Chapter 6 Elastc -Plastc Fracture Mechancs The hgh stresses at the crack tp cannot be sustaned b, practcall, an materal. Thus, f the materal does not fracture, a plastc zone (or damage zone or process zone) s formed around the crack tp. The damage s specfc to the materals but t can be sad that n general terms, for a ductle materal, the damage s n the form of plastc deformaton and for brttle materals n the form of mcrocrackng. Untl now s has been assumed that the sze of the plastc zone near the crack tp s relatvel small as compared to the specmen dmensons. Thus, the effect of ths zone has been neglected and the stran feld surroundng the crack tp s domnated b lnear elastc fracture mechancs asmptotc feld derved n chapter 4. In ths chapter the effects of a non-neglgble plastc zone s consdered n the case of Mode I onl. Frst, n order to determne at what pont t s necessar to nclude the nfluence of the plastc zone n the stress analss, two appromatons of the sze of the plastc zone are presented. Afterwar, two elastc-plastc fracture crtera are cussed, the crack tp openng placement and the J-ntegral. The J-ntegral s shown to be a generalzaton of the lnear elastc release rate to elastc-plastc fracture. The propertes of the J and the stress feld near the crack tp for elastc-plastc fracture are then derved. Lastl, calculatons of J for certan geometres are gven as eamples. 6.1 One dmensonal estmaton of Sze of Plastc Zone In order to obtan a frst estmate the sze of the plastc zone, we appl the von Mses eld crteron, ( ) ( ) ( ) + + = (6.1) 1 3 3 1 where ( =1,,3) are the prncple stresses. Thus, we assume that the regon n whch, ( ) ( ) ( ) + + 1 3 3 1 has plastcall deformed. Note that the von Mses eld crteron s dependent onl upon the prncple stresses. Ths eld crteron s applcable snce plastc flow s usuall a shear deformaton wth constant volume,.e. ndependent of the prncple stress components. Thus, we can defne a at the onset of plastc deformaton and then assume that all further plastc deformaton occurs at ths constant. For ths frst appromaton we consder onl the stress feld on the lne θ = 0, as shown n Fgure 6.1, and defne L as the characterstc length of the plastc zone. B comparng the stress feld wth the von Mses eld crteron, we wll determne the length L for whch the materal n the regon L has eceeded the eld lmt. The asmptotc stress feld near the crack tp are gven b (4.36). On the lne θ = 0 these reduce to, K θθ = = = πr I rr rθ 0 (6.) March 006 6-1
Note that θθ and rr are the prncple stresses 1 and for the lne θ = 0. Applng the plane stran condton (.e., 3 =ν( 1 + ); ν beng the materal's Posson rato) to eq. (6.1), one has, ( ) ( ) ( ) ( ) ( ) ( ( 1)( 1 ) ( 1) 1 = + 1 ν ν + 1 ν ν 1 1 1 = 4ν 1 ν + 1+ν +ν + ν+ 1 + 1+ν +ν ν+ 1 1 1 1 1 = ν ν+ + + ν ν ) (6.3) For the specal case of 1 = (such as along the lne θ = 0), eq. (6.3) s smplfed to, ( ) = ν ν + ν ν+ 4 4 1 ( ) = 8ν 8ν+ 1 ( 1 ) = ν 1 (6.4) Substtutng the prncple stress of eq. (6.) nto eq. (6.4), the eld crteron s gven b, ( ) I = 1 ν K πl (6.5) Thus, the length over whch plastc deformaton occurs s gven b, ( ) 1 ν K I L = (6.6) π Fgure 6.1 Schematc of plastc zone ahead of crack tp. The calculaton of L, s onl an appromaton snce the presence of damage wll modf the stress feld and thus the sze of the zone. However, t can serve as a parameter to compare wth the overall dmensons of the specmen, and determne the lmts of the lnear elastc soluton. The four prncple cases are summarzed n Fgure 6.. March 006 6-
