Fatigue Damage Modelling of Composite Material

8/9/2019 Fatigue Damage Modelling of Composite Material

1/97

Contents

Abstract ii

Resume iii

Acknowledgments iv

Content v

List of Figures ix

List of Tables xiv

I. Introduction I

1.1 Literature Review 2

1.2 Motivations and b!ectives 4

2. "roblem #tatement 6

2.1 Mec$anics of Fatigue Failure 6

2.2 %escri&tion of t$e "roblem 8

2.3 Re'uirement and Modeling #trateg( I I

2.) #ummar( I 2

*. #tress Anal(sis I 3

*.1 Free +dge +ffects, A Review I 4

*.2 "roblem #tatement I 6 *.* Finite +lement Anal(sis I 6

*.*.1 Material "ro&erties I -

*.*.2 Constitutive +'uations I 8

*.*.* Material onlinearities I 9

*.*.) Material rientation 22

*.*.# #$a&e Functions 2 4

3.3.6 %erivation of +lement Matrices 25

*.*.- Fast Formation of +lement Matrices 2 8

*.*./ Coordinate #(stems 2 9*.*.0 ound r( Conditions 3 0

*.*.13 Met$od of solution for a s(stem of nonlinear Algebraic +'uations 32

*.) #tress Anal(sis Results 32

*.).1 +ffects of Material onlinearit( 33

*.).1.1 Cross4"l( 53603787 33

v


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*.).1.2 Cross4"l( 50363787

*.).1.* Angle4"l( 59):6441:78.

*.).1.) #ummar( of t$e +ffects of Material onli11earit(

*.).2 +ffects of olt Load

*.).2.1 Cross4"l( Laminates 5360378. and 5036378.

*.).2.2 Angle4"l( Laminate 59):6441:787

*.: #ummar(

33

3-1

3-1

3535

3535

). Failure Anal(sis).1 An Introduction to Anal(sis

).2 #tatic Failure Criteria

).2.1 Fiber Tension #tatic Failure Mode

).2.2 Fiber Com&ression #tatic Failure Mode

).2.* Fiber4Matri; #$earing #tatic Failure Mode

).2.) Matri; Tension #tatic Failure Mode

).2.: Matri; Com&ression #tatic Failure Mode

).2.< ormal Tension #tatic Failure Mode

).2.- ormal Com&ression #tatic Failure Mode

).* Fatigue Failure Criteria

).*.1 Fiber Tension Fatigue Failure Mode

).*.2 Fiber Com&ression Fatigue Failure Mode

).*.* Fiber4Matri; #$earing Fatigue FailureMode

).*.) Matri; Tension Fatigue Failure Mode).*.: Matri; Com&ression Fatigue Failure Mode

).*.< ormal Tension Fatigue Failure Mode

).*.- ormal Com&ression Fatigue Failure Mode

).) #ummar(

-13

-13

-16

-18

-18

-18-19

-19

50

5()

51

5-1

5-1

55

5556

56

56

57

5. Material "ro&ert( %egradation

58

5.1 Ma

terial "ro&ert( %egradation rules 59

:.2 #udden Material "ro&ert( %egradation rules 60

:.2.1 Fiber Tension "ro&ert( %egradation 60

:.2.2 Fiber Com&ression "ro&ert( %egradation 61

:.2.* Fiber4Matri; #$earing "ro&ert( %egradation 61

:.2.) Matri; Tension "ro&ert( %egradation 62

5.2.5 Matri; Com&ression "ro&ert( %egradation

63


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5.2.6 ormal Tension "ro&ert( %egradation

63

5.2.7 ormal Com&ression "ro&ert( %egradation

64

vi


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5.2.8 #ummar( of #udden Material "ro&ert( %egradation Rules 65

5.3 =radual Material "ro&ert( %egradation rules 65

:.*.l ormali>ed #trengt$ %egradation Model 65

:.*.2 ormali>ed #tiffness %egradation rvlodel 71

:.) ormali>ed Fatigue Life Model 74

:.).I Mcdification of t$e Life model for #$ear Fatigue Conditions --

:.: =enerali>ed Material "ro&ert( %egradation Tec$ni'ue 78

5.6 Summary 79


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-.


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e List of Figures

Chapter 2

Fig.2.1

Fig. 2.2

=eometr( of a laminated com&osite &late wit$ a circular $ole.sub!ected to @a? &in loading7 and @b? bolt loading 9

%iffcrcn, failure mec$anisms of &in6bolt4loaded com&ositelaminates JO

Chapter 3

Fig.3.1

Fig.3.2

Fig.3.3

Laminate geometr( and edge effects @after "agano and "i&es5-08? 14

Finite element mes$ of t$e &roblem7 s$owing large number of clements at t$e $ole boundar( 16 T$ree4dimensional geometr( of a la(er of com&osite material 17

Fig.3.4

onlinear elastic in4&lane s$ear stress4strain @cr 'Y 4 E ;(

Fig. 3.5

Fig. 3.6

Fig. 3.7

Fig. 3.8

Fig. 3.9

Fig.3.10

Fig. 3.11

Fig.3.12

Fig.3.13

Fig.3.14

Fig. 4.1

Fig. 4.2

Fig. 4.3


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be$aviour ofA#)6*:314ed residual stiffness of a unidirectional &l( under longitudinal tensile fatigue loading conditions @using +'.:.1/? 92

ormali>ed residual strengt$ of a unidirectional &l( under longitudinal tensile fatigue loading conditions @using +'.5.16) 93

x


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xi

Fig. 7.31 Fig.7.32

Fig. 7.9 Gnidirectional 3D &lies7 failed under tensile static and fatigue loading in t$e fiber direction 93

Fig. 7.10 Fiber com&ression s&ecimen 95

Fig. 7.11 T(&ical stress4strain be$aviour of a unidirectional &l( under static com&ressive loading in t$e fiber direction @using back to

back strain gages? 95 Fig. 7.12 #tatic stiffness of unidirectional &l( under longitudinal

compressive stress 96

Fig. 7.13 #tatic strengt$ of unidirectional &l( under longitudinalcompressive stress 96

Fig. 7.14 ormali>ed residual strengt$ of a unidirectional &l( under longitudinal com&ressive fatigue loading conditions @using+'. :.1


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Fig. 7.33

Fig. 7.34

Fig. 7.35

Fig. 7.36

Fig. 7.37

Fig. 7.38

Fig. 7.39

Fig.7.40

Fig.7.41

Fig. 7.42

Fig.7.43

Fig.7.44

Fig.

7.45

Fig.7.46

Fig.7.47

Fig. 8.1

Fig. 8.2

Fig. 8.3Fig. 8.4

Fig. 8.5

T$ree4rail s$ear fi;ture modified for gri&&ing and e'ui&&edwit$ e;tensometer

Modified in4&lane s$ear s&ecimen

T(&ical s$ear stress4strain be$aviour of t$e unidirectionalmaterial

#tatic stiffness of notc$ed s&ecimens under in4&lane s$ear stress

"arameter of material nonlinearit( @ ? of notc$ed s&ecnnensunder in4&lane s$ear stress

#tatic strengt$ of notc$ed s&ecimens under in4&lane s$ear stress

ormali>ed residual stiffness of a unidirectional materialunder in4&lane s$ear fatigue loading conditions @using +'.:.1/?

ormali>ed residual strengt$ of a unidirectional materialunder in4&lane s$ear fatigue loading conditions @using +'.:.1


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107 1118

108

1119

llJ9

J/11

I Iii

I II

II I

J/2

J/3 J/3

J/4

J/5

J/5 J/6

J/7

J/9

/2()

122

124

126


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xiii

Fig. 8.6 Com&arison of calculated static failure load wit$ e;&erimentalresults for 5*37778 off4a;is s&ecimens 127

Fig. 8.7 Fatigue life @#4 ? curve of t$e 5*37778 off4a;is s&ecimens 127

Fig. 8.8 ff4a;is 5*377.8 s&ecimens7 failed under tensile static @to&? andfatigue @below? loading 128

Fig. 8.9 Fi;ture for &in6bolt4loaded com&osite laminates 131 Fig. 8.10 "rogressive static failure of a &in4loaded 5 i0378. cross4&l(

laminate 133

Fig. 8.11 Final static failure mec$anisms of t$e &in4loaded 5 9):764):787angle4&l( laminate @to&?7 and @03) 3787cross4&l( laminate@below? 134

Fig. 8.12 Final static failure mec$anisms of t$e bolt4loaded 5360378.cross4&l( laminate 134

Fig. 8.13 Final static failure mec$anisms of t$e &in4loaded laminateswit$ different geometries )e*# and +!#, 135

Fig. 8.14 Fatigue life curve of &in4loaded cross4&l( [ i90!". laminates 137

Fig. 8.15 Fatigue life curve of bolt4loaded cross4&l( 53603787 laminates 137 Fig. 8.16 Fatigue life curve of &in4loaded cross4&l( 5036378.laminates 138

Fig. 8.17 Fatigue life curves of &in4loaded @360378.and @0363787laminates 139

Fig. 8.18 Fatigue life curve of &in4loaded angle4&l( 5 9):64):787laminates 139

Fig. 8.19 T(&ical final failure mec$anisms of t$e &in4loaded 5360378.and 50363.87cross4&l( and f9):64):.87 angle4&l( laminates 140

Fig. 8.20 Residual fatigue strengt$ curve of &in4loaded cross4&l(@036378.laminates @simulation and e;&eriments? 141

Fig. 8.21 T(&ical normali>ed residual fatigue &ro&ert( @stiffness or strengt$? of a unidirectional &l( under unia;ial fatigueloading conditions @not to scale? 142

Fig. 8.22 ormali>ed residual stiffness of a unidirectional &l( under longitudinal tensile fatigue loading conditions @using +'. /.:? 143

Fig. 8.23 ormali>ed residual strengt$ of a unidirectional &l( under longitudinal tensile fatigue loading conditions @using +'. /.:? 144

Fig. 8.24 ormali>ed residual strengt$ of a unidirectional &l( under longitudinal com&ressive fatigue loading conditions @using+'. /.:? 144

Fig. 8.25 Residual fatigue strengt$ curve of &in4loaded cross4&l(@0363787laminates @modified simulation and e;&eriments? 145

Fig. 8.26 Fatigue life of &in4loaded 'uasi4isotro&ic 536J):60387 laminates@simulation b( t$e m$#%i and e;&eriments b( Kerrington and#abbag$ian 5:28? 146

Fig. 8.27 Fatigue life of bolt4loaded 'uasi4isotro&ic @36J):60387laminates @simulation b( t$e m$#e! and e;&eriments b(Kerrington and #abbag$ian @:28? 147

