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THE FORMATION OF EQUILIBRIUM CRACKS DURINGBRITTLE FRACTURE. GENERAL IDEAS AND HYPOTHESES.AXIALLY-SYMMETRIC CRACKS{O RAVNOVESNYKH TRESRCHINAKH ORRAZUSUSHCflIKS~A PRI

KRUPKOW RAZRUSHffNXI. OBSHCHIE PRRDSTAVLENIIAI BIPOTEZY. OSESIMHRTRICBNY& TRESHCHINY)PYM Yo1.23, No.3, 1959, pp. 434-444

G. 1. BABENBLATT(Moscor)

(Received 27 February 1959)

A lar ge nu mber of investigat ions ha s been devoted to th e problem of th eform at ion an d th e development of a cra ck dur ing britt le fra ctu re ofsolids. The first of th ese was th e well-known work of Griffith [l] de-voted to th e deter min at ion of th e critical length of a cra ck at a givenload, i.e. th e length of a crack at which it begins to widen cata str ophic-ally. Assumin g an elliptical form of a crack form ing in an infinit e bodysubjected to an infinitely homogeneous ten sion, Griffith obta ined a n ex-pres sion for the crit ical length of a crack as th at corr esponding to th etota l.of th e full increas e in ener gy (equa l t o th e sum of th e sur faceenergy plus the elastic energy released due to th e form at ion of th ecrack).

In recent years, in connection with th e nu merous technical applica-tions regar ding th e problem of cracks, th e nu mber of investigations ha sincrea sed, am ong th e first of which we ought to nam e th e works of Orowanan d Irwin , gen era l izing an d refining Griffith s th eory. A bibliograp hyand a short &urn6 of these works can be foun d in th e recent works ofOrowan [ 2 f , Irwin [ 3 1 , an d Bueckner [ 4 1.

The development of cra cks in brittle ma terials can be depicted in th efollowing fash ion. In th e ma ter ial th ere ar e a lar ge nu mber of micro-cracks. Upon an increase in load in a given spot of th e body, a str ess isrea ched sufficient for th e developmen t of th e micro-cra ck existin g atth at spot to a certa in size. The beginn ing of th e developmen t of th emicro-crack is deter min ed by some condit ion, becau se in view of th e factth at usua lly th e size of the micro-cra ck is small in ComPa rison with th echa ra cteristic linear dimension of th e str ess cha nge, th e sta te of stressin th e surr ounding ar ea of the micro-cra ck can be repr esented in accord-an ce with Griffith s scheme in th e form of a un iform in finite ten sion.

622


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Formation of equilibriun crocks during brittle fracture 623

I n th e cour se of its development, th e dimensions of th e crack increasean d finally become equal to th e chara cteristic dimension of th e str esschange. Under deter mined conditions (for example, when th e forces ar e nottoo large an d ar e applied sufficiently far removed from th e bounda riesof th e solid) th e developing crack is stopped upon rea ching its deter -mined length, an d th e solid can rema in in th is sta te a long time providedth ere is no change in th e loading.

In th e following study equilibrium cracks a re consid ered , i.e. cra cksform ing in a brit tle solid subjected t o a given syst em of forces, con-stant and not decreasing in time. Appar ent ly th e first ideas concerningequilibrium cracks ar e met in th e works of Mott 151 an d Fr enkel f6].These ideas are similar but were ar rived at independent ly. These workscont ain also a critical ana lysis of Griffiths th eory. Both th ese au th ors,however, limited th emselves to qualita tive considera tions, proceedingfrom th e assum ption of a crack of infinite length.

In our work th e question concerning equilibrium cracks form ing duringbritt le fractur e of a ma teria l is presen ted as a problem in th e classicalth eory of elasticity, based on certain very genera l hypotheses concern-ing th e str uctur e of a crack an d th e forces of inter action between itsopposite sides, and also on th e hypoth esis of finite str esses at th eends of th e crack, or. which amounts to th e same th ing, th e smooth ness ofth e joining of opposite sides of th e crack at its ends. Th e lat ter hypo-th esis was first put fort h by Khrist ianovich 171 in his considera tion ofcert ain problems in th e form at ion of cracks in rocks. Using this hypo-th esis it seemed possible to solve a series of problems relat ed to th edevelopment of cra cks in rocks [ 7-111 .

