^f bmjroc - solid mechanics at harvard...

Comparison of theory and experiment forelastic-plastic plane-strain crack growthL Hermann and J R Rice

Recent theoretical results on elastic-plastic plane-strain crack growth an~ reviewed and experimentalresults for crack growth in a 4140 steel are discussedin terms of the theoretical concepts The theory isbased on a recent asymptotic analysis of crack surfaceopening and strain distributions at a quasistaticallyadvancing crack tip in an ideally plastic solid Theanalysis is incomplete in that some of the parameterswhich appear in it are known only approximatelyespecially at large-scale yielding Nevertheless it issufficient for the derivation of a relation between theimposed loading and amount of crack growth prior togeneral yielding based on the assumption that ageometrically similar near-tip crack profile ismaintained during growth The resulting predictionsfor the variation of J with crack growth are found tofit well to the experimental results obtained on deeplycracked compact specimensL Hermann lng MS and J R Rice BS MS PhD are with theDivision of Engineering Brown University Providence USA

THEORETICAL ANALYSIS OF GROWINGCRACKSFigure 1a Shows the Prandtl slip-line field which describesthe plane-strain near-tip stress state (at least when largechanges in crack-tip geometry are neglected see Rice andJohnson3 and McMeeking4) for a non-growing cracksubjected to monotonically increasing load underconditions of well contained yielding This field may alsoapply at large-scale and general yielding in certain highlyconstrained configurations During processes of stable crackgrowth for which the applied load varies continuously withcrack length a a slightly different stress field results verynear the tip This was described by Rice et a1 and its slip-line interpretation is shown in Fig lb In this casean elasticunloading sector develops between the centred fan region Cand the back constant stress yield region B The angles 81and 82 depend somewhat on the Poisson ratio but are81 = 1150 and 82 = 1630 for v = 0middot3 Remarkably thestresses ahead of the crack differ only by about 1 fromthose for the Prandtl field of Fig 1a

Letting ~ be the opening displacement between the upperand lower crack surfaces the analysis shows that

J = ai(Jo+(pil(JoE)In [R(a-x)J (1)

A recent theoretical analysis by Rice et al1 described theplane-strain elastic-plastic deformation and stress field verynear the tip of a continuously growing crack in a non-hardening solidmiddot obeying the Huber-von Mises yieldcondition and associated flow rule (ie the Prandtl-Reussequations) By requiring as a possible criterion forquasistatic crack growth that a geometrically similar profileof crack surface opening be maintained very near the tip theanalysis led to predictions of the manner in which cracklength should vary with applied load during growth Thework described is an extension of earlier work by Rice andSorensen2 and includes a correction to the form of the near-tip deformation and stress field presented in the latter Inboth studies 12 the theoretical analysis is incomplete in thatcertain parameters or functions which appear in asymptoticexpressions for the near-tip deformation field must bedetermined from full numerical elastic-plastic solutions forgrowing cracks This has been done approximately bycomparison with finite element solutions for the case ofcrack growth under well contained plastic yielding Resultsfor large-scale and general yielding are not yet availablealthough some discussion of the fully yielded case has beenpossible 1 by use of dimensional considerations andcomparison with rigid-plastic slip-line solutions

Our purpose here is to present experimental results oncrack growth in a high-strength steel and to use these as abasis for comparison with the theoretical predictionsmentioned above

very near the tip of a continuously growing crack (ie asx - a where x is-the position along the crack) In thisexpression the superimposed dots denote time rates (Jo is theideally plastic tensile strength E the elastic tensile modulusand P is dependent on Poissons ratio having the value1

P = 5middot08 for v = 0middot3 Also the intensity of the appliedloading is without loss of generality phrased in terms of J(associated with the J-integral although as will be discussedequation (1) remains valid for different ways of defining jthe different choices imply different values for theparameter R)

In fact ex and R in equation (1) are not determined by theasymptotic analysis leading for example to the stress fieldof Fig lb These parameters must be determined bycomparison of equation (1) with complete elastic-plasticsolutions for growing cracks which are necessarilynumerical and subject to uncertainties as to accuracy andinterpretation very near the tip We discuss each parameterin turn First ex is dimensionless and might be expected to berather close in value to the parameter ex as defined by

d~tip = ex dJ(Jo (il = 0) (2)

for the crack tip opening displacement of a non-growingcrack subject to monotonic load increase This expectationseems to be confirmed by results of numerical solutions forcrack growth under well contained plastic yielding Lettingssy denote small-scale yielding it was found12 that towithin 5 to 10 accuracy exssy ~ exisy where exisy = 0middot65 Theparameter ex may vary from exssy as large-scale and generalyielding conditions are approached For example the

Metal Science August-September 1980 285

286 Hermann and Rice Elastic-plastic crack growth

definition of j it was suggested 1 that for a configurationsimilar to that in Fig 2 R might ultimately reach a limitingvalue of some definite fraction of the uncracked ligamentdimension b Suitable theoretical confirmation of thisexpectation (perhaps from numerical solutions) is lacking atpresent and our experiments suggest that the large-scaleyielding behaviour of R may be somewhat more

x complicated A possible criterion for stable crack growth has been

developed from equation (1) in the following way Byintegration the form of the crack profile very near the tipcan be written as

where e = 2middot718 is the natural logarithm base and r = a-xThe equation is asymptotically valid as r ~ 0 for acontinuously growing crack (ie when dJ Ida is finite) Theexpression can be rewritten as

