MICROMECHANICS-BASED STRENGTH AND LIFETIME

PREDICTION OF POLYMER COMPOSITES

Tozer Jamshed Bandorawalla

A dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

in

Engineering Mechanics

Scott W. Case, Co-Chairman Kenneth L. Reifsnider, Co-Chairman

Richey M. Davis Scott L. Hendricks

Demetrios P. Telionis

February 25, 2002 Blacksburg, Virginia

Keywords: Rupture, Static Fatigue, Creep, Unidirectional Composite, Model Composite, Monte Carlo Simulations

Copyright 2002, Tozer J. Bandorawalla

MICROMECHANICS-BASED STRENGTH AND LIFETIME

PREDICTION OF POLYMER COMPOSITES

Tozer Jamshed Bandorawalla

(ABSTRACT) With the increasing use of composite materials for diverse applications ranging from civil infrastructure to offshore oil exploration, the durability of these materials is an important issue. Practical and accurate models for lifetime will enable engineers to push the boundaries of design and make the most efficient use of composite materials, while at the same time maintaining the utmost standards of safety. The work described in this dissertation is an effort to predict the strength and rupture lifetime of a unidirectional carbon fiber/polymer matrix composite using micromechanical techniques. Sources of material variability are incorporated into these models to predict probabilistic distributions for strength and lifetime. This approach is best suited to calculate material reliability for a desired lifetime under a given set of external conditions. A systematic procedure, with experimental verification at each important step, is followed to develop the predictive models in this dissertation. The work begins with an experimental and theoretical understanding of micromechanical stress redistribution due to fiber fractures in unidirectional composite materials. In-situ measurements of fiber stress redistribution are made in macromodel composites where the fibers are large enough that strain gages can be mounted directly onto the fibers. The measurements are used to justify and develop a new form of load sharing where the load of the broken fiber is redistributed only onto the nearest adjacent neighbors. The experimentally verified quasi-static load sharing is incorporated into a Monte Carlo simulation for tensile strength modeling. Very good agreement is shown between the predicted and experimental strength distribution of a unidirectional composite. For the stress-rupture models a time and temperature dependent load-sharing analysis is developed to compute stresses due an arbitrary sequence of fiber fractures. The load sharing is incorporated into a simulation for stress rupture lifetime. The model can be used to help understand and predict the role of temperature in accelerated measurement of stress-rupture lifetimes. It is suggested that damage in the gripped section of purely unidirectional specimens often leads to inaccurate measurements of rupture lifetime. Hence, rupture lifetimes are measured for [90/03]s carbon fiber/polymer matrix specimens where surface 90 plies protect the 0 plies from damage. Encouraging comparisons are made between the experimental and predicted lifetimes of the [90/03]s laminate. Finally, it is shown that the strength-life equal rank assumption is erroneous because of fundamental differences between quasi-static and stress-rupture failure behaviors in unidirectional polymer composites.

iii

GRANT INFORMATION

This work was supported by the National Science Foundation and the Air Force Office of Scientific Research under NSF grant #CMS-9872331.

iv

ACKNOWLEDGEMENTS

As with most pieces of work, several individuals have contributed to this research. I have tried to name all those who were most closely tied to my work. I hope that if I have, inadvertently, neglected to acknowledge anyone that they know I am very appreciative of all they did in support of this effort. Dr. Scott Case – I have enjoyed working with Scott throughout the four years that he has been my advisor. He has, quite simply, been an excellent advisor. Initially, at my request, Scott gave me the unique opportunity to work on several research projects. The experience was invaluable and introduced me to different research groups across campus. Although, I am sure that at times I made little progress on some of these projects, Scott was patient and very helpful. When I decided it was time for me to identify a single research topic for my dissertation, Scott helped me arrive at this decision, and even contributed to the technical details of my earlier work. Scott has given me ample opportunity to grow professionally by encouraging me to present my work at conferences, by submitting articles to journals and by having me interact directly with corporate sponsors of our work. I am also extremely grateful to Scott for supporting me with a research assistantship throughout the four years I worked with him. In short, I have thoroughly enjoyed working with him. I thank him for everything he has done for me and wish him well in the future. Dr. Kenneth Reifsnider – I feel I had the best combination of advisors in Scott and Dr. Reifsnider. While Scott rightly deserves most, if not all, the credit for the guiding my work over the past four years, I deeply appreciated Dr. Reifnsider’s strong interest in my work, his insight about research directions, and most of all his constant encouragement and enthusiasm. I think all the students who have had the good fortune of interacting with Dr. Refisnider will agree that they always left his office “fired-up” about their work! Moreover, Dr. Reifsnider has often looked at my work from a different angle, and on more than one occasion pointed out important issues that needed more emphasis or further effort. Dr. Richey Davis, Dr. Demetrios Telionis, Dr. Scott Hendricks – I would like to thank Drs. Davis, Telionis, and Hendricks for serving on my graduate committee. I appreciate them making the time to attend my prelims, proposal, and defense. Their suggestions during my proposal and later during my defense are very much appreciated. I was also fortunate to have worked with Dr. Davis on an interdisciplinary project early in my Ph.D. program. I am grateful for what I learnt while a part of that project.

Marshal “Mac” McCord – Mac is the “hands-on” technical wizard in the Material Response Group (MRG) labs. Over the past 4 years Mac has contributed to several of the research projects I have been part of, and his ingenuity, creativity, and love for what he does never ceases to impress me. Mac has patiently listened to my experimental woes, and then come up with effective and often inexpensive solutions. Very often he has fabricated fixtures for me on his tabletop lathe and milling machines.

v

Blair Russell, Sneha Patel, Dr. Robert Carter, Dr. Nikhil Verghese, and other MRGers – The MRG is, undoubtedly, the best college research group to be a part of. Besides being very good friends, they have been excellent colleagues and I have learnt something from all of these folks. While all of the MRGers deserve thanks, four of them deserve special mention. Blair impressed upon me, by example, the need to be meticulous and very thorough with any experimental effort. Blair has also contributed to this work with experimental data on quasi-static strength and stress-rupture lifetime. Sneha explained to me the subtleties of high temperature testing, and introduced me to the “temperamental” environmental chamber. Sneha has also contributed to this work with creep data on neat resin specimens. Robert Carter taught me how to use the large Material Testing Systems’ (MTS) frame in Hancock 107, and even made suggestions on gripping methods for my macromodel composites. Nikhil impressed upon me the need to degas the large epoxy model composite castings to avoid voids. I also enjoyed Nikhil’s input on several other aspects of my work. David Simmons, Darrell Link – Dave and Darrell have energized the Engineering Science and Mechanics (ESM) Department machine shop. They are excellent additions to the Virginia Tech staff. They are very easy to work with, and get jobs done in a very timely manner. Dave has also made some very good design recommendations for my testing fixtures and methods. Dr. Judy Riffle and her research group in the Chemistry Department – I have used the facilities in Dr. Riffle’s laboratory on several occasions over the past four years. Dr. Riffle and her group have been very kind in allowing me free access to their labs at any time. In particular, I would like to thank Dr. Ellen Cooper, Linda Harris, and Mark Flynn. Ellen and Mark have even helped me on occasion with certain aspects of my work over the past four years. I am glad that Linda has finally forgiven me for her completely unfounded claim that I “bumped her head against a table.” I am glad that we are finally very good friends. Sandra Henderson – By virtue of being Scott’s sister, Sandra sometimes got roped into helping with my work, which she always did without the least hesitation. I appreciate Sandra teaching me to use the dilatometer and DMA in Dr. Thomas Ward’s laboratory in the Chemistry department. When I “broke” the computer that supported the DMA she was very patient and understanding, and even got the computer support technicians in the Chemistry department to fix it. Robert Simonds, Daniel Reed – I appreciate Bob teaching me the nuances of material testing and strain gage technology. He has helped me with testing in the Busting Laboratory in Norris Hall, and also provided me with valuable input for the other tests I have run. I thank Bob for the times he has taught me an approach that is different from what we normally do in the MRG. Danny taught me composite fabrication in the “Fab” lab. Over the past four years there have been numerous occasions when he has contributed to my work with his fabrication expertise. Dr. Eric Johnson –Dr. Johnson very kindly allowed me to use his MTS test-frame in Hancock whenever I please.

vi

Shelia Collins, Beverly Williams – This list of acknowledgments is in no particular order, because if it were, then Shelia and Bev deserve to be at the top. Shelia and Bev are a very important part of the MRG. From taking care of all our administrative needs, to making one of us feel special with a cake on our birthday, it is difficult for me to think of the MRG without them. I will miss talking to them during my “ham and cheese” lunch. They have informed me that I will be the second MRG alumni after Blair who will actually keep in touch after leaving Virginia Tech. They have that in writing now. Loretta Tickle, Wanda Robertson, Patricia Baker – My sincere thanks to the staff in the Engineering Science and Mechanics Department. Loretta was wonderful at making sure that I had all my graduate requirements submitted in a timely fashion and with having all the paperwork in place for me to get paid every month. Wanda, and later, Pat had to put up with my repeated requests for purchase orders, because much to Scott’s dismay I was continually ordering stuff. I am very grateful to my friends in Blacksburg and elsewhere who provided me with important diversions from work whenever I needed them. Finally, my heartfelt thanks to my dear parents who got me interested in everything and encouraged me to do almost anything I wanted.

vii

TABLE OF CONTENTS

GRANT INFORMATION..................................................................................................... iii ACKNOWLEDGEMENTS................................................................................................... iv LIST OF FIGURES ............................................................................................................... ix LIST OF TABLES ................................................................................................................xii 1 INTRODUCTION........................................................................................................... 1 2 LITERATURE REVIEW............................................................................................... 4

2.1 MODEL COMPOSITE EXPERIMENTS .............................................................. 4 2.2 LOAD-SHARING ANALYSIS.............................................................................. 8 2.3 QUASI-STATIC STRENGTH MODELING ....................................................... 12 2.4 STRESS-RUPTURE MODELING....................................................................... 17

3 MODEL COMPOSITE MEASUREMENTS ............................................................. 21 3.1 INTRODUCTION................................................................................................. 21 3.2 MATERIALS AND FABRICATION .................................................................. 22

3.2.1 Experimental Technique ........................................................................... 24 3.3 MODEL COMPOSITES WITH SINGLE FIBER FRACTURE.......................... 27

3.3.1 Seven Fiber Model Composites ................................................................ 27 3.3.2 Nineteen Fiber Model Composites............................................................ 30

3.4 MODEL COMPOSITES WITH MULTIPLE FIBER FRACTURES .................. 31 3.5 TIME-DEPENDENT MODEL COMPOSITE MEASUREMENTS.................... 33 3.6 SUMMARY AND CONCLUSIONS ................................................................... 39

4 QUASI-STATIC STRENGTH MODELING ............................................................. 41 4.1 INTRODUCTION................................................................................................. 41 4.2 FEM OF SINGLE FIBER FRACTURE IN MODEL COMPOSITE................... 43

4.2.1 Seven-Fiber Finite Element Analysis........................................................ 43 4.2.2 Seven-Fiber Model Composite Measurements ......................................... 44 4.2.3 Nineteen-Fiber Model Composite Measurements..................................... 45

4.3 FINITE ELEMENT-BASED NNLS..................................................................... 49 4.3.1 General Load-Sharing Concepts ............................................................... 49 4.3.2 Force Influence-Functions from FEM....................................................... 51

4.4 MULTIPLE FIBER FRACTURES IN MODEL COMPOSITE........................... 53 4.5 QUASI-STATIC STRENGTH SIMULATIONS ................................................. 56

4.5.1 Material Properties .................................................................................... 56 4.5.2 Strength Simulation Approach .................................................................. 57

4.6 MATERIAL VARIABILITY ............................................................................... 61 4.6.1 Shear-Lag NNLS....................................................................................... 61 4.6.2 Shear-Lag Versus Finite Element for Regular Hexagonal Fiber Packing 66 4.6.3 Effect of Material Variability on Strength Distribution ............................ 68

4.6.3.1 Fiber Volume Fraction............................................................... 68 4.6.3.2 Random Fiber Placement........................................................... 68 4.6.3.3 Initial Fiber Fractures ............................................................... 68 4.6.3.4 Results ........................................................................................ 69

4.7 STRENGTH PREDICTIONS WITH HVDLS ..................................................... 70 4.7.1 Hedgepeth and Van Dyke Load Sharing (HVDLS).................................. 70

4.7.1.1 Comparison with Shear-Lag NNLS............................................ 72

viii

4.7.2 Strength Simulation with HVDLS ............................................................ 72 4.8 SUMMARY and CONCLUSIONS ...................................................................... 74

5 STRESS-RUPTURE MODELING.............................................................................. 76 5.1 INTRODUCTION................................................................................................. 76 5.2 TIME-DEPENDENT LOAD SHARING ............................................................. 77

5.2.1 General Time-Dependent Load Sharing Concepts.................................... 79 5.2.1.1 Case I (i=1Lk-1) ...................................................................... 85 5.2.1.2 Case II (i = Lk-1+1Lk) ............................................................. 86

5.2.2 Time-Dependent NNLS ............................................................................ 88 5.2.3 Time-Dependent HVDLS ......................................................................... 94 5.2.4 Time-Dependent Load Sharing Based on Finite Elements ....................... 97

5.3 COMPARISON BETWEEN NNLS AND HVDLS ............................................. 98 5.4 STRESS-RUPTURE LIFETIME MODELING ................................................. 103

5.4.1 Rupture Simulation Approach................................................................. 103 5.4.2 Material Systems ..................................................................................... 108 5.4.3 Stress-Rupture Simulation Results.......................................................... 114

5.5 BUNDLE STRENGTH AND RUPTURE LIFETIME PREDICTIONS............ 119 5.6 VARIABILITY IN RUPTURE LIFETIME PREDICTIONS ............................ 120

5.6.1 Case I: Control Case................................................................................ 121 5.6.2 Case II: Narrower Fiber Strength Distribution........................................ 121 5.6.3 Case III: Shorter Perturbed Axial Length Due to Fiber Fracture............ 123

5.7 SUMMARY AND CONCLUSIONS ................................................................. 124 6 STRAIN RATE EFFECTS......................................................................................... 127

6.1 INTRODUCTION............................................................................................... 127 6.2 LITERATURE ON STRAIN RATE EFFECTS................................................. 127 6.3 MODELING ARBITRARY LOADING PROFILES......................................... 129

7 SUMMARY AND CONCLUSIONS.......................................................................... 131 7.1 FUTURE WORK ................................................................................................ 133

REFERENCES.................................................................................................................... 135 VITA..................................................................................................................................... 141

ix

LIST OF FIGURES

Figure 1-1. Schematic of fiber stress redistribution (load sharing) in unidirectional polymer composite loaded in the fiber direction ............................................................. 3

Figure 3-1. Schematic of model composite that is representative of a unidirectional composite material .......................................................................................... 23

Figure 3-2. Representative strain measurement from a gage mounted on an unbroken fiber (7-fiber model composite)............................................................................... 27

Figure 3-3. Model composite with two adjacent, coplanar fiber fractures (Model 8) ........... 32

Figure 3-4. Model composite with three adjacent, coplanar fiber fractures (Model 9) ......... 32

Figure 3-5. Schematic of fiber stresses under creep loading of model composite. (a) Broken fiber (b) Neighboring fiber.............................................................................. 34

Figure 3-6. Representative load versus strain curve obtained during stress relaxation tests . 36

Figure 3-7. Time-dependent strain concentration (Model 5) ................................................. 36

Figure 3-8. Time-dependent strain concentration (Model 6) ................................................. 37

Figure 3-9. Time-dependent strain concentration (Model 7) ................................................. 37

Figure 3-10. Time-dependent strain concentration (Model 8) ............................................... 38

Figure 3-11. Time-dependent strain concentration (Model 9) ............................................... 38

Figure 4-1. Geometry and boundary conditions of finite element model .............................. 45

Figure 4-2. Comparison between FEM and seven-fiber model composite. Broken fiber. (M = Model; G = Gage) ........................................................................................... 46

Figure 4-3. Comparison between FEM and seven-fiber model composite. Neighboring fiber, facing broken fiber. (M = Model; G = Gage).................................................. 47

Figure 4-4. Comparison between FEM and seven-fiber model composite. Neighboring fiber, facing away from broken fiber. (M = Model; G = Gage) ............................... 47

Figure 4-5. Comparison between FEM and nineteen-fiber model composite. Broken fiber. (M = Model; G = Gage) .................................................................................. 48

Figure 4-6. Comparison between FEM and nineteen-fiber model composite. Neighboring fiber. (M = Model; G = Gage)......................................................................... 49

Figure 4-7. Hexagonally packed array of fibers with fiber numbering scheme..................... 51

Figure 4-8. Weighted average finite element stress. (a) Broken fiber (b) Neighboring fiber......................................................................................................................... 52

Figure 4-9. Model composite with two adjacent, coplanar fiber fractures (Model 8) ........... 55

Figure 4-10. Model composite with three adjacent, coplanar fiber fractures (Model 9) ....... 55

x

Figure 4-11. Statistical strength of Grafil/PPS unidirectional composite (Gage length = 76 mm, Vf = 40%)................................................................................................. 59

Figure 4-12. Flowchart of Monte Carlo simulation for quasi-static strength......................... 60

Figure 4-13. Representative volume element (RVE) for quasi-static strength simulation .... 60

Figure 4-14. Composite strength of Grafil/PPS unidirectional composite obtained from simulation........................................................................................................ 61

Figure 4-15. Nearest neighbor load-sharing with random fiber placement ........................... 65

Figure 4-16. Stresses in nearest unbroken neighbors due to random fiber placement........... 65

Figure 4-17. Comparison between shear-lag and FEM for broken fiber ............................... 67

Figure 4-18. Comparison between shear-lag and FEM for nearest neighbor ........................ 67

Figure 4-19. Comparison between NNLS and HVDLS for broken fiber .............................. 73

Figure 4-20. Comparison between NNLS and HVDLS for neighboring fibers..................... 73

Figure 4-21. Comparison between strength predictions from NNLS and HVDLS ............... 74

Figure 5-1. Hexagonally packed array of fibers with fiber numbering scheme..................... 78

Figure 5-2. Break opening-displacements for breaks 1L1 due to first Lk fractures ............ 81

Figure 5-3. Break opening-displacements for breaks L1+1L2 due to first Lk fractures ...... 82

Figure 5-4. Break opening-displacements for breaks Lk-1+1Lk due to first Lk fractures .... 82

Figure 5-5. Fiber stresses at breaks 1L1 due to first Lk fractures........................................ 83

Figure 5-6. Fiber stresses at breaks L1+1L2 due to first Lk fractures .................................. 83

Figure 5-7. Fiber stresses at breaks Lk-1+1Lk due to first Lk fractures................................ 84

Figure 5-8. Stress in fiber (1,0) due to isolated break in shaded fiber at x = 0 computed with NNLS ............................................................................................................ 100

Figure 5-9. Stress in fiber (1,0) due to isolated break in shaded fiber at x = 0 computed with HVDLS.......................................................................................................... 100

Figure 5-10. Stress in fiber (2,0) due to isolated break in shaded fiber at x = 0 computed with HVDLS.......................................................................................................... 101

Figure 5-11. Stress in fiber (1,1) due to isolated break in shaded fiber at x = 0 computed with HVDLS.......................................................................................................... 101

Figure 5-12. Stress in broken fiber (0,0) due to isolated break at x = 0 computed with NNLS....................................................................................................................... 102

Figure 5-13. Model composite measurements of strain concentrations due to a single fiber fracture .......................................................................................................... 102

xi

Figure 5-14. Model composite measurements of strain concentrations due to a two adjacent coplanar fiber fractures.................................................................................. 103

Figure 5-15. Flowchart of Monte Carlo simulation for stress rupture lifetime.................... 107

Figure 5-16. Representative volume element (RVE) for rupture simulation....................... 108

Figure 5-17. Stress rupture lifetimes of Grafil carbon fiber/PPS unidirectional composite 110

Figure 5-18. Tabbing of specimens for tensile strength and stress rupture testing.............. 110

Figure 5-19. Failed specimens. (a) Grafil carbon fiber/PPS unidirectional composite (b) APC-2 [90/03]s laminate................................................................................ 111

Figure 5-20. Stress rupture lifetime of APC-2 [90/03]s specimens ...................................... 113

Figure 5-21. Master curve for shear creep compliance of PEEK......................................... 114

Figure 5-22. Shift factors for creep master curve of PEEK ................................................. 114

Figure 5-23. Rupture lifetime predictions for APC-2 composite at 125C (NNLS)............ 117

Figure 5-24. Rupture lifetime predictions for APC-2 composite at 140C (NNLS)............ 118

Figure 5-25. Rupture lifetime predictions for APC-2 composite at 125C (HVDLS)......... 118

Figure 5-26. Rupture lifetime predictions for APC-2 composite at 140C (HVDLS)......... 119

Figure 5-27. Lifetime distribution for control case .............................................................. 121

Figure 5-28. Lifetime distribution with narrower fiber strength distribution ...................... 122

Figure 5-29. Lifetime distribution with shorter perturbed axial length along a broken or neighboring fiber due to a fiber fracture ....................................................... 124

xii

LIST OF TABLES

Table 3-1. Constituent material properties for model composite12 ........................................ 24

Table 3-2. Quasi-static strain concentration measurements on seven fiber model composites......................................................................................................................... 28

Table 3-3. Quasi-static strain concentration measurements on nineteen-fiber model composites....................................................................................................... 30

Table 4-1. Comparison between shear-lag and FEM composite strength predictions (76 mm gage length) ..................................................................................................... 68

Table 4-2. Predicted composite Weibull shape parameter )(Xm , by including sources of material variability .......................................................................................... 69

Table 4-3. Predicted composite Weibull shape parameter )(Xm , from HVDLS and NNLS74

Table 5-1. Quasi-static strength of APC-2 [90/03]s (strengths reported at 76 mm gage length)....................................................................................................................... 112

Table 5-2. Quasi-static strength predictions of unidirectional APC-2 Vf = 54% obtained by applying two different load-sharing techniques (strengths reported at X = 0.47 mm) ............................................................................................................... 115

1

1 INTRODUCTION

The goal of this work is to develop models to predict the tensile strength and stress-rupture

lifetime of unidirectional carbon fiber/polymer matrix composite materials from the

properties of the fiber and matrix. Such a micromechanical model relies on an understanding

of the fiber stress redistribution near broken fibers, which is termed load sharing. The

unidirectional composites are loaded in the fiber direction as shown in Figure 1-1. If all the

fibers in the composite material are intact then a uniform state of axial fiber stress exists.

Once a fiber fracture occurs there is micromechanical fiber stress redistribution or load

sharing among the fibers in the vicinity of the fracture. In the absence of any matrix

material, a fractured fiber is totally ineffective in carrying load and its load is redistributed

equally to the remaining fibers in the bundle. This is not the case in composite materials

since the fibers are surrounded with matrix that adheres to them. Even in a composite

material, at the fracture location the broken fiber carries no axial stress. However, the matrix

feeds stress back into the broken fiber via shear on the cylindrical surface of the fiber, until at

a sufficient distance from the fracture the broken fiber is unaware of the break further down

its length. This is shown schematically in Figure 1-1. The matrix is also the medium by

which the load that the broken fiber is transferred to its nearest neighbors. Hence, there is an

overstressed length on the neighboring fibers as shown in Figure 1-1. This local fiber stress

redistribution is one of the primary interests of this work. Chapter 3 presents model

composite measurements of strain redistribution near single and multiple fiber fractures. The

measurements provide insight into load sharing and help develop models to describe it. The

fibers and matrix are treated as linearly elastic and a quasi-static version of load sharing is

developed for tensile strength models in Chapter 4. The fibers are treated as linearly elastic

and the matrix as linearly viscoelastic and a time-dependent version of load sharing is

developed for the stress rupture models in Chapter 5. Both the strength and rupture models

are Monte Carlo simulations that account for variability in fiber strengths. The tensile

strength simulations also incorporate initial fiber fractures, random fiber placement, and

distributed fiber volume fractions.

2

Chapter 3 through 5 form bulk of this dissertation, and they are compiled as three separate

articles. Chapter 6 very briefly discusses some of the issues associated with extending the

work in this dissertation to model arbitrary loading profiles, with special emphasis on strain

rate effects. Finally, in Chapter 7 the major contributions of this work are summarized and

avenues for future research are described.

From an experimental standpoint this work is important because direct measurements of

quasi-static and time-dependent load sharing are made on model composite systems with

three-dimensional fiber packing and high fiber content. The modeling work is a significant

contribution to the literature because it provides a technique for stress-rupture lifetime

prediction from the building blocks of a composite material, i.e. fibers and matrix. It is

hoped that micromechanical life-prediction efforts such as this one will eventually serve as

tools for making design decision with regard to choice of fiber, matrix, and interphase

systems for given lifetime requirements without the need for expensive testing programs. In

addition, prediction of stress-rupture lifetime is a fundamental step in future micromechanical

modeling efforts for composite lifetime that are general, in that they account for the

interaction of more complex mechanical and environmental loading profiles. The

micromechanical modeling technique also serves as a basis for comparison with other life

prediction techniques that are phenomenological in nature.

3

TENSILE LOAD ON COMPOSITEIN FIBER DIRECTION

FIBERS

MATRIX

TENSILE LOAD ON COMPOSITEIN FIBER DIRECTION

FIBERS

MATRIX

Figure 1-1. Schematic of fiber stress redistribution (load sharing) in unidirectional polymer composite loaded in the fiber direction

4

2 LITERATURE REVIEW

2.1 MODEL COMPOSITE EXPERIMENTS

Model composites provide a tool for studying the micromechanical events leading to failure

of composite systems. Knowledge gained from these experiments is used to support and

corroborate micromechanics models for load sharing and failure in real composite systems.

Several different types of model composites have been studied in the past, and the following

is a discussion of the significant works.

The simplest model composite systems consist of a single fiber embedded in matrix

material.1 These model composites are used to perform fragmentation tests wherein a tensile

load is applied to the model composite along the fiber axis until a saturation level of fiber

fragments is attained. The fragment lengths are used to calculate the fiber/matrix interfacial

shear strength. Huang and Young2 studied the dependence of interfacial shear strength on

curing temperature using a single carbon fiber/epoxy resin model composite. The fragment

length was detected using both optical microscopy with crossed polars and Raman

spectroscopy. Huang and Young showed that the composite cured at the higher temperature

showed significantly higher interfacial shear strength, lower levels of debonding during

fragmentation at high strain levels, and higher frictional shear stresses in the debonded

region. They explained this behavior by the existence of a higher radial pressure at the

fiber/matrix interface in the model composite cured at a higher temperature. In a later work,

Yallee and Young3 studied the micromechanics of single fiber fragmentation tests with a

model epoxy composite reinforced with α-Alumina fibers. The strain along the fibers was

obtained by luminescence spectroscopy, from which the distribution of interfacial shear

stress was derived using force-balance considerations. The effect of sized and unsized fibers

Nextel 610 α-Alumina fibers, and room and high temperature cured epoxy resin matrices

were studied. Pyrz and Nielsen4 monitored the residual thermal strains in a single carbon

fiber/polypropylene matrix model composite using micro-Raman spectroscopy. They

observed a non-linear accumulation of residual strains within the fiber as the temperature was

decreased. Residual strains in the fiber at room temperature were influenced by the cooling

5

rate with higher residual strains for the lower cooling rate. Cooling of the model composite

also caused wide spread fiber fragmentation with the fragment lengths and residual strains

dependent on cooling rate. This was attributed to the effect cooling rate has on thermoplastic

matrix crystallization and preferential crystal nucleation around the fiber.

Model composite systems have also been used to study the various mechanisms that occur

during fracture of unidirectional composite materials as a result of quasi-static and fatigue

loading. Zhao and Botsis5 constructed model composites with a compact tension specimen

geometry using an epoxy matrix and layers of long aligned glass or Kevlar fiber tows that

were equally spaced. The tows were aligned perpendicular to the plane of the notch. Direct

observation of the crack growth, debonding, crack opening displacements, possible

dissipative mechanisms in the matrix, and crack front changes due to reinforcement were

made owing to the simplicity of the model composite system. Their efforts were

concentrated on characterizing the crack initiation strength as a function of the fiber tow

spacing, and fatigue crack growth rate as a function of applied loads and tow diameter. They

showed that for a range of tow spacing and given matrix toughness, the crack initiation

strength was inversely proportional to the square root of tow spacing. The apparent

toughness of the composite specimen increased with decrease in tow spacing. A method was

presented to evaluate the stress intensity factors and tractions due to bridging fiber tows by

assuming a linear crack opening profile. A one-dimensional debonding analysis was used to

evaluate the debonding in the bridging zone for different tow spacing. Goutianos and Peijs6

used polarized light microscopy to study the influence of stress level and fiber/matrix

adhesion on fatigue failure process in carbon-epoxy multi-fiber model composites. The

model composite consisted of five parallel fibers embedded in an epoxy matrix with the

inter-fiber spacing adjusted to approximately three fiber diameters. They showed that

increasing the levels of tension-tension fatigue stress resulted in an increase in the total

number of fiber breaks due to stronger fiber-fiber interaction caused by stress concentrations

due to adjacent breaks in fibers. Improved fiber/matrix adhesion also caused an increase in

fiber-fiber interactions, while greater debonding was observed as a result of poor interfacial

adhesion. The experimental observations were explained by performing a quasi-static finite

element analysis.

6

Van Den Heuvel7 has presented a collection of studies using Raman spectroscopy and

polarized light microscopy to monitor fiber/matrix interaction in multi-fiber microcomposite

systems. This collection detailed work performed on the effects of inter-fiber distance,

fiber/matrix adhesion, and matrix properties on fiber-fiber interactions.

Chohan and Galiotis8 studied the interfacial and fracture characteristics of two-dimensional

microcomposite tapes, impregnated fiber tows, and full composite tensile coupons. The

composite system they studied comprised of high-modulus carbon fibers with an epoxy

matrix. The stress state in fibers was monitored by remote laser Raman microscopy.

Measurements were made on fibers in a planar array of uniformly distributed fibers, and on

fiber near the surface of full composites with randomly distributed fibers. For the full

composite system the measurements along any fiber were affected by shear field

perturbations due to the three-dimensional architecture that resulted in scatter in the

measured stresses. They derived an empirical relation for the strong dependence of stress

concentrations on inter-fiber distance and specimen geometry.

Model composites provide a simple material system to make measurements of statistical

material strengths and lifetime that are more amenable to modeling techniques. Phoenix et

al.9 obtained statistical strength and rupture lifetimes of unidirectional model carbon

fiber/epoxy matrix microcomposites. Their model composites consisted of seven parallel

carbon fibers forming approximate hexagonal packing embedded in an epoxy matrix. They

also presented statistical single fiber strengths at several gage lengths. The microcomposite

statistical strength and rupture lifetimes were interpreted by means of analytical models.

