1998 Msc Thesis Stress Intensity Factor Distributions in Bimaterial Systems - A Three-dimensional...

7/30/2019 1998 Msc Thesis Stress Intensity Factor Distributions in Bimaterial Systems - A Three-dimensional Photoelastic Inve

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STRESS INTENSITY FACTOR DISTRIBUTIONS IN

BIMATERIAL SYSTEMS - A THREE-DIMENSIONAL

PHOTOELASTIC INVESTIGATION

by

Eric F. Finla yson

Thesis submitt ed to the Fa culty of th e

Virginia Polytechnic Insti tute and State Universi ty

in partial fulfillment of the requirements for the degree of

MASTER OF SC IE NCE

I N

E N G I N E E R I N G M E C H A N I C S

APPROVED :

Char les W. Smith , Chairman

D a vid A. D illa rd Rona ld W. La ndgra f

February, 1998

Blacksburg, Virginia

Key Words: P hotoela stici ty , S tress I ntensity , Mode-Mixity , B imat eria l ,

In t er fa ce , Fracture


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STRESS INTENSITY FACTOR DISTRIBUTIONS IN BIMATERIAL

SYS TEMS - A TH RE E-DI MENS IONAL P H OTOEL ASTIC

INVESTIGATION

By

Eric F. Finla yson

Char les W. Smith , Chairman

Engineer ing Mechanics

(AB S TR ACT)

Stress- freez ing photoelas t ic exper iments are conducted us ing two

dif ferent sets of photoelast ic mat eria ls to investigat e stress int ensity behavior

near t o a nd coincident wi th bima ter ia l in t er faces . Homogeneous , bonded

homogeneous, and bonded bimaterial single edge-cracked tension specimens

a re ut ilized throughout th e investiga tion for compa ra tive purposes. The first

series of tests involves machined cracks obliquely inclined to the direction of

fa r f ield tensile loa ding. Mixed-mode str ess intensit y factors ar e observed

and quant i f ied us ing a s impl i f ied analy t ica l a lgor i thm which makes use o f

e x p e r i m e n t a l l y m e a s u r e d d a t a . I n t h i s s e r i e s o f t e s t s , t h e b i m a t e r i a l

specimens consis t o f a photoelas t ic mater ia l bonded to the same mater ia l

conta ining a moderate quant i ty o f a luminum powder ( for e las t ic s t i f fening

purposes). Moderat e yet s imi la r increases in s tress in tensi ty factors are

obser ved in bonded homogeneous and bonded b imater i a l spec imens ,

suggest ing the presence o f bondl ine res idual s t resses (ra ther than elas t ic

modulus misma t ch) a s the prima ry cont ribut ing factor. The second series of

tes ts involves the bonding o f mutual ly t r ans lucent photoelas t ic mater ia ls

w hose elast ic moduli differ by a r a tio of approxima tely four to one. Ma chined

notches are placed both near and co incident to the bimater ia l in ter faces .Mode-mixity and increases in stress intensity are found only in bimaterial

specimens whose cra cks a re placed close to th e bondl ine. U sing the

materials from the f irst ser ies of tests i t is shown that the increases in these

near-bondline experiments are due to thermally associated elastic mismatch

p r o p e r t i e s ( i n c u r r e d d u r i n g t h e s t r e s s f r e e z i n g c y c l e s ) r a t h e r t h a n

mechanical mismatch proper t ies .


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Acknowledgments

I w ould f irst l ike to tha nk Dr. C .W. Sm ith for the guidan ce, support , a nd

a tt ention tha t he ha s given to me throughout th is resea rch. I l ikew ise th a nk

him for providing me with the opportunity to work in his laboratory while

pursuing my degree. I w ould a lso l ike to acknowledge th e support of th e

other members o f my commit tee : Dr . D .A. Di l lard and Dr . R.W. Landgraf .

For his insight and assistance with experimental procedures, I would l ike to

tha nk Dr. Da vid H. Mollenhauer .

For their f inancial support I wish to thank the Phil l ips Laboratory who

provided funding t hrough the H ughes STX Corpora tion. I w ould also like to

thank the D epar tment o f Engineer ing Mechan ics a t Vi r g in i a Po ly technicInstitute for the use of their laboratory facilities.

L a s t l y , I w o u l d l i k e t o g r e a t l y t h a n k m y p a r e n t s f o r t h e i r

encouragement and support throughout my entire education.

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Table of Contents

ABSTRACT.........................................................................................................................ii

Acknowledgements.........................................................................................................iii

List of Tables......................................................................................................................vi

List of Figures....................................................................................................................vi

1.0 In t rodu ction. ....... ....... .... ............................................................................................1

2.0 Ana lytical And Experimenta l Considera tions.. . .. . .. . .. . .. . .. . . .. . .. . .. . .. . .. . .. . . .. . .. . .. . ..2

2.1 B ima teria l Fra cture........................................................................................2

2.1.1 In terfa ce And S ub-In terfa ce Fra cture...........................................3

2.1.2 Applicat ions - Rocket Motor G ra ins...............................................4

2.2 Ana lytica l Development.................................................................................6

2.2.1 Formulat ion Of St ress Int ensity R ela tions.. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ..6

2.2.2 Sm ith Ext ra pola tion Method...........................................................11

2.2.3 G enera lized Irw in Meth od..............................................................12

2.2.4 Mode I a nd Mode II SI F Determina tion... .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . ..14

2.2.5 Da ta Read ing Line Orient a tion......................................................16

2.3 Experiment a l P rocedures............................................................................24

2.3.1 Model Const ruction...........................................................................24

2.3.2 Therma l Cy cling................................................................................25

2.3.3 Loading Methods................................................................................27

2.3.4 S tr ess Freezing - 3D Ana lysis.........................................................27

2.3.5 Slice Ana lysis......................................................................................30

2.3.5.1 Ta rdy Meth od.......................................................................32

2.3.5.2 Fr inge Multiplicat ion........................................................33

2.3.6 Da ta Acquisition.................................................................................35

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3.0 Inclined Single Edg e-Notched Test Series.......................................................37

3.1 Model Configura tions...................................................................................37

3.2 Ma teria ls................................................. .........................................................38

3.3 Ana lytica l Considera tions...........................................................................47

3.4 Result s....................... ........................................................... ............................48

3.4.1 P re-Load St ress Fields......................................................................48

3.4.2 P ost-Load St ress Fields.....................................................................53

3.4.3 Norma lized St ress Int ensity Fa ctors............................................65

3.5 Discussion........................................................................................................72

4.0 P a ra llel C ra ck Test Series....................................................................................74

4.1 Model Configura tions...................................................................................744.2 Ma teria ls................................................. .........................................................77

4.3 Ana lytica l Considera tions...........................................................................81

4.4 Result s....................... ........................................................... ............................82

4.4.1 P re-Load St ress Fields......................................................................83

4.4.2 P ost-Load St ress Fields.....................................................................95

4.4.3 Norma lized St ress Int ensity Fa ctors..........................................107

4.5 Discussion......................................................................................................113

5.0 Su mm a ry an d Fut ure Work..............................................................................120

REFERENCES.................................................................................................................122

AP P E ND I X A: P hotogra phic Techniq ues..............................................................124

AP P END IX B : Ma teria l Fringe Va lue Determina tion... . . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . . ..131

VITA.................................................................................................................................137

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List of Tables

Table 3.1 Test specimen ident i f ica t ion , composi t ion , a nd geometry . .. .. .. ..44Table 3.2 Ar a ld ite a nd a luminum-ar a ldi te mater ia l proper t ies .. .. .. .. .. .. .. .45Table 3.3a N or mal ized s t r ess in tens ity va lues for the b= 45 degree test

series........................................................................................................66Table 3.3b N or mal ized s t r ess in tens ity va lues for the b= 30 degree test

series........................................................................................................67Table 3.3c N or mal ized s t r ess in tens ity va lues for the b= 15 degree test

series........................................................................................................68Table 3.3d N or mal ized s t r ess in tens ity va lues for the b= 0 degree test

series........................................................................................................69Table 4.1 Test specimen ident i f ica t ion , composi t ion , and geometry . .. .. .. .78Table 4.2 P LM-4B a nd P SM-9 mat eria l properties. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. . .. .. .. ..80Table 4.3 Mix ed-mode pr oper t ies measur ed f r om of f-bond line

bimaterial specimens........................................................................106Table 4.4a N or mal i zed s t r ess in tens ity va lues for t he homogeneouscont rol specimen .................................................................................110

Table 4.4b Norma l ized s tress in tensi ty va lues for the bondedhomogeneous specimens..................................................................111

Table 4.4c N or mal ized s t r ess in tensi ty va lues for the bondedbimaterial specimens........................................................................112

