Analysis of adiabatic heating in high strain rate torsion tests by an iterative method: application to an ultrahigh carbon steel

Transcription

1 Comutatonal Methods and Exerments n Materals Charatersaton III 219 Analyss of adabat heatng n hgh stran rate torson tests by an teratve method: alaton to an ultrahgh arbon steel J. Castellanos 1, I. Rero 1, M. Carsí 2, J. Muñoz 1 & O. A. Ruano 2 1 Deartment of Mathemats, Unversty of Castlla - La Manha, San 2 Deartment of Physal Metallurgy, Natonal Center for Metallurgal Researh (C.E.N.I.M.), San Abstrat An teratve algorthm has been develoed to establsh the adabat heatng orreton of flow urves for torson tests of an ultrahgh arbon steel ontanng 1.3% C. Hgh temeratures (1223 to 1473 K) and hgh stran rates (2, 5, 1 and 26 s -1 ) were used. The urves are orreted n a fnte and dsrete set of stran data by means of arametr dervatves and ntegraton on the ntal urve wthout orreton. The roess s reeated untl the termnaton tolerane for the stress s less than 1-2 MPa. Usually, four teratons are needed to reah ths tolerane. The orretons are bounded by the maxmum of mehanal energy avalable to be onverted nto heat. The orretons are arred out untl a true stran = 4 n order to avod the effets of flow loalzaton n the materal. Keywords: adabat heatng, torson test, modellng, smulaton, Garofalo equaton, hot workng. 1 Introduton Torson tests at hgh temeratures and stran rates of materals usually show a strong nrement of temerature durng the test above the rogrammed temerature that s attrbuted to adabat heatng [1]. The temerature orreton due to adabat heatng has been dsussed n varous works [1 6]. The followng exresson s usually onsdered: η. A T ( ) = σ ( ) d (1) ρ C max ISSN (on-lne) do:1.2495/mc7221

2 22 Comutatonal Methods and Exerments n Materals Charatersaton III where T s the test temerature, C s the sef heat aaty, ρ s the densty, η and A are effeny oeffents of the energet onversons and σ ( ) s the stress-stran relaton. Some authors assume a varable energy erformane n eqn (1) [2] or a onstant one [6]. Other authors use the relaton σ ( ) wthout onsderng the ntrns error due to the adabat heatng tself [3]. In general, t s not onsdered that determnaton of the term σ ( ) mles dervatves at onstant temerature, whh s not true under the effet of adabat heatng. These dervatves aear n the alulaton of the stran rate senstvty and the stran hardenng oeffents. In addton, onstant values for ρ and C are used n the entre workng range. In ths work, we onsder the followng exresson for determnng the true value of the orreted relaton σ, n ontrast to the exermental value σ :. σ σ ( ) = σ ( ) (,, T) T( ) + ξ, (2) where ξ s the error assoated to the exermental value of the stress that should be bounded to avod wrong answers. The goal of ths work s to desgn a modular and teratve algorthm method that guarantees the onvergene of the exermental funton σ ( ) to the nomnal funton σ ( ). Ths method s based n eqns (1) and (2). The valdty ranges of the algorthms are adjusted taken nto aount hysal fundaments on flow loalzaton [3,5] and bounds of the erformane for the onverson on mehanal energy nto heat. 2 Materal and exermental roedure The UHC-1.3%C steel studed n ths nvestgaton has the followng omoston: 1.3% C,.5% Mn,.6% S,.18% Cr and balane Fe [1]. The manganese was added to neutralze the deleterous effets of sulhur and hoshorus. The steel was obtaned at Sdenor Industry as a ast of 8 ltres by means of an nduton furnae. The as-ast ngot was ntally soaked at 15ºC and forged nto a bar of 6 mm x 55 mm ross seton. Smulaton of the formng roess of forged arts was arred out by means of torson tests. An nduton furnae heats the test samle and the temerature s ontnuously measured by means of a two-olor yrometer. A sla tube wth argon atmoshere ensures roteton aganst oxdaton. A helum atmoshere s used to obtan, after testng, a oolng rate of 325 K/s. The torson samles have an effetve gage length of 17 mm and a radus of 3 mm. The densty and sef heat are 78 kg m -3 and 67 J kg -1 K -1, resetvely. Stran rates n the range 2 to 26 s 1 were used. The temerature ISSN (on-lne)

