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fracture mechanics and boundary element methods
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7/18/2019 fracture bem aliabadi http://slidepdf.com/reader/full/fracture-bem-aliabadi 1/14  oundary element formulations in fracture mechanics MH liabadi Wessex Institute of Technology Ashurst Southampton S040 7AA UK This article reviews advances in the application of boundary element methods (BEM) to fracture mechanics which have taken place over the last 25 years. Applications discussed include linear, nonlinear and transient problems. Also reviewed are contributions using the indirect boundary element formulations. Over this period the method has emerged as the most efficient technique for the evaluation of stress intensity factors (SIF) and crack growth analysis in the context of lin- ear elastic fracture mechanics (LEFM). Much has also been achieved in the application to dy- namic fracture mechanics. This review article contains 289 references. 1 INTRODUCTION The BEM also known as the boundary integral equation (BIE) method is now firmly established in many engineering disciplines. The attraction of BEM can be largely attributed to the reduction in the dimensionality of the problem; for 2D analyses only the line-boundary of the domain needs to be discretized into elements and for 3D problems only the sur- face of the domain needs to be discretized. This means that, compared to domain type analysis, a boundary analysis re- sults in a substantial reduction in data preparation and a much smaller system of equations to be solved. Furthermore, this simpler description of the body means that regions of high stress concentration can be modeled more efficiently as the necessary high concentration of grid points in confined to one less dimension. This ability to model high stress gra- dients accurately and efficiently has been the main reason for the methods success in fracture mechanics applications. In- deed, fracture mechanics has been the most active special- ized area of research in the boundary element method and probably the one most exploited by industry. This paper reviews advances in the application of the boundary element method to fracture mechanics that have taken place over the last 25 years. Over this period the method has emerged as the most efficient technique for the evaluation of stress intensity factors (SIF) and crack growth analysis in the context of linear elastic fracture mechanics (LEFM). Much has also been achieved in the application to dynamic fracture mechanics. More work is required to achieve similar efficiency and robustness for nonlinear problems involving plasticity, etc. Nevertheless, for prob- lems involving small scale yielding the method is valuable due to the requirement of localized meshing. There have been partial reviews of BEM applied to the solution of crack problems by Atkinson [1], Smith [2] and Aliabadi and Brebbia [3]. Here, a more complete review of the method is presented. This paper reviews the modeling strategies that have been developed, as well as applications to: LEFM, SIF calculations, Dynamics, Anisotropic and Composite Materials, Interface Cracks, Non-metallic Materi- als, Thermoelastic Problems, Nonlinear Problems and Crack Identification Techniques. Special attention has also been given to Indirect BIE formulations, and in particular the Body Force Method and the Displacement Discontinuity Method (DDM). 2 CRACK MODELING STRATEGIES Straightforward application of the BEM to crack problems leads to a mathematical degeneration if the two crack sur- faces are considered co-planar, as was shown by Cruse [4]. For symmetrical crack geometries it is possible to overcome this difficulty by imposing the symmetry boundary condition and hence modeling only one crack surface. However, for non-symmetrical crack problems another way must be found. Cruse and Van Buren [5] explored the possibility of modeling the crack as a rounded notch with an elliptical clo- sure. However, this model required many elements to model the tip of the rounded notch. The reported accuracy for the stress intensity factor of the center-crack-tension-specimen was poor, with errors of around 14 . Snyder and Cruse [6] introduced a special form of fun- damental solution for crack problems in anisotropic media. The fundamental solution (Green's function) contained the exact form of the traction free crack in an infinite medium, hence no modeling of the crack surfaces was required. The crack Green's function technique although accurate, is lim- ited to 2D straight cracks. For kinked cracks, the region must be divided into segments with straight cracks (see Kuhn [7]). However, this approach is inefficient as it introduces addi- tional elements into the model The first widely applicable method for dealing with two co-planar crack surfaces was devised by Blandford et al [8]. This approach which is based on a multi-domain formulation is general and can be applied to both symmetrical and anti-symmetrical crack problems in both 2D and 3D configurations. The multi-region method introduces artificial boundaries into the body, which connect the cracks to the boundary, in such a way that each region Transmitled by Associate Editor Franz Ziegler ASME Reprint No AMR204 16 Appl Mech Rev vo150, no 2, February 1997 83 © 1997 American Society of Mechanical Engineers wnloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 11/24/2013 Terms of Use: http://asme.org/terms
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  ou nd ary e lem ent formulat ions in f racture m ech anics

M H l i a b a d i

Wessex Inst i tu te o f Technology Ashurst Southampton S0 40 7A A UK

T h i s a r t ic l e r e v i e w s a d v a n c e s i n t h e a p p l ic a t io n o f b o u n d a r y e l e m e n t m e t h o d s ( B E M ) t o f r a c t u re

mec han i cs w hi ch ha ve t aken p l ace ove r t he l a s t 25 yea r s . Appl i ca t i ons d i scussed i nc l ude l i nea r,

non l i nea r and t r ans i en t p rob l em s . A l so r ev i ewe d a r e con t r ibu t i ons us i ng t he i nd i r ec t bou ndary

e l em ent fo rmul a t i ons . Ov er t h is pe r i od the me t ho d has emerge d a s t he mos t e f f i c i en t t echn i qu e

for t he eva l ua t i on o f s tr e s s i n t ens i t y fac t o r s (S IF ) and c r ack g row t h ana l ys i s i n t he con t ex t o f l in -

ea r e l a s t ic f r ac t u r e mech an i cs (L E F M ) . M uc h has a l so been ach i eve d i n t he app l i ca t i on to dy-

nam i c f r ac t u r e mechan i cs . T h i s r ev i ew a r t i c le con t a i ns 289 r e f e r ences .

1 I N T R O D U C T I O N

T h e B E M a l s o k n o w n a s t h e b o u n d a r y i n t e g r a l e q u a t i o n

( B I E ) m e t h o d i s n o w f i r m l y e s ta b l is h e d i n m a n y e n g i n e e r in g

d i sc i p li nes . T h e a t t r ac t i on o f BE M can be l a rge l y a t t r i bu t ed

t o t he r educ t i on i n t he d i men s i ona l i t y o f t he p rob l em; fo r 2D

a n a l y s e s o n l y t h e l i n e - b o u n d a r y o f t h e d o m a i n n e e d s t o b e

d i sc r e t i zed i n t o e l ement s and fo r 3D prob l ems on l y t he su r -

f ace o f t he dom ai n nee ds t o be d i sc r e ti zed . T h i s means t ha t,

compared t o domai n t ype ana l ys i s , a boundary ana l ys i s r e -

su l t s i n a subs t an t i a l r educ t i on i n da t a p r epa ra t i on and a

m u c h s m a l l e r s y s t e m o f e q u a t io n s t o b e s o l ve d . F u r t h e rm o r e ,

t h is s i m p l e r d e s c r i p t i o n o f t h e b o d y m e a n s t h a t r e g io n s o f

h i gh s t r e s s concen t r a t i on can be m ode l ed m ore e f f i c i en t l y a s

t he necessa ry h i gh concen t r a t i on o f g r i d po i n t s i n conf i ned

t o one l e s s d i mens i on . T h i s ab i l i t y t o mode l h i gh s t r e s s g r a -

d i en t s accura t e l y and e f f i c i en t l y has been t he ma i n r eason fo r

t he me t hods success i n f r ac t u r e mechan i cs app l i ca t i ons . I n -

deed , f r ac t u r e mechan i cs has been t he mos t ac t i ve spec i a l -

i z e d a r e a o f r e s e a r c h i n t h e b o u n d a r y e l e m e n t m e t h o d a n d

p r o b a b l y t h e o n e m o s t e x p l o i t e d b y i n d u s tr y .T h i s pape r r ev i ews adv ances i n t he app l i ca t i on o f t he

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

t aken p l ace ove r t he l a s t 25 yea r s . Over t h i s pe r i od t he

m e t h o d h a s e m e r g e d a s t h e m o s t e f f i c ie n t t e c h n i q u e f o r th e

eva l ua t i on o f s t r e s s i n t ens i t y f ac t o r s (S IF ) and c r ack g rowt h

ana l ys i s i n t he co n t ex t o f l i nea r e l a s t ic f r ac t u r e mechan i cs

( L E F M ) . M u c h h a s a l s o b e e n a c h i e v e d i n t h e a p p l ic a t io n t o

d y n a m i c f r a c t u r e m e c h a n i c s . M o r e w o r k i s r e q u i r e d t o

ach i eve s i m i l a r e f f i c i ency and robu s t nes s fo r non l i nea r

prob l ems i nvo l v i ng p l a s t i c i t y , e t c . Never t he l e s s , f o r p rob-

l ems i nvo l v i ng sma l l s ca l e y i e l d i ng t he me t hod i s va l uab l e

d u e t o t h e r e q u i r e m e n t o f l o c a l i z ed m e s h i n g .

T h e r e h a v e b e e n p a r t i a l r e v i e w s o f B E M a p p l i e d t o t h eso l u t i on o f c r ack p rob l em s by At k i nson [1 ] , S mi t h [2 ] and

A l i a b a d i a n d B r e b b i a [ 3 ] . H e r e , a m o r e c o m p l e t e r e v i e w o f

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

s t r a t eg i e s t ha t have been deve l oped , a s we l l a s app l i ca t i ons

t o : L E F M , S IF ca l cu l a t i ons , Dynami cs , Ani so t rop i c and

Compos i t e M at e r i a l s , I n t e r f ace Cracks , Non-met a l l i c M at e r i -

a ls , T h e r m o e l a s t i c P ro b l e m s , N o n l i n e a r P r o b l e m s a n d C r a c k

Iden t i f i ca t i on T echn i ques . S pec i a l a t t en t i on has a l so been

g i ven t o Ind i r ec t B I E fo rmul a t i ons , and i n pa r t i cu l a r t he

B o d y F o r c e M e t h o d a n d t h e D i s p l a c e m e n t D i s c o n t i n u i t y

M e t h o d ( D D M ) .

2 C R A C K M O D E L I N G S T R A T E G I E S

S t ra i g h tf o r w a rd a p p l i c a ti o n o f t h e B E M t o c r a c k p r o b l e m s

l eads to a ma t hem at i ca l degen e ra t i on i f t he t wo c r ack su r -

f aces a re cons i de red co-p l ana r , a s was show n by C ruse [4 ] .

F o r s y m m e t r ic a l c ra c k g e o m e t r ie s i t is p o s s i b l e t o o v e r c o m e

t h is d i f fi c u l ty b y i m p o s i n g t h e s y m m e t r y b o u n d a r y c o n d i t i o n

a n d h e n c e m o d e l i n g o n l y o n e c r a c k s u r f a c e . H o w e v e r , f o r

n o n - s y m m e t r i c a l c r a c k p r o b l e m s a n o t h e r w a y m u s t b e

f o u n d . C r u se a n d V a n B u r e n [ 5 ] e x p l o r e d t h e p o s s i b i li t y o f

mod e l i ng t he c r ack a s a r ounded n o t ch wi t h an e l l i p t i ca l c l o -

s u re . H o w e v e r , t h is m o d e l r e q u i r e d m a n y e l e m e n t s t o m o d e l

t h e t ip o f t h e r o u n d e d n o t c h . T h e r e p o r t e d a c c u r a c y f o r t h e

s t r e s s i n t ens i t y f ac t o r o f t he cen t e r - c r ack- t ens i on- spec i men

was poor , w i t h e rro r s o f a round 14 .