List of Figures

Figure 2.1 G eneralized 2-D cracked-body stress field.. .. . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .7Figure 2.2 Fr inge order dependence upon the polar coordinates for use

in esta blishing Ir w ins criterion.....................................................13Figure 2.3 Nea r-t ip f r inge pat t ern dependency upon far f ie ld loading

cond it ions ... . ...........................................................................................15Figure 2.4 Representa tive photoela stic slice dat a .. .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. .17Figure 2.5 Mode-mixi ty dependence upon reading line or ienta t ion

a ng le........ . ....................................................... ........................................18Figur e 2.6 D efin it ion of angular va r iab les used for deter min ing the

orientation of the reading line angle...............................................20Figur e 2.7 Cur ve f it t ing of numerica l da ta f r om the incl ined cr ack

solution....................................................................................................21Figur e 2.8 Read ing line ang le var i a t ion as a funct ion of cr ack

in clina t ion. ... ... .......................................................................................23Figure 2.9 Time-temperat ure therma l cycle used for P SM-9/P LM-4B

a nd a ra ldite specimens......................................................................26Figure 2.10 Loading a ppara tus used for P SM-9/P LM-4B a nd a ra ld i te

specimen s.... ...... ...... .. .................................................... .........................28

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Figure 2.11 Orienta tion of slices ta ken near th e crack tip region of ea chphotoelastic specimen.........................................................................31

Figure 2.12 Fringe multiplica tion a ppara tus used for increasing thequa nt ity of photoela stic da ta obta ined from t hin slices.. .. . . . .. . . . ..34

Figure 3.1 b= 45 deg. inclined-crack specimens - geometry a nd loa ding... .39Figure 3.2 b= 30 deg. inclined-crack specimens - geometry a nd loa ding... .40Figure 3.3 b= 15 deg. inclined-crack specimens - geometry a nd loa ding... .41Figure 3.4 b= 0 deg. inclined-crack specimens - geomet ry a nd loa ding... . . .42Figure 3.5 v-notch dimensions a nd tip displacement in bima teria l

specimen A12.........................................................................................43Figure 3.6 P re-loa d residua l str ess distr ibution nea r bondline a nd

v-notch tip a rea s of bonded homogeneous specimen A5............50Figure 3.7 Residua l str ess fringe orient a tion a s a function of crack

inclination angle...................................................................................51Figu re 3.8 Specimen A1 post-load frin ge field........... .............. ............. .............54Figu re 3.9 Specimen A2 post-load frin ge field........... .............. ............. .............55

Fig ur e 3.10 S pecimen A3 post-load frin ge field....... ............ ........... ............ .........56Fig ur e 3.11 S pecimen A4 post-load frin ge field....... ............ ........... ............ .........57Fig ur e 3.12 S pecimen A5 post-load frin ge field....... ............ ........... ............ .........58Fig ur e 3.13 S pecimen A6 post-load frin ge field....... ............ ........... ............ .........59Fig ur e 3.14 S pecimen A10 post-load frin ge field....... ............ ........... ............ .......60Fig ur e 3.15 S pecimen A11 post-load frin ge field....... ............ ........... ............ .......61Fig ur e 3.16 S pecimen A12 post-load frin ge field....... ............ ........... ............ .......62Figu re 3.17 S pecimen A1 midd le slice ima ge......................................................64Figure 3.18 LMR va lues of norma lized mode I stress int ensity fa ctors

for various v-notch inclination angles............................................71

Figure 4.1 B onded homogeneous specimens - geomet ry a nd loading... . . . . .75Figur e 4.2 B onded bima teria l specimens - geomet ry a nd loa ding... . . . . . .. . . .76Figure 4.3 Loca tion a nd length of da ta zone used for SI F determin a tion..84Fig ur e 4.4 Specimen B pre-loa d fringe field.......................................................86Fig ur e 4.5 Specimen B 1 pre-loa d fringe field.....................................................87Fig ur e 4.6 Specimen B 2 pre-loa d fringe field.....................................................88Fig ur e 4.7 Specimen B 3 pre-loa d fringe field.....................................................89Fig ur e 4.8 Specimen B 4 pre-loa d fringe field.....................................................90Fig ur e 4.9 Specimen B 5 pre-loa d fringe field.....................................................91Fig ur e 4.10 S pecimen B 6 pre-load fringe field............ ................ .............. ...........92Fig ur e 4.11 S pecimen B 7 pre-load fringe field............ ................ .............. ...........93Fig ur e 4.12 S pecimen B 8 pre-load fringe field............ ................ .............. ...........94

Figu re 4.13 H omogeneous contr ol specimen B post -loa d fringe field...........96Figu re 4.14 S pecimen B 1 post -loa d fring e field...................................................97Figu re 4.15 S pecimen B 2 post -loa d fring e field...................................................98Figu re 4.16 S pecimen B 3 post -loa d fring e field...................................................99Figu re 4.17 S pecimen B 4 post -loa d fringe field.................................................100Figu re 4.18 S pecimen B 5 post -loa d fringe field.................................................101Figu re 4.19 S pecimen B 6 post -loa d fringe field.................................................102Figu re 4.20 S pecimen B 7 post -loa d fringe field.................................................103

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Figure 4.21 Specimen B 8 post-loa d fringe field.................................................104Figure 4.22 LMR values of norma lized mode I s tress intensity fa ctors . . .. .114Figure 4.23 Compar ison of a ra ld ite/a luminum-f i lled a ra ld ite n ear-t ip

fring e field w ith correspondin g P S M-9/P LM-4B specimen.....117

Figur e A.1 High ma gnifica tion crack-tip fringe fields... . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .125Figure A.2 Optica l a r ran gement used for close-up photography of

cra ck-tip regions.................................................................................126Figur e A.3 P la te g lass techn ique used for cr ea t ing a un i for m coa t ing

of index matching fluid.....................................................................129Figur e A.4 High ma gnifica tion crack-tip fringe fields... . . .. . . .. . . .. . . .. . . .. . . .. . . .. . . .130

Figure B.1 4-P oint bend specimen geometry a nd loa ding.... . .. . . .. . .. . .. . .. . .. . .. . . .134

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1.0 Introduction

An ex per imenta l inves t iga t ion i s under tak en which invo lves two

ser ies of b imat er ia l fr a ctur e tes ts . Measurements of s t ress in t ensi ty and

mode-mixity a re ta ken using t he st ress freezing t echnique of photoela sticity .

An analy t ica l development per ta ining to the work in both tes t ser ies is

presented in Ch a pter 2. Discussions relat ing to the experimenta l procedures

uti l ized throughout t he study ar e likewise present ed in Cha pter 2. Cha pter 3

dea ls wit h t he first series of experiments. This test series involved th e use of

inc l ined s ing le edge- cr ack ed spec imens in b imater i a l sys tems fo r the

qua nt i f ica tion of mixed-mode str ess int ensity fa ctors . Model configura tions

a s w ell a s ma ter ia l a nd a na ly t ica l considera t ions a re presented accordingly .

P hotogra phic a nd numerica l results a re f inal ly presented and discussed. An

investigation into single edge-cracked specimens containing cracks near to

and co inc iden t w i th b imater i a l in ter f aces i s cons ider ed in Chapter 4 .

Di f ferent crack geometr ies and photoelas t ic mater ia ls were used for th is

second series of experiments. An a na logous forma t t o th a t of Cha pter 3 ha s

been ut i l ized for the presenta t ion of procedures a nd resul ts . Ch a pter 5

summar izes the mater i a l pr esen ted in th i s s tudy and br ie f l y d i scusses

possible future work to be performed in the experimental area of bimaterial

fracture.The primar y int entions of this study w ere to approxima tely model a nd

quan t i f y the behav io r o f f l aws con ta ined near and w i th in the in ter f ace

between rocket propel lant and a rubber l iner wi thin so l id rocket motor

gra ins . This s tudy wa s in tended to be a supplement to invest iga t ions tha t

have been and are current ly being made us ing actual rocket motor gra in

ma terials . The photoelast ic ma teria ls used in this w ork were selected for th e

resemblance of their elastic properties to those of the rocket motor grain

ma ter ia ls . These rubber-l ike proper t ies were simula ted w i th t he use of

frozen stress photoelast ici ty . The photoela st ic a na lyses performed in th is

s tudy present quant i ta t ive resul ts which have been used for the qual i ta t ive

assessment o f the s igni f icance o f the bimater ia l f r acture scenar io for the

part icular cla ss of ma teria l properties found in solid r ocket m otor gra ins.

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2.0 Analytical And Experimental Considerations

Mater i a l con ta ined w i th in th i s chapter pr imar i l y per ta ins to the

m a t h e m a t i c a l d e v e l o p m e n t s a n d e x p e r i m e n t a l p r o c e d u r e s t h a t w e r e

required to generat e th e results found in Cha pters 3 a nd 4. Thea n a l y t i c a lcons ider a t ions d i scussed in th i s chapter a r e in i t i a l l y pr esen ted in the

general form from which several dif ferent algori thms for generating stress

intensity factors from photoelastic fr inge patterns have been proposed and

ut i l ized . From this general formula t ion, the a lgor i thm used in th is s tudy ,

known as the Smith Extrapola t ion Method, i s developed into a f inal form

sui t a b le for use in mixed-mode f r a c tur e a na lyses . F ina l l y , a r ecen t ly

formulated analys is which approximates the or ienta t ion f rom which data i s

t o be rea d in mixed-mode edge-cra cked specimens is presen t ed. This

par t icula r a na lys is ul t imat ely served as a n a id towa rds in terpret ing the near

tip photoelastic fr inge patterns found in the inclined-crack series of tests

considered in Chapter 3.