Fig. 8.28 e$aviour of a com&osite laminate under two different stresslevels 148


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List of Tables

Chapter 3

-a !e 3.1

-a !e 4.1

-a !e 4.2

-a !e 4.3

Material &ro&erties of A#)6*:314< 18

Chapter 4

%ifferent stress anal(sis tec$ni'ues and failure criteria used b(different aut$ors to anal(>e &in4loaded com&osite laminates

%ifferent magnitudes of c$aracteristic distances used in t$eliterature 45

A summar( of different load ratios utili>ed b( aut$ors 53

Chapter 5

-a !e 5.1

-a !e 7.1

-a !e 7.2

A summar( of different strengt$ degradation models 69

Chapter -

#&ecifications of test s&ecimens 88

Com&ression testing met$ods 94

Chapter 8

-a !e 8.1 %ifferent bia;ial test met$ods and t$eir in$erent difficulties 121

-a !e 8.2 Fatigue life results from t$e e;&eriments and t$e te%h i&ue 129

-a !e 8.3 Com&arison of failure initiation load b( linear t$eor(.nonlinear t$eor( and e;&erimental results 132

-a !e 8.4 Com&arison of final failure load b( linear t$eor(7 nonlinear t$eor( and e;&erimental results 132

-a !e 8.5 Final failure load for different &in4loaded com&osite laminateswit$ different geometries7 simulated b( model and measured

b( e;&eriments 135

-a !e 8.6 Residual fatigue life of &in4loaded 59):64):78. angle4&l(laminates7 measured b( e;&eriments and &redicted b( Miner rule and m$#e! 150

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Chapter 1

Introduction

T$e word fatigue is defined b( t$e A#M 5l8 as t$e &$enomenon leading to fracture under

re&eated or fluctuating stresses $aving a ma;imum value less t$an t$e ultimate static strengt$ of

t$e material. "er$a&s "oncelet in 1/*0 was t$e first &erson w$o called t$is &$enomenon fatigue

528. ne of t$e earliest &a&ers on fatigue of metals was &ublis$ed in 1/


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Chapter 1 ! r$#u%ti$ /////////////////////////////////////////////////////////////////////////////////////////////////////////2

1.1 Literature review

efore beginning t$is section. it s$ould be &ointed out t$at t$is t$esis covers numerous

to&ics. Kence7 combining a literature review of all t$e different to&ics covered b( t$e t$esis in t$is

section would make it lengt$( and confusing. Instead. t$is section reviews &a&ers related onl( to

t$e main sub!ect of t$e t$esis7 i.e.7 fatigue modeling of com&osite laminates wit$ or wit$out stress

concentrations. "a&ers related to s&ecific areas covered b( t$e t$esis arc rcvicwed and discussed in

detail in relevant c$a&ters.

Among t$e different t(&es of com&osite materials. &ol(mer matrices reinforced wit$

gra&$ite fibers are used e;tensivel( in aeros&ace industries. %ue to t$e large difference between

t$e mec$anical &ro&erties of gra&$ite fiber and e&o;( matri;7 t$e resulting com&osite materials

s$ow $ig$l( ort$otro&ic be$aviour. Also7 t$e failure modes of unidirectional com&osites under different fatigue loading conditions7 suc$ as tension. com&ression7 in4&lane s$ear or out4of4&lane

s$ear in fiber or matri; directions7 are all com&letel( different. Moreover a com&osite laminate7

consisting of &lies wit$ different orientations and different stacking se'uence7 $as a more

com&licated fatigue be$aviour. For instance7 delamination or se&aration between la(ers is a ver(

com&licated failure mec$anism w$ic$ is s&ecial to com&osite laminates. +;istence or stress

concentrations multi&lies t$e comit( of fatigue failure of com&osite laminates and conse'uentl(

t$e modeling is even more difficult. It is necessar( to mention $ere t$at t$e words stress

concentrations are e'uivalent to words suc$ as notc$es7 slots7 slits7 cracks7 o&en $oles and

&in6bolt4loaded $oles used b( aut$ors in t$e literature. ( considering all t$e aforementioned

comities7 some of w$ic$ are in$erent to com&osites7 t$e fatigue modeling of t$ese materials is

a ver( tedious task. #&ecificall(7 t$e fatigue of com&osite laminates wit$ stress concentrations is

one of t$e most com&licated sub!ects w$ic$ is still under researc$.

T$ere are some remarkable efforts in t$e literature stud(ing t$e fatigue be$aviour of

unidirectional &lies and laminated com&osites @see review &a&ers 5)408?. T$ree &rinci&al

a&&roac$es are used for &redicting t$e fatigue life of com&osite materials, residual strengt$7

residual stiffness and em&irical met$odologies. In eac$ categor(7 &$enomenological7 mec$anistic7statistical and mi;ed met$ods are utili>ed b( different aut$ors. Most of t$ese works arc devoted to

t$e fatigue be$aviour of unidirectional &lies or sim&le laminates7 w$ile t$e number of researc$

&a&ers on fatigue of notc$ed laminates is more limited. Gnfortunatel(7 t$e valuable information

&rovided b( different aut$ors for t$e fatigue of sim&le com&osites wit$out stress concentrations

can not be directl( used for t$e fatigue anal(sis of com&osite laminates wit$ stress concentrations.


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Chapter 1 lnroduction //////////////////////////////////////////////////////////////////////////////////////////////////////////// 3

Kowever7 t$e( can be considered a ba ic knowledge for understanding t$e mec$anics of fatigue

failure in order to develo& suitable models for t$e fatigue failure anal(sis of com&osites wit$ stress

concentrations.

ne of t$e earliest &a&ers in fatigue of com&osite laminates wit$ stress concentrations was &ublis$ed b( wen and is$o& 5I 8. Toda(7 t$ere are several e;&erimental 5114228 and anal(tical

52*4*08 researc$ &a&ers in t$is field. Kowever7 t$e &resent state of knowledge is still in t$e stage

of develo&ment and im&rovement. Among t$e e;isting models7 t$e semi4em&irical model of

ulkarni et a!. 52-4* I87 t$e critical element model of Reifsnider et a!. 5*24*)87 and t$e damage

growt$ model of #&earing et a!. 5*:4*08 are all s(stematic and met$odological7 t$erefore t$e( are

wort$w$ile for discussion. T$e semi4em&irical model of ulkarni et a!. 52-72/8 was develo&ed

ba ed on a mec$anistic wearout 5)38 framework. T$e wearout &$iloso&$( treats fatigue damage as

t$e growt$ of &re4e;isting flaws or discontinuities in a material. ( growing t$e flaws7 t$e

strengt$ of t$e material decreases and reac$es to t$e level of t$e state of stress and finall(7

catastro&$ic fatigue failure occurs. T$e semi4em&irical model was evaluated b( e;&erimental

tec$ni'ues 52087 but little correlation between t$e t$eor( and e;&eriments was found. Later7 t$e(

modified 5*38 t$eir model b( considering t$e effect of interlaminar s$ear stress in t$e failure

anal(sis. Kowever7 t$e e;&erimental results still did not s$ow an( correlation wit$ t$e results of

t$e anal(sis. In a subse'uent attem&t 5*187 t$e stress anal(sis &art of t$eir model was modified7

but unfortunatel( no attem&t was made to correlate t$e results of t$at anal(sis wit$ e;&erimental

data. Alt$oug$ t$e semi4em&irical model of ulkarni et a!. 52-4*18 never s$owed a correlation

wit$ e;&erimental results7 t$ere are $owever man( interesting &oints and valuable information in

t$eir work. T$e critical element model of Reifsnider et a!. @*24*)8 was develo&ed based on a

mec$anistic a&&roac$. T$eir mec$anistic a&&roac$ is based on micromec$anical re&resentations of

strengt$. #endeck(! 5


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Chapter I lnroduction ......... . .................. .. ..... ...... .. .. .... .. .....................6

t$e s$a&e of t$e notc$ is &re4defined. In addition. t$eir model is onl( ca&able of considering c(clic

tensile loading. T$us all e;isting models for fatigue anal(sis of com&ositN s $ave limitations w$ic$

make t$em unsuitable for general use.

1.2 Motivation and Objectives

+;&erimental c$aracteri>ation of fatigue be$aviour of com&osite materials is time

consuming and e;&ensive. Moreover. generali>ation b( e;tending and e;tra&olating of

e;&erimental results for com&osite laminates is not straig$tforward and sometimes. not &ossible.

T$erefore7 modeling is an attractive tool for saving time and e;&enses in t$e fatigue design of

com&osite laminates. ( considering t$e comit( of t$e fatigue failure of com&osite materials.t$e level of &resent knowledge and s$ortcomings of e;isting models. t$e necessit( of develo&ment

of more general models wit$ less limitations is 'uite obvious. T$e main o !cctivc of t$is researc$

is to establis$ a model to simulate t$e fatigue be$aviour of com&osite laminates under general

conditions @loading7 geometr(7 etc.?7 using t$e results of various t(&es of unia;ial fatigue

e;&eriments of unidirectional &lies. T$e model s$ould be ca&able of simulating c(cle4b(4c(cle

fatigue be$aviour of com&osite laminates wit$ or wit$out stress concentrations. T$ere arc several

&oints and &arameters w$ic$ s$ould be taken into account in order to ensure t$e gcncra.iit( of t$e

model. A general model must not be limited to a s&ecial geometr(7 la(4u&. loading condition7

boundar( condition7 loading ratio or loading se'uence.

"rogressive damage modeling7 w$ic$ is a widel( used failure anal(sis tec$ni'ue7 $as been

successfuli( utili>ed to stud( t$e be$aviour of com&osite laminates under static loading 5)14)*8. In

&rogressive damage modeling7 stress anal(sis7 failure anal(sis and material &ro&ert( degradation

are t$ree im&ortant com&onents. T$e states of stress in t$e com&osite materials arc found b( a

stress anal(sis tec$ni'ue7 e.g.7 finite element met$od. T$en t$e failure anal(sis is &erformed b(

evaluating t$e stresses using a set of failure criteria7 ca&able of distinction of different failure

modes. Finall(7 t$e material &ro&erties of failed regions are c$anged b( a set of degradation ru Oes.