3 Comutatonal Methods and Exerments n Materals Charatersaton III 221 range was 9 to 12 ºC. The samles were deformed n a SETARAM torson mahne at CENIM (Natonal Center for Metallurgal Researh) n Madrd, San. 3 Theoretal aroah Two man roesses lmt the onverson of mehanal energy nto heat n an adabat framework: hanges n the nternal energy of the materal and flow loalzaton. Both roesses are related to the start of atastroh falure [2 4]. A dfferental exresson for the frst law of thermodynams σ d = ρ dt + du mst where du mst s the varaton of the mrostrutural nternal energy an be onsdered. The last work arred out by the materal s transformed nto heat that s used to nrease the nternal energy of the materal. Some authors assume du mst = [3]. Other authors onsder du mst leadng to the general exresson [2]: σ d 1 dumst σ d dt = 1 = η(,, T ) (3) ρ σ d ρ where η(, T ) = ( 1 (1 σ ) [ du d ]), s the erformane of the onverson mst and t s varable. A onstant value for η of,9 or,95 may be taken but an teratve roedure would be neessary to elmnate the effet of ths aroxmaton. The aroah of Prasad et al. [4] s onvenent to estmate the uer lmt bound of the nrease of temeratures due to adabat heatng. A smle onsttutve equaton for the energy dssaton s the followng: σ = σ d + dσ = G + J σ (4) where G s the dssator ontent and J s the dssator o-ontent. Part of the ower dssated by the last flow, G, an be onverted nto heat. The quantty J s related to the roesses of form hange. The lmt for G s G max = ( σ max max) 2. The followng exresson an be dedued from eqn (4): η Gmax Tmax = (5) ρ C V that reresents the lmt for the adabat T n a volume V. The nrement n temerature an be exressed as [3]: η T = ρ ' σ d (6) ISSN (on-lne)

4 222 Comutatonal Methods and Exerments n Materals Charatersaton III where η s onstant and ' s the deformaton lmt where the last nstablty starts [3]. Therefore, a rtal deformaton an be onsdered above whh t s not ossble to aly ths knd of orretons [5]. Values of dt d = 165 K from eqn (6) are obtaned for the UHC-1.3%C steel. Assumng the analyss of Armstrong et al. [3] and onsderng the stressθ stran relaton σ = K 1., together wth the defnton of the stress exonent, n, n the Garofalo equaton and the defnton of onstant stran rate tests, = t, the ondton for last nstablty gves the followng equaton: dt 1 = θ + ( ' β) d n ' (7) Under stablty ondtons, d σ >. The nstablty starts at d σ =, that an be exressed as [7, 8]: σ σ σ dσ = d + d + dt =,,, (8) Under adabat heatng ondtons, and assumng onstant, by means of Q n R T t s obtaned [7] that the Garofalo equaton = A. e ( snh( α σ )) ( σ ) = θ σ 2 and ( ) ( ),T σ = σ Q R T. Taken these, exressons nto the last nstablty ondton, an exresson for the flow stress at whh the nstablty starts, σ, an be obtaned: n θ R T ρ σ n = η Q Ths exresson wll be aled later to the UHC-1.3%C steel. 2 (9) 4 Bas methodology The followng assumtons are used n the algorthm develoed n ths work: 1) adabat ondtons n the deformaton roess, 2) ρ and C do not vary wth T, 3) η and A are onstant wth stran, and 4) adabat heatng has an mortant effet from the eak stress of the urve σ ( ), to a value f. For a gven test at onstant j and T s, the ntal temerature, and for a gven value of, the unorreted stress, σ (exermental stress), an be exressed as a funton of the orreted stress, σ, as: ISSN (on-lne)

5 Comutatonal Methods and Exerments n Materals Charatersaton III 223 σ ( T) = σ ( T + T) (1) Alyng Taylor exanson of the funton σ n eqn (1) about the ont T: σ σ T ( ) σ ( ) ( ) T = T T (11) where t s assumed that σ σ T. On the other hand, the temerature nrement for eah measured stran, for a gven test at onstant j, an be exressed as: Substtutng eqn (12) n eqn (11): η A T( ) = σ (, j, T) d (12) ρ C = max (, j, T ) σ η σ,, T = σ,, T σ,, T d (13) A ( j ) ( j ) j ρ = max ( ) A sngle tme alaton of these equatons would result n a value of σ (,, T ) that s not aurate. Ths s due, as mentoned n the ntroduton, to the assoated error n the determnaton of σ and onsequently n the ntegral art of eqn (13). Furthermore, alulaton of σ (,, T) n eqn (13) s σ s a wared urve n the sae also not aurate sne the funton ( ) {,σ,t}. To mnmze the nauraes of the alulaton arred out n eqn (13), an teratve algorthm based n eqns (11) and (12) has been develoed usng as ntal value the stress σ (,, T ). It s a modular algorthm n three stes. In the frst ste, the value of Tk ( ) n the teraton k (from k=1 to the number of teratons) s alulated aordng to the followng exresson: η A T ( ) = ( σ σ ) d (14) k k k ρc max where σ k 1, s the unorreted stress n the teraton k-1, and σ k 1 s the orreted stress n k-1. The ntegraton nterval s dvded n suffently enough small arts. In ths work, we have worked ste by ste wth all the outut data ISSN (on-lne)

