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2 Engineering Frcture Mechnics 77 ( Contents lists ville t ScienceDirect Engineering Frcture Mechnics journl homepge: Effect of residul stresses on ductile crck growth resistnce X.B. Ren, Z.L. Zhng, *, B. Nyhus Deprtment of Structurl Engineering, Norwegin University of Science nd Technology (NTNU, N-749 Trondheim, Norwy SINTEF Mterils nd Chemistry, Deprtment of Applied Mechnics nd Corrosion, N-7465 Trondheim, Norwy rticle info strct Article history: Received 9 Octoer 9 Received in revised form 9 Ferury Accepted 5 Mrch Aville online 8 Mrch Keywords: Residul stresses Ductile crck growth resistnce Complete Gurson model Eigenstrin method It is well known tht residul stresses influence the ductile frcture ehviour. In this pper, numericl study ws performed to ssess the effect of residul stresses on ductile crck growth resistnce of typicl pipeline steel. A modified oundry lyer model ws employed for the nlysis under plne strin, Mode I loding condition. The residul stress fields were introduced into the finite element model y the eigenstrin method. A shrp crck ws emedded in the center of the weld region. The complete Gurson model hs een pplied to simulte the ductile frcture y microvoid nucletion, growth nd colescence. Results show tht tensile residul stresses cn significntly reduce the crck growth resistnce when the crck growth is smll compred with the length scle of the tensile residul stress field. With the crck growth, the effect of residul stresses on the crck growth resistnce tends to diminish. The effect of residul stress on ductile crck growth resistnce seems independent of the size of geometriclly similr welds. When normlized y the weld zone size, the ductile crck growth resistnce collpses into one curve, which cn e used to ssess the structurl integrity nd evlute the effect of residul stresses. It hs lso een found tht the effect of residul stresses on crck growth resistnce depends on the initil void volume frction f, hrdening exponent n nd T-stress. Ó Elsevier Ltd. All rights reserved.. Introduction It is widely ccepted tht the presence of residul stresses cn influence the filure chrcteristics of components nd structures. A significnt influence due to the residul stress on the filure nd lod ering cpcity of components under predominntly elstic conditions ws demonstrted y Ainsworth et l. [], who lso showed tht the effect on lod-ering chrcteristics reduced with the incresing plsticity within the component. A quntittive understnding of how residul stresses ffect the filure ehviour is thus required to provide ccurte integrity ssessment nd reduce the conservtism inherent in mny ssessment codes. Ductile crck growth plys n importnt role in the nlysis of the frcture ehviour of structures []. Crck extension reduces the lod-ering ligment nd will influence the cpcity of the structures. Ductile crck growth my lso chnge the ner-tip stress/strin fields nd promote the trnsition to unstle clevge frcture. The mechnism of ductile frcture in metllic mterils my e considered s three-stge process: nucletion, growth nd colescence of microvoids. The ductile frcture process is influenced y the locl conditions of stress trixility nd plstic strin within the vicinity of stress concentrtor such s notch or crck-tip [3]. Liu et l. [4] nd Ren et l. [5] hve demonstrted tht residul stresses cn induce n dditionl crck-tip constrint, nd prmeter R ws defined sed on the difference etween the full stress * Corresponding uthor. Tel.: E-mil ddress: zhiling.zhng@ntnu.no (Z.L. Zhng /$ - see front mtter Ó Elsevier Ltd. All rights reserved. doi:.6/j.engfrcmech..3.7

