AN UPPER BOUND SOLUTION OF BACKWARD TUBE EXTRUSION PROCESS THROUGH CURVED PUNCHES

Transcription

1 Acta Metallugica Slvaca, l., 4, N., p AN UPPER BOUND SOLUTION OF BACKWARD TUBE EXTRUSION PROCESS THROUGH CURED PUNCHES Heshatllah Haghighat *, Ghla Reza Asgai Mechanical Engineeing Depatent, Razi Univesity, Keanshah, Ian Glpayegan Paya Nn-Gvenental Institute Highe Educatin, Glpayegan, Ian Received:.. Accepted:.. * Cespnding auth: e-ail: hhaghighat@azi.ac.i, Tel.: , Mechanical Engineeing Depatent, Razi Univesity, Keanshah, Ian Abstact In this pape, an uppe bund appach is used t analyze the pcess backwad tube extu thugh abitaily cuved punches. Based n the assuptins pptinal angles and pptinal distances the andel suace in the deatin zne, tw kineatically adissible velcity ields ae ppsed and thse ae used in uppe bund slutin. By ug the develped uppe bund slutin, ptiu punch lengths which iniize the extu ces ae deteined a stealined punch shape and als a cnical punch. The cespnding esults ae als deteined by ug a inite eleent cde, ABAQUS, and by ding se expeients and cpaed with the analytical esults. This cpaisns shw a gd ageeent. Keywds: Backwad tube extu, Uppe bund, Expeient Intductin In backwad extu, thee is n elative veent between the initial billet and the cntaine and it is chaacteized by the absence ictin between the initial billet suace and the cntaine. This pcess needs lwe extu ce and als it is suitable pducing patially extuded pducts because the siplicity in ejecting the extuded pat the cntaine as cpaed with wad extu pcess. In this pcess, such as the etal ing pcesses, calculatin and ptiizatin extu ce ae iptant. Ang vaius analytical and nueical appxiate ethds slutin, the uppe bund technique and the inite eleent ethd have been widely used the analysis the extu pcess. One the liitatins st the cuent FEM slutin schees etal ing is that they d nt pvide paaetic analysis. Hence, any paaetic investigatin is usually dne anually by changing ne FE del t anthe until a easible slutin is btained. Establishent analytical slutins extu pcess acilitates paaetic study and ay help in undestanding the echanics behind the extu pcesses bette. Even thugh the inite eleent gives detailed inatin, it takes cnsideable CPU tie. Ug the uppe-bund technique has the eits saving cpute s CPU and it appeas t be a useul tl analyzing etal ing pbles when the bjective such an analysis is liited t pedictin deatin lad and/ t study etal lw duing the pcess. A nube peple have used the uppe bund ethd t analyze the extu pcess. Avitzu [-] develped dels wad d extu thugh cnical dies ug the uppe bund appach. Chen and Ling [4] develped a velcity ield axisyetic extus thugh ce, elliptic and DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