"rogressive damage modeling allows t$e detailed stud( of damage &rogression from damage

initiation to t$e final catastro&$ic state. Kowever7 so far t$is tec$ni'ue $as been used to stud(

com&osite laminates under static t(&es of loading. In t$is researc$7 t$e conce&t of &rogressi vc

damage modeling is e;tended and t$e framework of pr$gre''ive (atigue #amage m$#e!i g is

establis$ed and utili>ed for fatigue failure anal(sis of com&osite laminates wit$ or wit$out stress

concentrations


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Chapter I Jnroduction ///////////////////////////////////////////////////////////////////////////////////////////////////5

ne com&licated e;am&le of a com&osite laminate wit$ stress concentrations is a &in6bolt

loaded com&osite &late. e$aviour of &in6bolt4loaded com&osite laminates under static loading

situation $m been studied e;tensivel( @see 5)18 for a com&re$ensive review?. Kowever7 after a

t$oroug$ literature review in fatigue of com&osite laminates7 it is found t$at fatigue be$aviour of

&in6bolt4loaded com&osite laminates $as beer studied muc$ Pess e;tensivel(. All of t$e e;istinginvestigations 5))4:)8 are &rimaril( e;&erimental and restricted to a s&ecial t(&e of loading7 s&ecific

laminate configuration7 certain geometr(7 etc. T$erefore t$e &in6bolt4loaded com&osite &late under

fatigue loading conditions7 is selected as a reasonable and sufficientl( difficult e;am&le for

evaluating t$e m$#e! develo&ed in t$is stud(. Moreover7 b( considering t$e e;isting limited

researc$ conducted in t$e field of &in6bolt4loaded com&osite &lates7 t$e results obtained b( t$e

m$#e! will be a useful addition to t$e literature. T$e establis$ed m$#e! must be able to &redict t$e

residual strengt$7 residual life. final failure mec$anisms @direction of failure &ro&agation? and final

fatigue life of t$e com&osite laminates under general fatigue loading conditions.


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Chapter 2

Problem Statement

In t$is c$a&ter7 a general e;&lanation of t$e mec$anics of fatigue failure of com&osite

materials is &resented. Available information in t$e literature on t$e mec$anics of fatigue failure is

reviewed and discussed. %ifferent fatigue failure modes of unidirectional com&osite materials

under various unia;ial fatigue loading conditions are e;&lained. A &in6bolt4loaded com&osite

laminate is selected as a so&$isticated &roblem to be solved b( t$e m$#e!. Furt$ermore. a detailed

descri&tion of t$is &roblem is &resented and t$e &ertinent mec$anisms of failure arc discussed. T$e

difference between two terms used in t$is t$esis7 t$e modes and t$e mec$anisms of fatigue failure

are elucidated. T$e framework of t$e establis$ed fatigue modeling strateg(7 ca&able of simulating

t$e fatigue be$aviour of com&osite materials in general conditions is e;&lained.

2.1 Mechanics of Fatigue Failure

It is well known t$at t$e &ro&agation of a dominant crack is res&onsible for li11al fatigue

failure in metals7 w$ile accumulation of cracks causes failure in com&osite materials. Consider a

unidirectional com&osite material under fatigue loading conditions. Matri; cracking is t$e first

failure mode w$ic$ occurs at t$e first c(cles of fatigue loading of com&osite materials. Matri;

cracks occur at t$e interface of fiber and matri; as well as wit$in t$e matri;. ( increa ing t$e

number of fatigue c(cles7 cracks &ro&agate and accumulate. At $ig$er number of c(cles or stress

levels7 cracks initiate in fibers and finall(7 catastro&$ic failure occurs. T$is brief e;&lanation is a


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Chapter 2 r$ !em Stateme t -

sim&lified scenario of t$e unia;ial fatigue failure &rocess of a unidirectional com&osite wit$out

considering t$e directions @fiber or matri;? or t(&es @tensile7 com&ressive7 in4&lane or out4of4&lane

s$ear? of fatigue loading.

A &ro&er understanding of mec$anics of initiation and &ro&agation of damage in

unidirectional com&osite materials under c(clic loading is an im&ortant ste& in fatigue modeling of

laminated com&osites. %ue to non4$omogeneit( of com&osite materials7 t$e fatigue failure

be$aviour of a unidirectional &l( loaded in fiber direction is different from a &l( loaded in matri;

direction. Moreover7 t$e fatigue be$aviour of a unidirectional &l( under various t(&es of loading

suc$ as tensile7 com&ressive7 in4&lane s$ear and out4of4&lane s$ear loading is also different. T$e

fatigue be$aviour of a unidirectional &l( under various t(&es of loading is called mode of failure in

t$is stud(. Fatigue be$aviour of unidirectional &lies under longitudinal tensile 5::4< I87

longitudinal com&ressive 5


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Chapter 2 r$ !em Stateme t ........................................................................................... 8

fatigue failure modes to be incor&orated into a s(stematic model to &redict t$e global failure

be$aviour of a com&licated &roblem. T$is strateg( is under develo&ment 5*24*)8. $owever all t$e

essential information re'uired b( t$e model is not as (et available. Also b( considering t$e state of

our &resent knowledge. creation of suc$ information seems to be a ver( tedious task. Moreover.

t$e model establis$ed in t$is wa( would be too com&licated and e;&ensive. Instead of sacrificingt$e feasibilit( of t$e model because of its generalit( or vice versa. anot$er strateg( is ado&ted in

t$is researc$ and a modeling a&&roac$ is establis$ed w$ic$ satisfies bot$ generalit( and feasibilit(.

T$e &$enomenological be$aviour of unidirectional &lies under different t(&e of fatigue loading can

be measured. full( c$aracteri>ed and incor&orated into a global s(stematic model. T$e following

e;am&le $el&s to describe t$e im&ortant basis of t$e model.

2.2 escription of the !roblem

Among different e;am&les of com&osite materials under fatigue loading conditions. a

&in6bolt4loaded com&osite laminate is a com&licated &roblem. "inned6bolted !oints arc widel( used

in !oining of com&onents made of com&osite materials. T$e e;istence of stress concentrations.

edge effects7 different stacking se'uences and selection of various load and stress ratios arc some

difficulties w$ic$ e;ist in t$e &in6bolt4loaded com&osite laminate &roblem. Consider a com&osite

&late wit$ a &in6bolt4loaded $ole7 i.e.7 t$e &late $as a circular $ole tilled wit$ a rigid &in or $olt.

Load is a&&lied at one end of t$e &late and is resisted b( t$e rigid &in or bolt. T$e coordinate a;is7

dimensions and nomenclatures are s$own in Fig. 2.1. T$e e;istence of stress concentrations7

w$ic$ cause a non4uniform state of stress7 as well as t$e e;istence of singular state of stress near

t$e free edge of t$e $ole @edge effects?7 make t$e stress anal(sis more cumbersome. T$e &late is a

laminated com&osite made of la(ers of continuous fibers embedded in an organic matri;. +ac$

la(er of t$e laminate is called a &l( or unidirectional la(er . T$e geometr( @si>e and location of

t$e $ole7 lengt$7 widt$7 and t$ickness of t$e laminate?7 stacking se'uence of unidirectional &lies in

t$e laminate and t$e a&&lied fatigue loading ratio @load777777fload77777.?7 w$ic$ induces t$e fatigue

stress ratio @crm 6cr ? in t$e laminate can be selected arbitraril(. +;istence of all t$eses difficulties

makes t$e &in6bolt4loaded com&osite laminate a suitable &roblem for e;amining and evaluating t$em$#e!.

efore e;&laining t$e fatigue failure events in a &in6bolt4loaded com&osite laminate7 it is

wort$w$ile to e;&lain t$e ste&s of failure initiation and &ro&agation of t$is &roblem under static

loading conditions. T$e com&osite &late is loaded wit$ an in4&lane load " as s$own in Fig. 2.1.

( increasing t$e load monotonicall( to a certain value7 w$ic$ is called t$e first &l( failure load7


22/97

Chapter 2 r$ !em Stateme t........................................................... ...........................9

failure initiates at a location near t$e edge of t$e $ole. +;&erimental observations s$ow t$at matri;

cr.icking failure mode usuall( starts first. If after failure initiation7 t$e load is increased7 failure will

e

111 i22 (a)

I p x

..'

I

e

I(b)

Fig. 2. i "eometr# of a laminated composite plate with a circular hole$subjected to (a) pin loading$ and %b& bolt loading

&ro&agate in different directions. ( increasing t$e monotonic load7 delamination and fiber failure

occur after matri; cracking. Finall( at a $ig$er load called t$e ultimate strengt$ load7 damage will

&ro&agate to an e;tent t$at t$e &late cannot tolerate an( additional load. It $as been observed

e;&erimentall( t$at mec$anicall( fastened !oints fail under t$ree basic me%ha i'm'. T$e

x.3 L4eQ P

x. # I

P


23/97

Chapter 2 r$ !em Stateme t .......................................................... ....... JO

mec$anisms of structural failure are net tension. s$ear4out and bearing. T(&ical damage due to

eac$ mec$anism is s$own in Fig. 2.2. T$e magnitude of t$e first &l( failure load. t$e &osition of

t$e initial failure7 t$e direction of t$e failure &ro&agation @or mec$anism of failure? and t$e ultimate

strengt$ load de&end u&on t$e material &ro&erties. dimensions. laminate configurations and man(ot$er &arameters.

et4tension #$ear4out earing

Fig. 2.2 %ifferent failure mec$anisms of &in6bolt4loaded com&osite laminates

( noticing t$e failure modes and mec$anisms of a &in6bolt4loaded com&osite laminate

under static loading conditions7 t$e be$aviour of t$e com&osite under fatigue loading conditions

can be e;&lained easier. At t$e start of c(clic loading7 t$e strengt$ of t$e material is greater t$an t$e

stress state7 t$erefore t$ere is no static mode of failure an(w$ere on t$e &late. ( increasing t$e

number of c(cles from >ero7 based on t$e stress states at eac$ &oint of t$e &late7 material &ro&erties

@stiffness and strengt$? at t$ose &oints are degraded as functions of number of c(cles7 state of

stress and stress ratio. ( increasing t$e number of c(cles7 w$ic$ is accom&anied b( more

degradation of t$e material &ro&erties and redistribution of stresses7 failure begins in regions w$ere

t$e strengt$ of t$e material falls below t$e stress level at t$at &oint. After failure initiation7 stressesare redistributed around failed regions. ( furt$er increasing t$e number of c(cles7 failure

&ro&agates in different directions. Finall( after a certain number of c(cles called (atigue !i(e t$e

laminate cannot tolerate additional c(cles. At t$is &oint t$e ma;imum number of c(cles is reac$ed

and t$e laminate $as failed com&letel(. Alt$oug$ material &ro&erties of all &oints of t$e &in6bolt

loaded com&osite &late $ave been degraded as functions of number of c(cles7 e;&erimental


24/97

Chapter 2 r$ !em Stateme t ................................................................. I I

observations $owever7 s$ow t$at t$e final mec$anisms of failure in static and fatigue loading

conditions are t$e same.