F o r t h e p h o t o e l a s t i c w o r k c o n s i d e r e d i n t h i s s t u d y , m u l t i p l e

e x p e r i m e n t a l p r o c e d u r e s d e a l i n g w i t h m o d e l c o n s t r u c t i o n a n d

document a tion were uti l ized. Since relat ive compa risons a mong models w a s

a key feature in each series of experiments, a strict regimen of consistency in

model construct ion was pr ior i t ized in an ef for t to minimize ex traneoussources of er ror . Exper imenta l procedures are d iscussed sys t ema t ica l ly

beg inn ing w i th the in i t i a l cons t r uc t ion o f models and end ing w i th the

methods involved in extra cting da ta from these models .

2.1 Bimaterial Fracture

A br ief review o f theoret ica l and exper imental work that has been

per formed in the area o f in ter face and sub- inter face bimater ia l f r acture

mecha nics is present ed in th e follow ing sub-sections. The progress th a t ha s

been made in th is area is reviewed whi le s imul taneously d iscuss ing the

associa ted l imi ta t ions f rom both theoret ica l and exper imental s tandpoints .

Finally, the intended application of this study to rocket motor grains is briefly

discussed.

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2.1.1 Interface And Sub-Interface Fracture

For decades now ther e has been much in ter es t in the complex

si tuat ion concerning a crack located a t or near an in ter face between two

d i s s im i l a r m a t e r i a l s . F r o m a t h e or e t i ca l s t a n d p oi n t , t h i s b i m a t e r ia l

phenomenon presents a ra ther unique and complicated s i tuat ion in which

tw o inf in i te qua nt i t ies ex is t a t the sa me point : a sharp crack t ip and a jump

discontinuity in material properties. From a physical s tandpoint , analogous

di f ficul t ies exis t . I f tw o ma ter ia ls a re assumed to be joined a t a common

inter face and are l ikewise assumed to be able to res is t tens i le and shear

str esses, a dherent forces of some form must be present . In pra ctice, th ese

adherent forces are usually derived from either a third material or from the

ca s t ing and cur ing of one mat er ia l aga ins t a nother . For the case in wh ich a

third mater ia l i s used to fas ten two mater ia ls to one another , the devia t ion

from t he theoretical bima teria l model is obvious. For the case in which one

mater i a l i s cas t aga ins t ano ther , r es idua l s t r esses der ived f r om var y ing

coef f ic ients o f thermal expansion (and thus constra int agains t vo lumetr ic

changes in the cast material) represent just one of several considerations

w hich in pr a c t i ce ma y devia t e f r om the theor e t i ca l model . Al though

var ia t ions between theory and actual i ty wi l l inevi tably ex is t in problems

concerning bima teria l fracture, t he qua nt i f icat ion of these extra neous fa ctorsmay in some cases be made thr ough the combined use o f jud ic ious

experiments and analytical approximations.

One o f the f i r s t analy t ica l models concerning bimater ia l f r acture was

presented by Williams 1 in 1959. P erha ps the most common result from th is

work wa s th e a na lytica l ly derived presence of a n oscil la t ing st ress f ield in t he

high singular i ty zone nea r th e cra ck t ip. Although no experimenta l evidence

displaying this effect has been presented, several attempts to either include or

ma thema t ica l ly nega te th is fea ture have been ma de. Several a na lyses were

presented in the mid 1960s which dealt with the generalized stress field near

the t ip of a crack between dissimilar materials 2,3,4. These a na lyses however

did not ef fectively contain traction free crack surfaces: a property which at

t imes may no t co r r e l a te to the phys ica l r ea l i t y o f a f r ac tur e s i tua t ion .

C o m n i n o u 5 l a ter in troduced an analys is conta ining a f r ic t ionless contact

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zone which eliminated the oscil latory near-tip stresses found in previous

w ork. Additional a na lyses6,7,8 ha ve been presented m ore recently w hich dea l

with such factors as crack extension and direction, contact zone size and i ts

dependence on remote loa ding, a nd t he concept of a cha ra cter ist ic bima teria l

c r a c k l e n g t h w h i c h d e p e n d s u p o n t h e p a r t i c u l a r m a t e r i a l s u n d e r

cons ider a t ion . A par t i cu la r ly compr ehens ive a na lys i s a nd r ev iew of

publ ished works concerning bimater ia l f r acture phenomena is g iven by

Hutch inson and Suo9. Of par ticular interest to the work considered in this

present s tudy are the sect ions per ta ining to mixed-mode f racture and the

fra cture of sub-interfa ce bima teria l cracks.

Ex per imenta l inves t iga t ions pr imar i l y employ ing opt i ca l methods

ha ve more recently been a pplied to the f ield of bima teria l fra cture. Liecht i

a n d C h a i11 have demonstrated the use of optical interferometry methods for

measuring crack t ip opening displacements (CTOD) and fracture toughness

in an epoxy-gla ss bima teria l . Alterna te optical techniques have also been

recent ly used 11,12 for the measurement of mode-mixity and complex stress

intensi ty fa ctors . A photoelas t ic s tudy per formed by Chian g et . a l .13 is of

par t i cu la r in ter es t due to the i r a t tempt a t mak ing a d i r ec t compar i son

between analy t ica l and exper imental resul ts for a centra l crack loaded in

p u r e r e m ot e s h ea r . I n s t e a d of e v a l u a t i n g s t r e s s i n t e n s it y f a c t o r s, a

compar ison o f photoelas t ica l ly obta ined pr incipal shear s tress magni tudesa nd directions w a s ma de w ith an a lytical results . The a na lytical ly predicted

shear mode was indeed experimental ly found by observing the presence of

fringe loops in the cra ck tip vicinit y. The compa rison yielded mixed results

with good correlation in some regions while in others, deviations in excess of

30%were found.

2.1.2 Applications - Rocket Motor Grains

T h e c u r r e n t s t u d y u n d e r c o n s i d e r a t i o n i n v o l v e s t h e u s e o f

photoelast ici ty to qua lita t ively a nd qua nti ta t ively measure the stress intensity

conditions present a t a nd nea r t he bima teria l interface of s imila r rubber-l ike

ma ter ia ls . Quest ions ha ve been ra ised concerning th e s tora ge o f in ter-

continent a l bal l is t ic missi les (ICB M) a nd t he role tha t potent ial ga ps betw een

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rocket propel lant and i ts mat ing rubber l iner may play in their s t ructural

a nd chemical burn-ra te sta bili ty . The photoela stic study recently underta ken

was in tended to ser ve as an add i t iona l sour ce o f in fo r mat ion fo r the

assessment of the behavior of f laws contained within bimaterial interfaces of

rela t ively s imi lar e las t ic modul i. The s ta te of ma ter ia l incompress ibi l ity

(n= 0.5) found in both t he rocket propellan t a nd i ts m a ting ru bber liner wa s

l ikewise ef fect ively model led by us ing the s tress f reez ing technique o f

photoelasticity.

Depending on th e length wise location of a f law in a rocket motor grain,

va rying sta tes of elast ic pla ne stra in and plan e stress wil l l ikely exist . Single

edge-cracked photoelastic specimens of moderate thickness were utilized in

the photoela stic ana lysis t o provide a m easure of var ia t ion in stress intensity .

Throughout the analys is , an approximate s ta te o f general ized plane s tra in

w a s assum ed to ex is t in each specimen. Modera te elevat ions in s tress

intensi ty over the near sur face values were consis tent ly obta ined f rom the

pho toe las t i c ana lys i s o f th in s l i ces ex t r ac ted f r om the midsec t ion o f

specimens. Ela stic modulus ra tios in th e ra nge of 2:1 to 4:1 w ere used to

roughly s imulate the magni tudes o f modulus mismatch found between the

a ctua l rocket propellant a nd ru bber l iner .

The work presented in th is s tudy is ul t imately in tended to be o f a

qua l i t a t ive na tur e a l though many quan t i t a t ive measur es have been madethr oughout . A qua l it a t ive s t a ndpoin t ha s been adopted f r om th i s s tudy

pr imar i ly due to the fact that a l though the photoelas t ic mater ia ls used to

model the problem are mechanical ly similar , they indeed are not the actual

rocket motor gra in ma ter ia ls under invest iga t ion . As is the ca se for a l l

materials , unique peculiar i t ies are bound to exist in the specif ic propellant

ma ter ia l a nd rubber l iners . As such, the mea surements a nd conclus ions

pr esen ted in th i s wor k a r e in tended to pr ov ide ins igh t in to mater i a l

i n t e r a c t i o n s a n d b e h a v i o r s a s s o c i a t e d w i t h t h e c o m m o n e l a s t i c a n d

mechanical properties inherent in al l solid materials (e.g. s t i f fness) and are

specifica lly gear ed to t hose of th e rocket propellan t scena rio. This stu dy is

intended to be a supplement to exper iments that have been and wi l l be

conducted on the actual materials .

