), that for a
carbon fibre reinforced polymer matrix composite, the fibres responded non-linearly
upon compression to strain levels relevant for kink band development. Jensen Jensen
(1999a) developed a model where all constituents are allowed to behave non-linearly
as long as they are described by time independent elastic-plastic constitutive relations (the non-linearity is taken to be entirely due to the material behaviour). Similar to Lagoudas and Saleh Lagoudas and Saleh (1993), the states of deformation
inside and outside the kink band are determined solely by equilibrium, continuity
and balance of energy. Moreover, in the previous models (i.e see Refs. Moran
et al. (1995); Fleck (1997); Budiansky and Fleck (1993)) the following fibre lock-up
condition was used:
φ = 2β
(3.14)
34
corresponding to zero volumetric straining of the matrix material between the rigid
fibres. Experimental observations, however, indicate that Equation 3.14 is not satisfied precisely (Refs. Poulsen et al. (1997); Vogler and Kyriakides (1997)). In
Jensen’s model Jensen (1999a) the lock-up condition derives from energy balance
between the work done by external loads as the kink band broadens and the work
done by local stresses in the band as the fibres rotate from the initial to the final
state. The fibre rotations at lock-up were found to exceed the rotations given by
Equation 3.14, which is consistent with the experimental observations.
A quite different approach has been adopted by Sutcliffe and Fleck Sutcliffe and
Fleck (1993), and Sivashanker et al Sivashanker et al. (1996), who used a fracture
mechanics method (bridging analysis) to investigate the compressive failure. The
microbuckled region was treated as a cohesive zone with zero intrinsic toughness
at the tip of the microbuckle. In a subsequent work, Sutcliffe and Fleck Sutcliffe
and Fleck (1997) modelled the propagating microbuckle using the finite element
method with a tip process zone and a sliding crack behind the tip. The tip of the
microbuckle was modelled as alternate layers of fibre and matrix, while the existing
microbuckle was modelled as a sliding crack, with sliding resisted by a shear traction equal to the shear yield stress of the matrix. Predictions of the microbuckle
propagation direction were found to reasonable agree with the experimental data
in the case of polymer matrix composites. Using a similar approach, Fleck et al
Fleck et al. (2000) concluded that the presence of a free surface has little effect
on the compressive strength from a region of fibre waviness. Different results were
reported by Zhang and Latour Zhang and Latour (1997b,a), who conducted a fibre microbuckling analysis to investigate the effect of free-edge conditions. They
found that the free-edge fibres have significantly lower compressive strength then
35
the internal fibres. According to their predictions, free-edge fibre microbuckling
can be suppressed by the application of a matrix coating applied to free-edges of
the composite (i.e. for carbon-fibre/PEEK composites with uf =60%, it was found
that the matrix coating thickness should be equal or greater than 12 times the fibre
diameter). The cohesive zone model was also used by Soutis and Fleck Soutis and
Fleck (1990), Soutis et al Soutis et al. (1991, 2000) and Soutis and Curtis Soutis
and Curtis (2000) to predict the compressive strength of laminates with open holes.
Slaughter and Fleck Slaughter and Fleck (1993) studied the effect of matrix viscoelasticity in microbuckling of fibre composites. Their analysis is formulated in
terms of (a) a standard linear viscoelastic matrix, and (b) a logarithmically creeping solid inside the kink band, whereas the homogeneous material outside the kink
ban is assumed to behave elastically. Similar to the most of the previous analyses,
the kink band is assumed to pre-exist in the composite. They found that in some
cases (depending on the Ḡ value) viscoelastic microbuckling can occur at stress
levels below the plastic microbuckling stress.
Jensen Jensen (1999b) extended the analysis of Christoffersen and Jensen Christoffersen and Jensen (1996) to include the possibility of interface debonding between
the fibres and the matrix (tractions are not transmitted fully across the fibre/matrix
interface). For the case of uniaxial compression parallel to the fibres and full decohesion between the fibres and the matrix, the kink stress was found to be lower
by a factor u2m compared to the kink stress in the case of perfect bonding between
the fibres and the matrix. Furtherly, Jensen Jensen (1999b) refined his analysis by
taking account the spreading of microcracks in the matrix upon compressive loading (the effective moduli of the matrix is a function of the applied stress due to the
microcracking). The matrix was assumed to contain cylindrical voids arranged in a
36
perfect structure. For sufficiently large transverse compressive stresses, the microcracks remain closed, and the matrix moduli does not change relatively to the initial
value. For small or no transverse compressive stresses, however, the microcracks
spread stably. Thus, Jensen Jensen (1999b) concluded that the failure mode upon
superposition of transverse compressive stresses (i.e. hydrostatic pressure) changes
from matrix splitting to kink band formation, which is in qualitative agreement with
the experimental results of Weaver and Williams Weaver and Williams (1975) and
Parry and Wronsky Parry and Wronski (1982). The effect of matrix microcracking
has been also studied by Schapery Schapery (1995), who proposed that the kink
band initiation involves both matrix cracking and shear buckling in a band of initially misaligned fibres. The kink band angle was found to be quite sensitive to the
transverse strength of the composite and the stress normal to the fibres. Waas Waas
(1992) presented an analytical method for predicting the compressive behaviour of
composites in the presence of a finite thickness interphase. A single fibre in an
infinite matrix configuration was chosen. His most important conclusions were that
(i) for a fixed interphase thickness and for a given fibre/matrix modulus ratio, the
buckling strain shows little variation over a wide range of ratios of matrix to interphase modulus and (ii) a ’soft’ interphase, whose thickness is approximately (1/10)
of the fibre diameter or greater, will have a detrimental effect on the compressive
stability. Some drawbacks of this analysis are that all the phases are modeled as
2D-linear elastic continuum and that fibre interactions effects are not considered.
Niu and Talreja Niu and Talreja (2000) proposed a different mechanism for the
kink band formation based on the shear instability of the matrix, which was called
shear hinge. They assumed that the buckling state is in a transverse shear induced
buckling mode instead of a flexural buckling mode. In their analysis use is made of
37
the method of split rigidities Bijlaard (1951) where the global buckling load is given
by:
1
1
1
=
+
P
Pe Ps
(3.15)
where Pe is the Euler critical load and Ps is the shear buckling load. From Equation
3.15 an expression for the critical stress σc can be obtained as:
1
1
1
=
+
σc
σe σs
(3.16)
If the Euler critical stress is much larger than the critical stress, then Equation 3.16
is equivalent to Rosen’s microbuckling analysis. However, Niu and Talreja Niu and
Talreja (2000) assumed that the bending stiffness is infinitely large and thus only
the shear deformation is allowable. For a small initial fibre misalignment φ and a
load stystem with a small misalignment φ1 , the buckled state in the shear hinge is
given by:
σ=
τ (γ)
sin(γ + φ + φ1 )
(3.17)
The critical stress (for small angles) is given by:
σc =
τyc
γyc + φ + φ1
(3.18)
which is similar to Equation 3.10. Equation 3.18, however, includes also the effect of the misalignment of the loading system and more importantly the assumed
mechanisms taking place are somewhat different.
CHAPTER 4
EXPERIMENTAL
This Chapter contains information on the adopted experimental set-up and
procedure.
4.1
Introduction
Since a fully fibre-reinforced composite is far too complex system to study
the stress transfer mechanisms and interface integrity of carbon/epoxy composites
under compression, one has to revert to a model system.
4.2
Materials and Specimen Preparation
Surface treated high-modulus carbon fibers (M40-40B) were used as reinforce-
ment material. The mechanical properties of these fibers are given in Table 4.1.
Fibers had an effective diameter of 6.6µm and were embedded in a two part system Epikote 828/Ankamine 1618 (Shell). The resin (Epikote 828) and the hardener
(Ankamine 1618) were mixed at 50◦ C at a ratio 5:3 and degassed for 15min under
full vacuum.
Two different specimen geometries were employed, namely: a dogbone geometry
used in tension experiments and a prism geometry for the case of compression tests,
as shown in Figure 4.7. Prism length is twice its width/thickness to avoid buckling
according to ASTM D 695 standards. To produce the short-fiber specimens in both
cases the resin, after being degassed, was first poured into a silicon rubber mold
38
39
(dogbone specimens) or Teflon mold (compression specimens). Then the fibers
were carefully placed and aligned on top. Care was taken to embed the fibers at
a small distance away from the surface. The composite coupons were cured at
room temperature for one week; a low curing temperature was selected in order
to eliminate the development of residual stresses on the embedded fibers. Prior
to mechanical testing, the residual stresses on the embedded fibers were measured;
specimens in which fiber residual stresses were large or fiber ends were distorted
were discarded (Figure 4.8 depicts a typical fiber end). Fiber lengths varied from
∼ 1.8mm to ∼ 7.0mm.
4.3
Specimen Testing and Raman Spectra Acquisition
Raman spectra were taken with a Remote Raman Microprobe (ReRaM) de-
veloped by Paipetis et al. (1996). As shown in Figure 4.9 the collected Raman light
was guided through an optical fiber to a Spex 1000M single monochromator. The
Raman signal was collected via a Wright Instrument charge-coupled device (CCD)
and stored in a PC-computer. The laser used was a 514.5nm argon ion laser. An incident power of ∼ 1.2mW and an exposure time of 60s were chosen in order to avoid
fiber overheating. The spectral characteristics, i.e. peak positions were derived by
fitting the raw data with Lorentzian distribution functions.
The shift of the Raman wavenumber of the carbon fibers upon the application of
a tensile or compressive strain was measured with the cantilever beam test Vlattas
and Galiotis (1993). This was done by attaching individual filaments on the top
surface of a specially made cantilever beam which can be flexed up or down subjecting the fibers to compression or tension, respectively. Substantial matrix Raman
40
activity was observed in the 500-1700 cm−1 region, moreover with the application
of a compressive load, Raman peak shifts to higher wavenumber values, that is into
the matrix Raman peak. (Figure 4.10 shows the Raman band shift of a M40 carbon
fiber embedded in the epoxy matrix at an applied strain of 0.3% and -0.3%). However, at high wavenumbers the matrix material exhibits very little Raman activity
(see Figure 4.11), therefore the second order peak of the carbon fiber at 2760 cm−1
was employed for fiber strain measurements. The Raman wavenumber shift versus
strain is shown in Figure 4.12. The experimental data were fitted with a 6th order
polynomial of the form:
p1 ǫ + p2 ǫ2 + p3 ǫ3 + p4 ǫ4 + p5 ǫ5 + p6 ǫ6 = ∆ν
(4.19)
where pi are constants and ∆ν is the wavenumber shift of the Raman peak due to
strain ǫ. Equation 4.19 imposes continuity between tension and compression and
satisfies the natural condition:
∆ν = 0 for ǫ = 0
(4.20)
A 6th order polynomial was necessary in order to accurate fit the experimental data
and extrapolate them. Alternatively, the experimental data of Figure 4.12 can be
fitted with cubic splines (e.g. Refs. Chohan and Galiotis (1996); Galiotis et al.
(1999)). Since the experimental data are quite smooth, a polynomial fitting was
more preferable (in spite of its high degree) than cubic splines since polynomials is
simpler to use and the interpolation function is known.
As it can be seen in Figure 4.12 the M40-40B carbon fiber fails at a strain of approximately 1% in tension whereas its compressive failure strain is about -0.6%. The
41
calibration curve is in very good agreement with the data given in Ref. Narayanan
and Schadler (1999b). As expected, carbon fiber displays clearly a non-linear behavior in compression, the linear part extends up to a strain of -0.35% approximately.
In addition it can be observed that the compressive failure strain of the carbon
fibers is significantly lower than the tensile failure strain, which is in contrast with
the results reported by Vincon et al. (1998).
The short-fiber specimens were mounted on a Hounsfield universal testing machine
and strained at distinct strain levels up to ∼ -0.9% for the case of compression
and up to ∼ 1.25% in the case of tension. The applied tensile strain rate, used for
the transition form one strain level to the next one, was 2.1 10−5 s−1 , whereas the
compressive strain rate was 4.2 10−5 s−1 . Time consuming (several hours) point-topoint Raman measurements were taken along the fiber at each strain level. As a
result, measured fiber strains were affected by the viscoelastic nature of the epoxy
matrix, which was free to deform under constant load due to the experimental setup used in the compression experiments (see Figure 4.9). Matrix creep was present
even at low applied compressive strains, although as it is expected, was more severe
at high strain levels. To overcome this problem, we had to periodically relax the
load on the specimen, however, in some cases no strain correction was done or it
was not possible. This effect is only shown in Appendix B since time evolution of
stress profiles is not the main issue of the current work. Laser Raman sampling
was carried out in steps of 2µm at the vicinity of the fiber ends (from 0 to 100µm)
or fiber breaks, then in steps of 5µm (from 100µm to 300µm) and then in steps
of 10µm / 20µm until the middle of the fiber. The above protocol is necessary to
ensure detailed mapping of stress near the discontinuity where the first derivative
of the stress transfer function reaches a maximum value. The applied strain on the
42
specimens was also monitored by a strain gauge of gauge factor 2.09 attached to
the resin surface. In the case of compressive loading two strain gauges were used,
attached on opposite sides to detect any possible macro-buckling of the specimens.
4.4
Analysis of the Experimental Data
The mapping of the stress or strain distribution along an embedded fiber allows
the determination of all the important interfacial parameters (e.g. transfer length,
mode of failure, interfacial shear stress etc.) without resorting to an approximate
analytical solution of the stress transfer problem as is attempted by conventional
micromechanical analyses Cox (1952); Piggott (1978). To facilitate data manipulation polynomials are fitted to the raw fiber stress/strain values. However, if a
function is to be approximated on a larger interval the degree of the approximating
polynomial may have to be unacceptably large. The alternative solution is the use
of spline polynomial functions such as the B-cubic splines. Like most numerical
methods, spline interpolation requires an initial solution as an input, which is the
most arbitrary part of the analysis. Details of the procedure followed for the selection of the number of knots are given in the Appendix A. The selection of B-cubic
splines imposes ISS continuity along the fiber length irrespective of the local interface integrity. This seems to be a more reasonable approach compared e.g. to
Piggott Piggott (1978) or Nairn and Liu Nairn and Liu (1997) models in which ISS
discontinuity occurs between a damaged and an intact interfaces.
43
Figure 4.7: Geometry of single-fibre specimens: (a) prism geometry for compression
tests and (b) dogbone geometry for tension tests (all dimensions in mm).
Figure 4.8: Micrograph of a typical fiber end.
44
Figure 4.9: Schematic illustration of the experimental set-up of the remote Raman
microprobe.
Table 4.1: Mechanical properties
Mechanical properties
M40-40B carbon fibre
Epoxy matrix
ELf [GPa]
390
1.47
ETf [GPa]
14
-
GfLT [GPa]
20
0.56
f
νLT
[-]
0.20
0.3
νTf T [-]
0.25
-
αLf [10−6 C−1 ]
-0.36
-
αTf [10−6 C−1 ]
18
-
45
ε =0%
ε =0.3%
ε =-0.3%
1.0
Normalised intensity (-)
0.9
0.8
compression
tension
0.7
0.6
0.5
0.4
0.3
0.2
2700
2720
2740
2760
2780
2800
2820
2840
-1
Wavenumber (cm )
Figure 4.10: Raman spectra of the M40 carbon fiber embedded in the epoxy matrix
showing the shift of the 2760 cm−1 Raman band with applied strain.
46
300
M40 carbon fiber in epoxy
Intensity (Arbitrary units)
250
200
150
100
50
M40 carbon fiber
0
2600
2650
2700
2750
2800
2850
2900
2950
-1
Wavenumber (cm )
Figure 4.11: Raman spectrum of the M40 carbon fiber embedded in the epoxy
matrix.
15
compression
10
experimental data
polynomial fit (6th order)
-1
Wavenumber Shift (cm )
5
0
-5
-10
-15
-20
-25
tension
-30
-35
-40
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Strain (%)
Figure 4.12: Raman frequency shift versus strain for the M40-40B carbon fiber.
