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THE APPLICATION OF DEBOND LENGTH MEASUREMENTS TOEXAMINE THE ACCURACY OF COMPOSITE INTERFACE
PROPERTIES DERIVED FROM FIBER PUSHOUT TESTING
BY
VERNON THOMAS BECHEL
B.S., University of South Florida, 1991M.S., University of South Florida, 1993
THESIS
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Theoretical and Applied Mechanics
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 1997
Urbana, Illinois
iii
THE APPLICATION OF DEBOND LENGTH MEASUREMENTS TOEXAMINE THE ACCURACY OF COMPOSITE INTERFACE
PROPERTIES DERIVED FROM FIBER PUSHOUT TESTING
Vernon Thomas Bechel, Ph.D.Department of Theoretical and Applied MechanicsUniversity of Illinois at Urbana-Champaign, 1997
Nancy R. Sottos, Advisor
ABSTRACT
The interface failure sequence was observed during fiber pushout tests on several
model composites. Composites with varying fiber-to-matrix moduli ratio (Ef/Em), sample
thickness, interfacial bond strength, and processing residual stresses were tested to
determine which composites would debond from the top and which from the bottom. The
present pushout experiments combined with previous work in the literature indicate that
only composites with an Ef/Em ratio less than 3 and with negligible to moderate residual
stresses can be expected to debond from the top. The debond length as a function of force
and displacement was also measured in a polariscope for two of the model composites—
steel/epoxy and polyester/epoxy. The pushout data from a polyester/epoxy system that
debonded from the top was fit to a shear lag solution to obtain the fiber–matrix interfacial
toughness ( GIIc ). The resulting interfacial toughness was then used to check the predicted
debond length as a function of pushout force. The debond length calculated from the shear
lag model was less than the measured debond length by a nearly constant 1.5 fiber radii,
which may correspond to the thickness of the surface effects region for polyester/epoxy.
A procedure was then developed to determine accurately the debond length as a
function of force based on the model composite pushout data. A constant coefficient of
friction was calculated from the frictional portion of the experimental curve both with shear
lag theory and with a finite element analysis. Next, a relatively expeditious routine
involving iterative finite element analysis was applied to the progressive debonding portion
iv
of the pushout curve to compute debond length. The finite element simulation included
thermal and/or chemical shrinkage loads as well as the boundary conditions corresponding
to the exact probe and sample support dimensions. The resulting debond lengths
corresponded to within 10% to 200% of the measured debond lengths (depending on the
method used to calculate the coefficient of friction) for both top and bottom debonds.
Fracture toughness was also determined with the finite element method by computing
change in stored energy when incrementing the interface crack length by 0.1% of the total
crack length and subtracting the change in the energy dissipated by friction.
Finally, an experiment was designed and constructed to perform fiber pushout tests at
temperatures ranging from room temperature to 800˚C and in an environment of less than
one part per million oxygen to avoid oxidation of the interface at high temperatures. A
small DC motor and gear box were used to drive a flat faced diamond punch. The
composite sample and punch were aligned by observing the sample with a long distance
microscope through a recessed window in the vacuum chamber wall and were heated by an
infrared heater located outside another window on the vacuum chamber. Preliminary
pushout tests on an SiC/Ti-15-3 composite and both a pristine and a transversely fatigued
SiC/Ti-6-4 composite were conducted in atmospheric conditions to assess the capabilities
of this apparatus. A load drop was absent at total debond during room temperature pushout
testing of the fatigued composite, and a negligible force was required to slide the SiC fibers
at 400˚C, indicating that the fiber–matrix bond was broken by the fatigue loading. The
configuration of an experiment necessary to observe progressive debonding in metal matrix
composites is discussed.
v
To
Jill, Meagan, and Charlie
vi
ACKNOWLEDGMENTS
I would like to acknowledge the financial support of the Office of Naval Research
(under contract monitor R. Barsoum) and the Air Force Office of Scientific Research
(Senior Knight Program). The composites provided by 3M (Rob Kieshke and Herve
Deve), Wright Laboratory (Bill Kralic), and Textron Specialty Materials (Monte Treasure)
were critical to the success of this project. The time spent by fellow graduate student,
Pranav Shrotriya, measuring the coefficient of thermal expansion of various epoxies is also
appreciated.
I would like to thank Dr. Nicholas. J. Pagano from Wright Laboratory, Dr.
Gyaneshwar P. Tandon of Adtech Systems Research, and Professors Thomas J. Mackin,
K. Jimmy Hsia, James W. Phillips, and Philippe H. Guebelle from the University of
Illinois at Urbana–Champaign. They contributed many ideas to the theoretical and
experimental aspects of this project, and their efforts and enthusiasm are greatly
appreciated. Dr. Jeff I. Eldridge of NASA Lewis provided insight on the details of setting
up the high temperature fiber pushout apparatus. The discussions with Dr. Pochiraju V.
Kishore of Stevens Institute of Technology about the use of the finite element code
(ABAQUS) were also very helpful.
I also thank Dr. Nancy R. Sottos, my thesis supervisor, who not only guided me
through the technical aspects of this research, but in the process taught me about the
importance of patience, thoroughness, and creativity in research. Her example will guide
me in future ventures.
Finally, I would like to thank my wife and two children. Without their support and
encouragement the pursuit of this goal would not have been such a pleasure.
vii
TABLE OF CONTENTS
LIST OF TABLES ...........................................................................................................ix
LIST OF FIGURES.........................................................................................................x
LIST OF SYMBOLS .......................................................................................................xv
1. INTRODUCTION.....................................................................................................11.1 Introduction...................................................................................................11.2 Interface strength tests...................................................................................21.3 Complications inherent in the fiber pushout test............................................61.4 Project overview.............................................................................................8
2. DEBOND LENGTH MEASUREMENTS IN MODEL COMPOSITES...................112.1 Introduction...................................................................................................112.2 Fabrication of model composites...................................................................122.3 Experimental apparatus..................................................................................142.4 Measurement of debond length .....................................................................162.5 Top versus bottom debond ............................................................................202.6 Comparison with a shear lag solution............................................................25
2.6.1 Shear lag theory..............................................................................252.6.2 Determination of residual stresses..................................................292.6.3 Comparison of measured and shear lag theory debond lengths .....30
2.7 Additional experimental observations ............................................................342.8 Conclusions...................................................................................................352.9 Future work...................................................................................................36
3. FINITE ELEMENT DEBOND LENGTH PREDICTIONS .......................................373.1 Introduction...................................................................................................373.2 Finite element model......................................................................................403.3 Top debond—polyester/epoxy ......................................................................47
3.3.1 Modeling procedure .......................................................................503.3.2 Boundary conditions ......................................................................543.3.3 Debond length................................................................................553.3.4 Coefficient of friction .....................................................................653.3.5 Fracture toughness .........................................................................69
3.4 Bottom debond—steel/epoxy ........................................................................733.4.1 Modeling procedure .......................................................................743.4.2 Boundary conditions ......................................................................773.4.3 Results............................................................................................773.4.4 Importance of sample preparation...................................................81
3.5 Interface failure due to cutting .......................................................................923.6 Discussion.....................................................................................................953.7 Future work...................................................................................................98
4. HIGH TEMPERATURE FIBER PUSHOUT TESTS.................................................1004.1 Importance of interface strength versus temperature......................................1004.2 High temperature tests...................................................................................102
4.2.1 Sample preparation.........................................................................1034.2.2 Apparatus .......................................................................................104
4.3 SiC/Ti pushout tests ......................................................................................106
viii
4.3.1 Pristine SiC/Ti-6-4 .........................................................................1064.3.2 Fatigued SiC/Ti-6-4........................................................................111
4.4 Progressive debonding in SiC/Ti...................................................................1154.5 Discussion and future work...........................................................................128
5. CONCLUSIONS.........................................................................................................1295.1 Debond length measurements........................................................................1295.2 Finite element solution...................................................................................1305.3 Processing and fabrication.............................................................................1325.4 SiC/Ti pushout tests ......................................................................................133
APPENDIX A. MATRIX SHRINKAGE MEASUREMENT ........................................134
BIBLIOGRAPHY ............................................................................................................140
VITA.................................................................................................................................147
ix
LIST OF TABLES
Table 2.1. Model composite fiber-to-matrix moduli ratios and processing induced
residual strains..........................................................................................13
Table 3.1. Boundary and continuity conditions for matrix shrinkage of the mesh
with a fully bonded interface. (schematically shown in Figure 3.7a). .…..56
Table 3.2. Boundary and continuity conditions for matrix shrinkage of the mesh
with a top debond of length ld . (schematically shown in Figure 3.7b,
step 1). .......................................................................... ...........................57
Table 3.3. Boundary and continuity conditions for fiber pushout of the mesh
with a top debond of length ld . (schematically shown in Figure 3.7b,
step 2). .......................................................................... ...........................58
Table 3.4. Boundary and continuity conditions for differential shrinkage during
cool down after processing for the mesh with initial debonds of
length li1 (top) and li2 (bottom) produced by cutting and/or
residual stresses. (schematically shown in Figure 3.15a). ................ ........78
Table 3.5. Boundary and continuity conditions for differential shrinkage during
cool down after processing for the mesh with initial top debond of
length li1 and bottom debond length of ld (schematically shown in
Figure 3.15b step 1). ................................................................................79
Table 3.6. Boundary and continuity conditions for fiber pushout of the mesh
with initial top debond of length li1 and bottom debond length of ld
(schematically shown in Figure 3.15b, step 2). ............................ ............80
x
LIST OF FIGURES
Figure 1.1. Schematic of standard interfacial strength tests. ............................ ...........3
Figure 1.2. The fiber pushout test: (a) schematic of the fiber pushout test,
(b) schematic of a typical fiber pushout load–displacement curve. ...... .....5
Figure 1.3. Schematic of the interfacial shear stress near the top and bottom of a
sample for a pushout load, a thermal load due to processing, and
a combined pushout and thermal load. ...................................... ...............7
Figure 2.1. Schematic of the micromechanical test apparatus used to perform
pushout tests on model composites. .........................................................15
Figure 2.2. Pushout apparatus positioned in the circular polariscope. ................ ........15
Figure 2.3. Photoelastic images acquired during a steel/epoxy pushout test. ..............17
Figure 2.4. Force and debond length versus displacement curve from a steel/epoxy
pushout test. Debond was from the bottom. Fiber diameter = 1.65 mm,
sample thickness = 13 mm, support hole diameter = 2.05 mm, and
punch diameter = 1.4 mm. .................................................... ...................18
Figure 2.5. Photoelastic images acquired during a polyester epoxy pushout test. ... ...19
Figure 2.6. Force and debond length versus displacement curves from a
polyester/epoxy pushout test. Debond was from the top. Fiber
diameter = 1.9 mm, sample thickness = 5.3 mm, support hole
diameter = 2.05 mm, and punch diameter = 1.7 mm. ...................……… 21
Figure 2.7. Map of top and bottom fiber debonds as a function of fiber-to-matrix
stiffness ratio and residual thermal strain—independent of sample
thickness and interface strength. ............................................. .................22
Figure 2.8. Axial force and interfacial shear stress on a fiber element assumed by
shear lag theory. .......................................................................................27
Figure 2.9. Curve fit (Eq. (2.13)) of measured force–displacement data for
xi
polyester/epoxy. .......................................................................................31
Figure 2.10. Comparison of shear lag prediction and experimental measurement of ....
debond length. P* = 80.38 N, Pr = -45.83 N, µ = 0.52, and
GIIc = 389 J/m2. ............................................................... .......................33
Figure 3.1. Finite element mesh constructed to simulate processing and pushout
loads. ............................................................................ ...........................41
Figure 3.2. Schematic of the coordinate axes and relevant dimensions. ......................43
Figure 3.3. Finite element mesh over a region including the top of the interface. .... ...44
Figure 3.4. Finite element mesh over a region surrounding the debond tip. .......... .....45
Figure 3.5. Radial and shear stress along the interface for a typical
polyester/epoxy problem. .........................................................................48
Figure 3.6. Variation of interfacial stresses near the crack tip in a typical problem. ....49
Figure 3.7. Schematic of the modeling procedure for polyester/epoxy: (a) for
the first finite element run, the complete interface is bonded, and
(b) for the second finite element run, a debond is added at the top
of the interface. ................................................................. .......................52
Figure 3.8. Relative displacement at the top of the fiber for each phase of the
polyester/epoxy finite element analysis: (a) unloaded, (b) actual
deformation from matrix shrinkage, (c) matrix shrinkage with top
debond, (d) displacement from pushout test added. ....................... ..........53
Figure 3.9. Coefficient of friction versus force for fully slipping problem
computed by the LH&KP shear lag theory, finite element
analysis, and Pagano and Tandon’s model. ................................ .............60
Figure 3.10. Comparison of measured, shear lag, and finite element calculated
debond length as a function of force ( µ = 0.52, µ = 0.75). ....................61
Figure 3.11. Force–displacement curve from polyester/epoxy sample 1 and
predicted loads for various coefficients of friction. ........................ ..........66
xii
Figure 3.12. Force–displacement curve from polyester/epoxy sample 2 and
predicted loads for two coefficients of friction. ............................ ............67
Figure 3.13. Force–displacement curve from polyester/epoxy sample 3 and
predicted loads for two coefficients of friction. ............................ ............68
Figure 3.14. Fracture toughness versus debond length from shear lag theory
and finite element analysis. ................................................... ...................72
Figure 3.15. Schematic of finite element analysis boundary conditions for
steel/epoxy: (a) for the first finite element run, only the initial
debonds are present, (b) for the second finite element run, a
debond is added at the bottom of the interface. ............................. ...........75
Figure 3.16. Relative displacement at the top of the fiber for each phase of the
steel/epoxy finite element analysis: (a) unloaded, (b) actual
deformation from thermal shrinkage, (c) thermal shrinkage with bottom
debond added, (d) displacement from pushout test added. ............... ........76
Figure 3.17. Pushout curve from a steel/epoxy sample cut far from the ends of the raw
sample. Curve separation matches point when debond starts to
grow, and after initial curve separation the sample continues to
become more compliant as the debond grows. ............................. ............82
Figure 3.18. A comparison of the measured and finite element predicted debond
lengths for the steel/epoxy sample whose pushout curve is
shown in Figure 3.17. ........................................................ ......................83
Figure 3.19. Schematic of interface bonding for steel/epoxy as cool down
progresses during processing. ............................................... ..................84
Figure 3.20. The fiber extension measurement: (a) schematic of two samples with
different debond lengths, b) schematic of experiment. .................... .........87
Figure 3.21. Steel/epoxy pushout curve from sample cut from section B and C.
Debond grows 1.2 mm before the force–displacement curve
xiii
becomes nonlinear. ............................................................ ......................89
Figure 3.22. Pushout curve from a steel/epoxy sample cut from section B and C.
After initial nonlinearity, the slope of the force–displacement curve
remains the same over a significant additional displacement. ............. ......90
Figure 3.23. Top image shows a relatively long sample of steel/epoxy composite at
room temperature with photoelastic fringes near fiber ends. The bottom
image is of a sample cut from the center (section C) of the raw sample.
The stresses redistribute and small debonds form at the fiber ends. .........93
Figure 3.24. Steel fiber (200 µm diameter) in epoxy. After cutting, large debonds
are present at the top and bottom of fiber. .................................. ..............94
Figure 4.1. Schematic of the high temperature fiber pushout experiment. ............ ......104
Figure 4.2. Punch and top of a pushed out fiber in an SiC/Ti-15-3 composite. ..........107
Figure 4.3. Punch and bottom surface of an SiC/Ti-6-4 composite with a single
fiber pushed out. ............................................................... .......................108
Figure 4.4. Force–displacement curve for pristine SiC/Ti-6-4 tested at room
temperature. .................................................................... .........................109
Figure 4.5. Force–displacement curve for pristine SiC/Ti-6-4 tested at 400˚C. ...... ....110
Figure 4.6. Pushout curves obtained by Eldridge and Ebihara (1994) at various
temperatures for SiC/Ti-15-3. ................................................ 112
Figure 4.7. Two pushout tests on fatigued SiC/Ti-6-4 at room temperature. ......... .....113
Figure 4.8. A pushout test on fatigued SiC/Ti-6-4 at 400˚C. ........................... ...........114
Figure 4.9. Three fibers in an SiC/Ti-6-4 composite. Fiber A was not pushout
tested. Fiber B was pushed out and back at room temperature.
Fiber C was pushed out and back at 400˚C. ................................ .............117
Figure 4.10. Pushout curve with machine compliance removed for pristine
SiC/Ti-6-4 tested at 400˚C. ................................................... ...................119
Figure 4.11. Measurement of machine compliance of high temperature apparatus. ......120
xiv
Figure 4.12. Modified high temperature apparatus: (a) schematic, (b) fixture to
connect the stepper motor to the load cell. ................................................122
Figure 4.13. Modified high temperature apparatus: (a) schematic showing sample
transport required to place the sample and sample support into the
chamber, (b) top view of sample and sample support resting on the
sample transport. ......................................................................................123
Figure 4.14. Cross-section of a fiber in pristine SiC/Ti-6-4. ............................ ............125
Figure 4.15. Cross-section of a fiber in fatigued SiC/Ti-6-4. ............................ ...........127
Figure A.1. Geometry for the derivation of the relation between fringe order and
average interfacial radial stress. .............................................. ..................136
Figure A.2. Photoelastic fringe patterns surrounding the fiber in a steel/epoxy
pushout sample. ................................................................ .......................137
Figure A-3. Thickness average of radial stress in matrix as a function of distance
from the fiber center in a 7.8 mm thick polyester/epoxy fiber pushout
sample as calculated from the photoelastic fringe pattern. ................ ........139
xv
LIST OF SYMBOLS
Ef fiber Young's modulus
Em matrix Young's modulus
ν f fiber Poisson’s ratio
νm matrix Poisson’s ratio
GIIc mode 2 critical strain energy release rate
ld interfacial debond length
σN interfacial radial stress away from the fiber ends due to processing
Pr fiber axial force away from the fiber ends due to processing
P* axial fiber tensile force necessary to open the debonded portion of the
interface during a fiber pullout test
F force applied to the punch
d shear lag displacement, experimentally measured displacement
d1 a representative measured displacement minus machine compliance and
alignment in the progressive debonding section of the pushout curve
F1 the force that corresponds to d1
dt1 displacement of fiber top face from chemical or thermal shrinkage when no
debond due to a pushout load is present
dt2 displacement of fiber top face from chemical or thermal shrinkage when a
debond due to a pushout load is present
t pushout sample thickness
µ interfacial coefficient of friction
r f fiber radius
ro matrix radius
rs sample support hole radius
rp punch radius
xvi
KII mode II stress intensity factor
r radial coordinate
z axial coordinate
σrrf radial stress in the fiber
σrzf shear stress in the rz plane in the fiber
σ zzf axial stress in the fiber
σrrm radial stress in the matrix
σrzm shear stress in the rz plane in the matrix
σ zzm radial stress in the matrix
urf radial displacement in the fiber
uzf axial displacement in the fiber
urm radial displacement in the matrix
uzm axial displacement in the matrix
[urm ]1(r,0) radial displacement of the bottom of the matrix from chemical or thermal
shrinkage when no debond due to a pushout load is present
[uzm ]1(r,0) axial displacement of the bottom of the matrix from chemical or thermal
shrinkage when no debond due to a pushout load is present
li1 initial top debond after cutting
li2 initial bottom debond after cutting
RBSN reaction bonded silicon nitride
SFCL single fiber critical length test
LH&KP fiber pushout shear lag solution for isotropic fiber and matrix by Liang
and Hutchinson (1993) and Kerans and Parthasarathy (1991)
MMC metal matrix composite
U stored strain energy
U f frictional energy dissipated
1
1. INTRODUCTION
1.1 Designing composites for damage tolerance
The interface strength and friction coefficient, along with constituent elastic properties,
volume fraction, and residual stresses, determine the performance of a continuous fiber-
reinforced composite. Models developed to predict the uniaxial stress–strain curve (Daniel,
1993)* and the fracture toughness (Bao and Song, 1993) for a composite material require all
of the above information. Daniel showed that the uniaxial stress-strain curve for an
SiC/calcium aluminosilicate composite loaded along the fibers has two large changes in
slope due to matrix cracking and partial fiber debonding. Analytical models were proposed
to correlate the failure mechanisms with the measured monotonic load–strain curve. Bao
and Song showed that mode I fracture toughness for a composite is explicitly dependent on
the size of the region of intact fibers (bridging zone length) ahead of a crack tip present in a
composite under tensile loading parallel to the fibers. The bridging zone length was in turn
shown to be dependent on not only the fiber strength but also the fiber–matrix interface’s
capacity to transfer shear stress before and after debonding. The fiber and matrix elastic
constants as well as the sample geometry are usually known for a composite, and the
residual stresses away from the fiber ends can be calculated. Since the interface properties
are not known for most fiber–matrix–coating combinations as a function of processing
conditions, an experiment must be conducted to obtain the interface strength and friction
characteristics. Ideally, a simple test would be conducted on a composite to find the
fiber–matrix interfacial properties to be used as input for the models that are available to
calculate composite toughness and extension under uniaxial tension. The composite’s
response to loading could then be predicted for an arbitrary volume fraction.
* References are listed alphabetically by author, beginning on page 140.
2
1.2 Interface strength tests
Previous experiments for determining interfacial shear strength in composites involve
pushing or pulling on one fiber, several fibers, or a composite section in the direction
parallel to the fibers. A schematic of several of the better known of these experiments is
shown in Figure 1.1. In the single fiber critical length test (SFCL), a composite section is
strained axially until the embedded fiber breaks into equal lengths (Drzal, Rich, Camping,
and Park, 1980). The length of each broken fiber section is related to the interfacial shear
stress developed while the composite is strained. The matrix cracking test, which is used
when the matrix cracks under less strain than the fiber, as in many ceramic matrix
composites, is the inverse of the SFCL test (Aveston, Cooper, and Kelly, 1971). A
composite is strained axially and the distance between matrix cracks is related to interface
strength. Fiber pullout consists of pulling on one or both ends of a fiber embedded in a
matrix (Chou, Barsoum, and Koczak, 1991; Marshall, Shaw, and Morris, 1992). Load as a
function of displacement is required to calculate interface properties. Another form of fiber
pullout is the microdrop test in which a drop of the matrix material is cured around the fiber
(Miller, Muri, and Rebenfield, 1987). The inverse of the fiber pullout test is the fiber pushin
test (Marshall, 1987; Majumbdar, 1994). As shown in Figure 1.1, one or more fibers are
pushed into the matrix of a composite by a punch or a plate. The load–displacement curve
is again the information sought. Finally, the slice compression test has been developed
recently (Hseuh, 1994). A composite section is compressed axially between a rigid surface
and a relatively soft surface, and the depth that the fibers are forced into the softer material is
measured and related to interface strength.
One of the most popular tests to find interface properties is the fiber pushout test, which
consists of pushing a single fiber out of a thin slice of composite material while measuring
applied force and pushout tool displacement (Laughner, 1988; Netravali, Stone, Ruoff, and
Topoleski, 1989; Brun and Singh, 1988; Warren, Mackin, and Evans, 1992). Figure 1.2a
shows a schematic of the pushout test. The popularity of the fiber pushout experiment is
3
SFCL Matrix Cracking Pullout
Pullout Microdrop Pushin
Pushin Slice Compression
Figure 1.1 Schematic of standard interfacial strength tests.
4
derived from the ease of sample preparation, the wide range of composites to which the test
can be applied, and interface properties that can be computed from the test data.
