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

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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.