An analysis on overall crack-number-density of short-fatigue-cracks

Transcription

1 Mecancs of Materals 3 (999) 55±534 An analyss on overall crack-number-densty of sort-fatgue-cracks Yous Hong *, Yu Qao Lab for Non-lnear Mecancs of Contnuous Meda, Insttute of Mecancs, Cnese Academy of Scences, 5 ongguancun Road, Bejng 8, People's Republc of Cna Receved 8 August 997; receved n revsed form Marc 999 Abstract Te evoluton of dspersed sort-fatgue-cracks s analysed based on te equlbrum of crack-number-densty (CND). By separatng te mean value and te stastc uctuaton of lal CND, te equlbrum equaton of overall CND s derved. Comparng wt te mean- eld equlbrum equaton, te equlbrum equaton of overall CND as d erent forms n te expresson of crack-nucleaton-rate or crack-growt-rate. Te smulaton results are compared wt expermental measurements sowng te stastc analyses provde consstent tendency wt experments. Te dscrepancy n smulaton results between overall CND and mean- eld CND s dscussed. Ó 999 Elsever Scence Ltd. All rgts reserved. Keywords: Sort fatgue cracks; Crack number densty; Collectve damage; Stastc analyses. Introducton * Correspondng autor. Tel.: ; fax: ; e-mal: ongys@lnm.mec.ac.cn Generally, te fatgue press of metallc materals can be dent ed as two stages: sort-crack regme and long-crack regme (Hussan et al., 993; Fang et al., 995). In te sort-crack regme, cracks keep nucleatng and growng wtn gran domans, wc seldom may overcome granboundary obstacles and develop to be crossboundary cracks. Wt te progress of te fatgue press, te crack-nucleaton-rate keeps decreasng for cracks approacng gran boundares and a few sort-fatgue-cracks may grow nto negbour grans. Te appearance of a crack wt a sze large enoug to overcome gran-boundary obstacle effect caracterses te end of te sort-crack regme and te begnnng of te long-crack regme. Snce te sort-crack regme usually takes up a large porton of fatgue lfe, terefore t s n engneerng mportant to nvestgate te fatgue damage due to sort crack evoluton, so as to assess te relablty and fatgue lfe of materal under cyclc loadng. Durng te fatgue damage press, te ntaton and te growt of sort cracks are always randomly dstrbuted. In some lal areas, sort cracks may densely appear; smultaneously, tere may exst some oter areas even wtout any sort crack damage. Ts suggests tat te collectve damage of sort fatgue cracks s a stastc feature, wc calls for analyses to get a better understandng for suc a press. In te present paper, we descrbe te metod of crack-number-densty (CND) analyss wc s proposed to deal wt te collectve crack /99/$ ± see front matter Ó 999 Elsever Scence Ltd. All rgts reserved. PII: S ( 9 9 ) 4-9