S nyd er and Cruse [6] i n t roduc ed a spec i a l f o rm o f fun-

dament a l so l u t i on fo r c r ack p rob l ems i n an i so t rop i c medi a .

T h e f u n d a m e n t a l s o l u t i o n ( G r e e n ' s f u n c t i o n ) c o n t a i n e d t h e

e x a c t f o r m o f t h e t r a c t i o n fr e e c r a c k i n a n i n f in i te m e d i u m ,

h e n c e n o m o d e l i n g o f t h e c r a c k s u r fa c e s w a s r e q u i r e d . T h e

c r a c k G r e e n ' s f u n c t i o n t e c h n iq u e a l t h o u g h a c c u r a t e , i s l im -

i t ed to 2D s t r a igh t c r acks . F or k i nke d c r acks , t he r eg i o n mu s t

be d i v i ded i n t o segment s w i t h s t r a i gh t c r acks ( see K uhn [7 ] ) .

However , t h i s approach i s i ne f f i c i en t a s i t i n t roduces add i -

t i ona l e l ement s i n t o t he mode l T he f i r s t w i de l y app l i cab l e

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

d e v i s e d b y B l a n d f o r d et al [8 ] . T h i s approach whi ch i s based

on a mul t i - domai n fo rmul a t i on i s gene ra l and can be app l i ed

t o b o t h s y m m e t r ic a l a n d a n t i -s y m m e t r ic a l c r a c k p r o b l e m s i n

b o t h 2 D a n d 3 D c o n f i g u r a t i o n s . T h e m u l t i - r e g i o n m e t h o d

i n t roduces a r t i f i c i a l boundar i e s i n t o t he body , whi ch connec t

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

Transm itled by Associate Edi tor Franz Ziegler

ASME Repr in t No AMR204 16App l Mech Rev vo150, no 2 , February 1997 8 3 © 199 7 Ame rican Society of Mechanical Engineers

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84 A l iabad i : BE fo rmu la t ions in f rac ture mech an ics App l Mech Rev vo l 50 no 2 February 1997

conta ins a c rack sur face. The tw o reg ions a re then jo ined to -

ge the r such tha t equ i l ib r ium o f t rac tions and com pa t ib i li ty o f

d i sp lacem ents a re enforced . The m a in d raw back of th i s

m e tho d i s tha t the in t roduc t ion of a r t i f i c ia l boundar ie s a re

no t un ique , and thus cannot be im plem ented in to an au to-

m a t ic p rocedure . In add i t ion , the m e tho d gene ra te s a l a rge r

sys tem of a lgebraic equat ions than is s t r ic t ly required. De-

sp i t e these d rawbacks , the subreg ion m e thod has beenwide ly used fo r c rack prob lem s .

More recen t ly , the Dua l Boundary E lem ent Method

(DBEM) as deve loped by P or te la , Al iabad i , and Rooke [9 ]

for 2D prob lem s a nd Mi an d Al iabad i [10] fo r 3D prob lem s

has been shown to be , a gene ra l and eom puta t iona l ly e f f i -

c ien t way of m ode l ing c rack prob lem s in BEM. Genera l

m ixe d-m o de c rack prob lem s can be so lved wi th DBEM , in a

s ing le reg ion fo rm ula t ion , wh en the d i sp lacem ent BIE i s ap-

p l i ed on on e o f the c rack sur faces and the t rac tion BIE on the

o the r . In the con tex t o f the d i rec t BEM, the dua l equa t ions

were f i r s t p re sen ted by W atson [11], in a fo rm ula t ion based

on the d i sp lacem ent equa t ion and i t s norm a l de r iva t ive. Dua l

boun dary equa t ions have b een app l ied to so lve 3D po ten t i a ltheory by Gray and Gi le s [12] , Rudolph i e t a l [13], and in

3D e la s tos ta t ic s by Gray e t a l [14].

Assum ing con t inu i ty o f the d i sp lacem ents a t a boundary

poin t x , the boun dary in tegra l represen ta t ion o f the d i s -

p l a c e m e n t c o m p o n e n t u , i s g iven by

C o x ) u + ( x ) r o x , x ) u ( x ) a r : j ( x )

+ I n U o ' ( x " x ) b j ( x ) d f l

whe re i an d j d enote C artes ian com pone nts ; Tit and U,j repre-

sen t the Ke lv in t rac t ion and d i sp lacem ent fundam enta l so lu -

t ions , re spec t ive ly , a t a boundary po in t x . The sym bol

s tands fo r the Cauc hy pr inc ipa l -va lue in tegra l , and the coe f -

f i c ien t C o depends on the geom et ry and i s ob ta ined by r ig id

body cons ide ra t ion . A ssum ing the con t inu i ty o f s t ra ins a t x ,

the t rac t ion in tegra l equa t ion can be wr i t t en a s

C o . ( x ' ) t j ( x ' ) + n , ~ r V O'k x ' , x ) u k ( x ) a r

+ n i ~ n U i jk ( x " x ) b k ( x ) d ~ l

where Tko.a n d Uko.conta in de r iva t ives o f /~ j and U. , re spec -

t ive ly . The sym bol ~ s t ands fo r the Hadam ard pr inc ipa l -

va lue in tegra l and n , denotes the com ponents o f the ou tward

norm al a t the source po in t x .The m a in d i f f i cu l ty in the DBE M form ula t ion i s the de -

ve lopm ent o f a gene ra l and accura te m ode l ing procedure fo r

the in tegra t ion o f Cauchy and Hadam ard pr inc ipa l va lue in -

t egra l s appea r ing in the t rac t ion equa t ion . The neces sa ry

condit ions for the exis tence of these s ingular integrals , as-

sum ed in the de r iva t ion o f the dua l bound ary in tegra l equa -

t ions , im poses ce r ta in re s t r i c t ions on the cho ice o f shape

func t ions fo r the c rack sur faces . In the po in t co l loca t ion

m ethod of so lu t ion , the d i sp lacem ent in tegra l equa t ion re -

qu i re s the con t inu i ty o f the d i sp lacem ent com ponents a t the

nodes ( i e , col locat ion points ) , and the t ract ion integral equa-

t ion requ i re s the co n t inu i ty o f the d i sp lacem ent de r iva t ives a t

the nodes . These requirements were sat is f ied in [11] by

adopt ing the Herm i t i an e lem ents , however , the so lu t ions re -

por ted were no t ve ry accura te . Recen t ly , Watson [15] has

im proved the accuracy of th i s fo rm ula t ion . Rudolph i e t a l

[13] reported unexplained osci l la t ions in their resul ts , while

G r a y e t a [14] devised a scheme based on specia l integra-

t ion pa th a round the s ingu la r po in t fo r l inea r t r i angula r e le -ments . The formu lat ions in [13, 14] were appl ied to embed -

ded cracks only. In [9, 10 ], both crack surfaces were d iscre-

t i zed wi th d i s con t inuous quadra t i c e lem ents ; th is s t ra tegy n o t

on ly au tom at ica l ly s a t is f i es the neces sa ry condi t ions fo r the

ex i s tence o f the Hadam ard in tegra l s , bu t a l so c i rcum vents

the p rob lem of co l loca t ing a t c rack k inks and c rack-edge

com ers . S eve ra l exam ples inc lud ing em bedded , edge , k inked

and curved cracks were solved accurate ly in [9, 10]. For

o the r con t r ibu t ions in DBEM see , fo r exam ple , Gray and

Gi le s [12], Lu tz [16], H ong and Ch en [17], and Cha ng an d

Mear [18] . Recen t ly , You ng [19] has m odi f i ed the shape

func t ion represen ta t ion o f the d i sp lacem ent de r iva t ives to

a l low the use o f C ° e lem ents . A s im i la r p rocedure was ap-p l i ed to c rack prob lem s in geom echanics by Wi lde and A l ia -

badi [20].

The above procedures dea l wi th the m ode l ing of m acro-

c racks . R ecen t ly , Chandra e t a l [21] has deve loped a m ic ro-

m acro BEM for ana lys i s o f m ic ro-c rack c lus te rs . In the i r

work [21], the micro-scale effects are introduced into macro-

sca le BEM through an augm ented func t iona l so lu t ions ob-

ta ined f rom an in tegra l represen ta t ion o f the m ic ro-sca le

features .

De ta i l ed desc r ip t ion o f som e of the advanced BEM for -

m ula t ions can be fou nd in Al iabad i an d Brebbia [22] .

3 L I N E A R E L A S T I C F R A C T U R E M E C H A N I C S

The app l ica t ion o f the BE M to L inea r E las t ic F rac ture M e-

cham cs (LEF M ) i s now we l l e s tab l i shed and wide ly used in

prac t i ce . The m e thod of fe rs a c lea r advan tage ove r o the r

m e thods such a s the F in i te E lem ent M ethod fo r LEF M. O ne

of m a in reasons fo r th is advan tage i s the pos s ib i l i ty o f eva lu-

at ing the S tress Intens i ty Factors (SIF) accurate ly. There

h a v e b e e n m a n y m e t h o d s d e v i se d f o r t h e e v a l u a t i o n o f S I F s

us ing BEM. The m os t popula r a re pe rhaps the t echn iques

based on the qua r te r-po in t e lem ents , pa th ind ependent con -

tour in tegra l s , ene rgy m e thods , sub t rac t ion o f s ingu la r i ty

m ethod and the we igh t func t ion m e thods . A de ta i l ed de -

sc r ip t ion o f these m e thods can be fou nd in the t ex t book by

Aliabadi and R ooke [23].

The s tress f ie lds in the vic ini ty of the crack are s ingular

O(1/-¢rr)a t the crack t ip and the displacement f ie ld va r ie s

as O ( , f r ) . The re fore m ode l ing c rack reg ions wi th the usua l

shape f f im c tions , which a l low fo r po lynom ia l va r ia t ion on ly ,

does no t l ead to accura te so lu t ions un les s ve ry f ine m eshes

are used near the crack-t ip. Using the s tandard isoparametric

shape func t ions , the d i sp lacem ents and t rac t ions m ay be ap-

prox im a ted a s

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App l Mech Rev vo150 no 2 February 1997 A l iabad i : BE fo rmu la t ions in f rac tu re mecha n ics 85

r + a ( r ) 2

u i = a ° + a l 7 2 ~ 7

r + b ( r ) 2 't = b o bl 7

2 ~ -D

w h e r e ao, a, , a~, . . a n d bo , b , , b 2 . . . . a r e cons t an t s , r deno t es

t he d i s t ance t o t he c r ack t i p and I deno t es t he e l ement l eng t h .

By sh i f t ing t he mi d- s i de node o f a quadr a ti c e l emen t t o

qua r t e r - po i n t pos i t i on ( c l ose t o t he c r ack t i p ) , t he des i r ed

f i e l d va r i a t i ons a r e ob t a i ned , t ha t i s

U i = C ° e l C 2 7

l rt i = d o + d 1 + d 2

w h e r e c o, % e~, and do, di , d~, are constants.

