16. Family of crack-tip fields characterized by a triaxiality parameter—II. Fracture applications

8/6/2019 16. Family of crack-tip fields characterized by a triaxiality parameterII. Fracture applications

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1KE SOY6, Y2 $5.00 + 0.00( I YY2 Pergmm Prcw L.ld

FAMILY OF CRACK-TIP FIELDS CHARACTERIZED BY

A TRIAXIALITY PARAMETER-II. FRACTUREAPPLICATIONS

N. P. ODOWI, and C. F. SHIH~

Division of Eng ne c ring. Brow n linive rsity. Provid enc e . RI 02912. U.S.A

ABSTRACT

(LX K4L to the J-ba wd frac ture mec hanics ap proac h is the conc ept of J-dominance whereb y J alone setsthe stress Ic w l a s w e ll a s the six sc a le of the zon e of hig h strcssc s a nd strains. In Part 1 the ide a of II J Qannulus MU dcccloped. Within the a nnulus. the pla nt strain pla stic ne a r-tip lic lds arc me mb ers of :I IBmilyofsolu~ions pa ramc terized b y Q wh en dista nc es arc norma lized b y Jn,,. whe re (r,, is the yield stress. .I an dQ hu\ c distinct roles : .I sets the size sca le o wr wh ich large stresses an d strams d c vc lop while Q sc a lesthe wa r-tip stress distribution and the strw triarialitq ac hiewxi ahc ad of the c rac k. Spc c ilica lly, ncg nti\ c(po slti\ c ) Q \ nlues mun that the hydrwta tic stress IS reduc ed (increased ) b y PO,, from the p = 0 plnnc\ traili rcfwc nc e state. Thcreforc Q pro\ ides a qua ntitaticc mc asurc o f c rac k-tip co nstraint, a term w ide lytw xi in the litc raturc c onc erning gc om c tq an d six ellc c ts on a ma terials resista nc e to frac ture. Thesedcvel~~p~mx~ts a re disc ussed further in this p ap er. It is sho wn that the J Q ap proac h co nsiderab ly c xtcndsthe rang e of ap plica hllit!; of frac ture mec hanics for aha llow -crac k geo metries loa de d in tcnsion and be nding.and de ep -crac h getm xtries load ed in tension. The ,I Q theory p rovide s II franc wo rk to orga nwe toughneshda la ;I\ a function of constraint and to utili7c sue h da ta in eng ineering ap plications. Two method s forc\ timatins Q at fully yicldc d cond itions and an interpo lation sche me arc discussed . The effects of crac k4x an d spe c imen typ e 011 frac ture toug hness a re xid rcssed .

ODown and SHIH ( 1991). hereafter referred to as Part I, presented the J-Q approachto fracture mechanics. They showed that the full range of high- and low-triaxialityfields within the J-Q annulus are members of a Family of solutions parameterized byQ when distances arc normalized by J,o,,. where o,, is the yield stress. Specifically. the

near-tip stress level is governed by Q. a hydrostatic stress parameter, while J sets theske scale over which the large stresses and strains develop. These ideas have beenformulated within the context of plane strain deformations (.I scales the process ~oncsi7c).

In Part I the following two-term expansion for the small-displacement-gradientnear-tip tield is shown to prevail beyond the /one of finite strains :

+ i\ uthor to who m c orrespo nde nce should be ad dressed

93)


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940 N. P. ODowr) and


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Family of crack-tip fields-II 941

Taking note of the above features, a simplified form of (1.1) is

where 6,, is the Kronecker delta. The restriction on the domain of validity recognizes

that finite-strain effects are not accounted for in (I .2).To extend the range of validity of (1.2). we may choose the small-scale yielding

solution which matches the HRR stress distribution at v/(J/oo) = 2 and 0 = 0 as theQ = 0 distribution. Using this Q = 0 distribution as the reference solution, the memberfields are given by

oil = (~,,)~=o+Q~o6i, for I > J/O,,, 181 < 7r/2. (1.3)

Q is then defined as the difference between the full-field solution for ~~~~~ nd the HRRfield at r/(J/oo) = 2, 0 = 0, i.e.

(1.4)

The distance r/( J/CT) = 2 is chosen so that Q is evaluated outside the finite-strainregion but still within the J-Q annulus. Note that Q can be evaluated from any ofthe stress components and at any convenient angle within the forward sector. Theabove choice based on gli,, appears to be the most sensible one. The J-Q annulus isdefined as the region of outer boundary r, Q where the representation in (1.2) and(1.3) accurately describes the field.