5


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2.2 Analytical Development

A n a l g o r i t h m w h i c h c o m b i n e s t h e o r e t i c a l f o r m u l a t i o n s w i t h

exper imental ly obta ined informat ion is developed us ing class ica l f r acture

mecha nics together with th e theory of photoelast ici ty . A general formula tion

which rela tes photoelas t ic parameters to the c lass ica l f r acture problem is

f irst considered. Fr om th is formu la tion, th e development of a s implified

a lgorithm (The Smit h Ext ra pola tion Meth od) is outlined an d discussed. The

governing equat ions o f the ex trapola t ion method, which were used for

determining al l of the experimental s tress intensity factors presented in this

work, are f ina l ly presented in their general form.

2.2.1 Formulation of Stress Intensity Relations

In order to make use of experimental ly obtained data through the use

of photoelas t ic i ty , i t i s necessary to express the analy t ica l rela t ionships

der ived f r om theor e t i ca l f r ac tur e mechan ics in te r ms o f pho toe las t i c

par a meters . Consider th e plana r problem of a cra ck in a homogeneous

ma terial a s given in Fig. 2.1. The genera lized tw o dimensiona l equa tions14,15

relating mode I and mode II s tress intensity factors to the stress f ield near a

crack tip ar e given by:sx x =

1

2prF 1 K I + F 2 K I I

sy y =1


sx y =1


(2.1)

where KI

a n d KI I

are the respect ive mode I and mode I I s t ress in tensi ty

factors.

6


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sxx

syy

sxx

syy

t

xy

y

x

r

q

crack tip

Figu re 2.1: G enera lized 2-D cra cked-body stress field.

7


16/145

The t rigonometr ic functions (F 1 - F 5) found in E q. 2.1 are given by:

F 1 = cos

q

2

1 - s in

q

2

s in

3q

2

F 2 = - si n

q

2

2 + cos

q

2

cos

3q

2

F 3 = cos

q

2

1 + s in

q

2

s in

3q

2

F 4 = s i n

q

2

cos

q

2

cos

3q

2

F 5 = cos

q

2

1 - s in

q

2

s in

3q

2

(2.2)

The st ress field solution given by t his set of equa tions is for t he case in w hich

the far f ield loading is considered to be two-dimensionally hydrostatic. To

attempt to account for the possibil i ty that the non-singular stress f ield away

from the crack tip is not hydrostatic, a generalized non-singular stress field 16

i s super imposed over th e s ingula r s t ress f ie ld given in Eq . 2.1. This

considerat ion y ields the fo l lowing modi f ied form of the near t ip s tress

equations:

sx x =1

2prF 1 K I + F 2 K II - s

o

x x

sy y =1


o

y y

sx y =1


o

x y

(2.3)

where the functions F 1 through F 5 r emain unchanged and the ter ms soxx ,

soy y , a n d so

x y are referred to as the non-singular stress components. I t is

noted that for the case in which the stress f ield distant from the crack t ip is

in-plane hydrosta t ic , the non-s ingular s t ress components (soxx , so

y y , a n d

soxy ) must be identical ly zero in order to satisfy the 2-D hydrostatic stress

8


17/145

field relat ions given by Eq . 2.1. The individua l ma gnit udes of soxx , so

y y , and

soxy will in general not be equal in magnitude to the corresponding far f ield

loads (sx x ,syy , s x y ). These ter m s w i l l be car r ied a long thr ough the

development o f the a lgor i thm for conver t ing photoelas t ic data in to s tress

intensi ty factors . I t w i l l be seen that the inclus ion o f these non-s ingular

terms is o f pa ra mount impor ta nce over their actua l ma gni tudes when using

th e Sm ith Ext ra polat ion Meth od considered in S ection 2.2.2.

The theoretical s tress f ield solution can now be combined with the

stress-optic law of photoelasticity 17,24. In tw o-dimensiona l form, the str ess

optic law ca n be writ t en as :

tmax =

s1 - s2

2=

Nf

2t(2.4)

wher e s1 a nd s2 are the pr incipal in plane normal s tresses , tm a x i s the

principal in plane shear stress, and t is the model thickness perpendicular

to the transmission direction of l ight . The photoelastic parameters in this

rela t ion are character ized by the f r inge order N and the mater ia l f r inge

va lue f . The ma teria l fr inge va lue is an ea si ly determina ble const a nt for a

g iven pho toe las t i c ma ter i a l . Fr inge or der s a r e ob ta ined f r om v i sua l

inspection of the photoelastic model while in the polar iscope, and usually

document ed photogra phica l ly . Deta i ls of fr inge order and ma ter ia l fr inge

value determination are given in Appendix A.

Eq . 2 .4 can now be combined wi th the elas t ic i ty equat ion rela t ing

principal in-plane shear stress with the coordinate stresses sxx , syy , and sxy

to give:

tmax =Nf

2t=

(sxx- syy)2

+ 4sxy2

2(2.5)

Combining this relation with the general s tress f ield relations given by Eqs.

2.3 yields th e follow ing solut ion w hich relat es the photoela stic va ria bles to the

cracked body st ress field:

2tmax =Nf

t=

A

2pr+

B

2pr+ C (2.6)

9


18/145

wher e :

A= K I(F 1 - F 3 ) + K II(F 2 - F 4 )2+ 4 K IF 4 + K IIF 5

2

B = - 2(so

x x - so

y y) K I(F 1 - F 3 ) + K II(F 2 - F 4 ) - 8so

x y K IF 4 + K IIF 5

C = sox x - soy y

2

+ 4so1 2 2

Eva luat ion a nd simplif ica tion of the functions A, B , a nd C yields:

A= K Isin q + 2K IIcos q 2

+ K IIsin q 2

B = 2 sox x - soy y K Isin q sin

3q

2

+ K II

2sin

q

2

+ sin q cos

3q

2

- 4so

x y K Isin q cos

3q

2

+ K II

2cos

q

2

- sin q sin

3q

2

C = sox x - soy y 2

+ 2sox y 2

(2.7)

T h i s g e n e r a l r e l a t i o n s h i p p r o v i d e s t h e b a s i s f o r s e v e r a l

algori thms18,19,20,21,22 for use in the exper imental determinat ion o f s t ress

intensit y factors using photoelast ici ty . A fa ir ly comprehensive review and

compar i son o f these a lgo r i thms and o ther s i s pr esen ted by Smi th and

Olaosebikan 14. Ma ny of the algorithm s are shown to provide a ccura te results

w hen compa red wi th t heoret ica l solut ions . Ana lys is for use in th e workpr esen ted her e i s based on the Smi th Ex t r apo la t ion Method 1 8 for the

determina tion of stress intensity fa ctors using photoela stici ty . The rat iona l

for the choice of using this method l ies primarily in i ts relative analytical

simplici ty coupled w ith i ts int r insic a bili ty t o al low for a fair ly simple mean s

of a cqui r ing the necessa r y da ta . Fur t her mor e, c losed for m ana ly t i ca l

rela t ionships such as mode-mixi ty (K I I /K I ) are eas i ly der ived f rom the

solution to the extra polat ion method. I t w il l be show n tha t t he determina tion

of the mode-mixity for a given fringe pattern can be found by a single angular

mea surement ma de near t he region of a crack t ip.

10


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2.2.2 Smith Extrapolation Method

To elimina te h igher order term s in E q. 2.6, a polynomia l expa nsion ca n

be taken using the following general form of a binomial series:

(p + q)n= p

n+ np

(n - 1 )q +

n(n - 1)p(n - 2 )

q2

2!+

n(n - 1)(n - 2)p(n - 3 )

q3

3!+ . . . (2.8)

wher e :

p =A

2pr; q =

B

2pr+ C ; n=

1

2

Upon subst i tut ion and expansion o f the f i r s t two terms in the ser ies , the

rela tion betw een st ress intensity a nd t he photoelast ic quan ti t ies becomes:

2tmax =Nf

t=

A

2pr+

B

2 A+

C 2pr

2 A(2.9)

Retaining only the f irst two terms on the r ight hand side of this relation and

rearranging gives :

tmax =A

8p

1

r+

B

4 A(2.10)

Mult iplica t ion of Eq . 2.10 by 8pr/s pa produces the final desired form of

the S mith extra pola t ion equat ion:

K ap

s pa=

K

s pa+

2 B

2s K

r

a(2.11)

wher e :

K ap =Nf

2t8pr

K = A = K Isin q + 2K IIcos q 2

+ K IIsin q 2

s = nominal far field stress

r = distance from crack tip

a = length of crack

11


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Eq. 2.11 represents a l ine whose intercept is given by K and whose

slope is given by th e qua nt ity 2 B /2s K . In order for Eq . 2.10 to representa stra ight l ine, the slope must rema in consta nt . Inspection of the slope term

reveals dependence upon four factors: s , q, K I , a n d K I I . For a given fracture

scenario , the nominal far f ield stress as well as the opening mode and shear

mode stress intensit y fa ctors (K I and K I I ) w ill each be of const a nt va lue. Thus,

the polar angular coordinate q (as measured from the direction paral lel to

the crack) must remain constant . This d ic ta tes that the photoelas t ic data

must be read a long a st ra ight l ine w hose origin is at the crack t ip. Choosing

a f ixed polar a ngle q a long w hich t o read da ta yields da ta sets of the form: (N,

r). This ena bles Eq . 2.11 t o be plot t ed as a funct ion of r/a . A lea st squa res

cur ve f i t to a zone o f da ta tha t appear s r e l a t ive ly l inear can then be

extra polat ed to th e or igin. I t is seen from E q. 2.11 th a t t he intercept of this

l inear curve wi th the coordinate ax is K ap/ s pa yields a value for K/ s pa .

