1963-Paris, P.; ErdoganA Critical Analysis of Crack Propagation Laws

8/12/2019 1963-Paris, P.; ErdoganA Critical Analysis of Crack Propagation Laws

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P . P A R I S 1Assoc ia te P ro fesso r o f Mechan i cs andAssistant Di r ecto r o f the Inst i tu te o f Resea rch,Leh igh Un i ve rs i t y , Be th l ehem, P a ,

F . E R D O G A NPro fesso r o f M echan i ca lEng inee r i ng , Leh igh Un i ve rs i t y ,Be th l ehem, Pa . Mem . ASME

A C r i t i c a l A n a l y s i s o f C r a c k P r o p a g a t i o n L a w sTh e p r a c t i c e o f a t t emp t i n g va l i d a t i o n o f c r a c k - p r opaga t i o n l aw s ( i . e. , the l aw s o f H ead ,F r o st a n d D u g d a l e, M c E v i l y a n d I l l g , L i u , a n d P a r i s ) w i t h a sm a l l am o u n t o f d a t a ,s uch as a ew s i ng l e spec imen te st r e su l t s , i s quest i o ned . I t i s shown th a t a l l t he l aw s ,t h o u g h t h e y a r e m u t u a l l y c on t r a d i c t or y , c a n b e i n a g r e em en t w i t h t h e s ame sma l l s amp l eo f da t a . I t i s sugges ted th a t ag r eem en t w i t h a w i de sel e ct i o n o f da t a f r om m any spec i -m ens and over m any o r de r s o f m agn i t ud es o f c r a c k - ex tensi on r a t es m ay be necessa r y tov a l i d a t e c r a ck - p r o p a g a t i on l a i u s . F o r s u ch a w i d e c om pa r i s o n o f d a t a a n ew s imp l eem p i r i c a l l aw i s g i v en w h i c h f i t s t h e b r o a d t r e n d o f t h e d a t a .

s *I n t r o d u c t i o n

I V H L crack- propagat ion law s hav e been pr e-sented in the past few years, which claim to be verified by theexperimental data analyz ed in their respective papers. T heyare specifically the work of H ead [l], a F rost and Du gdale [2],M cEv i l j ' and I l lg [3] L iu [4, 5] and Paris, Gomez, and A nderson[6,7,8].

This paper will attempt to show that basing validation of acrack-propagation theory on a limited amount of data is a poortest of any theory. M oreover , a wide range of test data is nowavailable in a convenient form [9] which may be employed toanalyze critically these several crack propagation laws.

Therefore the paper will consist of three parts which will in-clude:

1 A review and comparison of existing crack -propagat ionlaws.

2 T he erroneous results obtain ed by compar in g each law wi tha limited range of test data.

3 T he results of comparin g crack-p ropagat ion laws wit h awi de range of dat a.

A R e v i e w o f E x i s t i n g C r a c k P r o p a g a t i o n L a w sCrack-propagation laws given in the literature take many

form s. I n general they tr eat cracks in infi nit e sheets subj ectedto a uniform stress perpendicular to the crack (or can be appliedto that configuration) and they relate the crack length, 2a, to thenum ber of cycles of l oad appli ed, N , w ith th e stress ran ge cr, andmaterial constants C,-.' T he single form in which all crack -propagation laws may be written is

d ad N = / ( f , a , C i ) ( 1 )

Therefore this format will be employed in the discussion tofollow.