T h e u s e o f q u a r t e r - p o i n t e l e m e n t s i n 3 D b o u n d a r y e l e -

men t ana l ys i s was r epor t ed by Cr use and W i l son [ 24] who

a l so i n t r oduced add i t i ona l modi f i ca t i ons f o r mode l i ng s i n -

gu l a r t r ac t i ons . S eve r a l ways o f eva l ua t i ng t he s t r e s s i n t en-

s i t y f ac t o r s f r om t he d i sp l acement s on t he c r ack su r f aces

have been p r oposed by r e sea r che r s [ 25 , 26 , 27] . S mi t h and

M a s o n [ 2 8 ] d e m o n s t r a t e d t h e u s e o f q u a r t e r -p o i n t e le m e n t

f o r c u r v e d c r a c k s . M a r t i n e z a n d D o m i n g u e z [ 2 7 ] p r o p o s e d

an a l t e r na t i ve way o f ob t a i n i ng t he S I F s f o r t he qua r t e r -

po i n t e l ement s . T he i r me t hod whi ch r e l a t e s t he so ca l l ed

t r ac t i ons a t t he c r ack t i p t o t he s t re s s i n t ens i t y f ac t o r s is m or e

e f f i c i en t t han t he d i sp l acem ent based f o r mul ae . A com par i -

s o n o f m e t h o d s o f e v a l u a t in g t h e S I F s f r o m t h e q u a r t e r-

po i n t e l emen t s has been r epo r t ed by S mi t h [ 26] . O t he r spe -

c i a l c r ack t i p e l ement s f o r mode l i ng t he nea r c r ack t i p be -

hav i o r a r e r epor t ed by A l i abad i [ 29] ; -J i a, S h i ppy , and R i zzo

[ 30] f o r 2D pr ob l ems ; L uch i and R i zzu t i [ 31] f o r 3D con-

t i nuous e l ement s ; and M i and Al i abad i [ 32] f o r 3D d i scon-

t i nuous e l ement s . Z amani and S un [ 33] have p r oposed a hy-

b r i d t y p e e l e m e n t . T h e i r p r o p o s e d e l e m e n t i s s i m i la r t o t h e

enr i ched e l ement used i n t he f i n i t e e l ement me t hod , wher e

t he c r ack t i p s t r e s s f i e l ds a r e added t o t he s t anda r d L agr an-

g i a n p o l y n o m i a l s .

T h e u s e o f p a t h i n d e p e n d e n t c o n t o u r i n t e gr a ls h a s a ls o

bee n popu l a r i n BE M , a s t he s t r e ss i n t ens i t y fac t o r s can gen-

e r a l ly b e e v a l u a t e d b y a p o s t - p r o c e s s in g p r o c e d u r e . O n e o f

t he m os t pop u l a r pa t h i nd epen den t i n t egr a l s is t he J - i n t egr a l ,

w h i c h c a n b e d e f i n e d a s

whe r e W i s t he s t r a i n ene r gy pe r un i t vo l ume , n j is t he com-

p o n e n t s o f t he o u t w a r d n o r m a l t o t h e p a th C i n t h e d i r e c ti o n

1 , A i s t he a r ea enc l o sed by t he co n t ou r C .

B o i s s e n o t , L a c h a t , a n d W a t s o n [ 3 4 ] r e p o r t e d t h e u s e o f J -

i n t egr a l f o r 3D symmet r i c c r ack p r ob l ems . L a t e r , K i sh i t an i ,

e t a l [ 35] and Ka r ami an d F enn er [ 36] r epor t ed i ts use f o r

seve r a l 2D sy mm et r i ca l p r ob l em s . A l i abad i [ 37] app l i ed t he

J - i n t e g r a l a n d B E M t o m i x e d - m o d e c r a c k p r o b l e m s a n d d e -

cou p l ed t he J i n t o i ts symm et r i ca l and an t i - symm et r i ca l

componen t s . I t was shown i n [ 37] t ha t accur a t e va l ues o f

mode I and mode I I s t r e s s i n t ens i t y f ac t o r s can be ob t a i ned

f r om t he J - i n t egr a l . M an , A l i abad i , and Rooke [ 38] u t i l i zed

t h e m i x e d - m o d e J - in t e g r a l t o s t u d y t h e e f f e c t o f c o n t a c t

f o r c e s o n t h e c r a c k b e h a v i o r . T h e a p p l i ca t i o n o f t h e J -

i n t e g r a l t o m i x e d m o d e 3 D p r o b l e m s w a s p r e s e n t e d b y -

R i gby and Al i abad i [ 39] and Huber and Khun [ 40] . S o l l e r o

a n d A l i a b a d i [ 41 ] p r o p o s e d a n a l te r n a ti v e m e t h o d f o r d e c o u -p i i n g th e m i x e d - m o d e J - in t e g r a l b a s e d o n c r a c k o p e n -

i ng / s l i d i ng d i sp l acement s r a t i o . S on i and S t em [ 42] and

S t e m e t a [ 43] deve l oped a pa t h i ndependen t i n t egr a l and

u s e d t h e B E M t o e v a l u a t e m i x e d - m o d e s t r e s s i n t e n s i t y f a c -

t o r s . M o r e r ecen t l y , W en , A l i abad i , and Roo ke [ 44] ha ve de -

ve l oped an a l t e r na t i ve pa t h i ndependen t i n t egr a l f o r t he

eva l ua t i on o f mi xed - mo de s t re s s i n t ens i t y fac t o r s . I n [ 44] , an

i n d ir e c t b o u n d a r y e l e m e n t f o r m u l a t i o n w a s u s e d t o e v a l u a t e d

t he i n t e r i o r va l ues o f d i sp l acement s an d s t r e s ses. B a i nbr i dge ,

A l ia b a d i a n d R o o k e [ 4 5 ] h a v e p r o p o s e d a p a t h i n d e p e n d e n t

i n t egr a l f o r 3D pr ob l ems . T he i r pa t h i nd epen den t i n t egr a l

u t i l i ze s so l u t i ons due t o po i n t f o r ces on s t r a i gh t f r on t ed and

p e n n y s h a p e d c r a c k s a s a n a u x i l i a r y f i e l d . M i x e d - m o d es t re s s i n t ens i ty f ac t o r s can be eva l ua t ed w i t h t h i s t echn i que .

A n o t h e r w a y o f c a l c u la t in g S I F s i s f r o m t h e u s e o f s t ra i n

ene r gy r e l ease r a t e G . How ever , t h i s me t ho d r equ i r e s s eve r a l

c o m p u t e r r u n s f o r 3 D p r o b l e m s . C r u s e a n d M e y e r s [ 4 6] p r o -

p o s e d a t e c h n i q u e f o r 3 D p r o b l e m s w h i c h l i m i t e d t h e c o m -

p u t e r ru n s t o tw o . T h e t w o c o m p u t e r r u n s c o n s i s t e d o f o n e

f o r t he o r i g i n a l c ra c k f r o n t a n d o n e f o r t h e p e r t u r b e d c r a c k

f r o n t , o b t a i n e d b y m o v i n g a l l t h e n o d e s o n t h e c r a c k f r o n t

r ad i a l l y a l ong l i nes nor ma l t o t he c r ack f r on t . C r use and

M eyer s [ 46] used l i nea r t r i angu l a r e l ement s . L a t e r , T an and

F enner [ 47] used quadr i l a t e r a l e l ement s w i t h quadr a t i c

v a r i a t i o n t o r e p r e s e n t b o t h t h e s u r f a c e a n d t h e u n k n o w n

f u n c ti o n s . F u rt h e r d e v e l o p m e n t o f B E M u s i n g t h e s t ra i n e n -e r gy r e l ease ra t e has been r epor t ed by B onn e t [ 48] .

T h e m e t h o d s d i s c u s s e d a b o v e a r e b a s e d o n a t t e m p t s t o

mode l t he s i ngu l a r behav i o r o f s t r e s ses nea r t he c r ack t i p . I n

con t r a s t , t he S ub t r ac t i on o f S i ngu l a r i t y M et hod ( S S T ) avo i ds

t he need f o r t h i s t a sk ; i t r em ove s t he s i ngu l a r f i e l ds com -

p l e t e l y . T h i s l eaves a non- s i ngu l a r f i e l d t o be mode l ed nu-

m e r i c a l l y . T h i s a p p r o a c h w a s f i r s t i n t r o d u c e d i n B E M b y

P apam i che l and S ym m [ 49] f o r ana l ys is o f symm et r i ca l s li t

i n po t en t i a l p r ob l ems . Xan t h i s e t a l [ 50] used t h i s f o r mul a -

t i on t o so l ve t he s ame pr ob l em of symmet r i ca l s l i t us i ng

quadr a t i c isopa r amet r i c e l ement s . T he e x t ens i on o f the

met hod t o 2D e l a s t i c i t y was p r esen t ed by A l i abad i e t a l [51,

52] , who ob t a i ned bo t h mode I and mode I I s t r e s s i n t ens i t y

f ac t o r s . T h i s f o r mul a t i on was ex t ended t o V- no t ch p l a t e s i n

[ 53]. T he app l i ca t i on o f t he m e t hod t o 3D p r ob l em s i s re -

p o r t e d b y A l ia b a d i a n d R o o k e [ 5 4 ] . A n e q u i v a l e n t t e c h n i q u e

t o th e S S T w a s d e v e l o p e d b y S m i t h a n d D e l l a - V e n u r a [ 5 5 ] .

I n t h i s s t udy , t wo- s t ep supe r pos i t i on me t hod i s used t o ob-

t a i n t he s t r e s s i n t ens i t y f ac t o r s . T he p r ocedur e i n [ 55] r e -

qu i r e s t he so l u t ion o f t he f u l l c r ack ed p r ob l em and t he so l u -

t i on t o a p r ob l em on t he s ame mesh a s sumi ng t he s i ngu l a r i t y

due t o a c r ack t i p i n a n i n f i n i t e r eg i on .

M et ho ds f o r t he eva l ua t i on o f s t r e s s i n t ens i t y f ac t o r s f r om

t h e c r a c k G r e e n s f u n c t i o n s h a v e b e e n p r o p o s e d b y M e w s

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86 A l iabad i : BE fo rmu la t ions in f rac ture mecha n ics

[56] fo r k inked c racks and Dowrick [57] and Young e t a l

[58] for s t i ffen ed panels . R ecent ly, Tel les e t a l [59] have

proposed to eva lua te the c rack G reen s fun c t ion num er ica l ly .

An a l t e rna tive m e th od to the usua l s t re ss ana lys i s fo r the

eva lua t ion o f s t re s s in tens i ty fac to rs i s the we igh t func t ion

m ethod . T he advan tage o f the we igh t func t ions l ie s in the ir

un ive rsa l i ty , tha t i s they a re independent o f the load ing .