They obtained a power-law relationship between stress level and lifetime with exponent that

depended on the Weibull shape parameter for fiber strength. Their model incorporated the

Weibull distributions for fiber strength, micromechanical stress transfer due to fiber

fractures, and power-law matrix creep around break locations that ultimately resulted in

creep-rupture. The important modeling parameters were the stress concentrations and

effective overloaded length, the Weibull shape parameter for fiber strength, and a creep

exponent for the matrix that governed growth of the overloaded region on fibers adjacent to

7

breaks. The model predicted Weibull composite strengths that compared favorably with

experimental measurements. Interpretation of the microcomposite lifetimes was not easy due

to dynamic overloads at the start of the rupture experiments. Phoenix et al. also

recommended more careful characterization of fiber strength, matrix viscoelasticity, and

time-dependent debonding of the fiber/matrix interphase in order to obtain better lifetime

predictions. In a later publication, Otani et al.10 addressed the shortcomings of the earlier

work by Phoenix et al.9 by paying special attention to clamping, specimen alignment, shock

isolation and accurate lifetime measurement. Otani et al. used a different epoxy matrix

system for their microcomposites which had a matrix creep exponent and effective load

transfer length that were about double and triple, respectively, the values from the earlier

work done by Phoenix et al. This resulted in slightly reduced strength, about one-half the

variability in lifetime, and one-half the exponent of the power-law relating stress to lifetime

when compared to the earlier study. A fractographic study of the microcomposite suggested

that time-depended fiber/matrix debonding is a key contributor to failure. Beyerlein and

Phoenix11 examined the statistics of size effect on strength by using unidirectional

microcomposites consisting of four carbon fibers embedded in epoxy matrices with

approximately square packing. They modified the model for statistical strength presented in

earlier works9,10 to account for a factor that reflects variability of the fiber diameter and

material texture from fiber to fiber as against variability along a single fiber. They then

compared the model predictions for size effects to the size effects observed in their

microcomposites.

Carman et al.12 developed an experimental procedure using a macromodel composite to

measure the perturbed strain field resulting from a failed fiber in a unidirectional composite

material. Glass rods with 3 mm diameter were used as fibers, and a birefringent epoxy was

used as the matrix. Since the fibers were large enough, strain gages were mounted directly

onto the fibers to make quantitative measurements of the fiber strains. Fabry Perot fiber

optic strain sensors embedded in the matrix yielded quantitative information about the matrix

strains. The stress gradients produced by the internal strain concentration produced different

hues of color in the matrix that provided qualitative information about the perturbed stress

field. A single controlled fiber fracture at a predetermined location was obtained by scoring

8

the fiber with a glass-cutting tool. Measurements of the stress concentrations and ineffective

lengths (i.e. size of perturbed stress field) were made for different fiber volume fractions and

fiber/matrix interphase coatings. Their findings suggested that the size of the matrix crack

propagating from a fiber fracture location significantly affected the stress transfer. The

interphase treatment was suggested as a factor that affected matrix crack propagation.

Carman and his co-workers12 studied a two-dimensional array of fibers (i.e. a single row of

fibers) with local fiber volume fractions of 20% and less. The expertise for fabricating and

testing the model composites described in this dissertation is drawn from work done by

Carman, et al.12 However, in this work three-dimensional hexagonal fiber packing is studied,

with a much higher fiber volume fraction of 40%.

In summary, several model composite studies have been performed for measuring load

sharing, strength and rupture lifetime. The work presented in this dissertation is a

contribution to the literature because quasi-static and time-dependent measurements of fiber

strain redistribution are made in three-dimensional model composites with high fiber volume

fractions. Polarized light microscopy and Raman spectroscopy present problems when

applied to three-dimensional fiber packing geometries since the presence of fibers in the line

of sight and directly behind the area of interest distort the obtained measurements. In this

work the effect of multiple fiber fractures is also studied, and the measurements are used in

Chapter 4 to evaluate the performance of influence-function superposition modeling

principles. Moreover, with the macromodel composites described in this work a mapping of

the three-dimensional strain field on the surface of the fibers is obtained. This provides a

unique insight into the complex strain field experienced by fibers in a composite material,

and as described in Chapter 4 provides important clues for modeling load sharing in the best

possible manner.

2.2 LOAD-SHARING ANALYSIS

As mentioned earlier, load-sharing analyses are key to any micromechanical strength

modeling effort. When one or more fibers break, a redistribution of fiber stresses in the

vicinity of the break is required in order to maintain equilibrium. The matrix makes this

stress redistribution possible. Hedgepeth and Van Dyke13 presented solutions for the fiber

9

stress state in the vicinity of failed fibers in a composite material. Their analysis was based

on the shear-lag assumption, which implies that the fibers are tension-carrying members

connected by matrix that carries only shear stress. The shear lag concept was originally

introduced by Cox14 to calculate the stresses in a broken fiber embedded in a composite

material. Hedgepeth and Van Dyke extended Cox’s idea to calculate the stresses in all fibers

due to coplanar fiber fractures occurring in a three-dimensional composite with regular fiber

packing. They also solved for the stress concentration factor adjacent to a broken fiber with

a matrix that exhibits linear elastic, ideally plastic behavior.

In subsequent works, several researchers borrowed Hedgepeth and Van Dykes influence

superposition approach to solve the problem of multiple breaks in a unidirectional composite

material with added degrees of modeling sophistication. Landis et al.15 addressed the

question of how to choose effective dimensions of the matrix springs connecting neighboring

fibers by modeling the matrix as three-dimensional finite elements and the fibers as

continuous one-dimensional springs. Their model also considered direct interactions of

broken fibers with the next to nearest neighbors, which was absent in the Hedgepeth and Van

Dyke formulation. They showed that the peak stress concentration on a fiber nearest to a

broken fiber is lower than predicted by Hedgepeth and Van Dyke which was in agreement

with three-dimensional and axisymmetric finite element calculations performed by Nedele

and Wisnom.16,17 Nedele and Wisnom also showed that the peak stress concentration on

fibers nearest to a broken fiber occurred slightly out of plane of the break. Landis and

McMeeking18 later extended the work of Landis et al.15 to account for the effects of interface

sliding, axial matrix stiffness, and uneven fiber positioning on stress concentrations

surrounding a single fiber break.

Beyerlein and Landis19 developed a technique to quickly compute load sharing in a two-

dimensional unidirectional fiber composite under shear-lag assumptions. They modified the

analysis so that both fibers and matrix were able to sustain axial loads. The governing

equations were derived based on the principle of virtual work. The primary objectives of

their work were computational speed and the ability to account for arbitrarily located fiber

fractures, with the ultimate goal of using this type of analysis in large-scale simulation codes

10

of failure in fibrous composites. An added advantage of their analysis was the ability to

account for misaligned fibers in a composite material.

For micromechanical stress-rupture modeling it is necessary to obtain the time-dependence

of load sharing due to matrix viscoelasticity. Lagoudas et al.20 modeled a monolayer of

unidirectional composite loaded in tension with linear elastic fiber properties and linearly

viscoelastic matrix response. Exact closed-form and approximate solutions for the time

evolution of stresses in the fibers were calculated under shear-lag assumptions and

comparisons were made to each other. All the solutions presented were for the simplified

case of clusters of coplanar breaks occurring simultaneously. The solution procedure for a

growing coplanar break cluster with pairwise fiber breaks that occur sequentially in time was

also outlined. Mason et al.21 considered the time evolution of stress transfer when the matrix

response is nonlinear, described by a power creep law. They considered a single broken fiber

in a planar composite consisting of three and five infinitely long fibers. The motivation for

their work was derived from the fact that linear viscoelastic matrix response may not be valid

in the highly constrained microscopic region between fibers. They suggested using a power

law to represent the large-scale yielding and subsequent strain hardening behavior of typical

epoxies in highly constrained geometries. They showed that the growth exponent of the

deformation zone was different from the exponent in the power law for the matrix. Beyerlein

et al.22 developed an efficient computational technique called viscous break interaction to

compute time-dependent stresses in fibers due to arbitrarily positioned fiber breaks in a

unidirectional composite material with a viscous or linearly viscoelastic matrix. They

developed this technique with the ultimate goal of being able to perform simulations to

model the statistical nature of creep-rupture in unidirectional composite materials. In order

to simplify the analysis, interface debonding was not permitted, and Hedgepeth and Van

Dykes shear-lag framework with influence superposition techniques was used. Several

interesting cases with interacting fiber break locations were presented to illustrate the general

applicability of this approach.

Nairn23 has presented a very detailed assessment of the accuracy of using shear-lag

assumptions to model stress transfer in unidirectional composite materials. Nairn started

11

with the full equations of transversely isotropic elasticity in axisymmetric coordinates, and

introduced a minimum number of assumptions to obtain the most commonly used shear-lag

equations. He suggested verifying whether shear-lag may be applied to a particular problem

by checking if these assumptions were valid. Nairn also derived a new shear-lag parameter

that yielded better agreement with elasticity solutions for the axial stresses in fibers and the

total strain energy. The shear-lag analysis yielded poor predictions of energy release rates

and shear stresses, and did not work well for problems involving low fiber volume fractions.

Shear-lag type analyses such as that developed by Hedgepeth and Van Dyke did not include

the effects of constituent properties, fiber volume fraction, and matrix crack size on the stress

concentrations experienced by neighboring fibers. Carman et. al24 and Case et al.25 proposed

an entirely different analysis technique to study the stress field in a general unidirectional

composite material containing fiber fractures. The model involved using an approximating

annular ring of fibers to represent the unbroken neighboring fibers. Multiple fiber breaks

were modeled by a fiber discount methodology. Direct comparison of the model results were

made to experimental data from model composite studies. In order to direct attention to

some of the features shear-lag models may be missing Case and Reifsnider26 addressed the

problem of a penny shaped crack in the center of multiple concentric cylinders. The problem

was solved by applying standard elasticity assumptions, with appropriate choice of stress

functions in each constituent. This solution was applied to the problem of a fiber fracture in

a unidirectional composite material by making a geometric assumption.

In summary, significant gains have been achieved with modeling load sharing in

unidirectional composite systems. However, two concerns predominate when using these

techniques in micromechanical models for strength and stress-rupture:

1. Many of the more sophisticated load-sharing approaches require experimentally

measurable input quantities that are difficult, or in some cases even impossible to

obtain. For example, the interfacial shear strength and fracture toughness are required

in order to calibrate load-sharing analyses that account for fiber/matrix debonding.

There is no consensus today on how these quantities should be measured.27,28

12

2. Computational speed and generality of the analysis technique are necessary when

using a Monte Carlo approach for strength/rupture modeling. The load-sharing

analysis needs to be performed several times for progressively increasing sets of fiber

fractures in each strength or lifetime computation. Hence, computational efficiency is

imperative. Also, the analysis should be able to allow for arbitrarily located fiber

fractures in a three-dimensional array of fibers and matrix. Hedgepeth and Van

Dykes influence-function superposition technique is most suited to strength/rupture

modeling with the Monte Carlo technique. This technique is adopted for much of the

modeling that will be presented in this dissertation.

In this work, model composite measurements are used to justify development of a new form

of load sharing called nearest neighbor load sharing (NNLS) wherein the load of a broken

fiber is redistributed only onto its nearest neighbors. Moreover, the quasi-static form of

NNLS is developed to include non-uniformities in fiber packing. It is also shown that for

modeling strength a shear-lag NNLS provides answers that are comparable with a more

sophisticated finite element load sharing. To the author’s knowledge, the time and

temperature dependent analysis developed in this work is the most general load-sharing

analysis in the literature in that it can account for an arbitrary sequence of fiber fractures and

matrix viscoelastic properties that can be expressed as a Prony series.

2.3 QUASI-STATIC STRENGTH MODELING

A very simple estimator of the tensile strength of unidirectional polymer composites loaded

in the fiber direction is obtained by calculating the bundle strength. The bundle strength is

the tensile strength of an unimpregnated fiber bundle.29 The matrix, however, does play a

critical role in the strength of a composite material and it would be remiss to ignore it. Since

not all fibers in a unidirectional composite are of the same strength, some of the weaker

fibers will fail first under tension in the fiber direction. Tsai and Hahn29 provide an excellent

explanation of the modes of further damage growth after initial fiber failures. They consider

two contrasting mechanisms. For a ductile matrix with weak interface the broken fibers are

separated from the intact ones as far as fiber load redistribution is concerned. This type of

failure is characterized by longitudinal splitting and this limiting value of composite strength

is given by bundle strength theory. The second failure mode occurs in a composite with a

13

brittle matrix with strong interface. A matrix crack propagates transversely across the

neighboring fibers leading to rapid composite failure. This failure mode is also not favorable

since the strength of the stronger fibers is not fully utilized. The optimum strength of the

composite material is realized for matrix and interface properties that are intermediate

between the above two extremes such that a combination of longitudinal and transverse

failure occurs in a localized region near the fiber tips. With this understanding, Tsai and

Hahn develop an elegant and simple mathematical framework for the tensile strength of a

unidirectional composite material that incorporates the fiber strength distribution and the

fiber/matrix interfacial yield strength.

Rosen30,31 is responsible for some of the early work in understanding tensile failure of

unidirectional composite materials loaded in the fiber direction. He experimentally showed

that random fiber fractures occur at loads below the ultimate composite strength level, and

the statistical accumulation of these flaws eventually leads to composite failure. The test

specimens on which he made these observations consisted of a single layer of glass fibers

embedded in an epoxy matrix. Microscopic evaluation of the internal failure process was

performed using photoelastic techniques. He also presented one of the first models for

tensile strength by assuming that the strength of the brittle fibers is statistically distributed.

He developed an approximate solution for the interface stresses and the stress concentration

on an adjacent fiber, which could result in adjacent fiber fracture or fiber/matrix separation,

respectively. The far-field load on the composite material was increased until a weak cross-

section of material could no longer sustain equilibrium. These ideas in essence have been

incorporated into most micromechanical models for tensile strength that have been developed

since then. Rosen attempted to make comparisons of the model with the experimental

observations.

Batdorf32,33 developed a probabilistic framework to track the growth of fiber fracture clusters

in a unidirectional composite material under the application of a tensile load. He considered

the probability of growing a cluster of fiber fractures based on the stress concentrations and

overloaded lengths seen by the nearest surviving fibers. The stress concentrations and

ineffective lengths are the material specific inputs to his analysis. Techniques such as

14

Hedgepeth and Van Dyke13 method can be used in conjunction with Batdorf’s analysis. Gao

and Reifsnider34 introduced a load-sharing analysis that they used in conjunction with

Batdorf’s probabilistic framework for tensile strength. The model is based on shear-lag

assumptions with a geometric assumption to smear the cluster of fiber fractures. They

incorporated shear yielding at the fiber/matrix interface by defining a shear parameter that

may be used to define complete fiber/matrix debonding or elastic, perfectly plastic matrix

behavior. They compared their strength predictions to experimental data in the literature. An

important aspect of their work was to show that changing the fiber/matrix interfacial shear

strength changes both the maximum overstress and the overstressed length on the nearest

unbroken neighbors in the presence of a fiber fracture. They showed that increasing the

interfacial shear strength increases the maximum overstress while decreasing the overstressed

length. Hence, an important conclusion of their work was to show that the quasi-static

strength of a composite material is not monotonically dependent on interfacial shear strength

and can attain a maximum at an optimum value of interfacial shear strength.

Smith et. al35 considered the strength of impregnated bundles of fibers with statistical

strength distribution and local load-sharing rules. Local load sharing implies that the load of

fractured fibers is redistributed only onto the nearest surviving neighbors. Two different

local load-sharing rules were developed, one geometrically motivated and the other

mechanically motivated. They showed that both approaches yielded approximately the same

strength although they were drastically different in nature. The probability distribution of

composite strength was calculated by asymptotic techniques.

Monte Carlo simulations have proved to be an extremely effective method for modeling the

strength distribution of unidirectional composites subjected to a tensile load in the fiber

direction. They provide a modeling framework to account for statistical fiber strengths,

without the simplifying assumptions that are necessary when developing approximate

analytical techniques. Zhou and Curtin36 developed a simulation technique for strength of

fiber-reinforced composites that utilized three-dimensional lattice Green’s functions to

calculate load sharing with fiber/matrix sliding. The technique allowed adjustment of the

zone of load-transfer to address effects that may be seen in real composites. They showed

15

that composite failure resulted from a complex arrangement of fiber fractures with a single

cross-sectional plane composed mainly of sliding fibers with a few intact strong fibers.

Ibnabdeljalil and Curtin37 studied the size effects of composite strength by using the lattice

Green’s function technique with Hedgepeth and Van Dyke’s load sharing used to obtain

stress concentrations in the plane of a fiber fracture and fiber/matrix interface sliding to

obtain out of plane fiber stress variations via a shear-lag model. They also developed

analytical asymptotical relations for the strength that agree well with the median strength and

high reliability tail of the distribution predicted by the Monte Carlo simulations.

Ibnabdeljalil and Curtin38 also used the Green’s function simulation technique to study the

strength of a composite material having a pre-existing cluster of fiber breaks. The

simulations were used to characterize the decrease in composite strength, but increased

reliability with increasing size of pre-existing cluster of fiber fractures. An analytical model

was developed for the strength and reliability of notched composites that agreed well with the

simulation results. Curtin and Takeda39 used the Green function simulation technique to

compare strength predictions obtained by considering square and hexagonal fiber packing.

They showed that fiber packing does not effect statistical strength predictions, and the size

scaling of strength significantly. Based on this, they were able to state that the analytical

results they derived by assuming square fiber packing37,38 were to a large extent independent

of fiber arrangement. In a companion paper,40 the analytic model was applied to predict the

tensile strength of AS-4 fiber/Epon828 matrix and T300 fiber/Epicote matrix composite

systems. The required constituent properties and composite data for comparison purposes

were available in the literature. Landis et al.41 developed a Monte Carlo based simulation

technique that utilized shear-lag load sharing15 along with fiber strengths described by a

Weibull distribution. Length scaling and the effect of number of fibers in the simulation

volume were investigated. The strength distributions of composite materials with different

fiber Weibull moduli were also computed. The work done by Landis et. al represents one of

the most complete strength simulations to date that incorporate three-dimensional load

sharing within the shear-lag framework. Mahesh et al.42 performed similar Monte Carlo

simulations. They, however, confined all breaks to occur within a two-dimensional plane

perpendicular to the fiber direction. With this simplification they were able to study much

lower values of fiber Weibull modulus and much larger composite sizes than addressed

16

previously. They showed that for very low fiber Weibull modulus the composite displayed a

transition to equal load sharing like behavior i.e. all the surviving neighbors carry the load of

the failed fiber equally. For narrower fiber strength distributions it was observed that the

fiber breaks grew in clusters with composite strength extremely sensitive to the size and

location of preexisting flaws. For high fiber strength variability the fiber fractures were

widely dispersed, and the composite strength was insensitive to initial flaws. They

investigated this transition from stress-driven to strength-driven composite failure in great

detail.

Phoenix and Beyerlein43 have compiled the most comprehensive review of strength theories

for composite materials. They discussed several past works on strength modeling and the

associated load sharing before proposing modifications to existing strength modeling

approaches, and even presenting new analytical techniques. New analytical approximations

for failure were developed that incorporated Hedgepeth and Van Dykes13 load sharing. For

small variability in fiber strengths the analytical model they developed compared favorably

with existing Monte Carlo simulations in Mahesh et al.42, Landis et al.41, and Ibnabdeljalil

and Curtin37,38. Comparisons of the various models were also made to experimental results,

with model prediction being in general agreement with the experimental observations.

The strength distribution of microcomposite systems have been modeled, and comparisons

made to experimental data9,10,11. These studies were used to assess whether the strength

models worked for simpler systems with fewer fibers and known constituent properties

before they were applied to more complex real composite systems.

In summary, Monte Carlo simulations provide the best technique for modeling the tensile

strength distribution of unidirectional composite materials by making a minimum number of

modeling assumptions. To date, however, the most sophisticated Monte Carlo techniques are

based on shear-lag load sharing assuming perfect fiber/matrix bonding or fiber/matrix

interface sliding. Incorporating matrix interfacial yielding, stress intensity factors due matrix

cracks propagating from a fiber fracture, or random fiber arrangements within a general

three-dimensional simulation framework is very challenging, and has not been addressed.

17

This work develops Monte Carlo simulations for tensile strength of composite materials

based on NNLS. Force influence-functions are computed from a nearest neighbor finite

element model and these influence functions are used in the strength computation. It is

shown that shear-lag based NNLS computes a strength distribution that is comparable to the

strength distribution obtained with the finite element approach. The effect of initial fiber

fractures, random fiber placement, and distributed fiber volume fractions on the computed

strength distribution is addressed by applying the shear-lag NNLS. Very good agreement

exists between the simulation predictions and the strength distribution of a carbon

fiber/polymer matrix composite.

2.4 STRESS-RUPTURE MODELING

Lifshitz44 has put together the earliest comprehensive review of mechanisms behind time-

dependent failure in unidirectional composites loaded in the fiber direction. He began with a

review of experimental data on the quasi-static and time-dependent strength of most

commonly used fibers. This was followed by a review of the time-dependent strength of

metal alloy matrices since the stress-rupture of composite systems with these matrices is to a

large extent controlled by creep of the matrix. Lifshitz then discussed the experimental and

theoretical studies on stress-rupture of composite systems with polymeric and metallic

matrices, and the theories on delayed failure of dry bundles of fibers. Of particular relevance

to this work is the time-dependent failure model for polymeric composites that was originally

published by Lifshitz and Rotem.45 They were the first to develop a micromechanical theory

for stress-rupture in composite materials consisting of brittle fibers with probabilistic

strengths embedded in a viscoelastic matrix. Their model was based on Rosen’s work30 for

the elastic strength of unidirectional composites loaded in the fiber direction. They used

viscoelastic arguments to calculate the growth of the portion of the broken fiber that becomes

ineffective in carrying load. It was widely believed that stress-rupture of polymer composites

resulted from the time-dependent strength of some fibers (e.g. glass fibers). Lifshitz and

Rotem used their model to show that delayed failure can occur even when the fiber strengths

are not time-dependent. Since most of the literature dealt with stress-rupture of glass-

reinforced materials, Lifshitz46 conducted some preliminary tests to show that the stress-

18

rupture phenomenon occurred in carbon-reinforced epoxies even though carbon fibers are

widely believed to be free of any creep response.

Christensen47 has developed a method for stress rupture lifetime of composite materials

based on crack growth in a homogeneous viscoelastic material. The theory did not

differentiate between fibers and matrix. The inputs to the method were the viscoelastic creep

function for the material, the surface energy of the crack, a characteristic failure dimension of

the material, the stress level, and a stress distribution parameter. The model was cast into a

statistical framework to consider data with large scatter in lifetimes. The results were

compared to stress rupture lifetimes of arimid composites.

Glaser et al.48 attempted to develop relations between the static strength and lifetime

distributions for composite materials. The composite was modeled as an equivalent

homogeneous medium. A pre-existing state of flaws is assumed in the material, which

yielded a Weibull distribution for static strength of the composite material. A theory of

kinetic crack growth was used to quantify the growth of these flows and in turn derive the

lifetime statistics for the material as a function of applied stress level. Comparisons were

made with experimental data.

Ibnabdeljalil and Phoenix49 have addressed the statistical aspects of creep rupture in

composite systems consisting of brittle fibers and brittle matrices. Delayed failure was

assumed to be a result of time-dependent strength of fibers. A primary objective of the paper

was to establish if composite lifetime mimics fiber statistical behavior to any extent. It was

shown that a power law scaling with respect to stress level was obtained for composite

lifetime with an exponent that differed from that of the fiber strength degradation. They also

showed that asymptotically the composite lifetime followed a log-normal distribution, in

contrast to composite strength that asymptotically followed a normal distribution. The size

effect of composite lifetime to volume of composite material was also addressed.

Iyengar and Curtin50 studied stress-rupture in fiber-reinforced metal and ceramic matrix

composites that occurred due to strength degradation of the fibers. Strength degradation in

19

the fibers was derived by assuming slow crack growth. Stress concentrations due to failed

fibers were ignored and the load of broken fibers is redistributed equally to surviving

neighbors. With these assumptions, the failure process was modeled both analytically and by

using a numerical simulation. Analytically, an approximate relation among applied stress,

time to failure, fiber Weibull modulus and slow crack exponent was derived. Good

agreement between the numerical and analytical technique was shown. The remaining

strength prior to failure was also studied, with the numerical technique predicting a more

sudden-death failure than the analytical method. In a later work, Iyengar and Curtin51 studied

time-dependent failure of ceramic and metal matrix composites resulting from matrix and

interface shear creep. Analytical representations of the time-dependent interfacial shear

relaxation were incorporated into a simulation model that tracked the evolution of fiber

damage and interfacial slip leading to composite failure. The failure time was obtained as a

function of strength and creep parameters of fibers and matrix. An explicit dependence of

failure time on specimen length was also obtained. Halverson and Curtin52 observed the

quasi-static and stress-rupture failure characteristics of Nextel 610 reinforced alumina-yttria

composite at elevated temperatures. They observed that a level of matrix cracking existed in

the virgin material, and it did not change significantly during stress-rupture. Fiber pushout

testing showed that the fiber/matrix interfacial frictional stress did not change significantly

during stress-rupture. This led them to deduce that fiber strength degradation was the

controlling mechanism for stress-rupture. The stress-rupture lifetime was modeled by

accounting for fiber strength degradation kinetics along with pre-existing matrix cracking.

Comparisons with experimental data revealed that although the creep deformation and trends

in rupture lifetime were accurately modeled, predicted lifetimes were less than experimental

values.

Models have also been developed to predict stress-rupture lifetimes of model composite

systems.9,10 Making comparisons to rupture lifetimes of simple systems provides valuable

insight into the merits and demerits of the stress-rupture modeling approach, before

proceeding to the more difficult task of looking at real composite systems.

20

In this work, we will develop a general three-dimensional Monte Carlo simulation for stress

rupture modeling based on shear-lag load sharing. The fibers are assumed to be free of any

creep deformation and the matrix is modeled as linearly viscoelastic. Rupture lifetimes are

computed by two different forms of load sharing: NNLS and a time-dependent version of

Hedgepeth and Van Dyke’s technique.13 Very broad distributions in rupture lifetime are

predicted. The reasons for large variability in computed lifetime are addressed. Encouraging

comparisons are made between predictions and measured rupture lifetime of a carbon fiber

polymer matrix system. Since the matrix properties are measurable as a function of time and

temperature, the rupture simulation predicts lifetimes at a desired stress level and

temperature. In this manner the model can be used to investigate the role of temperature in

accelerated testing of stress rupture lifetime.

21

3 MODEL COMPOSITE MEASUREMENTS

Abstract: When one or more fiber fractures occur in a composite material there is a micromechanical fiber stress redistribution in the vicinity of these fracture locations. A detailed understanding of this stress redistribution is essential when formulating models for tensile strength and stress-rupture lifetime of unidirectional composites loaded in the fiber direction. In this chapter we present a technique for in-situ measurement of strain concentrations due to single and multiple fiber fractures in three-dimensional composites. The method involves fabricating macromodel composites with glass rod fibers that are large enough that strain gages can be mounted directly onto the surface of the fiber. The technique is shown to be effective at obtaining both quasi-static and time-dependent measurements of strain concentrations. The results obtained here will be used in Chapter 4 to support and validate models of load redistribution in unidirectional fiber composites.

3.1 INTRODUCTION

In this work we expand on the expertise developed by Carman, et. al12 to fabricate and test

model composites with three-dimensional fiber packing and a local fiber volume fraction of

40%. In these macromodel composites the “fibers” are large enough that in-situ

measurements of strains can be made on the fiber surface using small strain gages. When

applying this approach to measure load redistribution the lateral orientation of the strain gage

in relation to the broken fiber is important. Hence, in this work a mapping of the three-

dimensional strain field on the surface of the fibers is obtained. This provides a unique

insight into the complex three-dimensional strain field experienced by fibers in a composite

material. The author believes that the technique presented here has two advantages over

model composite studies performed using polarized light microscopy and Raman

spectroscopy.6,7,8 Polarized light microscopy and Raman spectroscopy present problems

when applied to three-dimensional fiber packing geometries where the presence of fibers in

the line of sight and directly behind the area of interest distorts the obtained measurements.

A second drawback of polarized microscopy and Raman spectroscopy is that no information

is available on the three-dimensional variation of the strain/stress field across a fiber.

This chapter is organized as follows. In Section 3.2 the materials and construction of the

model composites is discussed. Three different fiber geometries were studied involving

22

seven, nineteen and thirty-seven fibers in hexagonal packing. In Section 3.3 we present the

results obtained from the seven and nineteen fiber model composites that have only one

break in the central fiber. Section 3.4 discusses the measurements made on model

composites with two and three coplanar fiber breaks in thirty-seven fiber model composites.

In Section 3.5 the time-dependent measurements obtained by loading the model composites

in stress relaxation are presented. The challenges associated with the time-dependent

measurements on the model composites are also outlined in this section. Finally, the results

are summarized together with a brief discussion on how they may be applied to develop and

verify models for load redistribution.

3.2 MATERIALS AND FABRICATION

Figure 3-1 shows the geometry of a typical macromodel composite containing seven fibers.

As discussed in the following sections model composites with up to thirty-seven fibers can be

fabricated. All the model composites are approximately 20 cm long and 3.81 cm in diameter.

The matrix material is a transparent photoelastic epoxy, PLM-9, supplied by Measurements

Group, Inc. The structural members in the model composites are borosilicate glass rods with

radius rf = 1.5 mm, which will henceforth be designated as fibers. For all the model

composites the distance between the centers of adjacent glass rods is 4.52 mm, which results

in a local fiber volume fraction, Vf, of approximately 40% in the central region.

23

The mechanical properties of the PLM-9 epoxy and the borosilicate glass rods are

summarized in Table 3-1. As discussed by Carman, et. al.12 the ratio of fiber to matrix

stiffness is representative of typical E-glass/epoxy material systems. The objective of the

model composite work is to measure the local fiber strain concentrations near a fiber break

location. In order to ensure that loading the model composite causes failure of certain glass

fibers at predetermined locations selected fibers are scored with a glass-scoring knife to

artificially introduce a weakness at that location. Local fiber strains are measured by

mounting strain gages directly onto the glass fibers before curing the epoxy matrix around

the fibers. The gages are mounted at various locations near the break on the broken or the

neighboring fibers, and are oriented to measure the axial strains in the fiber. These

embedded gages, designated SK-05-100GD-45C, are also supplied by Measurements Group,

Inc. They have grid dimensions of 2.54 mm × 2.03 mm. The high gage resistance of 4500 Ω

and a low bridge voltage of 0.5 V for the embedded gages are selected to minimize internal

Hexagonal packing of glass rods

Transparent epoxy (matrix)

Break in central glass rod

Strain gage with lead wires

Borosilicate glass rods;dia = 3 mm (fibers)

R19 mm

203 mm

LOADING DIRECTION

LOADING DIRECTION

x

Hexagonal packing of glass rods

Transparent epoxy (matrix)

Break in central glass rod

Strain gage with lead wires

Borosilicate glass rods;dia = 3 mm (fibers)

R19 mm

203 mm

LOADING DIRECTION

LOADING DIRECTION

Hexagonal packing of glass rods

Transparent epoxy (matrix)

Break in central glass rod

Strain gage with lead wires

Borosilicate glass rods;dia = 3 mm (fibers)

R19 mm

203 mm

LOADING DIRECTION

LOADING DIRECTION

x

Figure 3-1. Schematic of model composite that is representative of a unidirectional composite material

24

heating of the gages during testing, and to therefore minimize gage drift. The global model

composite strain is measured by mounting four 350 Ω gages equidistant on the outside

cylindrical surface of the model composite oriented so that they measure axial strain. An

excellent measurement of the global axial strain is obtained by averaging the strains from the

four strain gages to factor out any slight bending strains due to misalignment in the loading

fixtures. These external gages, designated CEA-06-125UW-350, were also supplied by

Measurements Group, Inc.