Case I, elastc fracture, occurs when L s much smaller than an other of the specmen dmensons. For ths case, the plastc zone s neglgble and the stress feld s domnated b lnear elastc fracture mechancs (LEFM). Ths s the case studed n chapter 4 and 5. Case II, contaned eldng, the effects of the plastc zone are no longer neglgble. For ths case, the stran feld s domnated b elastc-plastc fracture mechancs, the subject of ths chapter. We wll see later n ths chapter that the fracture can be well descrbed b the J c (non-lnear crtcal energ release rate). The fnal two cases wll not be consdered n these notes, however the are ncluded here to show other possbltes. Case III, full eldng, nvolves large deformatons due to the large plastc zone. Ths case s less useful snce tpcall the structure s no longer capable of supportng the appled loa. Case IV, dffuse spaton, s dfferent from the other three, as there are no tnct elastc and plastc zones. Instead, the fracture s domnated b non-lnear elastc, tme dependent phenomena, such as creep or vscoelastct. Fgure 6. Modes of fracture n specmens wth dfferent etend of plastc deformaton. March 006 6-3
6.1. Two dmensonal appromaton (D) Whle the length L gves us an ndcaton of the sze of the plastc zone relatve to the specmen dmensons, t s also useful to know the actual shape of the plastc zone around the crack tp. Usng the same approach as n the prevous secton, t s not dffcult to obtan ts shape. In clndrcal coordnates the prncple stresses 1 and for an arbtrar stress feld are gven b, θθ +rr θθ +rr 1, rθ = ± + (6.7) Near the crack tp, where the stress feld s domnated b the terms of eq. (4.36) we obtan after some algebrac manpulaton, K I θ θ 1, = cos 1± sn πr (6.8) and 3 ν( 1 +) = n the case of plane stran. Note that for the case of θ = 0, the last equaton reduces to (6.). Substtutng eq. (6.8) nto (6.3) and smplfng lea to the followng eld condton, = 1 KI θ θ cos 4( 1- )- 3cos r ν+ν π Thus, solvng for the radus of the plastc zone at whch ths condton s satsfed r p (θ), gves, 1 K I θ θ rp ( θ ) = cos 4( 1 ν+ν ) 3cos π (6.9) The shape of the plastc zone s plotted n Fgure 6.3(a) for ν = 1/3 and ν = 1/. Also plotted s the lmtng case of plane stress (.e. ν = 0 ). As the state of stress changes through the thckness of a thck specmen (.e. plane stress plane stran), the shape of the plastc zone also changes. Fgure 6.3(b) shows a tpcal shape of the plastc zone across a specmen. Note that the shapes of the plastc zone shown above do not take nto account the effects of the specmen boundares. Thus, the are derved for a crack n an nfnte plate. For an actual specmen, one must often consder the fnte wdth n order to determne the regon of plastc deformaton. Fgure 6.4 shows eamples of the shape of the plastc zone for three standard specmens. March 006 6-4
Fgure 6.3 Plastc zone shapes; (a) as a functon of ν,. (b) through the specmen thckness. Fgure 6.4 Effect of fnte wdth of specmen on shape of plastc zone; (a) double edge notch n tenson, (b) center cracked specmen n tenson, (c) edge crack n bendng. 6.1.1 Irwn s appromaton of the plastc zone It was mentoned earler that due to the materal s eldng at the crack tp the stress trbuton s not gven b the asmptotc fled (4.36) and shown n Fgure 6.3 b the curve (1). To account for the effects of plastc deformaton on the retrbuton of the stress feld and obtan a smple appromaton, perfect plastct s assumed thus, the stress ahead of the crack tp equals up to a tance r. After that tance, the trbuton s obtaned b a translaton of the asmptotc feld shown b the curve (). March 006 6-5
The two trbutons, before and after eldng, should result n equal forces snce equlbrum should be assured. Wth references to Fgure 6.5, force equlbrum results n, r 1 KI d =r 0 π (6.10) where the tance r, s the ntersecton of the fled (1) and the horzontal lne at gven b, 1 I K I = r 1 = π 1 K 1 πr (6.11) (1) r 1 r () Fgure 6.5 A crack n an nfnte plate wth plastc zones ahead of the crack tp. Integratng (6.10) and usng (6.11) one obtans, r 1 = tance equal to r1+ r = r1 gven b, r. Thus, the plastc zone eten over a r 1 K I 1 = π (6.1a) The smple analss shown above s for the case of plane stress.e., 3 = 0 whch s realstc for thn plates. In plane stran, eldng s confned due to the effects of 3 > 0 and a smaller plastc zone ahead of the crack tp s developed. In ths case, Irwn proposed the followng epresson, r 1 K I 1 = 6π (6.1b) March 006 6-6