Hence , once the we igh t func t ions a re eva lua ted fo r a g ivenc rack geom et ry , they can be use d to eva lua te the s t re s s in ten-

s i ty fac to rs fo r an y app l ied load ing . Bueckn er in t roduced the

concep t o f we igh t func t ions in ea r ly 70s . His we igh t func -

t ions s a t i s fy the l inea r equa t ions o f e la s t i c i ty , bu t have a

s t rong s ingu la r i ty a t the c rack t ip . He re fe rs to them as f u n

d a m e n t a l f i e l d s . Late r , R ice showed tha t the we igh t func t ions

could be equa l ly we l l de te rm ined by d i f fe ren t i a t ing known

elas t ic solut ions for displacement f ie lds with respect to the

c rack l eng th . F o r de ta i l s o f these two form ula t ions , reade rs

should consu l t A l iabad i and Rook e [23] .

Cruse and Besuner [60] and Besuner [61] deve loped a

BEM s t ra tegy fo r eva lua t ing the we igh t func t ions based on

Rices s de r iva t ion . In the i r work , a 3D BEM ana lys i s wasused to calculate average s tress intens i ty factors for each

per tu rba t ion o f the c rack f ron t . The ins tan taneous va lues a t a

spec i f i c po in t and the ave rage va lue a long the whole c rack

f ron t a re no t exac t ly equ iva len t fo r m os t 3D prob lem s s ince

the s tress intens i ty factors are not general ly constant . Fur-

ther, this technique requires many i tera t ions to obtain a s in-

g le s t re s s in tens i ty fac to r so lu t ion an d i s thus com puta t ion-

a l ly expens ive . Anothe r t echn ique us ing Rice s de r iva t ion i s

due to Heliot , Labbens , and Pel l is ier-Tannon [62]. This

techn ique i s the ex tens ion of the approx im a te po lynom ia l

d i s t r ibu t ion a s p roposed by Gran t ( s ee Al iabad i and Rooke

[23] ). In th i s work , the po lyn om ia l in f luence func t ions were

de f ined to cor re spond to the t e rm s of a po lynom ia l expan-s ion of the s t ress f ie lds act ing on the crack faces ; these influ-

ence func t ions a l so depended on the rad i i and dep th o f the

sem i-e l lip t i ca l c rack . Nu m er ica l c rack- face we igh t func t ions

were o b ta ined a f t e r f ive com pute r runs , one fo r each t e rm in

the po lynom ia l . La te r , Cruse and Ravendera [63] deve loped

a 2D B EM procedure based on the R ice s fo rm ula t ion . In

the i r work , the c rack G reen s fu nc t ion was used . Accura te

va lues o f s t re ss in tens i ty fac to rs were repor ted fo r sym m et r i -

ca l c rack prob lem s . Recen t ly , Wen, Al iabad i , and Rooke

[65 , 64] deve loped a BEM technique fo r eva lua t ing 2D and

3D we igh t func t ions . They used d i sp lacem ent d i s con t inu i ty

and f i c t i t ious s t re s s m e thods to ob ta in we igh t func t ions fo r

m ixed -m od e prob lem s accord ing to R ice s de r iva t ion .Car twr igh t and Rooke [66] showed tha t a boundary e le -

m ent ana lys i s p roduced m ore accura te s t re s s in tens ity fac to rs

m ore e f f i c ien t ly than the f in i te e lem ent ana lys i s. Th i s fo r -

m u l a t i o n w h i c h i s b a s e d o n B u e c k n e r s f u n d a m e n t a l f i e ld s

has been ex tend ed by A l iabad i , Ca r twr igh t , and Rooke [67]

to bo th m ode I and m ode I I de form a t ion which , in th i s fo r -

m ula t ion a re independent . The im provem ent to th i s m ode l

was repor ted by A l iabad i , Rooke , an d Car twr igh t [68] fo r 2D

problem s by em ploy ing the sub t rac t ion o f s ingu la r i ty t ech-

n ique . Ba ins , Al iabad i , and Rooke [69, 70] p resen ted a BEM

for eva lua t ing 3D we igh t func t ions based on the sub t rac tion

App l Mech Rev vo l 50 no 2 February 1997

of s ingu la r f i e lds . They de r ived and u t i l i zed fundam enta l

f i e lds fo r s t ra igh t f ron ted and penn y shaped c racks . The ap-

p l i ca t ion o f th i s m e thod was dem o ns t ra ted fo r a w ide range

of c rack prob lem s .

Othe r con t r ibu t ions can be found in Mende l son [71] ,

Boissenot and Serres [72], Rudolphi and Ashbaugh [73], Tan

and Fenner [74], Tanaka [75], Aliabadi and Rooke [76],

C h e i n e t a l [77], Porte la and Aliabadi [78], Aliabadi and

Porte la [79], Aliabadi [80], Zeng and Dai [81], Smith and

Aliabadi [82], Burs tow and Wearing [83], Liu and Tan [84],

Theocaris and Panagiotopoulos [85], Porte la , Aliabadi , and

Rooke [86], Giumaraes and Tel les [87], S turt , Nowell , and

Hil ls [88], Blackburn and Hall [89], Bonnet [90], Denda and

Dong [91] , Ventur in i [92] , Chen and Chen [93] , Yan and

Nguy en-Dang [94] , and F or th and K ea t [95] .

4 C R A C K S I N A N I S O T R O P I CA N D C O M P O S I T E M A T E R I A L S

One of the f i r st app l ica tion o f BEM to c racks in an i so t rop ic

m ate r ia l s was due to S nyder and Cruse [6 ] . In th i s work , the

c rack Green s func t ion was used a s a p rocedure fo r em bed -

d ing an exac t c rack m ode l ing in the bou ndary in tegra l repre -

sen ta t ion . Th i s approached proved popula r wi th s eve ra l

au thors fo r exam ple Konish [96] , Chan and Cruse [97] ,

Kam el and L iaw [98], and L iaw and Kam el [99] . How ever ,

as s ta ted earl ier this approach is l imited in i ts appl icat ion.

The m ul t i - reg ion m e thod and qua r te r-po in t s have been used

by Tan and Gao [100] to so lve s eve ra l c rack prob lem s in

orthotropic materia ls . Sol lero and Aliabadi [101] p resente d a

m ul t i - reg ion m e thod toge the r wi th a m ixed-m ode J - in tegra l

fo r c rack prob lem s in o r tho t rop ic and an i so t rop ic m a te r ia l s .

Doblare , Espiga, and Alcantud [102] have a lso used the

m ul t i - reg ion BEM form ula t ion . I sh ikawa [103] and S ladekand S ladek [104] p resen ted BEM resu lt s fo r 3D c rack prob-

lems in anisotropic materia ls . More recent ly, Sol lero and

Al iabad i [105] p resen ted a dua l bound ary e lem ent fo rm ula -

t ion for cracks in anisotropic materia ls . They ut i l ized a J -

integral formula t ion to obtain accurate s t ress intens i ty factors

for s eve ra l m ixed-m od e prob lem s .

The app l ica t ion o f BEM to c rack ing in com pos i t e m a te r i -

a l s has been repor ted by S h i lko and S heherbakov [106] , Tan

and Bige low [107] , Kam el e t a l [108], and Klingbei l [109].

Mo re recent ly, Bush [ 110] analyz ed fracture o f part ic le re in-

forced com pos i t e m a te r ia l s wi th BEM. Nonl inea r behav ior

of m e ta l m a t r ix f ibe r com pos i te s wi th dam age on the in te r -

face has been ana lyzed by S h ibuya and W ang [111]. S hanand Chou [112] have ana lyze d the p rob lem o f f ibe r /m a t r ix

interfacia l debounding. Chel la , Aithal , and Chandra [113]

s tudied a quas i-s ta t ic crack extens ion in f iber-re inforced

composi tes subjected to thermal shock. Sens i t ivi ty analys is

based on the ad jo in t fo rm ula t ion was deve loped in [113] to

eva lua te the ene rgy in tegra ls in c racked bodies .

S e lvadura i [122] has repor ted a s tudy of the beha v ior o f a

penny shaped m a t r ix c rack which m ay occur a t an i so la ted

fiber which is fr ic t ional ly constra ined. In this s tudy an in-

c rem enta l t echn ique was used to exam ine the p rogres s ion o f

the s e l f sim i la r g rowth o f the m a t r ix c rack . S uzuki and S ak i

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Appl Mech Rev vo150 no 2 February 1997 Aliabadi: BE formulations in fracture me cha nics 87

[ 1 2 3] p r e s e n t e d a B E M m o d e l f o r c r a c k b ri d g i n g o f a l a m i -

na l ca r bon- f i be r / ca r bon- mat r i x . Cosz et al [ 124] ca r r i ed ou t

a s t udy i n t o t he e f f e c t o f i n t e rphase p r op e r t i e s on t he t r ans -

ve r se s t r eng t h o f un i d i r ec t i ona l f i be r - r e i n f o r ced compos i t e s .

I n t he i r s t udy , Cosz et a [ 124] used a va r i a t i ona l l y coup l ed

B E M / F E M t o o b t a i n t h e m e c h a n i c a l r e sp o n s e o f a c e l l c o n -

t a i n i ng a cen t e r f i be r and pa r t s su r r oun d i ng t he f a i l u re .

O t he r con t r i bu t i ons can be f ound i n C l ement s and Hase l -

g r ov e [ I 14], K l i ngbe i l and K a t akaya [ 115] , T r an and W ang

[ 116] , S e l vadur a i [ 117] , S o l l e r o and Al i abad i [ 118] , Cor ad i

et a [ 119] , Chandr a et al [ 120] , and P an and Am ade i [ 121] .

5 I N T E R F A C E C R A C K S

T h e u s e o f H e t e n y i s f u n d a m e n t a l s o l u t io n i n B E M t o a v o i d

mo de l i ng t he i n t e r f ace o f t wo d i f f e r en t ma t e ri a l s was i n t r o -

d u c e d b y Y u u k i et al [ 125] and Yuuki and Cho [ 126] . I n

t hese pape r s s e ve r a l i n t e r f ace c r ack p r ob l em s wer e a l so

a n a l y z e d . T h e u s e o f m u l t i - r e g i o n m e t h o d f o r i n t e r f a c e

c r a c k s h a s b e e n r e p o r t e d b y L e e a n d C h o i [ 1 2 7 ] a n d T a n a n d

Gao [ 128] . T an and Gao [ 129] dev e l ope d a qua r t e r- po i n t

e l em ent t o mod e l i n t e r f ace c r acks be t w een d i s s i mi l a r ma t e r i -

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

dea l w i t h t he osc i l l a t o r y s i ngu l a r na t u r e o f t he s t r es ses . A 3D

BE M f or ana l yses o f i n t e r f ace c r acks and d i s s i mi l a r ma t e r i a l

j o i n t s h a s b e e n p r e s e n t e d b y Y u u k i a n d X u [ 1 3 0] . T h e a p p l i-

ca t i on o f t he v i r t ua l c r ack ex t ens i on and a con t our i n t egr a l

t e c h n i q u e t o i n t e r f a c e c r a c k s h a v e b e e n p r e s e n t e d b y M i -

y a z a k i et al [ 131 , 132] . Kown and Dut t on [ 133] ana l yzed

c r acks nor ma l t o b i ma t e r i a l i n t e r f ace us i ng t he mul t i - r eg i on

met hod . I n t he i r wor k a shape f unc t i on con t a i n i ng t he s ame

or de r o f s i ngu l a r i t y a s t ha t i n t he i n t e r f ace c r acks was used

f or t he i n t e r po l a t i on o f t r ac t i ons . T he app l i ca t i on o f t he d i -

r ec t d i sp l acement d i s con t i nu i t y me t hod t o i n t e r f ac i a l c r acksw a s p r e s e n t e d b y S e l c u k et al [ 134] . I n t he i r wor k squa r e

r o o t d i s p l a c e m e n t d i s c o n t i n u it y e l e m e n t s w e r e u s e d t o m o d e l

t he f i e l ds i n t he v i c i n i t y o f the c r a ck t ip . X i ao a nd H ui [ 135]

p r e s e n t e d a m u l t i -r e g i o n B E M b a s e d o n a m o d i f i e d c r a c k

c l osur e m e t hod t o o b t a i n t he ene r g y r e l ease r a t e f o r c r acks in

hom oge neo us ma t e r i a l s and a l ong a b i ma t e r i a l i n t e r face .