2 Acta Metallugica Slvaca, l., 4, N., p hypeblic dies. Ziean and Avitzu [5] als deled extu ug the uppe bund ethd with genealized shea bundaies. Mehta et al. develped a kineatically adissible velcity ield and cpaed it with the expeientally btained lw ield ug the visiplasticity technique [6]. Chang and Chi develped an uppe bund slutin tube extu thugh cuved dies [7]. Hatley ppsed a kineatically adissible velcity ield wad tube extu thugh a cnical die, which educes t the kineatically adissible velcity ield d extu in the liit as the andel diaete ges t ze [8]. Yang et al. [9] as well as Yang and Han [] develped uppe bund dels wad d extu with stealined dies. An uppe bund slutin stain hadening ateials has been develped by Yang et al. tube extu thugh stealined dies []. They extended the sae analysis thee-dienal wad extu abitaily shaped tubes []. Altan ppsed a deatin del tube extu thugh a lat die by assuing the lw lines t be staight []. Bakhshi et al. ppsed an ptiu punch pile in backwad d extu [4]. Sabi et al. studied the enegy cnsuptin in wad and backwad d extu [5]. A eatue based uppe bund del t analyze the backwad tube extu ppsed by Malpani and Kua [6]. The analysis was based n a kineatically adissible velcity ield t btain the ptial extu pessue by ptiizing the die length. Ebahii et al. [7] ppsed a kineatically adissible velcity ield wad tube extu thugh cnical dies. Gdn et al. wee develped an adaptable die design ethd wad d extu and descibed the in details in a seies papes [8-]. The pupse this pape is t develp a velcity ield that applicable t backwad tube extu thugh abitaily cuved punches. The ppsed velcity ield is used t ind ut an ptial stealined punch length and the cespnding extu ce a given pcess cnditins. The investigatin is als peed ug the inite eleent cde, ABAQUS and by ding se expeients. Uppe bund analysis Fig. shws a scheatic diaga the tube backwad extu thugh an abitaily shaped punch with ixed cylindical shaped andel. In this igue R and R ae ute adii the initial and the extuded tube and R is the andel adius. T analyze the pcess by ug the uppe bund ethd, the ateial unde deatin is divided int thee znes. In zne I, ateial is statinay and in zne III the ateial ves igidly with the velcity. Zne II is suunded by tw velcity discntinuity suaces S, S, andel suace and the punch suace. The punch suace, which is labeled as ( in Fig., is given in the spheical cdinate syste,,,, whee ( is the angula psitin the punch suace as a unctin the adial distance the igin. The igin spheical cdinate syste is lcated at pint O which is deined by the intesectin the axis syety with the line that ges thugh the pint whee the punch begins and the exit pint the punch. The spheical velcity discntinuity suace S is lcated at distance the igin and the spheical velcity discntinuity suace S is lcated at distance the igin. F Fig., we have R R, R R ( whee is the angle the line cnnecting the initial pint the cuved punch t the inal pint the punch and tan ( R R / L, whee L is punch length. The exit velcity is deteined by DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

3 Acta Metallugica Slvaca, l., 4, N., p R R ( R R Substituting Eq. ( int Eq. (, the exit velcity can be witten as Fig. Scheatic diaga shwing backwad tube extu thugh an abitaily cuved punch. ( R ( R ( R R ( F sall adius the andel, the abve equatin can be sipliied as ( (4 The ist step in the uppe bund analysis is t chse an adissible velcity ield the ateial undeging plastic deatin. The assuptin velcity ield will inluence the pedictin lad and etal lw. The velcity ield that has been deived incpessibility cnditin and satisies the velcity bunday cnditins is a kineatically adissible velcity ield.. elcity ield in the deatin zne Tw assuptins that wee used by Gdn et al. [8] wad d extu ae used hee backwad tube extu pcess. I the assuptin pptinal angles in the deatin zne is ade, then adial velcity cpnent is witten as DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

4 Acta Metallugica Slvaca, l., 4, N., p DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56 ( ( cs (5 Assuing the pptinal distance the andel suace then cs cs (6 The ull velcity ield the lw the ateial in deatin zne is btained by invking vlue cnstancy. lue cnstancy in spheical cdinate syste is deined as (7 whee ii is the nal stain ate cpnent in the i -diectin. The stain ates in spheical cdinates ae deined as ct ct (8 F the axisyetic extu (i.e. and the angula cpnent velcity, is btained by placing, Eqs. (5-(6 int Eqs. (7-(8, slving and applying apppiate bunday cnditins. Then, thee ae tw velcity ields depending n the entined assuptins: ( Assuing pptinal angles in the deatin zne: ( ( cs (

5 Acta Metallugica Slvaca, l., 4, N., p (9 ( Assuing the pptinal distances the andel suace: cs cs ( tan ( I the exit velcity is deteined by Eq. (4, then Eqs. (9-( satisy the incpessibility cnditin and the bunday cnditins n velcity discntinuity suaces S, S and andel suace as well as the punch suace. Theee, they ae deeed t be kineatically adissible ields. Based n the ppsed velcity ields, the stain ate ields deatin zne can be btained by Eq. (9. With the stain ate ield and the velcity ield, the standad uppe bund ethd can be ipleented. This uppe bund del invlves calculating the intenal pwe deatin ve the deatin zne vlue, calculating the shea pwe lsses ve the suaces velcity discntinuity, and the ictinal pwe lsses alng ictinal suaces. Since, n deatin ccus in znes I and III, theee, the stain ate cpnents ae ze.. Intenal pwe deatin The intenal pwe deatin in an uppe bund del is W i v ij ijd ( Intenal pwe znes I and III ae ze and the equatin t calculate the intenal pwe deatin in zne II is 4 ( W i ( ( R d d ( whee is the ean lw stess ateial and is given by d, ln R R R R (. Shea pwe lsses The equatin the pwe lsses alng a shea suace velcity discntinuity is W S ds (4 S v DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