whch s smaller than r 1 n (6.11) b a factor of 3. A smple comparson of the eqs (6.1) and (6.6) gves a plastc zone half n length of the one proposed b Irwn. Ths dfference s due to the fact that the analss of Irwn, beng more realstc, takes nto account the retrbuton of stresses due to plastc deformaton. In addton, Irwn argued that after the plastc zone the stress retrbuton due to eldng results n hgher stress than that gven b (6.). Ths hgher stress s reflected n an effectve SIF whch for a crack n an nfnte plates s epressed as, ( ) ( ) K I,a+ r1 = π a+ r 1 (6.13) 6.1. Dudgale s eld strp model In thn plates of certan metallc materals wth a central crack a narrow plastc zone eten from both crack tps as shown n Fgure 6.6. To predct the length c, of ths plastc zone, Dugdale (Hellan, 1984) assumed perfect plastct. Thus, the plastc zones are replaced b closng tractons over the length of the zone as shown n Fgure 6.6. Usng the prncple of superposton, the total stress ntenst can be calculated as shown net. c a c Fgure 6.6 Dugdale s eld strp appromaton of the plastc zone. There are two contrbutons to the total stress ntenst factor. One comes from the remote appled load and equals to, ( ) K,a+ c = π (a+ c) I (6.14) The other comes from the closng tractons on the crack faces. Usng eqs (5.3) and the prncple of superposton elaborated n chapter 5, one replaces Q= d as the force on the crack face, the total SIF at an of the two crack tps s gven b the followng ntegral, March 006 6-7
a + c (a + c) + (a + c) K I (,a+ c) = + d (a c) (a c) a (a + c) + (a + c) + + (6.15) c a c Fgure 6.7 The eld strp replaced b eld stress of the materal. Assumng perfect plastct, the eld stress s constant. Therefore, a+ c (a + c) d (a + c) 1 a KI = = cos π (a + c) π a+ c (6.16) a Accordng to ths model, the stress ntenst due to the remote stress and the stress ntenst due to the closure stresses must be zero. Ths s epressed as, ( ) ( ) K I,a+ c K I,a+ c = 0 (6.17) Ths last equaton serves as a condton to determne the length of the eld zone. Thus, ntroducng the epressons for the SIFs n (6.17) one easl obtans, a = cos π a c + (6.18) The last equaton ndcates that when the length of the eld strp c. When <<, eq. (6.18) can be smplfed b epandng t n a Talor s seres and keepng the frst two terms. Therefore, π a π K I(,a) c = = 8 8 (6.19) March 006 6-8
Interestngl, the predctons of the Irwn s analss and that of Dudgale s are ver close. Thus both models predct smlar plastc zone szes. 6. Crack Tp Openng Dsplacement as eld Crteron For lnear elastc fracture mechancs we have developed eld crteron such as K and G. We now develop smlar eld crteron for the case II, contaned eldng, of Fgure 6.. An ntal crteron was proposed b Wells (Anderson, 1995) who used the crack tp openng placement (CTOD) at the crack tp as a measure of the amount of plastc deformaton. Hs work was motvated after he attempted to measure K IC values for a seres of steels. He found that LEFM was not suffcent because the crack tp had blunted consderabl before the crack propagated. Thus, the plastc deformaton was not neglgble. Fgure 6.8 Irwn plastc zone correcton for blunted crack tp. One can estmate the sze of the CTOD δ t, usng the Irwn plastc zone correcton. Ths correcton s shown n Fgure 6.8. In order to model a blunted crack, the length of the crack s etended from a to r + a, as shown n Fgure 6.8. The crack openng u, at a pont along the center lne of the crack s gven b (4.40d) for θ = 0, KI r θ θ KI r u = sn κ+ 1-cos = µ π µ π θ=π For plane stress, the CTOD s then epressed as, ( ) κ+ 1 (6.0) 4K r I δ t = u = r = r µ ν+ 1 π ( ) (6.1) From the Irwn plastc zone correcton, the radus of the plastc zone s, r 1 K I = π (6.) March 006 6-9