(5)

(4)b = ((XrOo) dJda+(f3rOoE) In (eRr)

b = (f3rOoE) In (pr)

where

x

----a

3114 C

Y

L B

(a)

91

C

(b)

1 Slip-line representation of crack tip stress state afor non-growing crack b for growing crack

deeply cracked plane-strain bending configuration (Fig 2)leads at fully plastic (fp) conditions to156 (X~ = 0middot51Furthermore (Xfp= (X1p exactly for the rigid-ideally plasticmaterial model so that it may be presumed that(Xfp ~ ac~ = 0middot51 for elastic-plastic deeply cracked bendspecimens which are loaded well into the plastic range Infact estimates of (X from elastic-plastic solutions for non-growing cracks5 78 suggest that (X remains very close to(Xy( = 0middot65) until the applied moment M per unit thicknessapproaches closely to the fully plastic limit moment 16

Mo = 0middot3640ob2 and then (X decreases rapidly towards(X1p(= 0middot51) as deformation continues

The parameter R in equation (1) evidently has dimensionsoflength and it is reasonable to expect that R should have avalue in approximate proportion to the size of the plasticregion or at least to the size over which stress distributionssimilar to those of Fig 1 prevail This notion has limitedconfirmation from comparisons of equation (1) withnumerical simulations of crack growth under small-scaleyielding Such studies suggest that1

R ~ 0middot23EJ0~ (3)

for small-scale yielding where J has been identified as thefar-field value of the contour integral taken in elasticmaterial and hence equal to G == (1- v2)K2E in the small-scale yielding limit The value of R given by equation (3) isproportional to but approximately 15 to 30 larger thanthe maximum radius of the plastic zone under small-scaleyielding conditions The accuracy of equation (3) remainsan open question there may be some small dependence of Ron the ratio of the length of crack growth to the size of theyield zone but numerical solutions have not as yet beensufficient to resolve this

Certainly R must be expected to deviate from equation(3) as large-scale and fully plastic yielding conditions arereached As discussed by Rice et al1 the behaviour of R atlarge-scale yielding cannot be decoupled from the definitionof J if J is defined so that in the limit of rigid-plasticresponse (EO 0 ~ (0) j does not depend on a (but only on aand on the rate of imposed boundary displacement) then Rmight still be expected to have values in proportion with thesize of the fan-like stress region Hence with an appropriate

p = eR exp [((Xf3)(EO~) dJ Ida] (6)It is clear from equation (5) that the form of the crack profilevery near the tip is dependent only on the single parameterp Hence if one adopts as a possible fracture criterion thepremise that the crack maintains a steady near-tipgeometrical profile during growth thismiddot is seen fromequation (5) to be tantamount to assuming a crack growthcriterion in the form

p = constant (7)

Thus with p understood to be constant equation (6) isequivalent to the differential equation

dJda = (f3(X)(O~E) In (peR) (8)

which governs the variation of J with a during quasi staticcrack growth This is a differential equation in the sensethat R is dependent (in a way yet to be fully documented atleast beyond the small-scale yielding result of equation (3raquoon J and probably on the previous crack growth

Integrals of equation (8) have been presented in previouswork 1 for crack growth under small-scale yielding usingequation (3) for R In this case the J versus a relationship isdetermined fully by the initial value of J (ie JIC) at theonset of growth and bymiddot the parameter p J increases fromJIC with ongoing crack growth but ultimately approaches alimiting value (which makes R = pie so that dJda = 0) at

a~r-b

2 Deeply cracked plane-strain bending specimen

Metal Science August-September 1980

Hermann and Rice Elastic-plastic crack growth 287

with the reqUIsIte accuracy are available at presentNevertheless it is clear that equation (9) can be rephrased inthe form

EXPERIMENTSThe material used in the experiments was an AI-Si killedAISI4140 supplied as 19 mm thick plate by BethlehemSteel Corporation Compositional weight percentages are0middot42C 0OO4P 0014S 022Mo OOOIAsOOISb 0middotOO2SnThe material was austenitized in argon for 1 hat 1140 K oilquenched and tempered for 1 h at 770 K resulting innominal tensile yield and ultimate strengths ofOy = 1173 MN m-2 and ou = 1327 MN m-2 respectively

Four specimens were prepared each with in-planedimensions as shown in Fig 3 (the larger dimension30middot5 mm for the distance from the notch tip to specimenback surface applies for specimen 4 only) The specimenswere side-grooved in a shallow 90deg V-notch shape tosuppress shear lips and were pre-fatigue cracked so that theremaining uncracked ligament b had values at the start ofthe ductile crack growth tests as shown in Table 1 Thistable also shows the thicknesses and side groove depthsthicknesses listed are the average of those before and afterside-grooving and are used to reduce measured loads toloads per unit thickness for use in theoretical formulae Theuncracked ligaments b are small compared to overallspecimen dimensions and hence for analytical purposes thespecimens are modelled as deeply cracked bend specimens