CHAPTER 5
MODELING OF THE STRESS TRANSFER FROM THE MATRIX
TO THE FIBRE
5.1
Shear-Lag Analyses
Stress transfer analysis originated by the work of Cox Cox (1952) (shear-lag
analysis) in 1952. Cox derived a one-dimensional equation for fibre stress that can
be written as:
f
∂ 2 < σzz
>
2
f
2
f
− βcox
< σzz
>= βcox
< σzz,∞
>
2
∂ z
(5.21)
f
f
where < σzz
> is the average axial stress in the fibre, < σzz,∞>
is the average
axial stress in the corresponding infinitely long fibre embedded in an infinitely long
matrix, βcox is the shear-lag parameter given by Cox as:
2
βcox
=
2Gm
LT
2 f
rf EL ln Rrcox
f
(5.22)
f
where Gm
LT is the matrix shear modulus, EL is the fibre axial modulus, rf is the
fibre radius, and Rcox is the mean centre-to-centre separation of the fibres normal
to their length. For single-fibre composites Rcox can be defined as the radius of the
matrix cylinder beyond which there is no influence of the fibre upon the deformation
of the matrix Nairn (1997); Galiotis and Paipetis (1998). Cox, to derive Equation
5.21, started by an exact equilibrium relation between average axial fibre stress and
interfacial shear stress (τrz ):
47
48
f
2τrz
∂ < σzz
>
=−
∂z
rf
(5.23)
Equation 5.23 of equilibrium is the well known balance of forces argument, and its
derivation is given e.g. by Nayfeh Nayfeh (1977); Nayfeh and Abdelrahman (1998)
and Paipetis and Galiotis Paipetis and Galiotis (1999) and is based on axisymmetric
elasticity theory. It should be noted that Equation 5.23 is a quasi one-dimensional
equation and not a real one-dimensional equation as quite often is mentionned in
the literature. To eliminate shear stress, Cox introduced the assumption that:
τrz ∝
∂w
∂r
(5.24)
which is the fundamental shear-lag assumption Nairn (1997). Besides this assumption, the following assumptions must be made to derive Equation 5.21:
• The fibre and the matrix behave as linear elastic solids.
• The fibre/matrix interface is perfect.
• There is no load transfer through the fibre ends.
• Transverse stresses are ignored (< σrr + σθθ >= 0).
f,m
Additionally σzz
and w are weakly depend on r. By an axisymmetric elasticity
analysis and use of Equation 5.24 the axial fibre stress and the interfacial shear
stress are given by Nairn (1997):
κ21
κ1 z 2 κ2 z
+ κ3 ln r + κ4 ) − d(
+
− κ7 )
4
2d
d
(5.25)
r
1
τrz = (κ1 z + κ2 ) + (κ3 z + κ5 )
2
r
(5.26)
σzz = c(
49
where c, d, κi are constants. σzz is quadratic in z and τrz is linear in z. The next
key assumption is to relax the formulation of the ”exact” shear-lag shear stress
(Equation 5.26), and rewrite it in a more general form:
τrz =
f0 (z)r f1 (z)
+
2
r
(5.27)
where fi (z) are functions of z only. For more details on the assumptions required
to derive shear-lag analysis and its limitations the reader is referred to Nairn Nairn
(1997).
Shear-lag analyses have been extensively used to model stress transfer problems in
composites e.g. Refs.Landis and McMeeking (1999); Beyerlein and Phoenix (1996).
The importance of the stress transfer mechanisms in composites has resulted in
intense efforts and many refinements of Cox’s shear-lag analysis appeared in the
literature. Fukuda and Chou Fukuda and Chou (1981) modified the original shearlag theory by introducing partially axial load into the matrix. Hsueh Hsueh (1989,
1995), Nair and Kim Nair and Kim (1992) extended the shear-lag analysis to account
for bonded fibre ends. Zhao and Ji Zhao and Ji (1997) improved shear-lag analysis
to predict tensile and shear stresses in the matrix. Lagoudas et al Lagoudas et al.
(1989) modeled the matrix as a viscoelastic material. Cox’s treatment like all elastic
shear-lag analyses assumes that the shear stresses generated at the fibre/matrix
interface can always be accommodated, however it is well known today that at high
applied strains or stresses plastic yielding occurs van den Heuvel et al. (1997a);
Favre et al. (1995) or fibres may debond from the matrix van den Heuvel et al.
(1998). Piggott Piggott (1978) assumed that at high stresses slip occurs between
the fibres and the matrix near the fibre ends. At this region the stress is transferred
through friction, while in the unslipped region Cox’s treatment is applied. Thus the
50
fibre stress in the end region is given by:
f
2µ
∂ < σzz
>
=
< σrr >
∂z
rf
(5.28)
µ is the friction coefficient and σrr is given by:
< σrr >= −ν1 σm +
ν2 < σf > ELm
ELf
+ σr
(5.29)
where ν1 and ν2 are parameters depend on the matrix and fibre Poisson’s ratio respectively and σr are the residual stresses due to cure shrinkage of the matrix, or
differential thermal expansion of the fibres and matrix during manufacture. Heuvel
et al van den Heuvel et al. (1998) modified Piggott’s analysis to simplify the interfacial shear stress expression, the axial fibre stress is given by:
f
∂ < σzz
>
2τf ri
=
∂z
rf
(5.30)
where τf ri (interfacial frictional stress) assumed to be constant over the slipped
region, thus the axial fibre stress is linearly built-up over this region. In the case of
interfacial yielding, τf ri is replaced by τyint (interphase yield stress) van den Heuvel
et al. (1997b). Mendels et al Mendels et al. (1999) presented a shear-lag analysis
f,m
where σzz
variation in the r direction is not ignored, thus Equation 5.23 is written
in a more general form:
f
∂ < σzz
>
= Φ(r, z) = g(r)f (z)
∂z
(5.31)
where the function g is approximated as:
g(r) = 1 −
ELm
rf ELf
r
(5.32)
51
Their final shear stress formula is free of any adjustable parameter, instead it includes a structural parameter which is a function of the elastic moduli of the constituents and the given system geometry. In almost all the above shear-lag analyses
use is made of parameter β as given by Equation 5.22. However many researchers
Nairn (1997) and references therein showed that β should be given by:
2
βnairn
=
2
rf2 ELf ELm
um
4GfLT
ELf uf + ELm um
1
( 1 ln u1f − 1 −
2Gm um
LT
um
)
2
(5.33)
uf and um are the fibre and matrix volume fractions defined by:
uf =
rf2
2
rm
and
um =
2
rm
− rf2
2
rm
(5.34)
A shortcoming of shear-lag analyses is that they give no information on the radial stress terms, and they contain a shear-lag parameter which is unknown and
drastically effects the stress transfer rate. An experimental method to assess the
shear-lag parameter, βcox,nairn , was proposed by Galiotis and Paipetis Galiotis and
Paipetis (1998). They saw that βcox,nairn is really a true reflection of the elastic
response of the fibre/matrix model composites (see Equations 5.22 and 5.33) since
they found that the values of β did not change with applied strain. They used β
as a fitting parameter in Equation 5.21, which was further simplified for the case of
βcox,nairn lf > 10, where lf is the fibre length. The modified Equation 5.21 is given
by:
f
f
< σzz
>=< σzz,∞
> [1 − cosh(βcox,nairn z) + sinh(βcox,nairn z)]
(5.35)
52
5.2
Axisymmetric Analyses
An alternative solution is to abandon one-dimensional models and develop
three-dimensional or axisymmetric models (see Refs. Muki and Sternberg (1969);
Nairn and Liu (1997); Lenci and Menditto (2000); Lee and Mura (1994a,b)). These
models e.g. Refs. Nairn and Liu (1997); McCartney (1989) predict zero shear
stresses at the fibre ends (actually they force shear stresses to zero at fibre ends).
Tripathi et al Tripathi et al. (1996) modified Nairn’s variational analysis Nairn
(1992) to model stress transfer at high strains where matrix deforms plastically,
by simply replacing interfacial shear stresses higher than the matrix yield strength
with matrix shear yield strength. Nairn’s Nairn (1992) basic assumption is that axf,m
ial stresses (σzz
) are independent of the radial coordinate. The same assumption
was made by McCartney McCartney (1989) in his model. Additionally, similarly to
Cox, he introduced an equivalent matrix radius, rm (see Rcox ). Wu et al Wu et al.
(1997) developed an axisymmetric stress-transfer model based on minimization of
complementary energy. Their results agree better with finite element simulations
when compared to Nairn’s model Nairn (1992) but still are far away from FEA
results. Wu et al Wu et al. (1998) extended their model to account for a third
phase in between fibre and matrix (i.e the presence of fibre coating) but disagreement with FE still remained. Nairn and Liu Nairn and Liu (1997) introduced an
interface parameter(Ds ) to account for imperfect interfaces in a three-dimensional
axisymmetric analysis by using Bessel-Fourier series stress functions (see also Ref.
Parnes (1981) for the use of Bessel-Fourier series stress functions to solve stress
transfer problems). They collapsed the 3D interphase into a 2D interphase which
could be modeled by two interface parameters (Ds ∼ z direction and Dn ∼ r direction). In the case of tensile loading σrr is compressive the quality in the r direction
53
should not have important effects on the fragmentation process. Thus, the interface
could be modeled by a single one-dimensional parameter (Ds ) which physically is
a measure of the ability of the interface to transfer stresses from the matrix to the
fibre. Nairn and Liu Nairn and Liu (1997) and Paipetis et al Paipetis et al. (1999)
used this model to assess interface properties, and were able to characterise interfaces in the elastic regime - undamaged interfaces but still imperfect. Shear yielding
at the fibre ends was predicted despite τrz be zero by employing a Von Mises yield
criterion. It should be noted, however, that Dn should affect the fragmentation process when the loading is compressive and thus should be included in the analysis.
Weaver and Williams Weaver and Williams (1975) observed longitudinal cracks at
the fibre/matrix interface upon compressive loading of carbon/epoxy composites,
and suggested that this kind of failure was the result of tensile stresses. On the
other hand the complexity of the interphase may play a part as there might be a
gradient of mechanical properties across its thickness Pournaras and Papanicolaou
(1992); Jayaraman et al. (1992). Nayfeh and Abdelraham Nayfeh and Abdelrahman (1998) adopted McCartney’s treatment McCartney (1989) to solve the stress
transfer problem. They also treated the case of laterally constrained system. In this
case the radial displacements at the outer matrix borders of the concentric cylinder
model vanish instead of the radial stresses. This assumption represents better the
periodic structure of an unidirectional composite material.
CHAPTER 6
EXPERIMENTAL RESULTS
In this chapter the experimental results are presented.
6.1
Compressive behaviour
6.1.1 Elastic case
Figures 6.13(a) and 6.14(a) depict the axial fiber stress along the length of a
carbon fiber embedded in an epoxy matrix at different compressive strain levels. Figures 6.13(b) and 6.14(b) show the corresponding interfacial shear stresses. At these
low applied compressive strains it was relatively easy to control the macroscopic
matrix strain. That is creep effects were not present and this is well demonstrated
in Figs 6.13(a) and 6.14(a) where it can be observed that the fiber stress profiles
are almost perfectly symmetric. As it can be seen from Fig 6.13(a), the fiber is relatively stress-free in the unstrained case, with the residual compressive stress being
about -0.1GPa. When the matrix strain (applied strain) increases, the fiber stress
increases rapidly from the fiber ends to a plateau value along the middle of the
fiber. The stress is transferred from the fiber ends through shear at the interface
as in the case of tensile loading. A value of ineffective or transfer length of the
order of 350µm is observed. In the unstrained case the axial stress in the fiber ends
is ∼ -0.015GPa and -0.07GPa, respectively. At an applied strain -0.1%, the axial
fiber stress at the fiber ends increases to ∼ -0.20GPa and -0.14GPa, for the left and
right end, respectively. However further increase of the applied strain has almost
54
55
no effect on the left fiber end (σzz fluctuates about -0.20GPa) while at the right
fiber end the axial stress reaches a value of about -0.30GPa and then remains relatively constant. No interfacial failure is observed (see Figure 6.13(b) and 6.14(b))
for a strain lower than that required for fiber failure. Finally, it is worth noting
that there is no indication from Figure 7(a) of any direct load transmission through
the fiber ends. Similar results (stress transfer through shear activated mechanisms)
were conducted by Narayanan and Schadler (1999b); Mehan and Schadler (2000).
6.1.2 Compressive failure
In order to study the post-failure stress transfer mechanisms, the specimens
were compressed to sufficiently high strain levels to induce multiple fiber fracture.
Figures 6.15(a), 6.16(a) and 6.17(a) depict the stress profile in a fiber embedded in
an epoxy matrix when the microcomposite is loaded at average strains equal to 0%,
-0.34%, -0.56%, and -0.86%, respectively.
As it has been mentioned in the previous section, at high compressive strains, matrix
creep could not be avoided. This effect is clearly shown in Figures 6.15(a) and
6.16(a). For example, in Figure 6.16(a), we have started mapping the fiber from
the left fiber end towards to the right. Since this procedure can take several hours,
the second half fiber experiences higher strains as a consequence of the increase in
the matrix strain (more details are given in Appendix B).
Similarly to the results presented above, up to strains of about -0.35% no fiber
failure occurs, and the quality of the interface seems to be unchanged (see Figure
6.15(b)). However, at an applied strain of ∼ -0.6% multiple fiber fracture is observed
(Figure 6.16(a)). This strain level is equal to the fiber compressive failure strain as
it can be seen from Figure 4.12. Fiber break, however, occurs only in the second
56
half of the fiber due to matrix creep effects that induce high fiber strains leading to
local fiber failure. Further increase of the applied strain results almost in complete
fiber fragmentation (see Figure 6.17(a)).
It can be seen from Figure 6.16(a) and 6.17(a) that the cubic spline fit fails to
describe the experimental data in the neighborhood of fiber breaks. In order to do so,
we should significantly increase the number of knots used but this obviously would
affect the best of fit in the fiber ends regions. Moreover, as it will be discussed later,
it is meaningless to fit the experimental data at the vicinity of compressive fiber
breaks. Figure 6.18 depicts a micrograph of a typical shear compressive fiber break
observed in all the experiments performed. It can be easily seen that the broken ends
slide past each other and therefore compressive stresses can be transmitted as the
fiber fragments remain in contact. This is the reason for the very low compressive
ineffective length ∼ 40-80µm (Figure 6.19) compared to the tensile ineffective length
(see Figure 6.22(a)). Similarly, Amer and Schadler (1997) found that at a strain of
-0.6% the compressive average fragment length is about 85±10µm for a M40/epoxy
system.
Concerning the stress transfer from the fiber end regions it can be observed that
even at strains, exceeding the compressive failure strain of the fiber, no substantial interfacial damage occurs. Figures 6.16(b) and 6.17(b) clearly show that no
debonding or yielding occurs in these regions within experimental error.
6.2
Tensile behaviour
Figure 6.20(a) shows the stress profile of a fiber loaded in tension at applied
strains of 0%, 0.2% and 0.4%. As can be seen, the residual stress fluctuates around
57
zero (0% applied strain) although there is an indication of the presence of compressive stresses particularly around the right fiber end. At an applied strain of 0.2% the
stress builds from zero at the left fiber end to a maximum value of about 800MPa
and from -200MPa to 600MPa for the right half of the fiber. At an applied strain of
0.4% and at a distance of 0.3-0.4 from the left fiber end, an unexpected decrease of
the fiber stress is observed which might be associated with a fiber flaw and/or the
onset of fiber failure. The stress distortion of spread ∼ 200MPa is also observed at
applied strains of 0.6% (Figure 6.21(a)) and 0.8%. It is interesting to note that at an
applied strain of 1.25% (see Figure 6.22(a) fiber fracture occurs at the locus of the
observed fiber stress perturbation. The fiber stress at break location is equal to the
initial residual stress within the experimental and numerical errors. The ineffective
or transfer length (Lt ) is of the order of 450-500µm (see Figure 6.22(a)).
Since a kind of interfacial damage pre-exist in the left fiber end, only the right
fiber end will be used to assess the interfacial state during tensile loading. The ISS
increases from ∼ 1.5MPa at zero applied strain (Figure 6.20(b)) to ∼ 20MPa at
an applied strain of 0.8% (Figure 6.21(b)). At an applied strain of 1.1% interfacial
damage occurs (see Figure 6.22(b)), and the ISS profile displays a ”knee” which
is located 100µm from the fiber end. The maximum ISS value is somewhat less
than 20MPa. The existence of interfacial damage can also be easily identified from
the stress data (see Figure 6.22(a)). The growth of the interfacial damage zone
with applied strain is also very well depicted in Figure 6.22(a) and 6.22(b). The
maximum ISS value observed at an applied strain of 1.25% is located 185µm from
the fiber end. The maximum ISS values observed around the fiber break or the
left fiber end (all <20MPa) leads us to the conclusion that the interfacial shear
strength of the composite system examined is ∼ 20MPa. There are discrepancies
58
between the ISS values obtained at the left and right fiber ends which are within the
accuracy of the measurements performed (+5MPa). At the right side of the fiber
break (Figure 6.22(b)) shear yielding at the interface can also be identified. Finally,
another important conclusion is that fiber/matrix debonding does not occur even
at strains of 1.25% for this fiber/matrix combination.
59
0.2
: ε=0% &
,
: ε= -0.1%
,
Axial fiber stress (GPa)
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Norm. position along the fiber (-)
(a)
ε = 0%
ε = -0.1%
6
4
ISS (MPa)
2
0
-2
-4
-6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Norm. position along the fiber (-)
(b)
Figure 6.13: (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at strains of 0% and -0.1% respectively. The open circles/squares correspond
to experimental data whereas the solid lines represent the corresponding cubic spline
fits, (b) Corresponding interfacial shear stresses.