Chronologically the first crack-propagation law which drewwi de attent ion was tha t of H ead [1] in 1953. H e empl oyed amechanical model wh ich considered rigid-plastic work-har deningelements ahead of a crack tip and elastic elements over the re-main der of the infi nit e sheet. T he model required extensivecalculations and deductions to obtain a law which may bewritten as

d ad N

C ^ a ' *(C 2 - 0 >'/ ' (H ead's law ) 2 )

where Ci depends upon the strain-hardening modul us, the modu -lus of elasticity, the yield stress and the fracture stress of thematerial , and Ci is the yield strength of the materi al. H ead de-fined Wo as the size of th e pla sti c zone near the crack ti p and pr e-sumed it was constant during crack propagation. H owever,F rost [2] noticed that t he plasti c-zone size increased in directproportion to the crack length in his tests. M oreover, Ir win [10]has recently pointed out from analytical considerations that

Wo (3)for the configuration treated here which is in agreement withF rost's conclusions. Th erefore, though H ead adopted equation(2) with w0 considered constant as his crack-propagation law, weare forced here to introduce equation (3) into equation (2) toobtain a modifi ed or corr ected form of H ead's cra ck-pr opagati onlaw; i.e.,

1Al so Consultant to the Boeing Company, Tr ansport Divi sion,Renton, Wash.2N umbers in br ackets designate References at end of paper .3 In th is discussion a will impl y the stress-fluctuati on range and Ciwill vary slightly with mean stress in a general interpretation.Contri buted by the M etals E ngineering D ivision and presented atthe Winter Annual M eeting, New Y ork, N . Y ., N ovember 25-30,

1 9 62 , o f T H E A M E R I C A N S O C I E T Y O F M E C H A N I C A L E N G I N E E R S .M anuscript received at AS M E H eadquarters, Apr il 11, 1962. PaperN o. 62W A-234.

d a C 3o~ 2ad N = (C 2 - o) (H ead's corrected law) (4)

F rost and D ugda le [2] in 1958 presented a new approach tocrack-p ropagati on laws. T hey observed as was in tr oduced inequation (4), that H ead's law should be corrected for the vari a-tion of plasti c-zone size wit h crack length. T hey deduced th atthe corrected result, equation (4), depends linearly on the cracklength a . H owever, they also argued by dimensional analysis

N o m e n c l a t u r ea = h alf crack length

Qo = initi al half crack lengthB = a fu nct ion of stress ran ge (an d

mean stress)C = a numeri cal constantC ; constants (which vary slightly

with mean stress)D s = constants which depend onstress level

/( ) = a fun cti on of

F { } = a funct i on ofG { } = a funct i on of

k = stress-intensity factor (ran ge)IC lW = str ess-concentr ation factorm = a numeri cal exponentM = a material constantn = a numeri cal exponentN = cycle nu mber

Aro, N j, ini tial and final cycle n umber ,respectively

d a = crack extension per cycle ofd N loadA k = stress-int ensity-factor range

p , pi = cr ack-ti p radiu scr = appl ied stress (r an ge)U Q = stress at a cra ck tip (wi th

finite radius p)crnot = net section stress

w0 = pla sti c-zone size5 2 8 / D E C E M B E R 1 9 6 3 T r a n s a c t i o n s of t h e A S M E

Copyright 1963 by ASME

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that the incremental increase in crack length d a , for an in cremen-tal number of cyclesd N , should be directly proportional to thecrack length a . H ence th ey concluded th at (in dependent ofH ead's model)

d ad N B a (5)

following arguments: I rw in's [12] stress-intensity-factor k r e-flects the effect of external load and configuration on the in-tensit y of the whol e stress field around a crack tip. M oreover ,for various configurations the crack-tip stress field always has thesame form (i.e., dist ri buti on). T herefor e it was reasoned thatthe intensi ty of the crack-t ip stress field as represented b y kshould control the rate of crack extension.4 T hat is to say:

where B is a fun ction of the appli ed stresses. T hen th ey ob-served that in order to fit their experimental data: d N = ( ? { f c } (15)

B = C ;Combining equations (5) and (6) they obtained the law:

6)

da c r 3a = (F rost and Du gdale's law) (7)d N C iA bout the same time M cE vi ly and Il lg [3] modified a method

of analysis of stati c strength of plates wi th cracks used at N A S Ato obtain a th eory of crack propagati on. T heir arguments wereas follows: Pr esuming that a crack tip in a material has a charac-teristic (fictitious) radius pi, which allows computation of thestress (To, in the element at the crack tip using elastic stress-concentration factor concepts, the stress a 0i s

