SCIENCE CHINA Physics, Mechanics & Astronomy. An energy-equilibrium model for complex stress effect on fatigue crack initiation

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1 SCIENCE CHINA Phyic, Mchnic & Atronoy Articl My 014 Vol.57 No.5: doi: / z An nrgy-quilibriu odl or coplx tr ct on tigu crck initition ZHAO SiCong, XIE JiJi *, ZHAO AiGuo & WU XioLi Stt Ky Lbortory o Nonlinr Mchnic, Intitut o Mchnic, Chin Acdy o Scinc, Bijing , Chin Rcivd Dcbr 3, 01; ccptd Fbrury 18, 013; publihd onlin Mrch 14, 014 Bd on nk nd Mur tigu odl nd Griith thory or rctur, n nrgy-quilibriu odl w propod to xplin th coplx tr ct on tigu bhvior. Whn th ution o th ltic trin nrgy rl nd th tord trin nrgy o ccuultd diloction rch th urc nrgy o crck, th tigu crck will initit in tril. According to thi odl, or ultixil tr condition, th orinttion o th crck initition nd th initition li cn b dducd ro th nrgy quilibriu qution. For th unixil tigu loding with n tr, th rltion btwn th xiu tr or th iniu tr nd th tr plitud i in grnt with n llip qution on th contnt li digr. I th rtio o th n tr to tr plitud i l thn criticl vlu 0.17, nd th tr plitud kp contnt, th tigu crck initition li will dcr with th incr o th copr n tr. In thi odl, th n tr do not cu dg ccuultion with th tigu cycl in crck initition. For thi ron, th loding qunc o dirnt lod lvl would induc th cuultiv dg to dvit ro th Plgrn-Minr cuultiv dg rul. h procdur o titing th dg undr rndo loding i lo dicud. tigu initition, n tr ct, ultixil tigu, nk-mur odl, dg ccuultion PACS nubr(): , 6.0.Mk, Np Cittion: Zho S C, Xi J J, Zho A G, t l. An nrgy-quilibriu odl or coplx tr ct on tigu crck initition. Sci Chin-Phy Mch Atron, 014, 57: 91696, doi: / z Noncltur Hl grin iz ilt ngl o th norl dirction o pln ro th tnion dirction ΔU Accuultd diloction nrgy pr hl cycl W Spciic rctur nrgy K I, K II Str intnity ctor o od I nd od II crck G Eltic trin nrgy rl rt or crting unit r o crck S Ar o crck urc Norl tr n *Corrponding uthor (il: xijj@ln.ich.c.cn) Mn tnil tr Aplitud o tnil tr Endurnc liit ut Ultit tnil trngth Shr odulu A μw /[(1ν) k ] (, ) Orinttion ngl o n rbitrry pln p Fctor or lip irrvribility k Ritnc o diloction ovnt Mn hr tr Aplitud o hr tr Shr tr Shr tr rng Loding priod Ftigu cycl or crck initition N Scinc Chin Pr nd Springr-Vrlg Brlin Hidlbrg 014 phy.cichin.co link.pringr.co

2 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No Poion rtio Ph dirnc btwn th tnil tr nd th hr tr Prtr or cobintion o coplx loding. 1 Introduction Ftigu ilur i on o th ot coon od o ilur or chnicl coponnt. In gnrl c, coponnt in rvic r ubjct to coplx tr, uch loding with n tr or ultixil tr. Howvr, ot o th tigu dt r urd undr unixil loding condition. So, it i vry iportnt to dvlop th critrion o tigu ilur or pl undr coplx loding. hn th tr-li curv o unixil loding could b ud to th tigu li o coponnt undr coplx loding. h invtigtion o n tr ct o tigu bhvior r trcd to th rly ti o th tigu rrch [1 6]. h rltionhip o Goodn, Morrow, nd Grbr cn b ound in ny txtbook on tigu. Howvr, ll th thod r piricl thod. h icro chni o tigu dg with th n tr i not rlctd in th odl. Siilr to th invtigtion o n tr ct, ny dirnt thod [7 9] wr propod on th nt o th tigu li undr ultixil tigu. You nd L [8] cliid th thori into iv typ, i.. piricl orul nd odiiction o th Coin-Mnon qution; ppliction o tr or trin invrint; u o th pc vrg o tr or trin; criticl pln pproch; nd