2 56 Y. Hong, Y. Qao / Mecancs of Materals 3 (999) 55±534 evoluton problem. A stastc consderaton s ntroduced for bot lal eld and overall eld dervatons. Te smulaton results of stastc analyses and mean- eld teory are obtaned, and compared wt expermental measurements. Te dscrepancy between stastc analyses and mean- eld teory s dscussed.. Model of crack-number-densty analyss Consderng te collectve caracterstcs exstng n sort-crack regme, one may ntroduce crack-number-densty (CND) to analyse fatgue damage (Fang et al., 995; Qao and Hong, 998a). Omttng te d erence n te propensty of sort cracks at separate lal areas of materal, we wrte te non-dmensonal mean- eld equlbrum equaton of CND as o ot n c; t o A c n c; t ˆ N g n N c ; were n c; t s mean- eld CND, wt n c; t dc beng te number of cracks wt lengt between c and c + dc at tme t; A c s average crack-growtrate; and n N c s average crack-nucleaton-rate. All of tese pyscal quanttes are normalzed and te non-dmensonal coe cent N g ˆ n N d = n A, wt n N beng te caracterstc crack-nucleaton-rate, A te caracterstc crackgrowt-rate, n te caracterstc CND and d te caracterstc dmenson of te materal concerned (e.g. te gran dameter). Eq. () descrbes te equlbrum of CND n pase space. Te second term on te left sde descrbes te ow of CND, wc s attrbuted to crack growt; and te term on te rgt sde descrbes te contrbuton to CND made by crack nucleaton. Te evoluton press of mean- eld CND was derved by solvng Eq. () and tus te relevant damage parameters, suc as te maxmum crack lengt and te total number of sort-fatguecracks, were dscussed (Fang et al., 995; Qao and Hong, 998a; Hong and Qao, 998). Tese damage parameters were ntroduced to llustrate te extent of materal damage. Fg. sows a numercal result of te mean- eld equlbrum equaton. Troug te gure we see te dual-peak Fg.. Numercal result for CND evoluton of mean- eld teory, dased curve representng te saturaton dstrbuton. A d ˆ and d ˆ were used n te calculaton. dstrbuton of CND, wc s n agreement wt te dual-peak feature observed n experments (Fang et al., 995). Note tat te above analyses are fused on mean- eld case,.e. d erent parts of materal are assumed to ave te same lal mecancal caracterstcs. However, n te real fatgue damage press, tere always exsts stastc uctuaton of lal area beavour n materal (Prce, 988; Hong et al., 989, 99). Consderng te d erence n lal areas of materal, we ntroduced te concept of lal CND and obtaned te equlbrum equaton of lal CND (Qao and Hong, 998b): o ot n c; t; x~ o A c; t; x~ n c; t; x~ ˆ N g n N c; t; x~ ; were x~ s te poston of a small area of materal, n c; t; x~, A c; t; x~ and n N c; t; x~ are lal CND, lal crack-growt-rate and lal crack-nucleaton-rate, respectvely. Te lal area s treated as small enoug to be consdered as a pont at macroscale. On te oter and, te lal area also contans enoug sort-fatgue-cracks so tat te concept of CND can be appled. It s observed n

3 Y. Hong, Y. Qao / Mecancs of Materals 3 (999) 55± experments tat a vewed area on specmen surface may comprse a contnuous low-damage-extent area and dspersed g-damage-extent areas (Prce, 988; Hong et al., 997). Fg. s a potograp sowng te uneven dstrbuton of sort fatgue cracks. Ts was taken from te so-stress specmen of a low carbon steel subjected to te fatgue loadng of stress rato R ˆ, frequency f ˆ Hz and r max ˆ :7r y, were r max s te specmen surface maxmum stress and r y s te yeld stress of te materal. Fg. s a typcal example tat durng te progresson of fatgue damage, wtn some gran domans, sort cracks ave become densely dstrbuted, wereas some oter ferrte domans reman undamaged wtout a sngle sort crack appearng. Te expermental press and te results ave been descrbed elsewere (Hong et al., 989, 99, 997). Te stastc appearance of sort crack dstrbuton s scematcally llustrated n Fg. 3. Consderng tat te nteracton of sort-fatgue-cracks s weak n low-damage-extent area, we assume tat lal areas are ndependent of eac oter. By solvng te equlbrum equaton of lal CND, and takng nto account te stastc uctuaton of lal crack-growt-rate and lal crack-nucleaton-rate, te damage evoluton n d erent small areas of materal was studed. Accordng to expermental Fg.. Potograp sowng sort-fatgue-cracks developed n some ferrte grans and wtout crack damage n oter ferrte domans, vertcal drecton parallel to tensle stress. Fg. 3. Scematc llustraton of dspersed sort-fatgue-cracks. observatons, we assumed tat lal crack-growtrate and lal crack-nucleaton-rate ave te followng forms, respectvely (Qao and Hong, 998b): A c; t; x~ ˆ A c L c W x~ ad b l t; x~ ; 3 n N c; t; x~ ˆ n N c L c W x~ t; x~ ; pd q l 4 were A (c) and n N (c) are respectvely te mean values of lal crack-growt-rate and lal cracknucleaton-rate; D l s number of cracks n lal area; a, b, p and q are materal parameters; L and L are functons respectvely llustratng stastc uctuaton of A and n N ; and W and W are two wte-nose press. Lal CND was derved from Eq. (), and tus lal maxmum crack lengt and lal number of cracks were obtaned. Overall maxmum crack lengt and overall number of cracks were calculated by analysng te lal damage evoluton n eac lal area. Fg. 4 sows te results of overall maxmum crack lengt c max aganst normalzed tme t, were te datum ponts are te results of stastc numercal smulatons wt te dased lne beng te regressed curve of te stastc results and te sold lne s te numercal result of mean- eld teory (Eq. ()). Te d erence n te results between mean- eld equlbrum equaton and te stastc analyses becomes evdent wen t s beyond.6. In oter words, te result of stastc analyses s almost consstent wt tat of mean- eld equaton at te ntal stage of damage evoluton press and t becomes smaller tan te mean- eld