60
0.2
:ε= -0.2% &
,
:ε= -0.3%
,
0.0
Axial fiber stress (GPa)
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
-1.6
-1.8
-2.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Norm. position along the fiber (-)
(a)
16
ε= -0.2%
ε= -0.3%
12
8
ISS (MPa)
4
0
-4
-8
-12
-16
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Norm. position along the fiber (-)
(b)
Figure 6.14: (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at strains of -0.2% and -0.3% respectively. The open circles/squares correspond to experimental data whereas the solid lines represent the corresponding
cubic spline fits, (b) Corresponding interfacial shear stresses.
61
0.4
: ε=0% &
,
0.2
: ε= -0.34%
,
0.0
Axial fiber stress (GPa)
-0.2
-0.4
-0.6
-0.8
-1.0
-1.2
-1.4
-1.6
-1.8
-2.0
-2.2
-2.4
-2.6
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Norm. position along the fiber (-)
(a)
16
ε = 0%
ε = -0.34%
12
8
ISS (MPa)
4
0
-4
-8
-12
-16
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Norm. position along the fiber (-)
(b)
Figure 6.15: (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at strains of 0% and -0.34% respectively. The open circles/squares correspond to experimental data whereas the solid lines represent the corresponding
cubic spline fits, (b) Corresponding interfacial shear stresses.
62
: ε= -0.56%
,
1.0
0.5
Axial fiber stress (GPa)
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Norm. position along the fiber (-)
(a)
ε =-0.56%
20
15
10
ISS (MPa)
5
0
-5
-10
-15
-20
0.0
0.1
0.8
0.9
1.0
Norm. position along the fiber (-)
(b)
Figure 6.16: (a)Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at a strain of -0.56%. The open squares correspond to experimental data
whereas the solid line represents the corresponding cubic spline fit, (b) Corresponding interfacial shear stresses.
63
1.0
: ε= -0.86%
,
0.5
Axial fiber stress (GPa)
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Norm. position along the fiber (-)
(a)
25
ε = -0.86%
20
15
ISS (MPa)
10
5
0
-5
-10
-15
-20
-25
0.0
0.1
0.8
0.9
1.0
Norm. position along the fiber (-)
(b)
Figure 6.17: (a)Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at a strain of -0.86%. The open squares correspond to experimental data
whereas the solid line represents the corresponding cubic spline fit, (b) Corresponding interfacial shear stresses.
64
Figure 6.18: Micrograph of a typical shear compressive fiber break.
1.0
exp.data (ε = -0.86%)
0.5
Axial fiber stress (GPa)
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
3200
3400
3600
3800
4000
4200
4400
4600
4800
Position along the fiber (µm)
Figure 6.19: Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at a strain of -0.86%.
65
2.2
: ε=0%
: ε=0.2%
: ε=0.4%
,
,
,
2.0
1.8
Axial fiber stress (GPa)
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.9
1.0
Norm. position along the fiber (-)
(a)
20
ε=0%
ε=0.2%
ε=0.4%
16
12
ISS (MPa)
8
4
0
-4
-8
-12
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Norm. position along the fiber (-)
(b)
Figure 6.20: (a) Axial fiber stress profile of an embedded fiber in an epoxy
matrix loaded at strains of 0%, 0.2% and 0.4% respectively. The open diamonds/circles/squares correspond to experimental data whereas the solid lines
represent the corresponding cubic spline fits, (b) Corresponding interfacial shear
stresses.
66
: ε=0.6%
: ε=0.8%
,
,
3.0
Axial fiber stress (GPa)
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.9
1.0
Norm. position along the fiber (-)
(a)
24
ε =0.6%
ε =0.8%
20
16
12
ISS (MPa)
8
4
0
-4
-8
-12
-16
-20
-24
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Norm. position along the fiber (-)
(b)
Figure 6.21: (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at strains of 0.6% and 0.8% respectively. The open circles/squares correspond
to experimental data whereas the solid lines represent the corresponding cubic spline
fits, (b) Corresponding interfacial shear stresses.
67
4.5
: ε=1.1%
: ε=1.25%
,
,
4.0
Axial fiber stress (GPa)
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.9
1.0
Norm. position along the fiber (-)
(a)
30
ε =1.1%
ε =1.25%
25
20
fiber break location
15
ISS (MPa)
10
5
0
-5
-10
-15
-20
-25
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Norm. position along the fiber (-)
(b)
Figure 6.22: (a) Axial fiber stress profile of an embedded fiber in an epoxy matrix
loaded at strains of 1.1% and 1.25% respectively. The open circles/squares correspond to experimental data whereas the solid lines represent the corresponding
cubic spline fits, (b) Corresponding interfacial shear stresses.
CHAPTER 7
FINITE ELEMENT ANALYSIS
It is worth noting that there is no indication, from the experimental results
presented in the previous chapters, of any direct load transmission through the fibre
ends. To examine further this matter, an axisymmetric FE analysis, is presented in
this Chapter, in which it has been assumed that the fibre ends are perfectly bonded
to the matrix.
7.1
Finite element details
The single-fibre model composite used in the experiments performed was im-
plemented in the ANSYS finite element code. The ANSYS pre-processor was used
to generate an axisymmetric mesh of the microcomposite. As a result of the symmetry conditions, the mesh now needs to describe a quarter of the fibre only. A
schematic represenation of the model is depicted in Figure 7.23. To investigate the
stress transfer form the matrix to the fibre through axial stresses, an imaginary
fibre similar to the treatment of Hsueh Hsueh (1989, 1995) and Nair and Kim Nair
and Kim (1992), was adopted. The imaginary fibre (see Figure 7.23, refansus) is
actually a matrix material which pulls /or pushes (depending on the applied loading
conditions) the fibre.
68
69
7.1.1 Boundary conditions and applied loads
The FE analysis consisted of one load case. The applied load (compressive or
tensile) is applied to the microcomposite by prescribing a y-axis displacement to the
nodes at the far end of the model. A total strain of (1% in the case of tension and
-0.6% in the case of compression) with respect to the initial y-length was applied
in increments of 0.1%. Throughout the analysis the nodes at x=0 are fixed in the
x direction, and the nodes at y=0 are fixed in the y direction (see Figure 7.23).
As mentioned in Chapter 3, the model composites were cured at room temperature
(250 ) to minimise the residual thermal stresses. Indeed, from the results presented
in Chapter 5, the residual stresses in the fibres after specimens preparation are very
small, and thus in our FE analysis we didn’t apply any thermal load to account for
thermal residual stresses.
7.1.2 Material properties
The mechanical properties of the fibre and the matrix used are given in Table
4.1. Since our aim is to study only the stress transfer through the fibre ends an
elastic analysis was performed. This choice is also supported by the experimental
findings that the interface in compression at the fibre end regions does not fail even
at high compressive strains exceeding quite enough the compressive failure strain of
the fibres themselves.
7.1.3 FE Model’s dimensions
To investigate the amount of matrix surrounding the fibre, needed to describe
properly the stress situation in the model composite, a number of FE analyses has
70
been performed, where (a) the matrix cylinder width (in the x direction, see Figure
7.23), and (b) the imaginary fibre length, were varied.
7.1.3.1 Effect of the matrix width
First the influence of the matrix width of the stress situation in the model
composite will be addressed. In Figure 7.25 the dependence of the axial fibre stress
at the fibre end (Figure 7.25(a)) and of the far-field matrix stress (Figure 7.25(b))
on the matrix width is depicted. In the FE analysis performed in this case the
half fibre length was equal to 800µm, the imaginary fibre length was also equal
to 800µm, which is high enough to exclude any influence of this variable on the
stress situation. The applied strain to the microcomposite was ǫ=1%. As it can
be observed from Figure 7.25, the matrix cylinder radius should be at least higher
than 550µm approximately in order to have accurately results.
7.1.3.2 Effect of the imaginary fibre length
To investigate the influence of the imaginary fibre length on the stress state
in the model composite, a similar to the previous case procedure was followed. The
half fibre length and the applied strain are the same as above, but the matrix radius,
based on Figure 7.25, is now 550µm. In Figure 7.26 the independence of the axial
fibre stress at the fibre end (see Figure 7.26(a)) and of the far-field matrix stress (see
Figure 7.26(b)) on the imaginary fibre length are shown. Figure 7.26 most likely
suggests that the imaginary fibre should be at least 600µm long in order to have a
realistic stress situation in the model composite.
71
7.2
Results
The FEA results are presented in Figure 7.27 where the predictions of shear-lag
models of Hsueh Hsueh (1995) and Nair and Kim Nair and Kim (1992) are plotted
for both tension and compression loadings. As can be seen in Figure 7.27, the axial
stress imparted through the fibre ends per applied strain is low (∼ 80 MPa / 0.6%)
and thus it can be safely assumed that the predominant stress transfer mechanism,
at the interface, is shear.
72
Figure 7.23: Schematic representation of the axisymmetric finite element model. rf
and Rm are the fibre and matrix radii respectively, whereas lim is the length of the
imaginary fibres.
73
Figure 7.24: Finite element mesh
74
4.0
Far-field axial fibre
stress (GPa)
3.5
FEA
3.0
2.5
2.0
1.5
1.0
0
100
200
300
400
500
600
R cox (µm)
(a)
20
Far-field matrix stress (MPa)
19
Theoretical value
FEA
18
17
16
15
14
13
12
11
10
0
100
200
300
400
500
600
R cox (µm)
(b)
Figure 7.25: The independence (a) of the axial fibre stress at the fibre end and (b)
of the far-field matrix stress on the matrix width.
75
0.150
FEA
Axial fibre stress at the
fibre end (GPa)
0.148
0.146
0.144
0.142
0.140
0
100
200
300
400
500
600
700
800
900
Imaginary fibre length (µm)
(a)
24
Far-field matrix stress (MPa)
23
Theoretical value
FEA
22
21
20
19
18
17
16
15
14
0
100
200
300
400
500
600
700
800
900
Imaginary fibre length (µm)
(b)
Figure 7.26: The independence (a) of the axial fibre stress at the fibre end and (b)
of the far-field matrix stress on the imaginary fibre length.
76
0.3
tension
Axial fibre stress at the
fibre end (GPa)
0.2
0.1
0.0
Hsueh (shear-lag model)
Nair & Kim (shear-lag model)
FEA
-0.1
compression
-0.2
-0.8
-0.4
0.0
0.4
0.8
1.2
Applied strain (%)
Figure 7.27: Axial fibre stress at the fibre end versus the applied strain. Comparison
of the FEA predictions with the predictions of the shear-lag models of Hsueh Hsueh
(1995) and Nair and Kim Nair and Kim (1992) models.
CHAPTER 8
VALIDATION OF THE ANALYTICAL STRESS TRANSFER
MODELS
In this section the experimental data, presented above, will be used, to evaluate the efficiency of the analytical stress transfer models to describe accurately the
stress field in the fibre.
8.1
Shear-Lag Analyses
Figure 8.28 shows a general unit cell which is used in the Shear-lag analyses
and contains a cylindrical discontinuous fibre embedded in the continuous matrix.
The imaginary fibres are used in the models which assume that the fibre ends
are bonded to the matrix and thus they can transfer loads. Since the analytical
shear-lag stress transfer models do not account for residual thermal stresses an
additional compressive stress, fc , is superposed to their predictions, where fc equals
the magnitude of the stress experimentally observed at the fibre end examined.
First the original Shear-lag analysis will be addressed using the two different βcox,nairn
(see Equations 5.22 and 5.33) to fit the experimental data of Figure6.14(a) (applied
compressive strain equal to -0.3%). Only the left part of the fibre is considered.
Figure 8.29(a) shows the Shear-lag results for different Rcox /rf ratios, uf ranges
from 0 to 10%. A fibre volume fraction of uf =0 corresponds to a fibre in an infinite
matrix. In Figure 8.29(a) the solid lines represent the Shear-lag predictions using
βcox , whereas the dotted line using βnairn Shear-lag parameter. It is easy to see
that as uf →0 (or Rcox → ∞) Shear-lag analysis breaks down for both βcox,nairn
77
78
since it implies that the stress transfer length becomes infinite. In a correct analysis
the stress transfer should be independent of uf for low fibre volume fractions and
approach the stress transfer for the case of uf =0 Nairn (1997). Despite this shortcoming, Shear-lag analyses are very often used to interpret results from single-fibre
tests where βcox,nairn is treated as an adjustable parameter and not as a pre-defined
constant as mentioned in section 2. In this case it can be observed from Figure
8.29(a) that the use of βnairn gives always lower stress transfer lengths for a give
fibre volume fraction. Stress transfer length varies relatively little with fibre volume
fraction for the case of βcox compared to βnairn . This is more evident when the fibre
volume fraction is high. In these regions (which simulate better the situation in real
composite laminates) Shear-lag results, with the use of βnairn , give more reasonable
predictions about the stress transfer length and therefore should be used in favor of
βcox .
Although Shear-lag analyses do a qualitative good job in predicting the axial fibre
stress, its inaccuracies are more pronounced in the case of interfacial shear stress
predictions (see Figure 8.29(b)). Shear-lag results cross over cubic spline calculations which is indicative that the stress transfer is not exponential. Nairn Nairn
(1997) reached the same conclusions using the finite element method. Additionally,
if it is considered that i.e. a value of Rcox equal to 85µm (uf =0.15%) fits well to
the experimental data (see Figure 8.29(a), use of βcox ), the associated ISS values
are much more in error compared to the ISS experimentally derived with the use of
cubic splines (see Figure 8.29(b)). Thus, it can be suggested that Shear-lag analyses
can be used only qualitatively to describe the stress transfer from the matrix to a
fibre.
In the next graph (see Figure 8.30) a comparison of Mendels’s et al Mendels et al.
79
(1999) and Cox’s Shear-lag predictions is made. As previously mentioned, one of
the main advantages of Mendels’s et al Mendels et al. (1999) analysis is that it
does not include an adjustable parameter (like Rcox ) by properly defined bounds
to the stress transferred influenced matrix region. Instead of the Rcox , a structural
factor (depending only on Young’s moduli of the constituents, and on geometrical
parameters) is used. This is clearly depicted in Figure 8.30(a) where the effects
of the geometrical parameters are shown. For a fibre volume fraction of 0.15% (
Rcox /rf =855) the predictions are grossly in error, whereas for a uf =0.001% (this
uf value corresponds approximately to the fibre volume fractions in the single-fibre
specimens used, Rcox /rf =315) their results agree well with the cubic spline fit.
Similarly, to the results of the previous paragraph, Mendels’s et al analysis breaks
down for uf =0. In Figure 8.30(b) the differences between the calculated with cubic
splines ISS and Mendels’s et al analysis are depicted. Again, it can be observed
that Shear-lag analysis predicts higher values of ISS.
In the previous study the problem of load transmission through the fibre ends has
been discussed (case of compressive loading). At this point, this issue will be addressed again in some more detail and the experimental data will be examined with
modified Shear-lag analyses which consider fibre ends bonded to the matrix Hsueh
(1989, 1995); Nair and Kim (1992). Figs. 8.31 and 8.32 depict the axial fibre stress
at fibre ends as a function of the applied tensile and compressive strain, respectively.
In these plots the Shear-lag models of Hsueh Hsueh (1995) and Nair and Kim Nair
and Kim (1992) are used. It is shown that both analyses yield exactly the same
predictions for the fibre/matrix combination used. The experimental data in these
Figures is the average of three measurements in the fibre end region [0-4µm] and
not a single Raman result in order to reduce noise effects (see Appendix) although
80
substantial scatter still exists. It can be observed from Figure 8.31 that bonded
fibre ends is not the case in tension since fibre end stress clearly does not increase
with applied strain. On the other hand, the situation in compression is somewhat
different. A trend of fibre stress end increase with increasing applied compressive
strain can be identified despite the presence of high scatter. This scatter, however,
make it unable to evaluate the stress increase rate predicted by Hsueh Hsueh (1995)
and Nair and Kim Nair and Kim (1992).
The Shear-lag analyses examined above consider only elastic stress transfer from
the matrix to the embedded fibre i.e. they are limited in the case of failure absence.
In this part, Piggott’s Shear-lag model, as it has been modified by Van den Heuvel
et al van den Heuvel et al. (1998) to account for both interfacial yielding and
fibre/matrix debonding, will be considered. Figure 8.33(a) depicts the axial fibre
stress as a function of the distance from the fibre break (applied tensile strain,
ǫ=1.25%). The clear deviation between the experimental data and Cox’s (elastic
analysis) predictions is indicative that failure has occurred in the region next to
the fibre break. Van den Heuvel’s et al van den Heuvel et al. (1998) results are
also depicted in Figure 8.33(a). First it was assumed that there is only interfacial
yielding. According to their analysis the fibre strain in this region is given by:
< ǫfzz >=
2τyint
rf ELf
, |z0 | < |z| < |zy |
(8.36)
where τyint is the shear yield stress of the fibre/matrix interface, zy is the end of the
yielded zone and z0 is the fibre break coordinate; beyond this region the fibre axial
strain is given by Cox’s analysis. A yielded zone of ∼ 250µm has been found to fit
better the experimental data, the shear yield stress of the interface was assumed to
equal to the matrix shear yield stress (17.5MPa). The matrix yield stress, σym , was
81
measured as 35MPa and the τym was derived according to Tresca yielding criterion.
