Stress and Strain Analysis of Notched Bodies Subject to Non-Proportional Loadings

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1 World Acdmy of Scinc Enginring nd Tchnology Intrntionl Journl of Mchnicl nd Mchtronics Enginring Strss nd Strin Anlysis of Notchd Bodis Subjct to Non-Proportionl Lodings A. Inc Intrntionl Scinc Indx Mchnicl nd Mchtronics Enginring wst.org/publiction/ Abstrct In this ppr n nlyticl simplifid mthod for clculting lsto-plstic strsss strins of notchd bodis subjct to non-proportionl loding pths is discussd. Th mthod ws bsd on th Nubr notch corrction which rlts th incrmntl lstic nd lstic-plstic strin nrgy dnsitis t th notch root nd th mtril constitutiv rltionship. Th vlidity of th mthod ws prsntd by compring computd rsults of th proposd modl ginst finit lmnt numricl dt of notchd shft. Th comprison showd tht th modl stimtd notch-root lsto-plstic strsss strins with good ccurcy using linr-lstic strsss. Th prosd modl provids mor fficint nd simpl nlysis mthod prfrbl to xpnsiv xprimntl componnt tsts nd mor complx nd tim consuming incrmntl non-linr FE nlysis. Th modl is prticulrly suitbl to prform ftigu lif nd ftigu dmg stimts of notchd componnts subjctd to nonproportionl loding pths. Kywords Elsto-plstic strss-strin notch nlysis nonprortionl lodings cyclic plsticity ftigu. I. INTRODUCTION OTCHES nd othr gomtricl irrgulritis cus strss Nconcntrtions in mny nginring componnts. A strss incrs inducd by strss concntrtions strsss rsults oftn in loclizd plstic dformtion lding to ftigu filur. Thrfor th ftigu nd durbility stimtions of notchd componnts rquir dtil knowldg of notch-root strsss nd strins [3] [14] [15]. Th strss stt in th notch root rgion is in most css multixil in ntur. Axls nd shfts my xprinc for xmpl combind complx bnding nd torsion lods. Although Finit Elmnt (FE) commrcil softwr pckgs cn b usd to dtrmin notch root strsss nd strins in lstic nd lstic plstic bodis with high ccurcy for short loding historis such mthods r still imprcticl in th cs of long loding historis xprincd by mchins in srvic. Thrfor non-linr finit lmnt nlysis of long lod history would rquir prohibitivly long computing tim. For this rson mor fficint mthods of lstic-plstic strss nlysis r ndd for widly usd nginring pplictions. On such mthod suitbl for clculting multixil lstic-plstic strsss nd strins in notchd bodis subjctd to proportionl nd non-proportionl loding historis is proposd in this ppr. In rly rsrchs lsticity thory or photolstic nlysis ws usd to dtrmin thorticl strss concntrtion fctors. Nubr [1] proposd th most wll-known notch corrction A. Inc is with th School of Enginring Tchnology Purdu Univrsity Wst Lfytt IN USA (phon: ; -mil: inc@ purdu.du). mthod Nubr s rul stts tht th totl strin nrgy dnsity t th notch quls to th fictitious strin nrgy dnsity s if mtril hypothticlly bhvd lstic. Molski nd Glink [2] proposd th quivlnt strin nrgy dnsity (ESED) mthod. Th ESED mthod stts tht th strin nrgy dnsity t notch is qul s if body hypothticlly bhvd lstic. Similr pprochs wr proposd by [4]-[6]. All mthods consist of two prts nmly th constitutiv qutions nd th notch corrction rlting th psudo linr lstic strss-strin stt t th notch root with th ctul lstic-plstic strssstrin rspons. Rcnt notch corrction studis [7] [10] hv bn conductd to show tht th intgrtion of notch corrction mthod(s) nd th mtril constitutiv modl cn b usd to clcult th ctul lstic-plstic strins nd strsss. Th simplifid multixil strss-strin notch nlysis mthod is dvlopd hr to show tht th proposd modl is fficint nd ccurt by th compring computd locl strssstrin rsponss with ANSYS non-linr FE rsults. Simplifid nlyticl modls r prfrrd du to th simplicity computtionl fficincy nd low cost s comprd to complx nd long computing tim of th finit lmnt mthods. II. STRESS-STRAIN STATE AT NOTCH ROOT For th cs of gnrl multixil loding pplid to notchd body th stt of strss nr th notch root is tri-xil. Howvr th strss stt t th notch root is bi-xil bcus of th notch-root strss for fr surfc s shown in Fig. 1. Sinc quilibrium of th infinitsiml lmnt t th notch root must b mintind i.. 32 nd 32 thr r thr non-zro strss componnts nd four non-zro strin componnts. Svn fictitious linr lstic strss nd strin componnts ( ) cn b obtind from th linr lstic solution; howvr th ctul lstic-plstic strss nd strin componnts ( ) t th notch root r unknown. Thrfor thr r svn unknowns ll togthr nd st of svn indpndnt qutions is rquird for th dtrmintion of ll ctul lsto-plstic strss nd strin componnts t th notch root Th mtril constitutiv rltionships provid four qutions nd rmining thr qutions cn b obtind from th incrmntl form of th originl Nubr rul. (1) Intrntionl Scholrly nd Scintific Rsrch & Innovtion 9(6) scholr.wst.org/ /

2 World Acdmy of Scinc Enginring nd Tchnology Intrntionl Journl of Mchnicl nd Mchtronics Enginring th quivlnt strss incrmnt whr is Poisson s rtio E is lstic modulus G is shr modulus. Th function p q f q is idnticl to th plstic strin strss rltionship obtind xprimntlly from unixil tnsion tst Four notch strin incrmnts ( 11 ) nd thr notch strss incrmnts ( ) in form svn unknowns to b solvd. IV. NEUBER NOTCH CORRECTION STRESS-STRAIN RELATION Th Nubr rul [4] cn b writtn for th unixil nd multixil strss stt in th form of (7) nd (8) rspctivly. Intrntionl Scinc Indx Mchnicl nd Mchtronics Enginring wst.org/publiction/ Fig. 1 Strss nd strin stt t notch root III. MATERIAL CONSTITUTIVE MODEL Th Prndtl-Russ rltion [11] [12] [13] is considrd on of th most frquntly usd modls in th incrmntl plsticity. Th totl strin incrmnt cn b xprssd in th form of th Prndtl-Russ strin-strss rltionship. p 1 v v 3 q kk S (2) E E 2 Th ctul dvitoric strss componnts in th notch root cn nlogously b dfind s: 1 S kk 3 kk 11 whr r th ctul strss componnts nd is th Kronckr dlt. Th incrmntl dvitoric strss-strin rltions bsd on th ssocitd th Prndtl-Russ flow rul cn b subsquntly writtn s: whr: S S S S d d d d S S S S E 2 p q d q df ( ) p q q q d q 3 q SS 2 q (5) (6) (7) (8) It should b notd tht th originl Nubr rul ws dvlopd for bodis in pur shr strss stt. It mns tht th Nubr qution stts th quivlnc of only distortionl strin nrgis. In ordr to formult th st of ncssry qutions for multixil nlysis of lstic-plstic strsss nd strins t th notch root th qulity of incrmnts of th totl distortionl strin nrgy dnsity should b usd Inc t l. [10] proposd to us th quivlnc of incrmnts of th totl distortionl strin nrgy dnsity contributd by ch pir of ssocitd strss nd strin componnts i.. S S S S S S S S S S S S Fig. 2 Grphicl rprsnttion of th incrmntl Nubr rul (9) whr p q is th quivlnt plstic strin incrmnt q is Th qulitis of strin nrgy incrmnts for ch st of Intrntionl Scholrly nd Scintific Rsrch & Innovtion 9(6) scholr.wst.org/ /

3 World Acdmy of Scinc Enginring nd Tchnology Intrntionl Journl of Mchnicl nd Mchtronics Enginring Intrntionl Scinc Indx Mchnicl nd Mchtronics Enginring wst.org/publiction/ corrsponding hypothticl lstic nd ctul lstic-plstic strins nd strss incrmnts t th notch root cn b shown grphiclly in Fig. 2. Th r of dottd rctngls rprsnts th totl strin nrgy incrmnt of th hypothticl lstic notch root input strsss whil th r of th htchd rctngls rprsnts th totl strin nrgy dnsity of th ctul lstic-plstic mtril rspons t th notch root. Consquntly combintion of four qutions from th lstic plstic constitutiv in (6) nd thr qutions from th quivlnc of incrmnts of th totl distortionl strin nrgy dnsity in (9) yilds th rquird st of svn indpndnt qutions ncssry to compltly dfin lstic plstic notch-root strin nd strss rsponss for notchd componnt subjctd to multixil non-proportionl lods. Th combind st of qutions r writtn s st of svn simultnous qutions from which ll unknown dvitoric strin nd strss S incrmnts cn b clcultd. Th clcultd dvitoric strss incrmnts S cn subsquntly b convrtd into th ctul strss incrmnts using (10): S 1 3 S 1 3 S (10) V. RESULTS AND DISCUSSION Th st of dt ws obtind for multixil non-proportionl lod pths in ordr to vrify th prdiction ccurcy of th proposd mthod. For tht purpos svrl non-linr Finit Elmnt nlyss hv bn crrid out using two diffrnt lod pths. For non-proportionl strss pths th ccurcy of th mthod ws dmonstrtd by compring th clcultd notch root strss-strin rsponss to thos obtind from th finit lmnt mthod. Th lstic-plstic finit lmnt strss rsults wr obtind using th ANSYS finit lmnt softwr. Th isotropic strin-hrdning plsticity modl ws usd for clcultions. Th gomtry of th notchd lmnt ws tht of th circumfrntilly notchd br shown in Fig. 7. Th bsic proportions of th cylindricl notchd br wr /t = 1 nd R/t = 2 rsulting in th tnsil nd torsion strss concntrtion fctor K F = 1.41 nd K T = 1.15 rspctivly. Th rtio of th notch root hoop strss to th xil strss undr tnsil xil loding ws / = Th ctul rdius of th cylindricl spcimn ws R=25.4 mm. Th mtril for th notchd br ws SAE 1070 stl with cyclic strss-strin curv pproximtd by th Rmbrg- Osgood rltion in (11). Th mtril proprtis wr: E = 210 GP = 0.3 S Y = 242 MP n =0.199 nd K = 1736 MP. whil th nominl strsss in th nt cross sction wr dtrmind s: = n P R t 2 nd = n T R t 3 (12) Th first lod pth pplid to th notchd br wr monotoniclly incrsing torsion in th first phs nd thn incrsing tnsion in th scond phs whil th torsion lod bing kpt constnt s shown in Fig. 3. Th torqu T inducd th linr lstic shr strss t th notch root nd th xil lod F inducd th norml strss componnts nd. Th incrmnts of th hypothticl lstic strss componnts nd nd ssocitd strins wr usd s th input to th modl. Th psudo lstic quivlnt strss of th input t th notch root ws incrsing throughout th ntir loding procss to nsur monotonic loding pth. Th computd strss nd strin rsponss from th nlyticl proposd modl wr comprd with th lstoplstic strss-strin componnts obtind from th non-linr FE nlysis in in Figs. 4 nd 5. Fig. 4 shows th computd nd FE dtrmind strss componnts nd nd Fig. 5 shows th computd nd th FE modl dtrmind strin componnts nd. Not tht th clcultd strss nd strin rsponss nd th rsults of th finit lmnt nlysis r idnticl in th lstic rng. This is xpctd sinc th modl convrgs to th lstic solution in th lstic rng. Howvr byond th yild point t th notch root th strss nd strin rsults tht wr prdictd using th proposd modl nd th finit lmnt dt bgin grdully to divrg. As sn from Figs. 4 nd 5 tht th modl slightly undrstimts th ctul notch root shr strsss nd strins for th first lod pth but th prdictd strsss nd strins r vry clos to th numricl FE dt. Fig. 3 Th non-proportion torsion-tnsion lod pth 1 ' n ' E K (11) Intrntionl Scholrly nd Scintific Rsrch & Innovtion 9(6) scholr.wst.org/ /

4 World Acdmy of Scinc Enginring nd Tchnology Intrntionl Journl of Mchnicl nd Mchtronics Enginring strins for th scond lod pth but th prdictd strsss nd strins r in good grmnt with th numricl FE dt. Intrntionl Scinc Indx Mchnicl nd Mchtronics Enginring wst.org/publiction/ Fig. 4 Comprison of th clcultd nd FEM dtrmind strss pths for th non-proportionl torsion-tnsion lod pth Fig. 5 Comprison of th clcultd nd FEM dtrmind strin pths for th non-proportionl torsion-tnsion lod pth Th scond lod pth pplid to th notchd br wr monotoniclly incrsing tnsion in th first phs nd thn incrsing torsion in th scond phs whil th tnsion lod bing kpt constnt s shown in Fig. 6. Th xil lod F inducd th linr lstic norml strss componnts nd nd th torqu T inducd th linr lstic shr strss t th notch root. Th incrmnts of th hypothticl lstic strss componnts nd nd ssocitd strins wr usd s th input to th modl. Th psudo lstic quivlnt strss of th input t th notch root ws incrsing throughout th ntir loding procss to nsur monotonic loding pth. Th computd strss rsponss nd th strss rsponss from th non-linr FE nlysis r comprd in Fig. 7. Similrly th computd strin rsponss nd th non-linr FE strin rsponss r comprd in Fig. 8. As xpctd th clcultd strsss nd strins nd th rsults of th finit lmnt nlysis r sm in th lstic rng. Howvr ftr yilding t th notch root th strin rsults tht wr prdictd using th proposd modl nd th finit lmnt dt bgin grdully to divrg. Figs. 7 nd 8 show tht th modl slightly undrstimts th ctul notch root xil strsss nd Fig. 6 Th non-proportion tnsion-torsion lod pth Fig. 7 Comprison of th clcultd nd FEM dtrmind strss pths for th non-proportionl tnsion- torsion lod pth Fig. 8 Comprison of th clcultd nd FEM dtrmind strin pths for th non-proportionl torsion-tnsion lod pth VI. CONCLUSION In this ppr th simpl nd fficint nlyticl mthod for clculting lsto-plstic notch root strss nd strins inducd Intrntionl Scholrly nd Scintific Rsrch & Innovtion 9(6) scholr.wst.org/ /

5 World Acdmy of Scinc Enginring nd Tchnology Intrntionl Journl of Mchnicl nd Mchtronics Enginring Intrntionl Scinc Indx Mchnicl nd Mchtronics Enginring wst.org/publiction/ by multixil non-proportionl lod pths hs bn proposd. Th mthod hs bn formultd using with th mtril constitutiv rltion nd th incrmnts of th totl strin nrgy dnsity bsd on th Nubr corrction rul. Th proposd modl hs bn vlidtd with finit lmnt dt obtind for two non-proportionl loding pths. Bsd on th comprison btwn numricl dt from th non-linr FE nlysis nd clcultd strss nd strin rsponss from th modl for two diffrnt non-proportionl lod pths th simplifid modl clcultd notch strsss strins with rsonbl ccurcy using linr-lstic strsss. Th proposd modl provids mor fficint nd simpl nlyticl pproch to clcult notch root lstic-plstic strsss nd strins for th notch componnt undr th multixil non-proportionl lodings. Th clcultd notch root strsss nd strins cn b subsquntly usd to prform ftigu lif stimtions for th notchd componnts subjct to complx multixil lodings. REFERENCES [1] Nubr H. Thory of Strss Concntrtion for Shr Strind Prismtic Bodis with Arbitrry Strss Strin Lw Journl of Appl. Mchnics Vol. 28 pp [2] Molski K. Glink G. A Mthod of Elstic-Plstic Strss nd Strin Clcultion t Notch Root Mtrils Scinc nd Enginring Vol. 50 pp [3] Inc A. nd Glink G. A Gnrlizd Dmg Prmtr for Multixil Ftigu Lif Prdiction undr Proportionl nd Non- Proportionl Lodings Intrntionl Journl of Ftigu Vol.62 pp [4] Hoffmnn M. Sgr T. A Gnrlizd Mthod for Estimting Multixil Elstic Plstic Notch Strsss nd Strins Prt I: Thory Journl of Enginring Mtrils nd Tchnology Vol. 107 pp [5] Moftkhr A. Buczynski A. nd Glink G. Clcultion of Elsto- Plstic Strins nd Strsss in Notchs undr Multixil Loding Intrntionl Journl of Frctur Vol.70 pp [6] Singh M.N.K. Notch Tip Strss-Strin Anlysis in Bodis Subjctd to Non-Proportionl Cyclic Lods Ph.D. Dissrttion Dpt. Mch. Eng. Univrsity of Wtrloo Ontrio Cnd [7] Buczynski A. nd Glink G. Elstic-plstic Strss-Strin Anlysis of Notchs undr Non-Proportionl Loding Pths in Procdings of th Intrntionl Confrnc on Progrss in Mchnicl Bhviour of Mtrils (ICM8) Victori My ds. F. Ellyin nd J.W. Provn 3 pp [8] Kottgn V.B. Brky M.E. Soci D.F. Psudo Strss nd Psudo Strin Bsd Approchs to Multixil Notch Anlysis Ftigu of Enginring Mtrils nd Structurs 34 pp [9] Inc A. nd Glink G. A Numricl Mthod for Elsto-Plstic Notch- Root Strss-Strin Anlysis Journl of Strin Anlysis for Enginring Dsign Vol. 48 No. 4 pp [10] Inc A. Glink G. nd Buczynski A. A Computtionl Modling Tchniqu of Elsto-Plstic Strss-Strin Rspons for Notchd Componnts Intrntionl Journl of Ftigu Vol.62 pp [11] Grud Y.S. A Nw Approch to th Evlution of Ftigu undr Multixil Lodings Journl of Enginring Mtrils nd Tchnology 103 pp [12] Prndtl W. Spnnungsvrtilung in Plstischn Krprn Procdings of th First Intrntionl Congrss on Applid Mchncis pp [13] Russ E. Brucksichtigung dr lstischn Formndrungn in dr Zit- schrift fur Angwndt Mthmtik und Mchnik Vol. 10 pp [14] Inc A. nd Glink G. Innovtiv Computtionl Modling of Multixil Ftigu Anlysis for Notchd Componnts Intrntionl Journl of Ftigu 2015 doi: /j.ftigu [15] Inc A. A novl tchniqu for multixil ftigu modlling of ground vhicl notchd componnts Intrntionl Journl of Vhicl Dsign Vol. 67 No.3 pp doi: /IJVD Intrntionl Scholrly nd Scintific Rsrch & Innovtion 9(6) scholr.wst.org/ /