(10)yP = 188(2-v)(OoE) In (Cr)

where

C = Lexp [ml88(2-v)](E0~) dJda (11)and C measures the strength of the near-tip strain fieldHence the criterion for crack growth with a fixed strainstate prevailing very near the tip can be phrased as therequirement that C = constant and this leads to adifferential equation governing crack growth which issimilar in form to that of equation (8)

Fourth it is emphasized that the theoretical analysisleading eg to equations (14 9) is based on an ideallyplastic material We can account for actual strain hardeningin a very approximate manner by identifying 00 in equations(124-68) as the ultimate (nominal) tensile strength Ou

This is done in analysing the experiments (although inequation (3) for R we shall interpret 00 as the actual yieldstress Oy which is reasonable if R is to be associated with thesize of the plastic region) Nevertheless more subtle strain-hardening effects than are revealed in the uniaxial tensiletest can occur and may be relevant to the analysis of crackgrowth These are related to the development of severeshape changes of the plastic yield surface (in stress-space) Aspecial case is the development of a pointed vertex structureon the yield surface at the current stress state This greatlyreduced the stiffness of elastic-plastic response along non-proportional stressing paths compared to predictions basedon yield surfaces which do not change in shape and hencemight be expected to be critical to the analysis ofdeformation fields near growing cracks Remarkablyhowever a recent analysis of steady-state Mode III crackgrowth by Dean and Hutchinson9 shows very littledifference betweenthecrack surface displacements predictedaccording to a vertex yield model and to a model whichretains the Huber-von Mises shape of the yield locus (ieisotropic hardening) Whether effects of changes of yieldsurface shape are similarly without major effects forgrowing tensile cracks remains to be seen

6650middot8

P3 Specimen with dimensions shown in mm and clip

gauge

which continuing crack growth can occur without furtherincrease of J This limiting value in the steady state has beendenoted (J)ss in the previous work I 2 and (J)ss = 1600~ p E

A number of comments are pertinent First it is ofinterest to note that in order to maintain a steadygeometrical profile near the crack tip J must in general varyduring crack growth (unless the size parameter R hasreached the value pie) This is a plasticity effect traceableto the strain-path-dependent nature of elastic-plasticstressstrain relationships in particular to the difference inresponse stiffness for plastic loading versus elasticunloading It would not result in a non-linear elasticmaterial even if that material had the same stressstrainrelation for monotonic loading as does the elastic-plasticmaterial Instead in such a non-linear elastic material therequirement-of a constant near-tip crack profile duringgrowth would be essentially equivalent to the requirementof a constant value of J during growth

Second the crack growth criterion of equations (7) and(8) is based on maintaining in a sense a fixed deformationstate very near the crack tip during growth This may beexpected to provide a suitable model for ductile crackgrowth at least in circumstances for which significant voidnucleation occurs only very near the tip within the region ofthe fixed deformation state It cannot be expected to applywith much precision in cases for which microcrack or cavitynucleation is influenced by the size scale in the material overwhich high stresses act Obviously the growth criterion alsodoes not incorporate the physics of transition of a stableductile tearing mode fracture to a cleavage mode fracturewhich is presumably due to strain-rate elevations of localstress levels during quasi stable growth processes withconsequent change of microscopic fracture mode

Third it is of interest to note that the general form of thegrowth criterion seems to be independent of the manner inwhich the notion of a fixed near-tip deformation state ischaracterized For example Rice et all show that theequivalent plastic shear strain at points within the fan ofFig Ib at small distances r directly above or below thecrack tip is

yP = (mOo) dJda+ 188(2-v)(OoE) In (Lr) (9)for a continuously growing crack where the parameters m(analogous to (J of equation (4)) and L (analogous to eR) arealso undetermined by the asymptotic analyses Thedetermination of these parameters from numerical solutionsrequires that local strains be known accurately no results



6middot37middot07middot3

11middot8

Estimate from Visual observation ofcompliance changes mm load-cycle markings mm


12middot0

Table 2 Comparison of maximum amount of crackgrowth in each test as estimated from changes in elasticcompliance with actual growth observed visually frommarkings on fracture surface due to fatigue load cyclingat end of test

the values reported in the middle column are the totalcompliance-estimated growth at the end of the tests (egwhen only 3-4mm of ligament remained) Also at the endof each test several cycles of fatigue loading were applied tomark the position of the crack front and then the thickness-average growth was measured visually leading to results inthe last column The compliance-estimated growth is within0middot5 mm or less of the visually observed growth

Experimental results for specimen 2 are shown in Fig 5the trend of the curves shown is representative of the resultsfor all sp~cimens The horizontal axis denotes amount ofcrack growth and the curves labelled M and My denoterespectively the applied moment and the fully plasticmoment The latter is calculated from16 My = Omiddot3640yb2bull

Evidently the early portion of the test corresponds tocontained yielding and the later portion to general yieldingAlso shown in Fig 5 are the elastic energy releaserates G calculated from

G = 16(1-v2)M2Eb3 bull (13)