It s$ould be &ointed out t$at t$e failure modes of unidirectional com&osites under unia;ial

fatigue loading conditions are in$erent &ro&erties7 w$ile t$e mec$anisms of failure de&end u&on t$e &l( se'uence and geometr( of laminated com&osites. #ince t$ere are a limited number of failure

modes for a unidirectional &l(7 t$e( can be full( c$aracteri>ed b( e;&erimental tec$ni'ues and

mat$ematical models. Kowever t$e failure mec$anism is a case de&endent &ro&ert(7 t$erefore a

variet( of different mec$anisms of failure can e;ist for laminated com&osites wit$ various

geometrics7 stacking se'uence7 etc. T$is is an im&ortant &oint w$ic$ categori>es t$e fatigue

models into two classes, general and case de&endent.

2.' (e)uirements and Modeling *trateg#

It is desired to develo& a model to simulate t$e be$aviour of com&osite laminates under

general fatigue loading conditions. For t$is &ur&ose7 t$e framework of t$e pr$gre''ive (atigue

#amage m$#e! is establis$ed. T$e m$#e! is an integration of t$ree ma!or com&onents, stress

anal(sis7 failure anal(sis and material &ro&ert( degradation rules. T$e m$#e! determines t$e state

of damage at an( load level and number of c(cles7 from failure initiation and &ro&agation to

catastro&$ic failure. T$e m$#e! is able to &redict t$e residual strengt$7 residual life7 final failure

mec$anisms @direction of failure &ro&agation? and final fatigue life of t$e com&osite laminates

under general fatigue loading conditions.

Pn order to establis$ a model able to meet suc$ re'uirements7 a s&ecific strateg( must be

followed. Realistic stress anal(sis7 failure anal(sis and material &ro&ert( degradation rules are ke(

&oints in t$e fatigue failure anal(sis of com&osite laminates. First7 t$e stresses induced in t$e

com&osite laminate are anal(>ed b( utili>ing a finite element met$od. T$ere are some comities

w$ic$ make t$e role of a realistic stress anal(sis salient7 e.g.7 e;istence of bolt load7 non

$omogeneit( of com&osite materials before and after failure7 &resence of stress concentrations7

nonlinear s$ear stress4strain be$aviour of unidirectional &lies 5--8 and t$e e;istence of singular

states of stress near t$e free edges of com&osite laminates @see 5-/8 as a review?. Ac$ieving a

realistic stress anal(sis for different geometries7 la(4u&s7 loading conditions and boundar(

conditions in general forms7 is not &ossible wit$out using a t$ree4dimensional nonlinear finite

element tec$ni'ue. As mentioned earlier7 t$ere are different failure modes for unidirectional &lies


25/97

Chapter 2 r$ !em Stateme t 12

under unia;ial state of fatigue stress w$ic$ must be full( c$aracteri>ed and used as basic

information for t$e m$#e!. Also7 for &redicting t$e failure of a unidirectional &l( under multia;ial

states of stress7 t$e stresses are e;amined b( a set of stress4based failure criteria. Moreover7 on t$e

condition t$at failure e;ists7 material &ro&erties of t$e failed regions arc c$anged. For t$is

&ur&ose7 t$e m$#e! s$ould be ca&able of simulating t$e material &ro&ert( degradation of

unidirectional &lies @residual stiffness and residual strengt$? under unia;ial and multia;ial states of

fatigue stress. G&on increasing t$e number of fatigue c(cles7 t$e w$ole &rocess s$ould be

re&eated until catastro&$ic failure is ac$ieved. For t$is &ur&ose. a user4friendl( com&uter code

must be develo&ed to integrate t$e different com&onents of t$e m$#e!.

2.+ *ummar#

Alt$oug$ a general descri&tion of t$e modeling strateg( $as been given in a ver( com&acted

form in t$is c$a&ter7 t$e t$ree ma!or com&onents of t$e m$#e! stress anal(sis. failure anal(sis and

material degradation rules will be discussed in detail in t$e following t$ree c$a&ters. A detailed

e;&lanation of t$e m$#e! w$ic$ is an integration of t$e t$ree com&onents. will be &resented in

c$a&ter si;.


26/97

Chapter3

Stress nal!sis

In t$is c$a&ter t$e first com&onent of t$e m$#e! t$e stress anal(sis is e;&lained. T$efree edge effect @stress singularit(?7 a ver( im&ortant sub!ect in t$e stress anal(sis of

com&osite laminates7 is considered and discussed. A t$ree4dimensional finite element algorit$m

is develo&ed to anal(>e t$e t$ree4dimensional state of stress of a &in6bolt4loaded com&osite

laminate as a so&$isticated &roblem. A detailed e;&lanation of t$e finite element algorit$m and

t$e t$eoretical basis of t$e finite element formulation is &resented. For ac$ieving $ig$er

accurac(7 a twent(4node iso&aramctric 'uadratic solid element is used. T$ree differentconfigurations7 namel(7 cross4&l( 53603.P7 and 50363) 877 and an angle4&l( 59):64): ) 87 are

considered. ( using a large number of elements near t$e edge of t$e $ole and at la(er interfaces7 t$e edge effect is simulated. Also7 b(

noting t$e nonlinear s$ear stress4strain be$aviour of a unidirectional &l(7 t$e effect of material

nonlinearit( on t$e stress state near t$e edge of t$e $ole7 w$ic$ is a critical location for failure

initiation is considered. For t$is &ur&ose7 an e;isting model for t$e mat$ematical &resentation of

material nonlinear in4&lane s$ear stress4strain be$aviour of a unidirectional com&osite &l( is

e;tended to be also a&&licable for out4of4&lane s$ear stress4strain. Moreover7 an iterative sc$eme

is added to t$e algorit$m to &ro&erl( account for t$e e;istence of material nonlinearit(. T$e effectof e;istence of bolt load on t$e state of stress7 near t$e edge of t$e $ole is also studied and

discussed .


27/97

Chapter 3 Stre'' a!y'i' ................................................................. ... *4

'.1 Free ,dge ,ffects- (eview

%elamination is a well4known out4of4&lane mec$anism of failure w$ic$ usuall( occurs at

t$e free edge of com&osite laminates under static and fatigue loading conditions. It s$ould be

mentioned t$at delamination can also e;ist inside of t$e com&osite laminate. far from t$e edge.

resulting from im&act loading w$ic$ is not con idered in t$is stud(. In order to stud( t$e fatigue

be$aviour of com&osites under general conditions7 it is im&ortant to understand t$e t$rce

dimensional nature of t$e stress distribution of com&osite laminates. #tress anal(sis w$ic$

includes t$e t$ird or t$ickness direction is re'uired in order to understand t$e free edge effects.

T$e state of stress at free edges of a sim&le com&osite &late @wit$out an( sim&lif(ing assum&tions?.

was studied for t$e first time b( "i&es and "agano 5-08 in 10-3. T$e( used cl,L7sical linear t$eor(

of elasticit( to set u& t$ree cou&led7 elli&tic7 second4order7 &artial differential e'uations and a&&lied

finite difference tec$ni'ues to solve for dis&lacements and stresses. T$eir results revealed t$att$ere were significant magnitudes for normal and s$ear stresses on t$e free edges7 between

different la(ers of com&osite &lates @Fig. *.1?.

Fig. '.1 Laminate geometr# and edge effects %after !agano and !ipes /0 3

ecause of t$e im&ortant role of normal and out4of4&lane s$ear stresses in t$e creation of delamination7 numerous attem&ts $ave been made to obtain an accurate stress state on t$e edge of a

com&osite &late b( investigators. For t$is &ur&ose7 different tec$ni'ues like finite difference 5/387

finite element 5/14/-87 closed form solutions 5//413387 boundar( la(er t$eor( 5I3 I87 &erturbation

met$od 5132713*87 =alerkin met$od 513)8 and e;&erimental met$ods 50:713:713


28/97

Chapter 3 Stre' ' a!y'i' 15

tec$ni'ues7 suc$ aH com&uter time7 satisf(ing all boundar( conditions7 e;istence of a mat$ematical

singularit( in t$e stress field7 etc.7 a&&ro;imate tec$ni'ues were used b( "agano and "i&es 513-8.

T$e( assumed t$at t$e normal stress is constant inside of a com&osite &late and varies linearl( near

t$e edges. Kowever7 wit$ res&ect to t$e results t$at $ave been obtained b( ot$er tec$ni'ues7 t$isa&&ro;imation seems coarse. T$ere are ot$er attem&ts to sim&lif( t$is &roblem7 for e;am&le7 t$e

works of assa&oglou and Lagace 5I 3/87 and Rose and Kerakovic$ 5I 308 must be mentioned. In

10/37 Ra!u et a!. 5I P 8 reviewed man( &revious works and s$owed discre&ancies in t$e results

obtained b( different aut$ors. T$e( also investigated t$e reliabilit( of dis&lacement4formulated

finite element tec$ni'ues for anal(>ing t$e edge &roblem. A detailed review of different models for

stud(ing free edge effects is &resented b( "agano and #oni 5-/8.

Alt$oug$ in 5/:7/e effects on normal and interlaminar stresses

near t$e edges $ave been considered in t$eir works. B$ile in t$ese studies 511:41218 finite

element tec$ni'ues $ave been used7 in 512287 a sim&lified closed form solution was a&&lied to

com&ute t$e interlaminar stresses around a circular o&en $ole. T$ree4dimensional finite element

tec$ni'ues $ave been used in 512*412:8 to anal(>e t$e be$aviour of &in and bolt4loaded com&osite

laminates. evert$eless7 in t$ese works all stresses are not &resented. Also7 stress singularities

ncart$e edge of t$e $ole @edge effects? are not e;amined. In 512


29/97

Chapter 3 Stre'' a!y'i' .................................................................... 6e t$e

material &arameters @moduli7 strengt$s and "oisson s ratios? are reduced drasticall(. In t$is stud(7

A#)6*:314< material is used wit$ t$e material &ro&erties measured in Com&osite Materials

Laborator( of Mc=ill Gniversit(7 s$own in Table *.1.

>

>.........

y>

y;

y

Fig. *.* T$ree4dimensional geometr( of a la(er of com&osite material

w$ere in Table *.17 +77. +RR and " are t$e longitudinal7 transverse7 and normal modulus7

res&ectivel(. Also7 +77. +.7and + 7 are s$ear moduli7 v 7 v and v(> are "oisson s ratios7 and St.