In th e considera tion of problems relat ed to rocks, one ma y evident lyneglect th e effect of th e cohesive of th e ma teria l as compa red with th eeffect of rock pre ssu re, so th at t he neglect of th e cohes ive force of th ema teria l in referen ce [7-111 can be justified. However, in oth er problemsof britt le fractur e (for example, in problems of britt le fractur e ofmeta llic str uctur es) factors of th e type of rock pressu re ar e absent , andth e consider at ion of cohes ive forces of th e ma ter ial becomes absolut elynecessary. It seems th at the intens ity of th e cohesive forces an d th eirdistribut ion can be chara cterized with sufficient accur acy by some newuniversal propert y of th e ma teria l which we call th e modulus of cohesion.Moreover, the dimensions of the cracks and their oth er properties areuniquely deter mined by th e applied loads and th e cohesive modulus. To de-ter mine th e cohesive modulus of a ma teria l one can use compa ra tivelysimple tests.

l.Basic ideas and hypotheses. Statement of the problem.kt us consider e~ilibri~ cracks in a britt le ma terial, i.e. cracksma inta ining const an t dimensions un der th e influence of a given system of


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624 C. . Barenblatt

fo rces . Moreover , we sha l l be l im i ted here to th e cons idera t ion o f th es imp les t case , c r acks lying in one p lane (so tha t po in t s a t th e su r f aceo f t h e c r a c k i n t h e u n d e f or m e d s t a t e a r e lo c a t e d i n o n e p la n e , t h e p la n eo f th e c r ack ). Th is case occu r s when th e app l ied s t r e s ses a r e sym m etr ica lr e la t ive to th e p lane o f th e c r ack ; th e genera l case wil l be cons ideredsepara te ly .

Fig. 1. Fig. 2.

Thu s, th e fo l lowing problem is con sidered (Fig . 11 . A g iven break ingload syn rne t r ica l r e la t ive to th e p lane o f th e c r ack i s app l ied to an i so -t rop ic , b r i t t l e , e las t ic so lid whose d imen s ions we assum e a re la rge inc o m p a r i s on w ith t h e l en g th o f t h e c r a c k . Mo r e ov e r, i t i s a s s u m e d th a t t h ecom pos i te fo rce app l ied from each s ide o f th e c r ack i s l im i ted . In par -t icu la r th e load can be app l ied on par t o f the very su r f ace o f th e c r ack .Ihe s t r es ses in th e so l id away fr em th e c rack ev iden t ly converge to ze ro .

I f th e app l ied loac l i s su f fic ien t ly la rge , th e re occu r s a b r i t t l efr a c tu r e o f t h e m a te r i a l &ic h , i f le ft t o c h a n c e , m u s t o c c u r i n t h ep lane o f sy rnne t ry o f the app lied load , Moreover , a c r ack i s fo rm ed wh ichwidens an d r eache s ce r t a in d imen s ions . I&e p rob lem i s to find th e c r ackd im e n s io n s c o r re s p o n d in g t o t h e g iv e n l oa d a n d o th e r c r a c k p a r am e te r s .

l e t u s t u r n fi r s t t o t h e i n v e st i g a t i on o f a m o r e s im p le c a s e wh e n t h eload i s app lied to th e edge o f th e c r ack su r face . Thus , i t i s as sum edth a t i n t h e b od y t h e r e i s a c e r t a in o r igin a l d im e n s io n o n w h ic h i s im -posed som e rup t u r ing load , wh ich we assum e i s n o rmal to the p lane o f th ecrack . Br it t l e fr ac t u re occu r s a r t th i s fr ac t u re i s w idene d (r em ain ingfla t because o f sy rnne t ry o f th e lohd and th e i so t ropy o f the so l id ) toc e r t a in d e fin i t e m e a s u r e m e n t s .


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For-nation of equitibriun cracks during brittle racture 625

lhe cross-section (a) and the plane (b) of such a crack are shownschematically in Fig. 2. 'lhe rack is divided into two regions: region 1(the inner region) and region 2 (the terminal region). In the inner regionthe opposite sides of the cracks are at a significant spacing so thatinteraction between these sides does not occur. 'lhe nner region of thecrack falls into two sub-regions la and lb; in the first the applied loadacts on the opposite sides of the crack, and in the second the oppositesides of the crack are free from stress.

In the terminal region the opposite sides of the crack come close toeach other so that there are very large interaction forces attracting oneside of the crack to the other. As is known, the intensity of theattractive force acting in the material strongly depends upon the dis-tance, at first growing rapidly with an increase in the distance y betweenthe attracting bodies, from a normal interatomic distance y = b for whichthe intensity is equal to zero, up to some critical distance where itreaches a maximum value equal to the order of magnitude of Young's modulus,after which it rapidly falls with increasing distance (Fig. 3).

b

Fig. 3The accurate determination of the system of cohesive forces acting in

the terminal region is difficult. However, one can introduce certainhypotheses which permit limitation to one composite universal character-istic distribution of cohesive forces for a given material.