The reason to force interfacial shear yield stress equal to τym is that a τyint would
result a higher deviation from the experimental data (see Equation 8.36). However,
Van den Heuvel’s et al analysis still do not fit satisfactory the fibre axial stress (see
Figure 8.33(a)). The next step was assume that the fibre debonds from the matrix
after the fibre break. In the debonded region the fibre axial strain is given by:
< ǫfzz >=
2τf ri
, |z0 | < |z| < |zd |
rf ELf
(8.37)
where τf ri is the frictional stress at the fibre matrix interface and zd is the end of
the debonded zone which is followed by a yielded zone. As it can be seen from
Figure 8.33(a) Van den Heuvel’s analysis in this case is in a very good agreement
with the experimental data and the cubic spline fit. The fitting parameters were:
τf ri =10MPa, ld ∼100µm (debond length) and ly ∼180µm. Surprisingly, τf ri and
ld values are almost identical to the values given by Van den Heuvel et al van den
Heuvel et al. (1998) for the case of unsized-untreated fibres (which are expected to
debond relatively easy from the matrix). A similar analysis by means of the finite
element method was done by Nath et al Nath et al. (2000). In our experiments,
however, no fibre/matrix debonding was observed. A possible explanation is matrix
cracking. Sirivedin et al Sirivedin et al. (2000) have shown, using the finite element method, that a matrix crack, caused by the fibre failure, results in the lower
stress transfer rate from the matrix to the fibre. Ten Busschen and Selvadurai ten
Busschen and Selvadurai (1995); Selvadurai and ten Busschen (1995) have experimentally confirmed the formation of matrix cracks in cracked fibre locations in the
case of good fibre/matrix adhesion. The developed matrix cracks are usually conical
cracks and/or a combination of conical and penny-shaped cracks. Figure 8.33(b)
82
depicts the corresponding ISS profiles of Figure 8.33(a). It can be observed that in
general good agreement exists between spline results and Van den Heuvel’s analysis
although we should always have in mind the assumptions made. The ISS discontinuities observed are a direct consequence of the axial fibre strain profiles assumed
in the debonded and yielded zones (see Equation 8.36 and 8.37). Finally, we should
mention that in general the discrepancies between calculated cubic splines ISS and
Van den Heuvel’s predictions are usually higher, (see Figure in 6.22(b) the fibre end
regions and the region left to the fibre break).
Figure 8.28: Schematic illustration of the shear-lag model for stress transfer from
the matrix to the embedded fibre. Imaginary fibres are used in the case of fibre
ends bonded to the matrix
83
0.0
-0.2
Axial fibre stress (GPa)
exper. data
Fibre volume
fraction decrease
β cox
β nairn
-0.4
R cox / rf = 315
-0.6
-0.8
-1.0
-1.2
-1.4
-1.6
0
50
100
150
200
250
300
350
400
450
500
550
Position along the fibre (µm)
(a)
60
R cox / rf = 315
cubic spline fit
β cox
50
β nairn
ISS (MPa)
40
Fibre volume
fraction increase
30
20
10
0
0
50
100
150
200
250
300
350
400
450
500
550
Position along the fibre (µm)
(b)
Figure 8.29: Comparison of the Shear-lag predictions using two different β (βcox and
βnairn ). (a) Axial fibre stress profiles for a Rcox /rf ratio equal to 3, 10, 25, 315, ∞
(uf → 0), respectively. (b) Corresponding interfacial shear stresses (The ISS in the
case of βnairn and Rcox /rf =3 is not plotted, the maximum values are higher than
185MPa in that case).
84
0.0
Fibre volume fracton decrease
(Cox's Shear-lag analysis)
-0.2
exper. data
cubic spline fit
Cox
Manson
Axial fibre stress (GPa)
-0.4
-0.6
-0.8
R cox / rf = infinity
-1.0
R cox / rf = 25
-1.2
-1.4
R cox / rf = 315
-1.6
0
50
100
150
200
250
300
350
400
450
500
550
Position along the fibre (µm)
(a)
24
cubic spline fit
Cox
Manson
22
20
18
Manson's analysis
(R cox / rf = 315)
ISS (MPa)
16
14
12
10
Fibre volume fraction increase
(Cox's Shear-lag analysis)
8
6
4
2
0
0
50
100
150
200
250
300
350
400
450
500
550
Position along the fibre (µm)
(b)
Figure 8.30: Comparison of Mendels’s and Cox’s Shear-lag predictions. (a) Axial fibre stress profiles for a Rcox /rf ratio equal to 3, 10, 25, 315, ∞ (uf → 0),
respectively. (b) Corresponding interfacial shear stresses.
85
0.5
exper. data:
exper. data:
Axial stress at the fibre end (GPa)
0.4
l. fit,
l. fit,
Hsueh, Nair & Kim
Hsueh, Nair & Kim
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Applied strain (%)
Figure 8.31: Fibre stress at the fibre end upon tensile loading. Parameters used
in the Shear-lag models of Hsueh and Nair & Kim: Rcox /rf =315, l′ (length of
imaginary fibres)=100µm.
86
0.00
exper. data:
exper. data:
Axial stress at the fibre end (GPa)
-0.05
l. fit,
l. fit,
Hsueh, Nair & Kim
Hsueh, Nair & Kim
-0.10
-0.15
-0.20
-0.25
-0.30
-0.35
-0.40
-0.45
-0.50
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
0.00
Applied strain (%)
(a)
Axial stress at the fibre end (GPa)
0.0
exper. data:
exper. data:
l. fit,
l. fit,
Hsueh, Nair & Kim
Hsueh, Nair & Kim
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.8
-0.6
-0.4
-0.2
0.0
Applied strain (%)
(b)
Figure 8.32: Fibre stress at the fibre end upon compressive loading. Parameters
used in the Shear-lag models of Hsueh and Nair & Kim: Rcox /rf =315, l′ (length of
imaginary fibres)=100µm.
87
4.0
exper. data
cubic spline fit
3.5
Axial fibre stress (GPa)
3.0
Cox
2.5
2.0
Heuvel (interphase yielding)
1.5
1.0
0.5
Heuvel
(debonding followed by interphase yielding)
0.0
0
50
100
150
200
250
300
350
400
450
500
550
Distance from fibre break (µm)
(a)
5
0
-5
-10
ISS (MPa)
-15
-20
-25
-30
-35
cubic spline fit
Heuvel (interphase yielding)
Heuvel (debonding followed by
interphase yielding)
Cox
-40
-45
-50
-55
0
50
100
150
200
250
300
350
400
450
500
550
Distance from fibre break (µm)
(b)
Figure 8.33: Comparison of the cubic spline fit with Shear-lag models in the case
of interfacial failure (ǫ=1.25%). Heuvel’s analysis is used for two cases: (i) yielding
of the interphase and (ii) debonding followed by yielding of the interphase. The
parameters used to fit the results are: Rcox =85µm, σym =35MPa, τym =17.5MPa
(Tresca), case (i): yielding zone = 250µm, case (ii): debond length = 100µm and
yielding zone = 180µm. (a) Axial fibre stress profiles and (b) Corresponding ISS
profiles.
CHAPTER 9
DISCUSSION OF THE EXPERIMENTAL RESULTS
9.1
Compressive failure
The stress build-up emanating from a compressive fiber break (Figures 6.16(a)
and 6.17(a)) is quite different from that in tension (see Figure 6.22(a)). To start
with, the fiber stress at a compressive fiber break location does not necessarily drop
to zero or to the initial residual stress. In addition, the rate of stress transfer from a
fiber break is extremely high and therefore the corresponding transfer (ineffective)
length is extremely small when compared to that in tension (Figure 6.22(a)). As
mentioned earlier, compressive failure for the high-modulus fiber/epoxy system examined here leads to fiber ends sliding past each other. By further loading of the
system, stress is transferred in the fiber not only by interfacial shear but also by
fiber-fiber contact at the compressive failure location. Unlike previously reported
work (Narayanan and Schadler (1999b); Mehan and Schadler (2000)), an attempt
has been made here to distinguish between the two types of stress transfer so as to
assess the relative importance of interfacial adhesion in compression. As can be seen
in Figures 10a and 11a, the rate of stress transfer from the left or right fiber ends is
comparable with that observed in tension from a fiber break (or a fiber end) since
the prevailing stress-transfer mechanism at those locations is pure interfacial shear.
By limiting the true interfacial shear stress measurements in the case of compression
to the fiber ends reasonable values of maximum ISS of the order of 25MPa have
been obtained. As can be seen in Figures 6.16(b) and 6.17(b) the ISS calculations
88
89
from compressive failures in the middle of the fiber have been omitted since the
balance of forces argument is not valid there. As it has already been mentioned,
the derivation of Equation 5.23) is based on axisymmetric elasticity theory, however
when the fiber fails in compression and the broken fiber fragments slide past each
other the symmetry rule changes and, therefore, the use of Equation 5.23 can lead
to erroneous results. Narayanan and Schadler (1999a) in their earlier work have
employed Equation 5.23 to calculate the ISS in the case of compressive fiber breaks
and they reported ISS values in excess of 150MPa (Amer and Schadler (1997)) and
even 300MPa (Narayanan and Schadler (1999a)) for similar fiber/matrix systems.
Since these values were unacceptably high, an attempt was made (Narayanan and
Schadler (1999a)) to modify the balance of forces argument to account for the postfailure geometric configuration as given below:
f
2τrz
∂ < σzz
>
=−
sin θ
∂z
rf
(9.38)
where θ is the bending angle (see Ref. Narayanan and Schadler (1999a)) determined
to be approximately 8◦ . Based on the above formula the ISS value is reduced by
a factor of ∼ 7 and therefore a ”corrected” ISS value of approximately 51MPa was
obtained. The latter value is still too high for a high modulus carbon fiber/epoxy
system presumably due to the simplistic arguments employed for its derivation. As
stated by same authors (Narayanan and Schadler (1999a)), a more rigorous analysis
is needed to estimate accurately the interfacial shear stress in the neighborhood of
compressive fiber breaks. At the moment, the mapping of the ISS distribution from
the fiber ends is the only accurate method available for assessing the fiber/matrix
adhesion under compressive loading.
Another interesting feature observed at the loci of fiber failure in compression is
90
the fact that the stress fluctuates around zero and, in some cases (Figures 6.16(a)
and 6.17(a)), it reaches tensile values. This is again due to the deformation and
bending of fibers at the point of shear failure which depending on the laser Raman
sampling direction can be also tensile. These results are in distinct contrast with
those reported by Wood et al. (1995) for which the fiber strain profiles were completely unaffected by the fragmentation process. However, the photoelastic fringes
produced by the fiber failures clearly showed that stress perturbations did occur
(see Wood et al. (1994)) even in that case. It must also be noted that Wood et al.
(1995) identified a bulging mode of failure in compression for their high modulus
pitch-derived fibers in contrast with the shear mode of failure observed here. In
an earlier work, Hawthorne and Teghtsoonian (1975) also observed compression induced shear failure in carbon fiber produced from pitch, rayon and polyacrolonitrile.
Prior to fracture hair-like cracks in the fiber surface were observed. They concluded
that as fiber anisotropy increases the tendency is away from a single catastrophic
shear-like failure to what is often a series of fine, partial microcracks. They suggested that for high modulus fibers microcracking may initiate as a buckling of
microfibrils of well-ordered graphite crystallites. More recently, Boll et al. (1990)
found that intermediate modulus fibers embedded in an epoxy matrix fail by shear.
Compression failure initiates as a microcrack, which then propagates as a shear
failure. Subsequent post failure damage may take a variety of forms such as fiber
crushing, longitudinal splitting, bifurcation of shear fracture etc. Finally, Melanitis
et al. (1994) have performed systematic studies on PAN based carbon fibers of various moduli and observed that bulging is only present in low modulus fibers and as
the modulus increases shear failure clearly dominates.
91
9.2
Comparison of Compressive versus Tensile
Behaviour
In Figure 9.34 the far-field fiber strain versus the applied strain for both
loading conditions (tension & compression) has been plotted. At low strains an
approximately linear relationship between fiber strain and applied strain (matrix
strain) is obtained for both types of loading. However, for strains higher than 0.8%
in tension and -0.35% in compression a deviation from linearity is observed. Based
on earlier arguments, it is clear that the cause of the nonlinear behavior is different
in compression and in tension. In tension, the gradual deviation from linearity prior
to fiber fracture within the range of 0.8% to 1.1% strain is attributed to the onset
of matrix plasticity and hence the reduction of its shear modulus at high strains
(this early onset of matrix plasticity is due to the cold curing of the resin). In
compression, the significant reduction of the strain sustained by the fiber is due to
the multiple fiber failure that is observed at strains lower than -0.56%.
In terms of failure characteristics, it has been recognized quite earlier on Rosen
(1964) that composite tensile failure is governed by the statistical distribution of
fiber flaws or imperfections. For example as mentioned already the fiber break observed in Figure 6.22(a)) might be associated with a fiber flaw. Another, observation
is that the tensile fragment distribution is far from uniform i.e. see Refs. van den
Heuvel et al. (1997b); Paipetis and Galiotis (1997). On the contrary it can be observed from Figure 6.17(a) that the compressive fragmentation process exhibits a
clear uniformity. This is an indication that compressive fracture is not governed
by a random flaw distribution as in the case of tensile loading. Boll et al. (1990)
reached the same conclusion, by observing that for an AS4 fiber embedded in an
92
epoxy matrix the compressive average fragment size was 0.18mm. The size distribution was quite narrow and relatively symmetrical, whereas the average fragment
length in tension was found to be around 0.40mm and the distribution was highly
skewed (see also Ref. Favre and Jacques (1990)). Hawthorne and Teghtsoonian
(1975) examined numerous compression fractures and found only a few examples
where fracture might possibly have been associated with an observable flaw. Hence,
it seems that compressive carbon fiber failure is determined by its microcrystalline
structure rather than by random defects.
In Figure 9.35 the maximum ISS values obtained at various levels of applied strain
over both tension and compression regimes and for both fiber ends are presented.
As seen, the max. ISS takes up similar values for a given applied strain regardless
of the direction of loading. In fact, deviations from linearity are observed at strains
higher than 0.6% for both regimes, which indicate that (a) the interface holds well
in compression in spite of the shear failure of the fiber which occurs at lower strains
and (b) the origin of the slight drop of max. ISS should be attributed to the
reduction of shear modulus and not to interface failure since the maximum value
is obtained at the fiber ends. Further increase of input strain in tension shifts the
maximum towards the middle of the fiber, which indicates the onset of interface
failure followed by possible debonding or matrix cracking as the ISS at the fiber end
drops to zero (9.35). It is interesting to note that the new maxima generally follow
the overall curve indicating further gradual drop of shear modulus with increasing
strain.
93
1.50
1.25
tension
1.00
Fiber strain (%)
0.75
0.50
0.25
compressive
fiber collapse
0.00
interfacial damage,
onset of matrix
plasticity
-0.25
-0.50
-0.75
compression
fiber fracture
-1.00
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Applied strain (%)
Figure 9.34: Axial far-field fiber strain (ǫf iber = ǫmeasured − ǫresidual ) versus applied
strain. The dotted line represents the 1:1 relation between the applied strain and
the far-field fiber strain, whereas the solid line represents represents a 3rd degree
polynomial fit of the experimental data (◦).
94
40
ISS at the fiber end (MPa)
30
max ISS occurred at a
distance from the
fiber end
20
10
tension
0
compression
-10
-20
-30
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Applied strain (%)
max
Figure 9.35: Maximum ISS values (ISSmax = ISSmax
measured - ISSǫ=0% ) versus applied
strain. The different symbols in the left part (compression) of the graph correspond
to different experiments, whereas the solid squares in the tensile section of graph
represent measurements at a distance from the fiber ends. The solid line represents
the linear fir of the experimental data for -0.3%< ǫ <0.3%.
CHAPTER 10
NUMERICAL ANALYSIS OF THE STRESS TRANSFER IN
COMPRESSION IN THE CASE OF A BROKEN FIBRE
10.1
introduction
From the above, it can be argued that the fundamental issues, of the compres-
sive behaviour of fibrous composites still remain open. The problem of compressive
failure is more complicated than is generally appreciated (Lankford (1995)) and
appropriate models should include the role of fibre strength and its distribution,
matrix properties, and interfacial properties. In a previous work (Goutianos et al.