where s the eld stress of the materal. Substtutng eq. (6.) nto eq. (6.1) we fnd, KI 4 δ t = µπ 1 + ν And fnall, usng E = µ (1 +ν), 4KI 4G δ t = = Eπ π (6.3) The CTOD can thus be related unquel to K I and G, allowng one to determne the crtcal CTOD for crack propagaton for eample (.e. δ t at K Ic ). However, the CTOD s often dffcult to measure wthout large errors. In the net secton, a more general crteron s presented, based upon the energ released durng crack growth. 6.3 The J ntegral as eld crteron 6.3.1 Deformaton theor of plastct In ths secton we want to derve a fracture crteron for an elastc-plastc materal whch has a stress-stran behavor as shown n Fgure 6.9. For small loa, the stress-stran curve s lnear (1). Beond a crtcal load, the curve s non-lnear (). As the load s released, the stress-stran curve s once agan lnear (3), wth the same slope as durng the ntal loadng (1). The major dffcult n modelng ths materal s that the behavor s no longer reversble (.e. the stress-stran curve does not follow the same path for loadng and unloadng). Thus, the materal behavor s now dependent upon the stran hstor and the stress at a gven stran s no longer unque. However, ths effect does not become mportant unless unloadng occurs. Fgure 6.9 Stress-stran curves for (a) elastc-plastc and (b) non-lnear elastc materals. Thus, untl unloadng, the stress-stran curve s dentcal to that of a non-lnear elastc materal, also shown n Fgure 6.9. Note that the non-lnear elastc curve s reversble. March 006 6-10
Therefore, for the dervatons that follow, we wll replace the elastc-plastc behavor b the nonlnear case, specfng that we consder onl monotonc (non-cclc) loadng. Ths substtuton s called the Deformaton Theor of Plastct. Note that ths substtuton s not alwas vald for a trul 3D case of loadng, however t s vald for several cases. 6.3. Defnton of the J-ntegral In an earler chapter, the lnear elastc energ release rate G, was derved and shown to be the area under the load-placement curve for a gven bod between two load states. Smlarl, one can defne the nonlnear elastc energ rate J, as the area under the load-placement curve between two load states. Ths area s shown n Fgure 6.10. The bod s loaded followng the nonlnear curve untl the pont B. At ths pont the load s P and the appled placement u. Afterwar, the placement s ncreased to u+δu, or the pont B' n Fgure 6.10. At ths pont the load has decreased untl P-δP. The loadng s then removed, followng the nonlnear curve shown. The energ release rate between the ponts B and B' s gven b the area between the two curves. Fgure 6.10 Load-placement curve of non-lnear elastc materal. The shaded area gves energ, released from state B to state B', defned as J. In the same manner as G, J s used as a fracture crteron, meanng that for a gven materal at a certan crtcal value, J c, the crack wll propagate. Note that n the case of lnear fracture, J c reduces to G c. In practce, the nonlnear energ release rate J, s calculated usng the J-ntegral, wrtten as, J= wd T (6.4) where W s the stran energ, defned b, ε w = dε 0 j j and the T are the boundar tractons, gven b, March 006 6-11
T = n j j The contour over whch the ntegral s evaluated s shown n Fgure 6.11. The - aes are defned to have ther orgn at the crack tp and are n the drecton shown. The contour s evaluated n the counter-clockwse drecton, startng on one crack face and endng on the other crack face. It wll be shown n the net secton that the choce of s arbtrar. The normal to the contour n and the ncrement along the contour, are also shown n Fgure 6.11. As mentoned above, the ntegral epresson of eq. (6.4) defnng J s equvalent to the area under the curve of Fgure 6.10,.e. the nonlnear elastc energ release rate. Ths can be derved as shown n the followng. T n Fgure 6.11 Contour J- ntegral to calculate nonlnear energ release rate for elastc plastc fracture. The potental energ of a bod (wth or wthout a crack), n two dmensons, s epressed as, Π= wda A T u The frst ntegral s the stran energ of deformaton and the second on the potental energ of the appled loa. The change n potental energ due to a vrtual etenson of the crack da s then gven b, Π = a w a da A da T (6.5) As the crack s etended b da, the coordnate aes are also moved b da. Thus, a = and March 006 6-1