Table 3-1. Constituent material properties for model composite12 Material Young’s Modulus, GPa Poisson’s Ratio Thermal Expansion, /°C

7740 glass 62.7 0.2 3.3×10-6 PLM-9 epoxy 3.3 0.36 70.0×10-6

The model composite is cast inside an aluminum mold. The mold is coated with Frekote

4368 release agent, supplied by Loctite Corporation, to facilitate easy extraction of the

casting after curing. The manufacturer’s recommended mixing and curing instructions,

omitting the second-stage postcure, are followed for the PLM-9 epoxy. The motivation for

omitting the second-stage postcure is discussed in Section 3.2.1. All the fibers, including the

scored fibers and the fibers with gages and attached lead wires, are properly positioned in the

mold. The liquid resin and hardener mixture is poured into the mold around the fibers. The

epoxy mixture cures overnight at Tc = 42°C.

3.2.1 Experimental Technique

A tensile load is applied to the model composites using a Material Testing Systems (MTS)

servo-hydraulic load frame. The model composite extension is gradually increased until the

scored fibers fracture with an audible acoustic emission. This initial loading is done in

displacement control to cause the predetermined fiber fractures to occur in a stable fashion.

Fiber fracture is accompanied by a stable matrix crack that propagates a small distance

radially outward from the broken edge of the fiber before it arrests. The initial matrix crack

size cannot be controlled, but once the initial crack is formed further loading of the model

composite in displacement control may increase the matrix crack size in a stable fashion.

25

Hence, it is possible in certain cases to obtain different strain concentrations for different

matrix crack sizes.

Once the scored fibers are broken the model composite is used to make strain concentration

measurements loading it in displacement control to obtain post-fracture strain measurements.

Figure 3-2 shows a representative post-fracture strain measurement in a 7-fiber model

composite from a strain gage mounted near the break location on an unbroken neighboring

fiber. The strain is plotted with respect to the axial load on the model composite. It is

apparent that there are two linear portions to the curve with a distinct slope change at a load

level of 4626 N. The change in slope is explained as follows. After curing, the fibers and

matrix are in a stress free state at the cure temperature, Tc. On cooling the model composite

from Tc, to room temperature T, an axial strain εc, is produced in the model composite due to

the mismatch in thermal expansion coefficients. εc is given by

( ) ( )[ ]( ) m

glff

glf

mmglfff

glfc

c EVEV

EVEVTT

−+−+−

=1

1 ααε (3-1)

where f and m are the fiber and matrix thermal expansion coefficients, respectively, Ef and

Em are the fiber and matrix modulus, respectively, and glfV is the global fiber volume fraction

of the model composite considering the volume of all the fibers with respect to the total

volume of the model composite. The fiber and matrix expansion coefficients and moduli are

given in Table 3-1, and for the 7-fiber model composite %3.4=glfV . The thermally induced

strain εc is accompanied by an axial stress

( )[ ]cfcff TTE −−= αεσ (3-2)

in the fibers of an unloaded model composite. Similarly, there is an axial stress given by

( )[ ]cmcmm TTE −−= αεσ (3-3)

in the matrix of an unloaded model composite. Since m >> f, the axial fiber stress f is

tensile, while the axial matrix stress m, is compressive. During post-fracture loading of the

model composite there is no crack opening displacement in the broken fibers (and break in

the fiber is not visible) until the applied load is large enough to set up a tensile stress in all

the fibers. For the model composite shown in Figure 3-2, as the load is increased past 4626

26

N the stresses in the fibers change from compression to tension and the fiber fracture

becomes visible. The threshold strain εT, required to eliminate the compressive stress in the

fibers is calculated from

0=+ Tff E εσ (3-4)

The total load on the model composite, PT, at the threshold point is carried by the matrix. PT

is given by

mTmmT AEP εσ += (3-5)

where Am is the cross-sectional area of the matrix in the model composite. For the 7-fiber

model composite PT is calculated to be 4561 N, and is in very good agreement with the

experimental value. Naturally, it is only after the fibers are in tension that the strain

concentrations become apparent, and hence, the change in slope of the load-strain curves at

this point. The ratio of the initial slope of the load-strain response before fiber crack opening

to the slope after crack opening gives the strain concentration at the location of the strain

gage. In order to obtain strain concentrations for each matrix crack size it is necessary to

unload after creating the larger crack and then load again to make a post-fracture strain

measurement for the larger matrix crack size.

27

Carman, et. al.12 have reported that performing the second-stage postcure results in excessive

compressive stresses in the fibers, as a result of which it is not possible to break the central

scored fiber without failure of the entire model composite. Not performing the second-stage

postcure appears to result in a sufficiently cured matrix, and reduces these compressive fiber

stresses.

3.3 MODEL COMPOSITES WITH SINGLE FIBER FRACTURE

Seven and nineteen fiber model composites with a single central fractured central fiber are

fabricated. The results from the seven fiber model composites are discussed in Section 3.3.1,

and those from the nineteen fiber model composites in Section 3.3.2.

3.3.1 Seven Fiber Model Composites

The details of the quasi-static strain concentration measurements on seven fiber model

composites are outlined in Table 3-2. While performing the tests three distinct matrix crack

sizes, a, are observed which are referred to as ‘small,’ ‘medium,’ and ‘large.’ a, represents

the difference between the radius of the matrix crack and the fiber, which is the radial

0

2000

4000

6000

8000

10000

12000

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

Load

, P (

N)

PT = 4626 N

870)1064.6( 6 +×= εP

19)1014.8( 6 +×= εP

Tε

ε Strain,

Strain Conc. =1.23

0

2000

4000

6000

8000

10000

12000

0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 0.0016

Load

, P (

N)

PT = 4626 N

870)1064.6( 6 +×= εP

19)1014.8( 6 +×= εP

Tε

ε Strain,

Strain Conc. =1.23

Figure 3-2. Representative strain measurement from a gage mounted on an unbroken fiber (7-fiber model composite)

28

distance the crack propagates into the surrounding matrix from the surface of the broken

fiber. The matrix crack size a, is quantified in Table 3-2. The matrix crack is visible on the

fracture surface of some of the model composites. A hand held magnifier with a scale

attachment or an optical microscope is used to measure the matrix crack size from the

fracture surface of the model composite. Gages are mounted at different positions on the

broken central fiber or any one of the unbroken neighbors. The orientations of the gages in

relation to the broken fiber are shown in Table 3-2. The broken fiber is shaded. The axial

position of the strain gage is specified by the axial coordinate system, x, with an origin

located in the plane of the break as shown in Figure 3-1. The position x-coordinate locates

the center of the gage length from the plane of the break, and consequently may have a

positive or negative value depending on whether the gage center is above or below the break

plane. It should be pointed out that the stress or strain field in the model composite is

symmetric with respect to the break plane, and also shows cyclic symmetry about the central

broken fiber with a period of 60°.

Table 3-2. Quasi-static strain concentration measurements on seven fiber model composites

Location of Embedded Strain Gage Model #

Crack Size, a, rf Orientation Gage # Position, x, rf

Strain Concentration

1 0.33 1.22 1

0.47 (medium)

2 12 1

2 0.00 1.01

1 -0.21 1.19 2 (medium)

2 12 1

2 -0.28 1.05

1 1.83 0.53 0.16 (small)

1 21 2

2 2.37 1.09

1 1.83 0.34 0.7 (medium)

1 21 2

2 2.37 gage error

1 1.83 0.32

3

0.99 (large) 1 21 2

2 2.37 1.06

29

Location of Embedded Strain Gage Model #

Crack Size, a, rf Orientation Gage # Position, x, rf

Strain Concentration

1 3.07 0.52 (medium)

1 21 2

2 3.07 1.03

1 3.07 0.49 4

(large) 1 21 2

2 3.07 1.04

It is interesting to note that a strain gage mounted on a neighboring fiber at x=0 and facing

the outside of the model is not significantly affected by the presence of the break (see Gage 2

of Models 1 & 2). However, a strain concentration of approximately 1.2 exists at x = 0 on a

neighboring fiber facing the broken fiber (see Gage 1 of Models 1 & 2). This implies that a

gradient of strain exists across the unbroken fibers, and only the regions that face the broken

fiber experience the effects of a fiber fracture in the vicinity. In Chapter 4, finite element

analysis of the model composite domain is performed to reinforce this assertion. On the

broken fiber the axial strains gradually increase to the far-field axial strain for increasing x.

At x = 3.07 rf the axial strain on the broken fiber has increased to approximately half the far-

field axial strain. On the other hand, the axial strains in the unbroken fibers decrease to the

far-field value for increasing x. At x = 2.4 rf the axial strains on the unbroken fibers have

decreased to less than 1.09. This implies that the unbroken neighboring fibers recover the

far-field strain level over a much shorter axial distance than the broken fiber. In Section

4.2.2, finite element analysis will be used to obtain a better picture of the complex strains

experienced by neighboring fibers in the vicinity of a fiber fracture. It will be shown that the

neighboring fibers undergo local bending, and this behavior causes the axial fiber strains to

vary depending upon the lateral orientation on the surface of the fiber. Finally, the change in

measured stress concentration on the broken fiber is much greater when the matrix crack size

changes from ‘small’ to ‘medium’ than when the matrix crack size changes from ‘medium’

to ‘large.’

30

Table 3-3. Quasi-static strain concentration measurements on nineteen-fiber model composites

Location of Embedded Strain Gage Model #

Orientation Gage # Position, x, rf Strain Concentration

1 2.87 0.57 5 1 21 2

2 0.90 1.14

1 1.60 0.36 6 1 21 2

2 -0.13 1.21

1 4.13 0.65

2 2.07 1.11 7 1

2

3 1

2

3 1

2

3

3 1.53 1.10

3.3.2 Nineteen Fiber Model Composites

Table 3-3 describes quasi-static strain concentration results in nineteen fiber model

composites. For all the nineteen fiber model composite tests the matrix crack was estimated

to be nominally the same size as the medium sized crack of the seven fiber model

composites. At a distance of 4.13 rf from the plane of the fiber fracture the strain on the

broken fiber has increased to 0.65 of the far-field strain. The strain concentration seen by the

neighboring fiber in the plane of the fiber fracture is approximately 1.21 for a strain gage

facing the broken fiber. This is approximately the same as the in-plane strain concentration

measured in the 7-fiber model composite. Hence, addition of another ring of fibers around

the inner core of seven fibers does not influence the state of stress in the inner seven fibers,

which implies that the influence of an isolated fiber fracture is felt only by the broken fiber

and its nearest neighbors. At x=2.07 rf the strain concentration on the neighboring fiber

facing the broken fiber has decreased to approximately 1.10.

31

3.4 MODEL COMPOSITES WITH MULTIPLE FIBER FRACTURES

Green’s functions or influence-function techniques13,53 are used to model the fiber stress state

that result from multiple fiber fractures in unidirectional composite materials. Green’s

functions or influence-functions can be calculated form the stress state resulting from a single

fractured fiber in a unidirectional material. It is then assumed that the effect of multiple fiber

breaks can be calculated by superposition of the far-field stresses with the effect of each

individual fiber fracture. In order to verify this assumption strain concentration

measurements are made on model composites with more than one broken fiber. In Chapter 4

the influence-function technique is used to calculate the strain concentrations resulting from

multiple fiber fractures and a comparison is made to the measurements reported here. The

model composites fabricated for this study had thirty-seven hexagonally packed fibers. A

sufficient number of intact fibers are required in order to ensure that the multiple fiber breaks

occur in a stable fashion.

The results obtained by fabricating a model composite with two adjacent, coplanar breaks are

shown in Figure 3-3. The dashed lines are drawn to signify the lateral orientation of the

gages on the surface of the fibers. All the gages are mounted such that they lie nominally in

the same plane as the fiber fractures. It is interesting to note that the strain concentrations

measured by gages 1 and 2 are effectively the same. As would be expected, strain gage 3

sees the effect of both fiber fractures and consequently measures the highest strain

concentration of 1.48.

32

The strain concentration measurements obtained by testing the model composite with three

coplanar, adjacent fiber breaks is shown in Figure 3-4. Gage 1 is mounted such that it lies

nominally in the plane of the fiber fractures, but facing away from the break cluster as shown

in Figure 3-4. This gage does not show any evidence of strain concentration due to the break

1

2 3

1

2 3

1.48-0.073

1.29+0.102

1.28+0.101

Strain Conc.

x, rfGage

1.48-0.073

1.29+0.102

1.28+0.101

Strain Conc.

x, rfGage

Figure 3-3. Model composite with two adjacent, coplanar fiber fractures (Model 8)

1

2

3 4

1

2

3 4

0.401.703

0.676.674

0.383.532

1.000.001

Strain

Conc.x, rfGage

0.401.703

0.676.674

0.383.532

1.000.001

Strain

Conc.x, rfGage

Figure 3-4. Model composite with three adjacent, coplanar fiber fractures (Model 9)

33

cluster, which indicates a very sharp gradient in strain concentration across the unbroken

fibers near a cluster of fiber fractures. Gages 2 through 4 are mounted out of the plane of the

fiber fractures on the broken fibers. A slightly lower strain concentration is measured by

Gage 2, even though it is further away from the fracture plane than Gage 3. This may be

due to the fact that Gage 2 faces the center of the fractured cluster of fibers. Gage 3 on the

other hand faces the intact surrounding fibers. The gage furthest away from the fiber

fractures is Gage 4, and it sees a strain value that is two-thirds of the far-field strain.

3.5 TIME-DEPENDENT MODEL COMPOSITE MEASUREMENTS

While conducting long-term, time-dependent tests it is extremely important to ensure that the

thermal drift in strain gages output is as small as possible by selecting an optimum excitation

level for the Wheatstone bridge circuit. In all the work presented here a single active gage is

used in a quarter bridge arrangement. The Measurements Group website

(http://www.vishay.com/brands/measurements_group/) has recommendations for the

optimum excitation voltage based on the type of test conducted, heat sink conditions, gage

resistance, and accuracy requirement. For the external gages an excitation voltage of 0.7 V

produced minimal drift of output over 6 hours. However, for the embedded gages an

excitation voltage as low as 0.5 V was necessary to ensure minimal drift over 6 hours.

Once the scored fibers in the model composite are fractured, time-dependent strain

measurements can be made in either creep or stress relaxation loading. Creep tests can be

performed readily by ramping up the load on the model composite to a predetermined value,

and holding the load constant thereafter. In real polymer matrix composite materials with

high fiber volume fractions, the fibers would carry most of total load on the material because

their stiffness significantly higher than the matrix stiffness. However, in the model

composite system the global fiber volume fraction is much lower. For the 7-fiber model

composite the global fiber volume fraction is only 4%, while for the 37-fiber model

composite the global fiber volume fraction is 23%. Hence, the load carried by the matrix in

the model composite is significant. Under creep loading of the model composite the matrix

carries progressively less load due to viscoelasticity in the polymer. This load is transferred

into the fibers. Hence, in the model composites under creep loading the far-field fiber stress

34

changes appreciably with time. A schematic of the fiber stress in the broken and neighboring

fibers under creep loading of the model composite are shown in Figure 3-5. It is apparent

from Figure 3-5 that although there is an increase in the perturbed length of fiber near a

fracture20, the time dependent measurements under creep loading of the model composites

will be dominated by the global viscoelastic behavior of the polymer.

Based on the preceding discussion, it may be advantageous to conduct time-dependent tests

on model composites under stress relaxation conditions. In order to carry this out, the

servohydraulic machine is set up to use the global axial strain measurement from the external

strain gages as feedback in a control loop. The axial strain in the model composite is ramped

up to a predetermined value, after which it is held constant to achieve the stress relaxation

loading condition. Since the fibers are linearly elastic, the constant state of axial strain under

stress relaxation implies that the fibers see a constant far-field applied stress and the strain

gages measure the time-dependent strain redistribution due to a fiber fracture. Hence, stress

relaxation of the model composites is used to obtain all the time-dependent strain

measurements presented here. This loading condition closely mimics the creep behavior of

high fiber volume fraction real composites subjected to creep loading. Also, it is desirable to

conduct the time dependent tests on model composites with higher global fiber volume

fractions glfV , to maximize the load carried by the fibers. Hence, the time-dependent

measurements described in this chapter were performed on the nineteen and thirty-seven

fiber model composites. Figure 3-6 is a representative plot of the strains obtained during

Increasing Time Increasing Time

0.0

0.4

0.8

1.2

1.6

0 10 20 30 40 50

Distance along fiber (r f )

No

rmal

ized

Str

ess

0.0

0.4

0.8

1.2

1.6

0 10 20 30 40 50

Distance along fiber (r f )N

orm

aliz

ed S

tres

s

(a) (b)

Increasing Time Increasing Time

0.0

0.4

0.8

1.2

1.6

0 10 20 30 40 50

Distance along fiber (r f )

No

rmal

ized

Str

ess

0.0

0.4

0.8

1.2

1.6

0 10 20 30 40 50

Distance along fiber (r f )N

orm

aliz

ed S

tres

s

Increasing Time Increasing Time

0.0

0.4

0.8

1.2

1.6

0 10 20 30 40 50

Distance along fiber (r f )

No

rmal

ized

Str

ess

0.0

0.4

0.8

1.2

1.6

0 10 20 30 40 50

Distance along fiber (r f )N

orm

aliz

ed S

tres

s

(a) (b)

Figure 3-5. Schematic of fiber stresses under creep loading of model composite. (a) Broken fiber (b) Neighboring fiber

35

time-dependent measurements. The global strain is held constant for six hours after ramp up.

The strains of the embedded gages are monitored as a function of time. The time-dependent

strain concentration is defined as

Tff

Ttt

εεεε

−−=Ε )(

)( (3-6)

where ε is the strain measured by the strain gage, εT is the residual thermal strain obtained

from the kink in the load versus strain data as shown in Figure 3-6, and εff is the far-field

fiber strain which is equivalent to the global axial strain in the model composite. The time-

dependent tests are conducted on the nineteen and thirty-seven fiber model composites (

Model 5 through Model 9) after they were used for the quasi-static measurements.

The time-dependent strain concentration for tests on the nineteen-fiber model composites are

shown in Figure 3-7 - Figure 3-9. The curves are labeled according to the gage numbering

shown in Table 3-3. There is a marginal change in strain concentration over six hours. The

strain concentration shows a decreasing trend for gages mounted on the broken fibers and an

increasing trend for those on neighboring fibers. Only gage 2 of model 5 that is mounted on

a neighboring fiber does not follow this trend. The strain concentration for this gage actually

decreases slightly with time.

The thirty-seven fiber model composites have multiple adjacent, coplanar fractures. It was

hoped that the presence of multiple fractures would accentuate the change in strain

concentration with time. This appears to be the case, at least for certain gage locations, as

shown in Figure 3-10 and Figure 3-11. The gages that are mounted on neighboring fibers in

the plane of the fracture for model 8 show a fairly rapid increase in strain concentration over

the six-hour test period. On the other hand, gages that are mounted on the broken fibers do

not show an appreciable change in strain concentration with time as is apparent in Figure

3-11. The sole gage that was mounted on a neighboring fiber of model 9 does show a fairly

rapidly increasing trend even though it faces away from the fracture cluster of fibers (see

Figure 3-4).

36

0

2000

4000

6000

8000

10000

0.0000 0.0002 0.0004 0.0006 0.0008 0.0010Strain

Lo

ad (

N)

Global

BrokenNeighbor

Residual Thermal Strain (T)

Far-field fiber strain (ff )

Figure 3-6. Representative load versus strain curve obtained during stress relaxation tests

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

10 100 1000 10000 100000

Time (sec)

Str

ain

Co

nce

ntr

atio

n (

)

Gage #2

Gage #1

Figure 3-7. Time-dependent strain concentration (Model 5)

37

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

10 100 1000 10000 100000

Time (sec)

Str

ain

Co

nce

ntr

atio

n (

)

Gage #2

Gage #1

Figure 3-8. Time-dependent strain concentration (Model 6)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

10 100 1000 10000 100000

Time (sec)

Str

ain

Co

nce

ntr

atio

n

( )

Gage #1

Gage #3

Gage #2

Figure 3-9. Time-dependent strain concentration (Model 7)

38

1.2

1.3

1.4

1.5

1.6

10 100 1000 10000 100000Time (sec)

Str

ain

Co

nce

ntr

atio

n (

)

Gage #1

Gage #2

Gage #3

Figure 3-10. Time-dependent strain concentration (Model 8)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

10 100 1000 10000 100000

Time (sec)

Str

ain

Co

nce

ntr

atio

n

Gage #1

Gage #2

Gage #3

Gage #4

Figure 3-11. Time-dependent strain concentration (Model 9)

39

3.6 SUMMARY AND CONCLUSIONS

In this chapter, macromodel composites are shown to be an effective method to probe the

complex strain field that exists near fiber fractures in unidirectional composite materials.

Two reasons are cited for why the results obtained here are an improvement over competing

techniques such as Raman spectroscopy and polarized light microscopy. An unhindered line

of sight is not required for the approach presented here. Hence, using the macromodel

composites three-dimensional fiber-packing arrangements with relatively high local fiber

volume fractions can be studied. Also, the change in stress/strain with respect to lateral

position on the fiber surface is successfully investigated in this work. It should, however, be

pointed out that polarized light microscopy and Raman spectroscopy yield a continuous

stress profile with respect to position along a fiber. The method described here yields

discrete measurements at the locations strain gages are mounted on the fibers. These

measurements together with the load redistribution models discussed in Chapter 4 do yield

sufficient information about the stress redistribution near fiber fractures. To the author’s

knowledge, this work presents the first time-dependent strain concentration measurements in

model composites systems with single and multiple fiber fractures, three-dimensional fiber

packing, and high local fiber volume fractions.

Tests conducted on the seven fiber model composites show that a gradient of strain exists

across the unbroken fibers. As expected the highest strain concentration occurs facing the

broken fiber. The surface strains on the six unbroken fibers attain their far-field value over a

much shorter distance than the broken fibers. In Section 4.2.2 finite element analysis will be

used to show that the unbroken fibers see a complex state of strain with local bending and the

strain gage measurements capture this behavior. Addition of a hexagonally packed ring of

twelve fibers around the inner core of seven fibers did not change the peak stress

concentration on the nearest unbroken neighbor. This suggests that a ‘nearest neighbor load-

redistribution model’ may be adequate for systems with a high fiber volume fraction and

ratio of fiber to matrix stiffness. In order to experimentally validate the feasibility of using

influence-function superposition techniques for multiple fiber breaks, model composites with

two multiple break patterns are fabricated. The first one has two adjacent, coplanar breaks.

For this case the peak strain concentration on three of the nearest fibers is measured. As

40

expected, higher strain concentrations are obtained for two adjacent breaks. The second

multiple break pattern consists of three coplanar fiber breaks. For this model, measurements

are made on an unbroken fiber facing away from the break cluster, and on the three broken

fibers. The quasi-static measurements reported in this chapter will be used to support and

validate models for load redistribution to be developed in Chapter 4. Comparisons will be

made between finite element, shear-lag and the experimental measurements reported here.

The multiple break results will be used to verify whether influence-function superposition

techniques are applicable to determine the stresses/strains in unidirectional composites with

multiple fiber breaks. Eventually, the quasi-static load redistribution information feeds into

models for the strength of unidirectional composite materials.41

A series of time-dependent measurements are performed in stress relaxation. The author

believes that stress-relaxation is better than creep loading to obtain time-dependent strain

measurements in model composites with low global fiber volume fractions. For the nineteen-

fiber model composites with a single fiber fracture small changes in strain were observed

with time. For all but one gage the trends were as expected: increasing strain concentration

on the unbroken neighboring fiber and decreasing strain concentration on the broken fiber.

The change of strain concentration with time was appreciably larger, especially for the

unbroken neighbors, in the model composites with multiple fiber breaks. The increase in

strain concentration and hence stress in an unbroken neighbor with time is what eventually

leads to failure of unidirectional composites under stress rupture conditions. This occurs as a

result of viscoelastic deformation in the matrix. Hence, the time-dependent experimental

results presented here are a good confirmation of one of the underlying processes that lead to

failure of unidirectional composites under stress rupture loading.

41

4 QUASI-STATIC STRENGTH MODELING

Abstract: A natural approach for strength modeling of composite materials is to base predictions on constituent properties. One of the key aspects of this undertaking is micromechanical stress redistribution near a fiber fracture. In Chapter 3, strain concentration measurements made on macromodel composites with one or more broken fibers were presented. These measurements form the basis for load sharing models. It is shown that nearest neighbor load sharing (NNLS) describes the stress state in the model composites very well. The NNLS is incorporated into Monte Carlo simulations for tensile strength of unidirectional composite materials. The simulation methodology accounts for sources of material variability such as fiber strength distributions, distributions of fiber volume fraction, random fiber placement, and initial imperfections in the form of initial fiber fractures. There is very good comparison between the predicted and experimental strength distributions of Grafil/PPS unidirectional composites.

4.1 INTRODUCTION

The shear-lag14 approach is one of the most widely used methods to model fiber/matrix

interaction. When a fiber in a tension loaded composite material fails there is local fiber

stress redistribution termed load sharing. Hedgepeth54 and Hedgepeth and Van Dyke13

developed a load sharing technique to calculate fiber stresses due to an arbitrary number of

fiber fractures. Their work introduced the valuable concept of influence-functions. Nairn23

presented a very detailed assessment of the accuracy of using shear-lag assumptions to model

stress transfer in unidirectional composite materials. In this chapter we use a three-

dimensional finite element method to develop influence-functions. Comparisons are made to

the model composite measurements and the validity of this approach is experimentally

verified for single and multiple fiber fractures.

Batdorf32,33 developed a probabilistic framework to track the growth of fiber fracture clusters

in a unidirectional composite material under the application of a tensile load. This approach

yields a deterministic composite strength. More recently, Monte Carlo simulations36,37,38,39

have been gaining acceptance as a viable method for composite strength modeling since they

yield distributions for composite strength and hence reliability at a given load level. The

experimentally verified load sharing introduced in this work is incorporated into Monte Carlo

simulations for composite strength. Several important sources of material variability present

42

in ‘real’ composite systems are introduced into the simulation. Comparisons are made

between the results obtained and statistical strength of a carbon fiber/polymer matrix system.

The organization of this chapter is as follows. In Section 4.2 we present finite element

analysis results for a single break in a unidirectional composite material. The finite element

results are compared to strain concentration measurements made on seven and nineteen fiber

model composites with a single fiber fracture. It is shown that the state of micromechanical

stress for the 19-fiber model composite is very similar to the 7-fiber model composite. Based

on this observation the concept of Nearest Neighbor Load Sharing (NNLS) is introduced.

Section 4.3 discusses the calculation of force influence-functions from finite element stresses

due to a single fiber fracture. The approach for using the force influence-functions to obtain

fiber stresses resulting from an arbitrary combination of breaks is also presented. In Section

4.4 the finite element analysis results for a single fiber fracture are used to compute the strain

concentrations resulting from multiple fiber fractures. Comparisons are made to

measurements on model composites with multiple fiber fractures. The results presented in

this section provide further justification for using influence-function superposition principles

with NNLS to model multiple fiber breaks. In Section 4.5 the general finite element load

sharing developed in Section 4.3 is incorporated into Monte Carlo simulations for the quasi-

static strength distribution of a unidirectional carbon fiber/polymer matrix composite. It is

shown that the calculated Weibull strength distribution is much narrower than the

experimental strength distribution. In Section 4.6 certain aspects of material variability

present in ‘real’ material systems are modeled. The effect of random fiber placement,

distributed fiber volume fraction, and initial imperfections in the form of broken fibers on the

distribution of strength is studied. It is shown that a distributed fiber volume fraction results

in the greatest increase in variability of computed strengths. All the results presented in

Section 4.6 are obtained by applying a nearest neighbor shear-lag analysis. The validity of

nearest neighbor shear-lag assumption is evaluated by comparing stresses obtained by the

finite element and shear-lag analysis for regular hexagonal fiber packing. Section 4.7

presents the strength simulation with Hedgepeth and Van Dyke load sharing (HVDLS) where

the load of the broken fiber is distributed onto all the surrounding fibers. Results from

43

strength simulations based on HVDLS are compared to results obtained by applying NNLS.

Finally in Section 4.8, the major results and conclusions are summarized.

4.2 FEM OF SINGLE FIBER FRACTURE IN MODEL COMPOSITE

In order to further investigate the state of stress near a single fiber fracture finite element

analysis of the seven-fiber model composite domain is performed. The model composite

measurements of strain concentration due to a single fiber fracture are described in Chapter

3. The seven-fiber finite element results are compared to the measurements made on the

seven and nineteen fiber model composites. As expected, there is good agreement between

the seven-fiber model composite measurements and seven-fiber finite element results.

However, it is shown that there is excellent agreement with the seven fiber finite element

results and the nineteen fiber model composite measurements. This is an important

observation when developing models for load sharing in unidirectional composites.

Moreover, in Section 4.4 the seven fiber finite element results will be used in conjunction

with influence-function superposition to make very encouraging comparisons to the

measurements made on the 37-fiber model composites with multiple fiber breaks.

4.2.1 Seven-Fiber Finite Element Analysis

The finite element analysis is performed with ABAQUS/Standard, licensed from Hibbitt,

Karlsson & Sorensen, Inc. Figure 4-1 shows the finite element domain with appropriate

boundary conditions to exploit planes of symmetry in the model composite. The finite

element geometry is exactly the same as the seven-fiber model composite geometry. Only a

section of the model composite that subtends an angle of 30° at the central axis and has an

axial length from x = 0 to 5 cm is meshed. The choice of axial length for meshing is possible

because the model composite deformation is symmetric about the break plane and the gage

length of the model composite during testing is 10 cm (about 5 cm at each end is constrained

within fixtures for application of tensile load). Finite element results are generated for the

medium crack size, a = 1.1 mm. Linear elastic mechanical properties for the fibers and

matrix are used (Table 3-1). Perfect fiber/matrix adhesion is assumed. A very fine mesh is

used near the fiber fracture to capture the stress concentrations as accurately as possible. All

44

the finite element results presented in this and the following sections are obtained for a

domain similar to the one shown in Figure 4-1 i.e. with only the broken fiber and the nearest

neighbors modeled.