(2002b)) the Laser Raman Spectroscopy technique was used to get an insight into
the microstructural aspects of the compressive behaviour of carbon/epoxy composites. This was done by a comparative assessment of the stress transfer efficiency
in tension and compression in single fibre discontinuous model geometries. The
stress transfer mechanisms were completely described in the region around the fibre
ends, however, this was not the case in the vicinity of a compressive fibre break
since in this case the balance of forces argument (required to derive the interfacial
shear stresses) breaks down (Goutianos et al. (2002b)). Narayanan and Schadler
(Narayanan and Schadler (1999a)) have employed the balance of forces argument
to calculate the ISS in the case of a compressive fibre break, their reported ISS
values were in excess of 150 MPa or even 300 MPa (Amer and Schadler (1997))
for fibre/matrix systems similar with the system examined here and in our previous work (Goutianos et al. (2002b)). Since these values were unacceptably high,
95
96
an attempt was made (Narayanan and Schadler (1999a)) to modify the balance of
forces argument to account for the post-failure geometric configuration. Although
the new ISS values were reduced these values were still too high and as stated by
the same authors, a more rigorous analysis is needed to estimate accurately the interfacial shear stress in the neighbourhood of a compressive fibre break. As a result
in the current work, the finite element method is employed to study the complex
stress field (in both the fibre and the matrix) generated by a compressive fibre break
(shear failure).
Figure 10.36(a) shows a micrograph of a typical compression induced shear fibre
break, it can be easily seen that the broken ends slide past each other and therefore
compressive stresses can be transmitted as the fibre fragments remain in contact
(Goutianos et al. (2002b)). The same failure mode can be also observed in multifibre model composites (Figure ?? and ??), that is the fibres have failed in shear
at a certain (well defined) plane, whereas no fibre fractures were observed far from
the fracture planes (Goutianos et al. (2002a)) Fig. 1 clearly indicates the importance of understanding the stresses generated around a fibre break, which would
probably enrich our knowledge concerning damage development in compression. In
the current work only a single-carbon fibre embedded in an epoxy matrix will be
considered at first instance, while the case of a planar array of fibres will be reported
in a subsequent work.
10.2
FE model for fibre fracture in compression
Various analytical models (2D or 3D) have been developed over the past years
(e.g. Nath et al. (1996); Van den Heuvel et al. (1998); Sirivedin et al. (2000)) for
the derivation of stresses in both the fibre and the matrix in single or multi-fibre
97
composites. These analyses, however, are restricted in the case of tensile loading.
In contrast, very little work has been reported in the literature concerning numerical modelling of the compressive behaviour of composites at micromechanical level.
Usually the uniaxial composite is investigated using two or three-dimensional micromechanical models, which include a global (Hsu et al. (1998); Hsu et al. (1999))
or local (Vogler et al. (1999); Byskov et al. (2002)) imperfection by means of fibre
misalignment. These global or local imperfections are required in order to initiate
the kink band formation. Kink band initiation and growth are by a fibre microbuckling process. Fibre fracture is not considered in these analyses since it is assumed
that it might be a post-failure event, that is the fibres may fail due to excessive
buckling. To the best of our knowledge the compressive fragmentation process has
not yet been investigated by means of a numerical or even more an analytical model
mainly due to the complexity of the problem and lack of experimental data with
the exception of the work of Garland et al Garland et al. (submitted), which has
been already mentioned above. They have used a break-influence superposition
technique (Sastry and Phoenix (1993)) based on a simple shear-lag model to study
the development of compression damage zones. As almost every shear-lag model,
their model presents some major drawbacks such as it assumes that the matrix only
deforms in shear and its axial stiffness is ignored. More importantly the fibres are
not allowed to deform in the transverse direction, which is not the case as can be
seen in Figure ??. As a result of this fibre shearing restriction, their predicted stress
profiles are quite symmetric and similar to the tensile case, while according to the
experimental results () the fibre stress profiles are not symmetric along the fibre
length. The same will be shown later in the following section by means of FE analysis. Furthermore, the assumption that the fibre stress is zero at the break point is
98
not supported by the experimental data of Goutianos et al. (2002b). Finally, their
key assumption that the peak stress concentration produced by an angled break
(i.e. like the one shown in Figure ??) is the same as that of a straight break is
questionable. However, despite these drawbacks, their work shows nicely how a
damage zone of broken fibres could lead to kink band formation. Thus here we
attempt to model the compressive behaviour of a (at first instance)-single carbon
fibre embedded in an epoxy matrix by means of FEA, although this results in an
intense computation compared to the technique used by Garland et al. (submitted).
This material system is identical to the system experimentally studied in previous
works (Goutianos et al. (2002b); Goutianos et al. (2002a)).
10.2.1 Finite element details
10.2.1.1 The mesh
The single-fibre model composite was implemented in the ABAQUS finite
element code. The I-DEAS pre-processor was used to generate the mesh of the
microcomposite. Clearly the problem of the fibre fracture in compression (see Figure
??) is a three-dimensional problem. In this case, however, the complexity and size
of the model significantly increase which is beyond the scope of the current work.
Moreover, due to the shear-failure of the fibre, the problem cannot be considered
as an axisymmetric one. Thus, we have chosen to model the fibre fracture problem
as a plane stress problem. A schematic representation of the model is depicted
in Figure 10.37. Fibre fracture is assumed to occur in the centre of the model
composite. The fibre is then divided in two fragments, which are fully bonded up
to a certain applied strain level (equal to the fibre failure strain experimentally
determined in similar specimens (Goutianos et al. (2002b)). The angle between
99
the fibre fragments is 450, equal to the value observed in the experiments (see
Figure ??). In the subsequent step of the analysis the coefficient of friction between
the fibre fragments is reduced allowing in that way the fibre fragments to slide
past each other. The fibre fragments (near the fracture site) are also fully bonded
to the surrounding matrix using the TIED CONTACT option of ABAQUS. Due
to the multi-contact nature of the model, bilinear four noded elements had to be
used. The fibre close to the contact area (see Figure 10.37) was modelled using
incompatible modes elements in order to prevent shear locking or hourglass effects.
At this point, it should be noted that as the fibre fragments slide past each other,
bending of the fibre fragments occurs. Full integration elements were found to give
poor results in this case due to shear locking, whereas reduced integration elements
were sensitive to hourglass effects. Thus, the use of incompatible elements was
necessary although this resulted in significant increase of the computational time.
The fibre fragments far from the contact area (fibre fracture site) and the matrix
material were modelled using full integration elements. Finally, in order to facilitate
the multi-contact definition, node a and b belonging to the fibre fragments were
slightly moved as it can be seen in Figure 10.37. This kind of local imperfection
results in stress discontinuities at applied strains even much lower than the strain
level at which the fibre fragments are allowed to slide past each other. As it will
be shown, however, in the results section this imperfection has minor effects in the
overall analysis. Finally, it should be mentioned that the analysis performed was
quasi-static, that is any dynamic effect from the fibre fracture is excluded. This
is the case, however, also in the experimental work on the same material system
(Goutianos et al. (2002b)). Thus, comparison of the numerical predictions versus
experiments is still valid.
100
The axial fibre stress is calculated at the right hand side of the fibre as it is shown
in Figure 10.37, whereas the interfacial shear and transverse stresses are calculated
at the corresponding fibre/matrix interface.
10.2.1.2 Boundary conditions and applied loads
Through the analysis the nodes at x=0 (see Figure 10.37) are fixed in the
x-direction, whereas the far bottom right matrix node is also constrained in the
y-direction. A total strain of 0.8% with respect to the initial x-length of the model
is applied at the nodes at x=2000µm. Up to an applied strain of 0.5% the fibre
fragments are constrained to move relative to each other. Then, as it was mentioned
above, at an applied strain of 0.55% (fibre failure strain in the microcomposite) the
coefficient of friction between the fibre fragments is decreased allowing the fibre
fragments to slide past each other.
10.2.1.3 Material properties
The fibre is modelled as an anisotropic continuum with linear elastic behaviour
in both compression and tension. The fibre properties are listed in Table 1. Concerning the epoxy matrix, an elasto-plastic behaviour is assumed. Figure 10.38
depicts the experimental stress-strain curve for the epoxy matrix, whereas its mechanical properties are given in Table 1. It should be mentioned here that in the
results section, the effect of matrix modulus and the effect of matrix yield stress
on the stresses generated around the fibre fracture are investigated assuming an
elastic-perfectly plastic behaviour for the epoxy matrix.
101
10.3
Results & Discussion
10.3.1 Effect of fibre geometrical discontinuity on the
stress field
As it was referred in the previous section a geometrical imperfection was introduced in the model definition in the area where the fibre break is introduced. In
this section, the effect of this imperfection will be briefly investigated. Figure 10.39
depicts the axial fibre stress (σ11 ) at different applied strain levels (-0.3%, -0.4%
and 0.5%, respectively), which are lower than the applied strain at which the fibre
break is introduced (-0.55%). As it can be seen the fibre stress is constant along the
fibre length as expected. At x=1000µm, however, a fibre stress perturbation can
be observed due to the initially introduced imperfection. The same can be seen in
Figure 10.40, which shows the Interfacial Shear Stress (ISS) along the fibre/matrix
interface. ISS is zero everywhere, as it should be, except the area near the fibre
discontinuity with its absolute higher value being ∼ 17MPa.
Figure ?? depicts the shear stress contours (σ12 ) around the fibre break at three
different applied strains, a) ǫ11 =-0.465% (the fibre break is not yet introduced), b)
ǫ11 =-0.55% (fibre break introduction), and c) ǫ11 =-0.70%. It can be easily seen that
the shear stress perturbation due to the initial geometrical imperfection (Figure ??)
is much lower than the perturbation created by the fibre break (Figures ?? and ??).
Thus, it can be safely assumed that the initial fibre imperfection has minor effect
on the overall analysis, although the stress values are influenced exactly at the fibre
break location and for this reason stress values at these points will not be considered
in the analysis.
102
10.3.2 Effect of friction between the fibre fragments
Figure 10.42 shows the FEA predictions of the axial fibre stress profile in the
vicinity of a compressive fibre break. It can be seen that for a coefficient of friction
of 0.4 the numerical data agrees relatively well with the experimental values, except
at x ∼
= 0 due to the model definition as it was explained in the previous section. An
important observation is that in contrast to the tensile case (Van den Heuvel et al.,
1998; Nath et al., 1996), the fibre stress profile is quite asymmetric around the fibre
break. This is due to the fibre bending as the fibre fragments slide past each other
after the onset of fibre failure. These results also suggests that one-dimensional
analyses such as shear-lag models would fail to accurately capture the mechanics of
the problem as it is clear that this fibre failure mode causes also geometrical nonlinearities. Figure 10.43 depicts another set of experimental data of the same fibre
and applied strain (ǫ11 = -0.55%) but at different position along the fibre length.
It can be observed that in this case a higher coefficient of friction between the
fibre fragments is needed in order to match the experimental values. At this point
some reasons to explain the mismatch between numerical and experimental values
should be mentioned. Except the models imperfection at the fibre break location,
the experimental data presents themselves a high level of noise since a very detailed
mapping of the fibre stress was attempted (data sampling every 2µm near the fibre
fracture (Goutianos et al. (2002b))). Moreover, during data acquisition some parts
of the fibre were invisible to the laser light due to fibre overlapping as the fibre
fragments slide past each other (see Figure 10.36(a)). Furthermore, the laser spot
was of about 2µm, whereas the fibre diameter is 6.6µm and it was impossible to
control the point of the fibre fragment where the laser light was focused as it will
be shown later the fibre stress significantly varies across its section near the fibre
103
break position. Finally, as mentioned above the angle between the fibre fragments
was set to 45o in the model definition. However, in reality it is expected that more
probably there might be small deviations from this value, which would change the
effective coefficient of friction between the fibre fragments as a result of the change
of the normal force component in the plane of fracture. Considering the above, it
can be argued that models predictions agree well with the experimental findings.
The different coefficient of friction between the fibre fragments used in Figures 10.42
and 10.43 could be explained by the problems during data sampling previously mentioned. However, it might be possible that fibre breaks in the same fibre can have
different values of coefficient of friction. This is also supported by the experimental data reported by Goutianos et al. (2002b), where large differences between
types of breaks in the same fibre were observed, which cannot be explained by the
experimental error of the technique used.
10.3.3 Stress field in the vicinity of a compressive fibre
break
In this section the stress field around a compressive fibre break will be investigated. As it was mentioned above this problem concerning the axial fibre stress
was already experimentally investigated by Goutianos et al. (2002b). However,
as it is mentioned the derivation of the ISS, in the vicinity of a compressive fibre
break, from the fibre stress is not accurate since the balance of forces argument,
used successfully in similar problems in tension, is not valid. Thus here, after having validated the model so far by comparing the axial fibre stress obtained from
the finite element analysis with the experimental data, the interfacial shear and
transverse stresses will be derived.
104
Figure 10.44 depicts the axial fibre stress along the fibre length (the fibre break
position is at x=1000µm) at three different applied strains, -0.55% (fibre break
introduction), -0.6%, and 0.7%, respectively. It can be easily seen that the compressive ineffective length is of the order of ∼ 100µm, much smaller when compared
to the tensile ineffective length (∼ 500µm). This can be explained by the fact that
the fibre fragments remain in contact after fracture and thus stresses can still be
transmitted between the fibre fragments. Another important observation is that the
fibre stress profile doesnt change drastically after fracture with increasing applied
strain, which is in agreement with the experimental data reported by Goutianos
et al. (2002b). Finally, the far-field stress values (far from the fibre break location)
are much lower than predicted by applying Hookes law, i.e. for ǫ11 =-0.55%, σ11
should be around 2.1GPa, whereas from Figure 10.44 is only 1.25GPa. The fibre
fragments behave like a column that cant sufficiently support the applied load. Due
to the relative movement between the fibre fragments, the load applied to the fibre
is partially dissipated to the matrix material. This is in distinct contrast with what
is observed in the case of tensile loading, where the far-field fibre stress (or strain)
is always equal to the applied stress (or strain) unless the fibre fragment length is
less than the critical fibre length or excessive yielding and debonding takes place at
strain exceeding quite enough the failure strain of the fibres.
Concerning the interfacial shear stresses from Figure 10.45 it can be observed that
similar to the axial fibre stress ISS almost barely changes with increasing applied
strain. The same can also be seen in Figures ?? and ??, where the shear stresses
mainly change in the direction transverse to the fibre through the matrix material.
Furthermore, the ISS profile is quite different between the side of the fibre which is
under compression (maximum ISS ∼ 22MPa) and the side which is under tension
105
(maximum ISS ∼ 30MPa).
Figure ?? shows the transverse stresses at the fibre/matrix interface. The analysis
of these stresses in the case of tensile fibre fragmentation tests is usually ignored
since they are compressive and thus dont lead to transverse cracking. In the case of
compressive fragmentation, however, these stresses are positive (tensile) and hence
if they exceed the matrix failure stress (under the assumption that the interfacial
strength is equal to the matrix strength) failure will occur. From Figure ??, ignoring
the high values near the fibre break location (at x=1000µm), we can conclude that
transverse stresses dont cause transverse interfacial failure for this material system
within the applicability of the current model used.
From the above results some distinct differences between the compressive and tensile fragmentation process (Nath et al., 1996; Van den Heuvel et al., 1998) can be
immediately identified. As it was mentioned above, the ineffective length in compression is extremely small compared to that one in tension. Moreover, as it was
f
m
m
shown, the stress profiles (σ11
, σ22
, and σ12
) dont drastically change in the fibre
direction with increasing the applied strain after the event of fibre fracture. On
the other hand, from Figure 10.41 (shear stresses) it can be seen that this is not
the case in the direction transverse to the fibre axis. This is even more clear if
the transverse stresses are considered. Figure 10.47 depicts the transverse stress
contours at an applied strain level of 0.55%. It can be easily seen that the main
change in the stress field is in the transverse direction. Hence, it can be assumed
that a compressive shear fibre break results in a very local stress perturbation field
and the associated damage propagates in the transverse direction rather than in
the longitudinal direction. Additionally as it was shown by Boll et al. (1990), and
experimentally supported by our previous work (Goutianos et al. (2002b)), that
106
compressive failure of the fibres is not governed by fibre flaws. These suggestions
are in agreement with the results of Nakatani et al. (1999) who investigated in detail the distribution of the compressive and tensile strength of carbon fibres. They
found that the fibre length dependence of the average fibre compressive strength is
much smaller than those of the tensile strength. More importantly, they reported
significantly larger value of the shape parameter of the Weibull distribution for the
carbon fibre compressive strength (β=32) than the fibre tensile strength (β=6.1).
Based on these results, they also suggested that the scattering of the fibre tensile
strength comes from the stochastic nature of the existence and severity of defects
and irregularities on the fibre structure. On the other hand, the fibre compressive
strength, according to the same authors, depends mainly on intrinsic material properties of the fibre. This observation in conjunction with the current findings that
the stress field, around a compressive fibre break, changes at a higher rate in the
transverse direction could explain the sudden catastrophic failure observed in full
unidirectional composites tested in compression. That is, an initial fibre fracture
caused by a stress raiser such as a hole, cut-out or fibre misalignment results in an
intense stress perturbation in the transverse direction, and as soon as the stress in
the nearest adjacent fibre reaches its failure strength this fibre will instantaneously
break at this point (which is normally at a certain small angle with the initial fibre
failure as can be seen in Figure ??) since the failure pattern cannot be deflected by
the existence of fibre flaws as it happens in the case of tensile loading as suggested
above. As soon as more fibres fail then catastrophic failure takes place at a well
defined plane, which lies at an angle different from 90o to the load direction as also
shown by Garland et al. (submitted). This localised damage zone could then easily cause a local instability leading to kink band formation. This proposed failure
107
mechanism could explain the typical compressive failure modes shown in Figures
10.36(b) and 10.36(c).