A composite is fabricated exactly as it would be for service, and a slice is sectioned for
use as a pushout test specimen. Since multiple fibers may be present in a specimen, a
particular fiber is chosen and aligned over a hole so the bottom surface of the fiber is
traction free. The force–displacement curve recorded during the test is later related to
interface mode II toughness and the coefficient of friction between the fiber and matrix by
fitting it to one of the solutions available in the literature. A transparent matrix is not
required as in the SFCL test, a matrix with a lower failure strain than the fiber’s is not
required as in the matrix cracking experiment, and the problems that arise in gripping the
fiber when conducting the fiber pullout test are circumvented. Unlike the pushin and slice
compression tests, the fiber pushout test can be used to calculate the interfacial coefficient of
friction.
The typical profile of the experimental curve from a pushout experiment is plotted
schematically in Figure 1.2b. The force–displacement curve can be divided into three
distinct sections. In section I it is thought that the interface is completely bonded, while in
section II an interface crack is assumed to be growing. Finally, section III is believed to
correspond to a completely debonded interface. In Section III the fiber slides within the
matrix against the frictional force generated in the fiber–matrix interface.
Approximate solutions to the fiber pushout problem have been developed by Gao
(1988), Marshall and Oliver (1990), Liang and Hutchinson (1993), Kerans and
Parthasarathy (1991), and Hsueh (1990). The solutions by Liang and Hutchinson and
Kerans and Parthasarathy, subsequently referred to as the LH&KP solution, are identical
for a composite with an isotropic fiber and matrix and represent the most advanced solution
containing the shear lag assumption. No surface effects, uniform residual stresses in the
fiber and matrix with respect to axial position, and a critical compressive fiber axial stress
5
Displacement, d
I II III
F
A
B
Punch
Composite
Support
Fiber
Support
(a)
(b)
Loa
d, F
Figure 1.2 The fiber pushout test: (a) schematic of the fiber pushout test,
(b) schematic of a typical fiber pushout load–displacement curve.
6
as the criterion for debond are assumed. The LH&KP solution applies only to composites
that initially debond from the top face of the sample (punch side). The debond must
continue to grow along the interface toward the bottom without a bottom debond appearing
at any point during the fiber pushout test.
1.3 Complications inherent in the fiber pushout test
The fiber pushout test is not free from difficulties. Koss, Hellman, and Kallas (1993)
and Eldridge (1995) discovered that in selected metal matrix composites, the interface
initially debonded from the bottom during the pushout test. The location of initial
debonding was determined by interrupting the pushout test before the peak load and
examining the top and bottom surface. The superposition of thermal residual stresses due
to processing and mechanical pushout stresses produces a bottom debond for some
combinations of fiber and matrix elastic properties, interface strength, and sample height.
As shown schematically in Figure 1.2, the residual interfacial shear stress has the same sign
at the bottom face of the sample as the interfacial shear stress produced by a pushout load
and the opposite sign at the top of the sample. When the residual stresses are large enough,
the combined effect can lead to a bottom debond. Although the experimental
force–displacement curves from a pushout test on a system that initially debonds from the
top and from one that initially debonds from the bottom may have the same features
(compare Figures 2.4 and 2.6), the shear lag solutions available to date do not apply to the
bottom debonding composite.
In addition to interface strength and friction coefficient, the roughness of the fiber
surface, the differential shrinkage between the fiber and matrix due to cool down from
processing temperature, and the chemical shrinkage of the matrix during processing may
not be known accurately prior to the pushout test. These factors often significantly
influence the force–displacement curve and are sometimes derived from the pushout
experiment. Evidence that fiber surface roughness increases the radial compressive stress
7
F(τA )mech > (τB )mech
(τA )mech
(τB )mech
−(τA )th = (τB )th
(τA )th
(τB )th
∆Τ
F + ∆Τ
(τA )mech + (τA )th
(τB )mech + (τB )th
(τA )mech + (τA )th < (τB )mech + (τB )th
+
==>
Figure 1.3 Schematic of the interfacial shear stress near the top and bottom of a sample
for a pushout load, a thermal load due to processing, and a combined
pushout and thermal load.
8
on the fiber once the uneven fiber surface slides with respect to the matrix was found by
Jero and Kerans (1990) when pushing fibers back to their original position in a composite.
Also, Cordes and Daniel (1995) and Mackin, Yang, and Warren (1992) showed that the
amount of interface wear during the frictional pushout portion of the pushout test can cause
difficulties when modeling the coefficient of friction and the radial stress from processing
as constants.
Even when a model is developed that includes all of the above variables, several variables
may be fit to one curve. Fitting more than one property to one portion of the curve shown
in Figure 1.2b—although sometimes unavoidable—may produce inaccurate results from a
theory that is formulated correctly. In the pushout experiments reported in this dissertation,
only the interface strength is fit to the progressive debonding portion of the pushout curve
(section II), while the coefficient of friction is fit to the maximum load in the frictional part
of the pushout curve so that the effects of wear are minimized. The matrix compressive
radial stress away from the fiber ends due to processing is measured photoelastically before
progressive debonding occurs and after total debond so that the radial stress at the interface
due to fiber surface roughness is accounted for.
1.4 Project overview
The results of fiber pushout experiments were originally used to compare composite
interface strengths qualitatively by dividing the maximum load reached before total debond
by the surface area of the embedded fiber. With the advent of approximate solutions to the
fiber pushout problem, numerical values are now being reported for the interface coefficient
of friction and the interface strength in terms of a critical shear stress or fracture toughness.
The reported values from one author often do not correspond to the values reported by
another author for the same composite. Differences in interface properties determined by
testing identical composites with more than one of the strength tests from Section 1.2 also
appear in the literature (Herrara-Franco and Drzal, 1992). These inconsistencies illustrate
9
the need for a closer look at micromechanical interface strength tests and the corresponding
theories that are used to compute interface properties in composites.
This dissertation investigates several of the unresolved issues in the fiber pushout test
that were alluded to in section 1.3. In Chapter 2, the pattern of interface failure that is
thought to occur during the pushout test is verified by conducting fiber pushout tests on
model composites in a polariscope. Emphasis is placed on detecting the onset of debond
growth and its correspondence to a nonlinearity in the force–displacement curve. The
location of the initial debond is noted for several model composites, and this information is
combined with data in the literature to show how the fiber and matrix properties, as well as
the residual stresses, influence the pattern of interface failure during the fiber pushout test.
Finally, the evolution of the photoelastic fringe patterns also reveals the debond length as a
function of applied force. This function is compared with the debond length calculated by a
shear lag solution to the pushout problem.
Chapter 3 focuses on developing a numerical (finite element) solution that can be used
to calculate the debond length as a function of the load applied to the punch during pushout
testing. The finite element predictions of debond length are compared to the debond length
measurements from Chapter 2 for both top and bottom debonding model composites. The
important issue of whether or not debonds due to processing can be present in a composite
prior to pushout testing is addressed by identifying and measuring initial debonds in one of
the bottom debonding model composites (steel/epoxy). Inconsistencies in the
force–displacement curve for the steel/epoxy system and measurements of the portion of the
fiber extending from the matrix for various debond lengths leads to a discussion of how
differences in sample preparation can affect the derived force versus debond length.
Chapter 4 contains a detailed description of an apparatus that was designed and built to
push fibers out of metal matrix composites at elevated temperatures in a vacuum or
controlled atmosphere. Pushout tests are conducted on a fatigue loaded SiC/Ti-6-4
composite at room temperature and 400˚C to study the affect of fatigue on the fiber–matrix
10
interface bond. Recommendations are made for changes and improvements to the high
temperature fiber pushout apparatus that are necessary to make the experiment capable of
capturing the departure from linearity in the force–displacement curve for metal matrix
composites.
Finally, Chapter 5 contains a summary of conclusions that are based on the experiments
and modeling described in the previous chapters.
11
2. DEBOND LENGTH MEASUREMENTS IN MODEL
COMPOSITES
2.1 Introduction
Several previous investigations have been carried out with the intention of understanding
better the fiber pushout experiment and how to analyze the resulting data. Some of this
research employed the fiber pullout experiment in which a single fiber is pulled from a
composite rather than pushed out. Pullout experimental curves have the same general shape
as the schematic of a pushout curve shown in Figure 1.1b. Tsai and Kim (1991, 1996) used
a polariscope to study the stick–slip phenomenon in the frictional part of the pullout test
(section III) for an optical glass fiber in an epoxy matrix. Watson and Clyne (1992)
investigated the stresses produced during pushout before debond (section I) in an
epoxy/epoxy model composite also using a polariscope. Atkinson, Avila, Betz, and Smelser
(1982) pulled a glass rod from a polyurethane matrix. By tracking photoelastic fringes,
debond length was measured as a function of displacement and force. The measured
debond length was then used to produce the force–displacement curve analytically, but the
converse procedure of predicting debond length from the experimental results was not
attempted. Finally, Cordes and Daniel (1995) measured the debond length as a function of
force in a fiber pullout test for an SiC fiber in a glass matrix by observing a change in the
intensity of light reflected from the fiber surface when the interface debonded. The data
were fit to the LH&KP solution. Because interface wear was significant, a negative
coefficient of friction was predicted by the shear lag solution.
In the current study, the interface failure sequence is observed during fiber pushout tests
on model composites. Model composites consisting of various fibers in a birefringent
epoxy matrix are chosen because the interface failure sequence and the debond length are
determined by inspection of photoelastic fringe patterns in the matrix. This observation
allows the correspondence between the characteristics of the force–displacement curve and
12
the interface failure processes to be identified. The applicability of the LH&KP shear lag
solution is determined by noting the initial debond location and comparing measured to
predicted debond length.
2.2 Fabrication of model composites
Several types of model composites were fabricated for the pushout experiments. Many
combinations of materials, sample thicknesses, and support diameters were tried in an effort
to determine the most important parameters in producing a top or bottom debond. Tool
steel, borosilicate glass, quartz, nylon, glass particle reinforced nylon, and polyester fibers in
an epoxy matrix were fabricated. An epoxy matrix was chosen because of its high material
fringe constant for the photoelastic observations and ease of sample preparation. As shown
in Table 2.1, these constituents correspond to a composite Ef/Em range of 0.81 to 80.
Three different curing agents—diethylenetriamine (DETA), p–aminocyclohexylamine
(PACM), and bis–1–propanamine (Ancamin 1922)—were used to cure Epon 828 epoxy
resin and vary the differential shrinkage strain during processing from 0.0022 to 0.0084.
The DETA/Epon 828 and Ancamin 1922/Epon 828 systems were cured for 7 days at room
temperature, and the PACM/Epon 828 system was cured at 80˚C for 1 hour and at 150˚C
for 1 hour. To vary the interface strength some fibers were uncoated, some were coated
with a silane to increase the interface strength, and some were coated with a silicone release
agent to produce a very weak interface.
Samples were prepared by positioning a single fiber lengthwise in an 8 mm wide by 25
mm long mold and pouring a mixture of resin and curing agent into the mold. After curing,
the top and bottom faces of the sample were polished to a 15 µm finish. The pushout
samples were then cut from the bulk sample to thicknesses ranging from 6 to 20 fiber
diameters. Finally, the samples were placed on a steel support such that the fiber was
centered over a hole. The support hole diameter was varied from 1.08 fiber diameters to 5
13
Table 2.1. Model composite fiber-to-matrix moduli ratios and processing induced residual strains.
*Matrix chemical shrinkage **Processing ∆α∆T
Model fiber(# in Fig. 2.7)
Epoxy matrix Ef/Em Processing∆∆∆∆ΤΤΤΤ (˚C)
Residualstrain(εεεεr)
Polyester (1) EPON 828 + DETA 0.81 0 –0.0022*
Nylon (2) EPON 828 + DETA 1.70 0 –0.0022*
Glass reinf. nylon (3)Glass reinf. nylon (4)
EPON 828 + Anc.1922EPON 828 + PACM
3.063.31
0–125
–0.0030*–0.0056**
Borosilicate glass (5)Borosilicate glass (6)
EPON 828 + DETAEPON 828 + PACM
16.322.0
0–125
–0.0022*–0.0081**
Quartz (7) EPON 828 + PACM 29.0 –125 –0.0084**
Steel (8)Steel (9)
EPON 828 + DETAEPON 828 + PACM
50.083.0
0–125
–0.0022*–0.0072**
14
fiber diameters with most tests being carried out with a support hole diameter of 1.24 fiber
diameters to avoid sample bending.
2.3 Experimental apparatus
A micromechanical tester was designed for performing the pushout experiments in a
polariscope. Figure 2.1 shows a schematic of the pushout apparatus. The tester consists of
a Compumotor SX stepper motor and Daedal MS23 railtable. Displacement at the 0.9 fiber
diameter steel punch tip was measured by recording the commanded rotation of the stepper
motor armature as a function of time and then carefully subtracting machine compliance.
The punch velocity was maintained at 5 µm/sec. Load was measured by sampling a Kistler
piezoelectric charge transducer at 5 samples/sec. The load cell signal was conditioned by a
Kistler dual mode amplifier and digitized by a Tektonix TDS 420 oscilloscope.
The testing apparatus was positioned such that the pushout sample was entirely
illuminated by a circular polariscope as shown in Figure 2.2. The polariscope was
constructed using an argon laser (Lexel model 3500) as the light source. Since coherent
light can create interference fringes, a spinning ground-glass disk was used as a coherency
scrambler. A collimated beam of light passed through a polarizer (P), which vertically
polarized the light, and then through a quarter-wave plate (QW) with its fast axis at 45
degrees to the axis of the polarizer, producing circularly polarized light. The beam
continued on through the specimen, traversed a second quarter wave plate 90 degrees out of
phase with the first and through a second polarizer, eliminating the isoclinic fringes from the
resulting image. Field lenses (FL) were inserted into the beam before and after the sample.
The first field lens expanded the diameter of the beam from 2.5 mm to 25 mm, permitting a
larger field of view, and the second field lens focused the beam on the aperture of a CCD
camera (Panasonic BL200). A 640 pixel by 480 pixel frame grabber was used to store the
images at a maximum rate of 5 frames per second during the pushout test.
15
Stepper Motor
Load Cell
Punch
Pushout Sample
Sample Support
Figure 2.1 Schematic of the micromechanical test apparatus used to perform pushout
tests on model composites.
P QWFL PQWFL
CameraLaser
Figure 2.2 Pushout apparatus positioned in the circular polariscope.
16
2.4 Measurement of debond length
After the pushout test was completed, the individual images were inspected to determine
the debond length as a function of time so that the debond length could be plotted as a
function of force and displacement. Debond length was measured for only one bottom
debonding and one top debonding model composite of Table 2.1—steel/epoxy and
polyester/epoxy, respectively. Figure 2.3 shows a representative series of images taken
during a pushout test of a steel fiber in an epoxy (EPON 828 + PACM) matrix. The load
increases from frame A to B to C. The black rod in the center of each frame is the model
fiber. The gray areas to the left and right of the fiber containing the photoelastic fringes are
a portion of the matrix near the fiber. A small section of the punch can be seen near the top
of the fiber, and for clarity, this particular sample was mounted on an epoxy support so that
the support hole could be seen at the bottom of the frames. The tip of the interface debond
is the location along the interface of the greatest fringe density and is marked with an arrow
in each frame. The debond can clearly be seen to grow from the bottom face of the sample
toward the top face with increasing load.
Figure 2.4 shows the corresponding force–displacement curve for the steel/epoxy model
composite. This system debonded from the bottom, and therefore could not be fit to the
LH&KP shear lag solution to find interfacial toughness. The steel/epoxy pushout curve
contains a significant region of unstable debond growth between the peak load and total
debond that is not found in the pushout curve from a top debond (Figure 2.6). The large
stress gradients near the top surface apparently start interacting with the stress field around
the crack tip within approximately 4 fiber diameters of the top surface. Figure 2.4 and the
following pushout curves for polyester/epoxy (Figure 2.6) include a displacement from a
machine compliance of 0.42 µm/N.
A representative series of images is shown in Figure 2.5 for a polyester rod in an epoxy
(EPON 828 + DETA) matrix. Again, the load increases from frame A to C. For this
system, the debond progressed from the top face to the bottom. Only one small fringe
17
(A) (C)(B)
1 mm
Figure 2.3 Photoelastic images acquired during a steel/epoxy pushout test.
18
−0
−100
−200
−300
−400
−500
−600
−700
0
2
4
6
8
10
12
−0 −100 −200 −300 −400 −500
Forc
e (N
)
Deb
ond
leng
th (
mm
)Displacement (µm)
I II III
Force
Debond length
Figure 2.4 Force and debond length versus displacement curve from a steel/epoxy
pushout test. Debond was from the bottom. Fiber diameter = 1.65 mm,
sample thickness = 13 mm, support hole diameter = 2.05 mm, and punch
diameter = 1.4 mm.
19
(A) (C)(B)
1 mm
Figure 2.5 Photoelastic images acquired during a polyester/epoxy pushout test.
20
tracked the interface crack tip because the residual stresses were much smaller than in the
steel/epoxy system. The polyester/epoxy system offered an additional measure of debond
length. Before debond the fiber was transparent, but once the fiber–matrix interface
debonded, very little light passed through the debonded portion of the fiber. The
photoelastic fringes and the darkened portion of the fiber indicated the same debond length.
The corresponding load–displacement curve along with the measured debond length is
shown in Figure 2.6. These data are used in Section 2.6 to compare with the LH&KP
solution.
2.5 Top versus bottom debond
In order to determine which composite systems will debond from the top and which
from the bottom, the author tested model composites with varying fiber-to-matrix moduli
ratio (Ef/Em), sample thickness, interfacial bond strength, and processing induced residual
stresses (Table 2.1). Figure 2.7 shows a plot of the initial debond location as a function of
differential residual thermal strain and fiber–matrix moduli ratio. The upper right shaded
area contains the composite systems that initially debonded from the bottom during pushout
testing, and the lower left shaded area represents the top debonding composites. Results
from this work and from other pushout tests in the literature are plotted. The numbered data
points, as designated in Table 2.1, are from the current pushout experiments on model
composites. The points labeled Penn correspond to tests done by Koss, et al. (1993) on
alumina/niobium and sapphire/TiAl composites, and the NASA points correspond to tests
done by Eldridge (1995) on SiC fibers in a titanium alloy matrix and in a reaction bonded
silicon nitride matrix.
The steel/epoxy, quartz/epoxy, and glass/epoxy samples all initially debonded from
the bottom, and the debond grew from the bottom toward the top until total debond. A top
debond did not appear at any point during the tests. Decreasing the interface strength with a
silicone coating, or decreasing the residual stresses with a room temperature cure epoxy
21
−0
−100
−200
−300
−400
0
2
4
6
8
−0 −100 −200 −300 −400 −500 −600
For
ce (
N)
Deb
ond
leng
th (
mm
)Displacement (µm)
I II III
Force
Debond length
Figure 2.6 Force and debond length versus displacement curves from a
polyester/epoxy pushout test. Debond was from the top. Fiber
diameter = 1.9 mm, sample thickness = 5.3 mm, support hole
diameter = 2.05 mm, and punch diameter = 1.7 mm.
22
50 60 70 8015 20 25 30
Penn
Penn
NASA
NASA
Top debondBottom debond
−0.010
−0.008
−0.006
−0.004
−0.002
0.000
0.0020 1 2 3 4 5
1 2 3
4
5
6 7
8
9
Dif
fere
ntia
l Res
idua
l Str
ain
Ef / E
m
Figure 2.7 Map of top and bottom fiber debonds as a function of fiber-to-matrix
stiffness ratio and residual thermal strain—independent of sample thickness
and interface strength.
23
did not produce a top debond. Increasing the sample thickness to 20 fiber diameters did not
produce a top debond but did eventually cause the fiber end to be damaged under increasing
punch load. Since higher residual stresses would only increase the probability of a bottom
debond, the bottom debond region of Figure 2.7 is independent of residual stress as long as
the matrix shrinks more than the fiber during processing.
Ceramic fibers such as SiC and alumina can sustain a larger applied stress from the
punch than the steel or glass fibers used in these tests without damaging the fiber end, but
SiC and alumina fibers are also much smaller in diameter. Alignment difficulties and the
conical shape of some of the punches used on small fibers require the diameter of the punch
face to be a smaller percentage of the fiber diameter than the punches in the experiments
conducted here. Therefore, larger applied stresses from a punch smaller with respect to the
fiber cross-sectional area would most likely cause damage to even the ceramic fibers
available commercially. Thus, the shaded area corresponding to bottom debonds is
assumed to be independent of interface strength and sample thickness.
In the polyester/epoxy and nylon/epoxy samples a debond initiated from the top and
then grew toward the bottom until total debond. Samples as thin as 3 fiber diameters were
tested. This minimum thickness was chosen to avoid significant sample bending. A
debond initiated from the top, and then grew toward the bottom until total debond. Since
thicker samples promote top debonding, the shaded region of Figure 2.7 is expected to be
independent of sample thickness. As stated, a silicone coating was applied to the fiber to
produce a very weak interface in some of the polyester/epoxy and nylon/epoxy samples.
Since increasing the interface strength also promotes a top debond, this section of the graph
is independent of interface strength.
Most fiber pushout tests are not conducted on systems with transparent and/or
birefringent matrices, so the plot in Figure 2.7 may be useful when estimating whether the
system being tested will debond from the top or bottom. For low differential residual
24
thermal strain (and therefore, low residual stresses) as Ef/Em increases, there is a transition
from a top debond to a bottom debond at approximately Ef/Em = 3.0. As the residual
thermal strain due to processing increases at this transition point, the location of initial
debond again changes from top to bottom. The increase in residual strain for the
SiC/titanium and sapphire/TiAl composites over the epoxy matrix composites near Ef/Em =
3.0 corresponds to an increase in residual stresses because the constituents of the metal
matrix composites are stiffer.
Since the statement was made that Figure 2.7 is independent of sample thickness and
interface strength, the following discussion is provided to address how interface strength,
residual stresses, and sample thickness affect the location of initial debond. If the pushout
problem is modeled as linear elastic up to debond initiation, then the stresses at each point in
the body scale linearly with the applied load. Also, the stresses at each point for the pushout
problem and the thermal problem corresponding to a temperature drop during processing
can be superimposed. Since the asymptotic elastic solution at points A and B of Figure 1a
indicates that the shear stress is singular, the proper way to describe the stresses at A and B
is with stress intensity factors (Demsey and Sinclair, 1981). Finally, the stresses at each
point in the body scale with the applied load, so the stress intensity factors scale with the
applied load.
This argument leads to the following equation for the ratio of the stress intensity factors
at points A and B in Figure 1.1a:
KIIA
KIIB=
KIIA ∆F− KIIA ∆T
KIIB ∆F+ KIIB ∆T
. (2.1)
In Eq. (2.1), KIIA ∆F is the mode II stress intensity factor at point A that results from a load
F being applied to the punch, KIIB ∆T is the mode II stress intensity factor at point B that
results from processing induced residual stresses, and KIIA is the stress intensity factor
25
due to both the pushout and thermal loads. Also, KIIA ∆F/ KIIB ∆F
is constant with respect
to pushout load, F , and increases with increasing sample thickness. At point A, the stress
intensity factor due to residual stress decreases the total stress intensity factor. At point B,
the mechanical and residual stress intensity factors are additive. Assuming a failure
criterion proportional to K and that mode II drive the interface failure, several statements
can be made about the data in Figure 2.7:
1) Increasing the magnitude of the residual stresses (∆α∆T < 0 ) and holding all other
variables constant will favor a bottom debond.