(To = K n(T net 8 )

I C N = 1 + 2 (a /p ,) 'A (9)which is based on the elastic solution for an elliptical hole ofsemimajor axisa and end radiusp i .

which is the desired form in this discussion upon considering pi tobe a material constant, in likeness to the C s.M cE vi ly and I l lg go on in an empirical manner to obtain theform of the functionF { }, and suggest

logi 00509 ffiV crnel - 5. 472 - 34 (14)K N a n e, - 34(M cE vi l y and I l lg's law empir ical ly extended)

I t should be noted by the reader that M cE vi ly and Il l g's laws,equations (12) and (14), are not restricted to the special configura-tion here, as was the case for th e laws of H ead, and F rost andDu gdale. T he applicabili ty to other configurations and an addi-tional similarity to Paris' work [6, 7, 8] will warrant later com-ments.

I ndependent of M cE vi ly and Il lg, Paris [11] proposed a crack -propagat ion th eory at about th e same time. I t is based on th e

I n treatin g the special confi gurat ion of interest in this discussi on,it should first be observed that (11)

k = < r a ( 1 6 )whereupon equation (15) may be specialized to read:

d ad N = "A (17)

Somewh at later, Li u [4] restated F rost and Dugdal e's [2] di -mensional analysis in a much more elegant form and argued thatthe crack-growth rate should depend linearly on the cracklength; i.e.,

d a = B a (L iu's law)d N 18)

where K N is the stress-concentrat ion f actor and


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dimensions of any configuration considered, and C is a constantind ependent of the configura tion. F in all y, cr0in equation (24) m aybe interpreted to be the same asK N


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where \ is a constant for a given stress range a. I ntegrating thi sexpression givesrr = D i N + constn ' 2 (31)

T herefore H ead suggested plotti ng l / a ^ versusN and, observingequation (31), implied that obtaining a straight line for eachspecimen whose data are plotted this way indicates verification ofhis law.T he data of M arti n and Si nclair [13] are employed here as anunbi ased source. F rom their data specimens nos. 10, 14, and 15were chosen since those specimens were run at medium stresslevels and the greatest number of data points per specimen wasrecorded for them. F ig. 1 shows the type of pl ot suggested byH ead and it is noticed that portions of the data do form straightlines. Does this mean that H ead's theory is verified? T hereader is warned that this might be a hasty conclusion.F irst, consider the corrected law of H ead, Fr ost and D ugdale'slaw, and L iu's law, equations (4), (7), (18), and (19), respec-ti vely. F or a constan t stress-range test all these laws reduceto the form

d ad N = D * a

Integrating this result gives:loga = D 2N + const

(32)

(33)

(34)for a given material . T o test these theories, data from severalspecimens should be plotted on ak versusd a / d N (or logd a / d N )graph and these laws predict that the data of several specimenswill form a single curve. Again the same data of M arti n andSinclair are plotted in F ig. 3. N otice that the points which do

not fall on th e strai ght-l in e port ion s of F igs. 1 and 2 seem to beperfectl y acceptable here. Si nce the dat a were different iat edto obtaind a/ d N , an additional amount of scatter is introduced.T herefore the data also imply veri fication of these two laws aswell.H ence, all of the laws agree wi th th e data and, since these lawsare not identical, the method of verification of crack-propagationtheories from a limited amount of test data is evidently in error.An alternative approach employing a wider range of test datamust be employed.