6 224 Comutatonal Methods and Exerments n Materals Charatersaton III gven by the mahne. Therefore, a traezodal rule s used n order to omute the numeral ntegraton n eqn (14). In the seond ste, the artal dervatve wth reset to T of the unorreted stress, σk (, j, T) n the teraton k, s alulated by means of the, exresson: ( T, ) ( T, ) ( T, ) σ σ σ T T k s k s+ 1 + h k s + h s+ 1 s (15) where s + 1 refers to the test onduted at the same stran rate but at a temerature T k+1, belongng to the temerature set { T s} s = 1,, that s next n the N asendent sequene. Ths aroxmaton s good enough sne the dsretzaton ntervals are small. Fnally, the value of the orreted stress n the teraton k s gven as: (,, T ) σk j η A σ k (, j, T) = σk (, j, T) Tk( ) d (16) ρ The rteron adoted for stong the algorthm,.e. the termnaton tolerane, s σ 1 2 MPa for a gven ontrol stran. By means of ths roedure, the fnal measured temerature s reahed at a gven teraton for a value but the orreton s used only u to f, a value at whh the flow loalzaton s not onsdered mortant to dstort our orreton. 5 Results and onlusons 5.1 Analytal bass The results obtaned n seton 3 for the bound lmts are aled n our model to establsh the adabat orreton of the UHC-1.3%C steel. For ths steel, at = 1 s and T=1323 K, the eak stress s σ max = 1 MPa and, aordng to eqn (5), Tmax = 172 K. For = 2 s and T=1323 K, Tmax = 24 K. Ths gves an dea of the uer bounds of the unorreted values. Usng the data of Castellanos et al. for ths steel [9], the relaton,37 T σ ( T ) = K2 e s obtaned. The values of the Garofalo equaton are: Q=274,3 kj/mol and n=4,66. A value θ. 2 s obtaned for an ntegraton on all deformaton aths. For omarson, t was obtaned for = 1, 5, θ. 11 ' and dt ( ) d = 28 ' K and for next to the eak, θ and dt ( ' ) d = 58 ' K. ISSN (on-lne)

7 Comutatonal Methods and Exerments n Materals Charatersaton III 225 ' Table 1 shows the lmt strans,, for the start of last nstablty or flow loalzaton for the UHC-1.3%C steel. The values were alulated by means of eqn (9) usng θ =. 2 and η =, 9. The values ' are determned n the ont where the exermental urve σ ( ) ross the urve σ n ( ). The table shows that the last nstablty s delayed at hgh temeratures and low stran rates. It s worth notng that the loal nstablty develos rogressvely wth last stran. At =5 a lear hange of behavor of the flow urves s observed haraterzed by osllatons of the dervates of the stress wth reset to the stran. A value of f = 4 was hosen beause u to ths value the orretons were meanngful. Table 1: ' Lmt strans,, for the start of last nstablty for the UHC- 1.3%C steel as a funton of stran rate and temerature. T(K) from eqn (9) 1 = 2 s = 5s = 1 s = 26 s 1223 σ n = 52.7 / σ n = 57.1/ σ n = 61.7 / σ n = 66.4 / σ n = 71.3 / σ n = 76.4 / Correton of flow urves for the UHC-1.3%C steel The flow urves of the UHC-1.3%C steel have been modfed to onsder the adabat heatng. The urves were onduted at = 2, 5, 1, and 26 s -1 and T from 1223 to 1473 K, wth a varaton of 5 K. A maxmum of four teratons were onduted for the attanment of the fnal measured temeratures for [ 6,8] although the orretons were arred out u to f = 4. Fgure 1 shows true stress vs. true stran urves at varous stran rates and temeratures for the UHC-1.3%C steel. The sold lnes reresent the orreton for adabat heatng aordng to eqn (12). The orretons agree wth those arred out by other authors [1, 11]. However, somewhat dfferent results were obtaned when omared wth other nvestgatons where unrelable aroxmatons were onduted [12, 13]. Fgure 2 shows the evoluton of T, aordng to eqn (11), wth stran for = 26 s at varous temeratures. All the temerature nrements are nsde the bounds establshed for the maxmum nrements. The onvergene of the teratve algorthm was reahed at a maxmum of four teratons. ISSN (on-lne)