2) Increasing the interfacial strength and therefore increasing the pushout load required to
initiate a debond while holding all other variables constant will favor a top debond.
3) Increasing the sample thickness while holding all other variables constant will favor a
top debond.
4) If the fiber is damaged by the punch before debond initiation, then increasing the
sample thickness or increasing the interfacial shear strength will obviously not promote
a top debond before fiber damage is likely.
2.6 Comparison with a shear lag solution
The LH&KP shear lag solution was derived based on a debond from the top.
Consequently, the measured debond length data from model polyester/epoxy (curing agent
DETA) composites with no fiber coating was chosen to compare with the shear lag
predictions of debond length. The pushout curve in Figure 2.6 was used for the
comparison.
2.6.1 Shear lag theory
The shear lag assumption is the hypothesis that the change in axial stress in the fiber of
a composite is due solely to shear stress transferred through the interface from the
26
matrix—hence the name “shear lag”. Equilibrium of a fiber element in the axial direction
is formulated as
dP(z) = −τrz (r f , z)(2πr f )dz , (2.2)
where P(z) is the axial load on the cross-section of the fiber at the axial position z (see
Figure 2.8). The coordinate, z , originates at the top surface of the sample, and r f is the
radius of the fiber. Eq. (2.2) will produce accurate results only if the shear lag formulation
of the equilibrium of a fiber element is a good approximation to the point by point
equilibrium formulation of elasticity.
In the LH&KP shear lag solution to the pushout problem, the radial shear stress,
τrz (r f , z), is formulated as the sum of a term due to processing and a term due to Poisson’s
expansion of the fiber under the compressive punch load. Eq. (2.3) shows this relation.
τrz (r f , z) = µ σN − kP(z)
πr f2
(2.3)
The coefficient of friction is µ , and the constant, k , is a combination of the elastic
properties the fiber and matrix and is given by Kerans et al. The thermal residual radial
stress at the interface away from the ends, σN , is often calculated assuming plane strain, but
in the current work, σN was measured as described in Section 2.6.2. The assumption of a
constant σN would be most accurate for infinitely long samples. Eqs. (2.2) and (2.3) are
combined, and the resulting ODE is solved subject to P(z) = F (where F is the force
applied by the punch) at the top of the fiber to get and equation for the axial stress in the
fiber:
P(z) = (F − P*)e
−2µkz
r f + P* . (2.4)
The term Pd + Pr is then substituted into Eq. (2.4) as the load on the cross-section of the
fiber at z = ld where ld is the debond length. The following equation, which relates the
debond length to the applied load, results from the substitution:
27
z
P(z)
P(z) +dP(z)
dzdz
dz τrz (r f , z)r f
Figure 2.8 Axial force and interfacial shear stress on a fiber element assumed by shear
lag theory.
28
ld =r f
2µkln
P* − F
P* − (Pd + Pr )
. (2.5)
The quantity, P*, is the axial force in the fiber that would be necessary to open the
debonded portion of the interface if the fiber were being pulled from the composite, and Pr
is the thermal residual axial force in the fiber. The axial force necessary to open the
interface is related to (σN ) by the equation
P* =−πr f
2σN [Ef (1 + υm ) + Em (1− υ f )]
Emυ f, (2.6)
where Ef and Em are the elastic moduli, and ν f and νm are the Poisson’s ratios for the
fiber and matrix, respectively. The axial force, Pr , is calculated from σN according to the
following relationship:
Pr = −P*ν f
P* − 2σNπr2
P* + 2ν fσNπr2
. (2.7)
The calculations of Pr and P* assume plane strain conditions and an infinite matrix radius.
The extra displacement due to debonding is related to the applied load by assuming that the
amount that the debonded portion of the fiber is compressed can be calculated with the
following equation:
d =1− 2ν f k
πr f2Ef
(P(z) − Pr )dz0
ld
∫ , (2.8)
as
d =1− 2ν f k
2µkπr f E fF − Pd − Pr + (P* − Pr ) ln
P* − F
P* − Pd − Pr
(2.9)
Eq. (2.8) is based on the simple expression for deflection, δ = Fl / EA where the stiffness
of the fiber is greater than Ef because it is not allowed to expand freely in the radial
direction. The expression, Ef / (1− 2ν f k), is also based on the assumption of an infinite
matrix radius.
Finally,
GIIc =Pa
2
4πr f
d(d / F)dld
(2.10)
29
is used along with Eqs. (2.5) and (2.9) to calculate the following expression for mode II
interfacial fracture toughness, GIIc :
GIIc =(1− 2ν f k)Pd
2
4π2r f3 Ef
(2.11)
Eq. (2.11) assumes that the interface debonds to when the axial load in the fiber reaches a
critical value. When Eq. (2.11) is substituted into Eqs. (2.5) and (2.9), the following
equations for force as a function of debond length and displacement as a function of
debond length during progressive debonding are derived:
F = (C1GIIc1/2 + Pr − P*)eC3µld + P*, (2.12)
d =C2
µF − C1GIIc
1/2 − Pr + (P* − Pr ) lnP * −F
P* − C1GIIc1/2 − Pr
, (2.13)
where C1, C2 , and C3 are constants that depend on material properties.
In the frictional problem, the entire interface is debonded, and the embedded length, le ,
determines the load required to pushout the fiber. If the axial stress in Eq. (2.4) is set to
zero at z = le , the following relation between force and displacement during frictional
pushout results:
d = t −1
C3µln
P* − F
P*
. (2.14)
In Eq. (2.14), t is the sample thickness. Eq. (2.14) is free of the error associated with
calculating Pr . Also, the assumptions associated with calculating deflection from the
expression in Eq. (2.8) are not included in the derivation of Eq. (2.14). Eqs. (2.12)–(2.14)
are used in Section 2.6.3 to compare predicted debond length with experimentally measured
debond length.
2.6.2 Determination of residual stresses
The radial compressive stress at the fiber-matrix interface away from the fiber ends due
to processing and/or roughness was found photoelastically before and after debonding to
30
make sure that the additional radial compressive stress produced by fiber asperities after
interface debonding was not a significant factor. Samples twice as thick as the sample to
which the curve in Figure 2.6 corresponds were used so the surface effects region would be
a less significant portion of the total thickness of the sample. The photoelastic measurement
of the interfacial radial stress consisted of placing the pushout sample in the polariscope
such that the fiber was parallel to the laser beam. The bright field and dark field
isochromatic fringe patterns in the matrix were recorded and used to multiply the fringe
order digitally by two times according to the method described by Toh, Tang, and
Hovanesian (1990). Due to the axisymmetry of the sample, the isoclinic fringe patterns
were not required to separate the stresses. The radial stress at the interface away from the
fiber ends was calculated from the resulting isochromatic fringe pattern using the shear
difference method (Frocht, 1946). The pre- and post-debonding measurements of σN were
both –5.68 MPa from which Pr and P* were then calculated. See Appendix A for a more
detailed description of the measurement of σN .
2.6.3 Comparison of measured and shear lag theory debond lengths
The maximum load and corresponding displacement in section III of Figure 2.6, Pr ,
and P* were substituted into Eq. (2.14), and Eq. (2.14) was solved for µ . The result was a
coefficient of friction of 0.52. Once µ was known, force–displacement data derived from
section II was fit to Eq. (2.13) and an interfacial fracture toughness of 389 J/m2 was
obtained, which approaches the mode I toughness of the matrix. The appropriate
displacement to be used with Eq. (2.13) is the difference between the solid and dashed lines
in section II of Figure 2.6 since the displacement in the LH&KP solution is due to debond
growth only. The curve fit for GIIc is shown in Figure 2.9. The displacement in Eqs.
(2.13) and (2.14) is negative by convention; therefore, the shear lag displacement plotted in
Figure 2.9 is also negative. Although GIIc was iteratively least squares fit to the
force–displacement data, the natural log term in Eq. (2.13) remained small with respect to
31
−40
−35
−30
−25
−20
−15
−10
−5
0−420−400−380−360−340−320−300−280−260
Shea
r la
g di
spla
cem
ent (
µm
)
Force (N)
measured data
curve fit
Figure 2.9 Curve fit (Eq. (2.13)) of measured force–displacement data for
polyester/epoxy.
32
the other three terms on the right hand side. The small natural log term caused the curve fit
to be approximately linear with the slope equal to C2 / µ . Hence, GIIc was determined by
only the intercept of the line fit through the force-displacement curve with the force axis.
Since the slope of the line fit to the pushout data is reasonably close to the slope of the
measured force–displacement curve, the coefficient of friction calculated from the initial
frictional data point is consistent. Finally, the shear lag prediction of debond length was
calculated by substituting µ and GIIc into Eq. (2.12).
A comparison of predicted debond length and experimentally measured debond
length is shown in Figure 2.10. The measured debond length is a nearly constant 1.5 fiber
radii greater than the debond length predicted by shear lag theory. The under-prediction of
debond length is to be expected since the shear lag solution does not adequately account for
the large interfacial shear stress close to the top surface. After debond initiation, the debond
apparently grows easily through the surface effects region. Additional interface crack
growth from that point on follows the prediction of shear lag since the slopes of the
measured and predicted debond length curves are very close. Coincidentally, the interface
crack growth becomes unstable at approximately 1.5 fiber radii from the bottom of the
sample, so 1.5 fiber radii could be used as an estimate of the thickness of the surface effects
region for this system. If a method were available to measure or estimate the size of the
surface effects region for any composite system, the shear lag solution could be modified to
give a more accurate prediction of debond length, and therefore a more accurate prediction
of interface toughness. For a given load, if the debond length is smaller, a smaller portion
of the load is transferred to the matrix through friction in the debonded section of the
sample, and the resulting axial load in the fiber near the debond tip is greater. A larger
interface toughness will be needed to resist debond growth. Since the debond length for the
current system is under-predicted, the interface toughness will be over-predicted.
33
0
0.5
1
1.5
2
2.5
3
3.5
4
−400−380−360−340−320−300−280−260
Deb
ond
leng
th (
mm
)
Force (N)
Measured debond length
Shear lag predicted debond length
Figure 2.10 Comparison of shear lag prediction and experimental measurement of
debond length. P* = 80.38 N, Pr = –45.83 N, µ = 0.52 , and
GIIc = 389 J/m2.
34
If, instead of fitting the force–displacement data to Eq. (2.13) and checking the predicted
debond length, the measured debond length is fit to Eq. (2.12) and the displacement is
checked, the displacement is over-predicted. For the pushout curve shown in Figure 2.6,
this inverse procedure for fitting the experimental data to the shear lag model results in a
GIIc = 105 J/m2. Consequently, the toughness calculated from the shear lag theory varies
by a factor of three depending on how the pushout data is fit.
2.7 Additional experimental observations
The model composite experiments were also used to correlate interfacial failure
processes with points on the fiber pushout load–displacement curve. During the initial
linear part of the curve, there is no interfacial debond growth even if there is a debond
present from processing. If the sample is thick enough, a debond initiates before the peak
load is reached. At debond initiation there may be a small sharp load drop, but often,
especially for a bottom debond, smooth transition to a shallower slope is observed before
the load increases and the debond continues to grow. A similar nonlinearity in the
force–displacement curve can also be caused by yielding of the fiber if it is necessary to
apply an axial stress to the fiber that is larger than the local yield stress of the fiber. The
thickness of the sample and the interface strength were chosen to be small enough in these
tests so that no fiber damage occurred. The fiber top surface should always be inspected
after a pushout test to see if damage has occurred. Once the debond was present, it grew
toward the opposing surface until the debond growth became unstable within 1 to 4 fiber
diameters of the surface, depending on the fiber–matrix–coating combination tested.
In systems such as glass and steel fibers in a room temperature cure epoxy the failure
process was observed for very low residual stresses. The debond length was found at times
to be significantly longer on one side of the fiber than on the other side. Also, for low
residual stress composites, the debond tip often intermittently stopped and jumped
forward rather than growing smoothly with increasing load as in the systems with larger
35
residual stresses. This observation supports the findings of Eldridge (1995) who found
that high residual stresses promoted stable debonding during the pushout test. Therefore,
data from a system with large residual stresses that debonds from the bottom will most
likely have less scatter.
Progressive debonding was observed in the model composites tested for thicknesses as
small as 2.0 fiber diameters regardless of interface strength. Therefore, even qualitatively
comparisons of the average interfacial shear strengths from pushout testing of the model
composites would not have provided useful information.
2.8 Conclusions
Observations of the development of photoelastic fringe patterns in the matrix of several
model composites during fiber pushout tests demonstrated that the linear part of the
pushout curves corresponded to a fully bonded interface, the nonlinear part of the pushout
curves up to the maximum load corresponded to progressive debonding, and the portion
after the maximum load corresponded to frictional sliding of the totally debonded fiber.
This correlation between the interface failure and the force–displacement curve cannot
necessarily be extended to all composites based on the model composite pushout tests.
Further, composite systems with a fiber-to-matrix modulus ratio of greater than 3 should be
suspected of debonding from the bottom first during the pushout test, and therefore the
current shear lag models do not apply to these systems. Also, the debond length was
measured as a function of applied load for a system that debonded from the top. The
measured debond length was a nearly constant 1.5 fiber radii greater than the debond length
predicted by a shear lag solution. The under-predicted debond length is evidence that the
shear lag model will over-predict the interface toughness when used to analyze pushout
data. Finally, debond length versus displacement curves are shown for pushout tests on a
composite that debonds from the bottom. The debond length measurements can be used to
check the accuracy of theoretical solutions to the pushout problem if more advanced
36
solutions are developed that include surface effects and/or the possibility of a bottom
debond.
2.9 Future work
Debond length measurements during fiber pushout of model composites could yield
additional valuable information. The polyester fiber/epoxy matrix composite could be
exploited further. If a fiber coating were found that would make the interface strength
weaker than the interface strength for no coating (as was tested in this study) and stronger
than the interface strength for a silicon release agent coating (also tested in this study),
longer samples could be tested without damage to the fiber from the punch. Longer
debonds could be grown to determine whether the shear lag solution becomes more accurate
for crack tips further from the surface (as expected) or if the inaccuracies from shear lag are
from something other than surface effects.
The proper tailoring of the interface strength may be used to allow polyester fibers to be
pushed from an epoxy matrix with more substantial residual stresses. For a specific
coating, it may be possible to increase the residual stresses until the interface debonds from
the bottom instead of the top as it did in the pushout tests described in this chapter. This
test would be a further indication that the mechanism illustrated in Figure 1.3 is the
mechanism responsible for bottom debonds during fiber pushout testing. One problem
with the use of coatings to change the interface strength is that a coating cannot be explicitly
included in the LH&KP shear lag solution. Fiber pushout tests on fiber–matrix
combinations that would fill the gaps in Figure 2.7 still need to be done. And finally, a
study could be done of how the difference in Poisson's ratios between the fiber and matrix
influences whether a top or bottom debond occurs.
37
3. FINITE ELEMENT DEBOND LENGTH PREDICTIONS
3.1 Introduction
The experiments of Chapter 2 show that, although easy to apply, shear lag theory over-
estimates interfacial toughness for the polyester/epoxy model composite. The model
composite fiber pushout tests give evidence that many composites of current interest (metal
matrix composites (MMCs) near the Ef/Em = 3 transition) will debond from the bottom.
No shear lag solution exists for bottom debonds. The focus of this chapter is to present a
relatively simple procedure for converting the pushout data from experiment to interfacial
fracture toughness, GIIc . Although this method, which may include several finite element
runs, takes more effort than fitting one or two shear lag equations, the time required is not
prohibitive.
The shear lag solution’s inaccurate prediction of debond length has several possible
causes:
1. The shear lag approximation does not hold because the sample thicknesses that are
required to avoid fiber damage during the experiment are too thin.
2. The actual boundary conditions corresponding to a punch diameter smaller than the
fiber diameter and a support hole diameter larger than the fiber diameter are not modeled
accurately.
3. The assumption and calculation of constant residual stresses with respect to the axial
coordinate may not be as sophisticated as necessary.
4. The possibility of a portion of the interface separating is not allowed by the LH&KP
solution.
The finite element method, while an approximate technique, addresses all the concerns
noted above. The shear lag assumption is not necessary since the complete set of coupled
partial differential equations from the formulation of an isotropic linear elastic elasticity
problem is solved (approximately), and boundary conditions that more closely simulate
38
processing and fiber pushout can be implemented. Additionally, the interface elements used
in this study allow a loss of contact if a tensile radial stress results during loading. The
interface elements can be located anywhere in the mesh so an interface crack can be
included in the mesh at the top of the interface, the interior, or the bottom. Because of its
flexibility, the finite element method was used to try to improve upon the shear lag results.
Kallas, Koss, Hahn, and Hellman (1992) used the finite element method to calculate the
stress distribution for a fully bonded thin slice fiber pushout sample of sapphire fibers in a
niobium matrix under typical pre-debonding loads. Axisymmetric finite elements were
used. Kallas and his co-workers showed that the relative magnitudes of the peak in the
shear stress (an artifact of the mesh’s inability to capture the stress singularity) at the
bottom of the interface and at the top of the interface is affected by the sample thickness and
the support hole size relative to the fiber diameter. No interface strength parameter was
calculated since a crack was not included in the model. Ghosn, Kantzos, Eldridge, and
Wilson (1992) did similar work with a three dimensional finite element mesh to model the
linear groove in the pushout sample support more rigorously. A groove is often employed
instead of a circular hole to permit more fibers to be pushout tested in less time.
Kishore, Lau, and Wang (1992, 1993) developed a method to find the stress intensity
factor by matching a global finite element solution to the asymptotic solution at the top of
the interface for a fiber pullout geometry (fully bonded inteface). With the appropriate
asymptotic stress field, this operation could also be used to calculate the stress intensity
factor at the crack tip in a fiber pushout problem if the debond length has already been
determined. Since a focused mesh of circular rings of elements in the plane of the cross-
section and concentric with the crack tip is required, this method may be very time intensive
when checking the critical stress intensity factor at several debond lengths. Also, no
equation was provided by Kishore and co-workers to relate the mode II stress intensity
factor for points in a frictional interface to the mode II strain energy release rate. The
energy release rate may be the more useful fracture parameter for characterizing an interface
39
because the asymptotic stress field for the point at the top of the interface was shown to be
different for different material combinations with the same stress intensity factor.
Beckert and Lauke (1995) used a finite element analysis to determine the interfacial
energy release rate in a fiber pullout simulation in which an interface crack extended from
the top surface to a point along the interface below the surface. This method could be
applied to calculate interfacial fracture toughness in the pushout test if the sign of the
applied load was changed and the embedded fiber was allowed to extend through the entire
sample. Friction and residual stresses are not included in the model, and the length of the
debond must be determined by some other method. Chandra and Ananth (1995) included
residual stresses due to processing. Debond length was predicted by finite element
simulations of the fiber pushout test but was not compared with experimentally measured
debond length data. A maximum shear stress, (τrz )max , along the interface was chosen and
the interface in the model failed when (τrz )max was exceeded. Force–displacement curves
were generated by finite element analyses for several interface strengths and then compared
with the experimental force–displacement curve. The interface strength that corresponded to
the best curve fit was chosen. This method is potentially very powerful because the location
of initial debonding and even the possibility of debond growth occurring from both ends of
the sample at the same time could be predicted and included in the model. A drawback of
Chandra and Ananth’s formulation is their use of a maximum shear stress criterion for
characterizing resistance to growth of a sharp crack.
The concentric cylinders variational model developed by Tandon and Pagano (1996)
was used to study the interaction of an annular matrix crack with an interface crack that was
composed of opened, slipping, and sticking zones in a composite uniformly strained
longitudinally. Discretization was required only in the radial direction with the axial and
hoop stresses assumed to vary linearly in the radial direction within each element. The same
model was employed by Pagano and Tandon (1996) to solve fiber pushout problems
similar to the ones solved in this study. Pagano and Tandon’s formulation allows the exact
40
boundary conditions imposed on the top and bottom of the pushout sample during testing
to be imposed on the model. The interface can open if a tensile interfacial radial stress
develops, and Coulomb friction can be incorporated in the slipping zone of the interface. A
comparison was made between some of the results from the current work and results from
Pagano and Tandon’s model.
The finite element procedure described in this chapter includes residual stresses and
determines the debond length as a function of force, independent of a criterion for interface
decohesion. Both a top and a bottom debond can be included if necessary, and the
development of a frictional shear stress on the sliding debonded portion(s) of the interface
can be included in the interface continuity conditions. Finally, the critical strain energy
release rate can be computed to quantify resistance to interface rupture.
3.2 Finite element model
The commercial finite element code ABAQUS (Hibbitt, Karlsson, & Sorenson, Inc.)
was used for the finite element modeling described in this chapter. The finite element mesh
designed for the pushout problem is shown in Figure 3.1. The fiber–matrix interface and
top and bottom faces of the sample were densely meshed, since the stresses may change
rapidly in those areas and the shape functions (simple polynomials) representing the
variation of displacement within each element may not be able to approximate closely the
actual displacement. The stress concentrations at the punch and support hole edges also
required a large concentration of elements. Typical meshes contained 2500 to 3000
elements.
The minimum mesh refinement necessary was determined as follows:
1. For problems that did not involve a fracture toughness calculation, e.g. problems run to
calculate the sample compliance for a given debond length or fully slipping problems,
the mesh was considered dense enough when further refinement of any portion of the
41
Punch
Interface
Edge of samplesupport hole
Figure 3.1 Finite element mesh constructed to simulate processing and pushout loads.
42
model caused less than a 0.1% change in the resulting total load on the top of the fiber.
2. For problems in which stored strain energy and frictional energy dissipated were
extracted for use in computing fracture toughness, an additional criterion was
implemented that any further mesh refinement near the interface crack tip caused less
than a 1% change in the value of fracture toughness.
The fiber pushout sample is modeled with second order axisymmetric isoparametric
(CAX8) elements and second order axisymmetric frictional interface (INTER3A) elements.
The fiber and matrix materials are assumed to be isotropic and linear elastic. The interface
elements can sustain compressive radial stress σrr and shear stress τrz (magnitude less than
or equal to µσrr ) where r is the radial coordinate originating at the fiber central axis, and z
is the axial coordinate, which is zero at the bottom of the sample as shown in Figure 3.2.
The formulation of the interface elements allows not only separation, but also sliding of
finite amplitude and arbitrary rotation of the surfaces to (Hibbitt, Karlsson, & Sorenson
Inc., 1994). Figure 3.3 shows the deformation of the finite element mesh in the area near
the top of the interface and the outer edge of the punch under a typical pushout load. An
open zone develops at the top of the interface, and interface slippage is apparent below it. In
Figure 3.4 the mesh surrounding the debond tip is shown under the same pushout load.
The interface slips above the debond tip and sticks below it.
Although the materials were linear elastic, the solution was nonlinear due to contact
being lost across part of the interface as the punch load increased and due to the energy lost
from the nonconservative frictional shear stress generated by the debonded portion of the
fiber sliding with respect to the matrix. Within each finite element run the pushout problem
was solved in two steps. The thermal/chemical shrinkage load was applied in step one, and
43
FiberMatrix
Punch
Samplesupport
rp
ro
t
r
z
rs
r f
ld
Figure 3.2 Schematic of the coordinate axes and relevant dimensions.
44
Open zone
Fiber Matrix
Cornerof punch
Figure 3.3 Finite element mesh over a region including the top of the interface.
45
Cracktip
Fiber Matrix
Figure 3.4 Finite element mesh over a region surrounding the debond tip.