u o nrgy which h ccuultd on th tril undr conidrtion. O th dirnt thod, th thod o criticl pln pproch [9] i th ot populr or ultixil tigu nt bcu o it ctivn nd brod ppliction. Sin, Findly, McDirid, Dng Vn, tc. [7], propod th rpctiv tr-bd criticl pln odl or tigu dg critri. In bri, linr or nonlinr cobintion o hr tr nd norl tr cting on th criticl pln w clcultd to b n quivlnt o unixil loding. h orinttion o th criticl pln w lo dtrind with dirnt criticl pln concpt, uch th pln o xiu hr tr or th pln o xiu hr tr rng. Siilr to tho bd on tr nlyi, th pproch bd on trin nlyi wr lo propod by Brown nd Millr, Fti nd Soci, Sith, Wton nd oppr [7], rpctivly. All th thod uccdd in o c o ultixil tigu nt. Howvr, th odl wr lo piricl. Prtr in th odl hould b dtrind by uicint xprintl dt or dirnt tril. h chni o tigu dg or ultixil tigu i not vry clr in th odl. In gnrl, or polycrytllin tl or lloy, th icro chni o tigu dg i rltd to th xtruion or intruion long th priting lid bnd in th grin [,10,11]. Ftigu crck would initit t th it. Bd on diloction ccuultion with cyclic loding, Mur nd coworkr propod odl [1 14] or tigu crck initition. Chn [15] xtndd nk nd Mur odl to includ crck iz nd rlvnt icro tructurl prtr in th qution. Howvr, in th odl, th prtr o tr i only th hr tr rng on th lip pln. So, th odl cn not b ud to dcrib th tigu undr th loding c with n tr or ultixil tr. Although th tigu li vri with th loding condition, th icro chni o tigu dg ut b th. So, thr ut b uniid qution to dcrib th tigu li undr dirnt loding condition, including th n tr ct nd ultixil loding. Bd on thi point nd nk & Mur tigu odl, thi ppr ttpt to propo odiid Enrgy-Equilibriu thod or nt o tigu with coplx tr. hi nw odl provid uniid critrion or ultixil tigu. h n tr ct nd ultixil tr ct r dicud in dtil. Modl nd nlyi.1 Enrgy-quilibriu odl In 1981, nk nd Mur [1] propod diloction odl or tigu crck initition. hy pointd out tht i th tord trin nrgy du to diloction ccuultd tr n-cycl bc qul to th urc nrgy, crck with th lngth o grin iz would initit. N U 4 W, (1) whr N i th cycl o loding or crck initition, ΔU i th ccuultd nrgy pr hl cycl, i th hl lngth o th grin iz, nd W i th pciic rctur nrgy or unit r. Ltr, Mur nd coworkr [13,14] dvlopd thir odl bd on r nrgy conidrtion. h totl r nrgy o th yt incr with tigu cycl nubr du to th incr in tord ltic trin nrgy o ccuultd diloction. Whn th Gibb r nrgy chng ttin xiu t criticl nubr o cycl, icro crck would initit. In thi odiid odl, th chnicl nrgy rl w ddd in th nrgy quilibriu qution. Howvr, in thir ppr, th chnicl nrgy rl i rltd to th tr rng, but not th xiu tr. A dcribd in Griith rctur thory, thr i lwy ltic trin nrgy rl with th crck initition in olid. So th ltic trin nrgy rl with th oring o tigu crck i not ngligibl. Bd on thi point, w work out odiid nrgy blnc qution or tigu crck initition ollow:

3 918 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No. 5 S G S N U SW xd, () 0 whr, G i th ltic trin nrgy rl rt or crting unit r o crck. h irt it t th lt o th qution i th xiu vlu o virtul ltic trin nrgy rl in on cycl o loding or crting crck with n r o S. Obviouly, thi vlu hould b dtrind by th xiu loding intd o th loding rng in on cycl. h cond it t th lt id o th qution i th diloction nrgy ccuultd with th nubr o th cycl, which i th th it in nk-mur odl. h it t th right id o th qution i th urc nrgy or crting tigu crck with n r o S. ht i to y, i th u o th