(which is consistent with equation (12raquo) and two differentmeasures of quantities that can be associated with the J~integral and which are discussed below

The points denoted by the circular symbols in Fig 5 givevalues of the deformation theory value of J namely Jd in

1234

Specimennumber

o 234567 9CRACK GROWTH) mm

5 Applied moment M general yield moment My G Jd

and Jf verus crack growth for specimen 2

150140

J

130J

120J Jf

J110

IE 100J

JzJ 90 J 70

i ) JD 8

bull i0 0 o 0 oJd 60c

~ e bullcu bull bullbulln 50~0

(9~ bulllaquo G

a 402~J

30 -gspecimen 2 cu

202

10

Specimen Initial ligament Thickness Side-groovenumber size bo mm (average) mm depth mm

1 10-75 12middot19 0middot512 10middot42 15middot37 0middot513 10middot62 18middot54 0middot764 15middot86 18middot54 0middot76

Table 1 Specimen ligament sizes thicknesses andside-groove depths (for other dimensions see Fig 3)

(Fig 2) with M computed about the mid-point of theuncracked ligament Thus M = P(a + b2) and incrementsde of the rotation measure in Fig 2 are identified asde == dd(a + b2) where d is the opening displacementalong the load line This displacement is measured by theclip-gauge shown The gauge is of high sensitivity it is madeof high strength steel with strain gauge sensors and mountedin knife-edge supports (razor blades attached to thespecimen fit into sharp V-grooves in the clip gauge)

Tests were performed on an Instron closed-loopservohydraulic test machine Amounts of stable crackgrowth were estimated according to the elastic unloadingcompliance technique described by Clarke et al 10 elasticrotations were assumed to be given by their expression

ee = 16(1- v2)MEb2bull bull (12)

The accuracy of the measurement of small changes in cracklength (ie based on the ligament b inferred from equation(12) from the response to small increments of elasticunloading) is increased by the use of low-noise high-gainelectronic instrumentation amplifiers to process P versus drecords and by minimization of friction The latter isaccomplished partly through design of the clip gauge andpartly through the use of flexible hanger plates which allowthe loading pins to rotate in needle bearings This makespossible the resolution of inferred changes in crack size ofthe order of 0middot01 mm

Figure 4 shows a P versus d record for specimen 2 This isrepresentative of all the specimens and illustrates the elasticunloading and reloading to infer crack growth

The crack growth estimated by the compliance techniqueis compared to visually observed growth in Table 2 Here

DISPLACEMENT 1---+

4 Loaddisplacement diagram for specimen 2 showingelastic unloading and reloading



(Jd)SS=83 kN m-1

p=7middot444mm

(Jd)ss=80 kN m-1

p=7middot175mm

mmo

bull

~ 10

~ 90 Jd~80 -0-)------- 0--raquo bullbullbull

~ 70 I~ th(loryc ~en 60 q250ozlaquo 401E30

~20

8 Jd (circular symbols) versus crack growth forspecimen 3 and comparison with theoreticalprediction

(Jd)SS=80 kN m-1

p=7middot175mm

(Jd)SS=85 kN m-1

p=7middot623mmz 90L

bull80gt

2701J

~ 60250

~ 40laquo30IEz 20L

10)

o 345678CRACK GROWTH mm

6- Jd (circular symbols) versus crack growth forspecimen 1 and comparison with theoreticalprediction

(14)

(15)

the terminology of Rice et aZ 1 This is obtained byintegrating

dJd = (2b)M de- (Jdb) da= (2b)P dL- (Jdb) da~

throughout the test starting at Jd = 0 when L = O If thematerial were actually non-linear elastic then as shown byHutchinson and Paris 11 the above expression whenintegrated for deeply cracked bend specimens defines avalue J d which depends only on the imposed deformation eand the ligament size b and not on the path by which thecurrent values of e and b were obtained Further Jd wouldthen agree with the line-integral definition of J and reduce toJd = G in the limit of crack growth with a negligibly smallnon-linear zone Similar independence of J d from the pathby which cu~rent e and b values were obtained andagreement with line-integral values of J cannot be expectedfor actual elastic-plastic solids in such cases Jd is merelydefined by equation (14) On the other hand it remains truethat Jd = G when a crack grows in an elastic-plastic solidwith a negligibly small plastic zone Indeed it is clear fromFig 5 _that experimental results for Jd agree closely withthose for G until the applied moment M approaches closelyto My In view of the properties of Jd it seems mostappropriate to associate J in equations (3) and (8) with J d in

order to use those equations at loads extending slightlybeyond the limits of small-scale yielding