46
the mechanical load was applied to the fiber top face in step two. Within each step, the load
was applied in increments. For the first solution attempt, the entire load was applied. If the
solution of the field equations was not converging quickly enough according to criteria
placed on whether the field equations were satisified (peak force residual < 0.005 N) and the
largest correction to a nodal variable (largest change in incremental displacement < 0.01
µm), the load was applied in smaller increments. Within each increment ABAQUS solved
the governing balance equations (coupled with constraints written for the interface elements)
iteratively using a modified Newton’s method. If, at the end of an iteration, one or more of
the constraints in any of the interface elements was violated, the interface element(s) in
question was(were) allowed to open or slip, and a new iteration was started. Typically,
from 1 to 16 increments per step were necessary, depending on the debond length, and less
than 10 iterations per increment were needed.
The accuracy of the solution was investigated by checking whether the continuity
conditions at the interface and the boundary conditions were satisfied. In the bonded
portion of the interface, interface elements were not used. Consequently, slipping or
opening could not occur. Since the displacements along the interface side of an element (or
any side) depend only on the displacements of the nodes on the corresponding side of the
element, the displacements must be continuous across element boundaries and therefore
across the bonded part of the interface. In the closed and debonded part of the interface the
radial displacements of corresponding fiber and matrix nodes at the same position along the
interface in the undeformed mesh were within 0.001 µm of each other in the deformed
mesh. The axial displacements were also checked to ensure that the fiber displaced
downward more than the matrix at every point along the sliding region of the interface.
In the debonded part of the interface the radial and shear stress components in
corresponding nodes in the fiber and matrix were within 0.0001 MPa of each other. In the
bonded portion of the interface, the finite element code calculated the interface stresses only
at the nearest integration points and extrapolated them to the interfaces so the continuity of
47
the radial and shear stresses could not be checked because of the error associated with
extrapolation. Figure 3.5 shows a graph of the interface stress components from the
solution of a typical polyester/epoxy pushout problem. In Figure 3.5, the magnitude of the
shear stress equals the coefficient of friction ( µ = 0.52) times the magnitude of the radial
stress within 0.001 MPa over the debonded part of the interface ( z = 2480 µm to 5360
µm), and both the radial and shear stress are zero over a small open zone at the top of the
interface. In the bonded part of the interface the stresses were calculated at a particular
position along the interface by averaging the stress components extrapolated from the four
adjacent gauss quadrature integration points. ABAQUS used three integration points in
each interface element; therefore the stresses at each fiber and matrix node along the
debonded part of interface that were reported by the finite element code did not require
extrapolation or averaging and were used directly in Figure 3.5.
One difficulty that should be noted is displayed in Figure 3.6. Near the crack tip on the
debonded side of the interface crack, Coulomb friction was satisfied, but the stresses
oscillated significantly from node to node (element length in the z direction = 6.0 µm) as
the debond tip was approached. The oscillations were disregarded because as the mesh was
refined near the crack tip, the amplitude of the oscillations decreased. Although the stresses
oscillated as the singularity at the crack tip was approached, the interface stresses did not
oscillate as the singularity at the bottom of the interface (element length = 31.0 µm) was
approached. Kurtz and Pagano (1991) reported this same phenomenon near a singularity in
their elastic solution of a fiber embedded in a matrix under a thermal load.
3.3 Top debond—polyester/epoxy
Pushout of the top debonding system, polyester/epoxy, was modeled first. The pushout
curve for a representative test is shown in Figure 2.6. Although the epoxy matrix (EPON
828/DETA) was cured at room temperature around the fully cured polyester fiber,
significant chemical shrinkage of the matrix took place during cure. Evidence of the matrix
48
−100
−50
0
50
100
150
0 800 1600 2400 3200 4000 4800 5600
Inte
rfac
ial s
tres
s (M
Pa)
Axial coordinate, z (µm)
Radial stress
Shear stress
Top of sample at z = 5360 µmDebond tip at z = 2480 µm
Figure 3.5 Radial and shear stress along the interface for a typical polyester/epoxy
problem.
49
−100
−50
0
50
100
150
2400 2450 2500 2550 2600 2650 2700 2750 2800
Inte
rfac
ial s
tres
s (M
Pa)
Axial coordinate, z (µm)
Radial stress
Shear stress
Debond tip at z = 2480 µm
Figure 3.6 Variation of interfacial stresses near the crack tip in a typical problem.
50
shrinkage is seen in Figure 2.6, where a substantial load is needed to continue to slide the
fiber within the matrix after total debond. Also, as described in Section 2.6.2 and Appendix
A, an average radial stress at the interface was measured photoelastically and found to be
approximately equal before and after interface debonding, indicating that the compressive
radial stress distribution along the interface is due to processing and not due to fiber
roughness. The matrix shrinkage was determined with a finite element analysis by letting
the fiber have zero shrinkage and adjusting the matrix shrinkage until the average radial
stress along the interface was –5.68 MPa as was measured photoelastically. A matrix
shrinkage of 0.0022 was calculated this way. The boundary and continuity conditions for
the finite element analysis are described in the next section. Appropriate simulation of both
the chemical shrinkage and the pushout load was necessary to calculate the debond length
as a function of force.
After processing, the ends of the sample were cut to produce parallel top and bottom
faces so that the punch load could be applied perpendicular to the fiber end. The sample
cutting process is difficult to simulate because the exact curvature needed before processing
to result in flat faces after processing would have to be determined iteratively. Instead, the
finite element simulation in the current work starts with an unloaded sample that has flat top
and bottom faces and allows the faces to be deformed as the matrix chemical shrinkage is
applied.
3.3.1 Modeling procedure
The concept behind the computation of debond length is straightforward. A
displacement from the progressive debonding portion of Figure 2.6 is chosen and machine
compliance and alignment distance is subtracted. Machine compliance is the amount the
test fixture stretches during loading and was measured as 0.42 µm/N. Alignment distance
is the displacement required to generate a load large enough for all parts of the test fixture to
become seated and for the entire punch face to contact the fiber. Alignment distance can be
51
seen, for example, in Figure 2.6 as the point where the dashed line through the bonded part
of the pushout curve crosses the displacement axis at approximately 52 µm. The chosen
displacement, adjusted for machine compliance and alignment, and its corresponding force
from the force–displacement curve will be referred to as d1 and F1, respectively. The matrix
shrinkage and this displacement, d1 , are applied to the sample with an initial estimate of top
debond length, and the load at the top of the fiber is compared with F1. If the force
calculated is greater than the experimentally measured force, the debond length is increased
to create a more compliant structure, and the finite element analysis is conducted again for
the same matrix shrinkage and applied displacement. This routine is repeated until a
debond length is chosen that results in a load equal to F1 at the top fiber face. This series
of steps can be conducted for several force–displacement pairs in the progressive debonding
portion of the pushout data to obtain a plot of force versus debond length.
A schematic of the actual boundary conditions needed to simulate the pushout test is
shown in Figure 3.7. Since there are several important displacements of the punch nodes
(nodes at the top of the fiber where a displacement from the punch is applied) that must be
discussed, Figure 3.8 is provided to illustrate the relative magnitudes of these displacements.
Because the interface is assumed to be bonded along its entire length during processing, the
chemical matrix shrinkage is first applied to the mesh with no interface debond while the
bottom node located at the edge of the hole in the sample support is fixed in the axial
direction. The bottom of the sample is not forced to be flat in the simulation because the
bottom of the sample was not constrained during cure. The axial displacements of the
punch nodes dt1(r,t) and the axial and radial displacements of the bottom matrix nodes at
or beyond the support hole radius, [uzm ]1(r,0) and [ur
m ]1(r,0), are recorded. This finite
element calculation is shown schematically in Figure 3.7a.
The ABAQUS input file is then modified so that the mesh is the same except that
interface elements are added to simulate a debond. The length of the top debond that is
included is estimated as the measured debond length. If the measured debond length was
52
(b)
(a)
[urm ]1(r,0)
[uzm ]1(r,0)
[urm ]1(r,0)
[uzm ]1(r,0)
1 2
d1 + dt1(r,t)
dt1(r,t)
ld
ld
ld
t
Figure 3.7 Schematic of the modeling procedure for polyester/epoxy: (a) for the first
finite element run, the complete interface is bonded, (b) for the second
finite element run, a debond is added at the top of the interface.
53
Unloaded 2nd run, step 11st run 2nd run, step 2
(a) (c)(b) (d)
ld
ld
dt1(r,t) dt1(r,t)dt2 (r,t)
d1
Figure 3.8 Relative displacement at the top of the fiber for each phase of the
polyester/epoxy finite element analysis: (a) unloaded, (b) actual
deformation from matrix shrinkage, (c) matrix shrinkage with top
debond, (d) displacement from pushout test added.
54
not available, any initial estimate less than the thickness of the sample would work. Finally,
the matrix shrinkage is applied in step 1 of the finite element analysis of the new mesh (see
Figure 3.7b), and in step 2 the punch nodes of the matrix are moved to the positions where
they would be after applying the matrix shrinkage to a mesh with no debond, dt1(r,t), plus
the displacement applied by the pushout probe, d1 . The use of dt1(r,t) + d1 rather than
dt2 (r,t) + d1 in step 2 is necessary since the release of the axial stress in the fiber due to
the appearance of the interface debond in the new mesh occurs after the fiber pushout test
has commenced. If dt2 (r,t) + d1 is used instead of dt1(r,t) + d1 , the finite element
analysis under-predicts the slope of the fully bonded portion of the pushout data.
The application of a constant displacement to the punch nodes in step 2 of the second
finite element run produces a nonuniform axial compressive stress across the top surface of
the fiber that increases sharply with the radial coordinate from the center of the fiber to the
punch edge. The axial stress at the punch nodes is recorded along with the radii of the
punch nodes to calculate the total load applied to the fiber. Quadratic splines are fit through
these axial stress versus r data points. The resulting continuous function of σ zz (r,t) is
integrated with respect to the radial coordinate according to:
F = 2πrσ zz (r,t)dr0
r p
∫ , (3.1)
to determine the total load, F , on the fiber face which is compared to F1.
3.3.2 Boundary conditions
The boundary conditions and continuity conditions for the processes depicted in Figure
3.7 are listed in Tables 3.1, 3.2, and 3.3. In each case, r is the radial coordinate with r
= 0 at the center of the fiber, and z is the axial coordinate with z = 0 at the bottom of the
sample. The fiber radius is r f , and the matrix radius is ro . The actual fiber pushout
55
samples are rectangular since the front and back faces are parallel to the fiber and
perpendicular to the polariscope laser beam. The front and back faces must be flat and
parallel to each other for the laser beam to pass through them without distortion or partial
reflection. The matrix radius, ro , is half the distance between the front and back faces of the
sample. The length of the debond is ld , and the top of the sample is located at z = t before
the matrix chemical shrinkage is applied.
The friction between the punch face and the top of the fiber is neglected, and the friction
between the bottom face of the sample and the sample support is approximated by fixing the
bottom nodes in the r and z directions during the pushout phase of the simulation. The
bottom nodes were released in the r direction for a representative case to determine the
effect of representing a frictionless sample support surface. The results did not change
significantly. The fiber is modeled as a rod concentric within a hollow matrix cylinder, and
the problem is idealized as axisymmetric about r = 0 so that the hoop displacement is zero
everywhere within the sample.
3.3.3 Debond length
The coefficient of friction was calculated with both the shear lag solution (Section 2.6)
and the finite element analysis (Section 3.3.2). A load of 163 N and an embedded length of
5080 µm from section III of the polyester/epoxy pushout curve in Figure 2.6 resulted in a
coefficient of friction of 0.52 from shear lag theory. To find the coefficient of friction using
a finite element analysis, the boundary and continuity conditions of Tables 3.2 and 3.3 were
applied with ld = t and an applied displacement that caused slippage of the entire interface.
To suppress the solution corresponding to rigid body motion of the fiber, a linear spring
boundary condition was added at the center of the fiber along z = 0 with its axis parallel to
the z axis. A spring stiffness of 1 x 10-6 N/µm was used. For the displacement produced
at the point r = 0, z = 0 by displacing the top face of the fiber until
56
Table 3.1 Boundary and continuity conditions for matrix shrinkage of the mesh
with a fully bonded interface (schematically shown in Figure 3.7a).
z = 0 σrzf (r,0) = σ zz
f (r,0) = 0
σrzm (r,0) = σ zz
m (r,0) = 0
uzm (rs ,0) = 0
0 ≤ r ≤ r f
r f ≤ r ≤ ro
z = t σrzf (r,t) = σ zz
f (r,t) = 0
σrzm (r,t) = σ zz
m (r,t) = 0
0 ≤ r ≤ r f
r f ≤ r ≤ ro
r = 0 urf (0, z) = 0 0 ≤ z ≤ t
r = r f σrrf (r f , z) = σrr
m (r f , z)
σrzf (r f , z) = σrz
m (r f , z)
urf (r f , z) = ur
m (r f , z)
uzf (r f , z) = uz
m (r f , z)
0 ≤ z ≤ t
0 ≤ z ≤ t
0 ≤ z ≤ t
0 ≤ z ≤ t
r = ro σrrm (ro , z) = σrz
m (ro , z) = 0 0 ≤ z ≤ t
57
Table 3.2 Boundary and continuity conditions for matrix shrinkage of the mesh with
with a top debond of length ld (schematically shown in Figure 3.7b,
step 1).
z = 0 σrzf (r,0) = σ zz
f (r,0) = 0
σrzm (r,0) = σ zz
m (r,0) = 0
uzm (rs ,0) = 0
0 ≤ r ≤ r f
r f ≤ r ≤ ro
z = t σrzf (r,t) = σ zz
f (r,t) = 0
σrzm (r,t) = σ zz
m (r,t) = 0
0 ≤ r ≤ r f
r f ≤ r ≤ ro
r = 0 urf (0, z) = 0 0 ≤ z ≤ t
r = r f σrrf (r f , z) = σrr
m (r f , z)
σrzf (r f , z) = σrz
m (r f , z)
urf (r f , z) = ur
m (r f , z)
uzf (r f , z) = uz
m (r f , z)
σrrf (r f , z) = σrr
m (r f , z) ≤ 0
σrzf (r f , z) = σrz
m (r f , z) ≤ µσrrf (r f , z) = µσrr
m (r f , z)
urf (r f , z) = ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) < 0
urf (r f , z) ≤ ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) = 0
0 ≤ z ≤ t
0 ≤ z ≤ t
0 ≤ z ≤ t − ld
0 ≤ z ≤ t − ld
t − ld ≤ z ≤ t
t − ld ≤ z ≤ t
t − ld ≤ z ≤ t
t − ld ≤ z ≤ t
r = ro σrrm (ro , z) = σrz
m (ro , z) = 0 0 ≤ z ≤ t
58
Table 3.3 Boundary and continuity conditions for fiber pushout of the mesh with
with a top debond of length ld (schematically shown in Figure 3.7b,
step 2).
z = 0 σrzf (r,0) = σ zz
f (r,0) = 0
σrzm (r,0) = σ zz
m (r,0) = 0
urm (r,0) = [ur
m ]1(r,0)
uzm (r,0) = [uz
m ]1(r,0)
0 ≤ r ≤ r f
r f ≤ r ≤ rs
rs ≤ r ≤ ro
rs ≤ r ≤ ro
z = t uzf (r,t) = dt1(r,t) + d1
σrzf (r,t) = 0
σ zzf (r,t) = 0
σrzm (r,t) = σ zz
m (r,t) = 0
0 ≤ r ≤ rp
0 ≤ r ≤ r f
rp ≤ r ≤ r f
r f ≤ r ≤ ro
r = 0 urf (0, z) = 0 0 ≤ z ≤ t
r = r f σrrf (r f , z) = σrr
m (r f , z)
σrzf (r f , z) = σrz
m (r f , z)
urf (r f , z) = ur
m (r f , z)
uzf (r f , z) = uz
m (r f , z)
σrrf (r f , z) = σrr
m (r f , z) ≤ 0
σrzf (r f , z) = σrz
m (r f , z) ≤ µσrrf (r f , z) = µσrr
m (r f , z)
urf (r f , z) = ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) < 0
urf (r f , z) ≤ ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) = 0
0 ≤ z ≤ t
0 ≤ z ≤ t
0 ≤ z ≤ t − ld
0 ≤ z ≤ t − ld
t − ld ≤ z ≤ t
t − ld ≤ z ≤ t
t − ld ≤ z ≤ t
t − ld ≤ z ≤ t
r = ro σrrm (ro , z) = σrz
m (ro , z) = 0 0 ≤ z ≤ t
59
the entire interface just began to slip, the spring developed a negligible load of less than
0.0001% of the total load at the top of the fiber.
The results from shear lag, finite element analysis, and Pagano and Tandon (1996) are
plotted in Figure 3.9. A horizontal dashed line at 163 N was also included to show where
the data from each model achieved the applied load measured experimentally. The current
finite element analysis computed a coefficient of friction of 0.75 which is in agreement with
µ = 0.78 determined by Pagano and Tandon while the shear lag coefficient of friction was
significantly lower (0.52).
Force versus debond length was calculated from the polyester/epoxy pushout curve of
Figure 2.6 using both a coefficient of friction of µ = 0.52 and µ = 0.75. The finite
element results are shown in Figure 3.10 along with the shear lag predicted debond length
for µ = 0.52. The first measurable debond length was 1.1 mm at an adjusted displacement
of 43.7 µm, therefore only displacements greater than 43.7 µm and the corresponding load
from the pushout test of 267 N were modeled. The finite element calculated debond length
for µ = 0.52 not only shows a large improvement over the shear lag calculated debond
length but also tracks the measured debond length within a 7% error along most of the
debond growth. When a coefficient of friction of 0.75 was used in the finite element
analysis for the displacements and corresponding loads applied during progressive
debonding, the predicted debond lengths were larger than the measured debond lengths by
as much as a factor of 2.5 for loads from the initial part of progressive debonding. For the
later part of progressive debonding, debond lengths approaching the thickness of the sample
still produced loads larger than the loads measured experimentally so there is no plot of
predicted debond length from the finite element analysis ( µ = 0.75) for loading beyond
–276 N. The coefficient of friction from shear lag theory used in conjunction with the finite
element analysis of data from progressive debonding closely predicted the measured
debond lengths, but use of the coefficient of friction from finite element analysis
60
−400
−350
−300
−250
−200
−150
−100
−500.5 0.6 0.7 0.8 0.9 1
Shear lagBechel & SottosPagano & TandonExperiment
Forc
e to
slid
e fi
ber
(N)
Coefficient of friction
Figure 3.9 Coefficient of friction versus force for fully slipping problem computed by
the LH&KP shear lag theory, finite element analysis, and Pagano and
Tandon’s model.
61
0
1
2
3
4
5
−400−380−360−340−320−300−280−260
MeasuredShear lag (µ=0.52)FE (µ=0.52)FE (µ=0.75)
Deb
ond
leng
th (
mm
)
Force (N)
Sample thickness = 5.36 mm
Figure 3.10 Comparison of measured, shear lag, and finite element calculated debond
length as a function of force ( µ = 0.52, µ = 0.75).
62
in the finite element analysis of data from progressive debonding did not predict debond
length sufficiently close to the measured debond lengths.
Shear lag theory, which is a more simplified method of modeling fiber pushout than the
finite element method, was more accurate when modeling frictional data than the finite
element method. On the other hand, the finite element method was more accurate than shear
lag when modeling progressive debonding data. The issue of shear lag being better for
frictional data (Eq. (2.14)) than for progressive debonding data (Eq. (2.13)) will be
discussed first. Eq. (2.14) has the following advantages over Eq. (2.13):
1) During frictional pushout, the debond length does not have to be predicted since it is
already known as ld = t . Predicting debond length depends on using the force and
displacement from the experiment. To include the extra displacement due to debonding
Eq. (2.8) was used in deriving Eq. (2.13). Eq. (2.14) is free of the assumption of a
simple expression for the deflection at the top of the fiber.
2) As shown in Figure 2.8, the shear lag theory’s formulation of equilibrium depends on
an assumption of a nearly constant axial stress on each cross-section of the debonded
part of the fiber. Near the interface debond tip, the stresses change rapidly, therefore
the assumption of a constant axial stress on a fiber cross-section will be less accurate
when the debond tip is present. During frictional pushout a crack tip is not present as it
is during progressive debonding.
3) By definition, the debond length during frictional pushout is longer than during
progressive debonding (greater by a factor of at least 1.5 in the current pushout tests)
so the assumption of plane strain over the debonded part of the fiber used when
calculating residual stresses is approximated more closely.
4) The punch loads during frictional pushout are lower than during progressive debonding
(by less than half in the current pushout tests) so the length of the interface that is
opened is smaller. Shear lag theory does not allow the interface to open so a smaller
open zone will lead to a more accurate result.
63
5) The axial residual force in the fiber away from the fiber ends, Pr , was calculated based
on plane strain assumptions, but is not actually constant through the thickness of the
sample since it must be zero at the top and bottom face of the fiber. The quantity, Pr ,
is needed in the derivation of the Eq. (2.14) but is not used to derive Eq. (2.13).
Not only are the assumptions in the derivation of the shear lag equation for frictional
pushout approximated better than the assumptions for progressive debonding, previous
results are available in the literature to support the use of shear lag for modeling frictional
data. Shear lag was shown by Mackin, Yang, and Warren (1992) to predict both the
magnitude and the slope of fully slipping pushout data using a constant coefficient of
friction when sapphire fibers were pushed relatively far with respect to the surrounding
glass matrix so that over 50% of the fiber was exposed.
If the shear lag theory is accepted as being accurate when modeling frictional pushout,
then the coefficient of friction from finite element modeling, a more rigorous formulation,
should have agreed with the µ calculated with shear lag. It is possible that the coefficient of
friction increased as the polyester fiber slid. Immediately after total debond the fiber
instantly slid 150 microns due to the sudden loss of constraint. During this sliding,
asperities may have broken from the fiber surface and built up in the interface to cause a
coefficient of friction that is greater during frictional pushout than during progressive
debonding. Increasing friction by this mechanism was also observed during pushout tests
conducted on SiC/Ti alloy composites by Roman and Jero (1992) and Kantos, Eldridge,
Koss, and Ghosn (1992). If the coefficient of friction was greater during frictional sliding
than during progressive debonding, the shear lag theory under-estimated the coefficient of
friction during frictional pushout and the finite element result may have been correct.
Another possibility that must be explored is that the finite element solution of the
frictional problem was inaccurate. There were some difficulties associated with the finite
element solution of fully slipping problems that were not observed when modeling
progressive debonding problems:
64
1) If a constant load is applied to the top of the fiber, the fully slipping problem is an
unstable problem—the further the fiber is pushed, the less load is required.
2) If a constant displacement is applied to the top of the fiber, the solution that minimizes
potential energy is a rigid body movement of the entire fiber. No axial stress develops
in the fiber, and a zero punch load results.
3) To get a solution other than the rigid body solution, a relatively compliant spring was
attached from ground to the bottom of the fiber (Pagano and Tandon used a spring
also). Varying the stiffness of the spring by an order of magnitude did not change the
results, but this test is not rigorous proof that the spring technique yields a solution in
which potential energy is minimized and the entire interface is just beginning to slip.
For a given coefficient of friction, the punch load calculated by finite elements was
smaller than the load calculated by shear lag (which produce a µ that worked well in
progressive debonding). The finite element analysis computed a load between the load
from a rigid body solution and the load from shear lag.