7. !". S7. 7. 7. #7( #.7. and #(> are longitudinal tensile7 matri; tensile7 normal tensile7longitudinal com&ressive7 matri; com&ressive7 normal com&ressive7 in4&lane s$ear7 out4of4&lane

s$ear @;4> &laneU7 and out4of4&lane s$ear @(4> &lane? strengt$7 res&ectivel(. T$e ;7 ( and >

directions are material directions as s$own in Fig. *.*. T$e "oisson s ratio in (4> &lane @v(>? is

calculated b( using C$ristensen s formula 51:08.


31/97

Chapter 3 Stre'' r r.!y'i' .................................................................... IS

Table *.1 Material &ro&erties of A#)6*:314ed t$at

t$e mentioned nonlinearities are due to inelastic be$aviour of t$e material before failure

initiation. onlinearities due to failure are com&letel( different &$enomena and t$e( arc considered

b( failure anal(sis tec$ni'ues in t$is stud(.

'.'.+ Material Orientation

In t$e com&uter &rogram it is necessar( to determine t$e stiffness matri; or t$e material IE%.

wit$ res&ect to a coordinate a;is @;4(? @Fig. *.:?7 w$en t$e stiffness matri; or t$e material 5Cl is

known relative to a rotated coordinate s(st ,m @; 4( ? 5 1*28. #u&&ose we $ave t$e following

constitutive relation,

cr'/=#$% t'/ ,). '.11

w$ere @CPis a known transversel( isotro&ic stiffness matri; @+'. *.2?. In t$e rotated coordinate s(stem7 we $ave,

cr/ E (E% E/ ,). '.12


36/97

Chapter 3 Stre'' a!y'i' ............................................................................................... 23

y

..7.

;

Fig. *.: Original and rotated coordinate s(stems

T$e transformed stresses in rotated a;es arc,

O6 :

a8@m 'n '

n ' 3 3 3 2n1n cr77m ' 3 3 * 52mn cr((

cr8A7 3 3 * 3 3O67.Z

,). '.1'

cr81. 3 * 3 m -n 3 ay:r.cr8.: 3 3 3 n m 3 O6A:a8@ -mn mn 3 3 3 m 'B5n5 ' cr.7

w$ere m E cos e. and n E sin e.Also t$e transformed strains in rotated a;es are,

,). '.1+

#ubstituting +'s. *.1* and *.1) into +'. *.11. we obtain,

,). '.1C

, -w. m ' n ' 3 3 3 mn +77,8# n ' 0 ' 3 3 3 -mn +((e8.0. 3 3 3 3 3 +7ic#1. 3 3 3 0 -n 3 Ey:r., -w. 3 3 3 n 0 3 +77

c:Y 52rnn 2mn 3 3 3 m 'B5n5 '


37/97

Chapter 3 Stre'' a!y'i'.......................................................................................... 2 1

w$ere 5T 38 and 5T78 are t$e stress and strain transformations as defined b( +'s. *.1* and *.1) .

From +'. *. i:7 we $ave,

T$us7

,). '.1>

,). '.10

( substituting +'. *.2 into +'. *.1-. we obtain,

, 11 ;4 11 m4 =2m 2 n 2 %4

=24M3=4 22 n.-

+ 12 = m 2 n 2 @C11 9C 33 4)C


38/97


39/97


40/97

Chapter 3 Stre'' a!y'i'...........................................................................................26

a 3 3Cl;3 a 3Cl( a

3 35Cl8E a a

Cl>3

N ,, 3 33 Ni". 3

Cl( Cl; 3 3 N$ I 5 PE3 Cl> Cl(

l,l

Ni." +'. *.22

a a I"2. !,X

Cl>3

Cl; *!+ N 17?1 3

As s$own in +'. *.107 s$a&e functions are calculated in t$e . 11 and s coordinate s(stem.To obtain t$e derivatives of t$e s$a&e functions in +'. *.22 wit$ res&ect to ;. ( and >. t$e s$a&e

functions must be tra. lsformed from t$e &revious coordinate s(stem to t$e ;. ( and > coordinate

s(stem. Again b( utili>ing t$e iso&arametric conce&t7 we $ave,

:;L.Ni:i

y= N

A;LNiAi

+'. *.2*

and derivatives are given b(,

w$ere 5P8is t$e Pacobian matri;,

+'. *.2)

t$en derivatives of s$a&e functions wit$ res&ect to ;7 (7 and > arc found from +'. *.2),

+'. *.2:

a a 3 NN

3 N


41/97

Chapter 3 Stre'' a!y'i' ............................................................................................... 27

+'. *.2

8rIC 3 I 3EK6 (% I (%

GHKL (% 3 I


46/97

Chapter 3 Stre'' a!y'i'.......................................................................................... 32

*.*.13 Met$od of #olution for a #(stem of onlinear Algebraic +'uations

In order to solve +'. *.**7 different tec$ni'ues can be a&&lied. Kowever because of t$e

large number of degrees of freedom @d.o.f.? used in t$e finite element model. t$e stiffness matri;

5 8 is a large matri; @d.o.f. b( d.o.f.?. T$e costl( wa( to store t$e stiffness matri; is to save it in

form of an arra( of dimension @i.!?. Kowever. t$e stiffness matti; 5 8 is s(mmetric and t$ere

are man( unnecessar( >ero elements in it w$ic$ are never needed in t$e calculations. A more

effective met$od w$ic$ is called t$e 'e and mes$ distribution for ac$ieving a reliable stress state of t$e &roblem were found b( trial and error. ( refining t$e mes$es7 smoot$ stress distributions for

all si; stresses from t$e edge of t$e $ole to t$e far field region are ac$ieved. It s$ould be noted t$at

for A#)6*:314< wit$ a fiber volume of


47/97

Chapter 3 Stre'' a!y'i'...........................................................................................33

t$erwise7 t$e dimension of t$e smallest element would be smaller t$an t$e fiber diameter or

s&acing between fibers.

To stud( t$e effects of material nonlinearit( and bolt load7 all si; com&onents of t$e stress

tensor near t$e edge of t$e $ole and along t$e interface between t$e two la(ers wit$ differentorientations are e;amined. Alt$oug$ t$e stresses for all different &oints of t$e com&osite laminate

are used for failure anal(sis7 $owever t$e stresses far from t$e edge of t$e $ole are not discussed in

t$is c$a&ter. First t$e effect of material nonlinearit( on t$e state of stress of a &in4loaded com&osite

laminate is e;&lained. Moreover7 t$e effect of bolt load on t$e stress state near t$e edge of t$e $ole

is considered and discussed. As mentioned earlier7 two different cross4&l( laminates7 53)1 03)8 :

and 503)1 3)Ps7 and an angle4&l( laminate7 59):)1 4):)8 : 7 are considered.

'.+.1 +ffects of Material Nonlinearit#

3.4.1. 6 Cr$'' !y=0 *904>. .

All si; stn.,sses of a @3)1 03)87cross4&l( laminate7 along t$e interface between t$e two la(erswit$ different orientations at t$e $ole edge7 for linear and nonlinear cases7 are s$own in Fig. *.0.

Inducing material nonlinearit( for s$ear stress4strain res&onse in t$e ;4( and ;4> &lanes decreases

cr.7 and cr.77res&ectivel(. As s$own in Fig. *.07 cr.7 is decreased between *3D 0


48/97

Chapter 3 Stre'' a!y'i' .................................................................... 3.*

t$e normal stress @cr77? is different fort$e two cases. ( com&aring cr77 in Figs. *.0 and *.13. for t$e linear case @solid curves? it is seen t$at t$e magnitude of normal stress @cr77? at 0 = 3D. for

53.i603)Ps case is $ig$er t$an t$at of 503.i63) 87 case. T$is evidence 'ualitativel( confirms t$e

e;&erimental results re&orted in 5 1:/8. w$ere it was found t$at la(4u&s wit$ 03D la(ers located att$e surface7 under &in4load conditions. s$ow su&erior bearing strengt$. Kowever. it must be

em&$asi>ed t$at wit$out a&&l(ing a &rogressive damage model t$e difference in strengt$ between

53.i603)8: and 503P3)8s can not be s$own 'uantitativel(. ( considering Fig. *.13. it is seen t$at.

like t$e 5 .i603)87 case7 material nonlinearit( s$ifted cr77 at 0 = 3D. from t$e negative towards &ositive magnitudes w$ic$ again is not favoured in design.

3.4.I .3 6 g!e10!y = ?45 i(1454@A

T$e distribution of all si; stresses of an angle4&l( 5 9): ) 64): ) 87 along t$e $ole edge. for

linear and nonlinear cases7 are s$own in Fig. *.11. As s$own in Figs. *.11. material nonlinearit(decreased t$e magnitudes of cr and CP7R4 ( e;amining CP77.in Fig. *.11. it is clear t$at t$is stress

is decreased at ):D 0 123D. Also7 as s$own in Fig. *.117 CP77is reduced at 3D 0 *:D andat

:33 0 1*3Dfor t$e nonlinear case7 and it remains nearl( t$e same elsew$ere. ( considering

Fig. *.117 it is seen t$at t$e material nonlinearit( increased t$e longitudinal stress @CP77?locall( at eE ):3. ( considering cr77 and CP7>in Fig. *.117 it is clear t$at. for nonlinear case7 t$ese stressesare increased at 3D 0 1*:D and remain constant at 1*:D 0 1/3D. Interlaminar normal stress

@cr77?7 in Fig.*.117 is decreased at 3D 0 2:D and is increased at 2:D 0 c?ou and at 113D 0

1can create delamination w$ic$ is not favoured in t$e

design of com&osite laminates.

3.4.1.4 Summary $( the ((e%t' $( Bateria! ;$ !i earity

In general7 b( inducing t$e material nonlinearit( for s$ear stress4strain res&onse of @cr77 4

e77and cr7> 4 e7.>?7 relevant stresses @cr77 and cr77? are decreased for all t$ree different configurations@two cross4&l( laminates and one angle4&l(?. Kowever7 b( looking at t$e be$aviour of

longitudinal stress @cr ? for t$e t$ree configurations7 Fig. *.0 to *.117 it is clear t$at t$is str ss isnot strongl( effected b( inducing t$e mentioned material nonlinearit(. Moreover7 transverse stress

@cr((? and one of t$e interlaminar stresses @CP77?are significantl( increased at 3D 0 03D for all

t$ree configurations. Also7 interlarninar normal stress @CP77?is increased and s$ifted towards t$e &ositive magnitudes for all t$ree cases. T$us7 considering material nonlinearit( causes significant

decrease in magnitudes of some stresses w$ile ot$ers are increased to com&ensate .

2


49/97

Chapter 3 Stre'' a!y'i' 35

'.+.2 ,ffects of Jolt Load

As mentioned earlier7 t$e state of stress on t$e edge of t$e $ole @w$ic$ is a site for failure

initiation? and t$e effects of t$e bolt load on t$e stresses at t$at location are discussed $ere. T$estress state far from t$e edge of t$e $ole is also im&ortant and will be used for failure anal(sis later.