O t h e r c o n t r i b u ti o n s c a n b e f o u n d i n C h o et al [136] , Bat -

t a c h a y y a a n d W i l lm e n t [ 1 3 7 ], H e et al [ 138] , L ee [ 139] , and

S l adek and S l adek [ 140] .

6 D Y N A M I C FR A C T U R E M E C H A N I C S

T h e b o u n d a r y e l e m e n t s o l u ti o n s t o e la s t o d y n a m i c p r o b l e m s

ar e usua l l y ob t a i ned ( s ee M anol i s and Beskos [ 141] , Dom-

i nguez [ 142] , B r ebb i a and Nar d i n i [ 143] , I r s ch i k and Z i eg l e r

[ 1 4 4 ] , D i n e v a - V l a d i k o v a et al [ 145] ) by e i t he r t he t i me do-

ma i n , L ap l ace o r F our i e r t r ans f o r ms , t he dua l r ec i p r oc i t y

m e t h o d o r d o m a i n d i s c r et i z a ti o n t e c h n iq u e .

N i sh i mur a , Guo , and Kobayash i [ 146 , 147] used t he t i me

d o m a i n m e t h o d t o s o l v e c r a c k p ro b l e m s . T h e y u s e d t h e d o u -

b l e l aye r po t en t i a l f o r mul a t i on w hi ch con t a i ns t he hype r s i n -

gu l a r i n t egr a l s . T he equa t i ons wer e r egu l a r i zed by us i ng i n -

t egr a t i on by pa r t s t w i ce . T he cons t an t and l i nea r shape f unc -

t i ons wer e used f o r t he spa t i a l and t empor a l appr ox i ma t i ons

r espec t i ve l y . T he me t hod was app l i ed f o r s t a t i ona r y and

gr owi ng s t r a i gh t c r acks in 2D, and p l ane c r ack i n 3D i n f i n i t e

domai ns . T he dynami c s t r e s s i n t ens i t y f ac t o r s wer e ob t a i ned

f o r m t h e c r a c k o p e n i n g d i s p l ac e m e n t s ( C O D s ) . L a t e r Z h a n g

a n d A c h e n b a c h [ 1 4 8 ] i m p r o v e d t h e c r a c k m o d e l i n g u s e d i n

[ 147] by u t i l i z i ng cons t an t e l ement away f r om t he c r ack

f r on t and spa t i a l squa r e - r oo t f unc t i ons nea r t he c r ack t i p .

T hey ana l yzed co l l i nea r c r acks i n i n f i n i t e domai n . H i r ose

[ 149] and Hi r ose and Achenbach [ 150 , 151] app l i ed t he

f o r m u l a t io n b a s e d o n t r a c ti o n e q u a t i o n w i t h p i e c e w i s e l i n e a r

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

shaped c r acks . Z hang and Gr oss [ 152] used t he t wo- s t a t e

conse r va t i on i n t egr a l o f e l a s t odynami c , w hi ch l eads t o non -

hyper s i ngu l a r t r ac t i on i n t egr a l equa t i ons . T he un know ns i n

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

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

and squa r e c r acks i n i n f i n it e domai ns .

N i c h o l s o n a n d M e t t u [ 1 5 3 ] a n d M e t t u a n d N i c h o l s o n

[ 154] used t wo t ypes o f appr ox i ma t i on : i ) cons t an t e l ement s

f o r b o t h s p a ti a l a n d t e m p o r a l i n t e r p o l at i o n o f b o u n d a r y

quan t i t i e s ; and i i ) quadr a t i c i n space and l i nea r i n t i me . T he

m e t h o d w a s a p p l i e d t o s o l v e s e v e r a l o p e n i n g - m o d e c r a c kp r o b l e m s . D o m i n g u e z a n d G a l l e g o [ 1 5 5 ] u s e d a m i x e d

var i a t i on o f bound ar y va l ues i n whi ch t r ac t i ons w er e a s -

sumed t o be cons t an t and d i sp l acement s l i nea r i n t i me . T he

boundar i e s wer e d i v i ded i n t o quadr a t i c e l ement s . A t t he

c r ack t i ps o r d i na r y and t r ac t i on- s i ngu l a r qua r t e r - po i n t e l e -

ment s ( QP E s) wer e used . T he dynami c s t r e s s i n t ens i t y f ac -

t o rs w e r e d e t e r m i n e d u s i n g t h e C O D a n d t r a c t io n s o f t r ac -

t i on- s i ngu l a r e l ement s . T he me t hod was app l i ed t o f i n i t e

b o d i e s w i t h c r a c k s . M i x e d - m o d e c r a c k p r o b l e m s w e r e a n a -

l yzed us i ng t he subr eg i on t echn i que .

S i ebr it s and Cr ouch [ 156] have p r esen t ed a t i me - dom ai n

d i sp l acement d i s con t i nu i t y f o r mul a t i on f o r 2D pr ob l ems . I n

t he i r f o r mul a t i on l i nea r , con t i nuous i n t i me and p i ecewi sel i nea r i n space i n t e r po l a t i on f unc t i ons wer e a s sumed f o r t he

d i sp l acement d i s con t i nu i ti e s .

T h e d u a l b o u n d a r y e l e m e n t f o r m u l a t i o n i n t i m e d o m a i n

was p r esen t ed by F ede l i nsk i , A l i abad i , and R ook e [ 157]. T he

t e m p o r a l v a r ia t i o n o f b o u n d a r y d i s p l a c e m e n t s a n d t r a c ti o n s

w a s a p p r o x i m a t e d b y p i e c e w i s e l i n e a r a n d c o n s t a n t f u n c -

t i ons , re spec t i ve l y . T he dyna mi c s t r es s i n t ens i t y f ac t o r s wer e

ca l cu l a t ed us i ng t he c r ack open i ng d i sp l acement s and t he

pa t h i ndependen t .~ i n t egr a l . T h i s me t hod was used t o s t udy

dynam i c behav i o r o f s ta t i ona r y c r acks i n f i n i te and i n f i n i te

d o m a i n s i n 2 D a n a ly s is . B o t h m o d e I a n d m i x e d - m o d e c r a c k

pr ob l ems wer e cons i de r ed .

A n a p p l ic a t io n o f a L a p l a c e t r a n s f o rm m e t h o d w a s p r e -s e n te d b y S l a d e k a n d S l a d e k [ 1 5 8 ] , w h o a n a l y z e d a p e n n y -

shaped c r ack i n an i n f i n i t e e l a s t i c domai n sub j ec t ed t o ha r -

m o n i c a n d i m p a c t l o a d s o n c r a c k s u r fa c e s . T h e p r o b l e m w a s

so l ved us i ng t he t r ac t i on i n t egr a l equa t i on i n t e r ms o f t he

d i sp l acement d i s con t i nu i t y . A s i mi l a r me t hod was used by

Yi n and L i [ 159] t o ana l yze a r ec t angu l a r p l a t e w i t h a cen t r a l

c r ack . T he i n f l uence o f d i f f e r en t leng t hs o f t he c r ac k and d i f -

f e r e n t t i m e - d e p e n d e n t l o a d in g s w e r e s t u di e d . T h e d y n a m i c

s t r e s s i n t ens i t y f ac t o r s wer e ca l cu l a t ed f r om t he COD of

qua r t e r - po i n t e l ement s and t he ex t r apo l a t i on t echn i que .

T a n a k a et al [ 160] used a s i mi l a r me t hod t o ca l cu l a t e dy-

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88 Al iabadi : BE formulat ions in f racture mechanics Appl Mech Rev vol 50 no 2 February 1997

nam ic s t re s s in tens i ty fac to rs . The e f fec t o f va ry ing the

n u m b e r o f b o u n d a r y e l e m e n t s a n d t h e n u m b e r o f p a r a m e t er s

in the Durb in and Hosono m e thod of inve r t ing was inves t i -

ga ted . P o lyzos et a [161] ana lyzed rec tangula r p la te o f v i s-

coe la s t i c m a te r ia l wi th e i the r a cen t ra l o r an edge inc l ined

c rack . The m ixe d-m o de c rack prob lem was so lved us ing the

m ul t i - reg ion m e thod . Zhan g and S h i [162] used the Lap lace

t rans form B EM to s tudy the t rans ien t s t re s s in tens ity fac to rsof a s em i -c i rcu la r su r face c rack in a rec tangula r ba r .

Wen, A l iabad i , and Rooke [163 , 164] p resen ted a Lap lace

t rans form d i sp lacem ent d i s con t inu i ty and f i c t i t ious s t re s s

form ula t ions fo r 2D and 3D prob lem s . They ob ta ined the

s t re s s in tens i ty fac to rs fo r m any 2D and 3D prob lem s us ing

an equivalent s t ress approach.

Fedel insk i , Aliabad i , and Rooke [165, 1 66] presented a

Laplace t rans form dua l boundary e lem ent fo rm ula t ion fo r

2D prob lem s . S eve ra l m ode I and m ixed-m ode c rack prob-

lem s w ere so lved and the s t re s s in tens i ty fac to rs were eva lu-

a ted us ing the qua r te r -po in t e lem ents .

Wen, A l iabad i , and Rook e [167, 168] p resen ted a fo rm u-

la t ion fo r eva lua t ion dynam ic we igh t func t ions in two- and

three -d im ens ions . These we igh t func t ions a re independent o f

bo th spa t i a l d i s t r ibu t ion and t im e va r ia t ion o f the load ing .

These advan tageous fea tu res were dem ons t ra ted fo r s eve ra l

m i x e d - m o d e p r o b le m s .

The F our ie r t rans form m ethod was used by Chi r ino and

Dom inguez [169] to ana lyze c racks in an in f in i t e p lane , a

half-plan e and a f ini te dom ain. Hirose [170] used the t ract ion

equa t ion and the d i sp lacem ent d i s con t inu i ty m e thod to in -

ves t iga ted the s ca t t e r ing o f e la s t i c waves f rom a penny

shaped c rack .