4) When a displacement was applied to the top of the fiber which was much greater than
the displacement to initiate slipping in the entire interface (displacements as great as
20% of the sample thickness), the punch load that was calculated remained the same.
The load should have decreased as the embedded length decreased. The polyester
fibers were not pushed far enough in the current pushout tests to make it possible to
measure the slope of the pushout data during frictional pushout, but intuitively the slope
should be negative since less of the fiber surface is in contact with the matrix as the
fiber is pushed out. The finite element solution of the fully slipping problem predicts a
zero slope.
These points indicate that the finite element solution of the fully slipping problem should be
investigated further. The more rigorous formulation introduces additional problems. A
fully dynamic formulation may be necessary to overcome the problems associated with the
65
fully slipping problem. Finally, it may be possible that a friction law other than Coulomb
friction is necessary to describe slipping in the inteface of polyester/epoxy.
3.3.4 Coefficient of friction
Results from the analysis of the progressive debonding portion of three separate
pushout tests (all on polyester/epoxy) are presented in Figures 3.11, 3.12, and 3.13 to
resolve some of the ambiguity associated with determining the coefficient of friction from
section III of the force–displacement curve. The three pushout samples that were chosen
were used because their interface strengths and dimensions made them fundamentally
different from each other while still being composed of the same materials. The three
samples that correspond to Figures 3.11 to 3.13 will be referred to as samples 1, 2, and 3,
respectively. The debond length was measured during the progressive debonding phase of
each of the pushout tests and used in the finite element analysis to predict the
force–displacement curve for various coefficients of friction. The plots in Figure 3.11 show
that µ = 0.52 most closely reproduces the force–displacement data curve the pushout test
on sample 1.
Figure 3.12 shows the pushout curve from sample 2 which had the same dimensions as
sample 1. The only difference between samples 1 and 2 was the interface strength of
sample 1, which was greater than the interface strength of sample 2 as evidenced by the
greater loads required in sample 1 to grow debonds of equal length. Figure 3.13 contains
pushout data from sample 3, which was 20% longer than sample 1. Based on the measured
debond length versus displacement, forces were computed using µ equal to 0.52 and 0.75
for samples 2 and 3. The computed forces are also shown in Figures 3.12 and 3.13. A
coefficient of friction of 0.52 produced forces much closer to those experimentally
measured than µ = 0.75 for both a sample of different size than sample 1 and a sample of
identical size but different interface strength than sample 1.
66
−500
−400
−300
−200
−100
0−50 0 50 100
Experimentµ = 0.40µ = 0.52µ = 0.60µ = 0.75
Forc
e (N
)
Displacement (µm)
FE fullybonded
Figure 3.11 Force–displacement curve from polyester/epoxy sample 1 and predicted
loads for various coefficients of friction.
67
−500
−400
−300
−200
−100
00 50 100
Experimentµ = 0.52µ = 0.75
Forc
e (N
)
Displacement (µm)
FE fullybonded
Figure 3.12 Force–displacement curve from polyester/epoxy sample 2 and predicted
loads for two coefficients of friction.
68
−600
−500
−400
−300
−200
−100
0−20 0 20 40 60 80 100 120 140
Experimentµ = 0.52µ = 0.75
Forc
e (N
)
Displacement (µm)
FE fullybonded
Figure 3.13 Force–displacement curve from polyester/epoxy sample 3 and predicted
loads for two coefficients of friction.
69
If the results of Figure 3.11 are considered to be a measurement of coefficient of
friction obtained independent of frictional pushout data, and if the coefficient of friction is
assumed to be the same for all the polyester/epoxy samples, then Figures 3.11 to 3.13 show
that if the coefficient of friction is known accurately apriori, the finite element method can
predict the debond length relatively accurately—especially for shorter debond lengths.
Hence, this computational method can be used to compute debond length versus force from
tests in which the differential shrinkage during processing is known, the entire interface
remains bonded throughout processing, and a portion of the experimental data can be
identified in which progressive debonding is thought to be occurring.
Regarding whether shear lag theory or the finite element method can more precisely
compute the coefficient of friction from section III force versus displacement data, there is
evidence in Figures 3.11–3.13 to support both possibilities. Figures 3.11–3.13 show that
the coefficient of friction calculated from shear lag theory, 0.52, can be used to reproduce
the force–displacement data during progressive debonding for three fundamentally different
samples. On the other hand, in Figures 3.12 and 3.13 the force calculated based on a
coefficient of friction of µ = 0.52 becomes lower than the measured force as the debond
grows. A slowly increasing coefficient of friction would be necessary to produce consistent
loads as the debond length increases. Therefore, even before total debond, it appears that as
the debonded portion of the fiber slides with respect to the matrix, the coefficient of friction
is increasing. Thus, there is evidence that the coefficient of friction may actually be greater
(possibly 0.75) in section III of the pushout data than in section II.
3.3.5 Fracture Toughness
Finally, the fracture toughness of the polyester/epoxy interface was computed using the
above finite element analysis based on the debond length, force, and displacement data
corresponding to the plots in Figure 3.11. A coefficient of friction of µ = 0.52 (which
reproduced the force–displacement curve accurately) was selected for the analysis. The
70
finite element simulation was run for a particular displacement and its corresponding
debond length from the progressive debonding part of the data. The total strain energy and
the frictional energy dissipated during loading were recorded and used to approximate the
potential energy at the end of the finite element simulation. The debond length was then
increased by 0.1 to 0.5% (depending on the length of the debond), and an equal
displacement was applied. Strain energy and the frictional energy dissipated were again
recorded. The mode II critical energy release rate was then calculated with the equation:
GIIc =U1 − U2
2πr f (ld 2 − ld1)−
U f 2 − U f 1
2πr f (ld 2 − ld1). (3.2)
In Eq. (3.2) the subscripts 1 and 2 stand for the original debond length and the incremented
debond length, respectively. The symbol for strain energy is U , and U f is the frictional
energy dissipated. As in earlier equations, r f and ld are the radius of the fiber and the
debond length, respectively.
The definition of GIIc given in Eq. (3.2) is based on evaluating the rate of change of the
total potential energy in the model with respect to crack growth by numerical differentiation
(Anderson, 1991), with the addition of a term that has been included to compute the rate of
change of frictional energy dissipated with respect to crack growth. The calculation of
interfacial toughness is based on the notion that the debond length will increase by a
differential increment whenever the stress state is such that the decrease in strain energy
from an increment of debond growth is equal to the energy consumed by friction during the
increment of debond growth plus the energy required to debond an increment of the
surface.
Several assumption are made when Eq. (3.2) is used to compute GIIc . The debond
length is assumed to increase continuously and in a stable fashion during progressive
debonding so that each load during progressive debonding is the critical load required to
cause the onset of further debond growth. This assumption was not always satisfied during
the pushout test. At times, the debond crack tip stopped moving and later jumped
71
forward. This problem was overcome by calculating GIIc at several debond lengths. Eq.
(3.2) is also based on the assumption that the debond tip is loaded primarily under mode II
conditions. Finally, the difference between ld1 and ld 2 is assumed to be small enough to
approximate an infinitesimal increase in debond length. To satisfy the last assumption, the
difference between ld1 and ld 2 was reduced until any further reduction changed GIIc by less
than 1% (the finite element mesh around the crack tip was exactly the same for each debond
length).
The mode II fracture toughness was determined at several displacements during
progressive debonding and the results are shown as a function of debond length in Figure
3.14. If the interface strength was uniform in the pushout sample, the computed interface
toughness should also have remained constant as a function of debond length. As Figure
3.14 shows, the finite element computed fracture toughness increased from about 70 J/m2
to 180 J/m2. The fracture toughness (389 J/m2) obtained in Section 2.6.3 by fitting the
progressive debonding force–displacement data to Eq. (2.13) of the shear lag analysis is
plotted as a constant with respect to debond length. Also, shown in Figure 3.14 is the
fracture toughness (103 J/m2) obtained when the measured debond length versus force data
were fit to Eq. (2.12) of the LH&KP shear lag solution.
Several forces and the corresponding debond lengths were substituted into Eq. (2.12)
point by point to determine whether the shear lag calculation of fracture toughness also
varied with debond length or remained constant. The GIIc calculated point by point from
shear lag decreased from 150 J/m2 to 70 J/m2 as the debond length increased from 1.3 to
3.3 mm. When the measured debond lengths are used in shear lag theory instead of
allowing the theory to predict debond length, the average fracture toughness calculated is
nearly the same as the average fracture toughness calculated by the finite element analysis of
progressive debonding data. Both averages are approximately 110 J/m2. If the shear lag
theory could be modified to predict debond length more accurately, the computation of
interfacial fracture toughness would be more accurate.
72
0
100
200
300
400
500
1 1.5 2 2.5 3 3.5
Frac
ture
toug
hnes
s (J
/m2 )
Debond length (mm)
Shear lag curve fit (from F-d)
FE pt. by pt. (from F-d or F-ld)
Shear lag pt. by pt. (from F-ld)
Shear lag curve fit (from F-ld)
Figure 3.14 Fracture toughness versus debond length from shear lag theory and finite
element analysis.
73
3.4 Bottom debond—steel/epoxy
The finite element modeling of the steel/epoxy model composite was similar to the
modeling of the polyester/epoxy composite described in Section 3.3. The epoxy matrix for
this system (EPON 828/PACM) was cured at 150˚C resulting in a ∆T of –125˚C. The steel
fiber coefficient of thermal expansion was 12 x 10-6/˚C, and the matrix coefficient of
thermal expansion was measured with a thermal mechanical analyzer (Perkin Elmer) as 68
x 10-6/˚C. The coefficient of friction was first calculated with shear lag theory. The
differential shrinkage from processing was input into Eq. (2.6) along with the force
(–275 N) and embedded length values at the data point corresponding to the first peak in the
frictional pushout section of the pushout curve in Figure 2.4. A coefficient of friction of µ
= 0.33 for the steel/epoxy interface was calculated. When µ = 0.33 was used in the finite
element analysis of the fully slipping problem, a load of –236 N was determined.
For the same coefficient of friction, the finite element method obtained a load which is
close to the average of the loads at the first peak and first trough of the frictional portion of
the pushout curve while shear lag theory obtained a load that corresponds to the first peak.
Shear lag theory determined a coefficient of friction that predicted the progressive
debonding force–displacement curve accurately for polyester/epoxy. Also, the first peak in
the frictional data is the most appropriate load for modeling the fully slipping steel/epoxy
problem since the steel fiber did not slip until the first peak was reached. Therefore, µ
= 0.33 was used in the finite element modeling of progressive debonding in steel/epoxy.
Inspection of the pushout samples in the polariscope indicated that debonds 3 to 6 mm
long formed at the fiber ends of 30 mm long samples during cooling from the processing
temperature of 150˚C to room temperature. The ends of the sample containing these
debonds were removed, and the sample was inspected for debonds. Once again, from
cutting and residual stresses, debonds 1 to 3 mm long grew from the top and bottom
surfaces. These debonds were carefully measured and included in the finite element
analysis. The length of the initial top and bottom debonds will be designated as li1 and li2 ,
74
respectively. Similar initial debonds before or after cutting were not found in the
polyester/epoxy composites because the residual stresses generated by the chemical
shrinkage of the matrix in the polyester/epoxy composite were smaller than necessary to
break the bond between the fiber and the matrix.
3.4.1 Modeling procedure
The steps in the finite element simulation, illustrated in Figure 3.15, are similar to the
steps in the modeling of the top debonding polyester/epoxy composite except that the initial
debonds are included. The relevant displacements of the top of the fiber are compared
schematically in Figure 3.16. The thermal shrinkage in Figure 3.15a and step 1 of Figure
3.15b is the differential shrinkage between the fiber and the matrix from the temperature
drop during processing. The chemical shrinkage is zero, because it is assumed that the
chemical reaction is completed at the peak processing temperature when the matrix is above
its glass transition temperature and can sustain very little stress. Most of the stress that
develops at the peak temperature (150˚C) would be relaxed since the peak temperature is
held for one hour. For this bottom debonding system, dt1(r,t) < dt2 (r,t) where dt1(r,t) is
the downward displacement of the top of the fiber due to the temperature drop when only
the initial debonds are present, and dt2 (r,t) is the downward displacement of the fiber due
to the temperature when the top initial debond and the bottom debond from the pushout load
are present. The bottom debond caused by the pushout load (length ld ) is longer than the
initial debond (length li2 ) so the debond of length ld allows the fiber to relax downward
more under the thermal load from processing. As before, dt1(r,t) is used as the reference
displacement of the fiber top surface after processing because the bottom debond grows to
length ld only after the punch has applied some displacement to the fiber.
A simulation of the initial linear part of the force–displacement curve was conducted by
applying increasing loads in step 2 of Figure 3.15b without increasing ld ( ld = li2 ) since
75
(b)
(a)
[urm ]1(r,0)
[uzm ]1(r,0)
[urm ]1(r,0)
[uzm ]1(r,0)
d1 + dt1(r,t)
t
t
dt1(r,t)
ld ld ld
li1
li1li1
li1
li1
li2li2
1 2
Figure 3.15 Schematic of finite element analysis boundary conditions for steel/epoxy:
(a) for the first finite element run, only the initial debonds are present,
(b) for the second finite element run, a debond is added at the bottom
of the interface.
76
Unloaded 2nd run, step 11st run 2nd run, step 2
(a) (c)(b) (d)
ld ld
dt1(r,t)
d1
dt1(r,t) dt2 (r,t)
Figure 3.16 Relative displacement at the top of the fiber for each phase of the
steel/epoxy finite element analysis: (a) unloaded, (b) actual
deformation from thermal shrinkage, (c) thermal shrinkage with bottom
debond added, (d) displacement from pushout test added.
77
the bottom debond had not grown past its initial length for that part of the data. The slope
of the data (15.5 N/µm) in section I (loading before progressive debonding) was predicted
with less than a 1 percent error by the finite element solution. This comparison in the linear
region verifies the choice of dt1(r,t) rather than dt2 (r,t) and also illustrates the necessity of
including the initial debonds. The procedure for deriving force versus debond length from
force and displacement pairs chosen from the progressive debonding portion of the
steel/epoxy pushout data was the same as for polyester/epoxy except that ∆T = –125˚C was
applied to the entire sample containing the initial debonds instead of applying a matrix
shrinkage strain of 0.0022.
3.4.2 Boundary conditions
The boundary and continuity conditions for the calculation of dt1(r,t) and the radial and
axial displacements of the bottom nodes are shown schematically in Figure 3.15a and listed
in Table 3.4. The boundary and continuity conditions for the application of the thermal load
from processing and the punch load from pushout testing are shown schematically in
Figure 3.15b and listed in Tables 3.5 and 3.6.
3.4.3 Results
Although the method described to calculate debond length for the steel/epoxy composite
was similar to the method described for the polyester/epoxy composite and the predicted
force versus debond length function was accurate for polyester/epoxy, the results for the
steel/epoxy simulation were not as accurate. The debond length calculated was, generally,
only 20% of the measured debond length. This problem was solved by fabricating the
steel/epoxy model composite in a longer mold, which allowed the test samples to be cut
further from the ends of the raw composite. Modeling of the pushout data from the
samples cut further from the ends of the mold produced more accurate predictions of
debond length. A possible explanation of the improvement in the results for
78
Table 3.4 Boundary and continuity conditions for differential shrinkage during cool
down after processing for the mesh with initial debonds of length li1 (top)
and li2 (bottom) produced by cutting and/or residual stresses
(schematically shown in Figure 3.15a).
z = 0 σrzf (r,0) = σ zz
f (r,0) = 0
σrzm (r,0) = σ zz
m (r,0) = 0
uzm (rs ,0) = 0
0 ≤ r ≤ r f
r f ≤ r ≤ ro
z = t σrzf (r,t) = σ zz
f (r,t) = 0
σrzm (r,t) = σ zz
m (r,t) = 0
0 ≤ r ≤ r f
r f ≤ r ≤ ro
r = 0 urf (0, z) = 0 0 ≤ z ≤ t
r = r f σrrf (r f , z) = σrr
m (r f , z)
σrzf (r f , z) = σrz
m (r f , z)
urf (r f , z) = ur
m (r f , z)
uzf (r f , z) = uz
m (r f , z)
σrrf (r f , z) = σrr
m (r f , z) ≤ 0
σrzf (r f , z) = σrz
m (r f , z) ≤ µσrrf (r f , z) = µσrr
m (r f , z)
urf (r f , z) = ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) < 0
urf (r f , z) ≤ ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) = 0
0 ≤ z ≤ t
0 ≤ z ≤ t
li2 ≤ z ≤ t − li1
li2 ≤ z ≤ t − li1
0 ≤ z ≤ li2 & t − li1 ≤ z ≤ t
0 ≤ z ≤ li2 & t − li1 ≤ z ≤ t
0 ≤ z ≤ li2 & t − li1 ≤ z ≤ t
0 ≤ z ≤ li2 & t − li1 ≤ z ≤ t
r = ro σrrm (ro , z) = σrz
m (ro , z) = 0 0 ≤ z ≤ t
79
Table 3.5 Boundary and continuity conditions for differential shrinkage during cool
down after processing for the mesh with initial top debond of length li1 and
bottom debond length of ld (schematically shown in Figure 3.15b, step 1).
z = 0 σrzf (r,0) = σ zz
f (r,0) = 0
σrzm (r,0) = σ zz
m (r,0) = 0
uzm (rs ,0) = 0
0 ≤ r ≤ r f
r f ≤ r ≤ ro
z = t σrzf (r,t) = σ zz
f (r,t) = 0
σrzm (r,t) = σ zz
m (r,t) = 0
0 ≤ r ≤ r f
r f ≤ r ≤ ro
r = 0 urf (0, z) = 0 0 ≤ z ≤ t
r = r f σrrf (r f , z) = σrr
m (r f , z)
σrzf (r f , z) = σrz
m (r f , z)
urf (r f , z) = ur
m (r f , z)
uzf (r f , z) = uz
m (r f , z)
σrrf (r f , z) = σrr
m (r f , z) ≤ 0
σrzf (r f , z) = σrz
m (r f , z) ≤ µσrrf (r f , z) = µσrr
m (r f , z)
urf (r f , z) = ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) < 0
urf (r f , z) ≤ ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) = 0
0 ≤ z ≤ t
0 ≤ z ≤ t
ld ≤ z ≤ t − li1
ld ≤ z ≤ t − li1
0 ≤ z ≤ ld & t − li1 ≤ z ≤ t
0 ≤ z ≤ ld & t − li1 ≤ z ≤ t
0 ≤ z ≤ ld & t − li1 ≤ z ≤ t
0 ≤ z ≤ ld & t − li1 ≤ z ≤ t
r = ro σrrm (ro , z) = σrz
m (ro , z) = 0 0 ≤ z ≤ t
80
Table 3.6 Boundary and continuity conditions for fiber pushout of the mesh with
initial top debond of length li1 and bottom debond length of ld
(schematically shown in Figure 3.15b, step 2).
z = 0 σrzf (r,0) = σ zz
f (r,0) = 0
σrzm (r,0) = σ zz
m (r,0) = 0
urm (r,0) = [ur
m ]1(r,0)
uzm (r,0) = [uz
m ]1(r,0)
0 ≤ r ≤ r f
r f ≤ r ≤ rs
rs ≤ r ≤ ro
rs ≤ r ≤ ro
z = t uzf (r,t) = dt1(r,t) + d1
σrzf (r,t) = 0
σ zzf (r,t) = 0
σrzm (r,t) = σ zz
m (r,t) = 0
0 ≤ r ≤ rp
0 ≤ r ≤ r f
rp ≤ r ≤ r f
r f ≤ r ≤ ro
r = 0 urf (0, z) = 0 0 ≤ z ≤ t
r = r f σrrf (r f , z) = σrr
m (r f , z)
σrzf (r f , z) = σrz
m (r f , z)
urf (r f , z) = ur
m (r f , z)
uzf (r f , z) = uz
m (r f , z)
σrrf (r f , z) = σrr
m (r f , z) ≤ 0
σrzf (r f , z) = σrz
m (r f , z) ≤ µσrrf (r f , z) = µσrr
m (r f , z)
urf (r f , z) = ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) < 0
urf (r f , z) ≤ ur
m (r f , z) for σrrf (r f , z) = σrr
m (r f , z) = 0
0 ≤ z ≤ t
0 ≤ z ≤ t
ld ≤ z ≤ t − li1
ld ≤ z ≤ t − li1
0 ≤ z ≤ ld & t − li1 ≤ z ≤ t
0 ≤ z ≤ ld & t − li1 ≤ z ≤ t
0 ≤ z ≤ ld & t − li1 ≤ z ≤ t
0 ≤ z ≤ ld & t − li1 ≤ z ≤ t
r = ro σrrm (ro , z) = σrz
m (ro , z) = 0 0 ≤ z ≤ t
81
these samples is discussed in Section 3.4.4.
The pushout curve is shown in Figure 3.17 from a steel/epoxy test specimen cut from
the center of a sample cured in the longer mold (50 mm). The force–displacement curve
shows a drop in stiffness at the same displacement that the initial debond begins to grow.
Also, the slope of the pushout curve during progressive debonding continuously decreases
even in the initial stages of debond growth. The finite element simulation previously applied
to steel/epoxy pushout data (described in Sections 3.4.1 and 3.4.2) was applied to the data
in Figure 3.17 to calculate debond length.
The displacement (adjusted for alignment and machine compliance) versus debond
length function computed by the finite element analysis is compared to the measured
debond length in Figure 3.18. The finite element predicted debond length remains within
5% of the measured debond length for the first 1.5 mm of debond growth and within 10%
of the measured debond length for the final 1.7 mm of debond growth. The increasing
error for larger debond length may be caused by an underestimated value of the coefficient
of friction. The portion of the axial load generated by friction becomes more significant as
the debond length increases so an accurate coefficient of friction is more important at longer
debond lengths. Fracture toughness was not calculated using the finite element analysis and
could not be calculated with the LH&KP solution since the steel/epoxy composite
debonded from the bottom.
3.4.4 Importance of sample preparation
As described in Section 3.4.3, when steel/epoxy samples, obtained from the raw
composite by removing only the 5 to 6 mm debonds at the fiber ends, were pushout tested
and modeled, the debond length was under-predicted. The under-predicted debond lengths
and inconsistencies in the pushout curves from these samples can be explained with the
following hypothesis (presented in Figures 3.19a to 3.19e) about the events during the cool
down phase of processing for steel/epoxy. Figure 3.19a shows a schematic of a steel fiber
82
−600
−500
−400
−300
−200
−100
0 0
5
10
15
20
0 50 100 150 200 250 300 350
Forc
e (N
)
Deb
ond
leng
th (
mm
)Displacement (µm)
Debond length
Force
FE fullybonded
Figure 3.17 Pushout curve from a steel/epoxy sample cut far from the ends of the raw
sample. Curve separation matches point when debond starts to grow, and
after initial curve separation the sample continues to become more compliant
as the debond grows.
83
0
1
2
3
4
5
6
7
8
−600−550−500−450−400
Deb
ond
leng
th (
mm
)
Force (N)
Measured
Finite element (µ=0.33)
Figure 3.18 A comparison of the measured and finite element predicted debond lengths
for the steel/epoxy sample whose pushout data are shown in Figure 3.17.
84
==>
Α) ∆Τ = 0
) ∆Τ = 125
Α) ∆Τ = 0
Β) ∆Τ = 0 to 125
Β) ∆Τ = 0 to 125
(a)
(e)(d)
(c)(b)
T = T = 100° C T = 100° C150° C
75° CT = 25° CT =
C
° C
° C
° C
° C
° C
τ rz τ rz τ rz
τ rz
r
z
Figure 3.19 Schematic of interface bonding in steel/epoxy as cool down progresses
during processing.