The triangular symbols in Fig 5 denote values of the farfield value of J namely Jr obtained by integtating

dJr = (2b)M de = (2b)P dL= dJd + (Jdb) da

throughout the test The name far field arises because inthe particular case of a deeply-cracked rigidly-ideallyplastic bend specimen subject to monotonically increasingrotation Rice et al1 showed that Jr thus defined agrees withthe line-integral value of J as computed on a contourcoinciding with the outer boundary of the specimen Thereis no proof that a similar interpretation- would be valid forelastic-plastic solids or -even for rigid-plastic solids indifferent specimen configurations As noted above certaindefinitions of J are consistent at general yielding with a con-tinuing interpretation of R as a measure of the spatial extentof a region stressed in a fan-like manner Others are not Asshown by Rice et alf Jr has the proper features at fullplasticity whereas J d does not (Later we shall consideranother quantity which also exhibits the proper features atfull plasticity and which reduces to J ~ Jd ~ G at wellcontained yielding) We observe from Fig 5 that thedifference between Jr and Jd is insignificant in the very early



bullbullbull

(Jd)SS=80kN m-1

p=7middot175mm

M

o

~ 90

2gt80

-g 70en260

~50laquo40IEz3L

20)

10

o 2 3 4 5 6 7 8 9 10CRACK GROWTH mm


(Jd)SS=80kN m-1

p=7middot175mm

D-- -0---- -------- fYAJ OJ- bull bull bull (Jd)ss =75middot5 kN m-1

p=6middot950mm

z 90L

8021J 70cen 6020

50

~ 40

IE 30

~20

~10



3middot2

2middot8

2-4

2middot0EE

160

1middot2

0middot8

0-4specimen 3

o

0-8

0middot7

3middot2

2middot8

2middot4

2middot0EE

1middot6 0[

1middot2

0middot8

0middot4

specimen

R

o 0o ++

))LgtRb ~

+ EJ-0 023 _d ++) ~2b

0middot3 -tAt) y

0-2 -t5+

0middot1

0middot90middot8

0middot7

0middot6


10 Inferred values of R and Rib versus crack growthfor specimen 1 and comparison with predicted Ribvalue from small-scale yielding expression with Jidentified as Jd



stages of crack growth but increasingly more significantwith continuing growth Also it is evident from equation(15) that measurement of Jr versus a implies J d versus a andvice versa

Figures 6-9 show as circular symbols the experimentalresults for Jd as a function of crack growth for specimens 1-4 The open circles are data points obtained prior to generalyield (ie M lt My) and the solid circles are results obtainedafter general yield The solid and dashed curves representthe results of the theoretical prediction for well containedyielding

COMPARISON OF EXPERIMENTS WITHSMALL-SCALE YIELDING FORMULATIONThe theory is well developed only for crack growth underconditions of small-scale yielding We show predictions ofthe theory for this case in Figs 6-9 based on using Jd ratherthan G in the small-scale yielding formulation andsubstituting lJu or lJy for lJo as discussed earlier Thisformulation is given by combining equations (3) and (8) inthe form (using f3 = 5middot08 lI = lIssy = 065)

dJdda = 782(lJ~E) In (plJ023eEJd) (16)This equation has been integrated subject toJd = 35 kN m -1 at the onset of growth as suggested by thedata in Figs 5-9 and values of the one free parameter parechosen to best fit the data prior to general yield Results areshown by the solid and dashed curves in Figs 6-9

The solid curves in Figs 6-8 (specimens 1-3) are the sameas the best-fit curve in Fig 9 The value of p which generatesthis curve is p = 7middot175 mm which implies that(Jd)ss = 80 kN m -1 The dashed curves in Figs 6-8correspond to slightly different values of p giving the best fitfor each specimen although the theory implies of coursethat pis a specimen-independent material property

We conclude from Figs 6-9 that the theory is reasonablysuccessful in describing experimental data for crack growthprior to general yield

FORMULATION FOR ARBITRARY EXTENT OFYIELDINGHere we present what we believe to be a plausibleinterpretation of the terms in the growth criterion ofequation (8) for arbitrary amounts of crack growth andextent of yielding in deeply cracked bend specimens Theapproach will however almost certainly be subject torevision when and if sufficiently accurate numerical elastic-plastic solutions become available to predict from firstprinciples the relationship of terms in equation (8) toapplied deformation and crack length

First consider the interpretation of dJ in the theoreticalexpressions We can identify dJ as dG in the limit of growthwith an extremely small non-elastic zone On the otherhand dJ is subject to certain restrictions of interpretationas discussed earlier at large-scale and fully plastic yield-ing conditions In particular Rice et al1 show that for



) 0middot9

+ 3middot2 0middot8 3-2R

2middot8 0middot7 R + 2middot80

0 )+ 2middot4 0middot6 + 2middot4

)+ ~Rb ~ 2OE)

2middot0E+16 Ebull E)+ ~ EJd

~ 1middot6 o)+ 0middot23 - + 0)+ ~~b ) + Rb ~)+ 12 ) + EJd1-2

)+ 0middot23lt2h +specimen 2 08 ~yb 0-8

0middot1 0-4 specimen 4 0-4




rigid-ideally plastic bend specimens an expressiondJ = (2b)M dOP consistent with dJr of equation (14) isappropriate Here dOP is the plastic part of the rotation Wecan embody both limiting cases by writing

dJ = (2b)M dOP +dG (17)

This can be rearranged by noting that G = MOeb for adeeply-cracked bend specimen and that ee is given byequation (12) Thus

dG = d[Eb(oe)216J= (2Ebee 16) doe + [E(oe)216J db= (2b)MdOe-(Gb)da (18)

by equation (12) and by using db = - da Hence equation(17) can be written as

dJ = (2b)M d(oe +OP)-(Gb) daor

dJ=dJr-(Gb)da=dJd+[(Jd-G)bJda (19)