Π = + a w d A da T (6.6a) Applng the dvergence theorem on the frst ntegral one obtans, Π w = + = + a d da A da T wn T Therefore, J = wn T = wn T d d Π J= = wd T a (6.6b) Thus, the J-ntegral defned n eq. (6.15) s equvalent to the nonlnear energ release rate. When the materal s lnear elastc, the ERR GI s equal to J. Ths s evdent b comparng (3.34) and (6.5). 6.3.3 Propertes of J-ntegral The contour ntegral J has two mportant propertes, each of whch has a phscal sgnfcance. These propertes are, 1. J ntegrated along an closed contour s zero. Ths means that a contnut (such as a crack) whch nterrupts the contour s necessar for J to be non-zero. Thus J s trul a measure of the energ spaton strctl due to the crack.. J s path ndependent. Thus, the choce of the contour to calculate J s not mportant. One can choose a contour whch s convenent for calculaton purposes. Proof of propert 1 Consder the closed contour whch encloses the area A, shown n Fgure 6.1. Usng the defnton of J, we evaluate J along the contour. 1 w = = J wd T dd T A (6.7a) Net use the Cauch s formula T =jn j appl the dvergence theorem to the second term, 1 The notaton J ndcates the ntegral J evaluated along the contour. March 006 6-13
T = n = dd j j j A j (6.7b) Thus, J w = j dd A j (6.8) The frst term of eq. (6.8) s wrtten as, W w ε εj ε j = =j j (6.9) where, 1 j ε j = + j Thus, W j u j = + j (6.30) A Fgure 6.1 The closed contour enclosng the area A, for the evaluaton of J. Usng the smmetr of the stress feld,.e. j = j, we can smplf (6.30) to, w = j =j j j (6.31) March 006 6-14
Assumng zero bod forces and applng the condton of equlbrum wrte, j / = 0, one can j j j j j j = + = j j j (6.3) Substtutng eq. (6.3) nto (6.31) we have, w = j j (6.33) Replacng the frst term of eq. (6.8) lea to, J d d = j j A j j Therefore, J = 0 (6.34) Proof of propert Consder the two arbtrar contours 1 and, shown n Fgure 6.13(a). We want to show that J, evaluated on each of the two contours, s equvalent,.e. J = J. For ths we defne the 1 closed contour, shown n Fgure 6.13(b). The contour s composed of four ndvdual contour * segments = 1 3 4. Usng the propert 1, one can wrte (a) 1 (b) 3 1 4 * Fgure 6.13 (a) Independent contours 1 and for the evaluaton of J; (b) closed contour contanng 1 and. March 006 6-15
J = J + J + J + J (6.35) * 1 3 4 Note that, J = J (6.36) * as the drecton of the contour has been reversed to form the closed contour equalt and that the contour s closed one can wrte,. Usng the last J = J J + J + J = 0 1 3 4 (6.37) Consderng that the crack faces are tracton free, the contrbutons of the contours one has, and 3 4 T = jnj = 0 on and 3 4 In addton, the contours 3 and 4 are n the d drecton, (6.38) d = 0 on 3 and 4 (6.39) Applng eqs. (6.38) and (6.39) to eq. (6.37), we fnd J = J = 0 (6.40) 3 4 Fnall, substtutng eq. (6.40) nto eq. (6.37) gves, J = J 1 (6.41) 6.3.4 J-placement dagrams An elastc-perfectl plastc materal s a materal wth a load-placement behavor shown as a sold lne n Fgure 6.14(a). We consder ths case frst because t s the easest tpe of plastc behavor to model. The materal behaves lnear elastcall untl the crtcal placement u 0. At ths pont, the materal contnues to deform under constant load. The J-placement curve for ths materal s also shown as a sold lne n Fgure 14(b). In the elastc regon J s equvalent to G whch s proportonal to the placement squared. Once the materal s perfectl plastc, J s then proportonal to the placement. One can then consder the non-lnear elastc load-placement curve, shown as a dashed lne n Fgure 6.14(a), as a devaton from the elastc-perfectl plastc curve. When ntegrated, the correspondng J-placement curve s shown as a dashed lne n Fgure 6.14(b). Thus, n the lmts of small placements J u, and large placements J ~ u. However, n between these lmts the curve depen on the non-lneart. March 006 6-16