A reference distribution determined from a small-displacement-gradient for-mulation is adequate for many applications. However, some applications requireaccurate quantification of the field near and within the zone of finite strains, e.g.studies on the micromechanisms of ductile failure and cleavage fracture. For suchapplications the reference distribution should be obtained from a finite-deformationanalysis. Using the finite-strain distribution as the reference solution, the domain ofvalidity of (1.3) is extended for some distance inside the finite-strain zone. This canbe seen from the distributions given in Part I. However, we should point out that atdistances greater than r/( J/o,,) = 2 the difference between the finite and small-strainQ = 0 distributions is negligible (see Fig. 2 in Part I).

A comment about the representations in (1.2) and (1.3) is in order. Member fieldsof (1.2) and (1.3) with identical Q values have the same stress triaxiality at r/(J/a,,) = 2.At distance r/( J/go) # 2, however, the stresses given by (1.2) will differ slightly fromthose of (1.3). Our numerical investigations showed that (1.3) is the preferred rep-resentation for the Q-family of fields.

The interpretation of Q, made explicit by the approximate representations (1.2)and (1.3), simplifies its evaluation in finite-width geometries and its subsequent appli-cation. We should emphasize that though the present structure is largely based on thesmall-displacement-gradient expansion in (1. l), the existence of the Q-family of fields

under finite-deformation conditions can be deduced from dimensional considerationsas carried out in Part 1.


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Y42 N. I. OI~OWI) ;IIlcI ( I. 9llll

2. Q-~.Ahlll.Y 01. Frt.r.r,s

fi,, x Ki I; (!I) -t-72 I ,i ,,. (7. I )


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- n=lO. . n = 5

Q = F (T/o,; n!

For II = IO, ~1,) = -0.05. u, = 0.81, LI: = -0.54. and for f~ = 5, N,, = -0.1. (I, = 0.72.N? = -0.42. The Q -T relationships shown in Fig. 2 and the representation in (2.4)are small-scale yielding results. Of course. T has no relevance under fully yieldedconditions. However. the Q-family of fields can exist over the entire range of plasticyielding and does not depend on the existence of the elastic Geld (2.1).

BETNK)K and HANCOCK (1991) also llave shown that positive T-stresses havenegligible effect on the stress triaxiality. In contrast. negative T-stresses have a sig-nificant effect on the hoop stress ahcad of the crack. which they have approximated

by

(2.5)

For M = I3 they obtained U, = 0.64 and LI? = -0.4, Comparing (2.5) with (I .3). asimilar expression between 7;ia,, and Q can be obtained.

The crack-tip opening displacement ii, is defined by the opening where 45 linesdrawn backwards from the crack tip intersect the deformed crack faces. In Part I weshowed that d, has the form

This relation generalizes an earlier result derived from the HRR field alone ( S HI H,1 9 8 1 ) .The dependence of d on Q. determined from full-field solutions, is shown inFig. 3 for II = 5 and IO. For both materials, d increases considerably as Q becomesmore negative. This means that. at the same level of J. low constraint geometries(Q < 0) will undergo a greater amount of crack-tip blunting than high-constraintgeometries (Q 2 0).

It has been shown i n Part 1 that the plastic strain field has the form


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(4(b>

a

0.30~~..,...,~~.,~~.,..~,~~~,,,.-1.4-1.2-1.0-0.8-0.6-0.4-02 0.0 0.2 -1.2-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2

Q 4I-K;. 3. Depmiwc ol~~.xk opening d~splaccmcnt ~x~ramctcr tl 01, Q li,l- II = IO and 5

The first terln is the HRR singularity and the second tcrln involves both .I and Q.Observe that, if distances arc normalized by J;r j,,. the strain field in (3.7) is para-meterired by Q alonc, i.e. they are also members of the Q-fanlily. Figure 4 shows theQ-family of plastic strain fields for 17 = 10 generated by the modified boundary layetformulation (2. I ). The radial variation of the eflective plastic strain for a range of Q\:alues is shown at 0 = 0 and I) = 71/4 in Fig. 4(a) and (b). Ahcad of the crack (0 = 0).the plastic strain is insensitive to Q for ~/(.//a,,) > I. The weak dependence on Q isseen for 101 < 20 in Fig. 4(c) (f). In contrast, the plastic strain level along 0 = +4increases significantly as (_j becomes more ncgativc. Figure 4(c) and (d) show that theplastic strain increases and shifts touards the forward sector as Q becomes morencgativc. For increasingly positi\:e Q states sho\cn in Fig. 4(c) and (I), the plasticstrain in the for\\,ard sector decreases while that in the back sector. /01 > 7~ 3. increasz~.