C o m p a r i s o n o f C r a c k P r o p a g a t i o n L a w s W i t h a W i d eR a n g e o f T e s t D a t a

I n previ ous work [6, 8] it was observed that data fr om severalsources [3, 9, 13] ma y be plot ted in the form of F ig. 3 (semi-logarithmic) to obtain a single curve on a range of stress-in-tensity factor, Ak (corresponding to stress range) versus log(d a / d N ) diagram, whered a / d N covers as many as 6 log cycles.H owever, replotting these data on a log Ak versus log ( da / d N )graph reveals some perti nent results. T he data are replotted inF ig. 4 and the three specimens from M arti n and Si nclair's workused in the foregoing for the purpose of illustration are shown toindicate their concurrence with the general trend.

T he authors are hesitant but cann ot resist the temptation todraw the straight line of slope 'A thr ough the data in F ig. 4. T heequation of this line is observed to be

Therefore plotting loga versusN and obtaini ng straight lines hasbeen accepted as veri fyin g these theories. F ig. 2 is this type ofplot employing the very same test data point by point of M arti nand Sinclair shown in Fi g. 1. Again , the test data have straight-line portions. T herefore it now seems to veri fy both H ead's lawequation (30) and the other laws with the form of equation (32).B ut th e theories represented by equations (30) and (32 ) arenot the sameN ow let the laws of M cE vil y and I llg, and Paris, equations(12) and (15), be examined in the li ght of these same data. R e-calling equation (27) these two laws are equivalently expressedby

d ad N

( A k yM (35)

E quati on (35) fits the data almost as well as M cE vi ly and I ll g'sextended law, equation (14), and is considerably simpler in form.

T he advant age of thi s form is clarified by considering its a p-plication to the configuration employed earlier, in which case itbecomes

d ad N M (36)

T he laws of H ead, Fr ost and Du gdale, and Li u depend on a in amanner other than a 2. Cl early, their laws are at variance withthe data trend in Fi g. 4; i.e., Head predicts the slope of l/z andF rost and Dugdal e as well as L iu w ould predict '/* as indicatedin the figure.M oreover , if one migh t be so bold as to at tempt to in tegrateequation (36) it becomes

J _o M (N - N (37)

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ULeye. - 310 10

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where a and No are the initial crack size and cycle number, re-spectively. F or final failur e of a specimen, the number of cyclesmay be equated to N r , asa approaches infinity whereupon equa-tion (37)reduces to

N P = Mcr on (38)

whi ch does in fact resemble anS -N diagram. T he authors do notwish to imply that equation (38) is directly useful, but derive ithere to show that such considerations do not immediately lead tothe conclusion that equations (35) or (36) are in error.D a t a F r o m W e d g e F o r c e T e s t s

T he results of wedge-force tests of the confi gura tion sh own inF ig. 5 are useful in criti cally examini ng the dimensional analysesof Frost and Dugdale, and Liu in deriving their crack-propagationlaws. If th e sheet is infi nit e, the onl y characteri stic dim ensionof the problem is a . As the crack grows, strictly dimensionalarguments imply that for different crack lengths oi and a? theincremental rates of crack extension are

d N O i d N a .2which might be thought valid for all crack lengths or

5 = S a < 40 >NThis result is the basis of equations (5) and (18).I n wedge-force tests Don ald son and Ander son [9] observe that

the crack grows slower as it gets larger. T hi s is contr ar y to equa -

F i g . 5 W e d g e - f o r c e l e s t c o n f i g u r a t i o n

tion (40) and hence weakens any arguments based on dimensionalanalyses whi ch do not include fur ther considerations. T her eforethe crack propagation laws of H ead, F rost and Du gdale, and L iumust be considered as unclarified, if not totally in error.

M oreover , F ig. 6 is a graph, similar in f orm to Fi g. 4, but for adifferent material; i.e., 7075T6 aluminum alloy. T he data shownare from two independent sources [3 and 9] and in additionwedge-force test data fr om tests employi ng configur ation of F ig. 5are also plott ed. N o fur ther comment seems necessary.