4.2.2 Seven-Fiber Model Composite Measurements

Comparison between the finite element results and seven fiber model composite

measurements are shown in Figure 4-2-Figure 4-4. The quantity plotted is the axial strain

concentration. The curve in Figure 4-2 represents the average finite element axial strain

concentration as a function of x-coordinate. In order to make a proper comparison with strain

concentration measurements the finite element axial strains on the fiber surface are suitably

averaged over the arc described by the gage width when it is mounted on a fiber. For the

locations at which strain concentration measurements are made the finite element axial

strains are averaged over the gage grid area and the average value for that gage are plotted

together with the actual measurements as line segments at the correct x-coordinate. The

length of these line segments corresponds to the strain gage grid length. These line segments

are designated FEA (Model #, Gage #) and EXP (Model #, Gage #) for finite element and

measured strain concentrations, respectively. The model composite and gage numbering

scheme is presented in Table 3-2. The measured strain concentrations are for the medium

matrix crack size. It is apparent from Figure 4-2 - Figure 4-4 that the finite element and

measured strain concentrations are in excellent agreement for each strain gage location. The

maximum difference between the finite element and measured strain concentration in Figure

4-2-Figure 4-4 is 5.1%, and it occurs for one of the broken fiber strain gage locations (Model

4, Gage 1).

Figure 4-2 shows the strain concentrations for the broken fiber. As expected, aside from the

small region near the break the strain concentration increases with increasing distance from

the fracture location. Figure 4-3 shows the strain concentration on the neighboring fiber.

The results presented in this figure are for gages mounted facing the broken fiber. The peak

strain concentration occurs slightly away from the break plane as reported by Nedele and

Wisnom.16,17 There is an entirely different strain concentration profile seen by gages

mounted facing away from the broken fiber, as shown in Figure 4-4. This implies that the

45

nearest unbroken fiber sees local bending near the fracture location. The excellent agreement

between the finite element and measured strain concentration also validates the assumption

of linearly elastic behavior and good fiber/matrix adhesion in the model composite.

4.2.3 Nineteen-Fiber Model Composite Measurements

Similar comparisons between the finite element and nineteen-fiber model composite strain

concentration measurements are made in Figure 4-5 and Figure 4-6 for the broken and

neighboring fiber, respectively. It is important to recall that the finite element stress analysis

considers only the nearest six unbroken fibers. From Figure 4-5 and Figure 4-6 it is apparent

that there is good agreement between the finite element results and the measurements made

on the nineteen-fiber model composites. The maximum difference between the finite

element and measured strain concentration for the broken fiber in Figure 4-5 is 17.3%

(Model 5, Gage 1) and for the neighboring fiber in Figure 4-6 is 4.4% (Model 7, Gage 2).

Hence, an additional ring of 12 unbroken fibers around the inner core of 7 fibers does not

significantly influence the stress state in the inner 7 fibers. This observation leads us to

introduce the concept of nearest neighbor load sharing (NNLS) wherein the load of the

broken fiber is redistributed only onto the nearest neighbors. Further justification for NNLS

Matrix crack front

° plane, u q

° plane, u q

x = 0 plane, break plane, ux = 0 outside matrix crack

Fiber fracture location

Neighboring fiber

x = 5 cm plane, ux prescribed

Broken fiber center, uy = uz =0

Y

Z X

Figure 4-1. Geometry and boundary conditions of finite element model

46

is given in Section 4.4, where it is shown that NNLS predicts the strain concentrations in

model composites with multiple fiber fractures to a good degree of accuracy. In the

following section the nearest neighbor finite element stresses are used to generate force-

influence functions that form the basis for a NNLS framework for unidirectional polymer

composites.

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12x -coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEA

FEA (M3; G1)

EXP (M3; G1)

FEA (M4; G1)

EXP (M4; G1)

EXP FEAM3; G1 0.34 0.35M4; G1 0.52 0.49

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12x -coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEA

FEA (M3; G1)

EXP (M3; G1)

FEA (M4; G1)

EXP (M4; G1)

EXP FEAM3; G1 0.34 0.35M4; G1 0.52 0.49

Figure 4-2. Comparison between FEM and seven-fiber model composite. Broken fiber. (M = Model; G = Gage)

47

0.95

1.00

1.05

1.10

1.15

1.20

1.25

-2 0 2 4 6 8 10x - coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEAFEA (M1; G1)EXP (M1; G1)FEA (M2; G1)EXP (M2; G1)FEA (M3; G2)EXP (M3; G2)FEA (M4; G2)EXP (M4; G2)

EXP FEAM1; G1 1.22 1.21M2; G1 1.19 1.22M3; G2 1.08 1.04M4; G2 1.03 1.02

0.95

1.00

1.05

1.10

1.15

1.20

1.25

-2 0 2 4 6 8 10x - coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEAFEA (M1; G1)EXP (M1; G1)FEA (M2; G1)EXP (M2; G1)FEA (M3; G2)EXP (M3; G2)FEA (M4; G2)EXP (M4; G2)

EXP FEAM1; G1 1.22 1.21M2; G1 1.19 1.22M3; G2 1.08 1.04M4; G2 1.03 1.02

Figure 4-3. Comparison between FEM and seven-fiber model composite. Neighboring fiber, facing broken fiber. (M = Model; G = Gage)

0.98

1.00

1.02

1.04

1.06

1.08

1.10

-2 0 2 4 6 8 10x -coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEA

FEA (M1; G2)

EXP (M1; G2)

FEA (M2; G2)

EXP (M2; G2)

EXP FEAM1; G2 1.01 1.02M2; G2 1.05 1.02

0.98

1.00

1.02

1.04

1.06

1.08

1.10

-2 0 2 4 6 8 10x -coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEA

FEA (M1; G2)

EXP (M1; G2)

FEA (M2; G2)

EXP (M2; G2)

EXP FEAM1; G2 1.01 1.02M2; G2 1.05 1.02

Figure 4-4. Comparison between FEM and seven-fiber model composite. Neighboring fiber, facing away from broken fiber. (M = Model; G = Gage)

48

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

x -coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEAFEA (M5; G1)EXP (M5; G1)FEA (M6; G1)EXP (M6; G1)FEA (M7; G1)EXP (M7; G1)

EXP FEAM5; G1 0.57 0.47M6; G1 0.36 0.32M7; G1 0.65 0.59

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8 10 12

x -coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEAFEA (M5; G1)EXP (M5; G1)FEA (M6; G1)EXP (M6; G1)FEA (M7; G1)EXP (M7; G1)

EXP FEAM5; G1 0.57 0.47M6; G1 0.36 0.32M7; G1 0.65 0.59

Figure 4-5. Comparison between FEM and nineteen-fiber model composite. Broken fiber. (M = Model; G = Gage)

49

4.3 FINITE ELEMENT-BASED NNLS

A finite element model similar to the one shown in Figure 4-1 is used to develop a NNLS for

tensile loaded unidirectional composite materials. Prior to discussing the FEM-based load

sharing it is necessary to introduce some general load-sharing concepts.

4.3.1 General Load-Sharing Concepts

Consider an array of N M hexagonally packed fibers of length X, as shown in Figure 4-7.

The x-coordinate system is oriented perpendicular to the cross-sectional plane of the fibers.

This is the volume of material that will be considered for the Monte Carlo simulations

presented in Sections 4.5, 4.6, and 4.7. An average far-field tensile axial stress ff, is applied

to the fibers. Consider L fiber fractures in fibers (n1,m1), (n2,m2)(nL,mL) at axial locations

x1, x2xL, respectively. Due to the far-field tensile stress ff, the broken fiber ends are

separated by a displacement 2u1, 2u22uL. The quantities u1, u2uL will be referred to as

break opening-displacements. The stress in fiber (n, m), at location x, is given by13

EXP FEAM5; G2 1.14 1.17M6; G2 1.21 1.22M7; G3 1.10 1.10M7; G2 1.11 1.06

0.95

1.00

1.05

1.10

1.15

1.20

1.25

-2 0 2 4 6 8 10x - coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEAFEA (M5; G2)EXP (M5; G2)FEA (M6; G2)EXP (M6; G2)FEA (M7; G3)EXP (M7; G3)FEA (M7; G2)EXP (M7; G2)

EXP FEAM5; G2 1.14 1.17M6; G2 1.21 1.22M7; G3 1.10 1.10M7; G2 1.11 1.06

0.95

1.00

1.05

1.10

1.15

1.20

1.25

-2 0 2 4 6 8 10x - coordinate (r f )

Str

ain

Co

nce

ntr

atio

n

FEAFEA (M5; G2)EXP (M5; G2)FEA (M6; G2)EXP (M6; G2)FEA (M7; G3)EXP (M7; G3)FEA (M7; G2)EXP (M7; G2)

Figure 4-6. Comparison between FEM and nineteen-fiber model composite. Neighboring fiber. (M = Model; G = Gage)

50

( ) ( )∑=

−− −+=L

iiimmnnffmn uxxqx

ii1

,, σσ (4-1)

Equation (4-1) is simply an expression of superposition of the far-field stress and the

perturbation due to each fiber break in the composite material. The perturbation due to each

fiber break is expressed as the product of the force influence-function qn,m(x), and the

opening-displacement at the fiber fracture location. qn,m(x) is the axial stress produced in

fiber (n, m), at location x, due to a unit opening-displacement of a fracture in fiber (0,0) at x =

0. If uniform hexagonal packing is assumed for the composite material, then only a single set

of force influence-functions qn,m(x), with n = 1-NN-1, m = 1-MM-1, and x = -XX,

needs to be calculated. This is because for uniform hexagonal packing, every fiber fracture

perturbs its surroundings in exactly the same manner. The calculation of force influence-

functions from the finite element fiber stresses is discussed in Section 4.3. Before Equation

(4-1) can be used to obtain fiber stresses it is necessary to calculate the break opening-

displacements uj, j=1L, by solving the system of equations

( ) ( ) LjuxxqxL

iiijmmnnffjmn ijijjj

1, 01

,, =−+== ∑=

−−σσ (4-2)

51

4.3.2 Force Influence-Functions from FEM

As explained earlier, the model composite measurements show that the load of a single

broken fiber in a composite material is redistributed preferentially onto the nearest neighbors,

and the presence of other intact fibers surrounding the inner core of seven fibers does not

significantly influence this stress redistribution. Hence, if the shaded fiber in Figure 4-7 is

broken only the axial fiber stresses within the hexagonal area are perturbed due to this single

break. Within the context of the forgoing discussing, this would imply that all the force

influence-functions expect for q0,0, q1,0, q0,1, q-1,1, q-1,0, q0,-1, and q1,-1, are identically equal to

zero. Moreover, the perturbation due to a fiber fracture decreases rapidly for axial distance x,

from the fiber fracture plane. For distances greater than xp from the plane of a fiber fracture,

the stress perturbation vanishes. This length xp, is a function of the fiber and matrix stiffness

and the fiber volume fraction. Also, due to symmetry all the force influence-functions of the

nearest neighbors are equal i.e. q1,0 = q0,1 = q-1,1 = q-1,0 = q0,-1 = q1,-1, and qn,m(-x) = qn,m(x) for

all (n, m). Hence, it is only necessary to calculate q0,0, and q1,0 for x = [0, xp]. The other

force influence-functions for the neighboring fibers are set equal to q1,0. The discussion that

n

m

(0,0) (2,0)

(0,1)

(N-1,0)

(0,M-1)(N-1,M-1)

n

m

(0,0) (2,0)

(0,1)

(N-1,0)

(0,M-1)(N-1,M-1)

Figure 4-7. Hexagonally packed array of fibers with fiber numbering scheme

52

follows describes the calculation of q0,0, and q1,0 from the finite element model. Once q0,0,

and q1,0 are calculated for x = [0, xp] from a finite element model, the foregoing load sharing

is applicable for any axial length X > xp.

A finite element model similar to the one shown in Figure 4-1 can be generated for any

matrix crack size, local fiber volume fraction, fiber/matrix properties, and axial length. It

should be pointed out that the actual diameter of fibers in the finite element model is not

important if all the geometric quantities are non-dimensionalized with respect to a

representative length such as the fiber radius. FEM yields a three-dimensional variation of

stress within each fiber. The first step in calculating force influence-functions is to obtain a

single axial fiber stress fem(x), as a function of position along the fiber x. Once this is done

the fibers can be treated as one-dimensional filaments and not three-dimensional cylinders.

fem(x) is calculated from a weighted average of the finite element axial stress on the fiber

cross-section at a given x-position and is given by

( ) ( )[ ]m

m

A

xf

AzyxA

xf

1

femfem d,,1

∫= σσ (4-3)

Af is the cross-sectional area of the fiber, ),,(fem zyxxσ is the axial stress profile on the fiber

cross-section obtained by FEM, and m is the Weibull modulus for probability of failure of the

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30x (r f )

Str

ess

Co

nce

ntr

atio

n

0.96

1.00

1.04

1.08

1.12

1.16

0 10 20 30x (r f )

Str

ess

Co

nce

ntr

atio

n

(a) (b)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30x (r f )

Str

ess

Co

nce

ntr

atio

n

0.96

1.00

1.04

1.08

1.12

1.16

0 10 20 30x (r f )

Str

ess

Co

nce

ntr

atio

n

(a) (b)

Figure 4-8. Weighted average finite element stress. (a) Broken fiber (b) Neighboring fiber

53

fiber. Equation (4-3) is obtained by equating the Weibull failure probability of the fiber

cross-section due to fem(x) to the probability of failure resulting from the stress profile

),,(fem zyxxσ . fem(x) calculated in this manner is consistent with the Weibull probability

distribution for the strength of a fiber and hence is suited to strength modeling of

unidirectional composites. Figure 4-8 are typical fem(x) curves for the broken and

neighboring fibers. Applying Equation (4-1) to the finite element model with a single fiber

fracture yields

( ) fem0,0

femfem )( uxqx ffb += σσ , and

( ) fem0,1

femfem )( uxqx ffn += σσ for x = [0, xp] (4-4)

femffσ is the unperturbed FEM fiber stress, ufem is the FEM break opening-displacement, and

fembσ and fem

nσ are the weighted average axial stress for the broken and neighboring fiber,

respectively, calculated from Equation (4-3). The only unknown quantities in Equation (4-4)

are the force influence-functions. The force influence-functions calculated from Equation

(4-4) depend on the matrix crack size, local fiber volume fraction, fiber/matrix properties.

Using these force influence-functions the fiber stresses at any location within a unidirectional

composite with arbitrary fiber fractures can be calculated from Equations (4-1) and (4-2).

4.4 MULTIPLE FIBER FRACTURES IN MODEL COMPOSITE

In Chapter 3, measurements of quasi-static strain concentrations for model composites with

multiple fiber fractures are presented. In this section comparisons between finite element

results and the multiple break measurements are made.

The first step in making this comparison is to use the seven-fiber finite element model to

generate a set of force-influence functions by applying Equations (4-3) and (4-4). The force

influence-functions are used in Equation (4-2) to obtain break opening-displacements for the

multiple break model composites shown in Figure 4-9 and Figure 4-10. Two break opening-

displacements are calculated for Model 8 and three break opening-displacements are

calculated for Model 9. The finite element load sharing is a NNLS, and hence, the size of the

array of fibers is not important.

54

In order to proceed with the comparison for the multiple break model composites it is

necessary to introduce the strain analog of the force influence-function presented earlier.

Moreover, since the lateral orientation of the strain gage on the fiber is important, it is

necessary to develop axial strain influence-functions en,m( x

), that are three-dimensional in

nature. en,m( x

) gives the perturbation of the axial strain field on the surface of the broken

and neighboring fibers as a function of axial distance from the fracture plane and lateral

orientation on the fiber surface with respect to the location of the fracture. For brevity, the

vector x

, is used to represent both the axial distance and lateral orientation on the fiber

surface with respect to the location of the fracture. Hence, the three-dimensional state of

axial fiber strain ( )xmn

,ε , in the model composite is given by

( ) ( )∑=

−− −+=L

iiimmnnffmn uxxex

ii1

,,

εε (4-5)

where ffε is the far-field FEM axial fiber strain. The only seven non-zero strain influence-

functions are e1,0 = e0,1 = e-1,1 = e-1,0 = e0,-1 = e1,-1 and e0,0. As before, they need to computed

over x = [0, xp]. The seven-fiber finite element model with a single fiber fracture is used to

calculate e0,0 and e1,0 as given below

( ) ( )fem

femfem0,0

0,0 u

xxe ffεε −

=

, and

( ) ( )fem

femfem0,1

0,1 u

xxe ffεε −

=

for x = [0, xp]

(4-6)

( )xfem

0,0ε and ( )xfem

0,1ε are the FEM axial fiber strains for the broken and neighboring fiber,

respectively, ( )xff

femε is the unperturbed FEM axial fiber strain, and ufem is the FEM break-

opening displacement. Once the strain influence-functions are calculated from Equation

(4-6), Equation (4-5) is used to compute the strains in the model composites with multiple

breaks.

55

The comparison between the finite element and measured axial strain concentrations for the

model composites with multiple fiber breaks is shown in Figure 4-9 and Figure 4-10.

Excellent agreement is obtained for the double break configuration. It should be pointed out

that the strain concentration measured by gages 1 and 2 is almost identical, and this is in

agreement with the NNLS framework where only failures in the neighboring fiber influence

1

2 3

1

2 3

Strain Conc.

1.441.48-0.073

1.251.29+0.102

1.251.28+0.101

FEMEXPx, rfGage

Strain Conc.

1.441.48-0.073

1.251.29+0.102

1.251.28+0.101

FEMEXPx, rfGage

Figure 4-9. Model composite with two adjacent, coplanar fiber fractures (Model 8)

1

2

3 4

1

2

3 4

Strain Conc.

0.310.401.703

0.720.676.674

0.440.383.532

1.071.000.001

FEMEXPx, rfGage

Strain Conc.

0.310.401.703

0.720.676.674

0.440.383.532

1.071.000.001

FEMEXPx, rfGage

Figure 4-10. Model composite with three adjacent, coplanar fiber fractures (Model 9)

56

strains on a fiber. Fairly good agreement is obtained for the triple break configuration. Gage

3 shows the greatest difference between the measured and finite element strain concentration.

Gage 1 that is mounted facing away from the cluster of three broken fibers does not show

any strain concentration. This is further indication that NNLS is adequate. The finite

element method gives a strain concentration of 1.07 for Gage 1. This is may be due to

problems with mesh refinement and/or the fact that there are a greater number of intact fibers

surrounding the fractured fibers in the 37-fiber model composite than in the 7-fiber finite

element model. Based on the arguments presented in Section 4.2.3 and this section, a NNLS

framework is believed to be appropriate for modeling the tensile strength of unidirectional

polymer composites.

4.5 QUASI-STATIC STRENGTH SIMULATIONS

In this section, a Monte Carlo simulation to model the tensile strength of unidirectional

composite materials is developed within the framework of the finite element NNLS presented

in Section 4.3. Comparisons are made to the statistical static strength of Grafil 34-700

standard modulus carbon fiber/polyphenylene sulfide (PPS) pultruded unidirectional

composite tape.55

4.5.1 Material Properties

The probability of failure Pf, of a fiber of length l, at a stress level σ is given by the Weibull

distribution shown below

( )

−−=

m

oof l

llP

σσσ exp1, (4-7)

where σo is the Weibull location parameter, and m is the fiber Weibull modulus or shape

parameter. The quantity σo is interpreted as the stress level required to cause one failure on

average in a fiber of length lo. m is related to the variability in fiber strength, with a higher m

for a narrower distribution. Wimolkiatisak and Bell56 have studied the strength of Hercules

AS4 carbon fibers using the single-fiber fragmentation test. Their data can be used to

calculate the following parameters for the Weibull strength distribution of AS4 carbon fibers:

57

σo = 5.25 GPa, lo = 1 mm, and m = 10.65. In the absence of statistical fiber strengths of

Grafil fibers, the Weibull strength distribution of AS4 carbon fibers is used to compute

composite strength.

The experimental composite strengths of the Grafil/PPS composite shown in Figure 4-11 also

conform to a Weibull distribution with 0σ = 1.57 GPa and m = 29.4 at a gage length ol = 76

mm. The composites have a fiber volume fraction Vf , of 40%.

The axial Young’s modulus of the fiber Ef = 234.4 GPa, the shear stiffness of the PPS matrix

Gm = 1.1 GPa, and the fiber radius rf = 3.5 m. For this fiber and matrix stiffness and fiber

volume fraction the perturbed distance from the fiber fracture plane xp = 68 × rf.

4.5.2 Strength Simulation Approach

An outline of the strength simulation procedure is shown in Figure 4-12. Figure 4-13 shows

the representative volume of material with fibers and matrix that is considered for the

simulation. Uniform hexagonal fiber packing is assumed. Since the strength of the

composite material is fiber dominated the simulation will attempt to track the progression of

fiber breaks that leads to ultimate composite failure. For this purpose each of the fibers is

subdivided into elements along its length as shown in Figure 4-13. It is assumed that the

fiber and matrix constitutive relations are deterministic, but the fiber strengths are

statistically distributed. Values of strength are assigned to the individual elements by using

the fiber Weibull strength distribution described in Section 4.5.1. While the distribution of

fiber element strengths remains the same, the actual element strengths change for every

computation of strength. To begin the simulation process, the far-field axial stress level σff,

is increased to the strength value of the weakest element. This causes failure of the weakest

element. A break is positioned at a random location within each failed element. Landis et.

al.41 have reported that positioning a break at random within a fiber element improves

convergence of the simulation results with respect to the number of elements that must be

selected along a fiber. The finite element NNLS described in Section 4.3 is used to relate the

global composite stress level to the local fiber stresses. In order to expedite computing of

58

fiber stresses with random positions of breaks within fiber elements, force influence-

functions are computed and stored at a given number of equally spaced points over x = [0,

xp]. Linear interpolation is used to determine qn,m(x) at an arbitrary x. Local stress

redistribution may result in further fiber element breaks. σff is increased to cause failure of

the next weakest element only after no further fiber element breaks occur at the same global

far-field stress level. This process is repeated until the surviving fiber elements in a cross-

section of the simulation volume can no longer sustain the global load. This global stress

level is the calculated composite material strength.

All the simulation results presented in this chapter are obtained on a RVE consisting of a

10×10 array of hexagonally packed fibers. Increasing the number of fibers has a negligible

effect on the simulation strengths. For a 20×20 array of hexagonally packed fibers of length

X = 0.5 mm, the mean composite strength is 4% less than the mean for a 10×10 array of

fibers. The axial length of the simulation volume X, is progressively increased, and at each

length 100 composite strengths are calculated. Although the fiber elements conform to the

same Weibull strength distribution, each run of the simulation is performed with different

fiber element strengths. Hence, a different composite strength is calculated for each run of

the simulation program. The 100 composite strength values at each X conform to a Weibull

distribution with shape parameter ( )Xm , and location parameters ( )X0σ . Increasing the

simulation length reduces ( )X0σ . This is a reflection of the weakest link scaling for fiber

strengths given by Equation (4-7). Figure 4-14 shows the location parameter computed at

each length for the Grafil/PPS composite system. The relation between the Weibull location

parameter and the length scale, lo, is given by

m

o

o

X

X1

1

1

=

σσ

(4-8)

In order to obtain a set of Weibull parameters for the composite material a linear regression is

performed on the log( 0σ ) versus log(X) simulation results. From Equation (4-8) it is clear

that the slope of the linear regression is related to the shape parameter for the composite

material, and the location parameter for the composite material at X = 76 mm is easily

obtained by extrapolation of the linear regression in Figure 4-14. By applying this method

59

Weibull shape and location parameters of 51.1 and 1.72 GPa, respectively, are obtained for

the Grafil/PPS composite material at a gage length of 76 mm. It should be pointed out that a

very similar Weibull distribution would be obtained by running the strength simulation

directly for a length of 76 mm. However, the simulation would take a very long time if it

were run at 76 mm since each fiber would have to be divided into a very large number of

elements to ensure convergence of the predictions. The computed location parameter is less

than 10% greater than the experimental composite strength. The composite Weibull shape

parameter from the Monte Carlo simulation is much greater than the experimental Weibull

shape parameter implying less variability in the predicted composite strengths. In Section

4.6 some aspects of material variability that are unavoidable in ‘real’ composite systems are

modeled. The effect of random fiber placement, distributed fiber volume fractions, and

initial imperfections in the form of fiber fractures on the computed Weibull shape parameter

is discussed.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.2 1.3 1.4 1.5 1.6 1.7 1.8Strength (GPa)

Pro

bab

ility

of

failu

re sp

ace

Weibull distribution

Experimental

Figure 4-11. Statistical strength of Grafil/PPS unidirectional composite (Gage length = 76 mm, Vf = 40%)

60

Select simulation volume with fibers and matrix

Divide fibers into elements along length

Assign strengths to fiber elements with Weibull statistics

Ramp up far-field stress to break one more element

Re-distribute local stresses with load sharing

Check for fiber breaks

YES NO

Select simulation volume with fibers and matrix

Divide fibers into elements along length

Assign strengths to fiber elements with Weibull statistics

Ramp up far-field stress to break one more element

Re-distribute local stresses with load sharing

Check for fiber breaks

YES NO

Figure 4-12. Flowchart of Monte Carlo simulation for quasi-static strength

n

m

xX

n

m

x

n

m

xX

Figure 4-13. Representative volume element (RVE) for quasi-static strength simulation

61

4.6 MATERIAL VARIABILITY

In ‘real’ material systems the fibers are randomly packed, fiber volume fractions are

distributed rather than deterministic, and the composite may have initial imperfections in the

form of initially broken fibers. This section discusses the effect of including the above

sources of material variability on strength distribution predictions.

4.6.1 Shear-Lag NNLS

It would be very difficult to develop a finite element load sharing for random fiber placement

or distributed fiber volume fractions since several finite element models with different fiber

placements would need to be developed. Hence, all the results presented in this section are

obtained by applying shear-lag assumptions13 within a nearest neighbor load-sharing

framework. The general load-sharing framework described in Section 4.3.1 still applies.

However, different force influence-functions need to be calculated by applying shear-lag

assumptions. The analysis presented in this section is developed for random fiber placement.

1.68

1.72

1.76

1.80

1.84

1.88

1.92

1.96

0 1 10 100X , mm

( )Xοσ

m/1−

( )76oσ

GPa

1.68

1.72

1.76

1.80

1.84

1.88

1.92

1.96

0 1 10 100X , mm

( )Xοσ

m/1−

( )76oσ

GPa

Figure 4-14. Composite strength of Grafil/PPS unidirectional composite obtained from simulation

62

The centers of the fibers, marked by ‘+’ in Figure 4-15, are confined to rhomboidal cells of

side s, given by

ff

rV

so60sin

π= (4-9)

By placing this restriction on the location of fiber centers a certain degree of order in fiber

placement is maintained, and it is possible to identify the location of fibers in relation to each

other in a manner that is consistent with the (n, m) fiber numbering scheme introduced in

Section 4.3.1. It should be pointed out that for random fiber placement it is necessary to

calculate a set of force influence-functions, jimnq ,

, for each broken fiber (i, j). An analogous

form of Equation (4-1) is used to calculate the fiber stresses i.e.

( ) ( )∑=

−− −+=L

kkk

mnmmnnffmn uxxqx kk

kk1

,,, σσ (4-10)

Similarly the break opening-displacements uj, j=1L, are obtained by solving the system of

equations

( ) ( ) LjuxxqxL

kkkj

mnmmnnffjmn

kk

kjkjjj1, 0

1

,,, =−+== ∑

=−−σσ (4-11)

Consider a typical broken fiber (i, j), as shown in Figure 4-15. Fiber (i, j) has a break at x =

0. Under NNLS force influence-functions are required for the broken fiber i.e. jiq ,0,0 , and for

each of the neighboring fibers i.e. jiq ,0,1 , jiq ,

1,0 , jiq ,1,1− , jiq ,

0,1− , jiq ,1,0 − , and jiq ,

1,1 − . These are the only

non-zero force influence-functions and they are associated with fibers within the highlighted

cells in Figure 4-15. As described in Section 4.3, force influence-functions are calculated

over x = [0, xp], and ( )xq jimn −,

, = ( )xq jimn

,, holds. For notational convenience in the discussion

that follows, the broken fiber and its six neighbors are numbered from 1-7 as shown in Figure

4-15. Hence, jiq ,0,0 = q1,

jiq ,0,1 = q4,

jiq ,1,0 = q3, etc. Let v1(x), v2(x), v7(x) be the displacements

of fibers 1 through 7, respectively. Under shear-lag assumptions

63

Vx

EQ f d

d=

where

( )( )

( )

=

xq

xq

xq

Q

7

2

1

and

( )( )

=

)(7

2

1

xv

xv

xv

V

(4-12)

As shown in Figure 4-15, the distance between the centers of fiber i and fiber j is denoted by

dij, and wij = dij – 2rf. Shear-lag assumptions are applied to the seven highlighted fibers in

Figure 4-15 and the governing system of equations for fiber displacements is obtained as

[ ] 0dd

2

2

=+ VAVx

(4-13)

where

=)(

)(

7

1

xv

xv

V ; [ ]

=

77672717

67665616

56554515

45443414

34332313

27232212

17161514131211

/1000/1/1

/1/1000/1

0/1/100/1

00/1/10/1

000/1/1/1

/1000/1/1

/1/1/1/1/1/1

Awww

wAww

wAww

wAww

wAww

wwAw

wwwwwwA

CA ;

where

+++++−=

17161514131211

111111

wwwwwwA ;

++−=

23122722

111

wwwA ;

++−=

34132333

111

wwwA ;

++−=

45143444

111

wwwA ;

++−=

45155655

111

wwwA ;

++−=

56166766

111

wwwA ;

++−=

67172777

111

wwwA ;

ff

m

EA

GhC =

(4-14)

h is the thickness of the matrix shear spring that can be approximated as h = ( rf)/3. The

boundary conditions for calculating the influence-functions are

( ) 101 =v ; ( ) ( ) ( ) ( ) ( ) ( ) 0000000 765432 ====== vvvvvv ;

and 0d

d =∞=x

Vx

(4-15)

64

The eigenvalues of [A] are -(1)2, -(2)

2,-(7)2, and the orthonormal eigenvectors of [A] are

V1, V2,V7. Since [A] is a real symmetric matrix, it is always possible to find a set of

orthonormal eigenvectors. The solution of the system of Equations (4-13)-(4-15) is given by

[ ]( )( )

( )

−−

−−−−

=

xb

xb

xb

VV

777

222

111

exp

exp

exp

~

λλ

λλλλ

where [ ] [ ]721

~VVVV = , and

[ ]

=

0

0

0

0

0

0

1

~ T

7

6

5

4

3

2

1

V

b

b

b

b

b

b

b

(4-16)

[ ]V~

is a 77 matrix with the eigenvectors as columns. The force-influence-functions are

obtained from Equations (4-12) and (4-16).

An example of the effect of random fiber placement on the stresses in the six nearest

neighboring due to a single fiber fracture is shown in Figure 4-16. Very different stresses are

obtained for each of the neighboring fibers depending on the relative distance from the

‘central’ broken fiber. For regular hexagonal packing all the neighboring six fibers

experience the same stress profile. Similarly, the stress profile in the broken fiber changes

due to random fiber packing.