3.4.2. I Cr$''1p!y :ami ate' [0.i/904,+ a # [90.i/04 .!

In Figs. *.12 and *.1*7 si; different stresses for 536037Psand 503637Psunder bolt loadingconditions @b( considering t$e material nonlinearit(? are s$own7 res&ectivel(. ( com&aring t$e

results of stress anal(sis for a &in4loaded 536037Psin Fig. *.0 and a bolt4loaded 503637Psin Fig.*.127 it is clear t$at b( a&&l(ing bolt load7 all stresses are decreased dramaticall( b( some orders of magnitude. T$e decrease in t$e state of stress near t$e edge of t$e $ole is accom&anied b( anincrease in t$e magnitudes of stresses far from t$e edge of t$e $ole w$ic$ are certainl( not as

critical as t$e magnitudes of stresses rig$t at t$e edge. ( com&aring normal stress7 cru.7 andinterlaminar s$ear stresses7 cr.7.and cr(7 in Figs. *.12 and *.1*7 it is concluded t$at a&&l(ing boltload can decrease t$e normal and interlaminar s$ear stresses of t$e 536037Pscase mo e t$an t$e503637Pscase. ( com&arison between t$e &inned and bolted 53603) Ps cases in Figs. *.0 and

*.127 it can be seen t$at t$e effect of t$e bolted was$er decreases t$e interlaminar s$ear stresses7

cr and cr 7 b( a greater factor t$an t$e out4of4&lane norNial stress7cr77.

3.4.2.2 6 g!e1p!y :ami ate = 45.i/-45 4 " !.In Fig. *.1)7 si; different stresses for 59):64): ) 8s under bolt loading conditions @b(

considering t$e material nonlinearit(? are s$own. #imilar to cross4&l( laminates7 b( a&&l(ing boltload. all stresses are decreased dramaticall(. Furt$ermore7 as in t$e case of cross4&l( laminates7

bolt loading c$anged t$e sign of all &ositive normal stresses7 cru.7 to negative w$ic$ is favored int$e design of com&osite !oints.

'.C *ummar#

A t$ree4dimensional nonlinear finite element algorit$m is develo&ed in t$is c$a&ter. T$e

t$ree4dimensional stress state of t$e &in6bolt4loaded com&osite laminate is calculated. T$e edge

effect @stress singularit(? w$ic$ is one of t$e most im&ortant &$enomenon in fatigue be$aviour of

com&osite laminates is considered. T$e effect of material nonlinearit( and t$e bolt load on t$e

states of singular stress near t$e edge of t$e $ole and between la(ers wit$ different &l( orientations

are studied. T$erefore. t$e first com&onent of t$e pr$gre''ive (atigue #amage m$#e! isestablis$ed. #o far7 we are able to anal(>e t$e t$ree4dimensional stress state of a com&licated

lll y$ ...." C.4.


50/97


51/97

Chapter 3 Stre'' a!y'i' ............................................................................................... 37

2 34 '?? 4 + la).' 2?? Z .'1 %% +

3 "..F t&l"..+2 ? P

t&l.1 ? ? * 3

4 2 ? ?

4 ' ? ? ) 3

3 ) s 0 3 1 * s 1/3 3 ) s 0 3 1 * s 1 / 3e (


52/97

Chapter 3 Stre'' a!y'i' ............................................................................................... 38

)33 2 3 L.. *33 ..c.. 233 c..

&+&&& &4&&& 1 3

75 3H423

75 W7

4 2 ? ? * 3

5' ?? ) 3 +1++++

3 ) : 0 3 1*: 1/3 3 ) : 0 3 1 * : 1/3

e (


53/97


54/97

Chapter 3 Stre'' a!y'i' 40

0.4 ?.?>

m o.' ca o.o+a.. a..

&E 0.2 &+E0.02

-?.1 7 0."7

0 5?.?2

4 ?.$ 5? .?+0 4 5 9 0 1 3 5 180 0 4 5 9 0 135 1 8 0

e (


55/97

Chapter 3 Stre'' a!y'i' /////////////////////////////////////////////////////////////////////////////////////////////////// 41

a..

2 3.ation tec$ni'ues for strengt$ degradation7 stiffness degradation and fatigue

life of a unidirectional &l( under unia;ial state of stress are utili>ed. T$e $ a*i"e# 'tre !ith

#egra#ati$ and $ a!i"e#(atigue !i(e m$#e!' are ado&ted and modified from t$e works *f ot$er

investigators7 and t$e $ a!i"e# 'ti(( e'' #egra#ati$ m$#e! is develo&ed in t$is stud(. T$e

limitations of a&&lication of failure criteria in traditional forms7 discussed in C$a&ter four7 are

overcome in t$is c$a&ter. For t$is &ur&ose7 t$e ge era!i"e# materia! pr$perty #e ira#ati$

te%h i&ue is establis$ed b( t$e cou&ling of t$e $ a!i"e# 'tre gth #e ira#ati$ $rma!i"e#

'ti(( e'' #egra#ati$ and $ a!i"e# (atigue !i(e models. T$e develo&ed te%h i&ue simulates t$e

fatigue life and material &ro&ert( @strengt$ and stiffness? degradation of a unidirectional &l( under

multia;ial state of stress and arbitrar( stress ratio .


74/97

Chapter S Bateria! r$perty egra#ati$ ....................................................................59

C.1 Material !ropert# egradation (ules

In t$e &revious c$a&ter7 suitable failure criteria are establis$ to detect t$e sudden static and

fatigue failure modes of a unidirectional &l( under multia;ial state of stress. As failure occurs in a

&l( of a laminate7 material &ro&erties of t$at failed &l( are c$anged b( a set of sudden material

&ro&ert( degradation rules. #ome of t$e failure modes are catastro&$ic and some of t$em are not.

T$erefore7 for a unidirectional &l( failed under eac$ mode of static or fatigue failure7 t$ere e;ists an

a&&ro&riate 'u##e material &ro&ert( degradation rule.

T$e scenario of material degradation of a unidirectional &l( failed under static and fatigue

loading conditions before occurrence of sudden failure is different. For a unidirectional &l( under

multia;ial state of static stress before sudden failure initiation7 detected b( t$e set of static failure

criteria7 t$ere is no material degradation. Kowever7 for a unidirectional &l( under a multia;ial stateof fatigue stress before sudden failure initiation7 detected b( t$e set of fatigue failure criteria7 t$ere

is a gra#ua! material &ro&ert( degradation. To e;&lain t$is difference more clearl(7 consider a

laminated com&osite under static loading conditions. T$e load is increased monotonicall( and at a

certain load level7 failure initiation in a &l( of t$e laminate is detected b( t$e static failure criteria

@+'s. )41 to )412?. At t$is stage7 t$e mec$anical &ro&erties of t$e failed region of t$e

unidirectional &l( of t$e laminate must be c$anged. T$is t(&e of degradation is called 'u##e

materia! pr$perty #egra#ati$ . For a laminated com&osite under fatigue loading conditions7 in t$e

first c(c,Ies7 t$e strengt$ of t$e &lies can be $ig$er t$an t$e stress state. T$erefore7 during t$e first

c(cles7 t$e &ro&osed fatigue failure criteria @+'s. )41* to )410? do not detect an( sudden mode of fatigue failure. Kowever7 b( increasing t$e c(clic loading of t$e laminate7 material &ro&erties of

eac$ &l( are degraded. T$is t(&e of degradation is called gra#ua! materia! pr$perty #egra#ati$ .

( furt$er increasing t$e number of c(cles7 mec$anical &ro&erties of t$e &iles t,ventuall( reac$ to a

level w$ere different modes of failure can be detected b( t$e &ro&osed fatigue failure criteria @+'s.

)41* to )410?. At t$is stage7 t$e mec$anical &ro&erties of t$e failed material are c$anged b( 'u##e

materia! pr$perty #egra#ati$ ru!e'.

T$e 'u##e materia! pr$perty #egra#ati$ ru!e' for some failure modes of a unidirectional

&l( under a bia;ial state of static stress are available in t$e literature 5)14)*8. A com&lete set of

sudden material &ro&ert( degradation rules for all t$e various failure modes of a unidirectional &l(

under a multia;ial state of static and fatigue stress is develo&ed in t$is stud( and e;&lained in t$e

following section. Moreover. t$e gra#ua! materia! pr$perty #egra#ati$ ru!e' establis$ed in t$is

researc$ will be e;&lained t$ereafter.


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Chapter 5 Bateria! r$perty egra#ati$ 60

:.2 *udden Material !ropert# egradation (ules

In t$e following7 a com&lete set of sudden material &ro&ert( degradation rules is establis$ed

for eac$ mode of failure of a unidirectional &l( under a multia;ial state of static or fatigue stressdetected b( t$e static or fatigue failure criteria. T$e rules must be carefull( a&&lied to avoid

numerical instabilities during com&utation b( t$e com&uter &rogr.im.

Conventional finite element tec$ni'ues are b( definition limited to an intact continuum.

T$us7 after failure occurrence in a &l(7 instead of inducing a real crack. t$e failed region of t$e &l(

is re&laced b( an intact &l( of lower material &ro&erties @Fig. :.1?. T$erefore. conventional finite

element tec$ni'ues can be a&&lied for stress anal(sis even after failure initiation.

Fig. :.1 %egraded &l( is modeled b( an intact &l( of lower material &ro&erties

:.2.1 Fiber Tension "ro&ert( %egradation

Fiber tension failure mode of a unidirectional &l( @detected b( +'. ).2 for static and +'.

).1) for fatigue cases? is a catastro&$ic mode of failure and w$en it occurs7 t$e failed material

cannot sustain an( t(&e or combination of stresses. T$us7 all maierial &ro&erties of t$e failed &l(

are reduced to >ero7 as follows,

#tiffnesses and "oisson s ratios,

#E ;; E ## 'E AA6E :# 'E ;> E (> 9 :# 69 ;> 9 (> 9 (; 9 1.S 9-.(8

.L.0,0,0,0,0,0,0,0,0,0,0,0!

+'. :.1 @a?


76/97

Chapter " &aterial Propert! 'egradation 61

#trengt$s, "

0,0,0.0,0,0,0,0,0!

,). C.1 %b3

As mentioned7 t$is mode of failure is catastro&$ic7 t$erefore if it occurs7 t$e ot$er modes of

failure do not need to also be verified. %uring numerical com&utations b( t$e com&uter &rogram7

reducing material &ro&erties to >ero creates numerical instabilities. To avoid t$is difficult(7 for t$e

case of failure7 t$e material &ro&erties are reduced to ver( small values.