These strain fields constitute the Q-family of Gelds. Strain distributions for linitc-l\,idth crack geometries, prrscnted in subsequent sections. can be identified with thedistributions presented here.

In this and subsequent sections wt direct attention to the ellitcts of applied loadand crack geometry on crack-tip constraint as measured by Q. IJndcr large-rcalcyielding conditions the result (2.3) does not apply. Q now depends on the remote loadand crack geometry. This dependence can bc written in the form

(3.1)

where J.(Lo,,) is the dimensionless load: L is the relevant crack dimension of thefinite-width crack geometry.

WC examine tension and bending dominated gzeometrie:, with crack length (I a n dGdth W(ligamcnt length /I = F--u). In general ;I crack is designated a shallow cr:tck


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Family of crack-tip fields-11

0.100

(4 . **.0.075

-.- -1.07

__----- -0.70

P14) 0.050 - 0.0

- - 0.22

0.025

0.20

(b )

0.15

0.0000.0 1.0 2.0 3 .0 4.0 5.0

r/(J/a 0)0.25

Cc )0.20

0.05

0.00

0.20

(e )0.16

0.12P14)

0.03

0.04

0.00

0.0 45.0 90.0 135.0 180.0

8

Q- - 0.22 r/(J/o,) = 1

0.05

0.00

0.15

@)

0.10

PIW

0.05

0.000.0 45.0 90.0 135.0 180.0

0

0.106) :--0.22 c

zl

0.0 1.0 2.0 3.0 4.0 5.0

r/(J/o,)

r/(J/u,) = 2r Q\ - - -1.40. . -1.07

-- -0.70

--__-_. -0.35

- 0.0

0.0 45.0 90.0 135.0 180.0e

FIG. 4. Q-family of plastic strain fields. (a) and (b) radial distribution of effcctivc plastic strain at 0 = 0and 0 = ~4. (c) and (d) angular distribution for negative Q at r,(J/cr,,) = I and h(J,cr,,) = 2. (e) and (f)

angular distribution for pwtive Q.

when the relevant dimension is the crack length and a deep crack when the relevantdimension is the ligament. The material properties are those used in the analysesdiscussed in the preceding section and in Part 1. Only the IZ = 10 case will be discussedhere.

A schematic of the center-cracked panel loaded in biaxial tension was shown inPart I in Fig. l(b). The state of biaxiality is given by B E a:,/o,l;. By varying B,


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936 N. r. O~WI) d c.. t. SIIIII

difterent tcvcls of constraint at the crack tip arc obtained. Three b iaxiality raliox.

H = 0. 0.5 a nd 1. arc investigated.I-igure 5 shows the de pe nde nce of Q on the c xctent of plastic yielding mc asurc d bq

.J. (cm,,) for the center-cracked panel with cl,. M = 0. I. As before, Q is c \ ~aluaM h>ta king the difYercnc c b etwe en the full-field solution 1.01. he hoop slrcss and the HRR

field (or cqui\atentty the Q = 0 distribution) a( I (J ml,) = 2. 0 = 0. Fol- 13 = 0. fdl~yielded condilions are reached at ./(rrrr,,) 2 0.03 : for U = 0.5 and I .O, ILIII~ j ictdcdconditions arc reached al J.(tra,,) z 0. I anti 0.0. rcspe c li\ ety. J:or the highest hi-auiatity. B = I. the ptaslic /one does not spread across Ihe ligament but instcad iI

c ngulls the c rac k c om plete ly. The c ffezr of b i:lxialil! on p lastic / one \ in lhc ccn~cr-

c rac ked p ane l is disc ussed in Ja rt I.

For H = 0. Q drops r;Ipidty at lo\k, toads. indicaliny a loss olconslraint as ptasticil>

develops. At the fully yield& sta te. C;, seltlc ~ t o ;I c onstant value ola bo ut I..?. ;Isimilar Irend is been for B == 0.5 Q Iills rap idly :11 tirst and the n rcnia ins c fIIilcconstant as f~rtty yictdcd conditions arc rcachcd. b-or 8 = I. fJ rises graduatl~ ~iltl

increa sing load ~113 o .J(m,,) : 0.03 ~inc i tic n rises more steept>~ iis the toad incrcac~.F igIrc 6 show s the p lastic stra in and hydrostatic slrc ss in lhc c c ntc r-c rac kcd pa nel