3 36 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( field nd the reference field to quntify the residul stress-induced crck-tip constrint. It is thus interesting to further investigte how residul stresses influence the locl filure mechnisms nd glol ductile crck resistnce. Pnontin nd Hill [6] predicted the rittle nd ductile initition y micromechnicl models nd showed tht the effect of residul stress on the ductile frcture initition toughness is negligile. Sherry et l. [3] demonstrted tht high strength low toughness luminum lloy AL4-T35 showed mrked reduction in initition nd tering toughness for specimens contining mechniclly induced residul stress field. It should e noted tht the frcture ehviour of AL4-T35 remins in ductile frcture regime. Experimentl work undertken y Shrples et l. [7] on wide plte specimens hs lso demonstrted tht residul stress cn significntly influence the ductile tering ehviour of engineering mterils. However, the experiments performed y Mirzee-Sisn et l. [8] on the AISI Type 36H stinless steel indictes negligile impct on ductile tering toughness t lod rtio L r close to, i.e. close to the plstic collpse of the specimen. The oservtion is in contrst to their previous experiments performed on the A533B unstle clevge frcture in ferritic steels t 5 C [9]. A reduction of pproximtely 46% ws oserved in pprent frcture toughness for specimens contining residul stress field compred with specimens in the s-received condition. Mhmoudi et l. [] employed locl out-of-plne compression method to introduce residul stresses into C(T specimens for luminum lloys A65 nd A4. Their experimentl results showed tht tensile residul stresses reduced the initition clevge frcture toughness of A65 y out one hlf, nd the ductile tering resistnce of A4 lso decresed when tensile residul stresses present. To dte, the fundmentl understnding of the effect of the residul stresses on ductile frcture resistnce remins n open chllenge. To this end, we utilized micromechnicl dmge model in Modified Boundry Lyer (MBL model to investigte the prolem. The MBL model hs een widely used to investigte the crck prolems [], nd the trnsferility to rel components cn e well hndled y simply chnging the oundry conditions. In recent yers, dmge mechnics models hve ecome widely used in numericl simultion of ductile crck growth. The Gurson model [] is often used to ssess ductile tering ehviour under primry loding, where the effect of the void growth is tken into ccount in the constitutive model. The Gurson model hs lter een modified y Tvergrd nd Needlemn [3 5], thus, it is most often referred to s the Gurson Tvergrd Needlemn (GTN model. The GTN model is in fct void growth model. The effect of void colescence cn e considered with the introduction of so-clled criticl void volume frction, which is not physicl mechnism-sed colescence criterion. Thomson [6] proposed plstic limit lod model for colescence, which ws esy to implement nd very ccurte [7,8] compred with the finite elements results y Koplik nd Needlemn [9]. This work hs een further improved y considering the plstic strin hrdening []. By comining the GTN model for void growth nd Thomson s plstic limit lod model for colescence so-clled complete Gurson model hs een proposed y Zhng et l. [], with which ductile frcture is exclusively linked to the void nucletion prmeters nd the mesh size. Two scenrios hve een considered to study the effect of the residul stresses on the crck growth resistnce in this pper. Firstly, the effect of residul stresses on ductile crck resistnce in lrge round weld region ws investigted. In such wy, the finl crck growth length still loctes in weld region. Secondry, much smller rectngulr weld region ws constructed, where the crck cn propgte through the whole weld region. The effect of the weld zone size ws lso tken into ccount nd studied. In ddition, the effects of initil void volume frction nd mteril hrdening hve lso een studied.. Numericl procedure.. Prolem description This study concerns n idel prolem. A lrge cylinder with weld in the center ws studied. The cylinder ws simulted y D plne strin MBL model with remote oundry governed y the elstic K-field nd T-stress. A shrp crck ws emedded in the weld region. The nlysis procedure, s illustrted in Fig., consists of the following steps: ( enforce welding procedure in un-crcked ody, which introduces residul stress field; ( introduce shrp crck; nd (3 pply the externl lod. The contct etween the upper nd lower free surfces of the crck is considered when the residul stress ws introduced. Fig.. Illustrtion of the prolem. ( The round cylinder; ( welding t the center nd shrp crck is introduced; nd (c pply the externl lod.