First hypot~s~~, The di~nsion d Of the terminal region is small incomparison with the size of the whole crack.

Second hypothesis. The distribution of the displacement in the terminalregion does not depend upon the acting load and for the given materialunder given conditions (temperature, composition and pressure of the sur-rounding atmosphere and so forth) is always the same,

In other words, according to this hypothesis, the ends in all cracksin a given material under given conditions are always the same. bringthe propagation of the crack the end region merely moves over to anotherplace, but the distribution of the distortion in the sections of theterminal region with planes normal to the crack contour remains exactly


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626 G.I. Barenblat t

t h e s a n e . T h e c o h e s ive fo r c e s a t t r a c t i n g t h e o p p os it e s i d e s o f t h e c r a c kto e a c h o th e r d e p e n d o n ly o n t h e m u tu a l d i s t r ib u t i o n o f t h e s i d e s (i .e .on th e d i s t r ibu t ion o f th e d i sp lacem en t ); th e re fore , acco rd ing to th efo rm ula ted hypo thes i s , th ese s t r es ses w il l be th e sam e.

The f ixed shape o f th e te rm ina l r eg ion o f th e c r ack co r r esponds tot h e m a x i m u mposs ib le r es i s tance . We em phas ize th a t in v iew o f th e i r r e -ver s ibi li ty o f th e c r acks occu r r ing in t h e m a jo r it y o f m a te r i a ls , t h esecond hypo th es i s i s app l ied on ly to th ose equ i l ib r ium cracks wh ich a refo rm ed du r ing th e p r imary rup tu re o f th e in i t ia l ly unb roken b r i t t l e so l ida n d n o t t o t h o s e c r a c k s w h ic h a r e fo r m e d a t a n a r t i fi c ia l n o t c h wi th o u tsubsequen t p ropaga t ion o r du r ing a decrease in load r esu l t ing fromprev ious c r ack in g a t som e la rger load . Fo r th e la t te r types o f c r ack sth e s t r es s in th e te rm ina l r eg ion can be d i ffe r en t (sm al le r ); th ese typeso f c r acks a r e exc luded from th is d i scuss ion .

Third hypothesis. The opposite sides of the crack are snwothly joinedat the ends or, which amounts to the same thing, the stress at the endof a crack is finite.

(As we no t ed above ,th i s hypo th es i s was f ir s t exp ressed by Khr is t iano -v ich in r e la t ion to th e fo rm at ion o f c r ack s in rocks . )

Ihe ind ica t ed hypo th eses m ake poss ible th e so lu t ion o f th e p rob lemu n d e r c o n s id e ra t i o n . Le t u s m e n t ion t h e d i ffe r e n c e a p p e a rin g i n t h e c a s ewhere th e app l ied loads a r e no t on th e su r f ace o f th e c r ack bu t in s ideth e so lid . In p r inc ip le no t h ing i s chan ged in th i s case . Act ua l ly , wes h a l l r e p r e se n t t h e s t a t e o f s t r e s s a c t i n g i n t h e s o li d w ith a c r a c kund er th e ac t ion o f a ce r t a in load app l ied in s ide th e so lid , by th e sumo f tw o s t a t e s o f s t r e s s . &e o f t h e s e s t a t e s c o r re s p o n d s t o t h e s t a t e o ft h e c o n t in u o u s s o li d wi th o u t t h e c r a c k , a s t a t e a p p e a r in g u n d e r t h eac t ion o f a g iven load ing system. The o th e r s t a t e o f s t r e s s c o r re s p o n d sto t h e s t a t e i n a b od y w ith a lo a d a p p l ie d o n t h e s u r fa c e o f t h e c r a c k .lh e c o m p o s i t e n o r m a l a n d s h e a r s t r e s s e s o n t h e s u r fa c e o f t h e i n n e rr e gion o f t h e c r a c k m u s t b e e q u a l t o z e r o b e c a u s e t h e s u r fa c e o f t h einner r eg ion o f th e c r ack i s fr ee o f s t r e s s . Ihere fo re , th e load app l iedto t h e s u r fa c e o f t h e in n e r r e gion o f t h e c r a c k r e p r e s e n t s a c o m p r e s s iven o r m a l s t r e s s , e q u a l i n m a g n i tu d e a n d o p p os it e i n s ig n t o t h e t e n s i les t r es s appear ing in th e p lane o f synm et ry o f th e app l ied load in th e bodywi th o u t t h e c r a c k . In t h e t e r m in a l r e gion fo r t h e s e c o n d s t a t e o f s t r e s sth e n o x m a l s t r e s s e s a r e e q u a l to th e s t r e s s o f th e cohes ive fo rces , bydeduc t ion cor r espond ing to th e s t r e s s o f th e fi r s t s ta te o f s t r e s s . Inv ie w o f t h e f a c t t h a t t h e p la n e o f t h e c ra c k i s t h e p la n e o f s y m m e t r y o fth e a p p li e d lo a d , t h e s h e a r s t r e s s e s o n t h e s u r fa c e o f t h e c r a c k a r ea b s e n t . T h e n o r m a l d i s pla c e m e n t s o f t h e p o in t s o f t h e s u r fa c e o f t h ec r a c k a r e d e t e r m in e d o n ly by t h e s e c o n d s t a t e o f s t r e s s , i n s o fa r a sth e y a r e e q u a l t o z e r o i n t h e fi r s t s t a t e .