65

n = 0 1 2 i N-1

n

mm

= 0

1

2

j

M-1

s

s

60°

1

2 3

4

56

7

wij

dij

i j

n = 0 1 2 i N-1

n

m

n

mm

= 0

1

2

j

M-1

s

s

60°

1

2 3

4

56

7

wij

dij

i j

wij

dij

i j

Figure 4-15. Nearest neighbor load-sharing with random fiber placement

0.90

1.00

1.10

1.20

1.30

1.40

0 10 20 30 40 50 60x (r f )

Str

ess

Co

nce

ntr

atio

n 234567

23

4

5

1

6

7

Figure 4-16. Stresses in nearest unbroken neighbors due to random fiber placement

66

4.6.2 Shear-Lag Versus Finite Element for Regular Hexagonal Fiber Packing

Comparisons are made between the shear-lag and finite element results for load sharing and

between the shear-lag and finite element results for composite strength. Both these

comparisons are made for the case of NNLS in a regular hexagonal array of fibers.

The stresses on the broken and neighboring fiber due to a single fiber fracture are shown in

Figure 4-17 and Figure 4-18, respectively. The geometry and material properties of the 7-

fiber model composite are used for this comparison. The finite element weighted average

stress is given by Equation (4-3). For the broken fiber there is excellent agreement between

the FEM and shear-lag stresses. There is fair comparison between the FEM weighted

average and shear-lag stress for the neighboring fiber. The agreement between finite element

and shear-lag stresses for the neighboring fiber improves for higher fiber/matrix stiffness

ratios and fiber volume fractions as is to be expected based upon the shear-lag assumptions.

The axial finite element stresses at two diametrically opposite points on the neighboring fiber

are also shown in Figure 4-18. It is apparent that there is a steep gradient of axial stress

across the neighboring fiber. The stress variation with respect to cross-sectional position in

the fiber is not captured by the shear-lag technique that treats fibers as one-dimensional

filaments.

The random shear-lag load sharing developed in Section 4.6.1 is easily specialized for

regular hexagonal fiber packing and incorporated into the strength simulation framework

developed in Section 4.5. There is excellent agreement between the finite element and shear-

lag based strength predictions for the Grafil/PPS composite as shown in Table 4-1. Hence,

the simpler shear-lag load-sharing analysis yields strength predictions that are in excellent

agreement with the FEM load sharing for composites with a high fiber volume fraction and

fiber/matrix stiffness ratio.

67

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10 15 20 25 30 35x (r f )

Str

ess

Co

nce

ntr

atio

n

FEM (Weighted Avg.)Shear-Lag

Figure 4-17. Comparison between shear-lag and FEM for broken fiber

0.95

1.05

1.15

1.25

1.35

0 5 10 15 20 25 30 35x (r f )

Str

ess

Co

nce

ntr

atio

n p

p FEM (Weighted Avg.)

Shear-Lag

FEM (Position 1)

FEM (Position 2)

1 2

Figure 4-18. Comparison between shear-lag and FEM for nearest neighbor

68

Table 4-1. Comparison between shear-lag and FEM composite strength predictions (76 mm gage length)

oσ GPa m

Shear-lag with regular fiber packing 1.69 49.5 FEM 1.72 51.1

4.6.3 Effect of Material Variability on Strength Distribution

Only variability in fiber strength has been included in the simulation predictions presented so

far. This section discusses the effect of including additional sources of material variability

such as fiber volume fraction, random fiber placement, and initial fiber fractures.

4.6.3.1 Fiber Volume Fraction

The strength simulations consider a small representative volume of material, for example a

10 × 10 array of fibers less than 15 mm in length. At this small volume scale variations in

fiber volume fraction may be expected from one material location to another within a

specimen. Moreover, there is considerable variability in all fiber volume fraction

measurement techniques. The Monte Carlo simulation computes 100 strengths at for each

simulation length X. To account for variable fiber volume fractions a different fiber volume

fraction is used for each strength computation. The results presented here are obtained by

using a normal distribution of fiber volume fraction with a 40% mean, and a 1% standard

deviation.

4.6.3.2 Random Fiber Placement

The load sharing developed for random fiber placement in Section 4.6.1 is used in the

simulation. A different random configuration of the 10 × 10 fiber array is generated for each

of the 100 strength computations at a given length X.

4.6.3.3 Initial Fiber Fractures

Initial imperfections in a composite material in the form of fiber fractures can occur due to

processing methods such as pultrusion. Each of the 100 strength computations at length X, is

made with a given number, on average, of the weakest fiber elements initial failed. The

number of failed elements is determined such that there are, on an average, four breaks per

69

fiber per 25.4 mm. For any fewer than four breaks per fiber per 25.4 mm there is a negligible

effect on the computed strength distribution for X 15 mm.

4.6.3.4 Results

As mentioned in Section 4.5.2, the strength simulation without any material variability (other

than fiber strengths) yields composite strengths with a much narrower distribution than is

experimentally observed. A higher Weibull shape parameter m , implies a tighter composite

strength distribution. Table 4-2 shows how the composite Weibull shape parameter )(Xm ,

is effected by including each of the additional sources of material variability introduced

above. The first row in Table 4-2 is computed by including only the fiber strength variability

in the analysis. A distributed fiber volume fraction produces the most consistent decrease in

m for all lengths. Random fiber placement results in a decrease in m for only the smaller

lengths. On the other hand, initial fiber fractures produce a decrease in m for the longer

simulation lengths. This is consistent with the criterion of a given number of initial breaks,

on average, per fiber per unit length, which implies that there are more initial breaks for

longer lengths.

Naturally, broader composite strength distributions would be predicted by simultaneously

including the effect of all the sources of material variability described above. It should be

pointed out that the Weibull location parameters )(Xoσ , is essentially unchanged by

incorporated any of the additional sources of material variability described above.

Table 4-2. Predicted composite Weibull shape parameter )(Xm , by including sources of material variability

X, mm 0.5 0.9 1.9 3.8 7.6 15.1

Only Fiber Strength Variability 45.8 59.4 60.6 64.8 59.7 64.9 Distributed Vf 32.5 33.7 34.0 34.3 36.1 36.1

Random Fiber Placement 40.7 49.4 49.0 50.4 59.7 60.1 Initial Fiber Fractures 45.6 59.6 58.9 58.7 56.1 51.5

70

4.7 STRENGTH PREDICTIONS WITH HVDLS

Hedgepeth and Van Dyke13 developed a load sharing methodology under shear-lag

assumptions. The HVDLS redistributes the load of a broken fiber onto all the surrounding

fibers. In this section comparisons are made between the shear-lag NNLS and stress from

HVDLS. Also the strength simulation results obtained by using HVDLS are compared to

predictions from the shear-lag NNLS.

4.7.1 Hedgepeth and Van Dyke Load Sharing (HVDLS)

The HVDLS analysis considers an infinite array of fibers where all the breaks are confined to

a single cross-sectional plane i.e. the x = 0 plane. The general load-sharing framework

introduced in Section 4.3.1 accommodates fiber fractures at any arbitrary x locations.

Moreover, the analysis that is briefly explained here is developed for a periodically repeating

N M array of fibers. Hence, the state of fiber (n, m), at location x, is the same as fiber

(n+N, m+M), at location x. The strength simulation results shown later are actually

calculated using a load-sharing analysis that is periodic in the x-direction too. This is done

by an appropriate choice of boundary conditions when calculating force influence-functions.

However, for simplicity periodicity in the x-direction is not considered in the treatment for

force influence-functions presented below.

It is necessary to develop a new set of force-influence functions qnm(x). Consider regular

hexagonal fiber packing as shown in Figure 4-7. The displacement of fiber (n, m), at axial

position x, is denoted by vn,m(x). The distance between the centers of adjacent fibers is d. h

is the fiber diameter, and w = d – 2rf. Equation (4-17) is the system of equation for fiber

displacements obtained by applying shear-lag assumptions.

( ) 06d

d,1,11,11,,11,,12

,2

=−+++++′+ +−−+−−++ mnmnmnmnmnmnmnmn vvvvvvvC

x

v for all n, m

where wEA

hGC

ff

m=′

(4-17)

The boundary conditions for calculating influence-functions are

71

( ) 100,0 =v , and ( ) 00, =mnv for all other n, m

0d

d , =∞=x

mn

x

v for all n, m

(4-18)

A discrete Fourier transform is used solve Equation (4-17).

( ) ( )

−

−=∑∑

−

=

−

= M

jmi

N

knijkxvxv

N

k

M

jmn

ˆ2exp

ˆ2exp,,

1

0

1

0,

ππ

where 1ˆ −=i

(4-19)

The inverse Fourier transform is given by

( ) ( )

= ∑∑

−

=

−

= M

jmi

N

knixv

NMjkxv

N

n

M

mmn

ˆ2exp

ˆ2exp

1,,

1

0

1

0,

ππ (4-20)

Substituting Equation (4-19) into Equations (4-17) and (4-18) yields

0622

cos22

cos22

cos2d

d2

2

=

−

−+

+

′+

M

j

N

k

M

j

N

kvC

x

v ππππ (4-21)

with boundary conditions

( )NM

jkv1

,,0 = and 0dx

d =∞=x

v (4-22)

The solution to Equations (4-21) and (4-22) is given by

( )

−−

−

−′−= x

M

j

N

k

M

j

N

kC

NMjkxv

ππππ 22cos2

2cos2

2cos26exp

1,, (4-23)

Substituting Equation (4-23) into Equation (4-19) yields

( ) ∑∑−

=

−

=

−−

−

−′−=

1

0

1

0,

22cos2

2cos2

2cos26exp

1 N

k

M

jmn x

M

j

N

k

M

j

N

kC

NMxv

ππππ

−

−×

M

jmi

N

kni ˆ2exp

ˆ2exp

ππ (4-24)

The force influence-functions are obtained by

( ) ( )x

xvExq mn

fmn d

d ,, = (4-25)

It should be pointed out that the above method is not a nearest neighbor load-sharing

Equation (4-24) is the solution to a coupled system of NM ordinary differential equations

72

given by Equation (4-17). The NNLS is obtained by solving a set of seven coupled ordinary

differential equations for the broken fiber and its nearest six neighbors as shown in Section

4.6.1.

4.7.1.1 Comparison with Shear-Lag NNLS

Stresses calculated with HVDLS are compared to NNLS for a single fiber fracture in a

unidirectional composite with regular fiber packing. The geometric and material properties

are given in Section 4.5.1. The stresses for HVDLS are calculated with a 10 × 10 array of

fibers. Figure 4-19 shows that there is excellent agreement between the two load sharing

approaches for the broken fiber. However, there are significant differences in the stresses on

unbroken neighboring fibers as shown in Figure 4-20. The HVDLS technique predicts much

lower stresses on the nearest neighboring fibers. To illustrate that the influence of the single

break is felt beyond the nearest neighbors for the HVDLS analysis, stress profiles for fibers B

and C are also shown in Figure 4-20.

4.7.2 Strength Simulation with HVDLS

Very different strengths are predicted for the Grafil/PPS composite by using HVDLS and

shear-lag NNLS as shown in Figure 4-21. Lower composite Weibull location and shape

parameters that are in better agreement with the experimental distribution are calculated by

employing NNLS. This is to be expected based on the neighboring fiber stresses shown in

Figure 4-20. The NNLS yields much higher stresses on the nearest neighboring fibers that

translates into a greater probability of failure for the unbroken neighboring fibers and

consequently into lower composite strengths. Also as shown in Table 4-3, ( )Xm obtained

by NNLS is consistently lower than ( )Xm from HVDLS at each simulation length X. Since

NNLS results in a greater localization of stress redistribution the fiber strengths in the

immediate vicinity of the first few fiber fractures control the final strength of the composite

material. Hence, greater variability in computed composite strength is expected depending

on how the strengths of the fiber elements in the immediate vicinity of the first fiber fractures

change from one strength computation to another at a given X. The fact that the NNLS

strength predictions are in better agreement with experimental strength distributions lends

greater support to this theory.

73

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100x (r f )

Str

ess

Co

nce

ntr

atio

n

HVDLS

NNLS

Figure 4-19. Comparison between NNLS and HVDLS for broken fiber

0.96

1.00

1.04

1.08

1.12

1.16

1.20

0 20 40 60 80 100x (r f )

Str

ess

Co

nce

ntr

atio

n

HVDLS (A)HVDLS (B)HVDLS (C)NNLS (A)

A BC

Figure 4-20. Comparison between NNLS and HVDLS for neighboring fibers

74

Table 4-3. Predicted composite Weibull shape parameter )(Xm , from HVDLS and NNLS

X, mm 0.5 0.9 1.9 3.8 7.6 15.1

HVDLS 85.9 88.1 109.7 114.7 108.6 100.7 NNLS (Shear-Lag) 45.8 59.4 60.6 64.8 59.7 64.9

4.8 SUMMARY AND CONCLUSIONS

Chapter 3 and 4 present a systematic approach to modeling composite strength that involves:

1. Measurements of load sharing on macromodel composites

2. Using the measurements to support and validate models for load sharing

3. Incorporating the load sharing models into Monte Carlo simulations for composite

strength. Comparisons are made to the experimental strength distribution of a

Grafil/PPS unidirectional composite.

1.65

1.70

1.75

1.80

1.85

1.90

1.95

2.00

0 1 10 100X mm

( )XοσGPa

HVDLS

NNLS

84.71.83HVDLS

29.41.57EXP

49.51.69NNLS

84.71.83HVDLS

29.41.57EXP

49.51.69NNLS

GPaoσ m

Gage length is 76 mm

1.65

1.70

1.75

1.80

1.85

1.90

1.95

2.00

0 1 10 100X mm

( )XοσGPa

HVDLS

NNLS

84.71.83HVDLS

29.41.57EXP

49.51.69NNLS

84.71.83HVDLS

29.41.57EXP

49.51.69NNLS

GPaoσ m

Gage length is 76 mm

Figure 4-21. Comparison between strength predictions from NNLS and HVDLS

75

A finite element model of a single fractured fiber surrounded by its six nearest intact

neighbors is developed. Excellent agreement exists between the finite element model and

measurements on the model composites with six intact fibers around a single broken fiber.

The same finite element model yields strain concentrations that compare very well with

measurements on model composites with eighteen intact fibers around a single broken fiber.

This provides initial justification for developing a nearest neighbor load sharing analysis i.e.

the load of a broken fiber is transmitted to the nearest six hexagonally packed neighbors.

Moreover, NNLS predicts strain concentrations that are in good agreement with

measurements on model composites with multiple fiber fractures. In Section 4.7.2 a

comparison between composite strength predictions obtained by NNLS and HVDLS is made.

For HVDLS the influence of a single fiber fracture is felt beyond the nearest surrounding

neighbors. It is shown that NNLS yields much better composite strength predictions than are

obtained by the HVDLS technique.

The only material variability accounted for in the initial composite strength predictions is the

fiber strength distribution. The simulation predicts a Weibull location parameter for strength

that is within 10% of the experimental location parameter. However, the simulation predicts

far less variability in composite strength than is experimentally observed. To improve the

strength predictions additional sources of material variability such as distributed fiber volume

fractions, initial imperfections in the form of fiber fractures, and random fiber placement are

introduced. It is shown that a normal distribution for fiber volume fraction with a standard

deviation of 1% yields variability in strength predictions that is comparable to the

experimental variability. Variation in fiber volume fraction of this order may well be

expected in ‘real’ composite systems especially at the volume scales considered by the

simulation. Random fiber placement and initially fractured fibers have a marginal effect on

the computed distribution. In order to evaluate the effect of random fiber placement on

composite strength distribution, a NNLS framework is developed under shear-lag

assumptions.

76

5 STRESS-RUPTURE MODELING

Abstract: This chapter develops the Monte Carlo simulation as a technique for predicting the stress rupture lifetime of unidirectional polymer composites based on fiber and matrix properties. Matrix viscoelasticity is cited as the primary cause of rupture failure. Time-dependent matrix deformation leads to an increase in the overstressed length of unbroken fibers in the vicinity of a cluster of fiber fractures. A general time-dependent load-sharing framework that is able to account for an arbitrary sequence of fiber fractures is developed. Matrix deformations are based on the shear-lag assumption. The time-dependent load sharing is incorporated into a Monte Carlo simulation for stress rupture lifetime. The only material variability included in the simulation is the fiber strength distribution. It is shown that very broad lifetime distributions are computed. The reasons for broad rupture lifetime distributions are discussed. The author is able to avoid the necessity of adopting the strength-life equal rank assumption to develop the life-prediction methodology described in this work. Moreover, the fundamental reasons why the strength-life equal rank assumption does not hold for stress-rupture of unidirectional polymer composites are presented. Encouraging comparisons are made to the experimental rupture lifetime of carbon fiber/polymer matrix composites. Finally, recommendations for improving the testing procedures for stress rupture of unidirectional composite materials are made.

5.1 INTRODUCTION

The use of composite materials in engineering applications requires an understanding of their

behavior under various loading conditions. In particular, as composite materials are

deployed in applications where several years of reliable service life are required, predictions

of long-term durability are necessary. Because of the myriad of possible combinations of

fiber and matrix materials, the best-case situation would be to make such predictions in terms

of constituent properties. This is the goal of the present study for the case in which failure is

governed by tensile failure of the fibers in the composite. A key aspect of this work is use of

Monte Carlo simulations for lifetime prediction. There is ample experimental evidence that

the rupture lifetime of a material is not deterministic. Hence, it is extremely important for

life-prediction techniques to be able to determine component reliability at a given stress level

for a desired lifetime. The models presented in this work utilize probabilistic techniques to

account for variability in fiber strength that translate into variability in lifetime.

77

The outline of this chapter is as follows. The chapter begins with a development of time-

dependent load sharing. Section 5.2 discusses two different shear-lag based load-sharing

approaches. The two methods are compared to each other in Section 5.3. A qualitative

comparison is also made to time dependent measurements on model composites described in

Chapter 3. The reasons for stress-rupture failure of unidirectional polymer composites are

presented. Section 5.4 describes the stress rupture simulation framework. Initial

measurements of rupture lifetime are obtained on a Grafil carbon fiber/Polyphenylene

Sulfide (PPS) unidirectional composite. The problems associated these tests are discussed,

and an alternative material system is studied to overcome these challenges. The material

system selected is an AS-4 carbon fiber/polyetheretherketone (PEEK) laminate with [90/03]s

layup. Quasi-static strengths and rupture lifetime measurements on the AS-4/PEEK

composite system are presented. Comparisons between the rupture predictions and

measurements on the AS-4/PEEK system are also presented in Section 5.4. Section 5.5

describes the significance of bundle strength in stress rupture lifetime predictions. The

reasons for large variability in lifetime predictions are discussed in Section 5.6 by performing

a study of the effect of certain material parameters on lifetime. Section 5.6 also describes the

fundamental reasons why the strength-life equal rank assumption is not valid for modeling

stress-rupture of unidirectional polymer composites. Finally in Section 5.7, the major results

and conclusions are summarized.

5.2 TIME-DEPENDENT LOAD SHARING

In Chapter 4 the author introduced a quasi-static framework to calculate the micromechanical

fiber stress redistribution due to arbitrary fiber facture locations. The technique was based in

superposition of the effect of individual fiber fractures. In this chapter a similar framework is

required that is able to compute time-dependent fiber stresses due to an arbitrary sequence of

fiber fractures. The time-dependent version is considerably more complicated. The times at

which fiber fractures occurred and their locations need to be taken into account to calculate

the fiber stresses at any position and time. This general time-dependent framework for load-

sharing is developed in Section 5.2.1. The fibers are assumed to be linearly elastic, while the

matrix is assumed to be linearly viscoelastic. It is shown that the effect of each fiber fracture

is expressed as a convolution of the crack tip opening-displacement at the fiber fracture

78

location and a time-dependent force influence-function. The force influence-functions

depend on geometric and material properties of the composite, and assumptions of the

mechanism of load transfer (i.e. shear-lag, finite element) and number of neighboring fibers

involved in this transfer. Once again we draw on experience with modeling quasi-static load

sharing to compute the force influence-functions. Two different types of load sharing are

developed in Sections 5.2.2 and 5.2.3 depending on the technique used to calculate the force

influence-functions. They are called Nearest Neighbor Load Sharing (NNLS) and Hedgepeth

and Van Dyke Load Sharing (HVDLS). NNLS was introduced in Chapter 4 to model quasi-

static fiber stress redistribution in unidirectional polymer composites. NNLS assumes that

the load of a broken fiber is redistributed only onto the nearest neighbors. The author

believes that NNLS may be valid for time-dependent stress redistribution too. HVDLS is a

time-dependent extension of the traditional quasi-static technique introduced by Hedgepeth

and Van Dyke.13 In HVDLS the load of a broken fiber is transferred onto all the surrounding

fibers with preferential load transfer to the nearer fibers. A comparison between the time-

dependent stresses calculated by the two approaches is made in Section 5.3.

m =

0

1

2

j

M-1

2

1

n = 0 1 2 i N-1

n

m

3

4

56

7

w

d

m =

0

1

2

j

M-1

2

1

n = 0 1 2 i N-1

n

m

3

4

56

7

w

d

2

1

n = 0 1 2 i N-1

n

m

3

4

56

7

w

d

Figure 5-1. Hexagonally packed array of fibers with fiber numbering scheme

79

5.2.1 General Time-Dependent Load Sharing Concepts

Consider a regular array of N M hexagonally packed fibers of length X, as shown in Figure

5-1. The x-coordinate system is oriented perpendicular to the cross-sectional plane of the

fibers. This is the volume of material that will be considered for the Monte Carlo simulations

for stress rupture presented in Section 5.4. A far-field tensile axial stress ff H(t), is applied

to the fibers, where H(t) is the Heaviside unit step function.

The purpose of this section is to present a framework for calculation of axial fiber stress

( )txmn ,,σ , in fiber (n, m), at time t, due an arbitrary sequence of fiber fractures occurring

prior to time t. Let b1, b2, b3br fiber fractures occur at times t1, t2, t3tr, respectively, such

that 0 t1 < t2 < t3tr t. The total number of fiber fractures that have occurred by ti is Li =

b1++bi for i = 1,2,r. Hence, a total number of Lr fiber fractures have occurred by time t.

For convenience the fiber fractures are sequentially numbered as 1, 2,Lr. The fractures 1,

2,Lr occur in fibers (n1, m1), (n2, m2)(nLr, mLr) at axial locations x1, x2xLr, respectively.

At ti the fiber fractures designated Li-1+1 through Li occur. Naturally, L0 = 0. Due to the far-

field tensile stress ff, the broken fiber ends are separated by a displacement 2 ( )txu imn ii,, , i =

1,2,Lr. The quantities ( )txu imn ii,, , i = 1,2,Lr will be referred to as break opening-

displacements. ( )txktmn ,,σ , k = 1,2,r, represent the fiber stresses at t tr due to only the

Lk fiber fractures that have occurred by tk. Similarly, ( )txu it

mnk

ii,, , k = 1,2,r, represent the

break opening-displacements at t tr due to only the Lk fiber fractures that have occurred by

tk. The only non-zero ( )txu it

mnk

ii,, are for i = 1,2,Lk. Hence, the fiber stresses and break

opening-displacements at t are given by

( )txtx rtmnmn ,),( ,, σσ = (5-1)

and

( )txutxu it

mnimnr

iiii,),( ,, = (5-2)

80

respectively. ( )txrtmn ,,σ and ( )txu i

tmn

r

ii,, are calculated from the recursive relations

( ) ( ) ( ) ( )kkk

mnt

mnt

mn ttttxtxtx kk −−+= − H,,, ,,,1 σσσ (5-3)

and

( ) ( ) ( ) ( )kkik

mnit

mnit

mn ttttxutxutxuii

k

ii

k

ii−−+= − H,,, ,,,

1 (5-4)

( )kk

mn ttx −,,σ , k = 1r represents the change in fiber stresses produced due to the set of bk

fractures that occur at tk. Similarly, ( )kik

mn ttxuii

−,, , k = 1r represents the change in break

opening displacements produced by the set of bk fractures that occur at tk. Once again, the

only non-zero ( )kik

mn ttxuii

−,, are for i = 1Lk. If ( )kk

mn ttx −,,σ and ( )kik

mn ttxuii

−,, are

known the fiber stresses and break opening-displacements are readily obtained from

Equations (5-1) through (5-4). It should be pointed out that the virgin material state has no

fiber fractures. Hence when applying Equation (5-3) and (5-4) with k = 1, break opening-

displacements ( )txu it

mn ii,0

, are non-existent and ( )txtmn ,0

,σ = ff H(t). The remainder of

Section 5.2.1 outlines the approach to calculate ( )kk

mn ttx −,,σ and ( )kik

mn ttxuii

−,, . It is

convenient to calculate ( )kik

mn ttxuii

−,, and ( )kk

mn ttx −,,σ in Laplace domain since the

governing system of equations involve convolution integrals that are converted into a system

of linear algebraic equations by Laplace transformation. The time-domain results are then

obtained in an approximate sense by Schapery’s direct Laplace inversion57 given by

( )ts

sst2

1F)f(

=≈ (5-5)

where ( )sF is the Laplace transform of f(t) i.e. [f(t)] = ( )sF .

Representative curves for ( )txit

mnk

ii,,σ and ( )txu i

tmn

k

ii,, as given by Equation (5-3) and (5-4),

respectively, are shown in Figure 5-2 through Figure 5-7. The fiber stresses at t tr due to

only the Lk fiber fractures that have occurred by tk are given by

( ) ( ) ( ) ( )∑∫

=−− ∂

∂−−+=

kk

jj

jj

k

L

j

tj

tmn

jmmnnfft

mn

xutxxttx

1 0

,

,, d,

,QH, ββ

ββσσ (5-6)

81

Equation (5-6) is simply an expression of superposition of the far-field stress and the

perturbation due to each fiber break in the composite material. ( )txmn ,Q , is the axial stress

produced in fiber (n, m), at location x, calculated by applying a Heaviside unit step opening-

displacement, i.e. H(t), to a break in fiber (0,0) at x = 0. ( )txmn ,Q , is called the time-

dependent force influence-function. ( )txmn ,Q , depends on geometric and constitutive

properties of the fibers and matrix and the load sharing assumptions i.e. NNLS, HVDLS,

shear-lag, etc. Calculation of ( )txmn ,Q , is presented in Section 5.2.2 and 5.2.3.

t

( )1

,

1

,

Li

txu it

mnk

ii

=

t1 t2 t3 tk

( )txu it

mn ii,1

,

( )txu it

mn ii,2

,

( )txu it

mn ii,3

,

( )txu it

mn ii,0

,

t

( )1

,

1

,

Li

txu it

mnk

ii

=

t1 t2 t3 tk

( )txu it

mn ii,1

,

( )txu it

mn ii,2

,

( )txu it

mn ii,3

,

( )txu it

mn ii,0

,

Figure 5-2. Break opening-displacements for breaks 1L1 due to first Lk fractures

82

t

( )21

,

1

,

LLi

txu it

mnk

ii

+=

t2 t3 tk

( )txu it

mn ii,3

,

( )txu it

mn ii,2

,

( )txu it

mn ii,0

,

t

( )21

,

1

,

LLi

txu it

mnk

ii

+=

t2 t3 tk

( )txu it

mn ii,3

,

( )txu it

mn ii,2

,

( )txu it

mn ii,0

,

Figure 5-3. Break opening-displacements for breaks L1+1L2 due to first Lk fractures

t

( )kk

it

mn

LLi

txu k

ii

1

,

1

,

+= −

tk

( )txu it

mn ii,0

,

t

( )kk

it

mn

LLi

txu k

ii

1

,

1

,

+= −

tk

( )txu it

mn ii,0

,

Figure 5-4. Break opening-displacements for breaks Lk-1+1Lk due to first Lk fractures

83

t

( )1

,

1

,

Li

txit

mnk

ii

=

σ

t1

ffσ( )txi

tmn ii

,0,σ

t

( )1

,

1

,

Li

txit

mnk

ii

=

σ

t1

ffσ( )txi

tmn ii

,0,σ

Figure 5-5. Fiber stresses at breaks 1L1 due to first Lk fractures

t

( )21

,

1

,

LLi

txit

mnk

ii

+=

σ

t1 t2

( )txit

mn ii,0

,σ

( )txit

mn ii,1

,σ

t

( )21

,

1

,

LLi

txit

mnk

ii

+=

σ

t1 t2

( )txit

mn ii,0

,σ

( )txit

mn ii,1

,σ

Figure 5-6. Fiber stresses at breaks L1+1L2 due to first Lk fractures

84

To begin the solution process, we note that the Laplace transform of Equation (5-6) is given

by

( ) ( ) ( )∑=

−− −+=k

k

jjjj

k

L

jj

tmnjmmnn

fftmn sxussxx

ssx

1,,, ,,Q,

σσ (5-7)

From Figure 5-2 through Figure 5-4 it is apparent that Equation (5-4) can be simplified to

( ) ( ) ( )kkik

mnit

mn ttttxutxuii

k

ii−−= H,, ,, , for i = Lk-1+1Lk (5-8)

since these fiber fractures i = Lk-1+1Lk do not exist until tk. Comparing Equation (5-8) with

Equation (5-4) reiterates that

( ) 0,1, =− txu i

tmn

k

ii, for i = Lk-1+1Lk (5-9)

i.e. breaks i = Lk-1+1Lk do not exist before tk. Equation (5-4) applies as is for all breaks i =

1Lk-1 (i.e. the breaks that exist before tk). Substituting the Laplace transforms of Equations

(5-4) and (5-8) into Equation (5-7) yields

( ) ( ) ( ) ( )[ ]

( ) ( ) k

k

k

jjjj

k

k

jj

k

jjjj

k

stL

Ljj

kmnjmmnn

L

j

stj

kmnj

tmnjmmnn

fftmn

esxussxx

esxusxussxxs

sx

−

+=−−

=

−−−

∑

∑

−

−−

−

++−+=

1,,

1,,,,

1

1

1

,,Q

,,,Q,σ

σ

(5-10)

t

( )kk

it

mn

LLi

txk

ii

1

,

1

,

+= −

σ

t1 t2 t3 tk

( )txit

mn ii,0

,σ( )txi

tmn ii

,1,σ( )txi

tmn ii

,2,σ( )txi

tmn ii

,3,σ

t

( )kk

it

mn

LLi

txk

ii

1

,

1

,

+= −

σ

t1 t2 t3 tk

( )txit

mn ii,0

,σ( )txi

tmn ii

,1,σ( )txi

tmn ii

,2,σ( )txi

tmn ii

,3,σ

Figure 5-7. Fiber stresses at breaks Lk-1+1Lk due to first Lk fractures

85

We can rewrite Equation (5-7) to find ( )sxktmn ,1

,−σ

( ) ( ) ( )∑−

−−

=−− −+=

1

11

1,,, ,,Q,

k

k

jjjj

k

L

jj

tmnjmmnn

fftmn sxussxx

ssx

σσ (5-11)

Substituting Equation (5-11) into Equation (5-10) yields

( ) ( ) ( ) ( ) k

k

jjjj

kk stL

jj

kmnjmmnn

tmn

tmn esxussxxsxsx −

=−−∑ −+= −

1,,,, ,,Q,, 1σσ (5-12)

Comparing the Laplace transform of Equation (5-3) with Equation (5-12) shows that

( ) ( ) ( )∑=

−− −=k

jjjj

L

jj

kmnjmmnn

kmn sxussxxsx

1,,, ,,Q,σ (5-13)

Applying Equation (5-12) to the break locations 1Lk yields a system of equations given by

( ) ( )

( ) ( ) k

L

j

stj

kmnjimmnn

it

mnit

mn

Liesxussxx

sxsx

k

k

jjjiji

k

ii

k

ii

1,2,for ,,,Q

,,

1,,

,,1

=−+

=

∑=

−−−

−σσ

(5-14)

In order to calculate the unknowns ( )sxu jk

mn jj,, , j=1,2,Lk, in Equation (5-14) it is necessary

to consider two separate cases. For Case I, Equation (5-14) is applied to the location of

breaks that occurred before tk. For Case II, Equation (5-14) is applied to the location of

breaks that occur at tk.