ltic trin nrgy rl nd tord trin nrgy du to diloction ccuultd tr n-cycl bco qul to th urc nrgy, tigu crck will initit. For coplx condition, tr tnor vri with th ti. hi n tht not only th vlu but lo th dirction o th in tr chng with th ti. And th tr condition on n rbitrry pln lo vri with th ti. A hown in Figur 1(), or n rbitrry pln (, ), th norl tr n nd th hr tr will b olvd by uing coordintion trnor. According to th vlu o th norl tr n nd th hr tr, th xiu o virtul ltic trin nrgy rl in on cycl o loding or crting crck o r S in thi pln could b clcultd by uing linr ltic rctur chnic thod. Hr, i n i coprd, thr will b no od I typ ltic trin nrgy rl. h ccuultd diloction nrgy pr hl cycl cn b titd with th xiu rng o hr tr in thi pln. A hown in Figur 1(b), in th pln (, ) th hr tr chng with th ti. In on cycl, th trc o th hr orc vctor or clod curv in th pln. h xiu clipr ditr i th rng o th hr tr in thi pln. hn, ccording to q. (), th tigu crck initition li N in tr o th loding prtr n, nd on th pln (, ) could b producd. h iniu o th tigu li N with th chng o th orinttion (, ) i th tigu crck initition li o th pl. And th pln (, ) which i rltd to th iniu o th tigu li N i th orinttion o th tigu crck.. nion-orion tigu crck initition For 3-D crck initition, th nlyi i vry coplictd. Howvr, or nion-orion tigu, it' rltivly ipl. Fro q. (), w hv 1 v 0 1 vc n 1 K K d( ) N U 4W I II x x v k N 4W 8W C n N k, x 1v whr n i th norl tr on crck urc, i th hr tr on crck urc, k i th ritnc o diloction ovnt on th lid pln, i th rng o hr tr on crck urc in on cycl o loding, nd it ut b grtr thn th k. C i gotricl ctor o th crck. Hr, i n i coprd, thn thr will b no ltic trin nrgy rl o Mod I. Conidr pl ubjct to th tnion nd torion orc, hown in Figur. h wvor o loding i (3) Figur 1 (Color onlin) () Str cting on pln in th pl ignd pln (, ), whr, wr orinttion ngl o th norl dirction o thi pln in thr-dinionl coordint yt on th pl; (b) th trc o hr tr vctor on th pln (, ) in cycl or clod curv. h rng o th hr tr i th xiu clipr ditr o th r urroundd by th curv.

4 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No Figur A chtic illutrtion o tr cting on th urc o rod undr tnion-torion tigu loding. inuoidl nd th priod o tnion tr nd hr tr r th. h tr tnor i ollow: 11 1 ij 1 t t in in, (4) t in 0 whr, r th n tr nd th plitud o tnion tr rpctivly, nd, r th n tr nd th plitud o hr tr rpctivly. i th ph dirnc btwn th tnion tr nd th hr tr, nd i th priod o loding. For pln, in which th norl dirction tilt n ngl ro th tnion dirction, th norl tr nd th hr tr r clcultd ollow: t t in in, n in co t in coin t in co. h rng o hr tr on pln i co in co co co in. (7) (5) (6) Obviouly, th norl tr nd th hr tr in thi pln vry with th ti. h virtul ltic trin nrgy rl o od I nd II lo chng with th ti in on loding cycl. I th norl tr i tnil, th ltic trin nrgy rl includ od I nd od II prt. Othrwi, thr i only od II prt. A ntiond in q. (), th virtul ltic trin nrgy rl hould b th xiu o th poibl vlu in on loding cycl. hn, put th xiu vlu o th ltic trin nrgy rl nd th rng o hr tr in thi pln into th q. (3), nd th poibl tigu li o thi pln cn b dducd. Finlly, th tigu li o th pl i th iniu vlu o ll th tigu li with dirnt orinttion. In q. (3), both it t th lt id i rltd to orinttion ngl nd th irt it-ltic trin nrgy rl hould b th xiu o th poibl vlu in on loding cycl. Hnc, th nlyticl olution i vry coplictd, xcpt or pl undr unixil tigu