The dua l rec ip roc i ty m e thod w as used by Ba las, S ladek ,

and S ladek [171] fo r sym m et r ic c rack prob lem . P ekau andBa t ta [172] used the subreg ion m e thod fo r s t a t iona ry and

grow ing c racks in a rec tangula r p la te . A s im i la r m e thod was

used by C hi r ino , Ga l lego , S aez , and Dom inguez [173] , who

a l so used the subreg ion t echn ique and p resen ted a com para -

t ive s tudy o f d i f fe ren t approaches .

F ede l insk i , Al iabad i , and R ooke [174 , 175] p resen ted a

D B E M f o r m u l a t i o n f o r d y n a m i c s u s in g t h e d u a l r e c ip r o c it y

m ethod . The s t re s s in tens i ty fac to rs fo r m ixed-m ode prob-

lem s were ob ta ined us ing CO D and the J - in tegra l m e thod .

A c om par i son o f the t im e-dom ain , Lap lace trans form and

dua l rec ip roc i ty m e thod in t e rm s of com put ing t im e and

s torage a s we l l a s accuracy have b een presen ted by F ede l in -

ski , Aliabadi , and Ro oke [176].Othe r con t r ibu t ions can be found in Bonne t and Bui

[177]; Fedel inski , Aliabadi , and Rooke [178, 179]; and

Geubel le and Rice [180].

7 T H E R M O E L A S T I C F R A C T U R E M E C H A N I C S

Ear ly app l ica t ions o f BEM to the rm oe las t i c c rack prob lem s

are due to Ku hn [181] an d P rede leanu and S c repe l -F leur ier

[182] . La te r Tanaka , Togoh an d Kiku ta [183] used a BEM

form ula t ion wi th dom ain t e rm to represen t the t em pera tu re

f ie ld . Lee and Cho [ i 84] so lved s eve ra l sym m et r ica l c rack

prob lem s . S ladek and S ladek [185] so lved t rans ien t sym m et -

rical crack problems. Tanaka et al [186] appl ied the mult i -

reg ion m e thod wi th dom ain d i s c re t i za t ion . Raveendra and

Banerjee [187] ut i l ized the mult i -region formulat ion to ther-

m oe las t i c c rack prob lem s us ing a boundary on ly fo rm ula -

t ion. Quarter-point e lements were used in [187] and the

s tress intens i ty factors were evaluated from the crack open-

ing displacements . Liu and Alterio [188] presented a series

of so lu t ions to m ode I and m ode I I c rack prob lem s . Re-cent ly, Prasad, Aliabadi , and Roo ke [189, 190 ] have pre-

sen ted a BEM form ula t ion fo r m ixed-m ode c rack prob lem s

in s ta t ic and t rans ient thermoelas t ic i ty. In their formulat ion

two pa i r s o f boundary in tegra l equa t ions were em ployed .

One pa i r cons i s t s o f t em pera tu re and d i sp lacem ent , and the

other pair of f lux and t ract ion. The s tress intens i ty factors

were eva lua ted f rom bo th the qua r te r -po in t e lem ents and a J -

integral formulat ion. A specia l cons iderat ion to the thermal

s ingulari ty a t the crack t ip is add ressed in [191].

Appl ica t ion o f BEM to the rm a l ly s tre s sed in te r face c racks

was p resen ted by Ka tsa reas and Ani fan t i s [192] . They u sed a

m ul t i -dom ain fo rm ula t ion toge the r wi th spec ia l c rack t ips to

m ode l the s ingu la r behav ior o f t em pera tu re and d i sp lacem ent

fie lds in the vic ini ty of the crack t ip.

Othe r con t r ibu t ions can be found in Karam i and Kuhn

[193], Raveendra et a [194], Prasad and Aliabadi [195], and

Sladek and Sladek [196].

8 N O N L I N E A R F R A C T U R E M E C H A N I C S

One o f the f i r st a t t em pts in app ly ing e la s top las t ic boun dary

e lem ent fo rm ula t ions to f rac tu re m echanics was m ade by

Mor ja r i a and Mukher jee [197] and Banth ia and Mukher jee

[198, 199]. In their approach, the crack G reen s fu nct io n was

used to model the crack. Later , Cruse and Polch [200] a lso

used the G reen s func t ion approach toge the r wi th an im -

proved m ode l o f the c rack. Tan and Lee [201] use d the Ke l -

v in s fundam enta l so lu t ion and in t roduced the c rack b y

spec i fy ing appropr ia te boundary condi t ions . In the i r s tudy ,

the behav ior o f an in te rna l ly p res sur ized th ick-wa l led cy l in -

de r con ta in ing a rad ia l c rack was inves t iga ted . Yong and

Guo [202] have a lso s tudied a pressurized thick-walled cyl-

inde r wi th sym m et r ic rad ia l c racks . Recen t ly , Hantsche l ,

Busch , Kuna , and Maschke [203] m ode led the p la s t ic c rack

t ip f i e lds by the use o f spec ia l s ingu la r e lem ents an d the in -

t roduc t ion of the HRR f ie lds . Le i tao , Al iabad i , Rooke , an d

Cook [204 , 205] have used the BEM to s im ula te c rack

growth in p resence o f re s idua l s t re s s f i e lds in t roduced by

co ld expans ion t echn ique . Le i t ao and Al iabad i [206] dem o n-

s t ra ted the e f f i c iency of the BE M for eva lua t ing s eve ra l d i f -

ferent non l inear J - integrals .

Al iabad i and Car twr igh t [207] deve loped a BEM tech-

n ique fo r the eva lua t ion o f p la st i c zone s ize a rou nd a c rack

using the s t r ip yie ld model . In their analys is , a weight func-

t ion fo rm ula t ion was used .

Only a few publ ica t ions dea l wi th m ixed-m ode e la s to -

plas t ic problems. Rul3wurm [208] addressed the mixed-

m ode prob lem us ing the c rack G reen s fun c t ion approach ,

while Lei tao, Aliabadi , and Rooke [209] used an e las toplas-

t i c dua l boundary e lem ent fo rm ula t ion . The prob lem of

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App l Mech Rev vo150 no 2 February 1997 A l iabad i : BE fo rmu la t ions in f r ac tu re mech an ics 89

c r ack con t ac t and e l a s t op l a s t i c behav i o r was i nves t i ga t ed by

L e i t ao , A l i abad i , and Rooke [ 210] .

O t he r con t r i bu t i ons can be f ound i n I keda [ 211] , S l adek

and S l adek [ 212] ; L or enzo , Car t wr i gh t , and Al i abad i [ 213] ;

L e i t ao , A l i abad i , and Rooke [ 214] ; and Cor r ad i , A l i abad i ,

and M a r che t t i [ 215] .

N O N - M E T A L L I C M A T E R I A L S

T h e B E M h a s b e e n p o p u l a r f o r a n a ly s i s o f c r a c k s i n g e o m e -

chan i ca l p r ob l ems . I n pa r t i cu l a r , t he i nd i r ec t BE M f or mul a -

t i o n o f t h e d i s p l a c e m e n t d i s c o n t i n u i ty m e t h o d ( D D M ) h a s

b e e n a p p l i e d t o m a n y p r o b l e m s i n c l u d i n g m o v e m e n t o f

j o i n t s , f r ac t u r e r ocks and c r ack i ng due t o ea r t hquake r e -

s p o n s e [ 2 1 6 , 2 1 7 ] . O n e o f t h e e a r l y a p p l ic a t io n s o f B E M t o

s i mul a t e hydr au l i c f r ac t u r i ng i s due t o C l i f t on and Abou-

s a y e d [ 2 1 8 ] . L a t e r S o u s a e t a l [ 219] used t he DDM t o s t udy

t he coa l e scenc e and o r i en t a t i on o f mul t i p l e f rac t u r es p r opa -

ga t i ng f r om a we l l bor e wh i ch i s no t a li gned wi t h l i ne o f t he

pr i nc i pa l s t r e s s d i r ec t i ons . Har dy and Asg i an [ 220] used t he

DDM t o s t udy t r ans i en t f r ac t u r e f l u i d p r es sur e and f r ac t u r ewi d t h dur i n g r epr esen t a t i on o f hydr au l i c f r ac tu r e . A de t a i l ed

r ev i ew o f f o r m ul a t i ons used t o s i mul a t e hydr au l i c fr ac t u r i ng

i s p r esen t ed by Asg i an [ 221] . Hash i da e t a l [ 222] used t he

t ens i on- sof t en i ng mode l f o r t he ana l ys i s o f f r ac t u r e p r oc -

es ses o f r ock w i t h pa r t i cu l a r r e f e r ence t o t he e f f ec t o f con-

f i n i ng p r es sur e o n t he f r ac t u r e ex t ens i on .

T h e a p p l i c a t i o n o f B E M t o t h e a n a l y s i s o f c r a c k i n g i n

conc r e t e i s r e l a t i ve l y new and t he r e appea r s t o be on l y a f ew

p u b l i c a ti o n s o n t h e s u b j e ct . T h e u s e o f D D M t o g e t h e r w it h

t h e f i c t i t i o u s c r a c k m o d e l ( F C M ) w a s p u b l i s h e d b y H a r d e r

[ 223] , bu t no r e su l t s wer e r epor t ed . L i ang and L i [ 224] p r e -

sen t ed BE M ana l ys i s t o s i mul a t e t he non l i nea r f r ac t u r e zone

i n cement a t i ous ma t e r i a l s , us i ng F CM . L a t e r , Cen and M ai e r[ 2 2 5 ] u s e d t h e m u l t i - d o m a i n f o r m u l a t i o n w i t h F C M t o

s i mul a t e c r ack g r owt h i n conc r e t e . S a l eh and Al i abad i [ 226 ,

2 2 7 ] u s e d t h e d u a l B E M t o g e t h e r w i t h F C M f o r th e a n a l y s is

o f b o t h p l a i n a n d r e i n f o r c e d c o n c re t e . H o r i i a n d I c h i n o m i y a

[ 228] used Dugda l e - Bar enb l a t t mode l t o mode l t he f r ac t u r e

p r o c e s s e s z o n e i n c o n c r e t e . T h e y c o m p a r e d t h e i r re s u lt s w i t h

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

p l a c e m e n t s o b t a i n e d f r o m l a s e r s p e c k le t e c h n i q u e a p p l i e d t o

a m o r t a r a n d c o n c r e t e s p e c i m e n . R e c e n t l y A l e s s a n d r i a n d

D i e l o [ 2 2 9 ] p r e s e n t e d a 2 D B E M m o d e l f o r m o d e I I f a il u re

of p l a t ed conc r e t e spec i men . P ekau and Ba t t a [230] deve l -

o p e d a t im e d o m a i n B E M t o s t u d y c r a c k p r o p a g a t i o n in c o n -

c r e t e s t r uc t u r es sub j ec t ed t o s e i smi c l oad i ngs . Chahr our andOht su [ 232] s t ud i ed a mi xed- mode c r ack g r owt h i n a s ca l ed-

down mode l o f a conc r e t e g r av i t y dam. I n t he i r ana l ys i s ,

m u l t i - d o m a i n f o r m u l a t i o n w a s u s e d a n d b o d y f o r c e t e r m w a s

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

gr owt h .