5.2.1.1 Case I (i=1Lk-1)

From Figure 5-5 through Figure 5-7 it is apparent that Equation (5-3) can be simplified to

( ) ( )txtx it

mnit

mnk

ii

k

ii,, 1

,,−=σσ , for i = 1Lk-1 (5-15)

since these fiber fractures have occurred by tk-1. The state of stress at (ni, mi) and xi, i =

1Lk-1 is unchanged by any further fiber fractures that occur after tk-1. This is reinforced by

comparing Equation (5-15) with Equation (5-3) which implies that

( ) ( ) 0H,, =−− kkik

mn ttttxii

σ , for i = 1Lk-1 (5-16)

Substituting the Laplace transform of Equation (5-15) into Equation (5-14) yields the system

of equations given by

86

( ) ( )∑=

−− −=k

jjjiji

L

jj

kmnjimmnn sxussxx

1,, ,,Q0 , for i = 1Lk-1 (5-17)

Equation (5-17) requires that the additional break opening-displacements that occur at tk do

not alter the stress state at the location of breaks that occurred before tk.

5.2.1.2 Case II (i = Lk-1+1Lk)

Applying Equation (5-3) to the stress at breaks i = Lk-1+1Lk yields

( ) ( ) ( ) ( )kkik

mnit

mnit

mn ttttxtxtxii

k

ii

k

ii−−+= − H,,, ,,,

1 σσσ (5-18)

From Figure 5-5 through Figure 5-7 it is apparent that

( ) ( ) ( ) ( )kit

mnkkik

mn tttxttttx k

iiii−−=−− − H,H, 1

,, σσ , for i = Lk-1+1Lk (5-19)

since the stresses at (ni, mi) and xi, i = Lk-1+1Lk go to zero at tk. Substituting the Laplace

transform of Equation (5-18) into Equation (5-14) yields the system of equations given by

( ) ( ) ( )∑=

−− −=k

jjjijiii

L

jj

kmnjimmnni

kmn sxussxxsx

1,,, ,,Q,σ , for i = Lk-1+1Lk (5-20)

Equation (5-20) together with Equation (5-19) requires that the additional break opening-

displacements at tk cause the stresses at the location of breaks that occur at tk to vanish.

Collectively, Equations (5-17) and (5-20) represent Lk linear algebraic equations that can be

solved for ( )sxu jk

mn jj,, , j = 1Lk, provided ( )sxi

kmn ii

,,σ in Equation (5-20) is known. It

will be shown that Equation (5-19) can be used to obtain ( )sxik

mn ii,,σ , i = Lk-1+1Lk. Once

( )sxu jk

mn jj,, , j = 1Lk are calculated, the Laplace transform of Equation (5-4) and (5-2) can

be used to calculate the break opening-displacements in Laplace domain. Moreover,

Equation (5-13), and the Laplace transforms of Equations (5-3) and (5-1) can be used to

calculate the fiber stresses in Laplace domain. Hence, a complete solution to the problem is

readily available in Laplace domain. In order to calculate the time-domain solution Equation

(5-5) is used as described below.

Within the context of Schapery’s direct inversion

87

( ) ( ) ( )( )ats

sa ssats−=

−− −≈2

1

1 FHFe (5-21)

Hence, the approximate inverse Laplace transform of Equation (5-12) is

( ) ( )

( ) ( ) ( )( )

∑= −

=−− −−+

≈ −

k

k

jjjj

kk

L

j tts

jk

mnjmmnnk

tmn

tmn

sxussxxtt

txtx

1 2

1,2

,

,,

,,QH

,, 1σσ

(5-22)

Similarly, Equation (5-4) becomes

( ) ( ) ( ) ( )( )k

ii

k

ii

k

ii

tts

ik

mnkit

mnit

mn sxustttxutxu−

=−+≈ −

2

1,,, ,H,, 1 (5-23)

From Equations (5-22) and (5-23) it is apparent that ( )sxu jk

mn jj,, , j = 1Lk, is required at s =

1/[2(t-tk)]. Hence, the solution to Equations (5-17) and (5-20) is obtained at s = 1/[2(t-tk)] for

each k =1r. Equations (5-17), (5-20), (5-22), and (5-23) are used recursively starting with

k = 1 through k = r. The final time-domain fiber stresses and break opening-displacements

are given by Equations (5-1) and (5-2), respectively.

We still need to calculate ( )sxik

mn ii,,σ , i = Lk-1+1Lk in Equation (5-20). For t tk

Equation (5-19) is simply

( ) ( )txttx it

mnkik

mnk

iiii,, 1

,,−−=− σσ , for i = Lk-1+1Lk (5-24)

Making the change of variables t ′ = t – tk in Equation (5-24) gives

( ) ( )kit

mnik

mn ttxtx k

iiii+′−=′ − ,, 1

,, σσ , for i = Lk-1+1Lk, and t ′ 0 (5-25)

Within the context of Laplace inversion given by Equation (5-5)

( )

≈

sss

2

1f

1F (5-26)

Using Equation (5-26) to evaluate the Laplace transform of ( )txik

mn ii′,,σ

( )

≈

sx

ssx i

kmni

kmn iiii 2

1,

1, ,, σσ (5-27)

Using Equation (5-25) to evaluate the to evaluate the right hand side of Equation (5-27) gives

88

( )

+−≈ −

kit

mnik

mn ts

xs

sx k

iiii 2

1,

1, 1

,, σσ , for i = Lk-1+1Lk (5-28)

Hence, within the Laplace inversion approximation given by Equation (5-5), Equation (5-28)

is the left hand side of Equation (5-20). As mentioned earlier, Equations (5-17) and (5-20)

are solved for ( )sxu jk

mn jj,, , j = 1Lk, at s = 1/[2(t-tk)]. Evaluating Equation (5-28) at s =

1/[2(t-tk)] gives

( )( )

( ) ( )txttsx it

mnktt

sik

mnk

ii

k

ii,2, 1

,2

1,−−−≈

−=

σσ , for i = Lk-1+1Lk (5-29)

Equation (5-29) is substituted into Equation (5-20). Because of the recursive approach to

solving the problem starting with k = 1,2,…r, ( )txktmn ,1

,−σ on the right hand side of Equation

(5-29) is available for all (n, m) and x before ( )sxu jk

mn jj,, , j = 1Lk, is calculated.

5.2.2 Time-Dependent NNLS

In order to implement the general load-sharing framework discussed in Section 5.2.1, it is

necessary to obtain the Laplace transform of the force influence-functions i.e. ( )sxmn ,Q , . As

mentioned in Section 5.2.1, ( )txmn ,Q , is the axial stress produced in fiber (n, m), at location

x, due to a unit step opening-displacement, i.e. H(t), at an isolated break in fiber (0,0) at x =

0. If uniform hexagonal packing is assumed, every fiber fracture location perturbs its

surroundings in exactly the same manner. Hence, only a single set of Laplace-domain force

influence-functions, with n = 1-NN-1, m = 1-MM-1, x = -XX, needs to be calculated.

The analysis in this section is based on NNLS assumptions. Consider a typical broken fiber

(i, j), as shown in Figure 5-1. Fiber (i, j) has a break at x = 0. Under NNLS assumptions,

only the axial fiber stresses within the hexagonal area are perturbed due to this single break.

This would imply that all the force influence-functions expect for Q0,0(x,t), Q1,0(x,t), Q0,1(x,t),

Q-1,1(x,t), Q-1,0(x,t), Q0,-1(x,t), Q1,-1(x,t) are identically equal to zero. Moreover, the

perturbation due to a fiber fracture decreases rapidly for axial distance x, from the fiber

fracture plane. For distances greater than xp from the plane of a fiber fracture, the stress

perturbation vanishes. This length xp, is a function of the fiber and matrix stiffness and the

89

fiber volume fraction. Also, due to symmetry all the force influence-functions of the nearest

neighbors are equal i.e. Q1,0(x,t) = Q0,1(x,t) = Q-1,1(x,t) = Q-1,0(x,t) = Q0,-1(x,t) = Q1,-1(x,t), and

Qn,m(-x,t) = Qn,m(x,t) for all (n,m). Hence for NNLS, it is only necessary to calculate

( )sx,Q 0,0 and ( )sx,Q 0,1 for x = [0, xp]. Once ( )sx,Q 0,0 and ( )sx,Q 0,1 are calculated for x =

[0 xp] the foregoing load sharing is applicable for any axial length X > xp.

For notational convenience in the discussion that follows, the broken fiber and its six

neighbors are numbered from 1-7 as shown in Figure 5-1. Hence, Q0,0(x,t) = Q1(x,t), Q1,0(x,t)

= Q4(x,t), Q0,1(x,t) = Q3(x,t), etc. Let v1(x,t), v2(x,t),v7(x,t) be the displacements of fibers 1

through 7, respectively. Under shear-lag assumptions

Vx

EQ f d

d=

where

( )( )( )( )( )( )( )

( )( )( )( )( )( )( )

=

=

−

−

−

−

tx

tx

tx

tx

tx

tx

tx

tx

tx

tx

tx

tx

tx

tx

Q

,Q

,Q

,Q

,Q

,Q

,Q

,Q

,Q

,Q

,Q

,Q

,Q

,Q

,Q

0,1

1,0

1,1

0,1

1,0

1,1

0,0

7

6

5

4

3

2

1

and

=

),(

),(

),(

),(

),(

),(

),(

7

6

5

4

3

2

1

txv

txv

txv

txv

txv

txv

txv

V (5-30)

As shown in Figure 5-1, the distance between the centers of two adjacent fibers is denoted by

d, and w = d – 2rf, where rf is the fiber radius. Let Gm(t) be the shear relaxation modulus of

the matrix, Ef be the fiber axial Young’s modulus, and Af be the fiber cross-sectional area.

Shear-lag assumptions are applied to the seven highlighted fibers in Figure 5-1 and the

governing system of equations for fiber displacements is obtained as

90

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] 0d,6,,,,,,0

176543221

2

=−+++++∂∂−+

∂∂

∫t

m xvxvxvxvxvxvxvtGCx

v βββββββββ

β

( ) ( ) ( ) ( ) ( )[ ] 0d,3,,,0

231722

2

=−++∂∂−+

∂∂

∫t

m xvxvxvxvtGCx

v ββββββ

β

( ) ( ) ( ) ( ) ( )[ ] 0d,3,,,0

341223

2

=−++∂∂−+

∂∂

∫t

m xvxvxvxvtGCx

v ββββββ

β

( ) ( ) ( ) ( ) ( )[ ] 0d,3,,,0

451324

2

=−++∂∂−+

∂∂

∫t

m xvxvxvxvtGCx

v ββββββ

β

( ) ( ) ( ) ( ) ( )[ ] 0d,3,,,0

541625

2

=−++∂∂−+

∂∂

∫t

m xvxvxvxvtGCx

v ββββββ

β

( ) ( ) ( ) ( ) ( )[ ] 0d,3,,,0

651726

2

=−++∂∂−+

∂∂

∫t

m xvxvxvxvtGCx

v ββββββ

β

( ) ( ) ( ) ( ) ( )[ ] 0d,3,,,0

761227

2

=−++∂∂−+

∂∂

∫t

m xvxvxvxvtGCx

v ββββββ

β

(5-31)

where

wEA

hC

ff

= (5-32)

h is the thickness of the matrix shear spring that can be approximated as h = ( rf)/3. The

boundary conditions for calculating the influence-functions are

( ) )H(,01 ttv = ;

( ) ( ) ( ) ( ) ( ) ( ) 0,0,0,0,0,0,0 765432 ====== tvtvtvtvtvtv for t ≥ 0;

and ( )

0, =

∂∂

∞=x

n

x

txv, for n = 1…7 and t ≥ 0;

(5-33)

and the initial conditions are

( ) 00, =xvn , for n = 1…7, x = [0, ] (5-34)

The Laplace transform of Equation (5-31) is

( ) [ ] 0d

d2

2

=+ VACsGsVx m (5-35)

where

91

where

=

),(

),(

),(

),(

),(

),(

),(

7

6

5

4

3

2

1

sxv

sxv

sxv

sxv

sxv

sxv

sxv

V , and [ ]

−−

−−

−−

−

=

1100011

1310001

0131001

0013101

0001311

1000131

1111116

A (5-36)

The Laplace transform of the boundary conditions given by Equation (5-33) is

( ) ssv 1,01 = ;

( ) ( ) ( ) ( ) ( ) ( ) 0,0,0,0,0,0,0 765432 ====== svsvsvsvsvsv ;

and ( )

0d

,d =∞=x

n

x

sxv, for n = 1…7

(5-37)

Equations (5-35) through (5-37) represent a boundary value problem in x. The eigenvalues

of [A] are -1, -2,-7, and the orthonormal eigenvectors of [A] are V1, V2,V7.

Since [A] is a real symmetric matrix, it is always possible to find a set of orthonormal

eigenvectors. The solution of the system of Equations (5-35) through (5-37) is given by

[ ]( )[ ]( )[ ]( )[ ]

−

−

−

=

xCsGss

a

xCsGss

a

xCsGss

a

VV

m

m

m

77

22

11

exp

exp

exp

~

λ

λ

λ

where [ ] [ ]721

~VVVV = , and

[ ]

=

0

0

0

0

0

0

1

~ T

7

6

5

4

3

2

1 s

V

a

a

a

a

a

a

a

(5-38)

92

[ ]V~

is a 77 matrix with the eigenvectors as columns. The force-influence-functions in

Laplace domain are obtained from Equation (5-38) and the Laplace transform of Equation

(5-30). They are given by

( )( )

( )

( )( )

( )

[ ]

( )( )[ ]

( )( )[ ]

( )( )[ ]

−−

−−

−−

=

=

−

−

xCsGss

CsGsa

xCsGss

CsGsa

xCsGss

CsGsa

VE

sx

sx

sx

sx

sx

sx

m

m

m

m

m

m

f

7

7

7

2

2

2

1

1

1

7

2

1

0,1

1,1

0,0

exp

exp

exp

~

,Q

,Q

,Q

,Q

,Q

,Q

λλ

λλ

λλ

(5-39)

and ( )sxmn ,Q , = 0 for all n and m. As described in Section 5.2.1, Equations (5-17), (5-20),

(5-22), (5-23) need to be evaluated at s = 1/[2(t-tk)]. There are singularities in Equations

(5-17), (5-20), (5-22), (5-23), and (5-39) at t = tk, k = 1r. Hence, it is necessary to rewrite

these equations to remove the singularities so that stresses and break opening-displacements

can be computed even at the instants fiber fractures occur i.e. t = tk, k = 1r.

The shear relaxation modulus can be written as a Prony series

( ) ∑

−+= ∞

i iim

tGGtG

τexp (5-40)

Taking the Laplace transform of Equation (5-40) and multiplying by s gives

( ) ∑+

+= ∞i

i

im

s

GGsGs

τ1

1

(5-41)

For s = 1/[2(t-tk)], Equation (5-41) becomes

( ) ( )( ) ( )∑ −+

+==− ∞−

=i

i

k

i

tts

mk ttG

GsGsttk

τ2

1B

2

1 (5-42)

Evaluating Equation (5-39) at s = 1/[2(t-tk)] yields

93

( )( )

( )( )

( )

( )( )

( )( )kk tt

s

k

tts

sx

sx

sx

tt

sx

sx

sx

−=−

−

−=−

−

−=

2

10,1

1,1

0,0

2

10,1

1,1

0,0

,q

,q

,q

2

,Q

,Q

,Q

(5-43)

where

( )( )

( )( )

( ) [ ]( )( )( )( )

( )( )

−−

−−−−

−−=

−=−

−

xCtta

xCtta

xCtta

VCttE

sx

sx

sx

k

k

k

kf

tts

k

777

222

111

2

10,1

1,1

0,0

Bexp

Bexp

Bexp

~B

,q

,q

,q

λλ

λλλλ

(5-44)

and ( )sxmn ,q , = 0 for all other n, m. It should be pointed out that in the final numerical

solution of Equation (5-44) for ( )sxmn ,q , there is only one eigenmode because 5 of the 7 ai’s

and one of the i’s are zero. Evaluating Equation (5-17) and (5-20) at s = 1/[2(t-tk)] gives the

system of equations

( )( )

( )( )k

jj

k

k

jiji

tts

jk

mn

L

j tts

jimmnn sxussxx−

== −=

−−∑ −=2

1,1 2

1, ,,q0 , for i = 1Lk-1

( ) ( )( )

( )( )

kk-

tts

jk

mn

L

j tts

jimmnnit

mn

LLisxus

sxxtx

k

jj

k

k

jiji

k

ii

1 for ,,

,q,

1

2

1,

1 2

1,,1

+=×

−≈−

−=

= −=

−−∑−σ

(5-45)

Equation (5-45) represents Lk linear algebraic equations that are solved for the quantity

( )sxus jk

mn jj,, , j = 1Lk, at s = 1/[2(t-tk)]. In order to obtain Equation (5-45), it is

necessary to substitute Equation (5-29) into Equation (5-20). Finally, Equation (5-22) is

rewritten as

94

( ) ( )

( ) ( ) ( )[ ]( )

∑= −

=−− −−+

≈ −

k

k

jjjj

kk

L

j tts

jk

mnjmmnnk

tmn

tmn

sxussxxtt

txtx

1 2

1,,

,,

,,qH

,, 1σσ

(5-46)

The final time-dependent solution procedure is as follows. Equation (5-45), (5-46), and

(5-23) are solved recursively starting from k = 1 through k = r. Finally, the fiber stresses and

break opening displacements due to the arbitrary sequence of breaks are given by Equation

(5-1) and (5-2), respectively.

If the shear creep compliance Jm(t), is available instead of Gm(t) then B(t-tk) needs to be

redefined. The Prony series for the creep compliance is

( )∞

+

−−+= ∑ ητ

tt

GGtJ

i iim exp1

11

0

(5-47)

Taking the Laplace transform of Equation (5-47) and multiplying by s gives

( )∞

+

+−+= ∑ η

τs

sGG

sJsi

i

im

11

1

11

11

0

(5-48)

)(sJ m and )(sGm are related by

( ) ( )sJssGs

mm

1= (5-49)

Hence,

( ) ( )( )

( )( )

∞

−=

−+

−+

−+

==−

∑ ητ

k

i

i

ki

ttsm

k

ttttGG

sJstt

k 22

1

11

11

11B

0

2

1

(5-50)

5.2.3 Time-Dependent HVDLS

In order to implement the HVDLS, it is necessary to develop a new set of ( )sxmn ,q , for x =

[0, xp]. Consider regular hexagonal fiber packing as shown in Figure 5-1. The displacement

95

of fiber (n, m), at axial position x, is denoted by vn,m(x). Equation (5-51) is the system of

equation for fiber displacements obtained under shear-lag assumptions.

( ) ( ) ( ) ( ) ( )[( ) ( ) ( )] mnxvxvxv

xvxvxvxvtGCx

v

mnmnmn

t

mnmnmnmnmmn

, allfor , 0d,6,,

,,,,

,1,11,1

0

1,,11,,12

,2

=−++

+++∂∂−+

∂∂

+−−+

−−++∫ββββ

βββββ

β (5-51)

The boundary conditions for calculating influence-functions are

( ) ( )ttv H,00,0 = ;

( ) 0,0, =tv mn for all n, m other than n = m = 0 and t ≥ 0

and ( )

0,, =

∂∂

∞=x

mn

x

txv, for all n, m and t ≥ 0

(5-52)

and the initial conditions are ( ) 00,, =xv mn , for all n, m and x = [0, ] (5-53)

The Laplace transform of Equations (5-51) and (5-52) is

[ ] 06 d

d,1,11,11,,11,,12

,2

=−++++++ +−−+−−++ mnmnmnmnmnmnmnmmn vvvvvvvGCs

x

v for all n, m (5-54)

with boundary conditions

( ) ssv 1,00,0 = ;

( ) 0,0, =sv mn , for all n, m other than n = m = 0

and ( )

0,, =

∂∂

∞=x

mn

x

sxv, for all n, m

(5-55)

Equation (5-54) is solved by applying the discrete Fourier transform given by

( ) ( )

−

−=∑∑

−

=

−

= M

jmi

N

nlijlsxvsxv

N

l

M

jmn

ˆ2exp

ˆ2exp,,,~,

1

0

1

0,

ππ

where 1ˆ −=i

(5-56)

The inverse Fourier transform is given by

( ) ( )

= ∑∑

−

=

−

= M

jmi

N

nlisxv

NMjlsxv

N

n

M

mmn

ˆ2exp

ˆ2exp,

1,,,~

1

0

1

0,

ππ (5-57)

96

Applying the Fourier transform implies periodicity in the n and m directions. Hence, the

state of fiber (n, m), at location x, is the same as the state of fiber (n+N, m+M), at location x.

Substituting Equation (5-56) into Equations (5-54) and (5-55) yields

0622

cos22

cos22

cos2~d

~d2

2

=

−

−+

+

+

M

j

N

l

M

j

N

lvGCs

x

vm

ππππ (5-58)

with boundary conditions

( )sNM

jlsv1

,,,0~ = and ( )

0dx

,,,~d =∞=x

jlsxv (5-59)

The solution to Equations (5-58) and (5-59) is given by

−−

−

−−= x

M

j

N

l

M

j

N

lGsC

sNMv m

ππππ 22cos2

2cos2

2cos26exp

1~ (5-60)

Substituting Equation (5-60) into Equation (5-56) yields

( )

−

−×

−−

−

−−= ∑∑

−

=

−

=

M

jmi

N

nli

xM

j

N

l

M

j

N

lGCs

sNMsxv

N

l

M

jmmn

ˆ2exp

ˆ2exp

22cos2

2cos2

2cos26exp

1,

1

0

1

0,

ππ

ππππ

(5-61)

Finally, the force influence-functions in Laplace domain are given by

( ) ( )x

sxvEsx mn

fmn ∂∂

=,

,Q ,, (5-62)

The differences between NNLS and HVDLS are apparent in the solution approaches.

Equation (5-61) is the solution to a coupled system of NM ordinary differential equations

given by Equations (5-54) and (5-55). The NNLS is obtained by solving a set of seven

coupled ordinary differential equations for the broken fiber and its nearest six neighbors as

shown in Section 5.2.2.

For HVDLS the quantity ( )sxmn ,q , in Equations (5-45) and (5-46) is

97

( )

( )

( )

−

−×

−−

−

−−−×

−−

−

−−−×

=

∑∑−

=

−

=

−=

M

jmi

N

nli

M

j

N

l

M

j

N

lCtt

xM

j

N

l

M

j

N

lCtt

NM

E

k

N

l

M

jk

f

ttsmn

k

ˆ2exp

ˆ2exp

22cos2

2cos2

2cos26B

22cos2

2cos2

2cos26Bexp

q

1

0

1

0

2

1,

ππ

ππππ

ππππ

(5-63)

where B(t-tk) is given by Equation (5-42) or Equation (5-50) for shear relaxation modulus or

creep compliance, respectively. Similar to Equation (5-43)

( )( )

( ) ( )( )kk tt

smnktt

smn sxttsx−

=−

=−=

2

1,2

1, ,q2,Q (5-64)

5.2.4 Time-Dependent Load Sharing Based on Finite Elements

It is possible to generate force influence-functions in Laplace domain by finite element

analysis of a single fractured fiber surrounded by one or more hexagonally packed rings of

neighboring fibers. This approach was developed in Chapter 4 for quasi-static NNLS. For

time-dependent force influence-functions, a transient finite element analysis with linearly

elastic fiber properties and linearly viscoelastic matrix properties would be required.

Displacement boundary conditions would be applied in the fiber direction to produce a far-

field axial strain of ff H(t) in the fibers. Time-dependent axial fiber stresses ( )txmn ,fem,σ ,

would then be calculated for the broken and neighboring fibers. Let the broken fiber be

designated (0, 0) and the fracture be in the x = 0 plane. The time dependent break opening-

displacement ( )tu fem , of the single fiber fracture would also be available from the finite

element analysis. ( )txmn ,fem,σ and ( )tu fem could be fit to a Prony series. The Laplace

transform of these two quantities are related by

( ) ( ) ( )sussxs

sx mnff

mnfem

,fem, ,Q, +=

σσ (5-65)

98

from which ( )sxmn ,Q , , and hence ( )sxmn ,q , , could be calculated. The difficulties associated

with calculating influence-functions in Laplace domain by the finite element method outlined

above are:

1. Sufficient mesh refinement for spatial convergence of results.

2. Sufficiently small increments in time for temporal convergence of results.

3. Accounting for residual thermal stresses due to cure shrinkage in the macromodel

composites if quantitative comparison is to be made to time-dependent load-sharing

measurements of Section 3.5.

For the reasons cited above, the shear-lag load-sharing techniques described in Sections 5.2.2

and 5.2.3 are used for the stress-rupture lifetime predictions made in this dissertation.

5.3 COMPARISON BETWEEN NNLS AND HVDLS

Representative stress profiles in neighboring fibers caused by an isolated break in fiber (0, 0)

at x = 0 are shown in Figure 5-8 through Figure 5-11. The axial location along the fiber is

expressed in terms of the fiber radius, rf. Under NNLS only the stress in the nearest neighbor

is perturbed as shown in Figure 5-8. Although the far-field fiber stress is held constant,

matrix viscoelasticity causes the overloaded length on unbroken fibers adjacent to a fiber

fracture location to increase with time. Consequently, there is a greater probability of fiber

failure occurring in these unbroken fibers. This time-dependent fiber stress redistribution is

the primary cause of failure in a unidirectional polymer matrix composite under longitudinal

stress-rupture loading. Similar trends in the time dependence of stresses calculated by

HVDLS are seen in Figure 5-9 through Figure 5-11. The HVDLS results are computed for a

1010 array of hexagonally packed fibers. HVDLS predicts a lower peak stress

concentration than NNLS in the fibers closest to the fractured fiber. However, the HVDLS

approach also produces a small perturbation of the stresses in the next to nearest neighbors as

shown in Figure 5-10 and Figure 5-11. The peak stress concentration on the next to nearest

neighboring fibers is much smaller than on the nearest neighbors. An important consequence

of shear-lag assumptions is that the peak stress concentration due to an isolated fiber fracture

does not change with time. In fact this peak stress concentration is not a function of any

geometric or material properties of the composite if regular hexagonal fiber packing is

assumed. It should also be pointed out that since under shear-lag assumptions the matrix

99

does not carry any normal tensile stress there is no provision in the analysis to account for an

increase in the far-field fiber stress with time due a viscoelasticity-based decrease of the

tensile load carried by the matrix. This is not a serious source of error since for most

polymer matrix composites with high fiber volume fractions and high fiber to matrix stiffness

ratios the total tensile load carried by the matrix is negligible.

The stresses in the broken fiber decrease with time as shown in Figure 5-12. Although

Figure 5-12 shows stresses calculated by NNLS, very similar curves are obtained for

HVDLS. It is unlikely for another break to occur in the under-stressed region of a broken

fiber. Hence, the decrease of axial stress in a broken fiber with time is not the controlling

mechanism for stress-rupture failure in unidirectional polymer composites.

Time-dependent measurements on model composite systems described in Chapter 3 show

similar trends for both the broken fibers and the unbroken neighbors. In-situ measurements

of strain concentrations are made on macromodel composites with fibers that are large

enough that strain gages can be mounted directly onto the fibers at the locations shown in

Figure 5-13 and Figure 5-14. The strain concentration measurements due to a single fiber

fracture are shown in Figure 5-13. There is a decreasing trend with time for the gage

mounted on the broken fiber, and a slightly increasing trend with time for gages mounted on

neighboring fibers. The increase in strain concentration for gages mounted on unbroken

neighboring fibers is more pronounced due to two adjacent fiber fractures as shown in Figure

5-14. Both the fiber fractures are made to occur at the same axial x location. The model

composite measurements provide a qualitative verification for the load sharing philosophy

described in Section 5.2. A detailed time-dependent finite element analysis of the model

composite domain is necessary to make a quantitative comparison between the measurements

and modeling approach. The finite element model could then be used to establish whether

NNLS or HVDLS is more appropriate for modeling time-dependent micromechanical stress

redistribution in unidirectional composite materials. This procedure was followed in Chapter

4 to investigate the applicability of shear-lag models for quasi-static load sharing.

100

0.96

1.00

1.04

1.08

1.12

1.16

1.20

0 20 40 60 80 100 120 140x (r f )

Str

ess

Co

nce

ntr

atio

n

Increasing time

1,01,0

Figure 5-8. Stress in fiber (1,0) due to isolated break in shaded fiber at x = 0 computed with NNLS

0.96

1.00

1.04

1.08

1.12

1.16

1.20

0 20 40 60 80 100 120 140x (r f )

Str

ess

Co

nce

ntr

atio

n

Increasing time

1,01,0

Figure 5-9. Stress in fiber (1,0) due to isolated break in shaded fiber at x = 0 computed with HVDLS

101

0.996

1.000

1.004

1.008

1.012

1.016

0 20 40 60 80 100 120 140x (r f )

Str

ess

Co

nce

ntr

atio

n

Increasing time

2,02,0

Figure 5-10. Stress in fiber (2,0) due to isolated break in shaded fiber at x = 0 computed with HVDLS

0.996

1.000

1.004

1.008

1.012

1.016

0 20 40 60 80 100 120 140x (r f )

Str

ess

Co

nce

ntr

atio

n

Increasing time

1,11,1

Figure 5-11. Stress in fiber (1,1) due to isolated break in shaded fiber at x = 0 computed with HVDLS

102

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120 140x (r f )

Str

ess

Co

nce

ntr

atio

n

Increasing time

0,00,0

Figure 5-12. Stress in broken fiber (0,0) due to isolated break at x = 0 computed with NNLS

0.6

0.7

0.8

0.9

1.0

1.1

1.2

10 100 1000 10000 100000

Time (sec)

Str

ain

Co

nce

ntr

atio

n

( )

1

2

3

2.073

1.532

4.131

x, rfGage

2.073

1.532

4.131

x, rfGage

1

2

3 1

2

3 1

2

3

Figure 5-13. Model composite measurements of strain concentrations due to a single fiber fracture

103

5.4 STRESS-RUPTURE LIFETIME MODELING

A Monte Carlo simulation is used to predict the stress-rupture lifetime of a unidirectional

composite material. Micromechanical stress redistribution can be calculated by applying

either the NNLS or HVDLS described in Section 5.2. Initial rupture lifetimes are measured

on a unidirectional carbon fiber/polymer matrix composite.55 The difficulties associated with

measuring rupture lifetimes of unidirectional systems are discussed, and rupture lifetimes are

obtained for an alternate material system with 90 plies on the surface. Comparisons

between the simulation predictions and lifetime measurements on the alternate material

system are presented.