10.3.4 Effect of matrix modulus
It is well known that the tensile properties of fibrous composites have been
significantly improved over the past decades by improving the fibre properties, understanding the role of the interface and optimising the fibre/matrix adhesion, as
well as matrix material. However, this is not the case concerning the compressive
properties where not much improvement was achieved. In the following sections
a parametric analysis of the matrix properties on the stress field generated by a
compressive fibre break is performed with the final aim to improve the compressive
strength by using a matrix with the appropriate properties. The matrix behaviour
is now modelled as elastic-perfectly plastic material. First, the effect of the matrix
Youngs modulus will be addressed. A variation of the matrix Youngs modulus can
be seen also as a variation of its shear modulus.
Figure 10.48 shows the axial fibre stress profiles along the fibre length for three
different matrix moduli of 1.5, 2.0, and 2.5GPa, respectively. The matrix yield
stress is 45MPa in all the cases. It can be observed that the stiffer the matrix the
lesser the fibre fragments can bend and therefore lower maximum fibre tensile and
compressive stresses can be attained. On the other hand, the far-field fibre stress is
higher for a stiffer matrix.
Concerning the ISS profiles it can be observed from Figure ?? that for the side of
the fibre fragment which is under tension (Figure 10.49(b)) all three cases examined
show very similar behaviour (maximum ISS = 22MPa). In the case of a matrix
with a lower modulus of elasticity the stress build-up from the fibre break is slightly
108
lower. However, at the compressive side of the fibre (Figure 10.49(a)) the effect of
the different matrix moduli is more pronounced. A high matrix Youngs modulus
results in much higher shear stresses. For Em =2.5 GPa the maximum ISS is ∼
20MPa, whereas for Em =1.5GPa the maximum ISS is only ∼ 15 MPa.
The transverse stresses are investigated in Figure 10.50. Figure 10.50(a) depicts
the transverse stresses in the side of the fibre fragment, which is under compression
and Figure 10.50(b) the side, which is under tension. It can be seen from Figure
10.50(a) that the higher transverse stresses are observed for the stiffer matrix similar
to the case of the interfacial shear stresses. Concerning the side, which is under
tension, again not much difference between the three different matrix moduli can
be observed. In all the cases the maximum attained values are the same, while as
the matrix modulus decreases (read: the fibre can bend more easily) decreases at
lower rate at a small distance from the fibre break and then rapidly drops to zero.
From the results presented above it can be concluded that a low matrix Youngs
modulus (or shear modulus) results in larger deformations of the fibre fragments
after fibre fracture simply because the matrix provides less support to the fibre
fragments. In general, however, it can be assumed that a variation in the Youngs
modulus of the matrix doesnt drastically alter the stress field caused by the fibre
fracture. Additionally, the matrix shear stress patterns in the transverse direction to
the fibre break were almost identical for the three different matrix moduli examined.
At this point it should be mentioned that Rosens analysis (Rosen (1965)) treats
the problem of compression as a buckling process in an elastic foundation and its
predictions are much higher compared with experimental values. From the results
of this section, it can be argued that the matrix modulus within the range examined
109
has no direct effect on the compressive fragmentation process.
10.3.5 Effect of matrix yield stress
In this section the effect of the matrix yield stress will be investigated. For
this reason the matrix Youngs modulus is kept constant and equal to 2GPa. Figure
10.51 shows the axial fibre stress in the case of three different matrix yield stresses
30, 45, and 60MPa, respectively. It can be easily observed that a low matrix yield
stress results in lower maximum fibre stress values and higher ineffective length.
Additionally, it is interesting to observe that the effect of the matrix yield stress is
more pronounced of the compressive side of the fibre fragment.
The corresponding interfacial shear stresses are plotted in Figure 10.52, where it
can be seen that the shear stresses are strongly affected by changing the matrix
yield stress. Moreover, excessive yielding occurs for a low matrix yield stress (see
Fig. Figure 10.52(b)), whereas no yielding can be identified at the compressive side
of the fibre (Figure 10.52(a)).
The transverse stresses for the three different matrix yield stresses used are given
in Figure 10.53(a) (compressive side of the fibre fragment) and Figure 10.53(b)
(tensile side of the fibre fragment). In both cases a low matrix yield stress results
in a higher affected zone caused by the fibre break as the matrix is more compliant
and therefore the fibre fragments can bend more easily. The difference between the
different matrix yield stresses chosen are now more clear in the side of the fibre
fragment which is under tensile loading.
It is clear from the results presented that the matrix yield is a key parameter in
studying the compressive behaviour of composites. A high matrix yield stress is
essential in order to improve the compressive properties of fibrous composites. This
110
is in agreement with the results of i.e.
Argon (1972); Budiansky (1983); and
Fleck et al. (1995), the only basic difference with their analysis is that the primary
damage mechanism is the fibre fragmentation itself.
Next a comparison between an elastic (σym =0) and elastic-perfectly plastic matrix
(σym =45 MPa) will be performed to highlight the effect of matrix nonlinearity. Figure 10.54 depicts the axial fibre stress profiles along the fibre length (fibre break
location at x=1000mum) for the two different matrices. The ineffective length
doesnt show large differences although it is lower for the elastic matrix. The most
striking observation is that for the elastic matrix the stress profile is shifted downwards to higher compressive values, i.e. the far-field axial fibre stress is -1.9MPa,
whereas for the elastic-perfectly plastic matrix is only -1.5MPa. Concerning the ISS
profiles it can be seen from Figure 10.55 that the elastic interfacial shear stresses
are unacceptably high (100MPa) exceeding the shear yield stress of any commercial
epoxy matrix. In the elastic case the matrix is so stiff that the fibre fragments
display small lateral deformations and as a consequence the perturbation of shear
stresses around the fibre break is minor. On the other hand, if the numerical predictions for the elastic matrix concerning the axial fibre stress are compared with
the experimental data presented in Figures 10.42 and 10.43, then it can be observed
that plasticity needs to be taken into account in order to model the compressive
behaviour of fibrous composites.
10.4
Conclusions
A numerical investigation of the compressive behaviour of high-modulus car-
bon fibres embedded in an epoxy matrix was performed. First the FE model was
validated by comparing numerical predictions with experimental data. It was shown
111
that:
• The stress build-up emanating from a compressive fibre break is quite different
from that in tension. In contrast to the tensile case in compression the fibre
stress at a fibre break does not necessarily drop to zero.
• The rate of stress transfer from a compressive fibre break is extremely high,
since load transmission occurs between the fibre fragments, and therefore the
corresponding ineffective length is extremely small.
• The matrix modulus of glassy polymers has no drastic effects on the stresses
generated by the fibre fracture and thus enhancement of the compressive
strength cannot be achieved by increasing the matrix modulus (for an elastoplastic matrix).
• Matrix plasticity is a key parameter in modelling the compressive behaviour
of composites a high yield stress is required for compressive strength improvement.
112
(a)
(b)
(c)
Figure 10.36: (a) Micrograph of a typical compression induced shear fibre break, (b)
micrograph of a fracture site showing fibres failed in shear at a well defined plane,
(c) SEM picture showing again co-operative fibre shear failure in compression.
113
Figure 10.37: Schematic representation of the FE model used.
70
60
114
Stress (MPa)
50
40
30
20
10
0
0.00
0.02
0.04
0.06
0.08
0.10
Strain (-)
Figure 10.38: Stress-strain curve for the epoxy matrix.
115
0.0
Axial fibre stress (GPa)
-0.4
-0.8
-1.2
-1.6
-2.0
= -0.3%
= -0.4%
= -0.5%
applied strain:
-2.4
-2.8
0
500
1000
1500
2000
Position along the fibre length (µm)
Figure 10.39: Axial fibre stress of an embedded fibre (Lf = 2 mm) in an epoxy
matrix, loaded at strain of 0.3%, -0.4%, and 0.5%, respectively.
20
applied strain:
15
= -0.3%
= -0.4%
= -0.5%
ISS (MPa)
10
116
5
0
-5
-10
-15
-20
0
500
1000
1500
2000
Position along the fibre length (µm)
Figure 10.40: Corresponding interfacial shear stresses of Fig. 4. The applied strains
are 0.3%, -0.4%, and 0.5%, respectively
117
(a)
(b)
(c)
Figure 10.41: Shear stress field around the compressive fibre break. a) applied strain
= -0.465% (no fibre break is introduced), b) applied strain = -0.55% (introduction of
the fibre break), and c) applied strain = -0.7%. (all values in MPa). The coefficient
of friction between the fibre fragments is 0.4.
Axial fibre stress (GPa)
1
exper. data
fea
118
0
-1
-2
-3
-4
-200 -160 -120 -80 -40
0
40
80 120 160 200
Position form fibre break (µm)
Figure 10.42: Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=0µm). The applied strain is 0.55%, and the coefficient of friction between the
fibre fragments is 0.4.
119
Axial fibre stress (GPa)
1
0
-1
-2
-3
exper. data
fea, friction=0.6
fea, friction=0.4
-4
-200 -160 -120 -80 -40
0
40
80 120 160 200
Position from break (µm)
Figure 10.43: Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=0µm). The applied strain is 0.55%, and the coefficient of friction between the
fibre fragments is 0.4 and 0.6, respectively.
1.0
Axial fibre stress (GPa)
0.5
0.0
applied strain:
ε = -0.55%
ε = -0.60%
ε = -0.70%
120
-0.5
-1.0
-1.5
-2.0
-2.5
800
850
900
950 1000 1050 1100 1150 1200
Position along the fibre length (µm)
Figure 10.44: Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=1000µm) at different applied strains (ǫ11 ), -0.55%, -0.60%, and 0.70%, respectively. The coefficient of friction between the fibre fragments is 0.4.
ISS (MPa)
121
35
applied strain
30
-0.55%
ε=
25
ε=
-0.60%
20
-0.70%
ε=
15
10
5
0
-5
-10
-15
-20
-25
800 850 900 950 1000 1050 1100 1150 1200
Position along the fibre length (µm)
Figure 10.45: Interfacial shear stress (σ12 ) in the vicinity of a compressive fibre
break (x=1000µm) at different applied strains (σ11 ) -0.55%, -0.60%, and 0.70%,
respectively. The coefficient of friction between the fibre fragments is 0.4.
30
20
122
10
σ22 (MPa)
0
-10
-20
-30
-40
-50
-60
850
applied strain:
ε = -0.55%
ε = -0.60%
ε = -0.70%
900
950
Position along the fibre length (µm)
50
40
(a)
σ22 (MPa)
30
applied strain:
ε = -0.55%
ε = -0.60%
ε = -0.70%
20
10
0
-10
-20
1000
1050
1100
1150
Position along the fibre length (µm)
(b)
Figure 10.46: Transverse stress (σ12 ) at the fibre matrix interface at different applied
strains (σ11 ), -0.55%, -0.60%, and 0.70% respectively. The coefficient of friction
between the fibre fragments is 0.4. a) Transverse stresses at the left side of the fibre
break, and b) Transverse stresses at the right side of the fibre break.
123
Figure 10.47: Transverse stress contours around the compressive fibre break at an
applied strain of -0.55% (introduction of the fibre break). The coefficient of friction
between the fibre fragments is 0.4. (All values in MPa).
Axial fibre stress (GPa)
0
Em = 1.5 GPa
Em = 2.0 GPa
Em = 2.5 GPa
124
-1
-2
700
750
800
850
900
950
1000
Position along the fibre break (µm)
Axial fibre stress (GPa)
1
Em = 1.5 GPa
Em = 2.0 GPa
Em = 2.5 GPa
(a)
0
-1
-2
1000
1050
1100
1150
1200
1250
1300
Position along the fibre length (µm)
(b)
Figure 10.48: Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=1000µm) for three different matrix modulus, 1.5GPa,2 GPa, and 2.5GPa respectively. The coefficient of friction between the fibre fragments is 0.4 and the
applied strain is 0.55%.
0
125
ISS (MPa)
-5
-10
-15
-20
700
Em = 1.5 GPa
Em = 2.0 GPa
Em = 2.5 GPa
750
800
850
900
950
1000
Position along the fibre length (µm)
30
Em = 1.5 GPa
Em = 2.0 GPa
Em = 2.5 GPa
25
(a)
ISS (MPa)
20
15
10
5
0
1000
1050
1100
1150
1200
1250
1300
Position along the fibre length (µm)
(b)
Figure 10.49: Interfacial shear stress (σ12 ) in the vicinity of a compressive fibre
break (x=1000µm) for three different matrix modulus, 1.5GPa, 2GPa, and 2.5GPa
respectively. The coefficient of friction between the fibre fragments is 0.4 and the
applied strain is 0.55%.
20
126
σ22 (MPa)
0
-20
-40
850
Em = 1.5 GPa
Em = 2.0 GPa
Em = 2.5 GPa
900
950
1000
Position along the fibre length (µm)
60
Em = 1.5 GPa
Em = 2.0 GPa
Em = 2.5 GPa
50
(a)
σ22 (MPa)
40
30
20
10
0
-10
1000
1050
1100
1150
Position along the fibre length (µm)
(b)
Figure 10.50: Transverse stress (σ22 ) in the vicinity of a compressive fibre break
(x=1000µm) for three different matrix modulus, 1.5GPa, 2GPa, and 2.5GPa respectively. The coefficient of friction between the fibre fragments is 0.4 and the
applied strain is 0.55%.
1.0
σy = 30 MPa
m
Axial fibre stress (GPa)
0.5
0.0
127
σy = 44 MPa
m
σy = 60 MPa
m
-0.5
-1.0
-1.5
-2.0
-2.5
750
800
850
900
950
1000
Position along the fibre length (µm)
1.0
σy = 30 MPa
m
σy = 44 MPa
m
Axial fibre stress (GPa)
0.5
σy = 60 MPa
m
(a)
0.0
-0.5
-1.0
-1.5
-2.0
1000
1050
1100
1150
1200
1250
Position along the fibre length (µm)
(b)
Figure 10.51: Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=1000µm) for three different matrix yield stresses, 30MPa, 45MPa, and 60MPa,
respectively. The coefficient of friction between the fibre fragments is 0.4 and the
applied strain is 0.55%, and the matrix modulus is 2.0GPa.
0
128
-5
ISS (MPa)
-10
-15
-20
σy = 30 MPa
m
-25
σy = 44 MPa
m
σy = 60 MPa
m
-30
750
800
850
900
950
1000
Position along the fibre length (µm)
40
σy = 30 MPa
m
σy = 44 MPa
m
ISS (MPa)
30
σy = 60 MPa
m
(a)
20
10
0
1000
1050
1100
1150
1200
1250
Position along the fibre length (µm)
(b)
Figure 10.52: Interfacial shear stress (σ12 ) in the vicinity of a compressive fibre
break (x=1000µm) for three different matrix yield stresses, 30MPa, 45MPa, and60
MPa, respectively. The coefficient of friction between the fibre fragments is 0.4 and
the applied strain is 0.55%, and the matrix modulus is 2.0GPa.
20
10
129
0
σ22 (MPa)
-10
-20
-30
-40
-50
-60
-70
800
σy = 30 MPa
y
σy = 44 MPa
y
σy = 60 MPa
y
850
900
950
1000
Position along the fibre length (µm)
70
σy = 30 MPa
y
60
σy = 44 MPa
y
50
(a)
σ22 (MPa)
40
σy = 60 MPa
y
30
20
10
0
-10
-20
-30
1000
1050
1100
1150
1200
Position along the fibre length (µm)
(b)
Figure 10.53: Transverse stress (σ22 ) in the vicinity of a compressive fibre break
(x=1000µm) for three different matrix yield stresses, 30MPa, 45MPa, and 60MPa,
respectively. The coefficient of friction between the fibre fragments is 0.4 and the
applied strain is 0.55%, and the matrix modulus is 2.0GPa.
1.2
Axial fibre stress (GPa)
0.6
σy = 0 MPa
m
σy = 44 MPa
m
130
0.0
-0.6
-1.2
-1.8
-2.4
-3.0
600
800
1000
1200
1400
Position along the fibre length (µm)
Figure 10.54: Axial fibre stress (σ11 ) in the vicinity of a compressive fibre break
(x=1000µm) for a) an elastic matrix, and b) an elastic-perfectly plastic matrix
(σym =45 MPa) . The coefficient of friction between the fibre fragments is 0.4 and
the applied strain is 0.55%, and the matrix modulus is 1.5GPa.