loding. Nuricl iultion i or prcticl or pl undr coon tnion-torion tigu loding..3 nil tigu loding with n tr For th pl undr unixil tnil tigu loding, th tr tnor tk th or: t 11 1 in 0 ij Fro q. (5) (7), th norl tr nd th hr tr on th pln r clcultd ollow: t n in co, t in coin. And th rng o th hr tr on th pln i (8) (9) (10) in. (11) I thr i 0, th xiu vlu o ltic nrgy rl i t th /4, whn th norl tr i tnil in n rbitrry pln. So, in thi c, thr i joint od I nd od II ltic trin nrgy rl. Fro q. (3), w hv 1 C co N 4W 1 in k co in 8W 1 C N k

5 90 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No. 5 co in 1 1 A A C N x co in 1 1, A A C N (1) whr tr i norlizd by th twic rictionl tr or diloction ovnt. W A,,. k k 1 k hi qution dontrt tht, in ny pln, th xiu tr nd th plitud tr r in grnt with n llip qution. h tigu li N i dtrind ro q. (1) : N A C x co in 1. (13) Obviouly, th tigu crck initition li chng with th dirction o th pln. Lt th drivtiv o N qul zro, nd w hv N 0A tg C tg tg 0. (14) 4 3 x C x C x A Root o q. (14) giv out th dirction ngl o th crck initition urc. According to th rult o nuricl iultion with dirnt prtr, thi tilt ngl i l thn but clo to th 45 dgr, which i rltd to th pln with xiu hr tr. I, th norl tr in n rbitrry pln i lwy coprd in th loding cycl. So, in thi c, thr i only od II ltic nrgy rl nd th xiu vlu o virtul ltic trin nrgy rl ppr t th 3/4. Fro q. (3), w hv 1 C co in N 4W C co in N k 1 in k in 8W 1 in co in in 1 1 A A C N in in 1 1. A A C N (15) In thi c, th tilt ngl o tigu crck initition pln i lwy th 45 dgr. So, w hv 1 in A A C N 1. (16) I 0, th xiu vlu o virtul ltic trin nrgy rl i t th /4 or 3/4. In on cycl o loding, whn t = /4, th norl tr i tnil, o th ltic trin nrgy rl includ od I nd od II. Whn t = 3/4, th norl tr i coprd, nd th ltic nrgy trin rl includ only od II. So, i G G in K K 1. /4 I II /4 3 /4 KII 3 /4 (17) hn, th xiu vlu o virtul ltic nrgy rl ppr t th /4. ht i to y, i in 0 rcin, (18) th tigu li N () i olvd by uing q. (13), othrwi uing q. (16). A hown in Figur 3, q. (18) giv th boundry lin. At th lt id o th boundry lin, q. (16) i ud to clcult th tigu initition li. Bcu th tigu initition pln i lwy t or clo to th 45 dgr, criticl rtio o th n tr to th tr plitud cn b clcultd ollow. For 4, ro q. (18), w hv (19) I th rtio o th n tr to th tr plitud i l thn th criticl vlu, q. (16) will b ud to olv th tigu li. Othrwi q. (1) will b ud. Furthror, ccording to nk nd Mur odl

6 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No Figur 3 h boundry btwn th r o uing q. (1) nd th r o uing q. (16) or th clcultion o th ltic trin nrgy rl. [1], th rng o hr tr ut b grtr thn twic th rictionl tr o diloction on th lid pln. So, or th tilt ngl, w ut hv in rcin rcin. (0) ht n, or tigu undr unixil loding, th tigu initition pln will not ppr t th 0 dgr or 90 dgr tilt ro th loding dirction..4 orion tigu loding with n tr For th pl undr torion tigu loding, th tr tnor tk th or ij t 0 in. t in 0 (1) Fro q. (5) (7), th norl tr nd th hr tr on th pln r clcultd ollow: t n in in, t in co. And th rng o th hr tr on th pln i Fro q. (3), w hv () (3) co. (4) 1 C 1 k N 4 1 C N k 1, A A C N W 8W 1 (5) whr i 0, thr i ud t th lt id o thi qution. Othrwi, thr i. In thi condition, th tigu crck will lwy initit long th 0 dgr or 90 dgr, which i th pln o xiu hr tr. 3 Dicuion 3.1 h prtr in th odl In nk nd Mur odl [1], thr r o prtr o tril, uch : pciic rctur