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Format ion o f equ i l ib r ium cracks during brittle f r a c tu r e 627

T h u s , t h e c h a n ge in c or r p a ri s on w ith t h e special case consid erede a r li e r c o n s i s t s i n t h e fa c t t h a t t h e i n n e r region of th e c r ack now doesno t fa l l in to tm sub -reg ions , and th e load ing of t h e s u r fa c e o f t h ecrack occ u r s a long t he who le inner region.

The th r ee hypo th eses fo rm ula ted above a re app l ied a l so to th i s m oreg en e r a l c a s e . Wi th r e fe r e n c e t o t h e fi r s t a n d t h e t h i r d h y p oth e s is t h i sdoes no t r equ i r e e luc ida t ion . As fo r the second hypo th es i s , th e poss ibi l-i ty o f i t s app l ica t ion i s exp la ined by th e fac t th a t chan ges in th ea p p li e d s t r e s s e s i n t h e t e r m in a l r e gion u n d e r t h e i n fl u e n c e o f t h eapp l ied load a re o f a very m uch la rger o rder th an th e app l ied load . Aswas sh own ear l ier , t h e s t r e s s e s i n t h e t e r m in a l r e gion a r e o f t h e o r d e ro f Youngs m odu lus , i . e . th ey s ign i f ican t ly su rpass th e m agn i tud e o f th eapp l ied load . Therefo re , in th e genera l case too one can neg lec t chan gesin s t r e s s e s a p p li ed t o t h e s u r fa c e o f t h e c r a c k i n t h e t e r m in a l r e gionun der th e in f luen ce o f th e app l ied load an d cons ider on ly the d i s t r ibu t iono f s t r e s s e s a n d d i s pla c e m e n t s i n t h e t e r m in a l r e gion n o t d e p e n d e n t u p o nth e load , i . e . one can app ly th e second hypo th es i s .

2. Axially symmetric cracks. (1 ) le t th e c r ack have the fo rm o fa round s l it o f r ad ius R. On bo th s ides o f th e po r t ion o f th e su r f ace o fth e round s l it (F ig. 4 ) a t va lues o f th e runn ing r ad ius r , sm al le r th an

Fig. 4.

s o m e r a d iu s r O ( ra iu s o f th e r eg ion o f th e load app l ica t ion ), com press -ive norm al forces 2% = - p(r ) are act ive. The par t o f the su r face o f th es l it co r r espond ing to th e in te rm ed ia te r eg ion rO < r < R - d, is f ree ofs t r e s s , In th e te rm ina l r eg ion R - d < r < R, t e n s i l e s t r e s s e s C(r) c o n -t ro l led by cohes ive fo rces ac t .