In terms of this interpretation of dJ the growth criterion ofequation (8) may now be written as

dJdda+(Jd-G)b = (508C()laquo(j~E)ln (peR) (20)

Here P = 5middot08 has been used and it is recognized that C(

should in general be regarded as variableWe have used experimental data for Jd and G to evaluate

the left-hand side of this equation and by taking C( = 0middot65and using the value of p from the small-scale yielding fit ofthe data we can infer R as a function of a from it That is weinfer the variation of R with a on the assumption that theexperimental data fit equation (20) The results arepresented in Figs 10--13 where we show R (right ordinate)and Rb (left ordinate) versus crack growth Also we showthe value of Rb implied by the small-scale yieldingexpression of equation (3) when Jd is used for J

We observe that R is rather well predicted by equation (3)throughout the range of these tests This is probably apeculiarity of these experiments because equation (3) is notexpected to be valid usually at general yielding It wassuggested in earlier work that Rb should approach alimiting value as general yielding conditions are


approached Our experiments extend only modestly into thegeneral yielding range and cannot provide a good test of thisconcept However the data which we have are notsuggestive of an approach to a limiting value of Rb at leastover the range studied Furthermore the final Rb values inthe experiments range from 0middot6-0middot8 and these seem ratherlarge if R is to be interpreted as a size in the material overwhich fan-like stress fields such as those of Fig 1 prevailThis topic requires further study from both the standpointsof developing a suitable theoretical basis for growth in thegeneral yielding range and of experimental analysis of theresulting growth criterion

ACKNOWLEDGMENTSThe work was supported by the US Department of Energy

under contract EY-76-S-02-3084 A003 with BrownUniversity We are grateful to Professor R J Asaro forhelpful discussions and for his and Mr L C Majnosassistance with the heat treatment and we wish to thank MrR E Dean for assistance with specimen preparation

REFERENCES1 J R RICE W J DRUGAN and T L SHAM Proc ASTM 12th

Annual Symp on Fracture mechanics May 1979 ASTM-STP 700 1980 189-221

2 J R RICE and E P SORENSEN J Mech Phys Solids 1978 26163

3 J R RICE and M A JOHNSON Inelastic behavior of solids (edM F Kanninen et al) 641-671 1970 New YorkMcGraw-Hill

4 R M McMEEKING J Mech Phys Solids 197725 3575 R M McMEEKING and D M PARKS Elastic-plastic fracture (ed

J D Landes et al) ASTM STP 6681979 175-1946 J R RICE Mechanics and mechanisms of crack growth (ed

M J May) Br Steel Corp Phys Met Cent Pub 1973(issued 1975) pp 14-39

7 D M PARKS Compo Meth Appl Mech Engr 1977 12 3538 c F SHIH J Mech Phys Solids in press9 R H DEAN and J W HUTCHINSON Ref 1 383-405

10 G A CLARKE W R ANDREWS P C PARIS and D W SCHMIDTMechanics of crack growth ASTM STP 590197627-42

11 J W HUTCHINSON and P C PARIS Ref 5 pp 37-65

copy THE METALS SOCIETY 1980


Forthcoming publication

HEATTREATMENT79

Proceedings of a conference organized by the Heat Treatment Committee of TheMetals Society in association with the Heat Treating Division of the AmericanSociety for Metals and held at Birmingham on 22-24 May 1979

The seven technical sessions and resulting discussions gave comprehensivecoverage of the latest scientific technological and economic aspects of heattreatment the papers presented being in the following subject areas electroheatsurface treatments developments in thermochemical processing structure andproperty modifications hardenability sintering and post heat treatment of PMengineering components and fatigue of surface-hardened engineering components

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definition of j it was suggested 1 that for a configurationsimilar to that in Fig 2 R might ultimately reach a limitingvalue of some definite fraction of the uncracked ligamentdimension b Suitable theoretical confirmation of thisexpectation (perhaps from numerical solutions) is lacking atpresent and our experiments suggest that the large-scaleyielding behaviour of R may be somewhat more

x complicated A possible criterion for stable crack growth has been

developed from equation (1) in the following way Byintegration the form of the crack profile very near the tipcan be written as

where e = 2middot718 is the natural logarithm base and r = a-xThe equation is asymptotically valid as r ~ 0 for acontinuously growing crack (ie when dJ Ida is finite) Theexpression can be rewritten as

(5)

(4)b = ((XrOo) dJda+(f3rOoE) In (eRr)

b = (f3rOoE) In (pr)

where

x

----a

3114 C

Y

L B

(a)

91

C

(b)

1 Slip-line representation of crack tip stress state afor non-growing crack b for growing crack

deeply cracked plane-strain bending configuration (Fig 2)leads at fully plastic (fp) conditions to156 (X~ = 0middot51Furthermore (Xfp= (X1p exactly for the rigid-ideally plasticmaterial model so that it may be presumed that(Xfp ~ ac~ = 0middot51 for elastic-plastic deeply cracked bendspecimens which are loaded well into the plastic range Infact estimates of (X from elastic-plastic solutions for non-growing cracks5 78 suggest that (X remains very close to(Xy( = 0middot65) until the applied moment M per unit thicknessapproaches closely to the fully plastic limit moment 16