With this value known, what remains to be found is a means of determining

K I and K II individually.

2.2.3 Generalized Irwin Method

The general ized Irwin method 16 yields the addi t ional rela t ionship

between K I a n d K I I as wel l as the reading l ine or ienta t ion angle q for thecomput a tion of K I a n d K I I individua lly using th e Smith extra polat ion meth od.

Irwins criterion is given by the condition that:

Mtm

M q (rm

,qm

)

= 0 (2.12)

The validi ty of this relationship can be ra tionalized w ith t he a ssista nce

of Fig. 2.2 which represents a typical photoelastic shear stress fr inge f ield

near a cra ck t ip . For a ny ar bi t ra ry f ixed va lue of r w i thin the generalregion of the crack t ip, traversal through the angular coordinate q from q = 0

t o q = p/2 results in a n increa se in fr inge order un ti l a peak or ma ximum

value is reached at some value of q, referred to a s qm . As long as qm p/2 ,

the fringe order, which is proportional to tm , wil l then begin and continue to

decrease in ma gnitude. A ma ximum of th e shear st ress t (designa ted a s tm)

12


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fringe order " N"

N+ 1

N+ 2

N+ 3

(r , q=0)

(r, q=p/2)

x

y

3.0

4.0

5.0

6.0

p p4 2

0

N

q

mq )(r,

data reading line

(r, q=0)

mq )(r ,

(r, q=p/2 )

M

M

t

q

ma x= 0

Figu re 2.2: Fr inge order dependence upon the pola r coordin a tesfor use in est a blishing Irw in's criterion.

13


22/145

is thus established as a function of q for fixed values of r and can be found

ma thema t ica l ly by evaluat ing the par t ia l der iva t ive of t (a s given by E q. 2.5)

a nd sett ing the result equa l to zero. The resulting relat ion is given by:

K II

K I

2

-4

3

K II

K I

cot 2q m -

1

3= 0 (2.13)

In the high s ingular i ty zone near the crack t ip , the var ia t ion in qm from

fringe loop to fringe loop tends t o be sma ll. In fa ct, qm tends t o a symptotica l ly

a p p r o a c h a c o n s t a n t v a l u e a s r60 . As r eg ions fur ther aw a y f r om the

singular zone are considered, the non-singular stress components (soxx , so

yy,

a nd soxy ) tend to distort t he fringe loops (Fig. 2.3) in such a w a y a s to va ry qm

in rela tion to the dista nce from t he cra ck t ip. The singular i ty domina nt zone

is the impl ic i t bas is for the character iza t ion o f the f racture mechanicspar a meters . As such, it i s des irable to exper iment a l ly determine qm by

observing f r inges in the s ingular zone near the crack t ip in an ef for t to

minimize error introduced by non-singular stress effects.

2.2.4 Mode I and Mode II Stress Intensity Factor Determination

Eq. 2.13 can now be coupled with Eq. 2.11 to form a system of two

equat ions and two unknowns in KI

a n d KII

. Solving th is syst em yields th e

following expressions for K I a n d K I I :

K I =K

sin2 q m 1 + F q m

2

+ 4F q m cos q m F q m cos q m + sin q m K II = F q m K I

(2.14)

(2.15)

wher e :

F q m =2cot 2q m " 4cot

2

2q m + 33

(2.16)

B y obta ining photoela stic dat a (N,r) a long qm , a plot may be generated using

Eq . 2.10. The dat a to be used is selected over a ra nge in which a st ra ight l ine

ma y be w ell fitt ed. A represent a tive plot of dat a obta ined from a n off-bondline

14


23/145

(a) (b)

Figure 2.3: Near-tip fringe pat tern dependency upon fa r field loa ding conditions.(a ) P ure mode I singula r zone found from hy drosta tic far field loa ding.

(b) P ure mode I sing ula r zone found fr om uni-directiona l far fieldtensile loa ding.

15


24/145

slice is given in F ig. 2.4. The extra pola tion of this st ra ight line curve fit t o th e

K ap/ s pa a xis yields a va lue for K/ s pa . The da ta r eading l ine or ienta tion

a n g l e qm as wel l as the value o f K found f rom the ex trapola t ion are then

substit uted int o Eqs. 2.14, 2.15, and 2.16 to yield the mode I a nd m ode II str ess

intensit y fa ctors . The ma gnitu des of th e a ddit ional stress term s (sox x , so

yy,

a nd sox y ) are therefore unnecessary for the determinat ion o f the s tress

intensity fa ctors . I t is clear though tha t th ese stress terms are embedded in

the s lope o f the curve f i t and are thus accounted for in the ex trapola t ion

method. From these equa tions it is noted tha t t he mixed-mode ra tio is given

explicitly as a function of the reading line angle qm :

K II

K I=

F qm

=2cot 2q m " 4cot2 2q m + 3

3 (2.17)

A plot of this function is given in Fig. 2.5 as are the representative fr inge

pa tt erns corresponding t o situa tions of int erest. Alth ough not shown for plot

sca l ing pur poses , i t can be shown tha t as qm approaches zero , F(qm )

a sympt otical ly a pproa ches inf inity . Thus, for the ca se in which K I I= 0 (pure

opening mode), the reading line and thus fringe loop tips will lie along a line

perpendicular to the crack direction. Likewise for the case in which K I= 0

(pure shear mode), the fr inge loop t ips wil l l ie along a l ine paral lel and

coincident with the crack direction.

2.2.5 Data Reading Line Orientation

I t ha s been noted tha t t he determina tion of the reading l ine angle must

be made by viewing the photoelastic fr inges in the immediate vicinity of the

crack t ip . The clas s ica l f r a cture mecha nics th eory predicts a complete

closure of each singula r fr inge loop a t th e crack t ip. In a ctua li ty , such a

fringe field is never quite achieved due to the finite thickness and root radius

of real cracks found in ma terials . With t his being t he ca se, the fr inges found

close to the crack t ip of a machined f law in a photoelastic material tend to

possess a more gradual , circular curvature (see Fig. 4.13b) rather than the

more sharp and dis t inct ive curvature that i s predicted by the theoret ica l

a na lysis (Fig. 2.3). To ensure tha t subjective interpreta tions of the fr inge

16


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5

10

15

20

25

30

35

40

0.2 0.4 0.6 0.8 1.0

r/a

(a)

5

10

15

20

25

30

35

40

0.2 0.4 0.6 0.8 1.0

r/a

K I

(b)

Figure 2.4: Represent a tive photoelast ic slice da ta .(a ) P hotoelast ic da ta obta ined from t he middle slice of a bima teria l specimen.

(b) Lea st sq ua res curve fit t a ken over selected ra nge of da tafor stress intensity factor determination.

0

0

Kap.

[psi.

in.]

Kap.

[psi.

in.]

data zone

17


26/145

0p

4

p

2

1

2

3

4

5

KI I

KI

/

qm

Rea ding Line Angle

M

ixedModeRatio

4

3 p

8

p

8

q = pm

2

q = 11pm

64

KI I

KI

/ 4 q = 0m

;

= dat a reading line

Figu re 2.5: Mode-mixit y dependen ce upon rea ding line orienta t ion a ngle.

KI I

K = 0I

/KI I

K = 1I

/

18


27/145

f ie lds and thus the reading l ine angles found in th is work were accurate ,

experimental ly obtained reading l ine angles were compared to an analytical

development which combines the Smith Extrapola t ion equat ions wi th the

numerical solution for a semi-infinite single-edge cracked plate of arbitrary

crack inclina tion.