4 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( Fig.. Modified oundry lyer model, ( glol view nd ( crck-tip mesh... Finite element model The present nlyses were crried out for the conditions of smll-scle-yielding. The MBL model used for this study consists of weld region locted in the center of the model, n outer se metl region, nd shrp crck in the center of weld. The lod ws pplied to the remote edges of the model through displcement field (u, v controlled y the elstic symptotic stress field of plne strin Mode I crck: þ m r uðr; hþ ¼K I E p rffiffiffiffiffiffi þ m r vðr; hþ ¼K I E p p where K I ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EJ=ð m Þ rffiffiffiffiffiffi cos ð3 h 4m cos hþþt sin h ð3 4m cos hþ T m r cos h E mð þ mþ r sin h E under plne strin condition, E is Young s modulus, m is Poisson s rtio, nd r nd h re polr coordintes centered t the crck tip with h = corresponding to the crck tip. The finite element computtions were performed using ABAQUS []. Due to symmetry, only the upper hlf of the geometry is modeled. The mesh is shown in Fig.. The rdius of the MBL model is tken s mm to ensure the smll-scleyielding condition is fulfilled. Close to the crck tip, there is rectngulr region (9.4 mm hed of the initil crck tip nd.6 ove the symmetry line with uniform mesh sizes of..5 mm for the first lyer nd.. mm for the rest of the lyers re creted. In following context, l c =. mm represents the size of the uniform elements. Full integrtion four-node D plne strin elements re used. The finite element model hs 66 elements. Nonliner geometry effects (NLGEOM in ABAQUS re ccounted for in the nlyses. An initil opening of. mm is pplied for the upper hlf model. When the residul stresses re introduced into the model, the crck fces cn e prtly closed. Thus, rigid nlyticl plne is defined in the model to simulte the contct of the crck surfces..3. Mterils As descried in Section, the complete Gurson model is utilized to simulte the crck growth, which is implemented into ABAQUS using set of generlized-midpoint lgorithms developed y Zhng vi the mteril user suroutine UMAT [3 5]. Both the weld metl nd se metl re modeled s power-lw hrdening mteril: ðþ A free copy of the ABAQUS UMAT Fortrn code cn e otined from the corresponding uthor.

5 38 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( r f ¼ r þ e p n ðþ e where r f is the flow stress, e p is the equivlent plstic strin, the Poisson s rtio m is.3, Young s modulus is, MP, yield stress r is 4 MP, e = r /E is the yield strin nd n is the plstic strin hrdening exponent. The yield function of the Gurson model hs the following form: /ðq; r; f ; r m Þ¼ q þ q r fcosh 3q r m r f ðq f Þ ¼ where f is the void volume frction, r m the men mcroscopic stress, q the von Mises stress, r f the flow stress, nd q nd q re prmeters introduced y Tvergrd [3,4]. Fixed vlues of q =.5 nd q =. hve een used in this study. In this study, cluster void nucletion model hs een used []. The criticl void volume frction (f c is utomticlly determined y Thomson s plstic limit lod criterion [6,]. The post-colescence deformtion ehviour of the Gurson model is numericlly simulted y n rtificil ccelertion of void growth, s suggested y Tvergrd nd Needlemn [5]: ( f ¼ f for f 6 f c f c þ f u ð4þ fc f F f c ðf f c Þ for f > f c where f u ¼ =q nd f F is the void volume frction t the end of void colescence. Here, f F =.5 + f, where f is the initil void volume frction. When the colescence strts nd f > f c, f * replces f in Eq. (3. ð3þ.4. Eigenstrin method Different therml expnsion coefficients of the se metl ( nd weld metl ( w were used to introduce the residul stresses into the model y the so-clled eigenstrin method. It should e noted tht the therml expnsion coefficients here re not physicl therml coefficients, ut re used to introduce residul stresses into the FE model. The eigenstrin method ws lso clled inherent strin method when first introduced y Ued et l. [6]. The concept of the eigenstrin method is tht the source of residul stress is n incomptile strin field cused y plstic deformtion, therml strins nd phsetrnsformtion etc. [7]. Thus, if the distriution of the eigenstrin is known, the distriution of residul stresses cn e otined through liner elstic clcultion y using the finite element method. Our pproch is to set the eigenstrin vlues equl to the therml expnsion coefficients for different regions. The model is then loded y pplying unit temperture decrese, therey introducing the residul stress field of interest. Similr pproches were lso pplied in the literture [8,9], nd their results showed tht the distriutions of residul stresses otined y such pproches greed well with the experimentl results..5. The crck driving force In frcture mechnics, the J-integrl [3] or Crck Tip Opening Displcement (d, CTOD re used to chrcterize the crcktip driving force of crcked ody mde of n elstic-plstic mteril. The J-integrl is pth-independent integrl sed on the ssumption tht the strin energy density is single-vlued function of the strin (or stress [3]. However, non-proportionl loding my occur in the region where the J-integrl is evluted nd led to pth-dependence, for exmple, in the cse of ductile crck growth [], or in the presence of residul stress field [3]. In this study, the J-integrl is evluted on contours fr wy from the zone of highly non-proportionl loding, nd displys prcticlly pth independence in oth cses with nd without residul stresses. Residul stresses s n dditionl stress field induce n initil J-integrl. However, the residul stress-induced J-integrl is negligile compred to the finl J-integrl cused y the comintion of residul stress nd externl lod. Also, under the smll-scle-yielding conditions, the difference etween clculted J-integrl nd pplied J-integrl is less thn.78% for the cse without residul stress, nd 3.5% for the cse with residul stresses [5]. Thus, J -integrl should e pplicle in the present study, nd the J in the following context represents fr-field J-integrl. 3. Results nd discussion 3.. Residul stress fields As descried in Section.4, we used the eigenstrin method to introduce residul stresses into the finite element model. A round weld region ws introduced in the center of the model, s illustrted in Fig. 3. According to the eigenstrin method, the therml expnsion coefficients of oth the weld metl ( w nd se metl ( were ssumed to e isotropic nd equl to the eigenstrin vlues respectively. In this section, we ssume =. Four residul stress fields were generted y setting w =-.,.,., nd.3 nd designted s RsField, RsField,, nd respectively. Fig. 4 shows the distriution of the residul stresses oth efore nd fter the crck ws inserted. Note tht the stresses re normlized y the yield stress, nd the distnce from the crck tip x is normlized y the size of the uniform element l c.

6 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( Fig. 3. Illustrtion of the weld region (gry re. The rdius of the weld is mm. σ /σ 3 RsField crck RsField RsField crck RsField crck crck σ /σ 4 3 RsField crck RsField RsField crck RsField crck crck x/l c x/l c Fig. 4. Comprison of the residul stress distriution in MBL model with lrge round weld efore (old line nd fter (thin line the crck ws inserted, ( components prllel to the crck plne nd ( norml to the crck plne. Four different residul stress cses were considered, where RsField is compressive nd the remining 3 re tensile. =; w =.,.,., nd.3; l c =. mm. It cn e seen tht the negtive eigenstrin vlue introduces the compressive residul stress t the weld region while the positive ones generte tensile residul stresses. Both tensile nd compressive residul stresses prllel to the crck front converge to zero fr from the crck tip. The opening residul stresses re self-lnced hed of the crck tip. There is shrp turning point in the distriution of the opening residul stress, which is the region where eigenstrin discontinuity hs een introduced into the FE model, nmely the weld metl-se metl oundry. The tensile residul stresses lso show similrity, nd the level of the tensile residul stress increses with the incresing w. Due to the singulrity, r is out 8 MP nd r is out MP t the crck tip for. Fig. 4 lso shows tht residul stress components re smller thn the yield stress efore the crck ws inserted. 3.. Effect of residul stresses on ductile crck resistnce Ductile crck growth resistnce is importnt for structurl integrity ssessment, nd it is interesting to investigte the effect of residul stresses on it. In this section, effects of residul stress fields on the crck growth resistnce were studied. The initil void volume frction f is fixed to e.% nd the strin hrdening exponent n =.. The crck growth resistnce descried y the J-integrl is showed in Fig. 5. For the cses chosen, the residul stresses seem to hve significnt effects on the ductile crck growth resistnce. Fig. 5 shows tht the compressive residul stress enhnces the crck growth resistnce while the tensile residul stresses hve the