5.4.1 Rupture Simulation Approach

Failure of unidirectional composite materials loaded in tension in the fiber direction is

controlled by failure of fibers. The stochastic simulation approach attempts to track the

1.2

1.3

1.4

1.5

1.6

1.7

1.8

10 100 1000 10000 100000Time (sec)

Str

ain

Co

nce

ntr

atio

n (

)

1

2

3

1

2 3

1

2 3

-0.073

+0.102

+0.101

x, rfGage

-0.073

+0.102

+0.101

x, rfGage

Figure 5-14. Model composite measurements of strain concentrations due to a two adjacent coplanar fiber fractures

104

progression of fiber fractures leading to eventual composite failure. All material property

inputs to the simulation other than the fiber strength are assumed to be deterministic. A

Weibull distribution given by Equation (5-66) is used to describe the probability of failure Pf,

of a fiber of length l, at a stress level σ.

( )

−−=

m

oof l

llP

σσσ exp1, (5-66)

σo is called the Weibull location parameter, and m is the fiber Weibull modulus or shape

parameter. σo is interpreted as the stress level required to cause one failure on average in a

fiber of length lo. m is related to the variability in fiber strength, with a higher m for a

narrower distribution. Weibull parameters for the strength distribution of certain fibers are

available in the literature.56

An outline of the stress-rupture simulation procedure is shown in Figure 5-15. Figure 5-16

shows the representative volume of material with fibers and matrix that is considered for the

simulation. Uniform hexagonal fiber packing is assumed. In order to track the location of

fiber fractures every fiber is subdivided into the same number of elements along its length as

shown in Figure 5-16. A fracture is allowed to occur at a random location within each fiber

element. Landis et. al.41 have reported that positioning a break at random within a fiber

element significantly reduces the number of elements along each fiber required for

convergence of the simulation results. Values of strength are assigned to the fiber elements

by using the Weibull strength distribution of Equation (5-66). While the distribution of fiber

strength remains the same, the actual element strengths change for every computation of

rupture lifetime. To begin the simulation process, the far-field axial fiber stress σff, is

increased to the fiber stress level at which rupture lifetimes are desired i.e. ruptffσ . This initial

ramp up is assumed to occur instantaneously, and depending on the rupture stress level may

result in fiber element failures. During the initial ramp up, fiber stress redistribution is

calculated by applying the quasi-static version of load sharing using the instantaneous matrix

modulus since ramp up is assumed to be instantaneous and hence matrix viscoelasticity does

not play a role. The general time-dependent load sharing framework is easily specialized to

determine the instantaneous stress redistribution. This is achieved by using Equations (5-45),

105

(5-46), (5-23), (5-1), and (5-2) to compute fiber stresses at t = 0 due to a single set of b1

breaks that occur simultaneously at t1 = 0. The ramp up is carried out by increasing the far-

field stress to cause failure of the next weakest element only if no further fiber element

failures occur due to stress redistribution at the current far-field stress level. Once ruptffσ is

attained, the far-field fiber stress is held constant and the time level is incremented to cause

failure of fiber elements. The mechanism for tensile stress rupture of unidirectional polymer

composites is discussed in Section 5.3. The time level is incremented in a geometric

progression to maximum of t = tmax. This results in a linear increase in time on a logarithmic

scale. At each new time level fiber stresses are computed and a check is performed for

further fiber element failures. If a fiber failure is detected at a current time level, it is

assumed to have occurred at an intermediate time halfway between the previous time level

and the current time level. Following the notation developed in Section 5.2.1, the most

recent fiber failures occur at tr. Local stress redistribution may result in further fiber element

failures at the same time level. These additional failures are assumed to occur at tr, and

hence, br may increase due to stress redistribution alone. The time level is incremented only

after no further fiber element failures occur at the same time level. The process is repeated

until the surviving fiber elements in a cross-section of the simulation volume can no longer

sustain the global load i.e. stress rupture material failure is predicted. The time level at this

point is the calculated rupture lifetime of the simulation volume. It is also possible that stress

rupture failure does not occur by tmax, in which case a runout is predicted. In order to

expedite the simulation process two techniques are implemented:

1. If all the fiber elements are intact after the initial ramp up, then the time-dependent

load-sharing framework described in Section 5.2 will not predict any change in fiber

stresses with time, and hence, no fiber element failures at all. This is a consequence

of assuming that the fibers are linearly elastic and that the matrix is capable of

sustaining only shear stresses. Hence, it is not necessary to progressively increase the

time level as a runout will be predicted if no fiber element failures occur during the

initial ramp up.

2. Even if a few fiber failures do occur during the initial ramp up, no additional fiber

fractures may occur in tmax due to time-dependent fiber stress evolution. This is very

easily checked by directly computing the stresses at tmax due to only those fiber

106

fractures that occur during the initial ramp up. If the stresses at tmax are not large

enough to cause failure of any additional fiber elements then a runout is predicted

without having to progressively increase the time level.

3. A single set of ( )tsmn sx

21, ,q=

is calculated for all (n, m) at a discrete number of points

over t = [0, tmax] and x = [0, xp]. ( )tsmn sx

21, ,q=

is computed at equally spaced axial

positions over x = [0, xp]. The temporal variation is obtained by computing

( )tsmn sx

21, ,q=

at t = 0 and for additional equally spaced times on a logarithmic scale

over t = (0 tmax]. Linear interpolation in x and the logarithm of t is used to determine

( )tsmn sx

21, ,q=

at an arbitrary axial position and time, respectively.

Before the simulation procedure described above can be used to obtain lifetimes the far-field

fiber stress level ruptffσ needs to be established. The first step in this process is to compute the

composite strength distribution of the simulation volume (Chapter 4). This is achieved by

ramping up the far-field fiber stress instantaneously and calculating fiber stress redistribution

with the quasi-static version of load sharing as described earlier in this section. The

composite strength of the simulation volume corresponds to the far-field fiber stress at which

all the fiber elements in a cross-sectional plane fail. 100 strengths are computed in this

manner. The computed composite strengths of the simulation volume conform to a Weibull

distribution with a location and shape parameter given by sim~oσ and sim~m , respectively. The

second step in establishing ruptffσ is to measure strengths of the composite material under

consideration. The experimental composite strengths conforms to a Weibull distribution with

location and shape parameter exp~oσ and exp~m , respectively. The experimental rupture

lifetimes are measured at a composite stress level of Rexp exp~oσ . The composite stress level for

performing the rupture simulations is Rsim sim~oσ . Rsim is calculated by equating the

experimental instantaneous probability of failure of the composite at Rexp exp~oσ to the

instantaneous probability of failure of the simulation volume at Rsim sim~oσ . Thus,

107

( ) ( )

−−=

−−=

simexp ~sim

~exp exp1exp1

mm

f RRP (5-67)

Finally, the far-field fiber stress level for the rupture simulation is given by

f

off V

R simsimrupt

~σσ = (5-68)

Select simulation volume, and divide fibers into elements

Assign strengths to fiber elements with Weibull statistics

Ramp up far-field stress to desired level

Increase time level

Compute fiber stresses

Check for fiber breaks

YES NO

Select simulation volume, and divide fibers into elements

Assign strengths to fiber elements with Weibull statistics

Ramp up far-field stress to desired level

Increase time level

Compute fiber stresses

Check for fiber breaks

YES NO

Figure 5-15. Flowchart of Monte Carlo simulation for stress rupture lifetime

108

5.4.2 Material Systems

Initial measurements of rupture lifetime are made on a Grafil 34-700 standard

modulus/polyphenylene sulfide (PPS) pultruded unidirectional composite tape.55 The

experimental composite strengths of the Grafil carbon fiber/PPS composite conform to a

Weibull distribution with exp~oσ = 1.57 GPa and exp~m = 29.4 at a gage length ol

~= 76 mm. The

composites have a fiber volume fraction Vf , of 40%. The experimental rupture lifetimes of

the Grafil carbon fiber/PPS composite are shown in Figure 5-17. When the stress rupture

simulation was used to predict lifetime two significant inconsistencies between experimental

lifetimes and the predictions were observed:

1. The simulation methodology over-predicted stress rupture lifetimes, and

2. The rupture lifetime predictions had much greater variability than the experimental

lifetimes.

Measuring the tensile strength and rupture lifetime of purely unidirectional composite

materials poses certain challenges. Although the specimens are tabbed as shown in Figure

5-18, the application of grip pressure unavoidably causes material damage and fiber fractures

in the gripped section of the specimen. The time-dependent propagation of these defects in

the grip section during stress rupture loading dominates the failure behavior of purely

n

m

xX

n

m

x

n

m

xX

Figure 5-16. Representative volume element (RVE) for rupture simulation

109

unidirectional specimens. The tabbing materials used consisted of 100-count (100 wires per

linear inch) stainless steel screen and 1000 series aluminum sheet that was 0.02 inches thick.

A piece of screen is folded in half over the ends of the specimen, and an aluminum piece is

folder over the screen on the ends of the specimen as shown in Figure 5-18. It is very

difficult to epoxy any tabbing material such as Grade G-10 Garolite woven glass fiber

laminates to a PPS-based composite.

In order to alleviate the problems caused by grip-induced damage stress-rupture lifetimes are

measured on an alternate material system with 90 external plies. Composite panels with a

[90/03]s layup are compression molded from APC-2 prepreg supplied by Cytec Industries.

APC-2 prepreg consists of AS4 carbon fiber with a thermoplastic polyetheretherketone

(PEEK) matrix. The specimens have a fiber volume fraction of 54% and a gage length of 76

mm with a rectangular cross-section of 1 mm × 12.7 mm, nominally. The tabbing method

described earlier in this section is used to test the APC-2 composite. However, a finer 200-

count (200 wires per linear inch) stainless steel screen was used instead of the 100-count

screen. The finer tabbing screen and the 90 external plies protect the load carrying 0 plies

from damage in the gripped section. Figure 5-19 shows a failed Grafil/PPS unidirectional

specimen along with a failed APC-2 [90/03]s specimen. It is difficult to tell where failure

initiated for the Grafil/PPS unidirectional composite. On the other hand, the APC-2 laminate

shows a very well defined failure in the gage section. Similar gage section failure patterns

are observed for almost all the APC-2 specimens tested.

The relative stiffness of the 0 and 90 laminae in the direction of the tensile load is used to

determine that each of the 90 plies carry approximately 1.7% of the total load on the [90/03]s

laminate. Hence, it can safely be assumed that the contribution of the 90 plies to the

strength and lifetime of the laminate is negligible.

110

0.84

0.86

0.88

0.90

0.92

0.94

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Time to rupture (sec)

Rex

p

90°C test data

80°C test data

70°C test data

Figure 5-17. Stress rupture lifetimes of Grafil carbon fiber/PPS unidirectional composite

Aluminum

Screen

Specimen

Aluminum

Screen

Specimen

Figure 5-18. Tabbing of specimens for tensile strength and stress rupture testing

111

Two batches of the APC-2 laminate are fabricated. Although both batches had the same fiber

volume fraction and lay-up, very different strength distributions are measured as shown in

Table 5-1. The reason for this inconsistency is not clear, although, it may be the result of

unintended differences in the temperature cycle during processing. The quasi-static strengths

of the APC-2 specimens are measured at two temperatures: 125C and 140C. All the tests

are performed by using a Material Testing System (MTS) servohydraulic machine. The

strengths are measured with a loading rate of 445 N/sec. Since the primary interest is rupture

lifetimes, very few specimens are used to measure quasi-static strength. The quasi-static

strength of only three specimens is measured at each temperature for the Weibull parameters

of Batch I. The quasi-static strength of five specimens is measured at each temperature for

the Weibull parameters of Batch II. The rupture lifetimes of the APC-2 specimens are also

measured at two temperatures: 125C and 140C. The load profile for the tensile rupture

tests consists of an initial ramp at 445 N/sec and a subsequent hold at the desired load. All

the lifetime measurements from both Batch I and Batch II are displayed in Figure 5-20.

However, because of the marked difference in the strength of Batch I and Batch II they are

treated separately when calculating Rexp in Figure 5-20. The test is stopped after

approximately 4 days, and any specimen that does not fail in that period of time is treated as

a runout. The data points corresponding to instantaneous failures in Figure 5-20 are placed at

0.1 seconds. It is immediately apparent that there is a very large variability in rupture

lifetimes at each stress level. Hence, it is very important that a life prediction technique be

stochastic in nature, and be able to compute material reliability at a given stress level and

(a)

(b)

Figure 5-19. Failed specimens. (a) Grafil carbon fiber/PPS unidirectional composite (b) APC-2 [90/03]s laminate

112

temperature. It will be shown that the Monte Carlo simulation technique described here is

particularly well suited to do this.

In the author’s opinion, it is more meaningful to display rupture lifetime as a function of the

probability of instantaneous failure, rather than a fraction of quasi-static strength such as Rexp.

For consistency with the accepted practice, in this work rupture lifetimes are shown as a

function of Rexp as in Figure 5-20. However, experimental results shown here (and the

simulation predictions shown later) should always be interpreted in light of the quasi-static

strength distribution, and hence, the probability of instantaneous failure at that stress level.

Moreover, there is a significant variability in the quasi-static strength distribution for Batch

II. This translates into greater variability in rupture lifetime measurements on Batch II.

Hence, in addition to the probability of instantaneous failure, it is important to be aware of

variability in quasi-static strength when interpreting rupture lifetime measurements. These

two important pieces of information are not apparent by looking only at Figure 5-20 where

the rupture lifetimes are simply displayed as a function of a normalized stress level.

Table 5-1. Quasi-static strength of APC-2 [90/03]s (strengths reported at 76 mm gage length)

BATCH I BATCH II exp~

oσ (GPa) exp~m exp~oσ (GPa)

exp~m 125C 1.56 24.7 1.78 6.6 140C 1.42 32.3 1.69 9.0

113

The fiber strength statistics, fiber stiffness and geometry, and the viscoelastic shear properties

of the matrix are required to implement the stress rupture simulation described in Section

5.4.1. Wimolkiatisak and Bell56 have studied the strength of Hercules AS4 carbon fibers

using the single-fiber fragmentation test. Their data can be used to calculate the following

parameters for the Weibull strength distribution of AS4 carbon fibers: σo = 5.25 GPa, lo = 1

mm, and m = 10.65. The axial Young’s modulus of AS4 carbon fibers Ef = 234.4 GPa, and

the fiber radius rf = 3.5 m. The shear creep compliance master curve shown in Figure 5-21

is generated from short term creep data of neat PEEK at several temperatures. Creep tests

were conducted in a TA Instruments DMA 2920. The associated shift factors as a function

of temperature are shown in Figure 5-22. Figure 5-21 and Figure 5-22 together give the

shear creep compliance of PEEK over several decades of time at any temperature from

124C to 205C. With this information predictions of rupture lifetime can be obtained at any

temperature from 124C to 205C. Hence, the simulation technique described here can be

used to understand and predict the role of temperature in accelerated measurement of stress

rupture lifetimes.

0.6

0.7

0.8

0.9

1.0

1.1

0.1 1 10 100 1000 10000 100000 1000000

Time to rupture (sec)

Rex

p

125 C (Batch I)

125 C (Batch II)

140 C (Batch I)

140 C (Batch II)

Figure 5-20. Stress rupture lifetime of APC-2 [90/03]s specimens

114

5.4.3 Stress-Rupture Simulation Results

The Monte Carlo simulation approach is used to predict stress rupture lifetimes of the APC-2

composite at 125C and 140C. The material properties described in Section 5.4.2 are used

for the lifetime predictions.

0

1000

2000

3000

4000

5000

6000

-20 -15 -10 -5 0 5 10 15

Log (t/aT) sec

Sh

ear

Cre

ep C

om

plia

nce

(

m

m2 /N

)

Figure 5-21. Master curve for shear creep compliance of PEEK

-10

-5

0

5

10

15

90 110 130 150 170 190 210 230

Temp (deg C)

Lo

g(a

T)

Figure 5-22. Shift factors for creep master curve of PEEK

115

As mentioned earlier, the first step is to compute a quasi-static strength distribution. The

strength distribution is required at both 125C and 140C. The specialization of load-sharing

described in Section 5.4.1 for instantaneous ramp up would not predict different quasi-static

strengths at different temperatures since it is based on the instantaneous shear compliance of

the matrix. For temperature dependent strength predictions, it is necessary consider finite

ramp up rates with temperature and time dependent viscoelastic shear properties for the

matrix material. The general load-sharing approach developed in Section 5.2 cannot be

easily modified to account for a finite ramp rate , in the far-field fiber stress, because the

Laplace inversion approximation given by Equation (5-5) is developed for problems where

all inputs are step-functions in time applied at t = 0.57 A crude approximation is used to

compute load sharing with a finite ramp rate for far-field fiber stress. The stresses at time t

are computed by treating the far-field stress t as if it were a step-function in time applied at

t = 0 i.e. ( )ttff Hrupt ασ = . With this technique it is possible to use time and temperature

dependent matrix shear properties to compute temperature dependent quasi-static strength

distributions. However, the loading rate of 445 N/sec is high enough that time and

temperature depend matrix deformation does not play a significant role in the strength

predictions and the same strength distribution is computed at both temperatures. 100 strength

computations are performed on a simulation volume consisting of a 10 × 10 array of fibers

with axial length X = 0.47 mm. The Weibull parameters obtained from the 100 strength

values are shown in Table 5-2. Different strength distributions are obtained by applying

NNLS and HVDLS. It should be pointed out that sim~oσ is computed for unidirectional APC-

2, while the strengths reported in Table 5-1 are for the [90/03]s laminate. Since the 90 plies

may be assumed to carry no load, sim~oσ for the [90/03]s laminate may be assumed to be 3/4th

of the values reported in Table 5-2.

Table 5-2. Quasi-static strength predictions of unidirectional APC-2 Vf = 54% obtained by applying two different load-sharing techniques (strengths reported at X = 0.47 mm)

sim~oσ sim~m

NNLS 2.60 47.5 HVDLS 2.72 80.1

116

Figure 5-23 through Figure 5-26 show rupture lifetime predictions for the APC-2 material.

All the rupture lifetime predictions are performed with a tmax of 4 days. The simulation

volume consisting of a 10 × 10 array of fibers with axial length X = 0.47 mm.

Figure 5-23 and Figure 5-24 are the rupture lifetimes predictions at 125C and 140C,

respectively, calculated with NNLS. The rupture lifetime predictions are plotted along with

the experimental results. At each Rexp, 100 rupture lifetimes are computed. The numbers

displayed with the symbol ‘

The symbol ‘

days. The number of measured and predicted runouts may be regarded as the experimental

and predicted reliability of the material to withstand the given stress level and temperature

for a desired lifetime of 4 days. The horizontal brace and its associated number refer to the

measured and predicted rupture failures that occur within 4 days. Rsim, and hence, the far-

field fiber stress level for the rupture simulation is calculated by equating the experimental

probability of instantaneous failure with the probability of instantaneous failure for the

simulation volume as given by Equation (5-67). The number of instantaneous failures

predicted by the simulation as a fraction of 100 is approximately equal to the probability of

instantaneous failure at each stress level and temperature. Since more experimental

measurements are made for Batch II, a complete comparison is obtained at the lower Rexp in

Figure 5-23 and Figure 5-24. The simulations predict more runouts at the lower temperature

of 125C than at 140C. Consequently, far fewer rupture failures are predicted at 125C than

at 140C. Although, the experimental measurements are limited, they appear to have a

similar trend of longer lifetimes at 125C than at 140C. It is encouraging to note that at

fraction of instantaneous failures, rupture failures in 4 days, and runouts show a close

correlation between the measurements and predictions at 140C and the lower Rexp.

Similarly, the rupture lifetimes computed with HVDLS at 125C and 140C are shown in

Figure 5-25 and Figure 5-26, respectively. The HVDLS also predicts longer lifetimes at

lower temperatures. Comparing the rupture lifetimes obtained by NNLS and HVDLS, it

appears as if the HVDLS technique predicts shorter lifetimes. As explained in Section 5.3,

for HVDLS a single fractured fiber perturbs the stresses in all the neighboring fibers and

117

there is an increase in the overstressed region of all the unbroken fibers with time. For

NNLS, only the stresses in the nearest unbroken neighbors is perturbed due to single

fractured fiber. Hence, stress rupture lifetimes computed with NNLS are longer than

lifetimes computed with HVDLS.

0.6

0.7

0.8

0.9

1.0

1.1

0.1 1 10 100 1000 10000 100000 1000000Time (sec)

Rex

p

Experimental (Batch I)

Simulation (Batch I)

Experimental (Batch II)

Simulation (Batch II)

70

2

30

1

30 4 66

11

8

Figure 5-23. Rupture lifetime predictions for APC-2 composite at 125C (NNLS)

118

0.6

0.7

0.8

0.9

1.0

1.1

0.1 1 10 100 1000 10000 100000 1000000Time (sec)

Rex

p

Experimental (Batch I)

Simulation (Batch I)

Experimental (Batch II)

Simulation (Batch II)

6

20

245

50

2

30

2

33 22

1

Figure 5-24. Rupture lifetime predictions for APC-2 composite at 140C (NNLS)

0.6

0.7

0.8

0.9

1

1.1

0.1 1 10 100 1000 10000 100000 1000000Time (sec)

Rex

p

Experimental (Batch I)

Simulation (Batch I)

Experimental (Batch II)

Simulation (Batch II)

1

29 683

37 2 6181

2 1

Figure 5-25. Rupture lifetime predictions for APC-2 composite at 125C (HVDLS)

119

5.5 BUNDLE STRENGTH AND RUPTURE LIFETIME PREDICTIONS

The bundle strength theory29 gives the tensile strength of a bundle of fibers in the absence of

any matrix. In this theory, once a fiber fractures it does not contribute to the total load

carried by the bundle of fibers, and all the unbroken fibers carry equal loads. The bundle

strength is given by

mo

of mX

lV

1

bundle )1exp(

= σσ (5-69)

where X is the length of the bundle. The modulus of a completely relaxed fully amorphous

thermoplastic matrix vanishes, and hence, a composite material with such a matrix would

behave as a bundle of fibers at very long times. Hence, one may be lead to believe that the

bundle strength can be treated as a threshold stress level below which failure will not occur in

a unidirectional composite with a fully amorphous thermoplastic matrix. However, since all

engineering thermoplastics are semi-crystalline and typical use temperatures do not approach

the melting point of the matrix, a bundle strength based tensile stress level is a conservative

design threshold.

0.6

0.7

0.8

0.9

1.0

1.1

0.1 1 10 100 1000 10000 100000 1000000Time (sec)

Rex

p

Experimental (Batch I)

Simulation (Batch I)

Experimental (Batch II)

Simulation (Batch II)

40 12

31

62

24

2

45

2

48

1

Figure 5-26. Rupture lifetime predictions for APC-2 composite at 140C (HVDLS)

120

5.6 VARIABILITY IN RUPTURE LIFETIME PREDICTIONS

In this section the reasons for variability in computed lifetimes are addressed by changing

certain model input parameters and observing the effect on rupture lifetimes. Under shear-

lag assumptions only the fiber stresses near a break location can change with time. Hence,

the zone of influence for stress rupture modeling is more localized than for strength modeling

where the far-field fiber stress is continually increased. The computed rupture lifetime

distribution is largely dictated by two factors:

1. The number of fiber element failures after the initial ramp up, and

2. the strengths of the fiber elements in the neighborhood of these initial failure

locations.

For quasi-static strength modeling the far-field fiber stress is continually increased to cause

failure of fiber elements, while for stress-rupture modeling the time level is increased which

in turn increases the overstressed length on unbroken fibers in the vicinity of a fractured

fiber. It is apparent that the stress-rupture lifetime is very sensitive to the initial fiber

fractures caused by the stress ramp-up, since these initial fiber fractures nucleate further fiber

failures. With the quasi-static strength simulation described in Chapter 4, the critical cluster

of fiber fractures that ultimately causes failure of the material is not confined to a specific

location since the far-field fiber stress is continually increased. The foregoing discussion

underscores the very different mechanisms associated with quasi-static and time-dependent

failure, and explains why the strength-life equal rank assumption does not hold for stress

rupture of unidirectional polymer composites. The strength-life equal rank assumption states

that a stronger specimen will have a longer lifetime. It is an extremely laborious task to

experimentally prove or disprove the strength-life equal rank assumption. However, the

Monte Carlo simulation technique can be very easily used to show that the strength-life equal

rank assumption does not hold. The random number generator in the Monte Carlo simulation

is seeded such that the same set of fiber strengths are assigned to compute a set of composite

strengths and composite lifetimes. This method can be used to show that there is no direct

correlation between strength and lifetime.

In order to further investigate some of the subtleties involved in modeling stress rupture three

different sets of input parameters are studied as described below. A total of 100 rupture

121

lifetimes are computed for each set of input parameters. All the results obtained in this

section are computed by applying NNLS.

5.6.1 Case I: Control Case

The first set of input parameters is the same as the geometric and material parameters for the

APC-2 composite at 140C given in Section 5.4.2. This is the control case. However, for the

results presented in this section tmax is increased to 3.1×1018 sec so that a more complete

distribution of rupture lifetimes is calculated as shown in Figure 5-27. The results displayed

in Figure 5-27 are calculated with a far-field fiber stress level 5.0ffσ , which corresponds to a

50% probability of instantaneous failure.

5.6.2 Case II: Narrower Fiber Strength Distribution

The rupture lifetimes shown in Figure 5-28 are calculated with all the same input parameters

as the control case except for the Weibull shape parameter for fiber strength which is

57

0 04 4

7 84

64 3

1 0 1 0 0 0 0 0 0 1 00

10

20

30

40

50

60

1.0E

-01

1.0E

+01

1.0E

+03

1.0E

+05

1.0E

+07

1.0E

+09

1.0E

+11

1.0E

+13

1.0E

+15

1.0E

+17

1.0E

+19

Time (sec)

Fre

qu

ency

Figure 5-27. Lifetime distribution for control case

122

changed from m = 10.65 to m = 25.0. This corresponds to a narrower fiber strength

distribution than the control case. The results displayed in Figure 5-28 are also calculated

with a far-field rupture stress level 5.0ffσ , which corresponds to a 50% probability of

instantaneous failure. It is apparent that longer rupture lifetimes with greater variability are

computed with m = 25.0. Increasing the Weibull shape parameter for fiber strength from m =

10.65 to m = 25.0 does not change the composite strength distribution significantly. Hence,

the far-field fiber stress level for rupture 5.0ffσ , is essentially unchanged. 5.0

ffσ is at the tail

end of the fiber strength distribution, and since the Case II fibers have a much narrower

strength distribution, fewer fiber element failures occur due to the instantaneous ramp up

than in Case I. This results in longer rupture lifetimes, and greater variability in the

computed lifetimes.

47

0 02

0 13 3

14

20

2 3 2 1 1 02

0

26

00

10

20

30

40

50

60

1.0E

-01

1.0E

+01

1.0E

+03

1.0E

+05

1.0E

+07

1.0E

+09

1.0E

+11

1.0E

+13

1.0E

+15

1.0E

+17

1.0E

+19

Time (sec)

Fre

qu

ency

Figure 5-28. Lifetime distribution with narrower fiber strength distribution

123

5.6.3 Case III: Shorter Perturbed Axial Length Due to Fiber Fracture

For this case, a combination of geometric and material parameters is altered from its value in

the control case. The factor ( )Ctt k−B in Equation (5-44) is related to the perturbed length

along the broken or neighboring fiber due to a fiber fracture. ( )Ctt k−B is changed to 4

times its value in the control case which has the effect of halving the perturbed axial length at

all times. B(t-tk) is essentially the shear relaxation modulus of the matrix Gm(t-tk), and C is

related to the fiber radius, fiber volume fraction, and fiber axial modulus as given by

Equation (5-32). As with the previous two cases, the rupture simulation is performed at a

far-field rupture stress level 5.0ffσ , which corresponds to 50% probability of instantaneous

failure. As shown in Figure 5-29, there is no significant change in the computed lifetime

distribution calculated by increasing ( )Ctt k−B to 4 times its control value. Similar results

are obtained by increasing ( )Ctt k−B to 9 times its control value. These results may be

counterintuitive at first. There are, however, two competing effects that negate each other so

that the computed lifetime at 5.0ffσ is unchanged. These two competing effects are:

1. The overstressed length on an unbroken fiber near a fiber fracture location is reduced

by a given factor at all times when compared to the control case. By itself this would

tend to increase lifetimes.

2. Decreasing the overstressed length results in an increase in quasi-static strength, and

hence, the far-field fiber stress level at which the simulation is performed i.e. 5.0ffσ .

By itself this would tend to decrease lifetime.

124

5.7 SUMMARY AND CONCLUSIONS

This chapter develops a micromechanical technique for predicting the lifetime of

unidirectional polymer composites loaded under tensile stress-rupture conditions. It is

assumed that stress rupture in unidirectional composite materials occurs as a result of

viscoelastic deformation in the matrix. The time-dependent response of the matrix causes an

increase in the overstressed length on unbroken fibers near a cluster of fiber fractures. This

increases the probability of failure of the unbroken fibers, and consequently the probability of

failure of the composite material as a whole. The formulation presented in this work assumes

linearly viscoelastic matrix behavior.

The first step in this effort is to develop a general framework for micromechanical stress

redistribution due to an arbitrary sequence of fiber fractures. An approximate technique

using Boltzmann superposition of time-dependent influence functions is developed. Two

different sets of influence-functions are calculated based on different shear-lag load-sharing

59

0 0

61

57

57 7

0 2 1 0 0 0 0 0 0 0 0 00

10

20

30

40

50

60

1.0E

-01

1.0E

+01

1.0E

+03

1.0E

+05

1.0E

+07

1.0E

+09

1.0E

+11

1.0E

+13

1.0E

+15

1.0E

+17

1.0E

+19

Time (sec)

Fre

qu

ency

Figure 5-29. Lifetime distribution with shorter perturbed axial length along a broken or neighboring fiber due to a fiber fracture

125

assumptions. The first set of influence functions assume that the load of a broken fiber is

redistributed only onto the nearest neighbors. This form of load sharing is termed Nearest

Neighbor Load Sharing (NNLS) and it is developed because model composite experiments

and finite element analysis of micromechanical load redistribution show that NNLS is

applicable under quasi-static conditions (Chapter 4). The second set of time-dependent

influence functions are developed by extending the traditional quasi-static analysis of

Hedgepeth and Van Dyke.13 A favorable comparison is made between the time-dependent

load sharing analysis and measurements of the strain redistribution in model composites.