As was shown by Sleddon [ 12 1 , t h e d i s t r ib u t i o n o f n o r m a l d i sp la c e m e n t so f po in ts o f th e su r f ace o f th e round c rack in an in f in i t e e las t ic so l id ,i f a no rm al s t r es s - g (r ) ac t s on th i s su r face , and shear s t r es se s a r ea b s e n t , h a v e t h e fo r m :


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628 C. I. Barenblatt

uI= 4(1-+)I?XE (2.4)

Here E is Youn gs m odulus an d v is Poissons ra t io . Par t ia l in t egra-t ion p roduces t h i s exp ress ion in th e fo rmu= 4(1-+)R

xl?(2.2)

Accord ing to th e cond i t ion o f sm oo th join ing o f th e oppos ite s ides o fth e c r a c k a t i t s e n d s , t h i s e v id e n t ly le a d s t o t h e r e la t i o n s h ip

Dif fere nt ia t ing (2 .2 ) we obta inI 1 1

a w 4 (1 -9) Rap= ?TE ilhe second m em ber o f th e r igh t -han d s ide o f th i s r e la t ionsh ip , un der

b road assum pt ions r e la t ive to th e fun c t ion g , converges t o ze ro fo rp + 1 , from wh ich i t i s ev iden t th a t to fu l fi ll cond i t ion (2 .3 ) i t i snece ssary t o fu l fi ll th e r e la t ionsh ip

1 (2.4)In th i s sam e fo rm re la t ionsh ip (2 .4 ) was ob ta ined in r e f e rence [ 91 as

th e cond i t ion o f fin i ten ess o f th e s t r e s s a t the edge o f an ax ia lly -s y m n e t r i c c r a c k . In t h e c a s e w e a r e c on s id e r in g

IP (r) (0 < r < fo)

(ro < r d R-d)s(r)= [$I(,) (II--d


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Formation of cquilibriua cracks during br i t t l e fracture 629

& th e s t r en g th o f th e fi r s t hypo th es i s , d i s sm al l in com par i son withR, so th a t x in a l l r eg ions o f in t eg ra t ion o f 1 - d/R < x < 1 d if fersli t t l e fr o m u n i ty a n d o n e wa y t a k e

(2.8)Let u s pass over in th i s in teg ra l from th e d imen s ion less var iab le x

to t h e d im e n s io n a l c o o rd in a t e s = R - Rr, c a l c u la t e d fr o m th e e d ge o f t h ec r a c k a n d c h a n g in g wi th in t h e i n t e g r a t i on l im i t s fr o m z e r o t o d. We h a v e

s in c e Rx dif fers l i t t le f rom R w ith in t h e l im i t s o f i n t e g r a t i on .Subs t i t u t ing th i s in t o (2 .8 ) we ob ta in

(2.9)On th e s t r e n g th o f t h e s e c o n d h y po th e s i s t h e d is t r i b u t i o n F(s) does

n o t d e pe n d u p o n t h e a pp li e d l oa d , s o t h a t t h e i n t e g r a ld F(s)dss - vi

represents for a given material at given conditions a constant veluewhich we shall signify by K and shall call the cohesive modulus.

It i s e a s y t o s h o w th a t t h e d im e n s io n s o f t h e c o h e s ive m o d u lu s a r e[K] = FL-/ z

where F i s th e d imen s ion o f . fo rce and L th e d imen s ion o f leng th . Thecohes ive m odu lus r ep resen t s a p roper ty o f a m at e r ia l wh ich has ev iden t ly habas ic s ign i f ican ce in th e th eo ry o f b r i t t l e fr ac t u re ; i t en t e r s in to th ebas ic r e la t ionsh ips r egard less o f th e chara c te r o f load ing and th e geo -m e t r i c a l s h a p e of t h e c r a c k . T h u s w e h a v e

(2.10)Subs t i t u t ing th i s r e la t ionsh ip in to equa t ion (2 .6 ) we ob ta in , pass ing

from th e d ime ns ion less var iab le x to th e d imen s iona l var iab le r:


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630 G. I. Barenblatt

This e q u a t io n d e t e r m in e s t h e r a d iu s o f c r a c k R a s a fu n c t i o n o f t h eapplied load .

(2) Let us con sider severa l exan ples . Let . t he applied load p(r) bec o n s t a n t , p(r) = PO. Then t h e e q u a t i o n (2 . 1 1 ) t a k e s t h e fo r m :

P, [R - I/R2 - r:] = K I/; (2 .12)It i s c o n ve n i e n t t o b r in g t h i s e q u a t i on t o t h e d im e n s io n le s s fo r m

(2 .13)

A graph of th e relat ionsh ip in (2 .1 3) is g iven in Fig. 5. We see,that for Podrp < K/\ l2,equation (2.13) does not have real solutions.'lhis act permits a very sinple interpretation: there exists a certainminimum stress, the application of which on a circle of a given radiusguarantees the possibility of opening the crack. At stresses less thanthis minimum the crack does not open. The magnitude of the minimum stressdecreases in inverse proportion to the square root of the radius of theregion of the load application. Each load surpassing the minimum corres-ponds to one single radius of the crack. With an increase in load theradius of the crack naturally grows.