8 226 Comutatonal Methods and Exerments n Materals Charatersaton III Fgure 1: Flow urves for the UHC-1.3%C steel. Sold lnes are orreted urves for adabat heatng and dotted lnes are unorreted urves. Fgure 2: Evoluton of T wth stran for several T ' s at a =26 s-1. ISSN (on-lne)

9 Table 2: Values of σ at = 1 (a) and = 3 (b) for the orreted tests. (a) 1223 K 1273 K 1323 K 1373 K 1423 K 1473 K = 2s = 5s = 1s = 26s (b) 1223 K 1273 K 1323 K 1373 K 1423 K 1473 K = 2s = 5s = 1s = 26s Table 2 shows a summary of all the results obtaned n ths work. The aumulated values of σ are gven for eah ar {,T} at = 1 and 3. Values at = 5, 6 are hgher but were not onsdered due to flow loalzaton. It an be onluded that the method, and the mlemented algorthm, that we have develoed n ths work s relable and onvergent. The orreted stressstran urves are effent and relable and take all the exermental data set wthout the need of average aroxmatons. In addton, the method rovdes the detaled orretons at the dsretzaton level gven by the mahne. The man onlusons of ths work are: 1. A new teratve aroah for the adabat heatng orreton for torson tests has been establshed. It s a natural generalzaton of a revous aroah where the orreton was arred out n a sngle run. 2. The new aroah brngs an mrovement n the reson of the orreted flow urves. The relatve errors assoated to determnaton of the exermental stresses are mnmzed. 3. The temerature nrements obtaned for the UHC-1.3%C steel are nsde the bounds establshed for the maxmum nrements due to adabat heatng. Aknowledgement The work was arred out through the Projet PBC from JCCM (Castlla-La Manha, San). Referenes Comutatonal Methods and Exerments n Materals Charatersaton III 227 [1] Fernández-Vente, A., Carsí, M., Peñalba, F., Carreño, F. & Ruano, O.A., Deformaton behavor durng hot torson of and ultrahgh arbon steel ontanng 1.3 wt.% C. Zetshrft für Metallkunde, 94(8), , ISSN (on-lne)

10 228 Comutatonal Methods and Exerments n Materals Charatersaton III [2] Pantleon, W., Franke, D. & Klmanek, P., Modellng adabat heatng durng hgh-seed deformaton. Comutatonal Materals Sene, 7, , [3] Armstrong, R.W., Coffey, C.S. & Elban, W.L., Adabat heatng at a dsloaton le-u avalanhe. Ata Metallurga, 3, , [4] Prasad, Y.V.R.K., Gegel, H.L., Doravelu, S.M., Malas, J.C., Morgan, J.T., Lark, K.A. & Baker, D. R., Modellng of dynam materals behavor n hot deformaton: Forgng of T Metallurgal Transatons A, 15A, , [5] Lndholm, U.S., Mehanal Proertes at Hgh Rates of Stran. Conferene Seres nº 21, ed. J. Hardng, Insttute of Physs: London and Brstol,. 3-21, [6] Bhattaharyya, A., Rttel, D. & Ravhandran, G., Stran rate effet on the evoluton of deformaton texture for α -Fe, Metallurgal and Materals Transatons A, 37(A), , 26 [7] Sematn, S.L., Staker, M.R., & Jonas, J.J., Plast nstablty and flow loalzaton n shear at hgh rates of deformaton. Ata Metallurga, 32 (9), , [8] Staker, M.R., The relaton between adabat shear nstablty stran and materal roertes. Ata Metallurga, 29, , 1981 [9] Castellanos, J., Rero, I., Carsí, M, Muñoz, J., Ruano, O.A., Analyss of several methods for the data onverson and fttng of the Garofalo equaton aled to an ultrahgh arbon steel. Journal of Ahevements n Materals and Manufaturng Engneerng, 18(1-2), , 26. [1] We-Guo, G., Nemat-Nasser, S., Flow stress of Ntron-5 stanless steel over a wde range of stran rates and temeratures. Mehans of Materals, 38, , 26. [11] Zhou, M., Clode, M.P., Thermal analyss of the torson test under hotworkng ondtons. Comutatonal Materals Sene, 9, , [12] Holzer, A.J. & Wrght, P.K., Dynam lastty: a omarson between results from mehanal testng and mahnng. Materals Sene and Engneerng, 51, , [13] Venugoal, P., Venugoal, S. & Seetharaman, V., Some asets of the deendene of the flow urve of ommerally ure ttanum on the formng temerature and the stran-rate. Journal of Materals Proessng Tehnology, 21, , ISSN (on-lne)