Io r the b iau iat it~ tt372ts. R = 0. 0.5 a nd I. In all cases Ihc ctrcss and \Irain lic lcl\s~ronply rcsc mb tc nionibe rs of the Q-la tnity : ttic C)-l;iniity of hydrosMtic hIress andstrain lietds arc sho\\m in Fir. I and Fig. 1. respc c livc ly. The fields in Fig. 6 a rc p lo~tc d;tI ,.,(.I nTi,)= 2, hug WC note that [tic i-ewmhtance p c rsisls For d ista nc es up to aboutI (J,o,,) = 5. The fic td s li)r B = 0 and 0.5 co rrc~pp ond to nega li\ o Q-fields 1llC tl~tll-o-sta tic slress d c c rca sc \ white t he pla stic strain aticatt olltlc crac k incrcasei 2s ptasticil!dc\dops. The loss in stress 1riaxia tity co~ipted willi the increase in plastic strain uoutdttmi to t il\ or ductitc tracture. Jhc high-bi;l\ i;llity C;IX. B = I. produc es po si[l\ c Q-ticlds. i.e. a:, Ihe ptasbcily dcvclops the hb ,tll-osta tic stress inc rca sc s sligh tly wh ile ltrc

pla stic strain in the lorward ~wtor rema ins rc tatiwty wa tt [c onipa rc Fig. 6(1) \ zittl

Fig. 6(b)]. These c on d itions IIVOI- clcavagc Irac(urc.In Jart I, \ \ ;c have noted that cxccssiw crack-tip blun~in, (r may result in ;I IO\ \ 01

0.5

0.0

4 -0.5

B= 1.0 ,/-


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Family of crack-tip fields- II 947

(4

0.15

04

____--. -0.52P

14 - -0.16

0.0 45.0 90.0 135.0 180.08

0.100

(40.075

e

0.100

(f) Q*.,,......... .lg v(J/o ,,) = 20.075 --- 0.17

.I iJ dominance. This appears to be the cast for the H = 0 biaxiality state at cxtensivcyielding as seen in Fig. 6(b). The plastic strains near 0 =: 0 for Q less than - I ;lrewnsiderab ly higher than those in Fig. 4(d). At these Q values. J,o,, is grea ter thnnab out 0.05rr. i.e. the c rac k- tip opening displacement is a sizable fraction of the crackxi/e. Under thcsc conditions, the J Q tield dominatw over ;I small rcgisn. Thedistributions of the hydrostatic stress in Fig. 6(a) at these J levels indica te go odagreement with the Q-family of fields. Despite this. the lack ofagreement in the strains

is not unespected since a small de viatio n fwm the J Q stress field is ma gnified in thestrains li: * ((T:(T,,) ].


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The variation of Q with J!(La,,) for the center-cracked panel with LI; + = 0.7 isshown in Fig. 7. In this case J is normalized by ha,, since h. the length of the uncrackcdligament, is the relevant distance. For biaxiality ratios. B = 0 and 1, we obtain negativeQ values and, for B = 2, positive Q are obtained. The variation of Q with J:( Lo,,) forthese biaxiality ratios is similar to that observed for the shallow-crack geometry.

The hydrostatic stress and plastic strain fields are plotted in Fig. 8 for the threebiaxiality states, B = 0, I and 2. By comparing these fields with those in Figs I and4. we can conclude that the fields in Fig. X are members of the Q-family. However,there appears to be an exception. The strain distribution identified by Q = 0.19 inFig. S(f) does not appear to be a member of the Q-family. As previously explain&.excessive crack-tip blunting can result in the loss of J-Q dominance. and this may hcthe case here.

There is broad agreement that J alone characterizes the crack-tip fields in decp-crack bend geometries (MCMEEKINC; and PARKS. 1979; SHIH and GERMAN. I98 1 ;HUTCHINSON. 1983). Here we explore whether a two-parameter characterization cansignificantly extend a fracture mechanics approach for shallow-crack bend geometries.

For the purpose of comparison the hoop stress distribution for shallow and deeplycracked center-cracked panels is presented in Fig. 9(a) and (c). It can be seen that thehoop stress distribution shifts downwards (uniformly) by an amount Qa,, as plasticyielding develops. Thus the J Q annulus extends over a physically relevant length

scale in both shallow- and deep-crack geometries at contained yielding and fullyyielded conditions.