4 58 Y. Hong, Y. Qao / Mecancs of Materals 3 (999) 55±534 Fg. 4. Results of overall maxmum crack lengt aganst tme derved by mean- eld teory and stastc analyses. Data ponts sowng results of stastc smulatons wt dased lne as regressed curve and sold lne beng te result of mean- eld teory. Fg. 5. Relatonsp between wole eld maxmum crack sze and total number of sort cracks, sold curve representng result of mean- eld teory, data ponts representng result of stastc smulatons. value at a certan stage of te fatgue evoluton press. Fg. 5 sows te result of overall maxmum crack lengt c max versus overall number of cracks D, were te sold curve representng te result of mean eld teory and data ponts representng te result of stastc smulatons. It s seen tat te relatonsp between c max and D as te same trend for mean eld teory and stastc analyss. Fg. 5 also ndcates tat c max and D are approxmately lnearly correlated. 3. Analyses of overall CND 3.. Evoluton equaton of overall CND Te sort-crack damage evoluton s consdered an ergodc system. Terefore, te equlbrum equaton of overall CND can be derved by ntegratng te equlbrum equaton of lal CND. Let n c; t be CND n a unt area of materal. Accordng to te de nton of lal CND n c; t; x~, we ave: n c; t ˆ n c; t; x~ dx~; 5 S were S s te area of ntegraton part on specmen surface. If a unt area s regarded as a lal area, ten n c; t s just te average CND. It follows tat n c; t; x~ ˆ n c; t ~n c; t; x~ ; 6 were ~n c; t; x~ s te stastc uctuaton of lal CND n c; t; x~. Lal crack-growt-rate A c; t; x~ and lal crack-nucleaton-rate n N c; t; x~ can also be wrtten as: A c; t; x~ ˆ A c; t ~A c; t; x~ ; n N c; t; x~ ˆ n N c; t ~n N c; t; x~ ; 7 8 were A c; t and n N c; t are respectvely te mean values of A c; t; x~ and n N c; t; x~, and ~A c; t; x~ and ~n N c; t; x~ are respectvely te stastc uctuatons ofa c; t; x~ and n N c; t; x~. Substtutng Eqs. (6)±(8) nto Eq. () and ntegratng n, we ave

5 on c; t ot S o~n c; t; x~ ot dx~ o A c; t n c; t S S S ˆ N g n N c; t S o ~A c; t; x~ n c; t dx~ o ~A c; t; x~ ~n c; t; x~ o A c; t ~n c; t; x~ dx~ N g ~n N c; t; x~ dx~: Note tat for an ergodc system ~n c; t; x~ dx~ ˆ ; ~A c; t; x~ dx~ ˆ ; dx~ 9 ~n N c; t; x~ dx~ ˆ : Substtuton of Eqs. ()±() nto Eq. (9) gves on c; t o A c; t n c; t ot o ~A c; t; x~ ~n c; t; x~ ˆ N g n N c; A; 3 were o ~A c; t; x~ ~n c; t; A 8 9 ˆ < o ~A c; t; x~ ~n c; t; x~ = dx~: 4 S : ; Eq. (3) s te equlbrum equaton of overall CND, wc sows te equlbrum of overall Y. Hong, Y. Qao / Mecancs of Materals 3 (999) 55± CND n pase space wt lal CND uctuatng n d erent lal areas. 3.. Fluctuaton n uence term Comparng Eq. (3) wt Eq. (), we see tat tere appears, n te equlbrum equaton of overall CND, a new term o ~A c; t; x~ ~n c; t; x~ =, wc we call te uctuaton n uence term (FIT) n te followng. FIT represents te e ects of te stastc uctuaton of lal damage on overall damage and t s caused by te non-lnear responses n te damage evoluton press. Te comparson of Eq. (3) wt Eq. () suggests tat FIT s te cause of te devaton of stastc analyses from mean- eld teory sown n Fg. 4. Referrng to te de nton of Eq. (4), we may wrte FIT as o ~A c; t; x~ ~n c; t; A 3 ˆ o 4 ~A c; t; x~ ~n c; t; x~ dx~ 5: 5 S Accordng to Eqs. () and (), te mean values of ~A c; t; x~ and ~n c; t; x~ are zero. Tus te functon n te brackets on te rgt sde of above equaton s te covarance of ~A c; t; x~ and ~n c; t; x~. Here we wrte t as l c; t. In general, l c; t ˆ q c; t r A c; t r n c; t ; 6 were r A c; t s te standard devaton of lal crack-growt-rate A c; t; x~ ; r n c; t s te standard devaton of lal CND n c; t; x~ and q c; t s te correlaton coe cent between A c; t; x~ and n c; t; x~. Substtutng Eqs. (5) and (6) nto (3), te equlbrum equaton of overall CND becomes: on c; t ot o A c; t n c; t ol c; t ˆ N g n N c; t ˆ N g n N c; t o q c; t r A c; t r n c; t Š: 7