Fgure 6.14 (a) Load-placement and (b) J-placement curves for an elastc-perfectl plastc materal. Dashed lnes show equvalent load-placement and J-placement curves for a non-lnear elastc materal. 6.4 Stress Feld Near Crack Tp 6.4.1 HRR Theor As for the elastc case, t s useful to derve the form of the stress feld near the crack tp for elastc-plastc fracture. Ths secton presents the most commonl used descrpton of the stress feld. Hutchnson, Rce and Rosengren (HRR) (Anderson, 1995) assumed a power law relatonshp between the plastc stran and stress, ε = +α ε0 0 0 n (6.4) where s the eld strength, a s a dmensonless constant, ε 0 = 0 E, and n s the stran 0 hardenng eponent. Thus α and n are materal propertes. For a lnear elastc materal, n = 1. For a stran hardenng materal n > 1. The materal followng (6.4) s called a Ramberg-Osgood materal. Now, for J to reman path ndependent, t s necessar that the stress feld var as l/r near the crack tp. Thus, ( + ) ( + ) 1 n 1 n n 1 J J j = A1 ; ε j = A r r (6.43) where A 1 and A are constants. The sngular stress-stran feld of eq. (6.43) s called the HRR sngulart. Note that for a stran hardenng eponent n = 1, we recover the lnear elastc case., ε ~ r 1 j j (6.44) Thus a cracked bod can have two sngulart domnated zones. In the plastc zone, closest to the 1 / (n+ 1) crack tp, the stress vares as r (the HRR sngulart). In the elastc zone, surroundng the plastc zone, the stress vares as 1 r. March 006 6-17
As seen n eq. (6.43), J defnes the ampltude of the HRR sngulart, just as K does for the lnear elastc case. Thus, J s an effectve stress ntenst factor for elastc-plastc fracture. Fgure 6.15 shows a graph of a tpcal stress feld due to elastc-plastc fracture, as compared to the HRR theor and lnear elastc fracture mechancs. Also shown n Fgure 6.15 s a small regon near the crack tp where crack bluntng occurs (large stran regon). As before n the lnear elastc case, we gnore ths regon n assumng that t does not affect the global behavor of the regons surroundng the crack tp. The HRR theor s based upon small placements whch s not applcable n ths regon. To stud ths regon, one would need to appl a large deformaton analss, for nstance usng fnte elements. Fgure 6.15 (a) Sngulart domnated regons near crack tp n elastc-plastc fracture; (b) stress feld near crack tp, dashed lnes show stress calculated from LEFM and HRR theores. March 006 6-18
6.5 Applcatons of the J-ntegral 6.5.1 The J-ntegral along a specfc contour To llustrate the use of the J-ntegral, consder frst a cracked specmen loaded n Mode I and crack free surfaces. Wthout the need to know the form of the loadng, we can proceed to determne the eplct forms of the J-ntegral along a smmetrc contour ABCDC B A shown n Fgure 6.16. The contour shown can be ether along the border of the specmen or the contour around the crack n a larger specmen or component. The crack faces A O and AO are tracton free. It s also understood that the chosen contour can be chosen along the boundares of the specmen or n t nteror. B C T n n A A O D n B C Fgure 6.16 A smmetrc contour for the calculaton of J. For 1-dmensonal crack growth, along -as, = J wd- jn j,j =, (for a plane problem) (6.45) Here W s the stran energ denst, j s the stress tensor, n s the outward normal to the element of of the contour and u s the placement. The stress vector and the outward normal are related b the followng relaton, T= jn (6.46) j where T s the stress vector. We further assume smmetr of the specmen and loa wth respect to the -as. For a plane stress, lnear elastc problem, the stran energ denst s gven b, 1 1 w= dε = ε= ε + ε + ε j j j j ( ) (6.47) March 006 6-19