131
100
σy = 0 MPa
m
σy = 44 MPa
m
ISS (MPa)
50
0
-50
-100
600
800
1000
1200
1400
Position along the fibre length (µm)
Figure 10.55: Interfacial shear stress (σ12 ) in the vicinity of a compressive fibre
break (x=1000µm) for a) an elastic matrix, and b) an elastic-perfectly plastic matrix
(σym =45 MPa) . The coefficient of friction between the fibre fragments is 0.4 and
the applied strain is 0.55%, and the matrix modulus is 1.5GPa.
CHAPTER 11
CONCLUSIONS
A detailed investigation of the compressional behavior of high-modulus carbon fibers
embedded in an epoxy matrix was performed by means of a comparative assessment
of stress transfer efficiency in tension and compression. It was shown that:
1. the mapping of the ISS distribution from the fiber ends is the only accurate
method available for assessing the fiber/matrix adhesion under compressive
loading. The maximum values of ISS obtained in both cases (tension & compression) was of the order of 25MPa.
2. at low strains an approximately linear relationship between fiber strain and
applied strain (matrix strain) is obtained for both types of loading. At high
compressive strains the deviation from linearity is due to to the multiple fiber
failure. On the other hand, the deviation from linearity in tension is related
to the onset of matrix plasticity.
3. the stress build-up emanating from a compressive fiber break is quite different
from that in tension. The fiber stress at a compressive fiber break location
does not necessarily drop to zero or to the initial residual stress. .
4. the rate of stress transfer from a fiber break is extremely high and therefore the
corresponding transfer (ineffective) length is extremely small when compared
to that in tension
132
133
5. the compressive fragmentation process is not governed by a random flaw distribution as in the case of tensile loading.
CHAPTER 12
FUTURE WORK
12.1
Introduction
In the previous Chapters, the compressive behaviour of a single carbon fibre
embedded in an epoxy matrix was examined in detail. Expansion, however, of
the single fibre compressive response to the case of a full composite system is not
trivial. Effects of additional parameters need to be considered, such as fibre-fibre
interaction.
Current work concerns with the compressive behaviour of a bundle of carbon fibres
embedded in an epoxy matrix, as shown in Figure 12.56 (a ’short’ bundle is chosen
in order to avoid end effects upon application of compressive load). These series of
experiments aim to build a bridge between the micromechanics (single-fibre model
composites) and the macromechanics (full composites). Two kind of measurements
are undertaken: (a) fibre strain distributions within a large ’window’ of observation,
and (b) mapping of stress areas near a discontinuity (fibre break). In the next
Sections some preliminary results are presented.
12.2
Fibre strain distributions
It is well known that during tensile loading of a composite the weaker fibres fail
first, whereas the stronger fibres carry the applied load Rosen (1964). Normally,
the sites of fibre breaks are scattered randomly in the volume of the composite.
134
135
As the load is incremented, however, more fibers will break as a result of stress
concentrations in the vicinity of broken fibers Zweben and Rosen (1970); Smith
(1980); Harlow and Phoenix (1978). As the number of broken fibers increases with
applied load, multiple fibre fractures may occur, leading eventually to very high
values of stress concentration factors in the remaining fibres and to the failure of
the whole composite. Moreover, the fabrication process can also introduce random
flaws, which may also affect the composite strength. Concerning, however, the case
of compressive loading, there is very little information available in the literature in
this area of research Narayanan and Schadler (1999b).
In this Section, a statistical approach, introduced by Filiou et al Filiou et al. (1992)
and Filiou and Galiotis Filiou and Galiotis (1999), is employed. The embedded
fibres (see Figure **) are scanned with the laser Raman microprobe within a large
’window’ of observation. At a first approximation, the fibre strain distributions are
fitted with normalised Gaussian distribution functions.
A histogram of fibre strain values obtained from the mapping of a specimen, prior
to application of compressive load, is shown in Figure 12.57. The histogram of these
values was constructed at strain steps of 0.025%. As it can be seen, the axial fibre
strain values are spread considerably, owing to the superposition of a stress field
induced by the manufacturing procedure. Furthermore, the distribution exhibit a
certain degree of skewness. The specimen, then was subjected to an incrementally
increasing compressive load. Histograms of fibre strain values were obtained at
each level of applied compressive strain. The fibre strain distributions, obtained at
each increment of applied strain are given in Appendix C. Figure 12.58 depicts a
histogram of fibre strain values at an applied strain of -0.6% approximately. As it
can be seen from this Figure, the skewness of the fibre strain distribution has been
136
increased compared to that one of Figure 12.57, which is indicative of presence
of failure within the examined specimen volume. In Figure 12.59, the associated
Gaussian distributions are presented. By increasing the applied strain, the fibre
strain distribution shifts to lower values. In Figure 12.60, the mean fibre strain value
is plotted as a function of the applied strain, measures by means of the attached
strain gauges. As it can be observed, the mean value of the fibre strain distribution
follows, relatively closely, the 1:1 line up to an applied strain of about -0.4%. At this
applied strain level, the mean fibre strain is about -0.6% (due to the superposition
of the residual strains), which is equal to the fibre failure strain approximately.
Further increase of the applied strain does not result in an increase of the fibre
strains. More importantly, a slight decrease of the mean fibre strain value can be
observed. The width of the distributions, which is an index of the spread of the fibre
strain distribution (Figure 12.61, does not vary with the applied strain as it would
be expected. It is worth noting, however, that this may be an artefact, induced by
the Gaussian distribution functions used. More experiments need to be performed.
Additionally, instead of using normal distributions, other asymmetric distributions
have to be employed. Finally, the statistical method, presented in this Section, will
be applied in full composite materials.
12.3
Mapping of strains in adjacent fibres
Although the information, that can be extracted from the method reported
previously, is very useful, it is of great importance to examine the effect of a compressive fibre break in its adjacent fibers.
In the literature and in the case of tension, there exist many statistical models for
the failure process on unidirectional composite materials, which require knowledge
137
of the distribution for strength of the fibres and details of the load redistribution
in the vicinity of single and multiple fibre breaks. A common technique for examining these stress concentrations is the shear-lag approach, presented in a previous
Chapter. Hedgepeth Hedgepeth (1961) used this approach to construct a solution for stress concentrations in a two-dimensional composite with the fibre breaks
aligned transversely to the fibres direction. Hedgepeth and Van Dyke Hedgepeth
and Van Dyke (1967) extended this model to a three-dimensional array of fibres
with again aligned breaks. Since then, a lot of improved analyses have been proposed in the literature for the failure of fibre reinforced composite materials (see
Ref. Wagner and Eitan (1993); Sastry and Phoenix (1993); Steen and Valles (1998);
Ochiai et al. (1997); Landis et al. (2000)). It is worth noting, however, that there
are no extensions of these approaches in the case of compression, mainly due to
the complexity of the problem. A notable exception is the work of Garland et al
Garland et al. (submitted), who developed a model using a shear-lag based influence superposition technique. With this treatment, the stress concentrations in the
fibres and the matrix, produced by broken fibres whose failure is at an angle (shear
breaks), perpendicular to the loading (fibres) direction, could be evaluated. Some
of the drawbacks, however, of this approach are: (a) the assumption of linear elasticity for both the fibres and the matrix, and (b) the model does not really match
the experimental results in many cases. Thus, one of the purposes of the current
project, is to develop a similar approach, which will give more realistic predictions
of the compressive behaviour. For this reason, it is essential to study first experimentally the effect of a single or multiple compressive fibre breaks on the adjacent
intact fibres. In Figure 12.62, the strain profiles along a broken fibre and its first
adjacent (intact) fibre can be seen. It is interesting to observe that the fibre strain
138
of the adjacent fibre (intact) fibre follows very closely the strain profile of the broken
fibre. This might be an indication of the existence of stress concentration in the
intact fibre, generated by the broken fibre. This is a subject of on going work. In
Appendix C, some more strain profiles in adjacent fibres are given.
12.4
Failure mechanisms
Figures 12.63-12.66 show the typical mode of fibres failure in compression.
As it can be seen, the fibres fail in shear, similar to the case of single-fibre model
composites. These fibre shear breaks can cause the formation of damage nucleus,
which grows into a damage zone (kink band formation was finally observed in the
specimens) Narayanan and Schadler (1999b). However, more experiments need to
be performed in order to verify this assumption and to quantify the stress concentrations effects.
139
Figure 12.56: Geometry of multi-fibre specimens (all dimensions in mm).
140
residual fibre strains
Number of measurements (-)
40
30
20
10
0
-0.4
-0.3
-0.2
-0.1
0.0
Axial fibre strain (%)
Figure 12.57: Fibre strain histogram. Residual fibre strains (200 measurements).
40
applied strain 0.6%
Number of measurements (-)
35
30
25
20
15
10
5
0
-0.7
-0.6
-0.5
-0.4
-0.3
Axial fibre strain (%)
Figure 12.58: Fibre strain histogram. Externally applied strain -0.6% (200 measurements).
141
50
applied strain = 0%
applied strain = -0.6%
Gaussian distribution (-)
40
30
20
10
0
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Axial fibre strain (%)
Figure 12.59: Gaussian strain distributions. (200 measurements).
0.0
exper. values
theor. values
-0.1
Mean axial fibre strain (%)
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Applied strain (%)
Figure 12.60: Mean value of fibres strain (laser Raman measurements) as a function
of applied strain (strain gauge measurements).
142
Width of the Gaussian distribution (%)
0.5
0.4
0.3
0.2
0.1
0.0
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
Applied strain (%)
Figure 12.61: Width of the Gaussian distributions as a function of applied strain
(strain gauge measurements).
-0.1
broken fibre
first adjacent fibre
-0.2
fibre fracture plane
Axial fibre strain (%)
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
0
20
40
60
80
100
120
140
160
Position along the fibre direction (µm)
Figure 12.62: Fibre strain profiles. The spacing between the fibres is 3µm and the
applied strain -0.9%.
143
Figure 12.63: Optical micrograph of fibre fractures
144
Figure 12.64: Optical micrograph of fibre fractures
145
Figure 12.65: Optical micrograph of fibre fractures
146
Figure 12.66: Optical micrograph of fibre fractures
147
APPENDIX A
RAMAN SPECTROSCOPY
148
A.1
Raman spectroscopy
In the current project, Laser Raman Spectroscopy (LRS) is used as the main
experimental tool to study the stress transfer from the matrix to the fibres at a
microscopic level. Therefore, a brief review of the background and the principles of
LRS is given in this section.
A.1.1 Introduction
The prediction that radiation scattered from molecules can have a frequency
different from the incident frequency made by Smekal in 1923, was first observed
experimentally by Raman in 1928 Baranska et al. (1987). Since then Raman spectroscopy has had many ups and downs, and in chemical analysis it has been outstripped by infrared absorption and gas chromatography. However, it has been
reborn as Laser Raman Spectroscopy with the discovery of lasers as ideal source of
monochromatic radiation. Today the Laser Raman Spectroscopy is an important
method in the identification of molecules and their structure, the characterisation
of chemical reactions and the study of stress transfer in fibre-reinforced polymer
composites.
A.1.2 The Raman effect
When a beam of light is passed through a transparent material, a small amount
of the radiation energy is scattered, the scattering persisting even if all dust particles are rigorously excluded from the substance. If monochromatic radiation, or
radiation of a very narrow frequency band, is used, the scattered energy will consist
149
almost entirely of radiation of the incident frequency (the so-called Rayleigh scattering) but, in addition, certain discrete frequencies above and below that of the
incident beam will be scattered; it is this which is called Raman scattering.
A.1.3 Quantum theory of Raman scattering
The monochromatic radiation (laser light) of frequency ν consists of a stream
of particles (photons) having an energy of hνf where h is Planck’s constant. Photons
can be imagined to undergo collision with the molecules of the substance Banwell
and McCash (1994), if the collision is perfectly elastic, the photons will be scattered
unchanged. A detector placed to collect energy will thus receive photons of energy
hνf , i.e. radiation of frequency νf .
However, it also possible that energy is exchanged between the photon and the
molecule during the collision (inelastic collision). The molecule can gain or lose
energy only in accordance with the quanta laws, i.e its energy change must be the
difference in energy between two of its allowed states. In the other words, the energy
change must represent a change in the vibrational and/or rotational energy of the
molecule Banwell and McCash (1994). If the molecule gains energy, the photon will
be scattered with energy hνf -∆E/h. Conversely, if the molecule loses energy, the
scattered photon will be scattered with energy hνf +∆E/h
Radiation scattered with a frequency lower than that of the incident laser light
is referred to as Stokes’ radiation, while that at higher frequency is called antiStokes’radiation. We should note that Stoke’s radiation is generally more intense
than anti-Stokes’ radiation, however the both Stokes’s and anti-Stokes’ radiations
are extremely small compared to the Rayleigh radiation.
150
A.1.4 Classical theory of Raman scattering
When a molecule is put into a static electric field is distorted, the positively
charged nuclei being attracted towards the negative pole of the field and the electrons to the positive pole. This distortion of charge causes an induced electric dipole
moment to be set up in the molecule Banwell and McCash (1994). The size of the
induced dipole µdip , depends on the magnitude of the applied field, E, and on the
ease with which the molecule can be distorted. The induced dipole is given by:
µdip = αE
(A.39)
where α is the polarizability of the molecule. In general the polarizability of a
molecule is different for different directions (anisotropic polarizability).
When a sample of molecules is subjected to a beam of radiation of frequency νf the
electric field experienced by each molecule varies according to the equation:
E = E0 sin(2πνf t)
(A.40)
and thus the induced dipole also undergoes oscillations of frequency νf , from Equation A.39:
µdip = αE0 sin(2πνf t)
(A.41)
Such an oscillating dipole emits radiation of its own oscillation frequency (Rayleigh
scattering).
In addition, the molecule undergoes some internal motion such as vibration or rotation, which changes the polarizability periodically. For example, for a vibration
frequency of νvib the change of the polarizability can be given by:
151
α = α0 + βpol sin(2πνvib t)
(A.42)
where α0 is the equilibrium polarizability and βpol represents the rate of change of
polarizability with the vibration. Thus we have:
µdip = (α0 + βpol sin(2πνvib t))αE0 sin(2πνf t)
(A.43)
and thus we have:
µdip = α0 E0 sin(2πνf t) + 1over2βpol E0 (cos(2π(νf − νvib )t) − cos(2π(νf + νvib )t))
(A.44)
and thus the oscillating dipole has frequency components νf ± νvib as well as the
exciting frequency νf .
It is important to note that in order to be Raman active a molecular vibration or
rotation must cause change in a component of the molecular polarizability (βpol 6=
0)
A.2
Noise in Raman measurements
Noise in Raman measurements It is clear from the results presented above that
Raman measurements are quite noisy and the wavenumber shift (∆ν) obtained from
the experiments is actually the result of the convolution of many unknown functions,
and can be written in the form:
∆ν = ∆ν ∗ ∗ ∆µ1 ∗ ∆µ2
(A.45)
152
where ∆ν ∗ is the real wavenumber shift, ∆µ1 is a noise function related to the electrical and optical noise and ∆µ2 is an error function related to the numerical errors
introduced with the Lorentzian fitting procedure used to obtain the band profiles.
Furthermore, ∆ν ∗ depends on material inhomogeneties, i.e. the interface quality
is not the same along the fiber even in model single-fiber composites for a variety
of reasons such as voids, impurities etc. We believe that this is the main reason
for the existence of noise to the experimental data. Raman measurements taken
from the same point of selected fibers gave very little scatter compare to the scatter
along the fiber length. Similar results were reported by Galiotis Galiotis (1991).
Boll et al Boll et al. (1990) based on photoelastic experiments suggested that there
are might be macroscopic inhomogeneties on the fiber surfaces, i.e. nonuniform distribution of sizing. Figure A.1 shows the wavenumber shift distribution of Raman
measurements taken from a given fiber location. The experimental data have also
been fitted to a Gaussian distribution function, where the standard deviation found
is 0.45cm−1 or in terms of strain ∼ 0.02%.
We will show the difficulty and the complexity of noise removal from the experimental data using the FFT Method. To apply the FFT Method the data have to
be equispaced, thus interpolation splines have been constructed to connect the raw
data by creating sets of data sampled at an interval of 25µm. Although this procedure is quite abrupt for the middle of the fiber where measurements were taken at
an interval of 10µm, the effect on the overall analysis should not be critical since
any damage should appear at the fiber ends where the experimental sampling was
2µm. Figure A.2 shows the FFT spectra of the data depicted in Figure 6.22(a).
Although the Fourier transform should be expressed in wavenumber domain, we
preferred the frequency notation in order not to confuse it with the wavenumber
153
shift of the carbon Raman peak term. It is well known that noise gives high frequencies in the frequency domain, thus a low pass filter should be applied for noise
removal. On the other hand, any damage mechanism (represented by a change in
the stress profile along the fiber length) should also give a high frequency. That
is, there are two competing phenomena making noise removal a very challenging
task. In Figure A.3 the results for two different cut-off frequencies are presented.