C.2.2 Fiber 4ompression !ropert# egradation

Fiber com&ression failure mode of a unidirectional &l( @detected b( +'. ).) for static and

+'. ).1: for fatigue cases? is a catastro&$ic mode of failure and w$en it occurs7 t$e failed material

cannot sustain an( t(&e or combination of stresses. T$us7 all material &ro&erties of t$e failed &l(

arc reduced to >ero7 as follows,


#E :: 'E ##. E 6 .0.. E :# 'E :A. E #A. v :#. v :A6 v #A. v #:. v 0.K$ vA#

"

0,0,0,0,0,0,0,0.0,0,0,0!

,). C.2 (a)

#trengt$s,

5;7777 77S77 7. 77#;( s.77s(78 "

0,0,0,0,0,0,0,0,0!

,). C.2 %b3

As mentioned7 t$is mode of failure is catastro&$ic7 t$erefore if it occurs7 t$e ot$er modes of

failure do not need to also be verified.

C.2.' Fiber5Matri: *hearing !ropert# egradation

In fiber4matri; s$earing failure mode of a unidirectional &l( @detected b( +'. ).< for static

and +'. ).1< for fatigue cases?. t$e material can still carr( load in t$e fiber7 matri; and normal


77/97

Chapter " &aterial Propert! 'egradation ////////////////////////////////////////////////////////////////////62

directions7 but in4&lane s$ear stress can no longer be carried. T$is is modeled Gy reducing t$e in

&lane s$ear material &ro&erties of t$e failed &l( to >ero7 as follows,


5,- .:. ,## ,Al ,:# ,:-r. ,#$P 9 :# 9 $$P.9#$ 9 #:. 9 6.Q.9

/' 0

,). C.' (a)

#trengt$s,

5;7. 7. >7.;7. 7.>7.s .s .s778 " ,). C.' %h3

After detecting t$is mode of failure7 w$ic$ is not catastro&$ic. t$e ot$er modes or railure

must be verified. After t$is mode of failure occurs7 in order to c$eck t$e ot$er modes or failure b(

relevant failure criteria7 t$e terms containing #77 must not be furt$er cor.sidcrcd.

C.2.+ Matri: ension !ropert# egradation

For matri; tension failure mode of a unidirectional &l( @detected b( +'. )./ for static and

+'. ).1- for fatigue cases?7 t$e transverse modulus7 +77. t$e transverse tensile strengt$ 7. and"oisson s ratios v ,md v 77are reduced to >ero. T$is mode of failure is not catastro&$ic. andaffects onl( matri; direction &ro&erties7 t$erefore ot$er material &ro&erties arc lert unc$anged.


#trengt$s,


78/97

,). C.+ (a)


79/97


# " ' ""J"' "' ""!".8' .8"". 8"" l.5. +'. :.) @b?

After detecting t$is mode of failure. w$ic$ is not catastro&$ic7 t$e ot$er modes of failure

must be verified.

:.2.: Matri; Com&ression "ro&ert( %egradation

Matri; com&ression failure mode @detected b( +'. ).0 for static and +'. ).1/ for fatigue

cases? results in t$e same t(&e of damage to t$e com&osite &l( as t$e matri; tension failure mode.

T$us7 t$e transverse modulus + t$e transverse com&ressive strengt$ 77 and "oisson s ratios 7

and v7. are reduced to >ero. T$is mode of failure is not cata tro&$ic7 t$erefore7 ot$er material

&ro&erties arc left unc$anged.


#E ;; E yy E IJ. E ;( E ;> E (> 9 ;( 9 ;> (> 9 (; r.; 69>(8

5+;; ,%, E 12 7+;(7 +;> 7+(>7 ;( xi. %, , >;. i.y>

+'. :.: @a?

#trengt$s,

+'. :.: @b?

#imilar to matri; tension failure mode7 matri; com&ression is not a catastro&$ic mode of

failure. T$us. after detecting t$is failure mode7 t$e ot$er modes of failure must be verified.

:.2.< ormal Tension "ro&ert( %egradation

For normal tension failure mode of a unidirectional &l( @detected b( +'. ).11 for static and

+'. ).10 for fatigue ca es?7 t$e normal modulus +77. t$e normal tensile strengt$ 77 and "oisson s

ratios v7W7and v7 are reduced to >ero. T$is mode of failure is not catastro&$ic7 t$erefore7 ot$er material &ro&erties are left unc$anged. T$is is essentiall( t$e same t(&e of failure as matri; tension

failure mode.


80/97

Chapter 5 Bateria! r$perty egra#ati$ .............................................. 7.s77.s .s 8P.. +c1. :.< @b?

After detecting t$is mode of failure. w$ic$ is not catastro&$ic. t$e ot$er modes or failure

must be verified.

:.2.- ormal Com&ression "ro&ert( %egradation

For normal com&ression failure mode of a unidirectional &l( @detected b( +'. ).12 for

static and +'. ).23 for fatigue cases?. t$e normal modulus +77.. t$e normal com&ressive strengt$ 7.

and "oisson s ratios v and v77are reduced to >ero. T$is mode or failure is not catastro&$ic.t$erefore7 ot$er material &ro&erties are left unc$anged. T$is is essentiall( t$e same t(&e of failure

as matri; com&ression failure mode.


+'. :.- (a)

#trengt$s,

5S777 7 77 S77 77 7 7s77.s77W7#77

P.. +'. :.- @b?


81/97

Chapter 5 Bateria! r$perty egra#ati$ ....................................................................... 65

Arter detecting t$is mode of failure7 w$ic$ is not catastro&$ic7 t$e ot$er modes of failure

must be verified.

:.2./ #ummar( of #udden Material %egradation Rules

T$is com&lete set of sudden material degradation rules @+'s. :.1 to :.-? for all various

failure modes of a unidirectional &l( under a multia;ial state of static @detected b( +'s. )41 to )412?

and fatigue stress @detected b( +'s. ).1) to )423? will be used in the pr$gre''ive (atigue #amage

m$#e! e;&lained in t$e ne;t c$a&ter. Kowever. t$e material &ro&erties of a unidirectional &l(

bd ore sudden failure @detected b( +'s. ).1* to )410? degrade due to fatigue loading. In order to

simulate t$is &$enomenon7 t$e gra#ua! materia! pr$perty #egra#ati$ ru!e' are establis$ed in t$is

researc$ and e;&lained in t$e following section.

:.* "radual Material !ropert# egradation (ules

To a&&l( t$e fatigue failure criteria @+'s. )41) to )423?7 t$e r sidual material &ro&erties of a

unidirectional &l( under arbitrar( multia;ial state of fatigue stress and stress ratio must be modeled.

For t$is &ur&ose7 t$e ge era!i"e# materia! pr$perty #egra#ati$ te%h i&ue is establis$ed w$ic$

simulates t$e fatigue be$aviour of a unidirectional &l( under multia;ial state of fatigue stress and

arbitrar( stress ratio b( using t$e results of unia;ial fatigue e;&eriments. In t$is wa(. t$e severe

limitation of a&&lication of t$e fatigue failure criteria in traditional forms7 mentioned in t$e &revious

c$a&ter. is overcome. In t$e following7 after a review of different residual strengt$ models &ro&osed in t$e literature7 a suitable model to simulate t$e be$aviour of a unidirectional lamina

under a unia;ial state of fatigue stress is selected. T$en7 t$e normali>ation tec$ni'ue for stiffness

degradation is establis$ed. #ubse'uentl(7 a &rocedure to find t$e fatigue life of a unidirectional

lamina under a unia;ial state of fatigue stress7 wit$ an arbitrar( stress ratio7 is e;&lained. Finall(7

to simulate t$e be$aviour of a unidirectional &l( under multia;ial fatigue loading7 wit$ arbitrar(

state of stress and stress ratio7 t$e ge era!i"e# materia! pr$perty #egra#ati$ te%h i&ue is

estabI is$ed.

:.*.1 ormali>ed #trengt$ %egradation Model

T$ere are two ma!or a&&roac$es to simulate t$e residual strengt$ of laminat,.,u ,om&osites

under unia;ial fatigue loading 521 P7 w$ic$ are call d t$e statistical @&robabilit(4O.Pased damage? and

mec$anistic @em&$asis on damage mec$anics? a&&roac$es. Borks of Kal&in et a!. 5)3721*8 and

routman and #a$u 521)8 are t$e two earliest e;am&les of statistical and mec$anistic a&&roac$es7


82/97

Chapter 5 Bateria! r$perty egra#ati$


83/97


stress7 t$e residual strengt$ as a function of number of c(cles is nearl( constant and it decreases

dra ticall( at t$e number of c(cles to failure @Fig. :.*?. T$e 'u##e #eath m$#e! 521172128 is a

suitable tec$ni'ue to describe t$is be$aviour. Kowever at low level state of stress7 t$e residual

strengt$ of t$e lamina7 as a function of number of c(cles7 degrades graduall( @Fig. :.*?. T$e +ear $ut m$#e! 5)38 is a suitable tec$ni'ue to &resent t$is be$aviour. In &ractice7 designers must deal

wit$ a wide range of states cf stress var(ing from low to $ig$7 t$erefore a model to &resent t$e

be$aviour of com&osite materials under a general state of stress is essential.

R@n? tlow level stress

L . +++ ====&&&&&++++11 ear +ut

-11

curv$ig$ level stresssudden deat$

Mrnt&..........................................i......................................... .o. L

*.25 Nr I n

Fig. :.* #trengt$ degradation under different states of stress

%ifferent models $ave been &resented in t$e literature to simulate t$e residual strengt$ of

com&osite laminates under fatigue loading. In t$e following7 a com&arison between differentmodels is made. It s$ould be mentioned t$at t$ere are man( details in t$ese models w$ic$ are not

discussed $ere. T$erefore in t$e following7 t$e models and e'uations used b( ot$er aut$ors are

onl( considered from our &oint of view. Reali>ing t$at various notations $ave been used b(

different aut$ors. for sim&licit(. a unified notation $as been a&&lied $ere in order to &resent t$e

models of ot$er aut$ors in an informative manner. It must be mentioned t$at in t$is t$esis7 t$ere is

no attem&t to stud( t$e &robabilistic features of t$e residual strengt$ of com&osite materials onl(

t$e mec$anistic c$aracteristics are considered $ere.