85
embedded in an epoxy matrix at the peak processing temperature. The sample is stress free.
As the temperature drops below the processing temperature to 100˚C, a large τ rz shear
stress and a tensile radial stress develop near the fiber ends (Figure 3.19b). If the
fiber–matrix bond is not strong enough, the interface may open due the radial tensile stress
or stay closed and slip due to the shear stress. Debonds will form at the fiber ends as
shown in Figure 3.19c. These debonds were observed in the steel/epoxy samples. The
debonds form near the fiber ends partly because the bond between steel and epoxy is
relatively weak and partly because the radial clamping stress (zero in the open zone if an
open zone develops) is relatively small at 100˚C. The interface debonds grow to a length
that allows the friction along the closed portion of the debonded ends to relieve some of the
interfacial shear stress which develops at the debond tip from differential shrinkage in the
axial direction.
As the sample cools down further, the compressive radial stress increases, forcing the
matrix into better contact with the fiber surface and possibly closing some of the open zone
if an open zone develops (Figure 3.19d). The interface bond becomes stronger and can
sustain a larger shear stress and/or tensile a radial stress if one develops. At this point, it is
proposed, a portion of each of the debonds, toward the middle cross-section, stops
slipping. The process of the end debonds shortening during cool down is continuous until
room temperature is reached. If this mechanism actually occurs during cool down, at room
temperature the sample would have three sections as shown in Figure 3.19e. Section A is
the debonded portions at each end that can be identified by viewing the photoelastic fringes
after cool down. As stated, these debonds, which were present after processing, were cut
off before pushout tests were performed. Section B contains the portions of the interface
that stopped slipping at some temperature between 150˚C and room temperature
(approximately 25˚C). The magnitude of the axial residual stress stored in this section does
not correspond to a ∆T = –125˚C as is modeled by the finite element simulation described
in Section 3.4.2, but actually corresponds to a distribution of stress that varies from a
86
minimum at the limit of section B toward the fiber end to a maximum at the limit of section
B toward the fiber middle cross-section. In section C, the interface did not slip during the
entire cool down.
The experiment portrayed in Figures 3.20a to 3.20e was conducted to support the
debonding–during–cool–down theory. Steel/epoxy pushout samples were prepared by the
same steps as for the steel/epoxy pushout tests, which yielded poor finite element debond
length predictions. After cool down, the 30 mm long samples with 5 to 6 mm long debonds
had the debonded portions at the ends cut away. After cutting, 1 to 1.5 mm long debonds
appeared at the fiber ends and were measured in the polariscope. A schematic of the sample
at this stage is shown in Figure 3.20a. Since the ends were debonded, the fiber ends
extended a short distance from the matrix (approximately 5 microns).
The same pushout fixture that was used for the fiber pushout tests was capable of
measuring the length of exposed fiber (see Figure 3.20d). A sample was placed on the
sample support, and the punch was lowered until it was near the fiber surface but not in
contact with the surface as evidenced by a zero load on the load cell. Once near the fiber
surface, the punch was lowered in 2 µm increments until a load (less than 0.2 N) registered
on the load cell. This location was used as a reference height. The punch was moved
laterally until it was entirely over the matrix, and then it was lowered to the matrix surface.
This routine was repeated at four locations 90 degrees apart on the matrix within 20 µm of
the fiber (shown as X's in Figure 3.20e) to measure the distance between the matrix and the
reference location (bottom of the fiber). The four measurements were averaged, and the
average value was recorded as the length of exposed fiber to within ± 1 µm. This first
measurement of exposed fiber length was only used as a reference length and not compared
to the finite element prediction of exposed fiber length because the process of cutting off the
ends of the sample may have removed some of the exposed fiber end.
After the initial measurement of exposed fiber was conducted, the sample was turned
over so the measured end was over the sample support hole, and a compressive load was
87
δ1
==>
pushout load
δ2
Stepper motor
Load cell
Punch
Pushout sample
δ1 < FE calculation δ2 > FE calculation
X
X X
X
==>pushout load
(a) (b) (c)
(d) (e)
Figure 3.20 The fiber extension measurement: (a) schematic of two samples with
different debond lengths, b) schematic of experiment.
88
applied to the top of the fiber until the bottom debond grew to 3 mm as schematically shown
in Figure 3.20b. This debond grew from its original length (1 to 1.5 mm) to a length of 3
mm through part of or all of the section B region in Figure 3.19e. The composite sample
was turned over again so that the end with the 3 mm debond was upward. The exposed
length was measured as before allowing the measurement, δ1, which is the increase in
exposed fiber length from the additional 1.5 to 2 mm of crack growth, to be derived. Both
the sample with the interface conditions shown in Figure 3.20a and the interface conditions
shown in Figure 3.20b were loaded with ∆T = –125˚C in a finite element simulation. The
boundary and continuity conditions for the finite element analysis were the same as
described in section 3.4.2 and shown in Figure 3.15b, step 1 with ld equal to 3 mm, li1
equal to the length of the initial top debond, and µ = 0.33. The difference in the exposed
fiber length at the bottom was extracted from the finite element analysis and found to be
30% more than the measured difference, δ1.
Similarly, the debond was grown to 6 mm and the additional exposed fiber length, δ2,
was measured. The measured δ2, on average, was within 5% of the finite element predicted
δ2. These measurements and finite element simulations revealed that the amount of residual
axial stress unloaded by debond growth was less than expected in the early stages of
debonding and close to the expected amount as the interface crack grew further from the
sample surface. This difference between measured and calculated exposed fiber length
could be the result of less residual stress near the ends of the sample than expected as is
predicted by the present theory about the cool down phase of processing the steel/epoxy
model composite.
The theory about cool down also explains a commonly occurring phenomenon, shown
in Figure 3.21, that was observed in the steel/epoxy pushout data from the samples obtained
close to the ends of the raw composite. A dashed line through and extending from the
initial linear part of the force–displacement curve is provided to highlight where the sample
becomes more compliant from debond growth. Debond growth of 1.2 mm beyond
89
−700
−600
−500
−400
−300
−200
−100
0 0
5
10
15
20
0 50 100 150 200 250 300 350
Forc
e (N
)
Deb
ond
leng
th (
mm
)Displacement (µm)
Debond length
Force
FE fullybonded
Figure 3.21 Steel/epoxy pushout curve from sample cut from section B and C. Debond
grows 1.2 mm before the force–displacement curve becomes nonlinear.
90
−800
−700
−600
−500
−400
−300
−200
−100
0 0
5
10
15
20
25
30
0 100 200 300 400
Forc
e (N
)
Deb
ond
leng
th (
mm
)
Displacement (µm)
Debond length
Force
FE fullybonded
Figure 3.22 Pushout curve from a steel/epoxy sample cut from section B and C. After
initial nonlinearity, the slope of the force–displacement curve remains the
same over a significant additional displacement.
91
the initial debond length can be observed without any noticeable nonlinearity occurring in
the force versus displacement curve by referencing where the vertical dashed line crosses the
displacement versus debond length curve. The release of axial residual stress in the
debonded portion of the fiber and the relaxation in the axial direction of the matrix around
the debonded portion of the fiber contribute to the departure of the force–displacement
curve from linearity. If there was very little release of residual stress in the initial stages of
debonding, there would be less tendency of the force–displacement curve to turn over when
debond growth began.
A similar but slightly different phenomenon also occurred in the pushout curves from
some of the steel/epoxy specimens and is shown in Figure 3.22. The debond growth can be
seen to start at the instant the force–displacement curve separates from the dashed line
drawn through the pre-progressive debonding part of the experimental curve. From 350 N
to 475 N, the force–displacement curve actually has the same slope as the dashed line
through the initial linear part of the pushout data even though the debond length grows more
than 2 mm during this period. The sample should become more compliant as residual
stress is unloaded from the bottom end of the fiber during debonding, allowing the top face
of the fiber to relax downward, but the sample compliance remains nearly constant. A
smaller than expected (possibly near zero) axial residual stress in several millimeters of the
fiber near the ends of the sample would also help explain this situation.
For the debond length prediction results shown previously in Section 3.4.3, longer
samples (50 mm long) were fabricated allowing pushout samples to be cut further from the
end surfaces. The actual boundaries of cool down section (B) were not measured, so it
cannot be stated for certain that the samples were cut completely from within section C (see
Figure 3.19e). The initial 5 to 6 mm debonds were removed plus another 10 mm on each
end. The extra 10 mm that was removed may or may not have included all of section B.
The top image in Figure 3.23 shows the photoelastic fringe pattern from an uncut single
fiber steel/epoxy sample after cool down. The fringes near the fiber ends indicate the
92
presence of shear stress while the shear stress must be near zero away from the fiber ends
since the fringes disappear at points over 10 mm from the ends. A pushout sample, cut
from the zero shear stress region of the raw material in the top picture of Figure 3.23, is
shown at the same magnification in the lower image. The appearance of several photoelastic
fringes illustrates the redistribution of stresses, and the points of greatest fringe density near
each end show the tips of the initial debonds. The end effects apparently extend over the
entire sample since there is no cross-section of the sample without fringes.
3.5 Interface failure due to cutting
Another factor complicating the analysis of fiber pushout data was found when
fabricating steel/epoxy pushout samples using smaller diameter fibers. Figure 3.24 shows
an image of the photoelastic fringe pattern in a steel/epoxy pushout specimen that was cut
more than ten fiber diameters from the ends of the raw sample and prepared by the same
steps as the previous steel/epoxy test samples were. In this sample, the steel fiber had a
0.200 mm diameter instead of a 1.65 mm diameter as those described previously. The
sample thickness is approximately 12 fiber diameters as were some of the larger fiber
diameter (1.65 mm) samples. This sample represents a scaled-down version of the
steel/epoxy pushout samples studied earlier. Two distinct fringes are observed on either
side of the fiber along the middle one third of the fiber where the fiber is still bonded to the
matrix. For the larger fiber diameter of 1.65 mm, initial debonds of less than 0.5 fiber
diameter were found while for the same composite with a smaller fiber diameter, initial
debonds up to 5 fiber diameters long were measured as illustrated in Figure 3.24. Even
though the initial debonds are longer in terms of fiber diameters and constitute a larger
portion of the sample thickness, their lengths are nearly the same—0.5 to 0.7 mm. If the
initial debonds grew because of residual stress, their lengths should scale with the fiber
diameter if the interface strength is assumed to be the same for all fiber diameters. The
constant absolute length of the initial debonds indicates that the initial debonds are probably
93
2 mm
Figure 3.23 Top image shows a relatively long sample of steel/epoxy composite at
room temperature with photoelastic fringes near fiber ends. The bottom
image is of a sample cut from the center (section C) of the raw sample.
The stresses redistribute and small debonds form at the fiber ends.
94
0.5 mm
Figure 3.24 Steel fiber (200 µm diameter) in epoxy. After cutting, large debonds are
present at the top and bottom of fiber.
95
produced during cutting. The cutting process adversely affects a region at the surface of the
pushout sample, and the thickness of the region affected is independent of the sample
thickness. Thinner samples may be in greater danger of containing initial debonds that
extend over a large portion of the sample thickness.
The top and bottom faces of the pushout sample shown in Figure 3.24 were polished to
a 15 µm finish to determine if polishing would cause further fiber debonding. No change
in the photoelastic fringe pattern could be detected after polishing. Fiber pushout tests were
carried out on the smaller diameter steel/epoxy samples with the apparatus described in the
next chapter. After the pushout tests were conducted, the samples were placed in the
polariscope to observe the effect on the photoelastic fringe pattern. The original fringes that
were parallel to the fiber before pushout testing were not there afterwards, indicating that the
fringes were present because of interface cohesion. Also, the applied axial stress necessary
to debond a 0.200 mm diameter steel fiber was 50% greater than the axial stress required to
debond a 1.65 mm diameter steel fiber from a sample with the same thickness in units of
fiber diameters. The scaled-down steel/epoxy samples must have had a greater interface
strength than the 1.65 mm fiber diameter samples, which is evidence that the proportionally
longer initial debonds in the small diameter samples were not a result of a decrease in
interface strength.
3.6 Discussion
The finite element method was used to derive the interface debond length as a function
of force from the pushout data for a top debonding polyester/epoxy composite with
relatively small residual stresses and assumed perfect bonding over the entire fiber length
during processing. Debond length was computed as a function of force for samples as short
as three fiber diameters to within seven percent of the measured debond length when a
coefficient of friction of µ = 0.52 was used. Although the method is more time
consuming to apply than a closed form solution, the finite element analysis implemented for
96
the progressive debonding portion of the pushout data was able to capture the effects of the
open portions of the interface, the difference in size between the punch and fiber, and the
nonuniform residual stress field.
The finite element method was also used to compute a coefficient of friction from the
data following total debond in a representative pushout test of polyester/epoxy. The
computed value of coefficient of friction agreed with the coefficient of friction from the
variational model of Pagano and Tandon (1996), but was 44% greater than the coefficient of
friction determined with shear lag theory. The finite element analysis of the fully slipping
problem may be inaccurate, the coefficient of friction may have increased during the load
drop and sliding immediately following total debond, or τ = µσrr may not model the
model the actual friction in the interface of polyester/epoxy.
This finite element method was also used to analyze pushout data from a bottom
debonding composite with residual stresses large enough to debond the fiber ends during
processing. The position within the raw composite that the test specimen was cut from was
shown to be critical to the calculation of debond length for this steel/epoxy system. When
data was analyzed from samples cut near the ends of the raw sample, the debond length was
grossly underestimated. The force–displacement curves from these samples often showed
no decrease in compliance during the initial stages of debond growth. Force versus
displacement data from test samples cut at least 6 fiber diameters from the ends of the raw
composite became nonlinear at the onset of debond growth, as intuitively expected, with the
slope continuously decreasing for increasing debond growth. Application of the finite
element procedure to the data from the samples cut further from the ends of the raw sample
yielded calculated debond lengths within 10% of the measured debond lengths. A
coefficient of friction of µ = 0.33 calculated from shear lag theory was used for all of the
steel/epoxy finite element simulations.
A theory was proposed to explain why the proximity of the test sample to the ends of
the raw sample affected the pushout results. It was hypothesized that a portion of the fiber
97
opened or slipped during the first stages of cool down, and part of this debonded portion
closes and/or stops slipping due to increasing radial clamping stress as cool down
progresses. This process leaves a section of the fiber debonded, a section containing
residual axial stress developed during the entire temperature drop, and a section with less
residual axial stress than would be expected from the temperature drop of processing.
Measurements of the exposed fiber length indicated that a smaller residual axial stress is
acting near the ends of the fiber in samples cut close to the ends of the raw sample than is
calculated with a finite element simulation.
The existence of debonds in the steel/epoxy pushout samples at the fiber ends after
processing and after cutting was also shown. These debonds, that were present before any
pushout force was applied to the fiber, had to be included in the finite element simulation to
compute debond lengths close to the measured debond lengths. It may be difficult in other
composites, such as metal matrix composites with large residual stresses, to avoid initial
debonds such as the ones found in steel/epoxy from thermal loads and/or cutting . The
presence of these initial debonds significantly affects the slope of the initial linear portion of
the pushout curve. One solution to the problem of not being able to measure the length of
initial debonds may be to compare the slope of the section I pushout curve to the finite
element predicted slope. If the finite element calculation of the slope of the initial linear part
of the curve is too stiff, then it could be assumed that initial debonds of equal length at the
top and bottom of the fiber are responsible for the difference in stiffness. Initial debonds
could be incorporated in the model and their length adjusted until the finite element
computed slope and the measured slope match. Confidence in the measurement of machine
compliance would be necessary for this method to accurately calculate the initial debond
length.
Finally, the presence of initial debonds was also a problem in the steel/epoxy model
composites with smaller diameter fibers. The observation of initial debonds with a length
98
on the same order as in the larger diameter steel/epoxy composite indicates that the process
of cutting is responsible for the initial debonds.
Although the method described here can be used to accurately calculate debond length
for both a top and a bottom debonding system (even one with debonds present before
pushout testing), the level of knowledge required about the condition of the interface and the
residual stress state after processing is significant. There is also some question as to
whether the coefficient of friction remains constant during the entire pushout test and
whether Coulomb friction is a valid representation of friction during pushout testing. In
general, matrices of composites are not transparent and birefringent so the initial debonds, if
present, could not be measured in a polariscope prior to pushout testing. Care in sample
preparation must be taken, so that initial debonds are not present in the particular composite
that is being tested. If possible, tests should also be done to determine how far the test
samples should be cut from the ends of a raw sample. In fact, the difference in interface
strength calculated from different micromechanical interface strength tests on identical
composites (Herrara-Franco and Drzal, 1992), thought to be attributable to the assumptions
made in the analyses, may be partly attributable to the difference in how the samples were
cut and where in the composite they were cut from by the different researchers.
3.7 Future work
Interesting work remains to be done in the area of finite element simulation and other
analytical solutions of the fiber pushout test. If at least an approximate functional variation
of the temperature at which slipping stopped during cool down in the portion of the sample
depicted in section B of Figure 3.19e could be found, it may be possible to model test
specimens cut near the ends of the raw sample. This temperature function may be
determined by expanding on the fiber extension tests of Figure 3.20. The debond length
could be increased in small increments to locate the boundary of section B. This exercise
99
may not advance the cause of the fiber pushout test, but it could give some evidence to
support or negate the cool down theory proposed in Section 3.4.4.
Pushout tests could be done on the same composites tested in the current work with a
groove instead of a hole in the support as is often used in pushout testing (Eldridge, 1995;
Jero and Kerans, 1990). If the modeling procedure used to get the results shown in Figures
3.10 and 3.18 still gives accurate results, then evidence would be available that using
axisymmetric assumptions to model pushout tests done over grooves is acceptable.
Many questions remain about the discrepancy between the finite element calculated
coefficient of friction and the coefficient of friction that best reproduced the
force–displacement data from experiment. More finite element modeling of frictional
pushout data needs to be done to determine if the slope of force–displacement curve can be
predicted. The finite element mesh may have to be constructed with only a portion of the
fiber embedded in the matrix prior to loading to find the load necessary to fully slip the
fiber for embedded lengths less than the sample thickness.
100
4. HIGH TEMPERATURE FIBER PUSHOUT TESTS
4.1 Importance of interface strength versus temperature
Numerous research efforts have focused on the development of high performance, high
temperature, metal matrix composites. Emphasis has been placed on strength, thermal
stability, and oxidation resistance. Understanding the behavior of the fiber–matrix interface
over a range of temperatures is essential for designing composites that will have a high
service temperature. Several composites, such as silicon carbide fibers or alumina fibers
embedded in low density, high ductility titanium alloy, aluminum, or ceramic matrices, are of
current interest for high temperature applications. If the interface properties of these
composites could be assessed as a function of temperature, it may be possible to determine
why some fiber–matrix–coating combinations work well at high temperatures and why
others do not.
This motivation has been the initiative for several studies of composite interface
properties as a function of temperature. Chou, Barsoum, and Koczak (1991) performed
fiber pullout tests on SiC fibers in two different glass matrices over a range of temperatures
from room temperature to 500˚C. Chou and co-workers reported the interface strength as
the peak fiber pullout load divided by the fiber surface area. The model pushout
experiments of Chapter 2 showed that a debond often grows through a large portion of the
sample before the peak load is reached, making the average shear stress dependent on
sample thickness in the fiber pushout test. The same dependence on sample thickness is
probably also true for the fiber pullout test. Morscher, Pirouz, and Heuer (1990) performed
fiber pushout tests on an SiC reinforced reaction–bonded silicon nitride (RBSN) composite
with a high temperature micro–hardness tester. A Vickers indenter was used to apply the
pushout loads at temperatures up to 1350˚C. As in Chou, Barsoum, and Koczak’s work,
Morscher and co-workers reported average shear stress as a function of temperature.
101
Brun (1992) conducted pushout tests on SiC fibers in mullite, cordierite, and titanium
alloy matrices at temperatures up to 1100˚C in an argon atmosphere. The fibers were
loaded with a 75 µm diameter flat faced punch which was less likely to damage the fibers
than a sharp Vickers indenter, but the sample was supported only by the edges which were
several fiber diameters from the particular fiber being pushed out. Elevated temperature
fiber pushout tests were also carried out by Eldridge and Ebihara (1994) and Eldridge
(1995). Fiber pushout tests were performed in a vacuum chamber at temperatures that
ranged from room temperature to 1100˚C on two different SiC/titanium alloy composites
and an SiC/RBSN composite. Average shear stress was the criterion for interface failure.
In the work by Eldridge (1995), interrupted pushout tests showed that all of the composites
tested partially debonded at loads as low as 60% of the peak load so the average peak shear
stresses reported must be sample-thickness-dependent. This literature review emphasizes
the need for a more advanced analysis procedure than the calculation of the peak load
divided by the fiber surface area.
The method developed in Chapter 3 to evaluate the interfacial critical energy release
rate, GIIc , from pushout data can be applied to any continuous fiber composite composed
of materials that behave linear elastically (not necessarily near the interface crack tip) under
the loads imposed by the fiber pushout test. The interface must also be loaded
predominantly in shear to refer to the energy release rate that is obtained as mode II, and the
interface friction law must be either a constant shear stress or a Coulomb friction
formulation. As long as these requirements are realized, no additional restrictions are made
concerning the thickness of the sample, the length of the debonded region, the ratio of the
fiber and matrix moduli, or whether the debond grows from the top or bottom of the
sample. Therefore, the method of Chapter 3 could be used to determine the fracture
toughness of some of the metal matrix composites that have been shown to debond from
the bottom (Koss, Hellman, and Kallas (1993); Eldridge (1995)). Also, if the elastic
properties of the constituents remain nearly constant and linear elastic as the temperature
102
increases, pushout data at increased temperatures from these MMCs could be reduced to
GIIc versus temperature.
4.2 High temperature tests
The analysis method developed in this project could be applied to the pushout data cited
in the literature if the reported displacement does not contain the machine compliance and
the boundary conditions are the same as assumed in Sections 3.3.2 and 3.4.2.
Unfortunately, all the work mentioned in Section 4.1 reports only cross-head displacement
since the primary goal was to measure the peak load only. Further, the Vickers indenter
used in Morscher, Pirouz, and Heuer’s experiments may have burrowed into the fiber,
making their displacement measurement even less reliable for use in the current finite
element analysis. Sample bending may have been significant in Brun’s work since there
was no attempt to make the sample support span relatively close to the fiber diameter. In the
absence of usable pushout data from previous high temperature tests, an apparatus was
developed to conduct the fiber pushout experiment in an environment that would simulate
the service environment of MMCs so the interface strength of high temperature composites
could be studied as a function of temperature.
The remainder of the work described in this chapter centers on the development of this
experiment. The detailed design of the apparatus is discussed as well as results from
preliminary pushout tests conducted on an SiC/Ti-6-4 composite. Both pristine and
laterally fatigued SiC/Ti-6-4 samples were tested to determine if the fibers in the fatigued
samples were debonded. In these experiments, pushout curves containing identifiable
progressive debonding was difficult to obtain. Modifications to the apparatus that may
improve the measurement of the pushout data are outlined. With these improvements, the
high temperature apparatus may be able to obtain pushout data on MMCs (with a
sufficiently accurate displacement measurement) that could be reduced to an interface
toughness with the analysis of Chapter 3.