T h e a n g u l a r v a r i a b l e s u s e d i n t h e f o l l o w i n g d e v e l o p m e n t a r e

schema tica lly defined in Fig. 2.6. A ta ble of va lues obta ined from a n umerical

analys is which correla tes mode-mixi ty wi th crack angle or ienta t ion (b) is

ta bula ted by Sih 23 a nd list ed in Fig. 2.7. A second order polynomia l curve fit

was generated us ing th is table o f va lues to y ield an approximate analy t ica l

expression relating the mode-mixity ratio (K I I /K I) to the crack orientation

angle (b degrees) a nd is given by:

K I I

K I= 10

- 3 9 . 319 + 9 . 169b + 0 . 0481b2 (2.18)

Eq. 2.18 can now be combined with Eq. 2.17 from the Smith extrapolation

method. First , rearra nging Eq. 2.17 in terms of qm gives:

cot 2q m =3 F q m

2

- 1

4 F q m (2.19)

Solving Eq . 2.19 for t he rea ding line an gle qm yields:

q m =

90o- t a n

- 1

3 F q m

2

- 1

4 F q m

2

(2.20)

Since F(qm)= K I I /K I , Eq. 2.18 can now be substituted into Eq. 2.20 to give the

reading l ine angle qm a s a function of the cra ck a ngle b:

q m =

90o- t a n

- 1

3 10

- 6 9 . 319 + 9 . 169b + 0 . 0481b2 2 - 1

4 10 - 3

9 . 319 + 9 . 169b + 0 . 0481b2

2

(2.21)

19


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b

27

qm63y

90

s s

reading line

Figure 2.6: Definition of ang ula r var iables used for determingth e orienta tion of the read ing line a ngle.

20


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pK

0.2

0.4

0.6

0.8

1.0

1.2

15 30 45 60 75 900

b (deg.)

K/KII

I

80.0 0.16200 0.1740 1.074175.0 0.23550 0.2225 0.9448

70.0 0.30500 0.2710 0.888560.0 0.46200 0.3370 0.729452.5 0.58400 0.3620 0.619950.0 0.62510 0.3648 0.583645.0 0.70500 0.3642 0.516640.0 0.78180 0.3543 0.453230.0 0.92050 0.3059 0.332322.5 1.00500 0.2470 0.245820.0 1.02860 0.2243 0.218115.0 1.06900 0.1740 0.162810.0 1.09790 0.1188 0.10825.0 1.11600 0.0600 0.05380.0 1.12125 0.0000 0.0000

b FI

FI I

FI I

FI/

I = F sI a pK = F s aI I I I;

Figure 2.7: Cur ve fitt ing of numerica l dat a from th e inclined cra ck solution.

21


30/145

Finally , as seen in Fig. 2.6, the or ientation of the reading l ine relative to a

perpendicular to th e free edge ca n be defined as :

y = b + qm = b +90

o

- t a n- 1

3 10

- 69 . 319 + 9 . 169b + 0 . 0481b

2 2

- 1

4 10 - 3

9 . 319 + 9 . 169b + 0 . 0481b2

2

(2.22)

A plot of y(b) from Eq . 2.22 is shown in F ig. 2.8. In spection of the plot r eveals

that for a wide range o f crack or ienta t ion angles , the reading l ine remains

virtu a lly pa ra llel to th e direction of far field tensile loa ding. In fa ct, for a ll of

the crack orientation angles considered in this study (b=00, 150, 300, an d 450),

the appropr ia te reading l ine remains para l le l to the d irect ion o f far f ie ld

load ing to w i th in 3 degrees . Ima ges ta ken nea r the notch t ips o f th e

inclined-angle araldite specimens seemed to confirm these results . As such,

the pho toe las t i c da ta fo r a l l o f the a r a ld i te spec imens was t ak en a long

rea ding lines oriented pa ra llel to th e direction of fa r field tensile loading.

22


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85

90

95

100

0 15 30 45 60

b (deg.)

y(b)

Figure 2.8: Rea ding line an gle va ria tion as a function of cra ckinclination. For the crack inclinations considered, negligible

deviat ions betw een t he reading l ine or ienta tion a nd t hedirection of far field loa ding ( = 90 ) a re found .

23

(d

eg.)

y(0 ) = 88.9

y(15 ) = 87.9

y(30 ) = 88.7

y(45 ) = 92.6

o

o

o

o

o

o

o

o

b o


32/145

2.3 Experimental Procedures

Three commercial ly available photoelastic materials were used in this

study t o const ruct models of fra cture specimens. A plast ic a ra ldite ma teria l

was chosen for the tests in which a mixed-mode fracture state was induced

by select ing va r ious crack or ienta t ion an gles . For the of f-bondl ine tes t

specimens, two plastic materials available from Photolastic Inc. were used.

These mat erials , tra de na med P LM-4B a nd P SM-9, dif fered from one an oth er

in their e las t ic modul i and cr i t ica l temperatures (Tc ). Above th eir critica l

temper a tur es , bo th mater i a l s possessed the e l as t i c mater i a l pr oper ty o f

incompress ibi l i ty (n= 0 .5). Descr ipt ions concerning th e va r ious methods

incorporated for the preparation, construction, documentation, and analysis

o f the photoelas t ic models considered in th is work are presented in the

following sections.

2.3.1 Model Construction

T he models used fo r a l l t es t s wer e cons t r uc ted f r om pla tes o f

photoela stic mat erial supplied by their respective ma nufa cturers. A nominal

plate thickness of 0.500 in. (12.7 mm) was chosen in order to generate a

mea sura ble va r ia t ion in stress intensity fa ctors th rough th e th ickness. Forthe homogeneous control specimens, s ingle rectangular sections were cut

out of the photoelastic plates and subsequently machined for squareness on

th e i r ha l f inch th ick outer sur f a ces . Al l bond l ine specimens wer e

constructed us ing a method that produced uni formity and consis tency in

bondline th ickness. Ea ch complete bondline specimen consist ed of tw o ha lves

w hich were cut and ma chined from the or iginal pla tes . For th e bonding

process, use of a s imple j ig was employed for maintaining proper al ignment

of th e tw o pieces to be joined. In order to genera te a const a nt t hickness

bondline, a feeler gauge (0.012 in. [0.3 mm]) was inserted between the two

pieces to be joined. With t he feeler ga uge in pla ce, a piece of cellopha ne ta pe

was then placed over the feeler gauge and a long the ent i re length o f the

bondline to be form ed. The specimen w a s th en able to be gent ly tur ned over

whi le apply ing mild pressure to the ends o f the model to ensure that the

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feeler ga uge remain ed in place betw een th e tw o ha lves to be bonded. Removal

o f the feeler gauge a t th is point rendered a uni form gap between the two

specimen ha lves. The ta pe is now seen to serve a dua l purpose: it secures the

two halves properly from one another and i t provides a form or boundary

for holding t he ad hesive w hile curing. The model can n ow be bonded by

simply pouring a dhesive into the t hin ga p between t he specimen ha lves.

P LM-9 resin wa s used for th e bonding of a ll models. When mixed with

i t s c o m p l e m e n t a r y h a r d e n e r a n d c u r e d , t h e r e s u l t i n g b o n d l i n e w a s

compos i t iona l l y s imi l a r to tha t o f the PSM- 9 mater i a l suppl ied by the

ma nufa cturer in sheet form. After thorough mixing an d dega sif icat ion, the

adhesive is deposi ted centra l ly on the gap between the two halves o f the

specimen. To ensure th a t a ir bubbles do not become tra pped a t t he bott om of

th e bondline, adh esive is pla ced only on th e center of the top surfa ce. Thisa l lows the a dhesive to immedia tely f low to th e middle of th e lower surfa ce of

th e bondline a nd th en spread to the low er corners. The bondline is th en

rea dy t o be cured once th e adh esive ha s risen to th e top a nd t hus filled th e gap

initially created by the implementation of the feeler gauge.

2.3.2 Thermal Cycling

The post-cur ing cycle o f PSM-9 as d ic ta ted by the manufacturer

requires a controlled elevation in temperature fol lowed by a dwell during

w hich t he temperat ure is held f ixed . Fina l ly , th e post-cur ing process is

completed by decreas ing the temperature a t a constant ra te back down to

room t emperat ure. This t ime-tempera tu re therma l cycle is depicted in Fig.

2.9. The therm a l cycles used for th e ara ldite specimens ut i l ized the sa me

temper a tur e gr ad ien ts found in F ig . 2 .9 but used a h igher max imum

tem perat ure of 250 F due to th e ma teria ls higher critica l tem perat ure. After

the thermal cycle is completed, photoelastic photographs of the specimen are

taken to document the residual stress f ield near the bondline created by theshrinka ge a ssocia ted wit h curing. Once document a tion is complete, th e v-

notch (a ra ldite s pecimens ) or slot (PS M-9/P LM-4B specimens) is ma chined

in to p l ace . The specimen is th en a ga in pho togr a phed w hi le in the

polariscope to document the effects on the stress field, if any, induced by the

ma chining of the v-notch or slot.

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15 30 45 60 75

250

220

190

160

130

100

70

10 F/hr . 3 F /hr .

Time (hrs.)

4 hrs.

0

Tempera

ture(F)

Figu re 2.9: Time-t empera tu re th erma l cycle used for P S M-9/P LM-4B specimens.Ara ldite specimens w ere subjected t o a peak t emperta tur e of 250 F.