Ot he r i mpor t an t pub l i ca t i ons i n non- met a l l i c ma t e r i a l s

c a n b e f o u n d f o r e x a m p l e i n : S a lv a d u r a i a nd A u [ 2 3 1 ], B e e r

[ 233] , Hash i da [ 234] , and S a l vadur a i [ 235] .

10 C R A C K I D E N T I F I C A T I O N

T he BE M i s an i dea l t echn i que f o r i den t i f i ca t i on o f i n t e r na l

cav i t i e s and c r acks . Gener a l l y , t he f l aw i s i den t i f i ed by

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

r e sponses due t o gues sed f l aw i den t i t y and t r ue r e sponses ,

due t o t he ac t ua l f law i den t i t y . T h i s e r r o r i s mi n i m i zed us i ng

op t i mi za t i on p r ocedur es . S akagami e t a l [ 236] i n i t i a l ly used

a soph i s t i ca ted t r ia l and e r r o r p r o cedu r e f o r t he i den t i f i ca t i onof i n t e r na l s t r a i gh t l i ne c r acks i n 2D ana l ys i s , us i ng e l ec t r i c

po t en t i a l measur ement s , i e sa t i s f y ing L ap l ace s equa t i on .

T h i s m u l t i - r e g i o n s c h e m e w a s s u b s e q u e n t l y m o d i f i e d w i t h

t he i nc l us i on o f op t i mi za t i on p r ocedu r e f o r t he i den t i f i ca t i on

o f e d g e a n d c o m e r c r a c k s in 3 D . T a n a k a e t a l [237] a l so

u s e d m u l t i - r e g i o n f o r m u l a t i o n w i t h a n e l a s t o d y n a m i c B E M

and op t i mi za t i on p r ocedur e t o i den t i f y 2D i n t e r na l c r acks .

N i sh i mur a and Kobayash i [ 238] p r esen t ed a c r ack i den t i f i -

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

d i s p l a c e m e n t m e t h o d . T h i s m e t h o d h a s b e e n i m p l e m e n t e d

f o r p o te n t ia l a n d e l a s to d y n a m i c s y s t em s a n d u s e d i n b o t h 2 D

and 3D pr ob l ems . M el l i ngs and Al i abad i [ 239 , 240 , 241]

p r e s e n t e d D u a l B o u n d a r y E l e m e n t f o r m u l a t i o n s f o r 2 D

pr ob l ems i n po t en t i a l and e l a s t i c i t y and 3D po t en t i a l . F i r s t

o r de r op t i mi za t i on p r oce dur es wer e u t i l i zed , i n wh i ch des i gn

sens i t i v i t i e s a r e used t o eva l ua t e s ea r ch d i r ec t i ons . I n t he i r

me t ho d M el l ings and Al i abad i com put e d the des i gn sens i -

t i v i t i e s us i ng an I mpl i c i t o r D i r ec t D i f f e r en t i a t i on M et hod ,

w h i c h i n v o l v e s t h e s o l u t io n o f m a t r ix s y s t e m s f o r m e d b y t h e

d i f fe r e n ti a t io n o f t h e d i c r e t i ze d D B E M e q u a t i o n s w i t h r e -

spec t t o t he des i gn va r i ab l e .

O t h e r c o n t r i b u t i o n s c a n b e f o u n d i n N i s h i m u r a a n d K o -

bayash i [ 242] , Ohj i [ 243] , M e l l i ngs and Al i abad i [ 244] ,

Bonne t [ 245] , and T osaka e t a l [246] .

11 C R A C K G R O W T H A N A L Y S IS

T he f i r s t a t t empt t o au t omat i ca l l y mode l c r ack g r owt h i n

m i x e d - m o d e c o n d i t i o n s w a s b y I n g r a f f e a , B l a n d f o r d a n d

L i g g e t [ 2 4 7 ] f o r 2 D p r o b l e m s . T h e y u s e d t h e m u l t i - r e g i o n

met hod t oge t he r w i t h maxi mum c i r cumf e r en t i a l s t r e s s c r i t e -

r i on t o ca l cu l a te t he d i r ec t i on o f c r ack g r ow t h . T h e ex t ens i on

o f t hi s m u l t i -r e g i o n m e t h o d t o 3 D p r o b l e m s w a s p r e s e n t e d

by Gr es t l e [ 248] . C r ack g r owt h p r oces ses i n o r t ho t r op i c ma-

t e r i a l s was p r esen t ed by Dobl a r e e t a l [ 102] . T hey used t he

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

s i mul a t e c r ack g r owt h . T he app l i ca t i on o f mul t i - r eg i on

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

a n d D o m i n g u e z [ 2 4 9 ] . I n t h i s p a p e r , t h e t i m e - d o m a i n f o r -

mul a t i on was u t i l i zed t oge t he r w i t h qua r t e r - po i n t e l ement s .

Cen and M ai e r [ 225] a l so used t he mul t i - r eg i on me t hod t o

s i mul a t e c r ack g r owt h i n conc r e t e s t r uc t u r es . I n t he i r f o r mu-

l a t i on , t he cohes i ve c r ack mode l was used t o s i mul a t e t he

f r ac t u r e p r oces s zone i n conc r e t e . T he d i f f i cu l t y w i t h t he

mul t i - r eg i on me t hod i s t ha t t he i n t r oduc t i on o f a r t i f i c i a l

boundar i e s t o d i v i de the r eg i ons i s no t un i que , and t hus can-

no t be eas i l y i mpl ement ed i n an au t omat i c p r ocedur e . I n an

i nc r ement a l c r ack ex t ens i on ana l ys i s , t hese a r t if i c ia l bound a-

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

c r ack ex t ens i on .

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9 Aliaba di: BE formulations in fracture mechanics

P or t e l a , A l i abad i , and Rooke [ 250] and M i and Al i abad i

[ 251 , 252] p r esen t ed an a pp l i ca t i on o f t he DBE M t o t he

a n a l y s is o f m i x e d - m o d e c r a c k g r o w t h i n 2 D a n d 3 D l in e a r

e l a s t i c f r ac t u r e mechan i cs . C r ack g r owt h p r oces ses wer e

s i mul a t ed w i t h an i nc r ement a l c r ack ex t ens i on ana l ys i s

based on t he m axi m um pr i nc i pa l s t re s s c ri t e r i on f o r 2D and

mi n i mu m s t r a in ene r g y dens i t y c r it e r i on f o r 3D . I n [ 250], f o r

e a c h i n c r e m e n t o f th e c r a c k e x t e n s i o n , t h e D B E M w a s a p -

p l i ed t o pe r f o r m a s i ng l e r eg i on s t r e s s ana l ys i s and t he J -

i n t egr a l t echn i que used t o comput e t he s t r e s s i n t ens i t y f ac -

t o rs . W h e n t h e c r a c k e x t e n s i o n i s m o d e l e d w i t h n e w d i s co n -

t i nuous e l emen t s , r emesh i ng o f t he ex i s t ing bound ar i e s i s no t

r equ i r ed b ecause o f t he s i ng l e - r eg i on ana l ys is , an i n t r ins i c

f e a t u r e o f th e D B E M . F o r s u r f a c e b r e a k in g c r a c k s i n 3 D

c e r t a in a m o u n t o f r e m e s h i n g i s h o w e v e r r e q u ir e d . A n a u t o-

m a t i c p r o c e d u r e f o r th i s p r o c e s s h a s b e e n d e v e l o p e d b y A l i-

abad i and M i [ 253] . S a l gado and Al i abad i [ 254] p r esen t ed

t he app l i ca t i on o f DBE M t o s t i f f ened s t r uc t u r es . T hey s i mu-

l a t ed c r ack g r owt h i n a i r c r a ft pane l s r e i n f o r ced by s t i ff ene r s .

T h e e x t e n s i o n o f D B E M t o e l a s to p l a st ic f a t i g u e c r a c k

gr ow t h ana l ys i s has been p r esen t ed by L e i t ao , A l i abad i and

R o o k e [ 2 5 5 ]. T h e e x t e n s i o n o f D B E M t o s t at ic a n d t r a n si e n t

t he r moe l as t i c c r ack g r owt h i s p r e sen t ed by P r asad , A l i abad i

and R ook e [ 256 , 257] . I n t hese pape r s , t he e f f ec t o f t he r ma l

l oad i ng was s t ud i ed on t he c r ack g r owt h and d i r ec t i on , bo t h

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

s i de r ed . F ede l i nsk i , A l i abad i and Rooke [ 258] have a l so ex-

t e n d e d t h e D B E M f o r m u l a t i o n in t i m e d o m a i n t o a n a l y se s o f

mi xed- mode f a s t g r owi ng c r acks . S o l l e r o and Al i abad i [ 259]

u s e d D B E M f o r a n i s o t r o p i c m a t e r i a l s t o s t u d y m i x e d - m o d e

cr ack g r o wt h i n com pos i t e l ami na t e s . L a t i f and A l i abad i

[ 2 6 0 ] d e v e l o p e d a n o n l i n e a r c o h e s i v e c r a c k m o d e l w i t h

D B E M f o r s i m u l a ti n g c r a c k i n g i n b o t h p l ai n a n d r e i n f o r c e dc o n c r e t e s t r u ct u re s . T h e a p p l i ca t io n o f B E M t o c r a c k g r o w t h

i n bone has be en p r esen t ed by M ar t i nez and Al i abad i [ 261] .

12 I N D I R E C T B O U N D A R Y

E L E M E N T F O R M U L A T I O N S

I n d i r ec t b o u n d a r y i n t eg r a l f o r m u l a t io n s f o r c r a c k p r o b l e m s

h a v e b e e n a r o u n d f o r m a n y y e a r s a n d a p p e a r i n m a n y

br anches o f f r ac t u r e mechan i cs . He nce , i t w i l l be ex t r em e l y

d i f f i cu l t t o p r esen t a com pr ehe ns i ve r ev i ew o f t hem a ll . M os t

o f t he f o r mu l a t i ons a r e howev er , l i mi t ed to i so l a t ed c racks i n

i n f i n i t e domai ns . Never t he l e s s , t hese so l u t i ons p r ov i de a

va l uab l e i ns i gh t i n t o t he behav i o r o f c r acks . E a r l y app l i ca -

t i o n s o f i n t e g r a l e q u a t i o n f o r m u l a t i o n s to c r a c k p r o b l e m s c a n

b e f o u n d i n c l a s si c a l w o r k s o f S n e d d o n [ 2 6 2 ] a n d B i b l y a n d

E s h e l b y [ 2 6 3 ] . O t h e r i m p o r t a n t c o n t r i b u t i o n s c a n b e f o u n d

i n t h e w o r k o f , f o r e x a m p l e , E r d o g a n a n d G u p t a [ 2 6 4 ] . M o r e

r e c e n t l y , th e c o n c e p t o f e l e m e n t d i s c re t iz a t io n h a s b e e n i n -

t r o d u c e d t o t h e m e t h o d , w h i c h a l l o w s t h e s o l u t io n o f p r o b -

l e m s i n fi n it e d o m a i n s s e e M i r a - M o h a m a d - S a d e g a n d A l t e -

r i o [ 265] , L e Van and Royer [ 266] , F a r es and L i [ 267] ) . T he

a p p l i c a ti o n o f t h e m e t h o d t o k i n k e d c r a c k s a n d c r a c k c o n t a c t

p r o b l em s h a s b e e n p r e s e n te d b y Z a n g a n d G u d m u n d s o n

[ 268] .