Figure 9(b) shows that the stress redistribution in ;I shallow-crack bend bar.(I/W = 0. I. behaves like that in center-cracked panels. Thus we may make similarconclusions about the existence of the J Q annulus. Indeed, AI,-ANI and HAN(YKX

0.5

0.0

Q -0.5

-3.50 -2.75 -2.00 -1.25 -0.50

log(J/(Lu,))

B = 2.0


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Family of crack-tip fields-11 949

3.0

(e)

2.0

0.0%). This can be explained by the strong stress gradient acrossthe ligament-the hoop stress is compressive near the free surface and must becometensile as the crack tip is approached. Therefore the Q-term, which represents a state


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0.0 1.0 2.0 3.0 4.0 5.0

r/(Jbo)

- 0.0021.0 - 0.006..._.

- 0.016,,....... II 0,046

0.0 J.l0.0 1.0 2.0 3.0 4.0 5.0

r/(J/ @ o)

(b) 4.o3.0

0

+. 2.02

I-

1 . 0

1 a/w = 0.1 1

- 0.001-mm--. 0.009-I 0.024,.,..,...., 0.073

0.0 -0.0 1.0 2.0 3.0 4.0 3.0

r/(J/~o)

L____. 0.025

-. 0.055 ....,

of unifo rm hyd rosta tic stress in the Ihrward sec to r [see ( I .3)]. has ;I small domuitl 01va lidily. ix. the J Q annulua is not muc h larger than tlic ,/-annul~is. In any c;isc 1ticr.c

is ge neral ap rwnent that the one-p aram eter J-c h;llacte rization works ~.tll thr do c pl~

cracked bend gcometries ( MC.MHKIUG and PARKS. 1979: StiIH rind GI:I~MAX. 19% I :HUTCXINXNX, 19X3). The d istribut ions in Fi,. C)(d ) a lw SllOW that \VhCIl ./*fl,, > O.M/?

the glob al b iding stress d istrib ution p reva ils wa r the c rac k tip. Nc~c rthelcus, Q

/defined by (I .4)J still p rovides ii men surc ol nw r-tip stress triaxia lil).

The hydrosta tic stress a nd c ff ec tivc pla stic strain for c l: + = 0. I, 0.3 and 0.5 arcplotted in Fig. I I The Q vnluc s a re eva luate d al r/(,/:ci,,) = 2. In all cuscs the stressand strain field s resem ble the Q-fam ily. It ma y be note d tha t the angular distribution

of the fields fbr the de eply crac ked be nd ba r c ontinues to rcsem blc the Q ficltfs cwn

:it c onc iitiws well heyo nc f the on~ct of fully yic ldc c i c onditions.


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Family of crack-tip ticlds~~ll 05 I

(a) o.25

-3.0 -2.5 -2.0 -1.5 -1.0 -0

log(J&,))

(b) o.25

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5

log(J&,))

4. METHODS FOR EVALUATING Q

Thus far, values of Q for different crack geometries have been evaluated from full-field solutions. Simplified methods for determining and using Q are proposed in thissection.

Under small-scale yielding conditions Q depends on load and geometry only

through T (2.3). The T-stress, which scales linearly with the applied load, has beencalibrated for a number of crack geometries including the center-cracked panel andthe edge-cracked bend bar (LEEVERS and RADON, 1982; SHAM, 1991). T can beexpressed in terms of a dimensionless parameter C which depends on specimen typeand relative crack size ul W, i.e.

(4.1)

For the center-cracked panel loaded in remote tension and for 0 < L/i ct < 0.7.C ranges from -0.56 to -0.68. For a bend bar C ranges from -0.21 to 0.19for 0 < N/I/~ < 0.7. When a center-cracked panel is loaded in biaxial tension {B =o,,/a,,. [set Fig. l(b) of Part I]), T is given by

T = t o,:..Ju

(4.2)

Using the relationship between K and the remote stress a,, ,

K = CJ,*,&aF(a/ W) (4.3)

we can write Tin the form

T = o,,.($cCF+ ), (4.4)


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3.0

Cc)

2.0

= +\,3 z .- I .8. From our full-field analysis for the short and deeply crackedpanel (II = IO material) we obtained Q = - I .3 at fully yielded conditions [see Fig.I?(a) and (c)l. This compares favorably with the slip-line field value of - I .8.

Consider a biaxially loaded panel with normal tractions of magnitude (T appliedat the vertical boundaries of the panel. The stresses across the fully yielded ligamentare : f7,, = ~o,,/~I 3 + G,], G,, = oil{, and (T,, = 0. Therefore Q, ts = - rciV 3 + oHgi~,,. dotethat fully yielded solutions do not exist when nB is sufficiently large ; only contained-yielding states are possible [see Figs 7(c) and 13(c) in Part I].