nrgy W, grin iz nd rictionl tr o th diloction k. It diicult to dtrin ll th prtr dirctly in prctic. Howvr, i th pl i only lodd undr onotonic incring tnil tr untill brk, th xiu o poibl tnil tr i th ultit tnil trngth o th pl, nd th orinttion o rctur urc would b 45 dgr tilt to th loding xil or ductil tl. hn, ro q. (1), w hv ut A, (6) whr C=1. And th ultit tnil trngth ut i norlizd by twic th rictionl tr or diloction ovnt. For ductil tl, in ddition to th ultit tnil trngth, th tru rctur trngth ut or th tigu trngth coicint yb othr poibl choic in q. (6). h rictionl tr o diloction k giv tigu liit. ht n tht i th hr tr i l thn th rictionl tr, thr will b no tigu dg. So, ndurnc liit cn b ud to ubtitut it. In nk nd Mur odl [1], it i ud tht th diloction ord by ch cycl o loding r irrvribl. hi uption i not rtionl, pcilly or th c whn th hr tr rng i vry clo to twic th rictionl tr o diloction. hi uption ld to th clcultd tigu li bing l thn th xprint rult or vrl ordr. Mur nd Nkon [14] dicud th rvribility

7 9 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No. 5 o diloction nd propod prtr in thir xtnion odl whr, vri ro 0 to 1, rprnting coplt rvribility to coplt irrvribility o lip. Mughrbi [10,11] propod iilr prtr p or th cyclic lip irrvribility. p will b zro whn th lip i ntirly rvribl, nd it will b qul to on whn th lip i ntirly irrvribl. According to ll th conidrtion, q. (1) will tk th ollowing or C pn x ut co in. (7) In q. (7), th prtr p i contnt ctor or lip irrvribility. Howvr, ntiond by Mughrbi [11], th lip irrvribility w ound to vry igniicntly in gnitud ro c to c. Highr loding plitud would induc highr irrvribility. Obviouly, th loding rquncy nd th tprtur r lo iportnt or lip irrvribility. h rthr rch o th ct o ll th prtr r out o thi ppr, nd hould b invtigtd in th utur work. Au tht th rltion btwn th lip irrvribility nd th loding plitud i powr lw unction: p B 1. (8) Hr, B nd r contnt or lip irrvribility. hn q. (7) will tk th or ollow: x ut BN ut x N B. (9) Hr, C=1 nd th crck initit lon 45 dgr tilt pln which i th pln with th xiu hr tr. hi orul i iilr to Bquin qution. Hr, th tigu trngth coicint nd th Bquin xponnt tk th or ollow: 1 x A A C N 1. (31) In ddition, ro q. (7), th rltion btwn th n tr nd th plitud tr tk th or ollow: US US 1 1 i, 1 C PN US US C PN 1 1 i. 1 (3) (33) h contnt li digr o thi odl i hown in Figur 4. Coprd to othr odl, thi odl giv n xtndd contnt li digr including copr n tr. In thi digr, i th tr plitud kp contnt, th tigu initition li will incr with th dcr o 1 th n tr whn, nd will dcr 1 1 with th dcr o th n tr whn. 1 ht i to y, th copr n tr i not lwy bnicil to prvnting tigu crck initition, lthough it i bnicil to dcring th tigu crck propgtion rt. In prctic, thr would b riction btwn th crck urc whn thr i norl copr tr on thi pln. Du to th riction o crck urc, th vlu o th ltic 1 ut x B 1, b. (30) 3. Mn tr ct 1 For th c o, with th incr o th 1 norl tnil tr, th tilt ngl would dcr ro 45 dgr to llr dgr. Howvr, in th long li tigu rgi, thi ngl i vry clo to 45 dgr. So, ro q. (1), th rltion btwn th xiu tr nd th plitud tr gr with th llip curv. Figur 4 (Color onlin) A chtic illutrtion o contnt li digr o thi odl. I / 0.17, th contnt li curv would b dcribd by q. (3), th dh curv in thi igur. Othrwi, th curv would b dcribd by q. (33), th olid curv in thi igur. h curv dcribd by Grbr rltionhip nd Goodn rltionhip r lo hown coprion.