6 Acta Metallugica Slvaca, l., 4, N., p The shea pwe lsses alng the velcity discntinuity suaces S and S with assuing pptinal angles in the deatin zne, bece W S ( cs ( R d (5 W S ( cs ( R d (6 and with assuing the pptinal distances the andel suace, bece W S ( ( ( R d tan (7 W S ( ( ( R d tan (8.4 Fictin pwe lsses The geneal equatin the ictin pwe lsses a suace with a cnstant ictin act is W ds (9 S F punch suace S the dieential suace and the agnitude the velcity dieence ae ds ( R ( d ( ( ( cs cs ( whee is lcal angle the punch suace with espect t the lcal adial velcity cpnent and cs, ( ( ( W ( ( ( R d ( DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

7 Acta Metallugica Slvaca, l., 4, N., p The ictinal pwe lsses alng the suace S 4, shwn in Fig., with assuing pptinal angles in the deatin zne is calculated as W R 4 ( ( d (4 and with assuing the pptinal distances the andel suace, it beces W R 4 ( ( d (5 The ictinal pwe lsses alng the suace S 5 can be given by R (6 W 5 Finally, the pwe dissipated n the ictinal suace S 6, punch land, beces W 6 R ( L (7 whee L is the length the punch land. Based n the uppe bund del, the ttal pwe needs backwad tube extu pcess is btained by suing the intenal pwe and the pwe dissipated n all ictinal and velcity discntinuity suaces. Theee, the ttal uppe bund slutin the elative extu pessue is given by P W W W W W ave i S S 4 ( R R W 5 W 6 (8 A MATLAB pga has been ipleented the peviusly deived equatins and is used t study the backwad tube extu pcess dieent punch shapes and dieent pcess cnditins. It includes a paaete L, punch length, which shuld be ptiized. Cpaisn the velcity ields The develped velcity ields and the uppe bund del can be used backwad tube extu thugh punches any shape i the punch pile is expessed as equatin (. T cpae the uppe bund esults btained the tw velcity ields, the die pile intduced by Yang and Han [9, ] wad d extu is selected the pile the punch. They ceated a stealined die shape as a uth-de plynial whse slpe is paallel t the axis at bth entance and exit the die. Die shape Yang and Han can be expessed in spheical cdinate syste, shwn in Fig., as DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

8 Relative Aveage Extu Pessue Relative Aveage Extu Pessue Relative Aveage Extu Pessue Acta Metallugica Slvaca, l., 4, N., p C ( R R ( R R ( R R ( R R C R R ( R R whee C 4 ( C R R ( R R cs cs R R ( ( L L R R 6 L L 6( L L R R R R 4 ( ( cs cs cs cs (9 whee L L is the elative psitin the inlectin pint the die and can vay t []. 4 =. R/R = =. R/R =.5 8 Pptinal Angles 5 4 Pptinal Angles 6 4 Pptinal Distances Punch Length ( Pptinal Distances Punch Length ( (b 4 =.5 R/R =.5 8 Pptinal Angles 6 4 Pptinal Distances Punch Length ( (c Fig. Relative extu pessues extu thugh a Yang and Han punch shape ug the angula and e velcity ields: (a R / R. 4 and. ; (b R / R.5 and. ; (c R / R. 5 and. 5. DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

9 Acta Metallugica Slvaca, l., 4, N., p Fig. shws the elative aveage extu pessue calculated the tw velcity ields as a unctin the punch length. This igue includes tw dieent tube geety ( R /. 4 R Fig. a and R / R.5 Figs. b-c and tw dieent ictin acts (. Figs. a-b and. 5 Fig. c. These igues shw that the assuing the pptinal distances andel suace pvide a lwe uppe bund slutin. 4 Cpaisn analytical esults with the FEM esults and expeient The backwad tube extu pcesses is siulated ug the inite eleent stwae, ABAQUS. Due t the syety the pcess, tw-dienal axisyetic dels ae used FEM analyses. In each case, the whle del is eshed with CAX4R eleents. Fig. a illustates the esh used t analyze the deatin. Punch, andel and cntaine undeg elastic stains nly. Thus, it is nt necessay t use a ine esh in these tw pieces. Hweve, suiciently ine eshing is essential in ateial which undeges plastic deatin. The cntaine is ixed by applying displaceent cnstaint n its ndes while the punch del is laded by speciying displaceent in the axial diectin. Deed del is shwn in Fig. b. (a The inite eleent esh (b The deed esh Fig. The inite eleent esh and the deed esh. F cpaisn, the cespnding esults ae als btained ug expeient. A 5 kn STM univesal testing achine is used t pe the expeients, which is illustated in Fig. 4, tgethe with a die-set. A sdiu based gease lubicant is applied anually n the cntact suaces billets and die-sets. The initial tube was lead with length 4, R, R 6, R and the lw stess given by tensile test as (MPa ( DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