Case 3 : edge-cracked panel under bendingPlasticity is confined to the ligament for N/W greater than about 0.3. The solution

for the deeply cracked geometry is the Green and Hundy slip-line (GREEN and HUNDY.


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056 N . P. ODOWI) and C. F. SHIII

1956) which gives the stresses near the crack tip 21s: CT,,2 2.9lcr,,, g,, z 1.75a,,. and(T,,. = 0. Comparing with (4.7) we obtain Q,:,, = -0.06. From Fig. 13(b) we see thatthe value of Q for the deeply cracked bend bar is close to QI,, at deformation levelsJ < hrr,,/X. At larger Jvalues, Q falls off dramatically from Q,;,,. This difference arisesbecause Q is measured at r;!( J/g,,) = 2, where at these J values the global bending

stress distribution dominates as discussed in Section 3.3.For crack sizes smaller than (I: CV= 0.3, the plasticity spreads to the back foes of

the crack. Under these conditions. the Q values itre significantly lower than those 1.01the deeply cracked geometry (EWING. 196X).

Case 4 : edge-cracked panel under combined tension and bendingThe stress triaxiality that develop under combined remote tension and bending arc

lower than those under bending alone. An upper-bound solution for combined tensionand bending has been proposed by RICE (1972). Estimates of Q for these cases :trc inprogress (ODown and S H I H . 1992).

The value of Q under f ~dIy plastic conditions can be obtained from fully plasticsolutions for pure power law hardening materials of the type used in simplifiedengineering fracture analysis (KUMAR ct d.. 19XI). Fully plastic solutions are relativelyeasy to generate and QP,, obtained from such solutions is expected to be more accuratethan that estimated by slip-line field analysis.

The material of the cracked body is taken to be described by J: deformation theoryfor an incompressible pure power law stress--strain behavior. In uniaxial tension thematerial deforms according to

LL,, = r((T,.(T,,)~.

Under multi-axial stress states the strain is

(4.8)

~i:,,;i,~, x(3,:Z)(a,rr,,) .\.>,+7,,, (4.9)

where .s,, is the stress deviator and (T, = V 33,,.\,,,2.The solution of a traction boundary value problem based on (4.9) and involving

only a single load parameter which is increasing monotonically has the functiomtlform

(T,,,cc, = (a /0,,)f?,,(x,L; geometry. I?),

,,i(Xl:,,) = (0 ~o,,)C,,(x,;L; geometry. 12). (4. IO)

where (T is a representative stress magnitude and i, is a characteristic crack dimension.rF,, and f:#, arc dimensionless functions of spatial position x and depend on crackgeometry and the strain hardening exponent. Observe that the stresses ;trc linearlyrelated to the applied tractions. The form of the J Q field in (1.2) or (I .3) and the

linear dependence of the stresses on the applied tractions in (4. IO) lead immcdiatclyto the forms of J and Q at fully yielded conditions :


19/25

F a m i l y f c r a c k -t i p ields--II

J /(w ~E~L ) = (ax /o)+ h , (geometry, n),

95 7

Q = H&J/a,& geometry, n). (4.11)

The dimensionless functions, h, and H,, depend on n and dimensionless groups ofgeometric parameters. H, depends weakly on J/ooL for most crack geometries.

For a number of crack geometries, h, values have been extracted from fully plasticsolutions and catalogued in a fracture handbook (KUMAR et al., 1981). Values of h,can be extracted from the same fully plastic solutions and catalogued in a similarmanner. An efficient numerical method for generating fully plastic solutions isdescribed by SHIH and NEEDLEMAN I 984).

4.4. Intrrpolating betw een Q ssy and QFP

To evaluate Q for the full range of plastic yielding, a scheme to interpolate between

Q at small-scale yielding, Qssr, and Q at fully plastic conditions, QrP, is required.QssY can be calculated from T using (2.4) (with a,,, a, and a2 appropriate to the strainhardening). This value depends on T alone, i.e. crack geometry and load magnitudeis transmitted to Q only through T. Since the linear dependence of Ton the generalizedload 9 is known, the derivative dQssY/dY can also be calculated. Moreover, undersmall-scale yielding the relationship between J and P is known: J cx K,? c;c Y [see(2.2)]. Therefore dQsSY/dJ is available immediately.

QPP can be obtained from fully plastic solutions and has the form in (4.11). Thederivative dQ,,/dJcan be evaluated from the dependence of H, on the first argument.Moreover, we can evaluate dQPp/d6P since J cc SC+ I).