Mo = 0middot3640ob2 and then (X decreases rapidly towards(X1p(= 0middot51) as deformation continues

The parameter R in equation (1) evidently has dimensionsoflength and it is reasonable to expect that R should have avalue in approximate proportion to the size of the plasticregion or at least to the size over which stress distributionssimilar to those of Fig 1 prevail This notion has limitedconfirmation from comparisons of equation (1) withnumerical simulations of crack growth under small-scaleyielding Such studies suggest that1

R ~ 0middot23EJ0~ (3)

for small-scale yielding where J has been identified as thefar-field value of the contour integral taken in elasticmaterial and hence equal to G == (1- v2)K2E in the small-scale yielding limit The value of R given by equation (3) isproportional to but approximately 15 to 30 larger thanthe maximum radius of the plastic zone under small-scaleyielding conditions The accuracy of equation (3) remainsan open question there may be some small dependence of Ron the ratio of the length of crack growth to the size of theyield zone but numerical solutions have not as yet beensufficient to resolve this

Certainly R must be expected to deviate from equation(3) as large-scale and fully plastic yielding conditions arereached As discussed by Rice et al1 the behaviour of R atlarge-scale yielding cannot be decoupled from the definitionof J if J is defined so that in the limit of rigid-plasticresponse (EO 0 ~ (0) j does not depend on a (but only on aand on the rate of imposed boundary displacement) then Rmight still be expected to have values in proportion with thesize of the fan-like stress region Hence with an appropriate

p = eR exp [((Xf3)(EO~) dJ Ida] (6)It is clear from equation (5) that the form of the crack profilevery near the tip is dependent only on the single parameterp Hence if one adopts as a possible fracture criterion thepremise that the crack maintains a steady near-tipgeometrical profile during growth thismiddot is seen fromequation (5) to be tantamount to assuming a crack growthcriterion in the form

p = constant (7)

Thus with p understood to be constant equation (6) isequivalent to the differential equation

dJda = (f3(X)(O~E) In (peR) (8)

which governs the variation of J with a during quasi staticcrack growth This is a differential equation in the sensethat R is dependent (in a way yet to be fully documented atleast beyond the small-scale yielding result of equation (3raquoon J and probably on the previous crack growth

Integrals of equation (8) have been presented in previouswork 1 for crack growth under small-scale yielding usingequation (3) for R In this case the J versus a relationship isdetermined fully by the initial value of J (ie JIC) at theonset of growth and bymiddot the parameter p J increases fromJIC with ongoing crack growth but ultimately approaches alimiting value (which makes R = pie so that dJda = 0) at

a~r-b

2 Deeply cracked plane-strain bending specimen






(10)yP = 188(2-v)(OoE) In (Cr)

where




6650middot8


gauge









11middot8



12middot0





G = 16(1-v2)M2Eb3 bull (13)



1234

Specimennumber




150140

J

130J

120J Jf

J110

IE 100J

JzJ 90 J 70

i ) JD 8



(9~ bulllaquo G

a 402~J

30 -gspecimen 2 cu

202

10










DISPLACEMENT 1---+




(Jd)SS=83 kN m-1

p=7middot444mm

(Jd)ss=80 kN m-1

p=7middot175mm

mmo

bull

~ 10



~20


(Jd)SS=80 kN m-1

p=7middot175mm

(Jd)SS=85 kN m-1

p=7middot623mmz 90L

bull80gt

2701J

~ 60250

~ 40laquo30IEz 20L

10)



(14)

(15)










bullbullbull

(Jd)SS=80kN m-1

p=7middot175mm

M

o

~ 90

2gt80

-g 70en260

~50laquo40IEz3L

20)

10



(Jd)SS=80kN m-1

p=7middot175mm


p=6middot950mm

z 90L

8021J 70cen 6020

50

~ 40

IE 30

~20

~10



3middot2

2middot8

2-4

2middot0EE

160

1middot2

0middot8

0-4specimen 3

o

0-8

0middot7

3middot2

2middot8

2middot4

2middot0EE

1middot6 0[

1middot2

0middot8

0middot4

specimen

R

o 0o ++

))LgtRb ~

+ EJ-0 023 _d ++) ~2b

0middot3 -tAt) y

0-2 -t5+

0middot1

0middot90middot8

0middot7

0middot6















) 0middot9




)+ ~Rb ~ 2OE)









dJ = (2b)M dOP +dG (17)





























HEATTREATMENT79












(10)yP = 188(2-v)(OoE) In (Cr)

where




6650middot8


gauge









11middot8



12middot0





G = 16(1-v2)M2Eb3 bull (13)