Fig. 5.It is an especially curious particular case when a crack is formed by

oppositely directed concentrated forces equal in magnitude and appliedto opposite sides of the crack. lhis limiting case corresponds tor/R


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F c r m a t io n o f

Whence we obtain

equil ibr ium cracks dur ing b r i t t l e fr ac t u re 6 3 1

(2.2 5)

l J s ing th e P -th eo rem o f th e th eo ry o f s im i la r i ty (see th e hook bySedov [ 11 1 ), th e la t t e r r esu l t can he ob ta ined with an accu ra cy with inth e c o n s t a n t fa c to r o f o n e o f t h e c o n s id er e d d im e n s io n s .

(3) With c e r t a in a s s u m p t ion s o n e c a n s h o w th e e xi s t e n c e o f a u n iq u eso lu t ion o f equa t ion ( 2 .1 1) for an arb i t r ary load p(r ). Indee d , le t usa s s u m e t h a t

(2.16)

J t i s e v id e n t t h a t w ith gr o wth o f R t h e f u n c t i o n

d e c r e a s e s m o n o to n i c a lly , a n d t h e fu n c t i o n iti, = KdR /t/ 2 i n c r e a s e smono ton ica l ly . F rom (2.16) i t follows t h at m l(rO) > @z(r,) an d t he refore ,t h e r e e x is t s o n e u n iq u e v a lu e R > ro, for wh ich fiI(R) = Qz(R), i .e. forwhich ep_at ion (2 .1 1) is fu lfi l led . Ihe inequ al i ty (2 .1 6) is th e con di-t ion under wh ich th e app l ied load wil l su rpass th e m in is load , andth u s h e a b le t o o p en t h e c r a c k .

.7 . Axially-symmetric crack s (continuat ,ion). (1) Let us con-s ider n o rma l d i sp lacem en t s o f po in ts o f th e su r f ace o f th e c r ack , de t e r -m in ing i t s w id th . At a g iven load t he r ad ius o f th e c r ack R i s d e t e r m in e dh y th e r e ia t i o n s h ip f 2 .11 ). Knowing th e r a t fiu s R one can de t e rm ine th ed isp lacem en t s m en t ioned us ing t he fo rm ulas (2 .1 ) and (2 .5 ). We have

(3.1)

Let . u s cons ider th e inner in te g ra l in th e exp ress ion fo r q jz . S ince byis c lo se to un i ty th i s in t eg ra l i s c lo se to th e in te g ra l


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632 (2.1. Barenblatt

1

s G (~3) dxI-dlR - j/l--.2 (3.4)Mhen ce also f rom (3 .3 1 i t fo llows t ha tS ince d/R is sm all , +

p > E- , / R, but 1 - p >>is sm all in com par ison wit h $I . Now let

d R, i .e . t h e c o n s id e re d p o in t o f t h e s u r fa c e o fth e c r a c k r e m a in s a t a s i gn i fi c a n t d i s t a n c e fr o m th e e n d s i n c o m p a r i s onwi th t h e d im e n s io n o f t h e t e r m in a l r e gion d. In t h i s c a s e

(3 .6)

a n d $ a s b e fo r e i s d e t e r m in e d b y t h e r e la t i o n s h ip (3 . 3 1 a n d c o n s e q u e n t ly ,by th e r e la t ionsh ip (3 .51 . I t i s ev iden t th a t a l so in th i s case &


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Fornation of equilibrium cracks during brittle fracture 633

In F ig. 6 a i s g iven th e ac t ua l d i s t r ibu t ion o f d i sp lacem en t s ; in Fig.6b i s g iv e n t h e d i s t r i bu t i on o f d i s pl ac e m e n t a t t h e s a n e r a d iu s o f c r a c kbu t found with ou t c ons ider ing th e cohes ive fo rces .