8 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No nrgy rl o Mod II y b vry diicult to clcult. Although q. (33) i vry ipl in phyicl chni, th nlyi or tigu crck initition ight b or coplx or c with copr tr. Eq. (33) would b th low liit or crck initition with copr n tr. Dowling [3] collctd th xprint dt o thr dirnt lloy or th n tr ct in hi book. h coprion o th xprintl dt nd th prdiction by q. (3) i hown in Figur 5. Fro Figur 5, or ll th lloy, th iultion curv it th xprint dt vry wll, xcpt or i6al4v whn th tigu li incr to 10 5 cycl or longr. And it tht th iultion prtr p or cyclic lip irrvribility dcr with th incr o th tigu li. h rng o p i ro 10 5 to 10 8, which i uch lowr thn tho collctd by Mughrbi [11]. In hi ppr, th irrvribility p o o pur tl or ingl ph lloy including coppr, -br, nd -iron vri ro 1 to Obviouly, th icrotructur o tllic lloy i uch or coplx thn pur or ingl ph tl. h cyclic lip irrvribility o diloction would b vry nitiv to th icrotructur. Bcu o th iportnc o p in tigu dg ccuultion, th cyclic lip irrvribility or dirnt tl or lloy hould b invtigtd in th utur work. 3.3 Criticl pln thod or ultixil tigu Eq. (3) giv th gnrl rltionhip btwn th tigu initition li nd th norl tr, hr tr, nd th hr tr rng on n rbitrry pln. Subtituting q. (6) into q. (3) nd conidring th lip irrvribility or n rbitrry pln, i th norl tr i tnil, w hv C n pn k x ut ut C n x 4 pn, k (34) whr th it t th right id o thi qution i nonlinr cobintion o th tr prtr on thi pln nd th tril prtr. hi cobintion i rltd to th virtul li or n rbitrry pln, which vri with th pln orinttion. I w din prtr th cobintion o th loding condition, th coplx loding could b convrtd to th ctiv unixil tr plitud. hn, th tigu li could b titd with th S-N curv undr unixil loding. ut n x ut 1 ut. 1 (35) Figur 5 (Color onlin) Contnt li digr or thr dirnt lloy. h ybol in th igur r xprint dt ro rrnc [3]. h continuu curv r iultion rult or dirnt tigu li. () A517 tl. (b) i6al4v. (c) Hr, i th prtr which i rltd to th tigu li nd th crck orinttion. For n rbitrry pln, th orinttion with th iniu vlu o th i th tilt ngl o th crck pln. For ll th loding condition, th vlu o n th tigu li. Obviouly, thi odl blong to th nrgy odl or ultixil tigu. And ll th nrgi or crck initition could b clcultd with th norl tr nd hr tr on n rbitrry pln. Dirnt ro othr nrgy odl or ultixil tigu, th virtul ltic trin nrgy rl in thi odl do not cu ccuultd dg with th loding cycl or crck initition. hi odl cn b ud or proportionl or nonpropor-

9 94 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No. 5 tionl loding without ny chng in th prtr. Howvr, it cn only b ud to tit th tigu crck initition li or polycrytllin tl nd lloy, but not th totl li. Obviouly, tigu crck initition tnd to ppr on th pln with xiu hr tr, whil th crck propgtion tnd to ppr lon th pln with xiu norl tr. h chni o dg ccuultion would chng in th cour o crck growth. So, thi odl i not good nough to th tigu crck growth or ultixil tigu. 3.4 Dg ccuultion nd rndo tigu According to nk nd Mur odl [1], tigu crck will initit whn th ount o tord diloction nrgy o ch cycl rch th vlu o urc nrgy. Obviouly, nk nd Mur odl coincid with th Plgrn-Minr cuultiv dg rul, which n tht th lod qunc h no ct on th tigu dg ccuultion. Howvr, in thi nw odl, th linr dg rul i not corrct. Although th ount o tord diloction nrgy ccuult linrly with th tigu cycl, th virtul ltic trin nrgy do not produc ny dg with th tigu