The load sharing framework is incorporated into a Monte Carlo simulation to predict the

stress rupture lifetime of unidirectional composite materials. An encouraging comparison is

made between predicted and measured lifetimes of a [90/03]s APC-2 composite laminate at

125C and 140C. Long-term time and temperature dependent viscoelastic properties of the

matrix material are easily obtained by applying time-temperature superposition principles to

short-term creep or relaxation data measured in a dynamic mechanical analyzer at several

temperatures. This information is supplied to the simulation to predict long-term rupture

lifetimes at any temperature. In this manner the simulations help understand and predict the

role of temperature in accelerated measurement of stress rupture lifetimes. The extreme

variability in rupture lifetimes makes it very important for predictive techniques to be able to

assess composite reliability for a desired lifetime at a given stress level and temperature. The

Monte Carlo simulation approach is particularly well suited to determine reliability under

stress rupture conditions.

Measuring the stress-rupture lifetime of purely unidirectional composites is challenging

because initial damage occurs in the gripped section of the specimen. The rupture lifetime of

a unidirectional specimen is very sensitive to the initial fiber fractures, and hence, the initial

damage nucleates subsequent material failure in the gripped section. This chapter puts

forward recommendations to alleviate this problem by testing specimens with a [90/0n]s

layup. The surface 90 plies protect the 0 layers from damage in the grip section while at

the same time carrying a negligible fraction of the total load. Consistent gage section failures

are observed when testing the [90/03]s APC-2 composite specimens.

126

In order to understand the reasons for large variability in computed rupture lifetimes a

parametric study is performed by varying some of the input quantities to the model. It is

shown that decreasing the variability in fiber strengths produces longer and more variable

lifetimes. Unexpectedly, the rupture lifetime distribution is unchanged by altering the

perturbed axial length due to a fiber fracture. This latter study is performed by changing a

combination of geometric and material parameters such that the perturbed length is halved at

all times. The lifetimes computed at a stress level that yields a 50% probability of

instantaneous failure are unaffected.

A major conclusion of this work is to show that the strength-life equal rank assumption is not

valid. For the lifetime predictions discussed in this work, the author is able to avoid the

necessity of making the strength-life equal rank assumption. The fundamental differences in

mechanisms that produce material failure when the stress is continually increased (as with

measuring the fast-fracture strength) and when it is held constant (as with measuring stress-

rupture lifetime) indicate that there is no basis for the strength-life equal rank assumption for

longitudinally loaded unidirectional polymer composites. Moreover, the random number

generator in the Monte Carlo simulation can be seeded such that the same set of fiber

strengths are assigned to compute a set of composite strengths and composite lifetimes. This

method can be used to show that there is no direct correlation between strength and lifetime.

127

6 STRAIN RATE EFFECTS

6.1 INTRODUCTION

Although the immediate purpose of this work is micromechanics-based prediction of stress

rupture lifetime, a very attractive extension is to apply the techniques described here to

predict failure of composite materials under more complex time-dependent loading profiles

such as fatigue and arbitrary strain rates. The question of interest is: Have we captured the

physics of time-dependent problems sufficiently that the work here could be directly applied

to more complex loading profiles? Such a study would also help to expose the merits and

demerits of the models presented in this work. This chapter is a very brief discussion of

some of the challenges associated with applying these techniques to more general loading

profiles, and it begins with a review of the literature on strain rate effects in polymer

composites.

6.2 LITERATURE ON STRAIN RATE EFFECTS

Strain rate effects in composite materials arise from time (and temperature) dependent

constituent properties and damage mechanisms. Polymer matrices are viscoelastic, and glass

fibers are capable of time-dependent failure. High strain rate testing of specimens can give

rise to propagating damage mechanisms such as fiber/matrix debonding, ply delamination,

and matrix cracking. The development of micromechanics-based predictions of strain rate

and temperature dependent properties is extremely challenging. Reifsnider et. al.58 applied

the Monkman-Grant equation to develop relationships between material properties and time

to failure that described experimental behavior over a range of strain rates from quasi-static

to very high strain rate values. They also proposed an equivalence principle between strain

rates and temperature based on the Arrhenius relationship.

Xia et. al.59 introduced a statistical model for the strain rate dependence of fibers. The input

parameters to the model were obtained by tensile impact of bundles of E-glass fibers. They

showed that the model was reliable and the test method feasible. In a later publication, Xia

128

and Xing60 introduced a rate-dependent constitutive equation for tensile impact of

unidirectional composites from statistical analysis of loading unloading tests.

Hayes and Adams61 investigated the rate sensitive strength of unidirectional graphite/epoxy

and glass/epoxy materials under tensile impact loading. The micromechanics of fracture

were studied with scanning electron microscopy. An interesting result of their work was that

the strength of graphite/epoxy specimens decreased with increasing impact rates, while the

strength of glass/epoxy specimens increased with increasing impact rates.

It is difficult to conduct tests at high strain rates because of the inertial problems and dynamic

response of the jigs and fixtures. Okoli and Smith62 conducted tests at low strain rates and

showed that material properties were a linear function of the logarithm of the rate of strain.

They claimed that the material characteristics at high strain rates could be obtained by

extrapolation of the low strain rate data to high strain rates. They studied the Young’s

modulus, tensile and shear strengths, and the shear modulus of glass-epoxy composites.

Ger et. al.63 used a modified form of the block-to-bar type impact test to measure the

mechanical properties of carbon fiber, Kevlar fiber, and carbon/Kevlar hybrid composites at

strain rates up to the order of 103 /sec. They conducted tests on cross-ply and ±45 laminates

and measured tensile strength and peak strain to failure. The greatest increase in dynamic

strength was observed for the ±45 laminates.

Temperature influences the energy absorption during impact of composites. Dutta et. al.64

used a Hopkinson pressure bar apparatus to study impact of a quasi-isotropic graphite/epoxy

laminate over a range of temperatures. C-scanning was used to determine the extent of

damage after impact. They also made qualitative comparisons of the experimental data with

numerical analysis accounting for the extent of matrix stiffening due to temperature. Takeda

et. al.65 performed systematic studies of delamination in centrally impacted composite

laminates. They investigated the effect of impactor geometry, mass, energy, and the

orientation of fabricated laminates on failure. Dee et. al.66 used the Split Hopkinson Pressure

Bar to measure strain rate dependent properties of composites. Compressive yield stress and

129

strain, ultimate strength and strain, and modulus of elasticity were studied at strain rates from

183 /sec to 653 /sec.

6.3 MODELING ARBITRARY LOADING PROFILES

The major stumbling blocks in extending the time-dependent failure model presented in this

dissertation to arbitrary loading profiles are enumerated below:

1. The models presented here are developed for the fiber dominated strength and rupture

lifetime of unidirectional polymer composites. Much of the strain rate dependent data

is measured on laminates having several off-axis plies63 where matrix strength is

critical. In order to investigate off-axis behavior it would be necessary to develop a

time-dependent Monte Carlo simulation (or probabilistic analysis) at the ply or global

level with a methodology to redistribute time-dependent ply level stresses based on

shear-lag or finite element techniques. If fiber failures are still the ultimate failure

mode, then the global analysis could feed into a micromechanics-based simulation at

the local level.

2. The models presented here consider tensile loading only. Very different failure

modes (ex. kink bands67, fiber buckling) and associated stress redistribution occur

when fibers go into compression. Compression loading is an important consideration

for micromechanics-based fatigue life prediction where compressive stresses are

involved.

3. The only damage mechanism considered in this work is fiber failure. Lifetimes under

fatigue and high strain rates may be controlled by ply delamination, matrix cracks in

off-axis plies, fiber/matrix debonding or interphase plasticity. The time-dependent

propagation of these damage mechanisms would have to be modeled. Moreover, the

models have to be based on measurable input quantities. It should be pointed out that

at this time there no consensus on measurement techniques for fiber/matrix interfacial

shear strength and fracture toughness.27,28 Hence, even if models for time-dependent

fiber/matrix interphase behavior could be developed calibrating these models would

be very challenging.

4. Much of the strain rate and fatigue data in the literature deals with glass fiber

composites.62 It is well known that glass fibers undergo time-dependent

130

deformation.68,59 The models developed in this work assume that the fibers are

linearly elastic as is the case with carbon fibers. A load sharing with viscoelastic

fiber properties would be necessary in order to capture the role of glass fibers under

high strain rate and fatigue loading. Indeed, fiber creep may be an important

mechanism in stress rupture of unidirectional glass fiber composites.

5. In addition to the above reasons, it is mathematically very challenging to incorporate

arbitrary loading profiles into the general time dependent load-sharing framework

described in Chapter 5. The direct Laplace inversion approximation used in Chapter

5 works best when all the boundary loads and displacements are step functions of

time applied at t = 0. In order to model arbitrary loading profiles it is necessary to

implement an alternate approximate inversion procedure such as point collocation.57

131

7 SUMMARY AND CONCLUSIONS

This dissertation presents a systematic approach to modeling the tensile strength and stress-

rupture lifetime of unidirectional composite materials loaded in the fiber direction. The work

begins with measurements of quasi-static and time-dependent micromechanical fiber strain

redistribution near fiber fractures in macromodel composites. Single and multiple fiber

fractures are studied. From the model composite measurements the following major

conclusions are drawn:

1. The influence of an isolated fiber fracture is felt primarily by the nearest unbroken

neighboring fibers. Based on this observation, a new form of load sharing called

nearest neighbor load sharing (NNLS) is developed. NNLS is used to model tensile

strength and rupture lifetime.

2. The stress state resulting from multiple fiber fractures is well described by

superposition of the perturbation due to each individual fracture. This observation

justifies the use for influence-function superposition principles to model quasi-static

and time-dependent load sharing.

3. The axial strains on unbroken fibers adjacent to a broken fiber increase with time.

This is experimental verification of the mechanism that leads to failure of

unidirectional polymer composites under stress rupture loading in the fiber direction.

The next phase of work in this dissertation is the development of tensile strength models for

unidirectional polymer composites. A finite element form of NNLS is developed, and it is

incorporated into a Monte Carlo simulation for tensile strength modeling. A much simpler

shear-lag NNLS is also developed. The sources of material variability that are included in

the shear-lag based strength simulation are the fiber strength distribution, non-uniform fiber

placement, variable fiber volume fractions, and initial fiber fractures. The following key

conclusions are drawn from the quasi-static strength modeling effort:

1. The finite element load-sharing agrees very well with model composite

measurements.

2. Although, there are significant differences between shear-lag and finite element

NNLS, it is shown that the simpler shear-lag approach computes a strength

132

distribution that is very comparable to the strength calculated by finite element load

sharing.

3. Model predictions are compared to the experimental strength distribution of a carbon

fiber/polymer matrix composite. Initially, the only material variability included in the

prediction is the fiber strength distribution. For this case, the computed strength

distribution is much narrower than experimentally observed. Including additional

sources of material variability such as distributed fiber volume fractions, initial fiber

fractures, and non-uniform fiber placement in the modeling yield results that are in

excellent agreement with the experimental strength distribution. It is shown that of

all the additional sources of material variability considered, distributed fiber volume

fractions have the greatest effect on the computed strength.

4. A comparison between the strength distribution calculated by applying NNLS and the

more common HVDLS (Chapter 4) is made. It is shown that the NNLS approach

developed in this work yields results that are in better agreement with the

experimental strength distribution.

The final phase of this work deals with modeling the stress rupture lifetime of unidirectional

polymer composites loaded in the fiber direction. A general time-dependent load-sharing

framework is developed by applying shear-lag assumptions. Two different forms of load

sharing are considered: NNLS and HVDLS. The time-dependent load-sharing

methodologies are included into Monte Carlo simulations to compute stress rupture lifetime.

The simulation approach is best suited to address the critical question of material reliability

for a desired lifetime under a given set of external conditions. Comparisons are made to the

rupture lifetime of a [90/03]s carbon fiber/polymer matrix composite. The major conclusions

of the stress-rupture modeling effort are enumerated below:

1. There is qualitative agreement between the model composite measurements and the

time-dependent load-sharing results.

2. Grip section damage is cited as a primary cause of errors in measuring rupture

lifetime of purely unidirectional composites. It is suggested that stress rupture tests

be carried out on specimens with 90 plies on the surface so that the load carrying

core of 0 is not damaged in the grip section.

133

3. Encouraging comparisons are made between predictions and experimental lifetimes

of a [90/03]s carbon fiber/polymer matrix composite.

4. The strength-life equal rank assumption is not valid for modeling the stress-rupture

lifetime of unidirectional polymer composites. It is not necessary to make this

assumption for the lifetime models discussed in this work. Moreover, such an

assumption is incorrect since there are fundamental differences in the mechanisms

that cause fast-fracture and stress-rupture failure. Also, by seeding the random

number generator in the Monte Carlo simulation it can be shown that there is no

direct correlation between fast-fracture strength and stress-rupture lifetime.

5. The reasons for very broad distributions in computed lifetime are addressed. It is

difficult to make a-priori estimations of the effect of certain input parameters on

computed lifetime. It is shown that a narrower fiber strength distribution results in

longer lifetimes with greater variability. Also, changing a certain combination of

input parameters that controls the perturbed length of fiber due to a break does not

produce any change in computed rupture lifetime. Both these parametric studies are

performed at a far-field fiber stress level that results in a 50% probability of

instantaneous failure.

6. Since the load-sharing is time and temperature dependent, lifetime predictions can be

made at different temperatures. Hence, the method presented here can be used to help

understand and predict the role of temperature in accelerated measurement of stress

rupture lifetimes.

7.1 FUTURE WORK

The following directions of future effort are suggested:

1. Develop a finite element based time-dependent load sharing. Make quantitative

comparison of the time-dependent load sharing to experimental measurements on the

model composites presented in Chapter 3.

2. Include additional sources of material variability such as initial fiber fractures,

distributed fiber volume fractions, and non-uniform fiber placement in stress rupture

predictions, and study their effect in detail.

134

3. Make extensive measurements of the rupture lifetime of [90/03]s carbon fiber/polymer

matrix specimens at two different temperatures. Measuring the rupture lifetime at

two different temperatures is an excellent way to investigate the physics of the stress

rupture problem.

4. Include the effect of the interphase in load-sharing, and hence, strength and rupture

predictions. Preferably, this can be done such that measurable inputs can be used to

calibrate the model. The aspects of interest are repeatable and reliable measurements

of interfacial shear strength and fracture toughness, and development of an

experimental and theoretical understanding of the time-dependent propagation of

interfacial damage mechanisms.

5. Develop a ply level load-sharing and simulation approach so that predictions of

strength and rupture lifetime can be made at the laminate level.

6. Finally, model arbitrary loading profiles such as linear strain rates and fatigue. The

issues associated with this challenging task are discussed in Chapter 6.

135

REFERENCES

1 Kelly, A., Tyson, W. R. “Tensile properties of fiber-reinforced metals: copper/tungsten and copper/molybdenum,” Journal of the Mechanics and Physics of Solids, Vol. 13, 1965, pp. 329-350. 2 Huang, Y., Young, R. J., “Interfacial Behavior in High Temperature Cured Carbon Fiber/Epoxy Resin Model Composite,” Composites, Vol. 26, 1995, pp. 541-550. 3 Yallee, R. B., Young, R. J., “Micromechanics of Fiber Fragmentation in Model Epoxy Composites Reinforced with α-Alumina Fibers,” Composites Part A, Vol. 29A, 1998, pp. 1353-1362. 4 Pyrz, R., Nielsen, A. S. “In Situ Monitoring of Residual Strains in Model Composites,” in Carpinteri, A., Brebbia, C. A., Ed. Damage and Fracture Mechanics Computer Aided Assessment and Control, Proceedings of Fifth International Conference on Computer Aided Assessment and Control of Localized Damage and Fracture Mechanics, Italy, 1998, pp. 617-625. 5 Zhao, D., Botsis, J. “Experimental and Numerical Studies in Model Composites Part I: Experimental Results,” International Journal of Fracture, Vol. 82, 1996, pp. 153-174. 6 Goutianos, S., Peijs, T. “Fatigue Damage Mechanisms in Carbon-Epoxy Multi-Fiber Model Composites,” Advanced Composites Letters, Vol. 10, No. 1, 2001, pp. 21-32. 7 P. W. J. van den Heuvel, An Experimental and Numerical Investigation into Failure Phenomena in Multi-Fiber Microcomposites, Technische Universiteit Eindhoven, Eindhoven, 1998. 8 Chohan, V., Galiotis, C., “Effects of Interface, Volume Fraction and Geometry on Stress Redistribution in Polymer Composites under Tension,” Composites Science and Technology, Vol. 57, 1997, pp. 1089-1101. 9 Phoenix, S. L., Schwartz, P., Robinson IV, H. H. “Statistics for the Strength and Lifetime in Creep-Rupture of Model Carbon/Epoxy Composites,” Composites Science and Technology, Vol. 32, 1988, pp. 81-120. 10 Otani, H., Phoenix, S. L., Petrina, P. “Matrix Effects on Lifetime Statistics for Carbon Fiber-Epoxy Microcomposites in Creep Rupture,” Journal of Materials Science, Vol. 26, 1991, pp. 1955-1970.

136

11 Beyerlein, I. J., Phoenix, S. L. “Statistics for the Strength and Size Effects of Microcomposites with Four Carbon Fibers in Epoxy Resin,” Composites Science and Technology, Vol. 56, 1996, pp. 75-92. 12 Carman, G. P., Lesko, J. J., Razvan, A., Reifsnider, K. L. “Model Composites: A Novel Approach for the Evaluation of Micromechanical Behavior,” Composite Materials: Fatigue and Fracture, Fourth Volume, ASTM STP 1156, 1993, pp. 381-400. 13 Hedgepeth, J. M., Van Dyke, P. “Local Stress Concentrations in Imperfect Filamentary Composite Materials,” Journal of Composite Materials, Vol. 1, 1967, p. 294-309. 14 Cox, H. L. “The Elasticity and Strength of Paper and Other Fibrous Materials,” British Journal of Applied Physics, Vol. 3, 1952, pp. 72-79. 15 Landis, C. M., McGlockton, M. A., McMeeking, R. M. “An Improved Shear Lag Model for Broken Fibers in Composite Materials,” Journal of Composite Materials, Vol. 33, No. 7, 1999, pp. 667-680. 16 Nedele, M. R., Wisnom, M. R. “Three-dimensional Finite Element Analysis of the Stress Concentration at a Single Fiber Break,” Composites Science and Technology, Vol. 51, 1994, pp. 517-524. 17 Nedele, M. R., Wisnom, M. R. “Stress Concentration Factors Around a Broken Fiber in a Unidirectional Carbon Fiber-Reinforced Epoxy,” Composites, Vol. 27, 1994, pp. 549-557. 18 Landis, C. M., McMeeking, R. M. “Stress Concentrations in Composites with Interface Sliding, Matrix Stiffness, and Uneven Fiber Spacing Using Shear Lag Theory,” International Journal of Solids and Structures, Vol. 36, 1999, pp. 4333-4361. 19 Beyerlein, I. J., Landis, C. M. “Shear-Lag Model for Failure Simulations of Unidirectional Fiber Composites Including Matrix Stiffness,” Mechanics of Materials, Vol. 31, 1999, pp. 331-350. 20 Lagoudas, D. C., Hui, C., Phoenix, S. L. “Time Evolution of Overstress Profiles Near Broken Fibers in a Composite with a Viscoelastic Matrix,” International Journal of Solids and Structures, Vol. 25, No. 1, 1989, pp. 45-66. 21 Mason, D. D., Hui, C., Phoenix, S. L. “Stress Profiles Around a Fiber Break in a Composite with a Nonlinear, Power Law Creeping Matrix,” International Journal of Solids and Structures, Vol. 29, No. 23, 1992, pp. 2829-2854.

137

22 Beyerlein, I. J., Phoenix, S. L., Raj, R. “Time Evolution of Stress Redistribution Around Multiple Fiber Breaks in a Composite with Viscous and Viscoelastic Matrices,” International Journal of Solids and Structures, Vol. 35, No. 24, 1998, pp. 3177-3211. 23 Nairn, J. A. “On the Use of Shear-Lag Methods for Analysis of Stress Transfer in Unidirectional Composites,” Mechanics of Materials, Vol. 26, 1997, pp. 63-80. 24 Carman, G. P., Lesko, J. J., Reifsnider, K. L. “Micromechanical Analysis of Fiber Fracture,” Composite Materials: Fatigue and Fracture, Fourth Volume, ASTM STP 1156, 1993, pp. 430-452. 25 Case, S. W., Carman, G. P., Lesko, J. J., Fajardo, A. B., Reifsnider, K. L. “Fiber Fracture in Unidirectional Composites,” Journal of Composite Materials, Vol. 29, No. 2, 1995, pp. 208-228. 26 Case, S. W., Reifsnider, K. L. “Micromechanical Analysis of Fiber Fracture in Unidirectional Composite Materials,” International Journal of Solids and Structures, Vol. 33, No. 26, 1996, pp. 3795-3812. 27 Drzal, L. T., Herrera-Franco, P. J. “Comparison of methods for the measurement of fibre/matrix adhesion in composites,” Composites, Vol. 23, No. 1, 1992, pp. 2-27 28 Narkis, M., Chen, E. J. H., Pipes, R. B. “Review of methods for characterization of interfacial fiber-matrix interactions,” Polymer Composites, Vol. 9, No. 4, 1988, pp. 245-251. 29 Tsai, S. W., Hahn, H. T. Introduction to Composite Materials, Technomic Publishing Company, Inc., 1980. 30 Rosen, B. W. “Tensile Failure of Fibrous Composites,” AIAA Journal, Vol. 2, No. 11, 1964, pp. 1985-1991. 31 Rosen, B. W. “Mechanics of Composite Strengthening,” in Fiber Composite Materials, papers presented at a seminar of the American Society for Metals (Oct. 1964), 1965, pp. 37-75. 32 Batdorf, S. B. “Tensile strength of unidirectional reinforced composites—I,” Journal of Reinforced Plastics and Composites, Vol. 1, 1982, pp. 153-167. 33 Batdorf, S. B. “Tensile strength of unidirectional reinforced composites—II,” Journal of Reinforced Plastics and Composites, Vol. 1, 1982, pp. 165-176.

138

34 Gao, Z., Reifsnider, K. L. “Micromechanics of Tensile Strength in Composite Systems,” Composite Materials: Fatigue and Fracture, Fourth Volume, ASTM STP 1156, 1993, pp. 453-470. 35 Smith, R. L., Phoenix, S. L., Greenfield, M. R., Henstenburg, R. B., Pitt, R. E. “Lower-Tail Approximations for the Probability of Failure of Three-Dimensional Fibrous Composites with Hexagonal Geometry,” Proceedings of the Royal Society of London A, Vol. 388, 1983, pp. 353-391. 36 Zhou, S. J., Curtin, W. A. “Failure of Fiber Composites: A Lattice Green Function Model,” Acta Metallurgica et Materialia, Vol. 43, No. 8, 1995, pp. 3093-3104. 37 Ibnabdeljalil, M., Curtin, W. A. “Strength and Reliability of Fiber-Reinforced Composites: Localized Load-Sharing and Associated Size Effects,” International Journal of Solids and Structures, Vol. 34, No. 21, 1997, pp. 2649-2668. 38 Ibnabdeljalil, M., Curtin, W. A. “Strength and Reliability of Notched Fiber-Reinforced Composites,” Acta Materialia, Vol. 45, No. 9, 1997, pp. 3641-3652. 39 Curtin, W. A., Takeda, N. “Tensile Strength of Fiber-Reinforced Composites: I. Model and Effects of Local Fiber Geometry,” Journal of Composite Materials, Vol. 32, No. 22, 1998, pp.2042-2059. 40 Curtin, W. A., Takeda, N. “Tensile Strength of Fiber-Reinforced Composites: II. Application to Polymer Matrix Composites,” Journal of Composite Materials, Vol. 32, No. 22, 1998, pp.2060-2081. 41 Landis, C. M., Beyerlein, I. J., McMeeking, R. M. “Micromechanical Simulation of the Failure of Fiber Reinforced Composites,” Journal of the Mechanics and Physics of Solids, Vol. 48, 2000, pp. 621-648. 42 Mahesh, S., Beyerlein, I. J., Phoenix, S. L. “Size and Heterogeneity Effects on the Strength of Fibrous Composites,” Physica D, Vol. 133, 1999, pp. 371-389. 43 Phoenix, S. L., Beyerlein, I. J. “Statistical Strength Theory for Fibrous Composite Materials,” in Kelly, A., Zweben, C., Ed.-in-chief, Chou, T. W. Ed. Comprehensive Composite Materials, Vol. 1, Chapter 1.20, 2000. 44 Lifshitz, J. M. “Time-Dependent Fracture of Fibrous Composites,” in Broutman, L. J. Ed. Fracture and Fatigue, Vol. 5, Chapter 6, 1974, pp. 249-311. 45 Lifshitz, J. M., Rotem, A. “Time-Dependent Longitudinal Strength of Unidirectional Fibrous Composites,” Fibre Science and Technology, Vol. 3, pp. 1-20.

139

46 Lifshitz, J. M. MED Report No. 33, Department of Materials Engineering, Technion, Israel, 1971. 47 Christensen, R. M. “Lifetime Predictions for Polymers and Composites under Constant Load,” Journal of Rheology, Vol. 25, No. 5, 1981, pp. 517-528. 48 Glaser, R. E., Christensen, R. M., Chiao, T. T. “Theoretical Relations Between Static Strength and Lifetime Distributions for Composites: An Evaluation,” Composites Technology Review, Vol. 6, No. 4, 1984, p. 164-167. 49 Ibnabdeljalil, M., Phoenix, S. L. “Creep Rupture of Brittle Matrix Composites Reinforced with Time Dependent Fibers: Scalings and Monte Carlo Simulations,” Journal of the Mechanics and Physics of Solids, Vol. 43, No. 6, 1995, pp. 897-931. 50 Iyengar, N., Curtin, W. A. “Time-Dependent Failure in Fiber-Reinforced Composites by Fiber Degradation,” Acta Materialia, Vol. 45, No. 4, 1997, pp. 1489-1502. 51 Iyengar, N., Curtin, W. A. “Time-Dependent Failure in Fiber-Reinforced Composites by Matrix and Interface Shear Creep,” Acta Materialia, Vol. 45, No. 8, 1997, pp. 3419-3429. 52 Halverson, H. G., Curtin, W. A. “Stress Rupture of Nextel 610/Alumina-Yttria Composites,” Ceramic Engineering and Science Proceedings, Vol. 21, No. 3, 2000, pp. 556-566. 53 Curtin, W. A., Xia, Z., Okabe, T. “Stress Concentrations in Fiber Composites: Shear Lag vs. Finite Element Models,” 16th Annual Technical Conference, American Society for Composites, September 2001, Blacksburg, VA. 54 Hedgepeth, J. M. “Stress Concentrations in Filamentary Structures,” NASA TN D-882, Langley Research Center, 1961. 55 Russell, B. E. “Material Characterization and Life Prediction of a Carbon Fiber/Thermoplastic Matrix Composite for Use in Non-Bonded Flexible Risers,” Master’s Thesis, Virginia Polytechnic Institute and State University, 2000. 56 Wimolkiatisak, A. S., Bell, J. P. “Interfacial Shear Strength and Failure Modes of Interphase-Modified Graphite-Epoxy Composites,” Polymer Composites, Vol. 10, No. 3, 1989, pp. 162-172. 57 Schapery, R. A. “Stress Analysis of Viscoelastic Composite Materials,” Journal of Composite Materials, Vol. 1, 1967, pp. 228-267.

140

58 Reifsnider K., Mahieux C., Walther, B., Sun, F. “Strain rate and temperature effects in polymeric matrices for composite,” Recent Developments in Durability Analysis of Composite Systems, Cardon, Fukuda, Reifsnider & Verchery, Eds., 2000, pp. 229-236. 59 Xia, Y., Yuan, J., Yang, B. “A statistical model and experimental study of the strain-rate dependence of the strength of fibers,” Composite Science and Technology, Vol. 52, No. 4, 1994, pp. 499-504. 60 Xia, Y., Xing, W. “Constitutive equation for unidirectional composites under tensile impact,” Composites Science and Technology, Vol. 56, No. 2, 1996, pp. 155-160. 61 Hayes, S. V., Adams, D. F. “Rate sensitive tensile impact properties of fully and partially loaded unidirectional composites,” Journal of Testing and Evaluation, JTEVA, Vol. 10, No. 2, 1982, pp. 61-68. 62 Okoli, O. I., Smith, G. F. “High strain rate characterization of a glass/epoxy composite,” Journal of Composite Technology & Research, JCTRER, Vol. 22, No. 1, January 2000, pp. 3-11. 63 Ger G. S., Kawata, K., Itabashi, M. “The mechanical properties in high velocity tension of CFRP, KFRP and CF/KF hybrid composites,” Proceedings of the Japan-U.S. Conference on Composite Materials-V, Tokyo, 1990, pp. 95-102. 64 Dutta, P. K., Faran, K. J. L. “Influence of low temperature on energy absorption in laminated composites,” Proceedings of ICCM/9, Composites Behaviour. Vol. V, 1993, pp. 311-320. 65 Takeda, N., Malvern, L. E.; Sierakowski, R. L. “Studies of impacted glass fiber-reinforced composite laminates,” SAMPE Quarterly, 1981, pp. 9-16. 66 Dee, A. T.; Vinson, J. R.; Gibson, R. F. “The effect of high strain rate on the compressive mechanical properties of unidirectional E-glass/epoxy (3M Scotchply 1003),” Proceedings of the 12th Technical Conference of the American Society for Composites, 1997, pp. pp. 399-408. 67 Budiansky, B., Fleck, N. A. “Compressive failure of fibre composites,” Journal of the Mechanics and Physics of Solids, Vol. 41, 1993, pp. 183-211. 68 Lubin, G., Editor. Handbook of fiberglass and advanced plastics composites, 1969.

141

VITA

Tozer Jamshed Bandorawalla is the son of Jamshed and Katayune. He was born on November 21, 1973 in Bombay, India. Tozer attended high school at Jai Hind College in Bombay, and completed his high schooling in 1991. He subsequently enrolled in the Mechanical Engineering baccalaureate program at Mangalore University in Manipal, Karnataka, India. Tozer graduated with a Bachelor of Engineering degree in 1995. He decided to pursue graduate studies in the United States and received a scholarship to attend the University of Missouri-Rolla starting 1995. In 1997, Tozer obtained a Master of Science degree in Engineering Mechanics from the University of Missouri-Rolla. He began his doctoral program in Engineering Mechanics at Virginia Polytechnic Institute and State University the same year. He graduates with a doctoral degree in March 2002. Tozer’s work over the past six years has dealt with composite materials. His research involves durability of composite materials, micromechanics, experimental methods, material testing, and computational mechanics.