103
4.2.1 Sample preparation
Both pristine and fatigue loaded SiC/Ti-6-4 composites were received from 3M
Corporation. The loaded composite had been laterally fatigued for 25,000 cycles at 180
MPa with an offset of R = 0.1. Inspection of the fatigue samples under a microscope
revealed no outward sign of damage such as cracking or permanent deformation. The raw
materials were sectioned with a diamond wafering saw into bars approximately 15 mm long,
2 mm wide, and 0.3 to 1.0 mm tall with the fibers aligned vertically. The top and bottom
faces were ground parallel to each other with 40 micron diamond particle sandpaper and
polished to a 1 micron finish with diamond paste.
4.2.2 Apparatus
The high temperature fiber pushout micromechanical tester was modeled, with some
minor variations, after the apparatus described in Eldridge and Ebihara (1994). A schematic
of the tester is shown in Figure 4.1. Each major part of the setup is labeled with a capital
letter. A vacuum chamber (A), was designed to house the experiment in a controlled
atmosphere while allowing the test to be viewed in progress. A mechanical vacuum pump
(Welch model 1402) was available to reduce the pressure in the vacuum chamber to 10-3
Torr, after which the chamber could be flooded with high purity argon (1 part per million
impurity) to provide a virtually oxygen free atmosphere.
Bellows (B1 and B2) permit vertical motion of the punch and motion of the sample table
in three perpendicular directions. The punch is attached to a motorized actuator (B) on the
outside of the chamber while the sample table is attached to a three axis stage (I), which is
also outside the chamber. The actuator consists of a small DC motor turning a 10683:1
reducing gear box (Klinger, model BM4CC) that is mounted on a linear motion stage on the
top flange of the chamber. The actuator, when energized by the Newport PMC200
controller, drives the punch at a minimum velocity of 1.0 micron/second. Displacement
104
A
B
C
DE
F
G
H
I
B1
B2
Figure 4.1 Schematic of the high temperature fiber pushout experiment.
105
of the probe is derived from the revolutions of the DC motor which are measured by an
optical encoder on the DC motor's armature. The 100 to 110 µm diameter punches used on
the SiC fibers were machined from tungsten carbide (National Jet). The punch face that
contacts the sample is flat, and the diameter of the punch shank increases slowly away from
the flat face. A fiber can be moved at least 100 µm with respect to the matrix before the
sides of the punch touch the matrix depending on which SiC fiber is being tested.
Load is measured by sampling a Kistler piezoelectric charge transducer (C) at 5
samples/sec as in the model pushout tests of Chapter 2. Also, as in previous tests, the load
cell signal was conditioned by a Kistler dual mode amplifier and digitized by a Tektonix
TDS 420 oscilloscope. The force versus cross-head displacement data are post processed
to account for load cell drift and machine compliance.
The sample is heated by an infrared spot heater (Research, Incorporated, model 4085).
The heater (G) focuses thermal energy generated by a lamp, located at one focal point of an
elliptical reflector, into a one cubic centimeter area engulfing the sample (H) at the other
focal point of the reflector. Since the infrared heater must be air cooled, the heater and the
half of the reflector containing the lamp are located on the outside of the chamber next to a
quartz window built into the wall of the chamber. The other half of the elliptical reflector is
inside the chamber on the opposite side of the quartz window. A quartz window was
chosen because quartz is significantly more transparent to infrared light than glass and,
therefore, passes the infrared energy more efficiently into the chamber than a glass window
would. The sample temperature is measured by a thermocouple with its bead placed so it is
in contact with the steel sample support. The punch (D) is attached to the upper bellows by
an alumina ceramic rod to maintain thermal isolation of the sample. Similarly, an alumina
rod supports the sample (H) and rests on the bottom bellows.
A long distance microscope (Infinity model K2) with a parfocal doubler and a 15X
eyepiece (E) is positioned inside a recessed glass window on the outside of the vacuum
chamber opposite the infrared heater. The recessed window permits the microscope
106
objective to be within two inches of the sample and still be located outside the chamber. A
black and white, high resolution CCD (580 horizontal lines) camera attached to the
microscope obtains an image of the punch, the fiber(s) to be pushed out, and the
surrounding area. A four inch extension tube placed between the camera and the
microscope further enlarges the image. The image captured by the camera is displayed on a
12 inch monitor with 850 lines of resolution. The purpose of the video system is to aid in
aligning the punch with the particular fiber that is over the support hole prior to pushout
testing, and can be used to observe the pushout experiment, in progress, at temperatures
below 150˚C. Observation of the test at temperatures above 150˚C is prevented by the light
from the infrared heater which saturates the CCD camera.
Figures 4.2 and 4.3 show the capabilities of the long distance microscope and CCD
camera system on the high temperature apparatus. Figure 4.2 is an image taken of the
tungsten carbide punch and a fiber pushed below the surface in a SiC/Ti-15-3 composite.
Figure 4.3 is an image of the bottom face of a SiC/Ti-6-4 sample on which a fiber pushout
test had been conducted. A single fiber extends outward from the sample surface, and
several untested fibers can be seen nearby. The images in Figures 4.2 and 4.3 also show
that the SiC/Ti samples were cut from actual composites containing many fibers rather than
a single fiber as in the model composite fiber pushout tests, therefore if the data from the
pushout tests on the SiC/Ti samples were analyzed, the material surrounding the fiber to be
pushed out would have to be modeled with elastic properties that reflect the presence of
fibers distributed throughout the matrix.
4.3 SiC/Ti pushout tests
4.3.1 Pristine SiC/Ti-6-4
Pristine samples of SiC/Ti-6-4 were tested at both room temperature and 400˚C. An
example of cross-head displacement versus force at each temperature is shown in Figures
4.4 and 4.5. At room temperature a drop in load was observed well before the peak load
107
20 µm
Figure 4.2 Punch and top of a pushed out fiber in an SiC/Ti-15-3 composite.
108
20 µm
Figure 4.3 Punch and bottom surface of an SiC/Ti-6-4 composite with a single fiber
pushed out.
109
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100
For
ce (N
)
Displacement (µm)
t=0.39 mm
Figure 4.4 Force–displacement curve for pristine SiC/Ti-6-4 tested at room
temperature.
110
0
2
4
6
8
10
12
14
0 10 20 30 40 50 60 70
For
ce (
N)
Displacement (µm)
t=0.30 mm
Figure 4.5 Force–displacement curve for pristine SiC/Ti-6-4 tested at 400˚C.
111
was reached. The fiber became totally debonded at the load drop, and, thereafter the load
increased another 8 N due to increasing friction in the interface before the decreasing
embedded length caused the load to drop. This phenomenon was also reported by Roman
and Jero (1992) for SiC/Ti-6-4 and by Kantos, Eldridge, Koss, and Ghosn (1992) for
SiC/Ti-15-3. Kantos and co-workers determined the cause of the increasing friction after
debond in fiber pushout tests on SiC/Ti-15-3. Layers of carbon containing varying
concentrations of SiC surround the SiC fiber. As the debond grows along the interface its
path switches between these layers. Interlocking fiber and matrix surfaces are produced
which crumble as the fiber is pushed out causing the friction between the fiber and matrix to
increase for a period after total debond.
In the pushout curve shown in Figure 4.5 the peak load was followed by a relatively
large load drop, signifying total debond, and then the fiber was pushed out under a nearly
constant load which remained lower than the debond load. At 400˚C, the debond apparently
chose a path that did not switch between layers since the pattern of increasing friction was
not observed for SiC/Ti-6-4. Eldridge and Ebihara (1994) acquired similar results, which
are shown in Figure 4.6, for SiC/Ti-15-3. At 23˚C and 300˚C interfacial friction increased
after the peak load, and at 400˚C and above, the interface bond strength determined the peak
load.
Also, in Figures 4.4 and 4.5, the samples appear to stiffen when a load of approximately
8 N is reached. This apparent increase in stiffness can be seen in force–displacement
curves as an increase in slope. Machine compliance measurements revealed that the
increase in slope is due to an increase in fixture compliance at 8 N.
4.3.2 Fatigued SiC/Ti-6-4
Next, the laterally fatigued SiC/Ti-6-4 composite described in Section 4.2.1 was
pushout tested. The results are presented in Figures 4.7 and 4.8. At room temperature, the
force–displacement curve was qualitatively the same as for the pristine SiC/Ti-6-4 except
112
Figure 4.6 Pushout curves obtained by Eldridge and Ebihara (1994) at various
temperatures for SiC/Ti-15-3.
113
0
5
10
15
20
25
30
35
0 20 40 60 80 100
For
ce (
N)
Displacement (µm)
t=0.6 mm
Figure 4.7 Two pushout tests on fatigued SiC/Ti-6-4 at room temperature.
114
0
0.5
1
1.5
2
2.5
3
0 10 20 30 40
For
ce (
N)
Displacement (µm)
t=0.6 mm
Figure 4.8 A pushout test on fatigued SiC/Ti-6-4 at 400˚C.
115
for the absence of a load drop prior to the peak load. At 400˚C, a fiber in a 0.3 mm thick
sample slid from the matrix under less than 3 N of applied load. The peak load at 400˚C
from a pristine sample of the same thickness was approximately 13 N. The lack of a
sudden load drop indicated that the chemical bond between the fiber and matrix was
destroyed. Several fibers were pushout tested with similar results. All the fibers in the
sample had been completely debonded by the fatigue load, and when a portion of the
residual stresses were relieved at 400˚C, the interfacial friction produced by radial clamping
was negligible. This conclusion is in agreement with results from static tests done by
Jansson, Deve, and Evans (1991). Jansson and associates applied a static transverse load to
a SiC/Ti-6-4 composite while observing the exposed faces of the fibers under a microscope.
They found that at 200 MPa, a portion of the matrix surrounding some of the fibers
separated from the fibers and closed back around the fibers upon unloading.
Warren, Mackin, and Evans (1992) applied a cyclic longitudinal load of 300 MPa with
R = 0.1 to an SiC/Ti-15-3 composite. A matrix fatigue crack formed with the aid of a
starter notch. Unlike the current results for SiC/Ti-6-4 laterally fatigued at 180 MPa, the
pushout tests at room temperature conducted by Warren and co-workers showed that only
the fibers near the fatigue crack were debonded in the longitudinally fatigued Ti-15-3.
At room temperature, the debonds in the laterally fatigued SiC/Ti-6-4 may not be
detrimental to the performance of the composite unless, through loading and unloading of
the composite, the interface wears and the friction between the fibers and matrix reduces
with time. On the other hand, at high temperature, preservation of the interfacial bond is
critical since the chemical bond, and not Coulomb friction, determines the limit on the
magnitude of shear stress that can be transferred from the matrix to the fiber through the
interface.
4.4 Progressive debonding in SiC/Ti
For room temperature pushout of SiC fibers from either Ti-6-4 or Ti-15-3 matrices, the
116
issue of finding the interface toughness is, at best, ambiguous. The interface conditions on
the surface of the debonded section of the fiber during progressive debonding are some
combination of friction and deformation of relatively large interlocking asperities. Also,
asperities may break free and slide along the interface. The method described in Chapter 3
for calculating the interfacial toughness does not apply when these interface conditions are
present. Therefore, at room temperature, even if the force–displacement curve becomes
nonlinear over some interval prior to total debond, the interfacial toughness could not be
determined because the coefficient of friction may have increased significantly throughout
progressive debonding. Also, Coulomb friction assumes that the length spacing of the
asperities on the interface is uniform and much smaller than the distance that the fiber is
moved. This assumption would probably not be satisfied either since Kantos, Eldridge,
Koss, and Ghosn (1992) showed that the size of the interlocking asperities in the interface
of SiC/Ti-15-3 can be on the order of the fiber radius.
As described earlier, the load drop in the curve in Figure 4.4 signifies total debond, and
the force–displacement data after the sharp load drop are the frictional data. The frictional
portion of the curve in Figure 4.4 shows that, after total debond, the interface can carry a
greater load due to increasing friction and interlocking of the fiber and matrix surfaces. The
strength of the interface bond does not determine the maximum shear load that can be
transferred from the matrix to the fiber. Consequently, if the interface bond strength is
computed from the portion of the pushout data prior to the load at total debond (signified by
the load drop), it would have no significance since the interface can withstand a greater shear
stress than is applied at total debond.
Fortunately, calculating the interfacial toughness for SiC/Ti composites may be more
straightforward at elevated temperatures. Figure 4.5 shows that the peak load occurred at
total debond for SiC/Ti-6-4 tested at 400˚C. The frictional pushout curve increased slightly
over a displacement of 13 microns and leveled out as the fiber was pushed further. Figure
4.9 shows an image captured with the high temperature apparatus of three SiC fibers.
117
AB
C
10 µm
Figure 4.9 Three fibers in a SiC/Ti-6-4 composite. Fiber A was not pushout tested.
Fiber B was pushed out and back at room temperature. Fiber C was pushed
out and back at 400˚C.
118
Fiber A was not pushout tested. Fiber B was pushed out at room temperature and pushed
back at room temperature. Debris can be seen lying around the outer diameter of fiber B.
This debris consists of pieces of the carbon/SiC layers originally surrounding the fiber.
The interlocking asperities on the fiber crumbled during pushout and were scraped from the
fiber during pushback. In contrast, fiber C, which was pushed out at 400˚C and pushed
back at 400˚C, had very little debris surrounding it. The lack of debris and shape of the
force versus displacement curve indicate that at 400˚C the debond took a path between two
of the SiC/carbon layers and did not jump between layers. The result is a lack of
interlocking surfaces unlike the case at room temperature. The interface conditions on the
surface of the debonded portion of the fiber at 400˚C appeared to be closer to contact and
sliding of two relatively smooth surfaces. If progressive debonding could be identified in
the pushout curves from SiC/Ti-6-4 at 400˚C, then an interfacial toughness could be
computed assuming no debonds were present prior to pushout testing.
Figure 4.5 shows the raw force–displacement curve from pristine SiC/Ti-6-4 at 400˚C.
Figure 4.10 shows a portion of the same test with the machine compliance carefully
removed. The pushout apparatus, apparently, became aligned and all parts seated by 7.5 N
since the force–displacement curve was predominantly linear at loads higher than 7.5 N.
Two small detours from linearity can be seen at approximately 11 and 12 newtons. It is not
clear if the nonlinearity near 12 newtons is progressive debonding or if it is caused by a
nonuniform fixture compliance. Figure 4.11 shows a plot of the force–displacement curve
from a test consisting of the punch holder being forced into the sample support. A constant
machine compliance of 2.47 µm/N was subtracted from the displacement to produce a plot
that becomes vertical once alignment and seating has been completed as of 11 microns. In
Figure 4.11, two detours from a constant machine compliance can also be seen at
approximately 11 and 12 newtons. It is likely that all or part of the curve separation near 12
N in Figure 4.10 is due to the test fixture compliance changing and not due to progressive
debonding only. A stiffer and more uniform test fixture response to loading is
119
6
7
8
9
10
11
12
13
10 15 20 25 30 35 40
For
ce (
N)
Displacement (µm)
t=0.3 mm
Figure 4.10 Pushout curve with machine compliance removed for pristine SiC/Ti-6-4
tested at 400˚C.
120
0
5
10
15
5 10 15 20 25 30 35 40
For
ce (
N)
Displacement (µm)
Variation incompliance
Machine compliance = 2.47 µm / Nsubtracted from displacement
Figure 4.11 Measurement of machine compliance of high temperature apparatus.
121
needed to determine if progressive debonding occurs in the titanium matrix composites at
high temperatures.
The stepper motor used to move the punch in the model composite tests of Chapter 2 is
capable of a more uniform velocity than the DC motor actuator used in the high temperature
tests, and with a lead screw with a pitch of 0.5 mm/revolution, the stepper motor could
produce a linear motion at a constant velocity of 1 µm/second. The gear box of the DC
motor actuator also appears to be very compliant compared to the coupler between the
stepper motor and the railtable of the model composite test fixture. For these reasons, a
stepper motor should be used in place of the DC motor actuator on the high temperature
apparatus. Several measurements of the effect on machine compliance of removing various
parts of the structure of the high temperature apparatus also indicated that the three axis
stage was responsible for a large portion of the test fixture compliance. Removal of the
three axis stage also produced a much more uniform compliance beginning at a lower load.
Figure 4.12a shows a schematic of the high temperature apparatus with the stepper motor
replacing the DC motor and with the sample stages removed. The ceramic rod supporting
the sample rests on the bottom of the vacuum chamber. An enlargement of the required
coupling between the stepper motor and the load cell is shown in Figure 4.12b.
If the three axis stage were removed, an alternate method of aligning a fiber with the
punch and support hole would be necessary. Based on crude tests run with similar
configurations, the following procedure would align the punch, fiber, and support hole.
1. First, a sample transport would be constructed with the three axis stage and located
outside of the vacuum chamber. The sample transport would be used to align a fiber in
the sample with the support hole and then to place the sample support with the sample
on top of it inside the chamber through the door on the front of the chamber. Figure
4.13a shows a schematic of the modified high temperature apparatus with the front
door open and the sample transport placing a sample inside the chamber. A top view of
the transport is presented in Figure 4.13b.
122
A
B
DE
F
GH
B1
J
(a)
(b)
C
K
Figure 4.12 Modified high temperature apparatus: (a) schematic, (b) fixture to connect
the stepper motor to the load cell.
123
A
B
DE
F
GH
I
B1
J
I
(a)
(b)
H
H
J
J
Figure 4.13 Modified high temperature apparatus: (a) schematic showing sample
transport required to place the sample and sample support into the chamber,
(b) top view of sample and sample support resting on the sample transport.
124
2. To center a fiber over the hole in the sample support, the sample support would be
placed on the sample transport under a microscope similar to the one used to inspect the
polish of the MMC samples, and the center of the support hole would be aligned with
the eyepiece crosshairs. The sample would then be lowered onto the sample support
with the center of the desired fiber also aligned with the crosshairs.
3. Using the sample transport, the sample would be moved inside the chamber and lifted
such that the fiber is aligned with the end of the punch and is touching the punch. The
alignment would observed with the long distance microscope, and the contact between
the punch and the fiber would be identified by a small load measured with the load cell.
4. Once the sample is aligned and lightly pinched between the fiber and the punch, the
punch and sample transport would be lowered simultaneously until the sample support
rests on the ceramic rod. The fiber, punch, and support hole would all be aligned with
each other at that point.
These modifications to the apparatus and to the alignment procedure may yield an
experiment capable of identifying progressive debonding in MMCs at high temperature. If
progressive debonding data can be gathered, the interface strength as a function of
temperature could be determined at elevated temperatures. The exact temperature above
which debonding during pushout does not bear interlocking fiber and matrix surfaces is not
yet known for SiC/Ti-6-4 but could be determined by conducting pushout tests at several
temperatures between room temperature and 400˚C.
Finally, it may be possible to measure the debond length in a metal matrix composite as
was done in model epoxy matrix composites. Samples of the SiC/Ti-6-4 composite were
cut parallel to the fibers and then ground and polished so that the cross-section of several
fibers could be observed under a microscope. Approximately a third of the fiber was
polished away with two thirds of the fiber thickness remaining in the matrix. Figure 4.14 is
a photograph of the cross-section of a fiber in the pristine SiC/Ti-6-4 composite, and
125
10 µm
Figure 4.14 Cross-section of a fiber in pristine SiC/Ti-6-4.
126
Figure 4.15 is a photograph of the cross-section of a fiber in the laterally fatigued SiC/Ti-
6-4. The fibers were not pushout tested.
The color of the interface in the image of the fiber from the pristine (bonded) sample is
uniform, but in the photo from the fatigued (debonded) sample, sections of the interface are
darker than others. These dark areas may be caused by pieces of the carbon layers in the
interface falling out during cutting and polishing. The carbon layer pieces were apparently
broken loose by the fatigue load. If this explanation is correct, the areas where the pieces
fell out would not be polished, and as a result, would not reflect light. Since the appearance
of the interface was different in a bonded sample than a debonded sample, it was assumed
that cutting and polishing the cross-sections did not debond the bonded sample. If cutting
and polishing had debonded the pristine fiber, both the interfaces from the pristine samples
and the fatigued samples would have appeared similar.
Majumbdar and Newaz (1995) located a similar darkened area in the interface of
SiC/Ti-15-3 that had been debonded by a longitudinal fatigue load. Majumbdar and Newaz
photographed a continuous dark strip in the fiber–matrix interface extending away from the
surface of a matrix crack produced by the fatigue load. In contrast, the darkened areas in
Figure 4.15 are discontinuous.
Regardless of their source, the presence or lack of the darkened areas in the interface
between an SiC fiber and a titanium alloy matrix could possibly be used to measure the
length of a debond. If a pushout test could be conducted in which progressive debonding is
detected with the modified high temperature apparatus, and if the test were interrupted
before total debond, the composite could be sectioned and pictures taken similar to the ones
in Figures 4.14 and 4.15. The debond portion of the interface may be recognizable as it is
in Figures 4.14 and 4.15. If so, the prediction of debond length from the analysis of
pushout data from SiC/Ti composites could be verified.
127
10 µm
Figure 4.15 Cross-section of a fiber in fatigued SiC/Ti-6-4.
128
4.5 Discussion and future work
The pushout tests in this chapter were of a preliminary nature only. The intent of the
tests was to show the possible work that could be done with the high temperature pushout
apparatus. Pushout tests conducted on a laterally fatigued SiC/Ti-6-4 composite at room
temperature and 400˚C showed that all the fibers in the composite were debonded. The
debonded fibers demonstrated almost a total lack of resistance to sliding at high
temperature.
A mechanism capable of driving the punch at a more uniform velocity than the DC
motor can drive it is necessary along with a stiffer coupling between the drive motor and the
load cell to obtain the quality of pushout data needed to distinguish progressive debonding
in SiC/Ti alloy composites. The current measurement of displacement also suffers from the
use of linear motion stages that are used to align the sample with the punch. With
improvements to the apparatus, pushout data from which the fracture toughness of the
interface in SiC/Ti-6-4 at high temperatures could be calculated may be obtainable.
Much work remains in the area of pushout testing at high temperatures. The elevated
temperature pushout tests conducted in this section should be repeated in an oxygen free
environment to prevent oxidation of the interface. Also, further attempts at debond length
measurement through pictures of fiber cross-sections could lead to verification of or
evidence against the accuracy of current fiber pushout analyses. Debond length
measurements may also reveal whether debonds from cutting and residual stresses are
present before pushout testing.
129
5. CONCLUSIONS
Experiments and theoretical modeling were conducted to investigate the assumptions
often made about the shape of force–displacement curves from pushout testing, the accuracy
of theories available for analyzing pushout data, and the accuracy of an improved method
(finite element) of reducing pushout data to interfacial properties. Pushout tests were also
conducted to determine the affect of fatigue on an SiC/Ti composite and to determine the
applicability of the finite element solution to pushout data from the same composite over a
range of temperatures.
5.1 Debond length measurements
The photoelastic fringe patterns in the matrix of several epoxy matrix composites were
viewed during pushout testing. The fringe patterns revealed that the fiber/matrix interface
was fully bonded during the initial linear part of the pushout curve for a composite with an
epoxy matrix that was cured at room temperature. For epoxy matrix composites with
residual stresses from processing, the interface was partially debonded prior to testing, but
the debonds did not grow during the initial linear part of the pushout test curve. The
location of the interface debond tip during pushout testing was assumed to be at the greatest
stress intensity and was determined in the photoelastic fringe patterns as the position along
the interface where the fringe distribution was the most dense. During the nonlinear part of
the pushout curve up to the maximum load a debond was detected growing along the
interface. At the peak load the interface completely debonded, and therefore the portion of
the force–displacement curve after the maximum load corresponded to frictional sliding.