7 33 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( J/(σ l c 5 5 J/J without without residul stress RsField RsField 5 no/res RsField RsField Fig. 5. Effect of residul stresses on crck growth resistnce, ( solute crck resistnce curves nd ( normlized crck resistnce curves. f =.%; n =.; E/r = 5. Fig. 6. Illustrtion of the weld region (gry re. The length of the weld region long the direction is 9.6 mm, nd the length long the direction is 6.4 mm. opposite effect. With the increse of tensile residul stress, the crck growth resistnce decreses. The solute difference etween the cse with nd without residul stress increses with the increse of crck growth. Fig. 5 presents the normlized crck growth resistnce. The resistnces with residul stresses were normlized y the resistnce t the sme crck growth without residul stresses. The distnce from the originl crck tip ws normlized y l c. It cn e seen tht the tensile residul stresses significntly decrese the initition toughness while the compressive residul stress increses it. With dvncing crck growth, the effect of the residul stresses decreses nd pproches to constnt vlue. Note tht the current crck grows only to 3 mm, which is very smll compred to the size of the residul stress dominnt length scle shown in Fig. 4. As shown in Fig. 4, eyond the singulrity ffected zone (x/l c > 3, i.e. 3 mm, the residul stress fields pproch hydrosttic stress stte. Thus, the residul stresses cnnot e esily relesed y the crck growth, nd the effect of residul stresses retins. In order to etter understnd the effect of residul stress on crck growth resistnce, smller rectngulr weld region ws constructed, s illustrted in Fig. 6. In this section, =, nd w ws ssumed to e orthogonl nd chrcterized y nd in following context. The rtio / ws fixed to e. Four residul stress fields with =-.,.,., nd.3 were generted nd represented y RsField, RsField,, nd respectively, s shown in Fig. 7. It should e noted tht the eigenstrin vlues selected here re tken from experimentl mesurement results in literture [33,34]. The residul stress fields generted y the selected vlues hve similr distriution s the residul stress fields showed in Ref. [35]. To otin ccurte distriutions of the residul stress fields y the eigenstrin method, one should crry out the experiments to mesure the distriution of the eigenstrin. However, the min ojective of this study is to investigte the effect of the residul stresses, nd prediction of the rel distriution of the residul stress field is outside of the scope. The solute nd normlized crck growth resistnces re shown in Fig. 8. It cn e seen tht the compressive residul stress increses the crck growth resistnce while tensile residul stresses decrese the crck growth resistnce, s shown in

8 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( RsField RsField 3 RsField RsField σ /σ σ /σ 5 5 x/l c 5 5 x/l c Fig. 7. Residul stress distriution in MBL model with rectngulr weld fter the crck ws inserted, ( components prllel to the crck plne nd ( norml to the crck plne. Four different residul stress cses were considered, where RsField is compressive nd the remining 3 re tensile. / =; =.,.,., nd J/(σ l c 5 J/J without no/res RsField RsField RsField RsField Fig. 8. Effect of residul stresses on crck growth resistnce, ( solute crck resistnce curves nd ( normlized crck resistnce curves. f =.%; n =.; E/r = 5. Fig. 8. Fig. 8 shows tht the reduction of the crck growth resistnce converges with crck growth to the cse without residul stress. As known, the quntity r m /r e defines convenient mesure of trixility linked to the growth rte of micro-scle voids consistent with the susequently introduced dmge mesures [36]. Therefore, we investigted the distriution of trixility hed of the crck tip. Fig. 9 presents the trixility vlues on the ligment hed of the originl crck tip for different crck growth. The distnce from the originl crck tip re scled y J/r, or equivlently (K I /r, which defines pproximtely the vlue of crck-tip opening displcement nd provides physicl meningful length scle for normliztion. It cn e seen tht tensile residul stresses enhnce the trixility vlues while the compressive residul stress reduces the trixility vlue t the crck initition, see Fig. 9, which corresponds to our previous findings [4,5]. However, the effect of residul stress on trixility tend to e negligile when the crck dvnced to 3.5 mm, s shown in Fig. 9. Higher stress trixility corresponds to smller plstic zone. Thus, the energy dissipted y the plstic deformtion is smller, which in turn result in lower crck growth resistnce Effect of weld zone size on crck growth resistnce The length scle of the residul stress field my ply n importnt role on the effect of residul stress on ductile crck growth resistnce, s shown in Section 3.. To etter demonstrte this, four geometriclly similr rectngulr weld regions were constructed, s illustrted in Fig.. The size of the weld is designted s c. The orthogonl therml expnsion coefficient ws used with vlue of =.4 nd =. for ll welds. The residul stress fields designted s Size, Size, Size3, nd Size4 respectively re showed in Fig.. The residul stresses re tensile in the weld metl nd show similr fetures s the previous residul stresses shown in Fig. 7. With the increse of the weld size, the residul stresses prllel to the crck plne increse; the size of tensile dominted region of the opening stress lso increses.