Appl Mech Rev vo150, no 2, February 1997

12 1 B ody for c e me thod

T h e b o d y f o r c e m e t h o d i s o n e o f th e m o s t p r o m i n e n t i n d i r e c t

B E M f o r m u l at io n s . T h e m e t h o d w a s o r i g i n a l ly p r o p o s e d b y

Ni s i tan i [ 269] f o r t he so l u t i on o f 2D s t r e s s p r ob l ems , and

was l a t e r ex t ended t o t he so l u t i on o f no t ch [ 270] and c r ack

p r o b l e m s [ 2 7 1 ]. A s i n B E M , t h e b o d y f o r c e m e t h o d u s e s t h e

s t re s s f ie l ds due t o a po i n t f o r ce i n an i n f i n it e dom ai n a s f un-

dament a l so l u t i ons . T he p r esc r i bed boundar y cond i t i ons a r e

sa t i s f i ed by app l y i ng t he body f o r ce a l ong t he i magi na r y

boundar i e s i n an i n f i n i t e shee t and ad j us t i ng t he f o r ce den-

s i ty so a s t o s a t i s fy the bo unda r y cond i t i ons . T he b oun dar i e s

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

w i t h u n k n o w n s d e f i n e d a t t h e m i d - p o i n t s o f t h e e l e m e n t s .

T h e a p p l i c a ti o n o f th e b o d y f o r c e m e t h o d t o 3 D c r a c k p r o b -

l ems can be f ound i n pape r s by N i s i t an i and M ur ak ami [ 272]

a n d M u r a k a m i a n d N e m a t - N a s s e r [ 2 7 3 ]. F o r m o r e r e c e n t a d -

vance s i n bod y f o r ce me t hod , r eade r s shou l d consu l t N i s i tan i

[274] .

12 2 D i s p lac e me nt d i s c ont inui ty me thod

As d i scus sed ea r l ie r , t he s t anda r d app l i ca t i on o f t he bou nd-

a r y i n t egr a l f o r mul a t i on f o r c r ack p r ob l ems has an i nhe r en t

ma t hemat i ca l degene r acy due t o t he co- p l ana r c r ack su r -

f a ce s . T o o v e r c o m e t h is p r o b l e m C r o u c h [ 2 7 5] p r o p o s e d a n

i nd i r ec t i n t egr a l equa t i on i n whi ch t he un know ns a r e t he d i s -

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

sur f aces . F ur t he r mor e , t he f undament a l so l u t i ons a r e due t o

d i sp l acement s d i s con t i nu i ti e s . T he app l i ca t i on o f t he d i s -

p l a c e m e n t d i s c o n ti n u i ty m e t h o d D D M ) t o 3 D p r o b l e m s

h a v e b e e n r e p o r t e d b y W e a v e r [ 2 7 6 ]. T h e e x t e n s i o n o f t h e

m e t h o d t o d y n a m i c c r a c k p r o b l e m s h a s b e e n p r e s e n t e d b y

Das and Aki [ 277] us i ng t i me dom ai n f o r mul a t i on i n 2D and

D a s [ 2 7 8 ] f o r 3 D . A c o m p r e h e n s i v e r e v i e w o f t h e r e c e n t d e -v e l o p m e n t s i n D D M w i t h a p p l i c a t i o n t o g e o m e c h a n i c a l

p r ob l em s i s g i ven by M a ck [ 279] .

T h e D D M i s u s u a l ly u s e d t o m o d e l t h e c r a c k s u r f a c e s a n d

n o t t h e n o n - c r a c k e d b o u n d a r ie s . F o r g e n e r a l c r a c k p r o b l e m s

D D M i s c o m p l e m e n t e d w i t h a n o t h e r i n d ir e c t i n te g r a l e q u a -

t i on know n as t he F ic t i ti ous S tr e s s M e t hod F S M ) t o mod e l

t he non- c r acked boundar i e s . I n f i c t i t i ous s t r e s s me t hod , t he

r ea l p r ob l em i s tr ans f o r m ed to an i nd i r ec t p r ob l e m i n an i n -

f i n i t e body and t he ou t e r boundar y cond i t i ons i nc l ud i ng

t r ac t i ons and d i sp l acement s a r e mode l ed by a s sumed d i s t r i -

bu t i on o f loads f i c ti t ious ) . C r ou ch and S t a r f i e l d [ 280] g i ve

de t a i l ed desc r i p t i on o f bo t h f o r mul a t i ons . Recen t l y , W en ,

A l ia b a di , a n d R o o k e [ 2 8 1 , 2 8 2 ] d e v e l o p e d a D D M a n d F S Mi n L a p l a c e t r a n s f o r m s p a c e f o r b o t h 2 D a n d 3 D p r o b l e m s .

O t h e r c o n t r i b u t i o n s c a n b e f o u n d f o r e x a m p l e i n C h a n a n d

E i ns t e i n [283] . S i mi l a r f o r mul a t i ons t o DDM have a l so be en

pr oposed by Cr use [ 284] , Gui de r a and L ar dne r [ 285] , Bu i

[ 286] , Ba l as and S l adek [ 287] , and T akaud , Koi zumi , and

S hi buya [ 288] .

13 C O N C L U D I N G R E M A R K S

I n t h is p a p e r , a r e v i e w o f b o u n d a r y e l e m e n t f o r m u l a t i o n s f o r

f r ac t u r e mechan i cs p r ob l ems was p r esen t ed . T he r ev i ew , a l -

t hough no t exhaus t i ve , cove r s t he i mpor t an t con t r i bu t i ons

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Appl Mech Rev vo150 no 2 February 1997 Aliabadi: BE formulations in fracture me cha nics 91

a n d , i n p a r t i c u la r , t h o s e t h a t h a v e c o n t r i b u t e d t o t h e m e t h o d

b e i n g u s e d i n p r a c t i c e . M u c h h a s b e e n a c h i e v e d d u r i n g th e

l a s t 2 5 y e a r s a n d , g i v e n t h e e v e r - i n c r e a s i n g r e s e a r c h b e i n g

c a r r i e d o u t o n t h i s t o p i c , g r e a t e r a c h i e v e m e n t s a r e a n t i c i -

p a t e d . I t i s e s t i m a t e d t h a t o v e r 1 0 0 0 p a p e r s h a v e b e e n w r i t t e n

o n t h e a p p l i c a t i o n o f t h e B E M t o f ra c t u r e m e c h a n i c s . F o r a

m o r e c o m p l e t e l i s t o f B E M r e l a t e d r e f e r e n c e s , r e a d e r s

s h o u l d c o n s u l t [ 2 8 9 ] .

1 4 A C K N O W L E D G M E N T

T h e a u t h o r w o u l d l i k e to th a n k D R o o k e f o r h is c o n t i n u o u s

s u p p o r t a n d e n c o u r a g e m e n t o v e r t h e y e a r s . I w i s h t o t h a n k

m y f o r m e r s t u d e n t s a n d r e s e a r c h f e l l o w s ( A P o r t e la , R

B a i n s , K M a n n , V L e i t a o , P S o l l e r o , S M e l l i n g s , R R i g b y ,

Y a o m i n g M i , P i h u a W e n , A L S a l e h , N P r a s a d, P F e d e li n s k i ,

a n d N S a l g a d o ) f o r t h e i r c o n t r i b u t i o n s . T h a n k s a r e a l s o d u e

t o C B r e b b i a f o r h i s e n c o u r a g e m e n t a n d F Z i e g l e r f o r h is

h e l p f u l c o m m e n t s o n t h e m a n u s cr i p t.

R E F E R E N E S

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92 Al iabadi : BE formulat ions in f racture mecha nics

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M o h a m m a d H o s s i e n F e r a y d o n A l i a b a d i ( k n o w n t o h is f r i e n d s a n d c o l l ea g u e s a s F e r r 0 w a s b o r n i n T e -

h r a n, I r a n i n O c t o b e r 1 95 6. H e r e c e i v e d h is B a c h e l o r o f S c i e n c e D e g r e e o r m t h e D e p a r t m e n t o f M a t h e -

mat ics, S ta t i s t ic s . a nd C omp ut ing a t the Univers i ty q f Teess ide UK in 1979 . His Mas ter o f Ph i lo soph y and

D o c t o r a l D e g r e e w e r e o b t a i n e d i ~ om t h e s a m e u n i v e rs i ty i n N o v e m b e r 1 9 8 1 a n d M a r c h 1 9 8 5 , r e s p e c ti v e ly ,

b o t h i n a p p l i e d m a t h e m a t ic s . H e j o i n e d t h e E n g i n e e r i n g M a t e r i a l D i v i s io n o f t he D e p a r t m e n t o f A e r o n a u -

t i c s and As t ronau t ics a t Sou thampto n Univers it y, UK in 1984 as a Pos tdo c tora l Researc h Fe l lo w to work

o n t h e a p p l i ca t i o n o f B o u n d a r y E l e m e n t M e t h o d t o F r a c t u r e M e c h a n i c s . I n 1 9 8 6, h e s p e n t s o m e t i m e a t t h e

De par t men t o f Mec hanic a l Enginee~ 'ing, Buckn e l l Universi ty , USA, as a Ben Frank l in V is i ting Scho lar be -

. fore re turn ing to the Un ivers i t y o f Sou thampton . H e jo i ne d the W essex Ins t i tu te o f Techno logy , UK, in 1988 ,

~ w h e r e h e i s c u r r e n tl y R e a d e r a n d H e a d o f t h e D a m a g e T o l e r a n c e D i v is i on . A l i a b a d i m a i n t a i n s r e s e a r c h

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c h a n ic s . H e h a s a b o u t 2 0 0 t e c h n ic a l p u b l ic a t io n s , h a s c o - a u t h o r e d a t e x t b o o k o n f r a c t u r e m e c h a n i c s , d e -

v e l o p e d a d a t a b a s e f o r s t r e s s i n t e n si t y f a c t o rs , a n d e d i t e d 1 5 b oo ks . H e i s a m e m b e r o f t h e o r g a n i z i n g c o m m i t t e e o f t h e I n te r n a -

t i o n a l C o n f e r e n c e s o n L o c a l i z e d D a m a g e , S u r f a c e T r e a t m e n t, a n d C o n t a c t M e c h a n i c s . H e h a s a l s o s e r v e d o n s e v e r a l o t h e r c o n f e r-

e n c e c o m m i t t e e s i n c l u d i n g t h e W o r l d C o n f e r e n c e s o n B o u n d a r y E l e m e n t s a n d I n t e r n a ti o n a l C o n f e r e n c e o n B o u n d a r y E l e m e n t

T e c h n o l o g y . A l i a b a d i i s o n t h e e d i to r i a l b o a r d o f t h e I n t e r n a ti o n a l J o u r n a l o f F a t i g u e a n d c o - e d i t o r o f t h e I n te r n a t i o n a l J o u r n a l o f

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