(2 ) l e t u s n o w c o n s id e r o n t h o s e c h a n g e s w h ic h m u s t b e i n t r o d u c e d t o t h ep reced ing d iscuss ion i f th e b reak ing load i s app l ied in s ide th e so l idand no t on th e su r f ace o f th e c r ack . Acco rd ing to ou r as sunp t ion , s inceth e l oa d i s s y m m e t r i c a l r e l a t i ve t o t h e p la n e o f t h e c r a c k , s h e a rs t r e s s e s i n t h i s p la n e a r e a b s e n t , Le t u s a s s u m e , a c c o r d in g t o t h e fo r e -g oin g, t h a t t h e s t a t e o f s t r e s s a c t i n g i n t h e c o n s id e re d b r it t l e s o li dwi th a c r a c k i s i n t h e f or m o f t h e s u m o f tw s t a t e s o f s t r e s s ; o n e o fwh ic h r e p r e se n t s t h e s t a t e o f s t r e s s i n a c o n t in u o u s b od y w ith o u t a c r a c kund er th e ac t ion o f a g iven load , a n d t h e o th e r a s t a t e o f s t r e s s i n abody with a c r ack in the p resence o f a b reak ing load a t th e su r face o f acrack . Let p(r) r e p r e se n t t h e d i s t r ib u t i o n o f n o r m a l s t r e s s i n t h e p la n eo f s y m m e t r y u n d e r a give n l oa d i n t h e a b s e n c e o f a c r a c k . T h e n t h e c o m -p r e s s iv e fo r c e a t t h e s u r fa c e o f t h e c r a c k d e t e r m in in g t h e s e c o n d s t a t eo f s t r e s s i s equa l t o - pfr). Repea t ing th e fo rm er r eason ing , we ob ta inin p lace o f equa t ion (2 .11 ) an equa t ion de t e rm in ing th e r ad ius o f th ec r a c k i n t h e f o r m

R

(3 .7)

wh ich d i ffe r s from equa t ion (2 .11 ) on ly by th e fac t th a t in p lace o fr ad ius r o, we use th e fu l l r ad ius o f th e c r ack R.

S in c e t h e n o r m a l d i s pla c e m e n t o f t h e p o in t s o f t h e s u r fa c e o f t h ecrack co r r espond ing to th e fi r s t s ta te o f s t r e s s ; i s equa l to ze ro , a llc o n c lu s ion s r e la t i v e t o t h e d i s t r ib u t i o n o f t h e d i s pla c e m e n t o f t h e p o in t so f th e su r f ace o f th e c r ack , fo rm ula ted in th e fo rego ing sec t ion a l soh o ld t r u e .

(3 ) Le t u s cons ider as an exam ple th e p rob lem o f the fo rm at ion o f ac rack by tu n oppos ite ly d i r ec ted concen t r a t ed fo rces T, app l ied a t po in tssepara t ed by a d i s tan ce o f 2L (F ig. 7 ). I t is na tu ra l to suppose th a t t hecrack l ies in the p lane o f sy rnne t ry o f th e load . Sum m ar iz ing th e well -known so lu t ions o f Eouss inesq [ 14 1 , i t i s easy to ob ta in th a t

Cond i t ion (3 .7 ) can be r ewr it t en in th e fo rm(3 .8)

(3.9)


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634 C.I. Barenblatt

Subst i tu t ing (3 .8 ) in (3 .9 ) we obta in1

The g raph o f t he func t ion g (L/ R) has th e fo rm shown in F ig. 8 . (OnFig. 8 i s r ep resen t ed th e case with V= 0.5 ). TIere is physica l s ignif i-canc e on ly to th e le ft -hand par t o f t h e c u r ve u p t o t h e m i n i m u m p o in t ,

Fig . 8 .shown by th e so l id l ine . In fac t , th e r igh t -han d par t o f th e cu rve co r -r esponds to th e g rowth o f th e c r ack r a d ius du r ing decrease in load . Asis ev iden t f rom th e graph of Fig . I I, a t

3 < rc &J W~ (:I. I)equa t ion (3 .11 ) does no t have a so lu t ion . In comp le te an a logy to th efo regoing , th i s m eans th a t fo r such m nal l 7 th e c r ack i s no t fo rm ed . AtT = TIgoKLji2 a c rack o f a de t e rm ined fina l r ad ius R, i s foxm ed a t oncewith fu r t her increase in T t h e r a d iu s o f t h e c r a c k R i s inc reased . Theva lues go and R, a r e found by e lemen ta ry m eans : a t v = 0 .5 we have g , =0 . 9 4 5 , L/R0 = 2 . 3 5 , a t v = 0 we ob ta in g , = 1 .39 , L/A, = 2 . 0 5 .