Usng the stress-stran relatons for plane stress (eqs.15), W s epressed as, 1 1+ν w= ( + -ν ) + E E (6.48) The components of the outward normal defned as, n = cos(,n ), n = cos(,n ) are determned when d and d are postve,.e., on contour BC (d>0) and CD (d>0). Thus, d d = n = 1, = n = 1 Net, one can calculate each term of the ntegral (6.45) along the chosen contour shown n Fgure 6.16. In ths eample, W s consdered known and we deduce the second epresson of (6.45), jnj =n +n +n + n (6.49a) or, jnj = d- d+ d- d (6.49b) Therefore, for each part of the contour the result s, Along AB or A B d = 0 and d 0 jnj = d+ d Along BC or B C d 0 and d = 0 jnj =- d- d Along CD or C D d = 0 and d 0 jnj = d+ d Fnall, along AO or A O the value of J s zero snce d = 0 and T = 0. Recallng the propertes of the lne ntegrals and the smmetr of the contour one can wrte, B C D J= w- - d+ + d+ w- - d A B C (6.50) March 006 6-0
It s mportant to notce that the stresses n (6.50) are the tractons along the contour and related to eq. (6.46). 6.5. Eample The double cantlever beam of unt thckness wth a>>h s loaded under constant placement, along the two horzontal borders as shown n Fgure 6.17. Determne the J-ntegral along the u contour OABCDEO. Note that the crack tp s not ncluded n the contour. To smplf the calculatons, t s assumed that the materal above and below the crack s stress free and the rest of the specmen s subjected to a constant stress correspondng to a vertcal placement stran ε= u /h. u E D F A a O h B C u B Fgure 6.17 Double cantlever beam subjected to remote vertcal placement wth the contour of ntegraton. Soluton Contour OA It s free of tracton T = 0 and d = 0. Therefore JOA = 0. Contour OF For the same reasons as before J = 0. Contours AB OD It s tracton free T = 0. It s further assumed that due to the presence of a long crack these parts of the specmen (above and below the crack) are unloaded and thus w = 0. Therefore J = AB 0. Contour EF For the same reasons as before J = 0. EF March 006 6-1
Contour BC Wth these values, (6.49b) results n, jnj =n + n = d+ d Snce there s no shear stress on ths contour = 0. Also / =0because the vertcal placement there u s constant. In addton. d=0 along the contour and wd=0. Therefore, J = 0. BC Contour DE For the same reasons as before, J = 0. DE Contour CD It s free of tracton T = 0. It s assumed that the state of stress far ahead of the crack tp s unform and the plate s thn wth a constant stran energ w. Thus, CD D +h (6.51a) J=J = wd = w d = hw C -h The eact form of the w depend on the state of stress (plane stran or plane stress). - Note that the stress-stan relatons could be lnear or non-lnear elastc. - The contours OA and OF could be elmnated from the start snce T = 0 and d = 0. Assumng lnear elastc behavor, plane stran and the lgament sze ver long, one can wrte, 1 1 (1-ν)E u W= ε (1+ν)(1-ν) h = Here the stress component gven b (4.3b) has been recalled and the Posson effect has been neglected. Therefore, (1-ν)Eu J=hw= (1+ν)(1-ν)h (6.51b) 6.5.3 Eample Consder the double cantlevered beam specmen shown n Fgure 6.18. The beam s loaded b dpole forces P, appled on the end faces. Wth the smple beam theor and assumng that a >> h, evaluate the energ release rate as a functon of crack length a usng, March 006 6-
(a) the complance method (b) the J-ntegral method usng a contour along the specmen boundar as shown b the arrows n Fgure 6.18. To smplf the calculatons, t s assumed that the materal above and below the crack s subjected to bendng due to the force P appled at the center of gravt of the secton hb wth the correspondng placement. The other part of the specmen s assumed stress free wth w=0. u P, u E D D A a h P, u B C B Fgure 6.18 Double cantlevered beam subjected to end loa wth the contour of ntegraton. Soluton (a) Complance method The energ release rate G, s gven b, 1 C 1 C G = P = P A B a (6.5) From beam theor (Del Pedro, Gmür & Botss, 001), P Bh u = + a I= EI 6 1 u = a 3 3 3 3 3 Pa 4a = = 3EI EBh P (6.53) The total complance (due to both beams), u 8a 3 C = 3 P = EBh (6.54) Substtutng (6.54) nto (6.5) the G s, March 006 6-3