T$e 'u##e #eath m$#e! 5211.2128 is ver( sim&le and straig$tforward. T$e strengt$ of t$e

com&osite lamina is constant until t$e number of c(cles to failure @ r?7 w$ere t$e com&osite lamina

fails catastro&$icall(. In t$e +ear $ut m$#e!. w$ic$ was initiall( &resented b( Kal&in et a!.=40> it

is assumed t$at t$e residual strengt$ R@n? is a monotonicall( decreasing function of number of

=I


84/97

Chapter 5 Bateria! r$perty egra#ati$ ....................................................................68

c(cles @n?7 and t$e c$ange of t$e residual strengt$ is a&&ro;imated b( a &ower4law growt$

e'uation7

dR@n? = 4A@crl6m5R@nl8 41

dn +'. :./in w$ic$ A@cr? is a function of t$e ma;imum c(clic stress @cr?. ,md m is a constant. T$is model $as

been used b( man( aut$ors 521:422" is t$e static strengt$.

( considering t$at at t$e number of c(cles to failure @ r?. t$e residual strengt$ @R@n?? is e'ual to

t$e a&&lied stress @cr?7 +'. :.13 reduces to,

(m m(m%n3 ;(m 4 s 5cr n

, NI

+'. 3.11

+'. :. I I e;&resses t$e residual strengt$ @R@n??7 as a function of static strengt$ @RP.

number of c(cles @n?7 and number of c(cles to failure @ r?. Also7 m is a constant w$ic$ is found

e;&erimentall(. For different states of stress7 m $as different values7 t$erefore lo full(

c$aracteri>e a material7 large number of e;&eriments s$ould be &erformed.

For com&aring betweer. different models &ro&osed b( different aut$ors7 +'. :.1 I can be

rewritten in t$e following form7

+'. :.12

and b( t$e following algebraic o&eration7


85/97

Chapter S Bateria! r$perty egra#ati$ ......................................................................69

( 111 %n35(-11 5cr111 =?6111 n= +'. :.1*

+'. :.12 reduces to,

R @n?4cr n4

44El44(111 5cr 111 N.

' I

+'. :.1)

+'. :.1) is a normali>ed form of +'. :.11 w$ic$ can be used for t$e &ur&ose of com&aring different

odels. ( using a unified notation and a&&l(ing similar algebraic o&erations to t$e ot$er models &ro&osed in

e literature7 a list of t$e residual strengt$ models is &resented in Table :.1.

R7 cr 7.

1

References Models +;&lanationsKal&in et a!. 5)38Ka$n and im 521:8ang et a!. 5210422


86/97

Table :.1 A summar( of different strengt$ degradation models


87/97


As s$own in Table :.1. Ka$n and im 521:8. ang t a!. 5210422ed number of c(cles NNg is introduced. T$ere

is no effort in t$eir &a&er 522-8 to define t$is function @Table :.1?. In Table :.1. in t$e e'uation

&ro&osed b( Reifsnider and #tinc$comb 5*28 and Reifsnider 5**.*)8. k is a curve fitting

&arameter w$ic$ must be found e;&erimentall(. Karris et a!. 52*142*28 &resented a normali>ed

e'uation consisting of two curve fitting &arameters @Table :.1?. T$e( s$owed t$at NNa.and NN arc

stress inde&endent curve fitting &arameters w$ic$ must be found e;&erimentall(. T$e( em&$asi>ed

t$at stress4inde&endent models. like t$e model &ro&osed b( Fong 52**8. w$ic$ is based on t$e

assum&tion t$at t$e fatigue &rocess is controlled b( a single &rimar( damage mec$anism. is not

realistic. T$e( &ostulated t$at t$eir model &ermitted t$e incor&oration of all modes of damage.,

accumulation7 from +ear $ut to 'u##e #eath. b( t$e ad!ustment of t$e curve fitting &arameters a.and N In t$eir studies7 t$e e'uivalent number of fatigue c(cles for a static loading condition is

assumed to be 3.:7 $owever b( considering t$at a static loading is reall( a 'uarter of a c(clc. t$e

e'uivalent number of c(cles s$ould be c$anged to 3.2:.

( using t$e normali>ation tec$ni'ue7 all different curves for different states or stress. 111

Fig. :.* colla&se to a single curve @Fig. :.)?. For use in t$is researc$. t$e e'uation &resented b(

Karris et a!. 52*172*28 is c$anged to t$e following form @wit$ t$e e'uivalent number of fatigue

c(cles for a static loading condition c$anged from 3.: to 3.2:?,

I

R@n7cr?E5l4@ log@[email protected] ?? 8 @R7-a)a

log@ 1 )+ [email protected]:?

+'. :.1:

R@n?4cr R s4

cr

I

'++++++++++'+++, logn4log.2:3 I log r log.2:

i, ig. :.) ormali>ed strengt$ degradation curve


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Chapter 5 Bateria! r$perty egra#ati$ .............................................. Ii

( $aving static strengt$ @R7?7 state of stress @cr?7 number of c(cles to failure @ r? related to

tl !c state of stress7 and t$e curve fitting &arameters (a and ?. residual strengt$ @[email protected]??7 as a

function of number of c(cles @n? and t$e state of stress @cr?7 is found.

#ince in +'. :. I :7 a and are stress inde&endent &arameters7 t$is model is called t$e

$rma!i"e# 'tre gth #egra#ati$ m$#e!. Kowever. number of c(cles to failure @ r? is a function of t$e state of stress @cr? and t$e stress ratio @ E cr777 76cr77777. ?. T$erefore7 to simulate t$e residualstrengt$ of a unidirectional &l( under a general unia;ial fatigue loading @arbitrar( state of stress and

stress ratio?7 a suitable relations$i& between t$e fatigue life @ r?7 state of stress and stress ratio is

needed. For t$e general case7 i.e. arbitrar( stress ratiC 7 + . :.1: is c$anged to t$e following form,

I

R@n7cr7 ?E5I4@ log@[email protected]:? P"8 @R74cr?9cr log@ 7 ?4 [email protected]:? N

+'. :.1e t$e residual strengt$ of a unidirectional &l( under

arbitrar( state of stress and stress ratio7 an infinite number of e;&eriments must still be &erformed.

As mentioned in t$e &revious c$a&ter7 man( aut$ors restricted t$eir failure criteria to a certain stress

ratio to overcome t$is difficult(. ".Pwever7 as &reviousl( discussed7 assuming a certain stress

ratio for t$e fatigue anal(sis of com&osite laminates is not alwa(s a realistic assum&tion. eforeremoving t$is obstacle b( introducing t$e $ a*i"e# (atigue !i(e m$#e! t$e $rma!i"e# 'ti(( e''

#Ggra#ati$ m$#e! for a unidirectional &l( in a normali>ed form is e;&lained in t$e following.

:.*.2 ormali>ed #tiffness %egradation Model

T$e residual stiffness of t$e material is also a function of state of stress and number of

c(cles. As discussed in t$e &revious section7 t$ere are various number of strengt$ degradation

models. #imilarl(7 t$ere is muc$ researc$ in stiffness degradation 52*)42):8 of com&osite

materials. T$e stiffness degradation models are attractive to man( investigators7 because t$e

residual stiffness can be used as a nondestructive measure for damage evaluati..,.. . ( &erforming a

similar &rocedure used as t$at for normali>ing t$e residual strengt$7 a suitable e'uation for residual

stiffness of a unidirectional &l( can be obtained. ( using t$e normali>ation tec$ni'ue7 all different

curves for different states of stress can be s$own b( a single master curve. T$e idea of

Chapter 5 Bateria! r$perty egra#ati$

+7r W y


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norrnali>ing t$e residual stiffness curves and establis$ing a master curve $as been used b( man(

aut$ors 5071:72)-8. In t$is stud(. a new met$od of nonnali>ation is develo&ed.

Consider a unidirectional lamina under a constant unia;ial fatigue loading. Gnder static

loading7 or e'uivalentl( at nE3.2: c(cles @'uarter of a c(cle? in fatigue. t$e static stiffness of t$e

unidirectional lamina is +7 @Fig. :.:?. It s$ould be mentioned t$at t$e c$aracter NN+.. is used as a

re&resentative s(mbol for t$e stiffness of a unidirectional &l( w$ic$ $as different magnitudes indifferent directions suc$ as7 +777 +77. etc.

E(n)

residual stiffnesscurve .

catastro&$icfailure &oint

0. -' -..3.2: n

Fig. :.: #tiffness degradation of a unidirectional lamina under aconstant unia;ial fatigue loading

( increasing t$e number of c(cles7 under a constant a&&lied stress @cr?7 t$e fatigue

stiffness @+@nU? decreases. Finall(. after a certain number of c(cles w$ic$ is called number or

c(cles to failure (Nr). t$e magnitude of t$e stiffness decreases to a critical magnitude (E"). At t$is &oint7 t$e com&osite lamina fails catastro&$icall(. T$e stiffness degradation or a unidirectional &l(

is s$own in Fig. :.:. T$e aforementioned critical value for stiffness @+7?can be e;&ressed b( t$e

following e'uation7

+'. :.1-

w$ere7

+rE average strain to failure

T$e average strain to failure @+ 1) is assumed to be a constant and inde&endent on t$e state

of stress and number of c(cles. T$is assum&tion is used b( man( aut$ors 52*)42*-8 andverified


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e;&erimentall( in t$is stud(. It s$ould be mentioned t$at for different states of stress7 t$e stiffness

degradation of t$e unidirectional &l( is different. T$e same as for t$e residual strengt$ case. under

$ig$ level state of stress7 t$e residual stiffness as a function of number of c(cles is nearl( constant

and it decreases drasticall( at t$e number of c(cles to failure @Fig. :.


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Chapter " &aterial Propert! 'egradation .............................................. i.J

( using t$is nonnali>ation tec$ni'ue. all different curves for different states of stress in

Fig. :.ed strengt$ degradation e'uation @+'. :.1


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Chapter 5 Bateria! r$perty egra#ati$ NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN

Introducing non4dimensional stresses b( division of t$e mean stress @ m?. t$e alternating

stress @ a? and t$e com&ressive strengt$ @crc? b( t$e tensile strengt$ @ tlN w$ere 'Ecrml Pi. aEcraf Pi.

and cEcrcfcrt. an em&irical interaction curve ma( be derived 52:142:28,

aE f@l4'?F@c9'? +'. :.108ere,

( u and v E curve fitting constants

a E @ ma; 4 minl62 E alternating stress

m E @ Pma; 9 Pminl62 E mean stress

' E

n6 Pt aE

MPafMPtc E MPcfMP1

0.4

0.3

K= 7

NormaliAed l4;.+Caltcrnatini--

stress %a3 ?. 2

0.19 #

..=1#(1+ )(c,O)%"2"0

- 1 -0.5 0 0.5 1 1.5

NormaliAed mean stress

Fatigue Damage Modelling of Composite Material

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