9 33 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( σ m /σ e.5 σ m /σ e.5.75 no/res RsField RsField x/(j/σ.75 no/res RsField RsField x/(j/σ Fig. 9. Trixility vlues hed the crck tip for different crck propgtion, ( D =. mm; ( D = 3.5 mm. f =.%; n =.; E/r = 5. Fig.. Illustrtion of weld size. The size increses y scle of. Fig. presents the solute crck growth resistnce. In Fig., crck growth length ws normlized y ł c. It cn e seen tht the residul stresses generted in the lrger welds influence more significntly the crck growth resistnce. With incresing crck growth, the effect of the residul stresses on the crck growth resistnce converges to the cse without residul stresses. However, n interesting pttern emerges when normlizing the crck growth length y the weld zone size c, s shown in Fig. c. Surprisingly, the normlized crck growth resistnces collpse into one curve, which indictes tht the effect of the residul stresses on the crck growth resistnce is nerly independent of the weld zone size. For the cses specified in this section, when the crck grows pproximtely to 3 times of c, the effect of residul stresses cn e neglected. For geometriclly similr welds, the uniform curve cn e used to roughly predict the length scle of residul stress-ffected region eyond which the effect of the residul stress cn e neglected. In our previous study [5], it hs een shown tht residul stress fields generted in round weld regions with three sizes respectively cn e normlized y the weld zone size nd collpsed into uniform field. Thus, the mster curve s shown in Fig. c cn e expected Effect of residul stresses on crck growth resistnce for different hrdening exponents It hs een shown tht the effect of strin hrdening on the ductile crck resistnce is not fully understood. Xi nd Shih [37] demonstrted tht the ductile resistnce increses with incresing hrdening cpcity. However, Eikrem et l. [38] nd Østy et l. [,39] reported tht decresing the hrdening exponent will significntly rise the resistnce curve. In this study, the effect of strin hrdening on the crck growth resistnce ws investigted for cses oth with nd without residul stress. It should e noted tht in this prticulr study, the initil volume frction f ws fixed to e.5%, nd the residul stress field ws introduced y the rectngulr weld with orthogonl therml expnsion coefficient =.4 nd =., i.e. in Fig. 7. The crck growth resistnce ws plotted s function of hrdening exponent n in Fig. 3. Fig. 3 shows tht with decresing hrdening exponent the crck growth resistnce increses for oth with nd without residul stress cses, which corresponds to the results demonstrted in Ref. [,38,39]. For given hrdening exponent, residul stresses reduce oth the initil toughness nd crck growth resistnce. Also, the effect of residul stress on the ductile resistnce ecomes stronger for stronger hrdening mteril, s shown in Fig. 3. With incresing crck growth, the reduction of the crck growth resistnce decreses nd tends to converge to the cse without residul stress. Residul stresses reduce the equivlent plstic strin significntly t crck initition, which indictes smller plstic deformtion. Hence,

10 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( Size Size Size3 Size4.5 Size Size Size3 Size4 σ /σ σ /σ x/l c x/l c Fig.. Residul stress distriution in MBL model with geometriclly similr welds fter the crck ws inserted, ( components prllel to the crck plne nd ( norml to the crck plne. =.4, = J/(σ l c 8 no/res Size 6 Size Size3 Size J/J without.9.85 Size.8 Size Size3 Size c.5.95 J/J without.9.85 Size.8 Size Size3.75 Size4 3 4 Δ/c Fig.. Effect of weld size on crck growth resistnce, ( solute crck growth resistnce, ( crck growth normlized y l c, nd (c normlized y the weld zone size c. f =.%; n =.; E/r = 5. lower crck growth resistnce curves cn e expected. With the crck growth, the effect of residul stresses on the equivlent plstic strin ecomes negligile for different hrdening exponents Effect of residul stresses on ductile crck resistnce for different initil void volume frction The initil void volume frction f represents the degree of dmge in the mteril. The lrger the initil void volume frction is, the lrger dmge the mteril hs. In this section, the effect of residul stress fields on crck growth resistnce ws