1 4a 1P a G = P = (6.55) 3 3 B EBh EB h (b) J-ntegral method Snce the materal s lnear elastc, J = G. Thus, we can use the J-ntegral to calculate G. Dvde the contour shown nto 5 segments, as numbered n Fgure 6.18. We want to evaluate, J= wd jnj (6.56) Where s the contour ABCDEF. Contours BC or DE T = 0 and d=0. Thus, JBC = JDE = 0 Contour CD T = 0 and W=0 due to the assumpton that the part of the beam far ahead of the crack tp does not deform. Thus, J = CD 0 Contour AB or EF Consder frst the term wth the W. To evaluate the ntegral along these contours we proceed as follows P = = 0,, hb 1 ε = + 1 Thus w= jdε j= ε= j j ε To determne the shear stran ε of a cantlever beam (Del Pedro, Gmür & Botss, 001) to obtan, one can use the placement feld from the theor of elastct 1, / ε = + = = a = h 0 Therefore, w=0. Now, consder the term, on AB or EF, March 006 6-4
=n +n = d+ d jnj =n +n +n +n d The contrbuton of each arm of the specmen s calculated as follows. From beam theor one obtans the vertcal placement of the gravt center of the secton hb (Del Pedro, Gmür & Botss, 001), s For the upper arm For the lower arm P = 0;, HB n P HB = j j j = a P = 0;, HB n P HB = j j j = a u P a = + EI 6 3 u P a = + EI 6 3 P Pa = + a ; = EI EI = a P Pa = + a ; = EI EI = a j jn j = Pa EIHB j jn j = Pa EIHB Fnall, j Pa Pa Pa jnj = d d= IEhB IEhB EIB h 0 0 h or u 1P a j jnj = and 3 EB h J 1P a = (6.57) 3 EB h Comparng the result wth the epresson (6.55) t s clear that the two metho above are equvalent means to calculate the energ release rate. March 006 6-5
6.4.4 Relatonshp between J and CTOD A smple et mportant relatonshp can be derved between the CTOD and the J-ntegral for the eld strp model. The bases of the calculatons are shown n Fgure 6.19. δ t a c Fgure 6.19 The J-ntegral contour for the crack and plastc zone of the eld strp model. The contour s chosen around the crack tp as shown n Fgure 6.19. Note that the sngulart s cancelled due to the presence of the plastc zone. Therefore, the onl stresses appled are the remote stress and the eld stress at the crack faces over the tance c. To calculate the J-ntegral along the contour t s notced that when c >> δt, d 0. Accordngl, ( 6.45) reduces to, J = n = n j j j j u Wth the onl non-zero component of stress beng =, the remanng of the ntegrand becomes, and = = jn j n d J= d = du = du = δ δt / δt t, δt / 0 J =δ (6.58) t March 006 6-6
In comparng (6.58) and (6.3) one sees that the latter result s hgher b about 5%. For a lnear elastc materals, or for small scale eldng (case 1 n Fgure ), the J-ntegral, ERR, SIF and the CTOD are equvalent parameters. K J= G I = =δ t (6.58) I E ' If the materal cannon be appromated b a perfect plastc materal, a smlar relaton can be obtaned between J and δ b usng the Ramberg-Osgood materal (6.4) (Anderson 1995), t J= n δ t where s the eld stress and n s the stran hardenng eponent n (6.4). 6.4.5 The J-ntegral along a rectlnear nterface Consder two homogeneous bodes Π 1 and Π whch are perfectl bonded along the as as shown n Fgure 6.0. a bomateral nterface and a crack along t (Fgure 6.0). The ntegral along the contour = epressed b (6.45) s, 1 J= wd- jn j,j =, (for a plane problem) (6.45) Π 1 Π 1 Fgure 6.0 A crack along a b-materal nterface and the contours for the J-ntegral. It s also path ndependent provded that the nterface lne s straght. Thus, the standard J- ntegral s etended to a bomateral nterface when the latter s straght. Its nterpretaton s also March 006 6-7
as the energ release rate durng crack advance. For a curvlnear crack the J-ntegral s not path ndependent because the conservaton law gven b (6.34) s not vald. R. E. Smelser and M. E. Curtn, On the J-ntegral for b-materal bodes, Internatonal Journal of Fracture, Vol. 13, 1977, pp. 38-384. March 006 6-8