cycl, until th lt cycl o th crck initition. For xpl, pl i ttd undr two unixil tr condition on tr nothr. h irt tr condition i t th xiu tr-tr plitud cobintion ( x1, 1 ), nd th tigu li undr thi tr condition i N 1. h othr tr condition i t th ( x, ), nd th tigu li undr thi tr condition i N. According to th nrgy-quilibriu q. (3), ipliid or i ollow: E N U E (36) i i i whr E i i th virtul ltic trin nrgy rl, which i rltd to th xiu tr lvl xi. ΔU i i th ccuultd diloction nrgy in on cycl, which i rltd to th tr plitud i. E i th urc nrgy or producing, crck with lngth o grin ditr. A hown in Figur 6, thr r two c. In th irt c, th pl i lodd undr ( x1, 1 ) or n 1 cycl, thn undr ( x, ) or n cycl to brk. In th cond on th pl i lodd undr ( x, ) or n cycl, thn undr ( x1, 1 ) or n 1 cycl to brk. Lt x1 < x, o w hv E 1 <E. Fro q. (36), or th irt c, w hv E n U n U E n E E n For th cond c, w hv N1 E E N n1 n 1. (37) N1 N E n U n U E n n E E N1 N E E1 n n 1 1. (38) N1 N Obviouly, bcu th poibl ltic trin nrgy rl i dirnt or th dirnt loding condition, only i x1 qul x, th loding qunc will induc th cuultiv dg to dvit ro th Plgrn-Minr cuultiv dg rul. Howvr, thi concluion i contrry to ny rrchr work [16,17] nd th xprint rult. A ronbl xplntion i tht thi odl i or tigu crck initition, but not or th crck propgtion. Othrwi, in nk nd Mur odl, thr i no lod qunc ct on th dg ccuultion o diloction. Actully, dicud rlir, th cyclic lip irrvribility would vry with th loding condition. So, th ct o loding hitory on volution o diloction hould b invtigtd in th utur or dply. For rndo tigu, th gnrl proc includ cycl Figur 6 A chtic illutrtion o th nrgy ccuultion or two dirnt lod qunc. h rrow lin dcrib th nrgy ccuultd with th tigu cycl. h dh lin dcrib tht o th pl undr only ( x1, 1 ), th point lin dcrib tht o th pl undr only ( x, ). Hr, lt x1 < x, o E 1 <E. () Spl lodd undr ( x1, 1 ) or n 1 cycl, thn undr ( x, ) or n cycl to brk; (b) pl lodd undr ( x, ) or n cycl, thn undr ( x1, 1 ) or n 1 cycl to brk.

10 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No counting nd ubqunt dg ccuultion clcultion. hrough cycl counting, rndo tr hitory could b rolvd to t including ri o plitud-n tr cobintion ( i, i ). hn, thod propod by Goodn, Grbr, Morrow tc. r ud to convrt ch plitud-n cobintion to quivlnt copltly rvrd tr plitud ri rpctivly. Finlly, ro n S-N curv or tr rtio R=1, th ccuultd dg cn b titd by uing Plgrn-Minr rul. Howvr, dicud rlir in thi ction, th n tr do not bring ccuultiv dg with th tigu cycl. According to thi odl, th ccuultiv diloction nrgy i only rltd to th tr plitud lvl. h convrion o plitud-n cobintion ( i, i ) to ri would bring rror in dg clcultion. Howvr, th xiu tr MAX ( i + i ) or th iniu tr MIN ( i i ) in th whol loding hitory i iportnt, which dtrin th xiu poibl ltic trin nrgy rl in th tigu crck initition. hr r ny dirnt thod ud or cycl counting. Rinlow thod [,3] ight b th ot populr on. Fro our odl, th cycl counting would b uch iplr. Wht w hould do i jut rcording th rng btwn ch pk nd th djcnt vlly. A hown in Figur 7, th lod hitory would b rolvd to t o ipl rng Δ i. For xpl, Δ 1 = B A. Hr, i Δ i i l thn twic th riction tr o diloction, thi gnt hould b nglctd. For xpl, i Δ 6 = G F < k, th gnt F-G hould b nglctd, nd thr i only gnt E-H rthr thn E-F, F-G, nd G-H. h xiu tr t point D would b ud to clcult th virtul ltic trin nrgy rl, o it hould b rcordd in th cycl counting. Bd on q. (9), th ccuultiv dg or rndo tigu could b ollow: n x i ut N B i (39) Hr, n i th nubr o th tr rng in th rndo loding pctru. N i th rptition nubr o th rndo loding pctru or tigu crck initition. x i th xiu tr in th rndo loding pctru. 