It is obvious that a high cut-off frequency fails to remove noise sufficiently, whereas
a low cut-off frequency results in loss of information in the area of fiber ends or
breaks. The FFT Method appears to be very promising for Raman measurements
and Equation A.45 can be written in the frequency domain as:
∆N = ∆N ∗ ∆M1 ∆M2
(A.46)
where ∆M1 and ∆M2 could be evaluated and thus making possible ∆N ∗ to be
calculated.
A.3
Cubic spline fitting of Raman measurements
Splines are piecewise polynomial functions defined by a set of discrete points
(control points), used to approximate an (unknown) function given by discrete function values (experimental data). There exist a variety of spline functions to choose
from but only B-cubic splines will be examined here. Splines can be constructed
in any order of polynomials. However, the higher the order, the more wiggles and
overshoot must be expected in the resulting function. B-splines of order n are C n−1
continuous and since we want the interfacial shear stress to be a continuous function,
we need at least a cubic polynomial.
154
A Matlab program was made to fit the experimental data with B- cubic splines.
The program required as an input the number of cubic polynomial functions used
to fit the data. The knots introduced in this way were equispaced, then it was
possible to move the knots so that we achieve the best least-square approximation.
The most trivial part of the analysis is the initial number of polynomials used,
this is shown in Figure A.4, but it is more clear in Figure A.5 which is derived
by differentiation of the data of Figure A.4 (the error of a derivative of a function
is one order of magnitude higher). Each number in these Figures is referred to
the number of polynomials used. The most striking observation is that a different
number of polynomial used can lead to different conclusion for the state of stress,
i.e. by using 16 polynomials, we conclude that the interface in the left fiber end has
failed whereas by using a smaller number of knots, no interfacial failure is predicted.
Clearly, a small number of polynomials affects detrimentally the best fit, whereas a
large number of polynomials is affected by the presence of noise.
The question, which arises, is, which is the minimum necessary number of polynomials needs to sufficiently describe the stress transfer problem. We have fit the data
at every strain level using a range of polynomials from 2 up to 40 (depending on the
applied strain). Then by comparing the stress and ISS profiles of different applied
strain levels we were able to choose quite safely the number of knots. Simple but
physically sound criteria were used, i.e. as the applied strain increases the necessary
number of knots to describe the phenomenon should also increase (or at least be
equal to the number of knots used in the immediate previous strain step) since the
stress field becomes more complex. When at a distinct strain level an interfacial
failure is predicted then at a higher strain interfacial failure should also occur. This
155
analysis is quite similar in principle with a rigorous knot removal method of Schumaker and Stanley Schumaker and Stanley (1996), they developed an algorithm
for removing knots from an interpolating spline without perturbing the spline more
than a given tolerance.
24
Frequency count (-)
20
16
12
8
4
0
2764.0
2764.5
2765.0
2765.5
2766.0
2766.5
-1
Wavenumber shift (cm )
Figure A.1: Raman frequency shift distribution.
156
0.5
Amplitude (-)
0.4
0.3
0.2
0.1
0.0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Frequency (Hz)
Figure A.2: FFT spectra of Raman measurements (applied tensile strain, ǫ=1.1%).
1.0
0.9
0.8
Axial fiber strain (%)
0.7
0.6
0.5
0.4
0.3
0.2
exper. data
cutoff frequency:
0.02 Hz
0.002 Hz
0.1
0.0
-0.1
-0.2
0
200
400
600
800
1000
1200
1400
1600
1800
Position along the fiber (µm)
Figure A.3: Filtered Raman measurements (applied tensile strain, ǫ=1.1%).
157
0.50
number of splines:
0.25
Axial fiber stress (GPa)
0.00
exp.data
2
4
10
12
6
14
8
16
-0.25
-0.50
-0.75
-1.00
-1.25
-1.50
-1.75
-2.00
0
500
1000
1500
2000
2500
3000
Position along the fiber (µm)
Figure A.4: Cubic spline fitting (applied compressive strain, ǫ=-0.3%).
number of splines:
16
2
10
4
12
6
14
8
16
12
8
ISS (MPa)
4
0
-4
-8
-12
-16
-20
0
500
1000
1500
2000
2500
3000
Position along the fiber (µm)
Figure A.5: ISS distribution obtained from Figure A.4.
158
APPENDIX B
MATRIX CREEP EFFECTS ON THE FIBRE STRESS PROFILES
159
B.1
Introduction
It is well known that polymer-based composites exhibit viscoelastic properties,
which in the general case are non linear (see i.e. Refs. Brinson (1999); Papanicolaou
et al. (1999); Ivengar and Curtin (1997)). This is due to the inherent time dependent
nature of the mechanical properties of the matrix material.
B.2
Results
As it has been referred above, creep effects were observed during the com-
pression tests. This was mainly due to the experimental set-up used, which did
not impose an axial kinematic constraint in the microcomposites during loading.
Secondly, the selected matrix system behaves strongly as a viscoelastic solid. Here,
we will provide some experimental evidence of fiber stress profiles evolution with
time caused by matrix creep.
Matrix creep is depicted in Figure B.1 for a low applied strain (load). It can be
clearly seen that after ∼ 17h matrix strain remains constant (measurements from
the strain gauges attached to the matrix surface). In the case presented in Figure
B.2 we have started mapping the fiber from the left end towards to the right. In
the same Figure, the macroscopic matrix strain versus time is also depicted. The
vertical line corresponds to the time and distance along the fiber length where a
strain adjustment was done by relaxing the applied strain (load). Fiber strain
(stress) increases with increasing matrix strain is clearly seen. Moreover, the fiber
strain, far away from the end regions, is higher than the macroscopic matrix strain.
This effect supports the idea that local matrix shear creep strains are much greater
160
than the globally measured matrix creep strains (Beyerlein et al Beyerlein et al.
(1998)). However, uncertainties exist concerning the constitutive local shear matrix
creep behavior. Lagoudas et al Lagoudas et al. (1989) used a power-law creep
compliance model for the matrix, given by:
t
m
J(t) = JLT
[1 + ( )am ]
tc
(B.47)
m
m
where JLT
is the instantaneous elastic compliance in shear of the matrix (JLT
=
1/Gm
LT ), and tc and am are material constants. Mason et al Mason et al. (1992)
considered the shear-lag model of Lagoudas et al Lagoudas et al. (1989), but with the
more realistic assumption that the matrix creeps according to a nonlinear, powerlaw with memory model.
Although matrix creep effects on the fiber stress profiles have been studied analytically by many authors (see also Refs. Otani et al. (1991); Ivengar and Curtin
(1997); Beyerlein (2000)), there exist very little experimental information regarding
this issue. Miyake et al Miyake et al. (1998) measured the stress relaxation in broken fibers embedded in epoxy using Raman spectroscopy. In the case on an intact
interface, they found that relaxation in the broken fiber is very little and very slow
in comparison with that of the matrix strain. That is, matrix shear stresses around
the fiber break relaxed much less than the matrix global axial stress. However, in a
subsequent work, Ohno and Miyake Ohno and Miyake (1999) concluded that the interfacial shear stresses around a fiber break relax much more than the normal matrix
strain. In our compression tests we have observed clear matrix creep effects on the
fiber stress. Fiber breaks introduced by matrix creep are very well demonstrated in
Figure B.3. At time t=0, no fiber breaks were observed, while global matrix strain
was -0.43%. At a time between t=13h (ǫm =-0.52%) and t=17h (ǫm =-0.55%) five
161
fiber breaks occurred. Moreover, matrix relaxation time is much greater than at
lower applied load (see Figure B.1). On the other hand, in the present case the
far-field fiber strain is almost equal or smaller (for large time values) to the global
matrix strain in contrast to the previous case (see Figure B.2). However, this is due
to the non-linear mechanical behavior of the fiber themselves (see Figure 4.12).
-0.15
Global axial matrix strain (%)
-0.16
-0.17
-0.18
-0.19
-0.20
-0.21
-0.22
0
5
10
15
20
25
Time (hours)
Figure B.1: Matrix relaxation at a low applied strain (load).
162
0
1000
2000
3000
4000
5000
6000
0.0
-0.200
(4.5h)
-0.1
-0.2
Axial fiber strain (%)
-0.210
-0.3
-0.4
-0.215
-0.5
-0.220
-0.6
-0.225
-0.7
(7.5h)
(4.1h)
-0.8
-0.230
(9.7h)
(11h)
Global axial matrix strain (%)
-0.205
(12h)
-0.9
-0.235
0
1000
2000
3000
4000
5000
6000
Position along the fiber (µm)
Figure B.2: Fiber stress evolution with time and Global matrix creep strain.
0
1000
2000
3000
4000
5000
6000
0.4
-0.42
-0.46
0.0
-0.48
-0.50
-0.2
-0.52
-0.54
-0.4
-0.56
-0.6
-0.58
(17.8h)
-0.8
0
1000
2000
3000
-0.60
(22.7h)
(25h) (28h)
4000
5000
Global axial matrix strain (%)
Axial fiber strain (%)
-0.44
fiber breaks
appearance
0.2
-0.62
6000
Postion along the fiber (µm)
Figure B.3: Fiber stress evolution with time and Global matrix creep strain.
163
APPENDIX C
FIBRE STRAIN DISTRIBUTION IN CARBON FIBRE-EPOXY
COMPOSITES UNDER COMPRESSIVE STRESS FIELD
164
C.1
Fibre strain histograms
Figures C.1-C.5 depict the histograms of fibre strain values obtained from the
mapping of the specimen presented in Chapter 11, for the intermediate applied stain
levels (from -0.1% to -0.5%, respectively).
C.2
Strain distribution in adjacent fibres
In Figures C.6-str93c strain profiles in adjacent fibres are presented. Fig. C.6
shows strain perturbation areas at z=80µm and z=100-120µm (in accordance with
optical observations), but no safe conclusions can be made due to the presence of
noise in the experimental data. Similar conclusions can be made for the other strain
profiles. From these results, it is obvious that more detailed mapping of the fibres
need to be performed, together with careful optical observations.
165
50
Number of measurements (-)
applied strain -0.1%
40
30
20
10
0
-0.55
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
Axial fibre strain (%)
Figure C.1: Fibre strain histogram. Externally applied strain -0.1% (200 measurements).
166
Number of measurements (-)
50
applied strain -0.2%
40
30
20
10
0
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
Axial fibre strain (%)
Figure C.2: Fibre strain histogram. Externally applied strain -0.2% (200 measurements).
Number of measurements (-)
50
applied strain -0.3%
40
30
20
10
0
-0.60
-0.55
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
Axial fibre strain (%)
Figure C.3: Fibre strain histogram. Externally applied strain -0.3% (200 measurements).
167
40
applied strain 0.4%
Number of measurements (-)
35
30
25
20
15
10
5
0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
Axial fibre strain (%)
Figure C.4: Fibre strain histogram. Externally applied strain -0.4% (200 measurements).
applied strain 0.5%
Number of measurements (-)
50
40
30
20
10
0
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
Axial fibre strain (%)
Figure C.5: Fibre strain histogram. Externally applied strain -0.5% (200 measurements).
axial fibre strain (%)
168
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0
fibre 1
0
20
40
60
80
100
120
140
160
-0.5
-0.6
-0.7
-0.8
fibre 2
-0.9
0
20
40
60
80
100
120
140
160
-0.6
-0.7
-0.8
fibre 3
-0.9
0
20
40
60
80
100
120
140
160
Position along the fibre direction (µm)
Figure C.6: Fibre strain profiles. The spacings between the fibres are: 8µm (fibre
1-fibre 2) and 4µm (fibre 2-fibre 3). The applied strain -0.6%.
axial fibre strain (%)
-0.4
-0.5
-0.6
-0.7
-0.8
fibre 1
-0.4
-0.5
-0.6
-0.7
-0.8
0
-0.4
-0.5
-0.6
-0.7
-0.8
0
-0.4
-0.5
-0.6
-0.7
-0.8
0
20
40
60
80
100
120
140
160
fibre 2
20
40
60
80
100
120
140
160
fibre 3
20
40
60
80
100
120
140
160
fibre 4
0
20
40
60
80
100
120
140
160
Position along the fibre direction (µm)
Figure C.7: Fibre strain profiles. The spacings between the fibres are: 4.5µm (fibre
1-fibre 2), 3µm (fibre 2-fibre 3), and 3µm (fibre 3-fibre 4). The applied strain -0.78%.
169
-0.3
-0.4
-0.5
-0.6
fibre 1
axial fibre strain (%)
-0.7
-0.3
0
20
40
60
80
100
120
140
160
180
-0.4
-0.5
-0.6
fibre 2
-0.7
-0.3
0
20
40
60
80
100
120
140
160
180
-0.4
-0.5
-0.6
fibre 3
-0.7
0
20
40
60
80
100
120
140
160
180
Position along the fibre direction (µm)
Figure C.8: Fibre strain profiles. The spacings between the fibres are: 4µm (fibre
1-fibre 2) and 4µm (fibre 2-fibre 3). The applied strain -0.9%.
-0.3
-0.4
-0.5
fibre 1
axial fibre strain (%)
-0.6
-0.3
0
20
40
60
80
100
120
-0.4
-0.5
fibre 2
-0.6
-0.3
0
20
40
60
80
100
120
-0.4
-0.5
fibre 3
-0.6
0
20
40
60
80
100
120
Position along the fibre direction (µm)
Figure C.9: Fibre strain profiles. The spacings between the fibres are: 4µm (fibre
1-fibre 2) and 4µm (fibre 2-fibre 3). The applied strain -0.9%.
170
APPENDIX D
NOMECLATURE
171
Table D.1: Nomeclature
*
*
Symbol
Definition
ǫ
strain
wk
kink band width
β
kink band angle
φ
fibre rotation angle
φ1
load misalignment angle
ǫT
transverse strain
γ
shear strain
φ
initial fibre misalignment angle
Tg
glass transition temperature
ELf
axial fibre Young’s modulus
ELm
axial matrix Young’s modulus
uf
fibre volume fraction
um
matrix volume fraction
σc
composite compresssive strength
Gm
LT
matrix shear modulus
E¯T
composite transverse modulus
Ḡ
composite effective shear modulus
σcf
fibre compresssive strength
σym
matrix yield strength
ǫm
c
matrix failure strain
σcm
matrix failure strength
172
Table D.1: (continued)
*
*
Symbol
Definition
τyc
composite shear strength
vm
matrix poisson’s ratio
Ω
standard deviation of the normal distribution
of the fibre misalignment angles
χ
dimensionless parameter
σ0 , τ0
applied stresses
rf
fibre radius
df
fibre diameter
Rcox
Cox radius
gf
fibre critical energy release rate corresponding to
the fibres flexural toughness
P
critical buckling load
Pe
euler buckling load
Ps
shear buckling load
σe
euler buckling stress
σs
shear buckling stress
pi
constants
∆ν
wavenumber shift of the Raman peak
f
< σzz
>
average axial stress in the fibre
< ǫfzz >
average axial strain in the fibre
f
< σzz,∞>
average axial stress in the corresponding infinitely long
173
Table D.1: (continued)
*
*
Symbol
Definition
fibre embedded in an infinitely long matrix
βcox
Cox’s shear-lag parameter
βnairn
Nairn’s shear-lag parameter
ETf
transverse fibre Young’s modulus
GfLT
fibre shear modulus
f
νLT
fibre poisson ration in the LT direction
νTf T
fibre transverse poisson ration
αLf
fibre thermal expansion coefficient in the fibre direction
αTf
transverse fibre thermal expansion coefficient
τrz
interfacial shear stress
z
fibre direction
r
direction transverse to the fibre
w
axial displacement in the fibre
< σrr >
transverse stresses (r direction)
< σθθ >
transverse stresses (θ direction)
c
constant
d
constant
ki
constants
fi (z)
functions of z
µ
friction coefficient
ν1 , ν2
parameters depend on the matrix and fibre Poisson’s
174
Table D.1: (continued)
*
*
Symbol
Definition
ratio respectively
σr
residual stresses due to cure shrinkage of the matrix
τf ri
interfacial frictional stress
τyint
interphase yield stress
rm
equivalent to Rcox
g
function of r
lf
fibre length
Ds
interfacial parameter in z direction
Dn
interfacial parameter in r direction
Lt
ineffective length
fc
constant equal to the magnitude of the stress experimentally
observed at the fibre end examined
zy
end of the yielded zone and
z0
fibre break coordinate
ld
length of the debonded zone
ly
length of the yielded zone
′
length of imaginary fibres
l
θ
bending angle
νf
frequency
h
Planck’s constant
µdip
induced dipole
175
Table D.1: (continued)
*
*
Symbol
Definition
E
magnitude of electric field
α
polarizability of a molecule
βpol
rate of change of polarizability with
the vibration
∆ν ∗
’real’ wavenumber shift
∆µ1
noise function related to the electrical and
optical noise
∆µ2
error function related to the numerical errors introduced
with the Lorentzian fitting procedure used to obtain the
band profiles
m
JLT
instantaneous elastic compliance in shear of the matrix
tc
material constant
am
material constant
176
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