10 Acta Metallugica Slvaca, l., 4, N., p Based n Eq. ( a ean lw stess 55.5 MPa was estiated the ateial used. The initial tube and the extuded saple cespnding t the cnical punch, punch length 8, ae shwn in Fig. 5. Expeiental lad displaceent cuve the cnical punch is shwn in Fig. 6. By cpaing the expeiental esults the cnical punch with the FEM siulatin esults, it is und that the shea ictin act the expeient is abut.5 and it is used in uppe bund del. Fig. 4 The 5 kn STM univesal testing achine with a die-set unted. (a initial tube (b cnical punch (c extuded tube Fig. 5 Initial tube and extuded saple cnical punch. DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

11 Extu ce (kn Extu ce (kn Acta Metallugica Slvaca, l., 4, N., p Expeient Displaceent( Fig. 6 Expeiental ce-displaceent cuve cnical die. In Fig. 7, the analytical, expeiental and FEM ce-displaceent cuves cnical punch is cpaed. The esults shw gd ageeent between the analysis and expeient. As shwn in this igue, the theetically pedicted extu ce is highe than the expeiental and FEM esults, which is due t the natue the uppe bund they. 6 5 Uppe bund Expeient FEM 4 R/R =.5 = Displaceent ( Fig. 7 Cpaisn between the analytical, expeiental and FEM ce-displaceent cuves cnical punch. Extu ce vaiatins vesus punch length backwad tube extu thugh cnical punch and the Yang and Han punch shape btained the uppe bund del, R 6, R, R and., ae cpaed with each the in Fig. 8. As can be seen the igue, the tend in the tw cuves is siila. Als, at any length the punch, the equied extu ce in the ptiu Yang and Han punch shape is less than that in the ptiu cnical punch. DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

12 Extu ce (kn Extu ce (kn Acta Metallugica Slvaca, l., 4, N., p Cnical Punch Yang & Han Punch Shape Fig. 8 Cpaisn between the extu ce vaiatin vesus punch length cnical punch and Yang and Han punch shape btained the uppe bund ( R 6, R, R and Punch Length ( In Fig. 9, the extu ce displaceent cuve ptiu Yang and Han punch shape btained the uppe bund slutin and the FEM siulatin is cpaed with each the. The esults shw gd ageeent between the analysis and FEM. As shwn in this igue, at the ealy stage extu, unsteady state deatin ccus, and the ateials have nt yet illed up the cavity the punch cpletely. Thus, the extu ce inceases as the extu pcess pceeds. Ate the ateials have illed up the cavity the punch cpletely, the extu ces ae cnstant. That is because the ictinal suaces and shea suaces ae n change as the punch is advanced. 6 5 Uppe bund FEM 4 R/R =.5 = Displaceent ( Fig. 9 Cpaisn analytical and FEM ce-displaceent cuves ptiu Yang and Han punch shape. DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

13 Extu ce (kn Acta Metallugica Slvaca, l., 4, N., p The eect ictin act upn extu ce is shwn in Fig.. As shwn in these igues, at a punch length that called the ptiu length, the extu ce is iniized. As shwn in this igue, the extu ce inceases with inceag the ictin act. Als, with inceag the ictin act, the ptiu length punch is inceased. 8 Fictin act.4.. Optiu Punch Length 6 4 R/R = Punch Length ( Fig. Eect ictin act upn the extu ce Yang and Han punch shape. 5 Cnclus In this pape tw velcity ields and thei pwe tes backwad tube extu pcess thugh punches any shape wee pesented. Deivatins thee ain cpnents the cnsued pwe including deatin, discntinuity and ictinal pwes wee pesented. The esults uppe bund dels tw velcity ields wee cpaed t each the extu thugh a stealined die shape. The esults denstated that assuing pptinal distances the andel suace in the deatin zne was bette than assuing the pptinal angles. Cpaisn the easued extu ce with that estiated by the ppsed uppe bund slutin shwed a gd ageeent. The analytical esults wee als in gd ageeent with the FEM data. The develped uppe bund del can be used inding the ptiu punch length which iniizes the extu ce a given punch shape and pcess paaetes. Reeences [] B. Avitzu: ASME Junal Engineeing Industy, l. 85, 96, p [] B. Avitzu: ASME Junal Engineeing Industy, l. 86, 964, p. 5 6 [] B. Avitzu: ASME Junal Engineeing Industy, l. 88, 966, p. 4 4 [4] C.T. Chen, F.F. Ling: Intenatinal Junal Mechanical Sciences, l., 968, p [5] Z. Ziean, B. Avitzu: ASME Junal Engineeing Industy, l. 9, 97, p. 9 9 [6] H.S. Mehta, A.H. Shabaik, S. Kbayashi: ASME Junal Engineeing Industy, l. 9, 97, p. 4 4 DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