11 334 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( investigted for three initil void volume frctions, f =.5%,.% nd.%. The residul stress ws introduced into rectngulr weld with =.4 nd =., i.e. in Fig. 7. The crck growth resistnce curves re shown in Fig. 4. As shown in Fig. 4, for oth with nd without residul stress cses, the crck growth resistnce increses with the decrese of initil void volume frction, which cn e expected ecuse the ductility ecomes etter when initil void volume frction decreses. Also, it cn e oserved tht the residul stress reduces the crck growth resistnce for fixed f. Normlized crck resistnce curves shown in Fig. 4 indicte tht with incresing crck growth, the effects of residul stresses decrese nd ecome less dependent on f. The residul stress enhnces the opening stress eyond the lrger strin effect region compred with the cse without residul stress t crck initition, which induces n increse of crck-tip constrint [4,5], nd lower frcture toughness cn then e expected. However, it hs een found tht the effect of the residul stress on opening stress ecomes negligile when crck growth ecomes lrger Effect of residul stresses on ductile crck resistnce for different T/r Crck tip constrint effects on frcture toughness hve received considerle ttention recently. In our erlier work [5], we hve found tht the residul stress-induced crck-tip constrint is lower for higher geometric constrint. Xi nd Shih [37], Tvergrd nd Hutchinson [4] studied the effect of T-stress on the crck growth resistnce nd showed tht negtive T-stress results in rpidly rising resistnce curve while the positive T-stress lowers the frcture resistnce. It is thus interesting to investigte how residul stress comined with vrying T-stress ffect the crck growth resistnce. In this pper, T/ r =.5,, nd.5 were studied. The initil void volume frction is fixed to e.%, nd the residul stress with =.4 nd =., i.e. in Fig. 7, ws introduced. The solute nd normlized resistnce curves re shown in Fig. 5. It cn e seen tht the tensile residul stress reduces the crck growth resistnce for ll T/r cses, s shown in Fig. 5. Fig. 5 shows tht the normlized crck growth resistnce is lower for smller T-stress. However, the differences etween 3 5 n=.5 no/res n=.5 with/res n=. no/res n=. with/res n=.5 no/res n=.5 with/res.5.95 J/(σ l c 5 J/J without n=.5 n=. n= Fig. 3. Effect of residul stresses on crck growth resistnce for different hrdening, ( solute crck growth resistnce curves nd ( normlized crck resistnce curves. =.4; =.; f =.5%; E/r = 5. J/(σ l c f =.% no/res f =.% with/res f =.5% no/res f =.5% with/res f =.% no/res f =.% with/res 5 5 J/J without f =.% f =.5% f =.% Fig. 4. Effect of residul stresses on crck growth resistnce for different initil void volume frction, ( solute crck growth resistnce curves nd ( normlized crck resistnce curves. =.4; =.; n =.; E/r = 5.

12 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( J/(σ l c T/σ =.5 no/res T/σ =.5 with/res T/σ = no/res T/σ = with/res T/σ =.5 no/res T/σ =.5 with/res 5 5 J/J without T/σ =.5 T/σ = T/σ = Fig. 5. Effect of the residul stress on normlized crck growth resistnces for different T/r, ( solute crck growth resistnce curves nd ( normlized crck resistnce curves. =.4; =.; f =.%; n =.; E/r = 5. the normlized crck growth resistnces re insignificnt. It should e noted tht reltively smll initil void frction ws used in this study, i.e. f =.%. With incresing crck growth, the effect of residul stresses on crck growth resistnce tends to e independent of the T-stress. The positive residul stresses reduce the plstic strin compred to the cse without residul stresses. Lower equivlent plstic strin indictes smller plstic zone nd thus the energy dissipted y the plstic deformtion decreses. In turn, the ductile crck toughness decreses. When the crck growth ecomes lrger, the effect of residul stresses on the equivlent plstic strin ecomes negligile. 4. Conclusions This study descries systemtic computtionl investigtion of the effect of residul stresses on ductile crck growth resistnce. D plne strin finite element nlysis hve een crried out in modified oundry lyer model with the remote oundry condition governed y K-field nd T-stress. The lrge strin effect hs een tken into ccount in the nlysis. The complete Gurson model ws utilized to predict the crck growth resistnce for oth with nd without residul stress situtions. Residul stresses were generted y the eigenstrin method nd introduced into the FE model. Both the tensile nd compressive residul stress fields were introduced. However, tensile residul stress ws known to e detrimentl for the