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2.3.3 Loading Methods

Once documentation of the residual stress f ield has been completed,

th e model i s rea dy for loa ding. A fa i r ly s impl is t ic dead w eight loa ding

scheme is used wi thin a large oven so that the model can be loaded whi le

cont rolling i ts tem pera tu re (Fig. 2.10). All fra cture test s considered here

ma de use of a purely tensile far field load. The a ra ldite series of tests ut ilized

a th ree pin loading syst em w hile the P SM-9/P LM-4B series of test s ut ilized a

single pin loa ding system. Due to the a dded kinema tic freedom of th e str ing

syst em of Fig. 2.10, free rota tion of th e loa ded models due t o the inherent la ck

of sym met ry of s ingle edge-notched specimens wa s possible. Du ring th e

comm encement of the la ter test s conduct ed on t he P SM-9/P LM-4B specimens,

i t became apparent that for the specimen geometry and loads being used, a

single pin loading scheme would suffice without causing failure due to stress

concent ra tions near t he pin holes. The three pin loa ding scheme, a l though

capable o f handl ing more load , required more adjustment o f the s tr ing

a ppara tus t o ensure a uniform far f ield tensile stress. The simplici ty of th e

single pin scheme provided consistently uniform far f ield tensile stresses

w hile requir ing minima l set-up t imes. I t is noted tha t mult iple pin schemes

become a necessity w hen th e rela tive a mount of loa d necessa ry t o genera te a

suf f icient ly h igh f r inge order f ield near a crack t ip becomes lar ge. Anex ample o f such an occur r ence would be tha t o f a spec imen w i th a

pa rt icular ly low cra ck lengt h vs. w idt h r a t io (a /W -> 0).

2.3.4 Stress Freezing - Three-Dimensional Analysis

The technique commonly referred to as the frozen stress method 24,25

w a s e m p l o y e d f o r e v e r y p h o t o e l a s t i c e x p e r i m e n t c o n d u c t e d .

Phenomeno log ica l de ta i l s o f pho toe las t i c mater i a l behav io r a t e leva ted

temperatures is covered in detail by Cernosek26. From a physica l s ta ndpoint ,

the f r ozen s t r ess method o f pho toe las t i c i ty invo lves the load ing o f a

pho toe las t i c model whi le above i t s c r i t i ca l temper a tur e (Tc ). This

temperature is dependent upon the material being used and often t imes the

a ge of the ma teria l as well . When the photoela stic model reaches i ts Tc , it

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lead shotcanister

loa ding plates

swivels

str ing

Figu re 2.10: Loa ding a ppa ra tu s used for a ra ldite specimens. P S M-9/P LM-4Bspecimens were loa ded simila rly wit h th e exception th a t a single

loa ding pin w a s used on ea ch specimen end.

edge-crackedspecimen

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becomes quite soft a nd rubbery. The ma teria l fringe value drops significa nt ly

th us y ield ing a n increased sensi t iv i ty to s tr ess . A tw enty to th ir ty fold

decrease in material fr inge value is of ten observable with most photoelastic

ma ter ia ls . Once th e model i s completely a bove i ts cr i t ica l tempera tur e

regime, the desired mechanical load is placed on the model and al lowed to

st a bilize. While ma int a ining th e loa d on the specimen, a cont rolled cooling

cycle of the model back down to room temperature will effectively lock or

freeze the photoelast ic str ess field in place. Once room tempera tur e ha s

been reached, th e model ma y be unloa ded and subsequently an a lyzed. Since

the f r inge sensi t iv i ty o f the mater ia l a t room temperature is s igni f icant ly

lower t ha n i t is a bove its Tc, the effect of unloading the stress frozen model at

room tempera tu re is negligible. The t ime temperat ure therma l cycle used

for t he loading process is very similar to t ha t of t he post-cure cycle depicted in

Fig. 2.10. For both t he ara ldite an d P SM -9/P LM-4B specimens, the hea tin g

a nd cool ing ra tes used were the same. For the ara ld ite specimens , a peak

soaking tem perat ure of 250 F w a s employed wh ile for t he P S M-9/P LM-4B

specimens, a peak soa king tempera tu re of 220 F wa s used. These va ria tions

ar e s imply due to the d i f f e r ences in c r i t i ca l temper a tur es be tween the

ma ter ia ls . The hea t ing a nd cool ing ra t es employed for a l l tes ts w ere th e

sa me since al l models were of the sa me nominal thickness.

P hotogra phs of the photoela stic shear stress f ield in ea ch specimen a reta ken wh ile th e specimen is in th e pola r iscope. The frozen stress met hod

proves par t icular ly benef icia l in tw o wa ys . For one, th e a mount of loa d

necessary to generate a h igh densi ty f r inge f ie ld is twenty to th ir ty t imes

lower than would be required i f the model were loaded at room temperature.

Typical loads used for experiments considered in this study were in the

neig hborh ood of 6 t o 14 pound s (2.7 - 6.3 kg.). At r oom t emper a t ur e, th ese

loads would therefore need to be increased by a factor of twenty to thir ty to

gener a te s imi l a r f r inge dens i t i es as those ach ieved above the cr i t i ca l

tempera tur e. Secondly, a st ress frozen photoela stic model can inherently be

photographed and analyzed in the polar iscope without the use of a loading

a ppar a tu s wit hin or nea r the pola riscope itself . This as pect of th e procedure

proves to be very important when considering the photographic techniques

r equ i r ed to pr oduce h igh ma gni f i ca t ion c lose-up ima ges . Ver sa t i l e

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positioning of the stress frozen photoelastic model within the polariscope is

intr insical ly more simplist ic than the posit ioning of a model undergoing a

l ive loa d wit hin a loa ding r ig. The photogra phic techniqu es employed for

producing reso lute high magni f ica t ion images require a c lose and f ixed

dista nce betw een the ca mera lens and th e model i tself . On the oth er ha nd,

wide view images require a specif ic posit ioning of the model relative to the

pola riscope. These cha nges in posit ion of t he model rela t ive to the ca mera

and the polar iscope would inherently require movement of ei ther the entire

loading r ig or the entire polar iscope i f l ive loads were to be used at room

temperature.

Perhaps the most appeal ing aspect o f the f rozen s tress method is i ts

abil i ty to al low for the extraction of photoelastic data from within a model .

Pr incipal shear s tress values a t po ints wi thin a photoelas t ic model can be

determ ined by dissecting t he model into thin slices. The cut tin g an d removal

of int erior sections of a photoela stic model does not dist urb t he st a te of frozen

st ress previously locked into the model. B y removing thin slices from plan es

paral le l to the upper and lower sur faces o f a model , one can ef fect ively

quant i fy the var ia t ion in pr incipal shear s tress as a funct ion o f posi t ion

th rough the thickness. For fracture problems, these thin sl ices ca n be ta ken

in the r eg ion o f the cr ack and ind iv idua l ly ana lyzed pho toe las t i ca l l y to

generat e stress intensity factors . Ea ch fracture specimen ana lyzed had threeth in sl ices removed from its cra ck t ip a rea. Fig. 2.11 il lustra tes th eir ty pica l

rela t ive posi t ions th rough the model th ickness . S l ice th icknesses w ere

a pproxima tely 0.04 in. (1 mm ). For each model, th e first t w o slices near t he

outer surfa ces (designa ted a s edge slices) w ere ta ken a pproxima tely 0.04 in.

(1 mm ) from t heir respective surfa ces wh ile th e third sl ice wa s ta ken from

th e middle of the specimen. All s lices were extr a cted using a t hin circula r

blade diamond sa w tha t w a s cooled in a bat h of f luid during opera tion.

2.3.5 Slice Analysis

Once thin sl ices have been extracted from a photoelastic model , data

from ea ch slice ma y be ta ken using the polar iscope. Since the ma gnitud e of

the principal shear stress in a photoelastic model is inversely proportional to

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P

Pedgeslice

edgeslice

middleslice

Figure 2.11: Orienta tion of slices ta ken near t he cra ck tip regionof ea ch photoela st ic specimen.

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the th ickness o f the model , the magni tude o f the pr incipal shear s tress

present in an ex tracted s l ice wi l l be propor t ional ly lower based on the

thickness of the sl ice as compared to the or iginal thickness of the entire

model . For the specimens considered in t his stud y, ty pica l thicknesses were

nea r 0.500 in. (12.7 mm) wh ile slice th icknesses w ere a pproxima tely 0.04 in. (1

mm). This thickness ratio thus yielded on average a twelve fold decrease in

fr inge order and thus fr inge quanti ty when viewing sl ices in the polar iscope.

The accuracy o f most i f no t a l l ex is t ing a lgor i thms used for conver t ing

photoela stic dat a into stress intensity fa ctors rely on a reasonable qua nt i ty of

da ta to be ta ken nea r th e region of the crack t ip. A method of improving the

qua nt i ty of fr inges as seen in the individua l sl ices w a s th us needed to a chieve

a ccur a te s t r ess in tens i ty f ac to r s . The fi r s t s tep in ach iev ing such a n

improvement is to maximize the in i t ia l s t ress f reez ing load such that the

maximum stress intensity factor found at the crack t ip of the model is only

moder a te ly lower than the cr i t i ca l s t r ess in tens i ty f ac to r (K I C ) for the

ma teria l . Other considera tions such a s str ess concent ra tions a t the loa ding

pins may a l so d ic t a te the max imum load tha t can be sa fe ly used w i thout

causin g da ma ge or fa i lure o f th e model . For most photoela s t ic s t r ess

f reez ing mater ia ls , the maximum al lowable damage f ree load for a g iven

scenar io wi l l s t i l l require that the s l ices ex tracted for analys is be ra ther

th ick. Over a given model cross section, thicker slices wil l tend to moregrea tly discretize the distr ibution of informa tion. This undesira ble effect

dictates that s l ice thicknesses remain fair ly small in relation to the overal l

model th ickness.