These observations support the common interpretation in the literature of the shape of
typical fiber pushout force–displacement curves.
The model composite pushout tests also indicated that the interface debond in
composites with a fiber-to-matrix moduli ratio of greater than 3 will initiate at the bottom
130
and grow toward the top of the sample during a fiber pushout test. Current shear lag
models in the literature do not apply to these systems. Also, the debond length was
measured as a function of applied load for a composite that debonded from the top. The
measured debond length was approximately 1.5 fiber radii larger than the debond length
calculated by the most rigorous of the shear lag solutions in the literature for progressive
debonding data. Since the debond length was under-predicted the shear lag solution over-
predicted the interface toughness. The inaccuracy of the shear lag solution is a result of the
plane strain assumption not being satisfied, the axial stress not being constant over the top
cross-section of the fiber and the cross-section of the fiber at the debond tip, and the
residual interfacial radial and fiber axial stresses not being constant.
In addition, a debond length versus displacement curve was presented in Chapter 2 from
a pushout test on a composite that debonded from the bottom. The debond length
measurements reported in this paper may be useful when checking the accuracy of
theoretical solutions to the pushout problem when and if more advanced solutions are
developed which include surface effects and a bottom debond if one is present.
5.2 Finite element solution
Such a solution to the fiber pushout problem was described in Chapter 3. The finite
element method was used to derive the interface debond length as a function of force from
the progressive debonding portion of the pushout data for a top debonding composite.
Polyester/epoxy pushout data were analyzed, so a comparison between the debond length
predicted by shear lag theory and by the finite element method could be made. Debond
length was calculated by adjusting the debond length in the finite element simulation until
the chosen punch displacement from the pushout test produced the corresponding punch
load from experiment. The debond length versus force calculated using this method agreed
with the measured debond length to within a seven percent error when a coefficient of
friction of µ = 0.52 (calculated from shear lag theory) was used. Although this technique,
131
which requires several consecutive finite element calculations, is more time consuming to
apply than a closed form solution such as the shear lag solution, it is apparently necessary.
The finite element method presented in the current work for analyzing the progressive
debonding portion of the pushout data captures the effects of the open portion(s) of the
interface, the difference in size between the punch and fiber, the difference in size between
the support hole and the fiber, and the nonuniform residual stress field. This type of
analysis is required to avoid under-predicting the debond length and, consequently, over-
predicting the interface strength.
The finite element method was also used to compute a coefficient of friction from the
data following total debond in a representative pushout test performed on polyester/epoxy.
The computed value of coefficient of friction was in agreement with the coefficient of
friction computed by Pagano and Tandon (1996) but was 44% greater than the coefficient
of friction determined with shear lag theory. When the coefficient of friction from finite
element analysis, µ = 0.75, was used in the finite element solution for progressive
debonding, debond length twice as long as measured were predicted. However, when the
shear lag value of µ = 0.52 was used in the finite element solution for progressive
debonding, debond length was closely predicted when the measured force and displacement
were used as inputs and closely predicted force when the measured debond length and
displacements were used as inputs. Either the finite element analysis of the fully slipping
problem was inaccurate, or the coefficient of friction increased during the load drop and
sliding immediately following total debond. The computation of loads from measured
displacement versus debond length information, for µ = 0.52, indicated that the coefficient
of friction increased with fiber sliding in some of the pushout tests. No method was
determined to measure how much the coefficient of friction had increased by the time the
interface was totally debonded. The difficulties in determining the coefficient of friction
from the frictional pushout part of pushout data highlight how difficult it is to fulfill the
assumption of a constant coefficient of friction in an actual pushout test and how difficult it
132
is to model the fully slipping problem which is not as constrained as the progressive
debonding problem.
The finite element method was also used to analyze pushout data from a bottom
debonding composite with residual stresses large enough to debond the fiber ends during
processing. The exact steps taken during fabrication of the steel/epoxy test samples proved
to be critical to the calculation of debond length from pushout data. When pushout data
were analyzed from samples cut near and including the ends of the raw sample, the debond
length was underestimated by 80% of the measured debond length. Cutting the samples at
least six fiber diameters from the ends of the raw sample solved this problem. A coefficient
of friction calculated from the LH&KP shear lag solution of the fully slipping problem in
conjunction with finite element analysis of the progressive debonding interval of the
pushout data, once again, yielded reasonably accurate results. Debond length as a function
of force was computed within 10% of the measured debond length.
5.3 Processing and fabrication
A hypothesis was proposed to explain why the proximity of the test sample to the ends
of the raw sample affected the pushout data. It was hypothesized that a portion of the fiber
debonds during the first stages of cool down, and part of this debonded portion sticks and
rebonds as cool down progresses. This process would leave a section of the fiber debonded
(a combination of an open region and a closed and slipping region), a section containing
residual axial stress developed during the entire temperature drop, and a section with less
residual axial stress than would be expected from the temperature drop during processing.
Measurements of the length of fiber exposed at the bottom of steel/epoxy pushout samples
versus debond length indicated the residual axial stress near one end of the fiber in samples
cut close to the ends of the raw sample was less than that predicted by a finite element
simulation of cool down after cure.
133
The existence of short debonds extending along the interface from the fiber ends after
processing and cutting in the steel/epoxy pushout samples was proven. These initial
debonds had to be included in the finite element analysis to predict the slope of the initial
linear portion of the pushout data. They also had to be included in the finite element
simulation of progressive debonding to compute debond lengths close to the measured
debond lengths. The problem of initial debonds present prior to pushout testing became
more significant as the diameter of the steel fiber was decreased.
The finite element analysis described here can be used to calculate accurately the debond
length for both a top and a bottom debonding system. Even samples with debonds present
before pushout testing can be analyzed. However, the condition of the interface, the residual
stress state following processing, and the length of initial debonds must be accurately
known prior to pushout testing. The coefficient of friction must remain constant for the
composite tested if it is computed from the frictional pushout part of the test data. These
requirements may be difficult to meet for many composites.
5.4 SiC/Ti pushout tests
Finally, Chapter 4 described the development of an apparatus for conducting fiber
pushout tests at temperatures up to 800˚C. From tests on metal matrix composites (multi-
fiber) of current interest, an attempt was made to obtain pushout data that would be
appropriate for the finite element analysis of Chapter 3 . A stiffer structure was necessary
to identify progressive debonding in the force–displacement curves from SiC/Ti-6-4
pushout tests. Pushout tests conducted on the SiC/Ti-6-4 composite at room temperature
and at 400˚C indicated that the calculation of interfacial toughness may have meaning only
for pushout data obtained at elevated temperatures. Also, pushout tests conducted on a
laterally fatigued SiC/Ti-6-4 composite at room temperature and 400˚C showed that all of
the fibers in the composite had been debonded by the fatigue load.
134
APPENDIX A. MATRIX SHRINKAGE MEASUREMENT
The Epon 828 resin and DETA curing agent mixture was cured at room
temperature. Both chemical shrinkage of the matrix and volume changes due to heat
generated by the chemical reaction occurred during cure. The chemical shrinkage of the
fiber during cure could not be found in the literature; therefore, the average radial stress at
the interface for the polyester/epoxy (DETA) system was measured photoelastically and
used to calculate indirectly the differential shrinkage between the polyester fiber and epoxy
matrix. The fiber and matrix elastic properties and the dimensions of the photoelastic
sample were used in a finite element simulation of processing. A differential shrinkage
strain of 0.0022 was calculated in the finite element simulation by isotropically shrinking
the matrix around the fiber until an average interfacial radial stress of –5.68 MPa was
produced. The derivation of the relation between the photoelastic fringe order and the
average radial stress at the interface is presented below.
The coordinate system, the relevant dimensions, and a schematic of the circular
photoelastic fringes that were observed in the epoxy matrix are shown in Figure A.1. The
derivation draws on one of the two dimensional equilibrium equations in Cartesian
coordinates:∂σ xx
∂x+∂σ xy
∂y= 0. (A.1)
The Fundamental Theorem of Calculus was used to write an expression relating the known
value of σ xx at one position (xo , yo ) to the unknown value of σ xx at another position
(x, yo ) :
∂σ xx (x, yo )
∂xdx = σ xx (x, yo ) −
xo
x
∫ σ xx (xo , yo ) (A.2)
Substitution of Eq. (A.2) into (A.1) yields one of the shear difference equations from
Frocht (1946):
135
σ xx (x, yo ) = σ xx (xo , yo ) −∂σ xy (x, yo )
∂yxo
x
∫ dx . (A.3)
Also, from Frocht (1946), the photoelastic fringe order relate to the principal stresses
according to the relation
σ1 − σ2 =Nf σh
, (A.4)
where N is the fringe order, h is the sample thickness (7.8 mm), f σ is the material fringe
constant (10.49 N/mm for Epon 828/DETA, (Kline, 1995)), and σ1 and σ2 are the
maximum and minimum principal stresses. The principal stresses are related to the σ xy
shear stress by the Mohr’s Circle relation
σ xy =σ1 − σ2
2sin(2θ ), (A.5)
where θ is the angle between the x axis and the direction of the minimum principal stress.
Eqs. (A.4) and (A.5) are combined and θ is written in terms of the x and y coordinates:
σ xy (x, y) =N(x, y) f σ
2hsin 2 tan−1 y
x
. (A.6)
The partial derivative of σ xy (x, y) is computed with respect to y and evaluated at y = 0:∂σ xy (x,0)
∂y=
f σh
N(x,0)
x. (A.7)
Along the x axis, the radial stress is the same as σ xx , and both are zero on the x axis at x
= ro . This information and the substitution of Eq. (A.7) into Eq. (A.3) yields an expression
for the radial stress (averaged through the sample thickness) at the interface:
σrr (x = r f , y = 0) =− f σ
h
N(x, y = 0)
xro
r f
∫ dx . (A.8)
The fringe order as a function of the radial coordinate must be measured to calculate the
radial stress at the interface.
A light field image of the photoelastic fringes near the fiber surface in the matrix of
a steel/epoxy sample is shown in Figure A.2. A steel/epoxy image is shown rather than a
polyester/epoxy image because in the latter the fringes are sparse and the image must be
136
Photoelasticfringes
Sampleedges
Fiber
Matrix
rox
y
r f
Figure A.1 Geometry for the derivation of the relation between fringe order and average
interfacial radial stress.
137
1 mm
Figure A.2 Photoelastic fringe patterns surrounding the fiber in a steel/epoxy pushout
sample.
138
magnified near the fiber to distinguish the fringes from the fiber. The steel/epoxy
photoelastic fringe pattern in Figure A.2 illustrates the concept more clearly.
Light and dark field images were recorded of the photoelastic fringe pattern in a
7.8 mm long polyester/epoxy sample. The pixel intensities of the dark field image were
subtracted from the pixel intensities of the light field image to produce an image with twice
the fringes to increase the resolution. No significant difference in fringe order as a function
of radius could be determined between images taken before and after total debond. The
fringe order could not be measured exactly at the fiber surface since the closest fringe to the
fiber was a finite distance from the fiber surface. Quadratic splines were fit through the
fringe order versus radius data points to obtain a continuous function of fringe order versus
radius that was then extrapolated to the fiber surface. This function was used in Eq. (A.8)
and an average radial stress at the interface of –5.68 MPa was calculated. The variation of
radial stress from the outer edge of the sample to the fiber surface is shown in Figure A.3.
139
−6
−5
−4
−3
−2
−1
0
0 2 4 6 8 10 12 14 16
Rad
ial s
tres
s (M
Pa)
Radial distance from center of fiber (mm)
Figure A.3 Thickness average of radial stress in matrix as a function of distance from
the fiber center in a 7.8 mm thick polyester/epoxy fiber pushout sample as
calculated from the photoelastic fringe pattern.
140
BIBLIOGRAPHY
Ananth, C. R., and N. Chandra. 1996. Elevated Temperature Interfacial Behavior of
MMCs: A Computational Study, Composites Part A, 27A: 805-811.
Anderson, T. L. 1991. Fracture Mechanics Fundamentals and Applications, CRC
Press, Boca Raton, Florida, 669-670.
Atkinson, C., J. Avila, E. Betz, and R. E. Smelser. 1982. The Rod Pull Out Problem,
Theory and Experiment, Journal of the Mechanics and Physics of Solids, 30: 97-120.
Aveston, J., G. A. Cooper, and A. Kelly. 1971. The Properties of Fibre Composites,
Conference Proceedings (IPC Science and Technology Press Ltd.,Teddington, U.K., 15.
Bao, G., and Y. Song. 1993. Crack Bridging Models for Fiber Composites with Slip-
Dependent Interfaces, Journal of the Mechanics and Physics of Solids, 41: 1425-1444.
Beckert, W., and B. Lauke. 1995. Fracture Mechanics Finite Element Analysis of
Debonding Crack Extension for a Single Fibre Pull-Out Specimen, Journal of Material
Science Letters, 14: 333-336.
Brun, M. K., and R. N. Singh. 1988. Effect of Thermal Expansion Mismatch and Fiber
Coating on the Fiber/Matrix Interfacial Shear Stress in Ceramic Matrix Composites,
Advanced Ceramic Materials, 3: 506-509.
Brun, M. K. 1992. Measurement of Fiber/Matrix Interfacial Shear Stress at Elevated
Temperatures, Journal of the American Ceramic Society, 75: 1914-1917.
141
Chandra, N., and C. R. Ananth. 1995. Analysis of Interfacial Behavior in MMCs and
IMCs by the Use of Thin-Slice Push-Out Tests, Composites Science and Technology, 54:
87-100.
Chou, H. M., M. W. Barsoum, and M. J. Koczak. Effect of Temperature on Interfacial
Shear Strengths of SiC-Glass Interfaces, Journal of Material Science, 26: 1216-1222.
Cordes, R. D., and I. M. Daniel. 1995. Determination of Interfacial Properties from
Observations of Progressive Fiber Debonding and Pullout, Composites Engineering, 5:
633-648.
Daniel, I. M., G. Anastassopoulos, and J.-W. Lee. 1993. The Behavior of Ceramic
Matrix Fiber Composites Under Longitudinal Loading, Composites Science and
Technology, 46: 105-113.
Dempsey, J. P., and G. B. Sinclair. 1981. On the Singular Behavior at the Vertex of a
Bi-material Wedge, Journal of Elasticity, 11: 317-327.
Drzal, L. T., M. J. Rich, J. D. Camping, and W. J. Park. 1980. Interfacial Shear
Strength and Failure Mechanisms in Graphite Fiber Composites, 35th Annual Conference,
Reinforced Plastics/Composites Institute: 20-C.
Eldridge, J. I., and B. T. Ebihara. 1994. Fiber Push-Out Testing Apparatus for Elevated
Temperatures, Journal of Materials Research, 9: 1035-1042.
142
Eldridge, J. I. 1995. Elevated Temperature Fiber Push-Out Testing, Material Research
Society, Symposium Proceedings, 365: 283-290.
Frocht, M. M. 1946. Photoelasticity, Wiley, New York, New York, 252-285.
Gao, Y. C., Y.-W. Mai, and B. Cotterell. 1988. Fracture of Fiber Reinforced Materials,
Journal of Applied Math and Physics, 39: 550-572.
Ghosn, L. J., P. Kantos, J. I. Eldridge, and R. Wilson. 1992. Analysis of Interfacial
Failure in SCS-6/Ti-Based Composites During Fiber Pushout Testing, HITEMP Review
(NASA CP-10104), 2: 27-1 to 27-12.
Herrara-Franco, P. J., and L. T. Drzal. 1992. Comparison of Methods for the
Measurement of Fibre/Matrix Adhesion in Composites, Composites, 23: 2-27.
Hibbitt, Karlsson, and Sorenson, Inc. 1994. ABAQUS Theory Manual, 1994, Hibbitt,
Karlsson, and Sorenson, Inc., 5.1.1-5.1.3.
Hsueh, C. H. 1990. Interfacial Debonding and Fiber Pull-Out Stress of Fiber-
Reinforced Composites, Material Science and Engineering, A123: 1-11.
Hseuh, C. H. 1994. Slice Compression Tests Versus Fiber Push-In Tests, Journal of
Composite Materials, 28: 638-655.
Jansson, S., H. E. Deve, and A. G. Evans. 1991. The Anisotropic Mechanical Properties
of a Ti Matrix Composite Reinforced with SiC Fibers, Metallurgical Transactions , 22A:
2975-2984.
143
Jero, P. D., and R. J. Kerans. 1990. The Contribution of Interfacial Roughness to
Sliding Friction of Ceramic Fibers in a Glass Matrix, Scripta Metallurgica et Materialia,
24: 2315-2318.
Kallas, M. N., D. A. Koss, H. T. Hahn, and J. R. Hellmann. 1992. Interfacial Stress
State Present in a ‘Thin-Slice’ Fibre Push-Out Test, Journal of Material Science, 27: 3821-
3826.
Kantos, P., J. I. Eldridge, D. A. Koss, and L. J. Ghosn. 1992. The Effect of Fatigue
Loading on the Interfacial Shear Properties of SCS-6/Ti-Based MMCs, Materials Research
Society Symposium Proceedings, 273: 135-142.
Kerans, R. J., and T. A. Parthasarathy. 1991. Theoretical Analysis of the Fiber Pullout
and Pushout Tests, Journal of the American Ceramic Society, 74: 1585-1596.
Kishore, P. V., A. C. W. Lau, and A. S. D. Wang. 1992. The Fiber Pull-Out Problem:
Matching of Singular and Complete Stress Fields, Proceedings of the ASM, 6th Technical
Conference, 1054-1063.
Kishore, P. V., A. C. W. Lau, and A. S. D. Wang. 1993. On Fiber-Matrix Interfacial
Stresses During Fiber Pullout with Thermal Stressing, Proceedings of the ASM, 7th
Technical Conference, 827-836.
Kline, G. E. 1995. Photoelastic Determination of In–plane Stresses in Shape Memory
Alloy/Polymer Matrix Composites, M.S. thesis, Department of Theoretical and Applied
Mechanics, University of Illinois at Urbana-Champaign.
144
Koss, D. A., J. R. Hellman, and M. N. Kallas. 1993. Fiber Pushout and Interfacial
Shear in Metal Matrix Composites, Journal of Materials, 45 March: 34-37.
Kurtz, R. D., and N. J. Pagano. 1991. Analysis of the Deformation of a Symmetrically
Loaded Fiber Embedded in a Matrix Material. Composities Engineering, 1: 13-27.
Laughner, J. W., N. J. Shaw, R. T. Bhatt, and J. A. Dicarlo. 1988. Simple Indentation
Method for Measurement of Interfacial Shear Strength in SiC/Si3N4 Composites.
Ceramics Engineering and Science Proceedings, 7: 932.
Liang, C., and J. W. Hutchinson. 1993. Mechanics of the Fiber Pushout Test,
Mechanics of Materials, 14: 207-221.
Mackin, T. J., J. Yang, and P. D. Warren. 1992. Influence of Fiber Roughness on the
Sliding Behavior of Sapphire Fibers in TiAl and Glass Matrices, Journal of the American
Ceramic Society, 75: 3358-3362.
Majumbdar, B. S. 1994. Personal conversation.
Majumbdar, B. S., and G. M. Newaz. 1995. Constituent Damage Mechanisms in Metal
Matrix Composites Under Fatigue Loading, and Their Effects on Fatigue Life, Materials
Science and Engineering, A200: 114-129.
Marshall, D. B., and W. C. Oliver. 1987. Measurement of Interfacial Properties in
Fiber-Reinforced Ceramic Composites, Journal of the American Ceramic Society, 70: 542-
548.
145
Marshall, D. B., and W. C. Oliver. 1990. An Indentation Method for Measuring
Residual Stresses in Fiber-Reinforced Ceramics, Material Science and Engineering, A126:
95-103.
Marshall, D. B., M. C. Shaw, and W. L. Morris. 1992. Measurement of Interfacial
Debonding and Sliding Resistance in Fiber Reinforced Intermetallics, Acta Metallurgica et
Materialia, 40: 443.
Miller, B., P. Muri, and L. Rebenfield. 1987. A Microbond Method for Determination
of the Shear Strength of a Fiber/Resin Interface, Composites Science and Technology, 28:
17-32.
Morscher, G., P. Pirouz, and A. H. Heuer. 1990. Temperature Dependence of Interfacial
Shear Strength in SiC-Fiber-Reinforced Reaction-Bonded Silicon Nitride, Journal of the
American Ceramic Society, 73: 713-720.
Netravali, A. N., D. Stone, S. Ruoff, and L. T. T. Topoleski. 1989. Continuous Micro-
Indenter Push-Through Technique for Measuring Interfacial Shear Strength of Fiber
Composites, Composites Science and Technology, 34: 239-303.
Pagano, N. J., and G. P. Tandon. 1996. Unpublished work.
Roman, I., and P. D. Jero. 1992. Interfacial Shear Behavior of Two Titanium-Based
SCS-6 Model Composites, Materials Research Society Symposium Proceedings, 273: 337-
342.
146
Tandon, G. P., and N. J. Pagano. 1996. Matrix Crack Impinging on a Frictional
Interface in Unidirectional Brittle Matrix Composites, International Journal of Solids and
Structures, 33: 4309-4326.
Toh, S. L., S. H. Tang, and J. D. Hovanesian. 1990. Computerized Photoelastic Fringe
Multiplication, Experimental Techniques, 14 July/August: 21-23.
Tsai, K.-H., and K.-S. Kim. 1991. A Study of Stick Slip Behavior in Interface Friction
Using Optical Fiber Pull Out Experiment, SPIE, Speckle Techniques, Birefringence
Methods, and Applications to Solid Mechanics, 1554A: 529-541.
Tsai, K.-H., and K.-S. Kim. 1996. The Micromechanics of Fiber Pull-Out, Journal of
Mechanics and Physics of Solids, 44: 1147-1177.
Warren, P. D., T. J. Mackin, and A. G. Evans. 1992. Design, Analysis, and Application
of an Improved Push-Through Test for the Measurement of Interface Properties in
Composites, Acta Metallurgica et Materialia, 40: 1243-1249.
Watson, M. C., and T. W. Clyne. 1992. The Use of Single Fibre Pushout Testing to
Explore Interfacial Mechanics in SiC Monofilament-Reinforced Ti—I. A Photoelastic
Study of the Test, Acta Metallurgica et Materialia, 40: 131-139.
147
VITA
Vernon Thomas Bechel was born in Durand, Wis., on July 9, 1965. He spent one
year at Rose-Hulman Institute of Technology, from 1983 to 1984, during which he was the
recipient of an Outstanding Student of Mathematics Award. He then repaired electrical and
hydraulic test equipment during a four year enlistment in the U. S. Air Force at MacDill Air
Force Base, Florida. He received his Bachelor of Science in mechanical engineering cum
laude from the University of South Florida in 1991. In 1993 he received a Master of
Science degree in mechanical engineering from the University of South Florida, and
received the Sigma Xi Outstanding Masters Thesis Award.
During the summer of 1992, he worked as a visiting scientist for Wright Laboratory
at Wright-Patterson Air Force Base, Ohio. In August 1992 he entered the doctoral program
in the Theoretical and Applied Mechanics Department at the University of Illinois at
Urbana-Champaign, where he has been a teaching assistant for two semesters and a
research assistant under the guidance of Professor Nancy R. Sottos. He was hired by
Wright Laboratory in August 1994 through the Senior Knight program, and he and his
family will be relocating to Dayton, Ohio, upon graduation.