6 53 Y. Hong, Y. Qao / Mecancs of Materals 3 (999) 55±534 One may notce tat f jq c; t j ˆ, Eq. (7) reduces to te form of Eq. () and te result of stastc analyses s te same wt tat of mean- eld teory. In order to dscuss te response of FIT wt te evoluton equaton of overall CND, we de ne te e ectve crack-growt-rate as ^A c; t ˆ A c; t; x~ n c; t; x~ n c; t ˆ A c; t n c; t ~A c; t; x~ ~n c; t; x~ ; 8 n c; t were A c; t; x~ n c; t; x~ ˆ S A c; t; x~ n c; t; x~ dx~: Substtuton of Eq. (8) nto (3) gves anoter form of te equlbrum equaton of overall CND: on c; t o ^A c; t n c; t ˆ N g n N c; t : 9 ot Eq. (9) mples tat te devaton of stastc analyses from mean- eld teory may also be attrbuted to te d erence between e ectve crackgrowt-rate ^A c; t and average crack-growt-rate A c; t Comparson wt expermental results Our prevous expermental nvestgaton of a low carbon steel ndcated tat te progresson of sort-fatgue-cracks s a collectve evoluton press wt te gradual ncrease n crack number per unt area wt ncreasng number of fatgue cycles (Hong et al., 997). Our recent expermental work agan revealed te collectve damage evoluton of sort fatgue cracks for a stanless steel. Fg. 6 sows te varaton of te total number of sort cracks D wt te normalzed number of fatgue cycles N=N f for two knds of specmen condtons wt average gran szes of 7 and 7 lm. Te two data sets gve a smlar trend between D and N=N f n spte of ter large d erence n gran sze. Te determnatons also ndcated tat te sort crack regme takes up as muc as 87% of total fatgue lfe. Te detaled Fg. 6. Total number of sort cracks D versus normalzed number of fatgue cycles N=N f for a stanless steel wt two grades of gran sze. predure and results were reported recently (Hong et al., 998, 999). Referrng to te lnear correlaton between c max and D ndcated by Fg. 5 and notng tat te normalzed tme t s assated wt normalzed fatgue cycles N=N f, one may evaluate tat te varaton trend of c max wt t s equvalent to tat of D wt N=N f. Tus, we see te evdence tat te varaton tendency between c max and t gven by stastc smulatons sown n Fg. 4 s consstent wt tat of te expermental results sown n Fg. 6. Te sold curve, plotted n Fg. 4 obtaned by mean- eld teory, overestmates te collectve sort crack evoluton wen t tends to a large value. 4. E ect of uctuaton n uence term Subtractng Eq. (3) from Eq. () gves te evoluton equaton for stastc uctuaton of lal CND: o~n c; t; x~ o A c; t ~n c; t; x~ ~A c; t; x~ n c; t ot ˆ N g ~n N c; t; x~ : It s clear tat, from Eq. () and Eq. (3), te mean value and te stastc uctuaton of overall CND nteract eac oter. To obtan tem one needs to study Eqs. (3) and () togeter,