2.3.5.1 Tardy Method

Two methods for ef fectively increasing the quanti ty of readable data

from thin sl ices were used in conjunction for the results presented in this

s tud y . The f r inge mul t ip li ca t ion techn ique 27 coupled wi th the Tardy

method24,28 provided a n a dequa te a mount of photoela stic dat a from ea ch slice

without the use of large slice thicknesses.

The Tardy method involves par t ia l ro ta t ions o f the analyzer o f the

po lar i scope which in tur n gener a te f r inge f i e lds whose f r inge o r der s

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correla te neith er t o the da rk (N= 0, 1, 2, 3, . . .) or light fields (N= 0.5, 1.5, 2.5,

3.5, . . .) comm only dea lt wit h in photoelast ici ty . In other words, fra ctiona l

ro ta t ions of the an a lyzer wi l l genera te f ract iona l f r inge order f ie lds . For

example, a rotation of the analyzer through 45 degrees relative to i ts dark

field posit ion wil l produce a fr inge f ield whose fr inge order magnitudes are

given by N= 0.25, 1.25, 2.25, 3.25, etc. For full field a na lysis of a photoela stic

model , the Ta rdy m ethod can prove cumbersome due t o the requirement th a t

the isoclinics (fringes of constant principal shear stress direction) need to be

placed over each point from which partial fr inge order data is to be taken.

Only th e da rk a nd l ight photoelast ic fields a re exempt from this requirement.

For a thorough explana tion of these phenomena , see Dally a nd R iley19.

For tuna te ly , the Smi th ex t r apo la t ion method fo r f r ac tur e ana lys i s

requires tha t da ta be ta ken along a stra ight l ine wh ose origin lies a t t he cra ck

tip. This requirement coupled w ith t he rela tively sma ll zone over which dat a

is taken usual ly a l lows for most o f the data to be read f rom just a s ingle

se t t ing o f th e i soc lin ic f r inge . This gr ea t l y f ac i li t a tes ma nua l da ta

acquis i t ion and r educes po ten t i a l e r r o r assoc ia ted w i th the o ther w ise

repet i t ive but necessa ry a djustment o f the a ppa ra tus . Al l da ta used in th is

stu dy wa s ta ken using ta rdy increments of 0.1. Thus, in doing so, fr inge

order and positional data (N,r) were found for nine discrete locations between

each set of adjacent dark field fringes.

2.3.5.2 Fringe Multiplication

U n l i k e t h e T a r d y m e t h o d , t h e f r i n g e m u l t i p l i c a t i o n t e c h n i q u e

developed by P ost 27 provides a mean s for increa sing the photoela stic dat a in a

given slice or model without making repetitive adjustments to the elements of

th e pola r iscope. Specif ics concerning the ma th ema tics a nd optimiza tion of

the t echnique a re not given h ere but a re well documented in t he l itera ture.

The technique is based on the placement o f a photoelas t ic model

between two par t ia l mir rors posi t ioned a t s l ight ly incl ined angles to one

a noth er. Referring t o Fig. 2.12, a collima ted source of monochroma tic light

enters f rom the lef t and passes through the f i r s t par t ia l mir ror which is

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1 multiple

5 multiple

st

3 multiplerd

t h

incidentbeam

partia l mirrors

photoelasticslice

expander

collimator

P A lenspinhole

card

fog glass

image

slice

quar te r waveplates

5

31

laserlight

source

Figure 2.12: Fringe multiplica tion appar a tus used for increa sing the

qua nt ity of photoela stic dat a obta ined from thin slices.

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placed perpendicula r to th e direction of th e incoming light source. The light

next passes through the th in photoelas t ic s l ice and then approaches the

incl ined pa r t ia l mir ror . A por t ion of the light i s t r a nsmit t ed through the

inclined pa rtia l mirror while the rest is ref lected ba ck along a n inclined pat h

towa rds th e slice. I f the a ngle of inclinat ion is small , the l ight w il l pass ba ck

th rough the slice at n early t he same posit ion th a t i t f irst entered. The l ight

wil l now encount er the f irst par t ial mirror with simila r results . A portion of

the l ight w ill be tra nsmitt ed while the rest w il l tra vel back through the model

a nd once a ga in encount er th e inclined pa rt ia l mirror . This sequence th us

yields a ser ies of s l ightly disposit ioned l ight beams which have each passed

th rough th e photoelast ic slice a n increasing n umber of t imes.

By inspect ion i t i s seen that l ight beams that penetra te and emerge

outwardly f rom the incl ined par t ia l mir ror must have t raveled through the

slice a n odd num ber of tim es (i.e. 1,3,5,7, .. .). A light beam tr a velling t hrough

the sl ice in a small region, say, 5 t imes, wil l accordingly sum the principal

shear s tress in that region f ive t imes s ince the pr incipal shear s tress , as

measured photoelastical ly , is inversely proportional to the model thickness.

This in turn must generate a f ive fo ld increase in the number o f f r inges

a ppea ring in th e considered region. B y using a coll imat ed monochroma tic

l ight source, the images are able to be focused and expanded as necessary,

thus allowing for the multiplication effect to take place over the entire regionof th e slice. Fur th ermore, each order of mult iplicat ion can be sepa ra ted from

one an oth er by focusing each emerging l ight beam down t o a point . Due to

th e sl ight inclina tion of the second pa rtia l mirror , a l l exit ing l ight beams w ill

be slight ly dispositioned from one a nother. A pinhole car d ca n th en be used

to select the multiplication of choice while preventing the transmission of all

other multiplied ima ges.

2.3.6 Data Acquisition

In preparation for analysis , a s l ice is coated with index matching f luid

a nd placed between the tw o par t ia l mir rors . P lacing the s lice a nd pa r t ia l

mirrors into the rest of th e a ppara tus (Fig. 2.12) a l lows for t he simulta neous

use of both Ta rdy a nd fr inge multiplicat ion techniques. Da ta correlat ing the

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posit ion of photoelastic fr inges relative to the crack t ip in a given sl ice is

ob ta ined ma nua l ly us ing a t r a ns la t ion s t age . In tegr a l w i th the t r ans l a t ion

stage is a micrometer that enables the user to vary the posit ion of the sl ice

a nd par t ia l mir rors rela t ive to the ent i re appara tus . Remaining f ixed to the

a ppara tu s is a piece of fog glass cont a ining thin crosshairs . Since th e f inal

image is displayed on the fog glass , distances between fr inges and the crack

t ip can be measur ed by ad jus t ing the pos i t ion o f the base w i th the

micrometer .

Due to exponential decreases in output l ight intensity as a function of

the desired fringe multiple, a fairly intense light source is generally required

to genera te a view a ble ima ge of highly mult iplied fr inge f ields. With th e

avai lable equipment , f r inge mul t ipl ica t ion f ie lds o f the f i f th order were

consistent ly achievable. Seventh order fields and higher w ere generally not

v is ible due to substant ia l losses in l ight in tensi ty caused pr imar i ly by the

pa rt ia l mirr ors. Ta rdy increment s of 18 degrees (w hich correspond to a 0.1

increase in the values of the fr inge orders, N) were made while using the

fifth order fringe mult iplica t ion field. The combina tion of th ese t echn iques

typical ly generated a to ta l o f th ir ty to for ty data points f rom which ten to

f i f teen were eventual ly used for l inear curve f i t t ing in the ex trapola t ion

algor i thm.

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3.0 Inclined Single Edge-Notched Test Series

A s e r i e s o f p h o t o e l a s t i c e x p e r i m e n t s w e r e u n d e r t a k e n u s i n g

modera tely t hick single edge-notched tension specimens. The str ess freezing

method o f photoelas t ic i ty was employed to faci l i ta te subsequent through-

thickness analyses by extracting thin sl ices near the v-notch t ip regions in

each model . Four d i f ferent angles o f v-notch incl inat ion rela t ive to the

uniform far -field tensile loa ding direction were stud ied. For ea ch inclina tion

angle considered, a three-specimen tes t ing procedure was incorporated

w hich uti l ized a homogeneous, bonded homogeneous, a nd bonded bima teria l

specimen. The bonded bima ter ia l specimen s w ere comprised of tw o

elas t ica l ly d iss imi lar i so tropic mater ia ls which were bonded together a t a

comm on in ter f a ce. The v-no tches w er e ma ch ined d i r ect l y in to ea ch

specimen so that their t ips lay wi thin the th in , uni form bondl ines which

a djoined the ma t ing ha lves . Af ter s t ress freez ing, s t ress in t ensi

1998 Msc Thesis Stress Intensity Factor Distributions in Bimaterial Systems - A Three-dimensional...

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