Gene ra l iz ing su i ta b ly th e d i scuss ion o f Sec t ion 3 , par t (2 ) to equa-t ion (3 .71 , o n e c a n s h o w th a t s u c h a c o n d i t io n i s t y p ic a l a t a n a p p li c a-t ion o f load in s ide th e so l id fo r a body o f in f in i t e d imen s ions . Thefo l lowing is t r ue: i f t he load , a ppl ied ins ide t he body, is propor t ionalt o s o m e p a r am e t e r , th en a t su f fic ien t ly sm al l va lues o f th i s param ete ran equ i l ib r ium crack in genera l does no t fo rm . i lpon r each in g som e c r i t i ca lva lue o f th e param et er , a c r ack o f fin i te r ad ius i s imn ed ia te ly fo rm ed .Upon fu r t her increase o f th e param ete r th e c r ack g rows con t inuous ly .


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Formation of equilibrium cracks during

Let us a lso no t e th e pecu l ia r l im i t ing casevergence o f L t o zero, i.e. t h e convergence ofo f fo rces . F rom th e r e la t ionsh ip (3 .11 ) a t L/ Rre la t ionsh ip

brittle fracture 635

co r respond ing to th e con -th e po in ts o f app l ica t ion+ 0 we ob ta in aga in th e

co inc iden t w ith th e r e la t ionsh ip (2 .15 ), as was t o b e e x p e c t e d .Ih e case of p lane c racks wil l be cons idered in a separa t e paper .

BIBLIOGRAPHY1. Griffith , A. A., The phen omenon of ru pt ur e an d flow in solids, Ph il.

Tra ns . Roy. Sot. A, Vol. 221, 1920.2. Orowan, E., En ergy criteria of fra cture. W elding J. Res. Suppl. March

1955.3. Irwin , G. R. , Analysis of str esses an d str ains nea r th e end of a crack

tra versing a plate. J. Appl. Mech. Vol. 24, NO. 3, 1957.4. Bueckner , H. F., The propagation of cracks and th e ener gy of elastic

deformation. Trans. Am er. Sot . Mech. Engs. Vol. 80, No. 6, 1958.5. Mott, N. F., Fra ctur e of metals: theoretical considerations. Engi-

neering. Vol. 165, No. 4275, 4276, 1948.6. Frenkel , Ya. I., Teoriya ohra timykh i neohra timykh treshchin v tver-

dykh telak h (Theory of reversible and irreversible cracks in solidbodies). Zh. tekh. fiz. Vol. 22, No. 11, 1952.

7. Zhel tov, Yu. P. an d Khr istia novich, S. A., 0 mekh an izme gidra vliches-kovo ra zryva neftenosnove plasta (The mechanism of hydra ulic fra c-tu re of oil-bear ing sheet s). IZU. Ak ad. Nauk SSSR, Otd. Tckh. NaukNo. 5, 1955.

8. Baren blatt , G. I. and Khristia novich, S.A., Ob obru shen ii kr ovli prigorn ykh vyra botk akh (Concernin g th e collapse of a roof durin gmining). Izv. Akad. N auk SSSR, Otd. Tekh. Nauk NO . 11, 1955.

9. Baren blatt , G. I., 0 nekotorykh zadachak h tebrii upr ugosti, vozni-kaiush chikh pr i issledovanii mekha nizma gidravlicheskovo ra zryvaneftenosnovo plasta (Concernin g several problems in th e th eory ofelasticity appea ring du ring th e investigation of th e mechanism ofhydr au lic fractur e of oil-bear ing she ets ). PMM Vol. 20. No.3. 1956.


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636 G. I. Barenblatt

10. Baren blatt , G. I., Ob obrazovan ii gorizont alnykh tr eshchin prigidravlicheskom ra zryve neftenosnovo plasta (Concerning th e form-at ion of horizont al cracks during hydrau lic fractur e of oil-bearingsheet). Itv. Akad. Nauk SSSR, Otd. Tekh. Nauk NO. 9, 1956.

11. Zhelt ov, Yu.P ., Ob obrazovan ii vertika lnykh tr eshchin v plaste pripomoshchi filtr uiu shcheisya zhidkosti (Concern ing th e form at ionof vertical cracks in a sheet dur ing liquid filtra tion). IZV. Akad.Nauk SSSR, Otd. Tekh. Nauk NO. 8, 1957.

12. Sne ddon, J .D., Fourier Transforms. New York, 1951.13. Sed ow, L. I., Metody podobiya i razmernosti v nekhanike (Methods of

Similarity and Dimensions in Mechanics). GITTL, Moscow-Leningrad,1957.

14. Love, A., Matematicheskaya teoriya uprugosti (Mathematical Theoryof Elasticity). ONTI, Moscow-Lenin gra d, 1935.

Translated by A.M.T.

barenblatt 59_0

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