4 Concluion Bd on nk nd Mur odl, n nrgy-quilibriu odl or coplx tr ct on tigu crck initition i propod in thi ppr. Fro thi odl, o concluion rchd in thi invtigtion r ollow: (1) h ltic trin nrgy nd th tord diloction nrgy both contribut to th tigu crck initition. Whn th ution o th ccuultd tord diloction nrgy nd th virtul ltic trin nrgy rl rch th urc nrgy or producing crck with th iz o grin ditr, th tigu crck will initit. () For pl undr unixil tigu loding, th rltion btwn th xiu tr or th iniu tr nd th tr plitud i in grnt with n llip qution on n rbitrry pln. (3) hr i criticl rtio o n tr to tr plitud, which qul bout I th rtio o n tr to tr plitud i l thn th criticl vlu, nd th plitud kp contnt, th tigu initition li will dcr with th dcr o th n tr. Copr n tr i not lwy bnicil to prolonging th li o crck initition. (4) Bcu o th ct o ltic trin nrgy rl, th Plgrn-Minr rul will yild incorrct rult in dg ccuultion or tigu crck initition. (5) h trditionl procdur or titing th dg undr rndo loding ight b incorrct, or th n tr do not bring ccuultiv dg with tigu cycl. h uthor wih to thnk Pro. HONG Y S nd Dr. SUN C Q o th Intitut o Mchnic, Chin Acdy o Scinc or hlpul dicuion in thi work. hi work w upportd by th Ntionl Bic Rrch Progr o Chin (Grnt No. 01CB937500), nd th Ntionl Nturl Scinc Foundtion o Chin (Grnt No nd 11010). Figur 7 A chtic digr o rndo loding hitory with th ti. h ipl rng btwn th pk nd djcnt vlly hould b rcordd in th cycl counting, o do th xiu tr or iniu tr in th whol wv. h ipl rng l thn k hould b nglctd. 1 Sndckyj G P. Contnt li digr hitoricl rviw. Int J Ftigu, 001, 3: Surh S. Ftigu o Mtril. Nw York: Cbridg Univrity Pr, , Dowling N E. Mchnicl Bhvior o Mtril: Enginring Mthod or Dortion, Frctur, nd Ftigu. Englwood Cli, NJ: Prntic Hll, Nihi M, Hulr P, Bollr C, t l. Evlution o n tr ct on tigu li by u o dg prtr. Int J Ftigu, 1986, 8(3): Dowling N E, Clhoun C A, Arcri A. Mn tr ct in tr-li tigu nd th Wlkr qution. Ftigu Frct Eng Mtr Struct, 009, 3: Dowling N E. Mn tr ct in trin li tigu. Ftigu Frct Eng Mtr Struct, 009, 3:

11 96 Zho S C, t l. Sci Chin-Phy Mch Atron My (014) Vol. 57 No. 5 7 Soci D F, Mrqui G B. Multixil Ftigu. Wrrndl, P.: Socity o Autootiv Enginr, Inc., You B R, L S B. A criticl rviw on ultixil tigu nt o tl. Int J Ftigu, 1996, 18(4): Krolczuk A, Mch E. A rviw o criticl pln orinttion in ultixil tigu ilur critri o tllic tril. Int J Frctur, 005, 134: Mughrbi H. On th li-controlling icrotructurl tigu chni in ductil tl nd lloy in th gigcycl rgi. Ftigu Frct Eng Mtr Struct, 1999, : Mughrbi H. Cyclic lip irrvribiliti nd th volution o tigu dg. Mtll Mtr rn A, 009, 40A: nk K, Mur. A diloction odl or tigu crck initition. J Appl Mch, 1981, 48: Vnktrn G, Chung Y W, Nkon Y, t l. Fr nrgy orultion o tigu crck initition long pritnt lip bnd: Clcultion o S-N curv nd crck dpth. Act Mtll Mtr, 1990, 38(1): Mur, Nkon Y. A thory o tigu crck initition in olid. J Appl Mch, 1990, 57: Chn K S. A icrotructur-bd tigu-crck-initition odl. Mtll Mtr rn A, 003, 34A: Mcqu G, Grcib S, Arouch A, t l. Squntil lw in ultixil tigu, nw dg indictor. Int J Ftigu, 005, 7: Košút J. Hitory inlunc xponnt in cuultiv tigu dg dtrind uing two-tp loding xprint. Ftigu Frct Eng Mtr Struct, 00, 5: