NUMERICAL SIMULATION OF THERMAL WARPING AND BUCKLING IN ENAMELLED STEEL PARTS

Transcription

1 NUMERICAL SIMULATION OF THERMAL WARPING AND BUCKLING IN ENAMELLED STEEL PARTS 337

2 XXI Intrnational Enamllrs Congrss Numrical Simulation of Thrmal Warping and Buckling in Enamlld Stl Parts Filip Van dn Abl, Eric Hofrlin, Lod Duprz, Philipp Gousslot ArclorMittal Rsarch Industry Gnt John Knndylaan 3, B-9060 Zlzat Kywords thrmal strsss, warping, buckling, simulation, stl Abstract Whn an namlld stl part is coold down aftr firing, rsidual strsss ar inducd du to th diffrnc in thrmal dilatation btwn th naml and th stl grad. Ths thrmal strsss can giv ris to buckling and warping of namlld stl parts. Espcially slndr dsigns, lik baking trays (houshold applianc) or flat architctural panls (building industry) ar pron to ths problms. Hr, a finit lmnt modl is prsntd to prdict ths phnomna and, corrspondingly, driv som guidlins to avoid thrmal warping and buckling in namlld stl parts. First, an namlld cantilvr bam is studid, and th calculatd strss distribution is compard with th traditional Timoshnko thrmostat thory. Th assumptions of th finit lmnt modl ar validatd by prforming xprimntal Klotz tsts. Thn, this modl is applid to simulat ral-scal namlld stl parts. Two cas studis ar considrd: a warpd baking tray, and a flat architctural panl, whr buckling occurs aftr namlling. It is shown that finit lmnt simulations can prdict ths distortions, and can b usd to driv dsign guidlins for namlld stl parts. Strss dvlopmnt in stl/naml composit systms Whn firing an naml, th stl/naml composit systm is subjctd to a typical tmpratur cycl lik th on shown in Fig. 1: Th naml may b applid ithr as a wt slurry or dry powdr onto th dgrasd sampl. Th naml layr will fus at a rlativly low tmpratur in th firing cycl, and start to sal th surfac of th mtal (to rtard oxidation). Th viscous naml layr flows to form a uniform film on th mtal substrat. During th hating of th stl/naml combination, th systm is not undr strss up to th point of pak fir, which is typically around 850 C. As th fusd glass and mtal start to cool, th diffrnt thrmal xpansions of th naml and th stl now bcom important in dvloping th charactristics of th fird part. Th stl xpands and contracts at an almost uniform rat within th rang of firing. Th naml, howvr, will hav diffrnt xpansion and contraction rats as compard to th stl, lik xplaind in Fig.. From th dilatomtric softning point T E to th glass transition tmpratur T G, th naml has a highr thrmal xpansion than th stl. At th dilatomtric softning point, th naml is vry soft, and any strsss ar immdiatly rlivd. Howvr, as th naml cools, it bcoms 338

3 Numrical Simulation of Thrmal Warping and Buckling in Enamlld Stl Parts mor viscous (stiffr), and th ability to rliv strsss dcrass significantly. Morovr, th viscosity will ris to such a lvl that th naml can b tratd as a solid. Hnc, at high tmpraturs (typically btwn~600 C and~350 C), th (viscous) naml layr is undr a tnsil strss stat. Any stl/naml composit systm xhibits a nutral tmpratur T N, whr thr is no diffrnc btwn thrmal dilatation of stl and naml. Thus, thr ar no thrmal strsss in th systm at th nutral tmpratur (in most cass around ~350 C, dpnding on th naml composition). Btwn nutral tmpratur (~350 C) and room tmpratur (~0 C), th stl wants to shrink fastr than th naml. As a rsult, th naml layr is undr comprssiv strsss at room tmpratur, and th composit bam buckls in th dirction of th naml. Fig. 1 Schmatic full firing cycl showing strss dvlopmnt in th naml layr Hnc, thrmal warping and buckling issus ar triggrd by th strsss inducd upon cooling from nutral tmpratur to room tmpratur. As sn on Fig. 3, th composit bam buckls in th dirction of th naml layr at room tmpratur. Howvr, th naml is still undr a comprssiv strss stat! Th rsidual strsss ar dtrmind incrmntally upon cooling by invoking th following analytical logic for ach tmpratur incrmnt [1]: Thr ar no rsidual strsss at th nutral tmpratur. Stl and naml xprinc an unconstraind diffrntial shrinkag, whr stl wants to shrink fastr. To achiv displacmnt compatibility at th intrfac, uniform tnsil and comprssiv strsss ar imposd on th stl and th naml rspctivly. Finally, bnding is allowd to occur, in ordr to balanc th bnding momnt inducd by th asymmtric strsss in th prvious situation. 339

4 XXI Intrnational Enamllrs Congrss Fig. Diffrnc in thrmal dilatation btwn stl and naml Fig. 3 Strss dvlopmnt during cooling from nutral to room tmpratur Calculating thrmal dflction with Timoshnko bimtal thory Th thrmal dflction of an namlld cantilvr stl sampl can b calculatd with th traditional Timoshnko bimtal thrmostat thory []. Th radius of curvatur ρ for a cantilvr bam with lngth L is connctd with th tangntial dflction δ = L ρ 340

5 Numrical Simulation of Thrmal Warping and Buckling in Enamlld Stl Parts Hnc, with th convntions of Fig. 4, and aftr som algbraic manipulations [], th thrmal dflction at tmpratur T can b writtn as δ K L H ( T ) = ( α α )( T T ) s whr T N is th nutral tmpratur, α s - α is th diffrnc btwn th thrmal cofficints of stl (s) and naml (), H = h + h s is th total hight, and N K = 3 3 ( 1 + m) ( 1 + m) + ( 1 + m n) m + m n 1 is a function of th dimnsionlss stiffnss n = E E s and th rlativ naml thicknss m = h h s Thus, th thrmal dflction δ(t) is linarly dpndnt on th tmpratur drop (T T N ), and th diffrnc in dilatation (α s - α ) btwn th stl and th naml. Whil th influnc of th stiffnss ratio n is not pronouncd (n 1/3, which can hardly b nginrd), th rlativ naml thicknss m is vry important, spcially for 0 < m < 0.3 (s Fig. 5). Th rsulting through-thicknss strss distribution shows a suprposition (Fig. 4) of th uniform comprssiv strss in th naml th uniform tnsil strss in th stl substrat th linar bnding strsss, intrscting th nutral fibr and th corrsponding comprssiv strss stat in th naml layr is givn by [3] σ δ L ( z) = σ E z max whr σ max = δ h L E 1 + m n 3m n ( m) ( 1 + m) is th maximum comprssiv strss at th naml intrfac. Not that this valu for σ max is, again, highly dpndnt on th rlativ naml thicknss m. 341

6 XXI Intrnational Enamllrs Congrss Fig. 4 Strss distribution in stl/naml systm at room tmpratur Fig. 5 Influnc of rlativ naml thicknss on thrmal dflction Finit lmnt simulation of Klotz tst In ordr to prdict thrmal warping, a simpl Klotz tst is simulatd first. In this procdur, an namlld stl sampl is hatd by an lctric currnt, and th rsulting thrmal dflction is masurd with an inductiv displacmnt snsor, whil th tmpratur is monitord with an infrard pyromtr. This acclratd tst mthod givs, within lss than 15 minuts aftr namlling, a summary of th complx physical proprtis of th stl/naml composit systm. Th masurd paramtrs compris th various typical tmpraturs (i.. T G, T N and T E ), and th dflction valus ar dirctly rlatd to th naml strsss. Basd on ths paramtrs, th quality of an namlld systm can b judgd. In our thrmomchanical modl, th composit shll thory was applid for a spcific st of naml proprtis, with 34

7 Numrical Simulation of Thrmal Warping and Buckling in Enamlld Stl Parts tmpratur dpndnt sitffnsss E (T) and E s (T) dilatomtric data for α (T) and α s (T) nutral tmpratur T N = 350 C as initial condition stl substrat thicknss hs = 1.5 mm total naml layr thicknss h = 0.3 mm Fig. 6 shows that th simulatd dflction of th spcimn is in vry good agrmnt with th xprimntal data, and on can obsrv that th simulation/xprimnt starts with an namlld (and buckld) spcimn h comprssiv strss stat at room tmpratur is vry wll prdictd th simulatd curv xhibits som ovrshoot at highr tmpraturs, whr thrmal dilatation masurmnts ar most difficult Fig. 6 Comparison btwn xprimntal and simulatd Klotz tst dflction Numrical prdiction of thrmal warping and buckling Th validatd finit lmnt modl can b applid to simulat ral-scal namlld stl parts, spcially to prdict thrmal buckling, which is a stabl dformation mod, rlatd to th diffrnc in thrmal xpansion btwn stl and naml. This procss is basically govrnd by th sam mchanism as th stl/naml cantilvr bam. Uniform buckling (in th dirction of th naml layr) can nvr b avoidd, sinc α < α s for T < T N. thrmal warping, which is an unstabl dformation mod, connctd with th frquncy rspons of th namlld part. Modal analysis lads to th blif that unstabl collaps might occur undr thrmal loading. This might xplain why asymmtric warping is obsrvd aftr namlling. 343

8 XXI Intrnational Enamllrs Congrss Fig. 7 Thrmal buckling and warping in namlld stl parts Fig. 7 shows th simulation of thrmal buckling (for a hob-top) and thrmal warping (of a baking tray) for houshold appliancs. Espcially slndr dsigns, with a low sht thicknss, ar pron to ths distortions. For instanc, whn a flat architctural panl movs through an namlling furnac (Fig. 8), buckling can hardly b avoidd [4]. Nonthlss, w now hav both analytical and numrical tools to simulat and prdict ths problms. This allows us to driv dsign guidlins for namlld stl parts. Fig. 8 Distortion of flat architctural panls aftr namlling Rfrncs [1] Fundamntal thrmal dflction analysis of naml-mtal systms, J.H. Lauchnr, R.L. Cook and A.I. Andrws, Journal of th Amrican Cramic Socity 39(8), pp. 88-9, (1955). [] Analysis of bimtal thrmostats, S. Timoshnko, Journal of th Optical Socity of Amrica 11, pp (195). [3] Rsidual strsss in mtal-cramic bondd strips, C.H. Hsuh and A.G. Evans, Journal of th Amrican Cramic Socity 68(5), pp (1985). [4] Prdicting thrmal buckling in namlld architctural panls, F. Van dn Abl, E. Hofrlin, L. Duprz and P. Gousslot, accptd for th 1 st Intrnational Enamllrs Congrss, Shangai (18-/05 008). 344