7 Y. Hong, Y. Qao / Mecancs of Materals 3 (999) 55± wc s relatvely complcated and d cult. If FIT can be smpl ed by smulaton metods, ten overall CND may be derved from Eq. (3) ndependently. In our prevous analyses (Qao and Hong, 998a, b), damage extent was assessed avng lttle e ect on crack-growt-rate. Hence, we assume tat lal crack-growt-rate s ndependent of tme t: A ˆ : On te oter and, Eq. (3) mples tat overall CND s not n uenced by te stastc uctuaton of lal crack-nucleaton-rate, because te crack-nucleaton-rate s a lnear term n te equlbrum equaton. Terefore, te stastc uctuaton of n N c; t; x~ wll be omtted n te followng dervaton. Note also tat te crack-growt-rate tends to a constant level wt te progress of fatgue press (Su et al., 99). Consequently, Eq. (3) reduces to: o A c n c; t ˆ N g n N on c; t ot o ~~n c; t; x~ A: Te second term on te rgt sde of above equaton may be nvestgated by usng te analytc soluton of Eq. () (Ke et al., 99): n c; t; x~ ˆ c g c;t;x~ N g n N c ; x~ dc ; 3 were te de nton of lower ntegral boundary g c; t; x~ s tat for a crack wt an ntal lengt of g c; t; x~ at t ˆ, ts lengt wll advance to c at tme t under te growt rate of. Eq. (3) suggests tat tere s a saturaton dstrbuton n te evoluton press of lal CND,.e. wt te progress of te fatgue press, te dstrbuton of lal CND gradually tends to a lal saturaton curve from a small to a large value of crack lengt (also see Fg. ). It s obvous tat te saturaton curve presents te stable dstrbuton of CND. Referrng to Eq. (3), we may wrte te lal saturaton curve as: n c; x~ ˆ c N g n N c ; x~ dc : 4 At te stage tat te fatgue damage fully developed, assume lal CND be of te trend wt ts dstrbuton approacng to te saturaton curve. Lettng te mean value and te stastc uctuaton of saturaton lal CND be n c and ~n c; x~, respectvely, from Eq. (4), one may wrte c ~n c; x~ ˆ ~B c; x~ N g n N c dc ; 5 were ~B c; x~ s te stastc uctuaton of =. Ten, we ave c ~~n c; x~ ˆ ~ ~B c; x~ N g n N c dc ; 6 were ~~n c; x~ s te mean value of ~~n c; x~ and A c; ~ x~ ~B c; x~ s te mean value of ~ ~B c; x~. Note tat! ~ ~B c; x~ ˆ ~ A A ~! ˆ A c : 7 Replacng lal CND n c; t; x~ wt te saturaton curve n c; x~ (Eq. (4)) n Eqs. (6) and (7), and referrng to Eq. (5), we can sow FIT n te followng form:! 3 ol ˆ o 4 A c n N c 4 A c 5 c n N c dc! 3 5: 8 Assume tat te dstrbuton of lal crackgrowt-rate s a logartm normal functon,.e. te probablty for ˆ m s