14 Acta Metallugica Slvaca, l., 4, N., p [7] K. T. Chang, J.C. Chi: ASME Junal Engineeing Industy, l. 94, 97, p. 8 [8] C.S. Hatley: Intenatinal Junal Mechanical Sciences, l. 5, 97, p [9] D.Y. Yang, C.H. Han, B.C. Lee: Intenatinal Junal Mechanical Sciences, l. 7, 985, p [] D.Y.Yang, C.H. Han: ASME Junal Engineeing Industy, l. 9, 987, p [] D. Y. Yang, H.S. Ki, C.M. Lee: CIRP Annuls, l. 6, 987, p [] D.Y. Yang. H.S. Ki, C.M. Lee, C.H. Han: Intenatinal Junal Mechanical Sciences, l., 99, p. 5-7 [] S.B. Altan: Junal Mateial Pcesg Technlgy, l. 4, 994, p. 5 [4] M. Bakhshi-Jybai, M. Sabi, S.J. Hsseinipu, M. Shakei, A. Gji: Junal Mateial Pcesg Technlgy, l. 77, 6, p , DOI:.6/j.jatptec [5] M. Sabi, M. Bakhshi-Jybai, M. Nani-Azad, A. Gji: Junal Mateial Pcesg Technlgy, l. 77, 6, p. 6 66, DOI:.6/j.jatptec.6.4. [6] M. Malpani, S. Kua: Junal Mateial Pcesg Technlgy, l. 9, 7, p. 6 74, DOI:.6/j.jatptec...5 [7] R. Ebahii, M. Reihanian, M. Kanaani, M.M. Mshksa: Junal Mateial Pcesg Technlgy, l. 99, 8, p. 4, DOI:.6/j.jatptec [8] W.A. Gdn, C.J. an Tyne, Y.H. Mn: Intenatinal Junal Mechanical Sciences, l. 49, 7, p , DOI:.6/j.ijecsci.6.7. [9] W.A. Gdn, C.J. an Tyne, Y.H. Mn: Intenatinal Junal Mechanical Sciences, l. 49, 7, p. 96, DOI:.6/j.ijecsci.6.7. [] W.A. Gdn, C.J. an Tyne, Y.H. Mn: Intenatinal Junal Mechanical Sciences, l. 49, 7, p. 4 5, DOI:.6/j.ijecsci.6.7. Nenclatue * J extenally supplied pwe deatin cnstant ictin act k ateial yield stength in shea L punch length L beaing length,, spheical cdinate syste adial psitin the velcity discntinuity suace S adial psitin the velcity discntinuity suace S S aea ictinal velcity discntinuity suace,, adial, angula and thid cpnents velcity vlue integatin velcity inal tube velcity punch W W, W W ictinal pwe lsses alng the ictinal suaces S, S 4, S 5 and S 6, W i, 4 5, 6 espectively intenal pwe deatin DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56

15 Acta Metallugica Slvaca, l., 4, N., p W S, W S shea pwe lsses alng the velcity discntinuity suaces S and S, espectively angle the line cnnecting the initial pint the punch t the inal pint the punch velcity dieence,, nal stain ate cpnents,, shea stain ate cpnents lcal angle the punch suace with espect t the lcal adial velcity cpnent lw stess the wkpiece ateial ean lw stess the wkpiece ateial ictinal shea stess angula psitin the punch suace as a unctin adial psitin DOI.776/as.vi.4 p-issn 5-5 e-issn 8-56