Using the procedures discussed above, we can determine Qssr, dQssr/d.Y and QfiP,dQl;p/dY, i.e. the Q values and slopes at the limiting load states are known. The valueof Q at intermediate load states can be obtained by interpolation. Alternatively wecan interpolate for Q at various J levels using Qssr, dQssv/dJ and QPP, dQ,,/dJ. Bothapproaches are under investigation (ODown and SHIH, 1992).

5 . FRACTURE TOUGHNESS L o c u s AN D S I ZE E F F E C T S

It is widely accepted that a single parameter J,(- quantifies the material toughnessunder high constraint. Experiments with low-constraint crack geometries have shownthat J at the onset of fracture (particularly cleavage fracture) can be considerablylarger than J,c. In other words a single value is inadequate for characterizing fracturetoughness over a range of crack-tip constraints. A toughness locus, namely Jc vsconstraint, is more relevant for a broad range of engineering applications. The J-Qtheory provides a framework which allows the toughness locus to be measured andutilized.

For the purpose of demonstrating the effect of constraint on fracture toughness weadopt a fracture criterion based on the attainment of a critical normal stress, cZ2 = (T,,at a critical distance, r = r,. ( R I T C H I E t al., 1973). Within the J-Q annulus the normal


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95x N. P. ODowu and C. F. SHIH

stress ahead of the crack is given by ( 1.3) or, more accurately, by (I .3). For simplicitywe choose the closed-form representation in (I .2) :

(5.1)

Assume that r, is within the J--Q annulus. Apply the fracture criterion to (5. I ) to get

(5.1)

Therefore we can solve for J, as a function of Q for various values of g, and I,It is helpful to introduce J,' . which in this context refers to the value of .J, fhr a

long crack, N/T, ---t 'YJ. We will show later that J, corresponds to the value of J, undersmall-scale yielding conditions with T = 0. Using (5.2) the ratio J,/ J, is

(5.3)

Observe that the ratio J, /J, is independent of I,. The variation of J, /J( with Q isplotted in Fig. 14(a) for an II = 5 material with cr/o,, = 2, 3 and 4. It can bc seen thattoughness rises rapidly as Q becomes more negative, corresponding to a loss ofconstraint. Figure 14(b) shows the dependence of J, ! J, on II. ANIEKSON rt rrl. ( I99 I )have constructed similar plots of J, ,!J,, vs Q for stress-controlled and strain-controlledfracture using results in Part I and in this paper (J,, z J, ). Their approach also takesinto account the statistics of cleavage fracture.

HARLIN and WILLIS (1988) have discussed crack size eflects on fracture toughnessand on the ductile- brittle transition temperature, and have shown that this can beaccounted for by the T-stress. We will demonstrate how size effects under plane strainconditions can be interpreted using the J-Q held and how they can be phrased interms of the T-stress. We rewrite (5.2) in a form suitable for investigating size cllects.Using dimensionless groups. .?, = J,((I.,(T,,). Fc = r:L, f,. = o-/u,, and using (3. I) WChave

(5.4)

Crack size effects enter through the dimensionless argument ],I; in G. We can solvefor J< as a function of r< for various 6, and geometries. The values of G(j(F


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Family of crack-tip fields-11 959

Q-1.00 0.75 -0.50 -0.25 0.00 0.25

.

-1.00 0.75 -0.50 -0.25 0.00 0.25

QFE. 14. (a) Normalized toughness locus for n = 5 and u,/uo = 2, 3 and 4. (b) Normalized toughness locus

for a,/~,, = 3 and n = 5 and 10.

(2.3) ; moreover T/a = KC/(o,~) f rom (4.1). Making use of the above in (5.2) andnoting that Jr = (1 - r)K;?/E, we get

2 = [!!~Y~]Cfl+161&9 = ,)+F(~J (5.5)

Now use the dimensionless groups, EC = K,/(cJ,,J~~~), & = t -,/a, and 6, = D~/c,,, to get

, _$,2 _,

( 1

101+ I)

5, = -z--K; 62>(6) = 0)+F-(RJJr, Tq. (5.6)n

The above equation can be solved for EC for different values of FC, eC and C. Note that

Z depends only on relative crack size and specimen type. We define K, as the valueof Kc for a long crack which is consistent with our definition of J:. Observe that the


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Y60 N. P. ODOWD and C. F. SHIH

4.0 -

(a)

8< 2.0 -+J

0.0 -2.0 3.0 4.0 5.0

log(a/r,)

(b) 4o :

3.0 - c = -0.2

8 -0.1

0.0 s k2.0 3.0 4.0

~ofzb/r,)

5.0

FIG. 15. (a) Crack size effects on fracture toughness for bend bar with 0: W = 0.1.0.3 and 0.5, for m,:(~~, ~~ 3and PI = 5. (h) Crack size effects on fracture toughness under small-scale yielding for (T, v,, = 3 and II - 5

for three specimen types. C = -0.2. pO. I and O.Oi.

argument of F in (5.6). ~?-,df( X, , vanishes in the long-crack limit FC -) 0. Since theargument of F is T/ g,, [see (2.3)], it follows that the long-crack constraint is given by

Q r= o.The variation of K,./K: with log (u/r) is shown in Fig. 15(b). The cases examined

are C = 0.03, -0.1 and -0.2. These values of C correspond to a bend bar withu/W = 0.5, 0.3 and 0. I, respectively. Observe that for E = -0.2 and -0. I, thetoughness increases rapidly as crack size decreases. In contrast when C = 0.03 thetoughness is almost unaffected by crack size. In the latter case the argument of F in(5.6) remains close to zero for all meaningful values of a/~,, so the toughness isessentially given by the long-crack toughness, K, The dashed portion of the curvefor 2: = 0.03 indicates the range of crack size where small-scale yielding conditions,i.e. u >> r,, > I,, may not be satified. In this regime results based on the large-scaleyielding solutions are relevant.


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Family of crack-tip fields-II

6. DISCUSSION

961

We have elaborated upon the distinct roles ofJand Q in the J-Q fracture approach :J sets the size of the process zone over which large stresses and strains develop whileQ scales the near-tip stress distribution and the stress triaxiality achieved ahead ofthe crack. Approximate representations for the full range of stress distributions interms of Q are given in this paper.

We have shown that the J-Q annulus for short-crack geometries, loaded by tensionand by combined tension and bending, is considerably larger than the J-annulus. Thisalso holds for deep-crack geometries loaded in tension. Therefore the J-Q approachallows us to extend the range of applicability of fracture mechanics for these importantgeometries. For deeply cracked bend geometries we have observed that the J-Qannulus is not much larger than the J-annulus.

The available data on cleavage fracture indicate that a slight decreast in the hydro-static stress level can result in a significant increase in the fracture toughness [e.g.

IWADATE et ul. (1983), ANDERSON and WILLIAMS (1986) and INGHAM et al. (1989)].In other words, cleavage fracture toughness depends strongly on crack-tip constraint.BETEG~N and HANCOCK (1991) have proposed that constraint be quantified in termsof the elastic T-stress. Their approach can overestimate or underestimate the actualhydrostatic stress by as much as 0.5~~. Since stress-controlled cleavage fractureis sensitive to the hydrostatic stress level, an error of this magnitude may beunacceptable.

The J-Q theory provides a framework which allows the toughness locus to bemeasured and utilized. Within this framework we can systematically develop tough-ness loci based on cleavage mechanism and ductile failure mechanism. Thus, the

competition between cleavage and ductile fracture [e.g. RITCHIE et ul. (1973, 1979),IWADATE: ef ul. (1983), LIN et al. (1986). ANDERSON and WILLIAMS (1986), TOWERSand GARWO~D (1986), HARLIN and WILLIS (1988), INGHAM et al. (1989) and HACKETTet al. (1990)] can also be investigated within the J-Q framework.

Detailed near-tip stress distributions are required for the evaluation of Q. We haveproposed two alternative methods for evaluating Q under fully yielded conditions:slip-line field and fully plastic analyses. The first method should provide a goodestimate of Q for low-hardening materials. The second method is more general andcan be applied to high- and low-hardening materials. A scheme to interpolate for Qat various J and load levels has been discussed.

ACKNOWLEDGEMENTS

This investigation is supported by the Office ofNaval Research through ONR Grant NOOOl4-86-K-0616. The computations were performed at the Computational Mechanics Facility ofBrown University supported in part by grants from the U.S. National Science Foundation(Grant DMR-8714665), the Office of Naval Research (Grant NOOOl4-88-K-0119) andmatching funds from Brown University.

REFERENCES

AL-AN, A. M. andHANCOCK. J . W. 1991 J. Mech. Phys. Solids 39, 23.


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962 N. P. ODowi, and (. F. SHIH

ANIXRSON. T. L. and WILLIAMS, S 1986

ANIXRSON. T. L., VANAPARTHY. Sand Douns, R. H.. JR

BETE&N, C. and HANCOCX. J. W.

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PARKS. D. M.

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Documents

16. Family of crack-tip fields characterized by a triaxiality parameter—II. Fracture applications