C o n c l u s i o nT he results here indi cate that the practi ce of using data fr omsingle test specimens is not a sensitive evaluation of a crack-

propagation law's validi ty. R andoml y chosen data from singlespecimens analyzed here leads to an apparent agreement ofseveral contradictory laws to the same test data.

F or th at reason the authors suggest that laws which corr elatea wide range of test data from many specimens are perhaps the"corr ect" laws. T he results at least indicate that hasty conclu-sions have been drawn in many earlier works which should be re-examined before any given crack-propagation law is accepted asvalid.

A c k n o w l e d g m e n tThis work was supported by the Boeing Company, Transport

Di vision, Rent on, Wash. T heir encouragement, as well asfinancial aid, is gratefu ll y ackn owledged.

R e f e r e n c e s1 A. K . Head, "T he Growth of Fatigue Cr acks," T h e Ph i l o -s op h i c a l M a g a z i n e,vol. 44, series 7, 1953, p. 925.2 N . E . F rost and D. S. Dugdale, "T h e Pr opagation of F atigueCracks in Sheet Specimens,"J ou r n a l o f the M echan i c s and Phy s i c s ofSo l i d s ,vol. 6, no. 2, 1958, p. 92.3 A . J. McE vily and W. I l lg, "T he Rate of Crack Propagation inT wo Aluminum A ll oys," NA C A T echnical N ote 4394, September,1958.4 H . W. Liu, "Cr ack Pr opagation in Thin M etal Sheets UnderRepeated L oading," T H E J O U R N A L OF B A S I C E N G I N E E R I N G , T R A N S .A S M E , Series D , vol. 83, 1961, p. 23.5 H . W . L iu, "F ati gue Crack Propagation and Appli ed StressRangeAn Energy Approach," J O U R N A L OF B A S I C E N G I N E E R I N G ,

TR A N S . A S M E , S e r i es D , v o l . 85, 1963 , p . 116 .6 P . C. Paris, M . P. Gomez, and W . E . An derson, "A Rati onalAnalytic Theory of F atigue,"T h e T r en d n En g i n e er i n g , vol. 13, no. 1,J anuar y, 1961, p. 9.7 W. E . Anderson and P. C. Paris, "E valuation of Air craft M a-terial by Fracture,"M et a l s En g i n e er i n g Qua r t e r l y , vol. 1, no. 2, M av,1961, p. 33.8 P . C. Paris, "C ra ck Pr opagation Caused by Flu ctuatingL oads," A SM E P aper N o. 62M et-3.

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9 D. R. Donaldson and W. E. Anderson. Crack PropagationBeha'ior of Some Airframe Materials. presented at the Crack Propagation SympoRium. Cranfield. England. September, 1961: proceedings to be published. Also available on request to The Boeing Company. Transport Division. Document No. D6-7888.10 G. R. Irwin . rac ture Mode Transition for a Crnck Traversing 1\ Plate. JO URNAL OF BASIC ENGINEERINO TRANS. ASME, Seriesn. yol. 82. no . 2, Jun e. 1960. p. 417.11 P. C. Paris, A Note 011 th e Variables Effecting the Rate ofCrack Growth Due to C:vclic Loading, The Doeing Company,Document No. 0-17867, Addendum N, September 12. 1957.12 G. R. Irwin. Analysis or Stresses and Strains Near the End offl Crark Tr avers in g n Plate. urnal 0/ Apph ed lIfechanin vo l. 24.TRANS. ASME. Y01. 79,1957 , p . 361.

13 D. E Martin and G. M. Sinclair. Crack Propagation Undernepf'aled Loading, Proceedings of t he Third U.S. National Congressof Applied M echan ics, June. 1958. p. 595.14 H. F. Hardrath and A. J . McEvily. Engineering Aspects ofFatigue Cmck Propug-aUon, presented nt th e Crnck PropagationSymposium , Cran field , England, September, 1961.

Journal of asic Engineering DECEM ER 963 I 5

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1963-Paris, P.; ErdoganA Critical Analysis of Crack Propagation Laws