1234

Specimennumber




150140

J

130J

120J Jf

J110

IE 100J

JzJ 90 J 70

i ) JD 8



(9~ bulllaquo G

a 402~J

30 -gspecimen 2 cu

202

10










DISPLACEMENT 1---+




(Jd)SS=83 kN m-1

p=7middot444mm

(Jd)ss=80 kN m-1

p=7middot175mm

mmo

bull

~ 10



~20


(Jd)SS=80 kN m-1

p=7middot175mm

(Jd)SS=85 kN m-1

p=7middot623mmz 90L

bull80gt

2701J

~ 60250

~ 40laquo30IEz 20L

10)



(14)

(15)










bullbullbull

(Jd)SS=80kN m-1

p=7middot175mm

M

o

~ 90

2gt80

-g 70en260

~50laquo40IEz3L

20)

10



(Jd)SS=80kN m-1

p=7middot175mm


p=6middot950mm

z 90L

8021J 70cen 6020

50

~ 40

IE 30

~20

~10



3middot2

2middot8

2-4

2middot0EE

160

1middot2

0middot8

0-4specimen 3

o

0-8

0middot7

3middot2

2middot8

2middot4

2middot0EE

1middot6 0[

1middot2

0middot8

0middot4

specimen

R

o 0o ++

))LgtRb ~

+ EJ-0 023 _d ++) ~2b

0middot3 -tAt) y

0-2 -t5+

0middot1

0middot90middot8

0middot7

0middot6















) 0middot9




)+ ~Rb ~ 2OE)









dJ = (2b)M dOP +dG (17)





























HEATTREATMENT79










11middot8



12middot0





G = 16(1-v2)M2Eb3 bull (13)



1234

Specimennumber




150140

J

130J

120J Jf

J110

IE 100J

JzJ 90 J 70

i ) JD 8



(9~ bulllaquo G

a 402~J

30 -gspecimen 2 cu

202

10










DISPLACEMENT 1---+




(Jd)SS=83 kN m-1

p=7middot444mm

(Jd)ss=80 kN m-1

p=7middot175mm

mmo

bull

~ 10



~20


(Jd)SS=80 kN m-1

p=7middot175mm

(Jd)SS=85 kN m-1

p=7middot623mmz 90L

bull80gt

2701J

~ 60250

~ 40laquo30IEz 20L

10)



(14)

(15)










bullbullbull

(Jd)SS=80kN m-1

p=7middot175mm

M

o

~ 90

2gt80

-g 70en260

~50laquo40IEz3L

20)

10



(Jd)SS=80kN m-1

p=7middot175mm


p=6middot950mm

z 90L

8021J 70cen 6020

50

~ 40

IE 30

~20

~10



3middot2

2middot8

2-4

2middot0EE

160

1middot2

0middot8

0-4specimen 3

o

0-8

0middot7

3middot2

2middot8

2middot4

2middot0EE

1middot6 0[

1middot2

0middot8

0middot4

specimen

R

o 0o ++

))LgtRb ~

+ EJ-0 023 _d ++) ~2b

0middot3 -tAt) y

0-2 -t5+

0middot1

0middot90middot8

0middot7

0middot6















) 0middot9




)+ ~Rb ~ 2OE)









dJ = (2b)M dOP +dG (17)





























HEATTREATMENT79









(Jd)SS=83 kN m-1

p=7middot444mm

(Jd)ss=80 kN m-1

p=7middot175mm

mmo

bull

~ 10



~20


(Jd)SS=80 kN m-1

p=7middot175mm

(Jd)SS=85 kN m-1

p=7middot623mmz 90L

bull80gt

2701J

~ 60250

~ 40laquo30IEz 20L

10)



(14)

(15)










bullbullbull

(Jd)SS=80kN m-1

p=7middot175mm

M

o

~ 90

2gt80

-g 70en260

~50laquo40IEz3L

20)

10



(Jd)SS=80kN m-1

p=7middot175mm


p=6middot950mm

z 90L

8021J 70cen 6020

50

~ 40

IE 30

~20

~10



3middot2

2middot8

2-4

2middot0EE

160

1middot2

0middot8

0-4specimen 3

o

0-8

0middot7

3middot2

2middot8

2middot4

2middot0EE

1middot6 0[

1middot2

0middot8

0middot4

specimen

R

o 0o ++

))LgtRb ~

+ EJ-0 023 _d ++) ~2b

0middot3 -tAt) y

0-2 -t5+

0middot1

0middot90middot8

0middot7

0middot6















) 0middot9




)+ ~Rb ~ 2OE)









dJ = (2b)M dOP +dG (17)





























HEATTREATMENT79









3middot2

2middot8

2-4

2middot0EE

160

1middot2

0middot8

0-4specimen 3

o

0-8

0middot7

3middot2

2middot8

2middot4

2middot0EE

1middot6 0[

1middot2

0middot8

0middot4

specimen

R

o 0o ++

))LgtRb ~

+ EJ-0 023 _d ++) ~2b

0middot3 -tAt) y

0-2 -t5+

0middot1

0middot90middot8

0middot7

0middot6















) 0middot9




)+ ~Rb ~ 2OE)









dJ = (2b)M dOP +dG (17)





























HEATTREATMENT79









dJ = (2b)M dOP +dG (17)





























HEATTREATMENT79









HEATTREATMENT79








^f bmjroc - solid mechanics at harvard...

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