frcture. Thus, the effect of the tensile residul stress on crck growth resistnce ws minly presented. Residul stresses s n dditionl field do not lter the mteril constitutive reltion, however, the residul stress my influence the nertip stress stte nd enhnce the level of dmge long the ligment hed of n dvncing crck tip. The following conclusions cn e mde:. Tensile residul stresses reduce the crck growth resistnce while compressive residul stress cn enhnce the crck growth resistnce. With the increse of crck growth, the effect of residul stresses decreses nd converges to the cse without residul stress. The initil dmge of the ligment elements cused y the residul stress is negligile for the cses studied.. Under certin circumstnces, the effect of the residul stresses on ductile crck growth resistnce cn e normlized y the size of the geometriclly similr weld, nd the normlized crck growth resistnce curves collpse into single curve. One cn use this curve to evlute the effect of the residul stress on the structurl integrity nd simplify the ssessment procedure. It cn lso e used to predict the length scle of the residul stress-ffected rnge eyond which the effect of the residul stresses cn e neglected. For the cses specified in this study, when the crck grows to length of 3c the effect of the residul stresses cn e neglected. 3. The reduction of the crck growth resistnce cused y residul stresses is smller for weker hrdening mterils, nd the influence of the residul stresses decreses with crck propgtion. 4. Residul stress reduces the crck growth resistnce more significntly for the mterils with lrger initil void volume. The influence of the residul stresses decreses with the crck growth nd ecomes independent of f. 5. The reduction of the crck growth resistnce induced y the residul stress increses with incresing T/r t the erly stge of crck growth nd then tend to e negligile when the crck growth is lrger. The results re in line with our erlier findings tht the effect of residul stress on the crck-tip constrint ecomes weker for higher T/r. The present study indictes tht residul stress s n dditionl stress field cn lter the stress stte ner the crck tip nd further influence the ductile crck growth resistnce. Trixility or crck-tip constrint ws shown to ply n importnt role in the ductile frcture ehviour. Therefore, the residul stress-induced crck-tip constrint is very importnt fctor to consider in structurl integrity ssessment. In our previous studies [4,5], we hve defined prmeter R to quntify the residul

13 336 X.B. Ren et l. / Engineering Frcture Mechnics 77 ( stress-induced crck-tip constrint. However, the effect of residul stress on the ductile crck resistnce hs not een linked with R y quntittive wy in this study. The reson for this is tht the residul stress field in relity is very complicted, hence one should find n efficient wy to stndrdize the distriution of the residul stresses for typicl weld joints. To reduce the conservtism of current integrity ssessments, proper description of the residul stress field is very importnt. Once proper description of the residul stress is otined, the mster curve, though otined from the MBL model, cn e pplied to rel engineering structures nd components tht contin similr residul stress fields under smllscle-yielding conditions. The trnsferility of the results from MBL studies to rel structures should e verified in further studies. Acknowledgment The funding from the Reserch Council of Norwy through the STORFORSK Project No /V3 is gretly cknowledged. The first uthor would like to cknowledge Dr. Erling Østy nd Dr. Sigmund Kyrre Ås of SINTEF Mterils nd Chemistry for their vlule comments nd discussions. References [] Ainsworth RA, Shrples JK, Smith SD. Effects of residul stresses on frcture ehviour experimentl results nd ssessment methods. J Strin Anl Engng Des ;35(4:37 6. [] Østy E, Thulow C, Zhng ZL. Numericl simultions of specimen size nd mismtch effects in ductile crck growth Prt I. Tering resistnce nd crck growth pths. Engng Frct Mech 7;74:77 9. [3] Sherry AH, Wilkes MA, Shrples JK, Budden PJ. The ssessment of residul stress effects on ductile tering using continuum dmge mechnics. J Pressure Ves Technol 8; [4] Liu J, Zhng ZL, Nyhus B. Residul stress induced crck tip constrint. Engng Frct Mech 8;75(4: [5] Ren XB, Zhng ZL, Nyhus B. Effect of residul stress on crck-tip constrint in oundry lyer model. 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