8 53 Y. Hong, Y. Qao / Mecancs of Materals 3 (999) 55±534 8 >< p f m ˆ p exp ln m l c Š m P ; r c m r c >: m < ; 9 were r c and l c are dstrbuton parameters determned by crack lengt c. Te mean value of f m s E c ˆ A c ˆ exp l c r c 3 and te varance of f m s D c ˆ exp l c r c exp r c : 3 Troug Eqs. (9) and (3), one derves! ˆ ˆ exp f m dm m l c : 3 r c ˆ A c exp r c Substtuton above equaton nto (8) gves: ol c ˆ n N c e r c 8 < : r c e r c c 9 = n N c dc ; or c : 33 Tus, te equlbrum equaton of overall CND (Eq. ()) can be solved ndependently by usng Eq. (33). Here we assume tat r c and n N (c) ave te followng forms: r c ˆ a exp bc ; 34 n N c ˆ c c 6 ; 35 c > ; were a and b are constants. Eq. (35) comes from prevous expermental observatons and t sows tat smaller fatgue cracks nucleate easer (Qao and Hong, 998a). Snce te d erence between stastc analyses and mean- eld teory comes from te term of FIT n Eq. (3), te followng dscussons wll concentrate on te caracterstcs of FIT. Fg. 7 llustrates te tendency of FIT varyng wt d erent values of c and b. It s seen tat FIT s slgtly larger tan zero at te negatve part of b-axs. If crack lengt c s relatvely large, tere appears a negatve peak of FIT at te postve end of te b-axs. It s derved tat r c as a functon of A c and D(c) from Eqs. (3) and (3): v " # u r c ˆ t ln D c : 36 A c De ne a parameter Q as Q ˆ oln D c Š= : 37 oln A c = Accordng to Eq. (37), we see tat f Q >,.e. te varaton extent of D c s faster tan tat of A c, te varaton tendency of r c s of te same trend wt tat of A c. Tus, because of te deceleraton±acceleraton pattern of sort-fatgue-crack evoluton (Fg. 8; Lankford, 98), one may determne tat n te sort-crack regme or c = < (b < ) and n te long-crack regme or c = > (b > ). Terefore, n Fg. 7, te press tat b vares from to corresponds to te press Fg. 7. Varaton of FIT as a functon of c and b.

9 Y. Hong, Y. Qao / Mecancs of Materals 3 (999) 55± Conclusons Fg. 8. Deceleraton±acceleraton pattern of sort-fatgue-crack propagaton. (Gran < Gran < Gran 3 Gran 4.) tat fatgue damage develops from te sort-crack regme nto te long-crack regme. At te ntal stage of te fatgue press, fatgue damage s wtn te sort-crack regme, tus FIT s at te negatve part of b-axs. From Fg. 7, t s observed tat FIT s somewat larger tan zero, wc mples tat te result of stastc analyses sould be slgtly larger tan te result of mean- eld teory. Wt te progress of te fatgue press, dspersed fatgue-cracks develop nto long-crack regme, tus FIT develops to te postve part of b-axs, were te negatve peak of FIT n te regon of large values of crack lengt c may ntensvely suppress te propagaton of cracks. Terefore, at te end of te sort-crack regme and n te wole long-crack regme, te damage extent derved by stastc analyses may be muc weaker tan by mean- eld teory. Te above dscusson nterprets te d erence between stastc analyses and mean- eld teory sown n Fg. 4, n wc te data ponts represent random presses wt Q >. By ntegratng te equlbrum equaton of lal CND, te equlbrum equaton of overall CND s obtaned and te followng conclusons are drawn: () Overall damage evoluton s n uenced by te stastc uctuaton of te lal crackgrowt-rate. Te n uence extent may be descrbed by te standard errors of lal crack-growt-rate and lal CND, and ter correlaton coe cent. () Te d erence between stastc analyses and mean- eld teory comes from te uctuaton n uence term related to te stastc uctuaton of crack-growt-rate. Overall damage evoluton s ndependent of te stastc uctuaton of cracknucleaton-rate. (3) At te ntal stage of fatgue press, te damage extent derved by stastc analyses s slgtly larger tan tat derved by mean- eld teory and at te followng stage of fatgue press te result of stastc analyses s smaller tan tat of mean eld teory. Te tendency of smulaton results by stastc analyses s consstent wt expermental measurements. Acknowledgements Ts paper was supported by te Natonal Outstandng Yout Scent c Award of Cna, te Natonal Natural Scence Foundaton of Cna and te Cnes Academy of Scences. References Fang, B., Hong, Y.S., Ba, Y.L., 995. Expermental and teoretcal study on numercal densty evoluton of sort fatgue cracks. Acta Mecanca Snca (Engls edton), 44±5. Hong, Y.S., Gu,.Y., Fang, B., Ba, Y.L., 997. Collectve evoluton caracterstcs and computer smulaton of sort fatgue cracks. Pl. Mag. A 75, 57±53. Hong, Y.S., Lu, Y.H., eng,.m., 989. Intaton and propagaton of a sort fatgue crack n a weld metal. Fatgue Fract. Engrg. Mater. Struct., 33±33. Hong, Y.S., Lu, Y.H., eng,.m., 99. Orentaton preference and fractal caracter of sort fatgue cracks n a weld metal. J. Mater. Sc. 6, 8±86.

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