Analytical Model for the Characterization of the Guiding Zone Tribotest for Tube Hydroforming

Transcription

1 Gracious Ngaile Chen Yang Deparmen of Mechanical and Aerospace Engineering, Norh Carolina Sae Universiy, Campus Box 7910, Raleigh, NC Analyical Model for he Characerizaion of he Guiding Zone Triboes for Tube Hydroforming Common par failures in ube hydroforming include wrinkling, premaure fracure, and unaccepable par surface qualiy. Some of hese failures are aribued o he inabiliy o opimize ribological condiions. There has been an increasing demand for he developmen of effecive lubricans for ube hydroforming due o widespread applicaion of his process. This paper presens an analyical model of he guiding zone riboes commonly used o evaluae lubrican performance for ube hydroforming. Through a mechanisic approach, a closed-form soluion for he field variables conac pressure, effecive sress/ srain, longiudinal sress/srain, and hoop sress can be compued. The analyical model was validaed by he finie elemen mehod. In addiion o deermining fricion coefficien, he expression for local sae of sress and srain on he ube provides an opporuniy for in-deph sudy of he behaviof lubrican and associaed lubricaion mechanisms. The model can aid as a quick ool for ieraing geomeric variables in he design of a guiding zone, which is an inegral paf ube hydroforming ooling. DOI: / Keywords: riboes, fricion coefficien, ube hydroforming, closed-form equaions 1 Inroducion Tube hydroforming THF is a process of manufacuring inricae shapes from ubular blanks. A hydraulic fluid is pressurized inside he ube o a yield poin, hence forcing he blank o conform o he die shape. To increase shaping capabiliy, he ubular blank ends are usually fed oward he die caviy during pressurizaion. Tube hydroforming has gained wide accepance in he auomoive and aerospace indusries due o is advanages over samping, such as par consolidaion, weigh reducion, higher par qualiy, fewer secondary operaions, improved srucural srengh, and increased siffness 1 4. Beer undersanding of ribological aspecs in THF is imperaive for advancemen of his echnology. Common failures in THF include wrinkling, premaure fracure, and unaccepable par surface qualiy 5,6. Some of hese failures are aribued o eiher uilizing ineffecive lubricaion/lubrican or failure o opimize ribological condiions in he process design. THF can be caegorized ino hree fricion zones: he guiding zone, he ransiion zone, and he expansion zone Fig. 1. Research has shown ha he hree fricion zones exhibi differen saes of sress 7,8. These sress differences, which also imply differen lubricaion mechanisms, have led o he developmen of riboess ha can be used o sudy numerous ribological aspecs in THF This includes a screening of new lubricans ha can perform well in he hree fricion zones, b deerminaion of fricion coefficien via riboes for use in numerical modeling, and c deerminaion of wear characerisics of he dies used. Three varians of riboess are normally used for he guiding zone, as shown in Fig. 2. All of hese ess involve pressurizing he ubular specimen o he required pressure level and pushing he ube hrough a cylindrical die. Varian I was originally developed a he Universiy of Darmsad in collaboraion wih Schuler 8. In his es, he normal load Conribued by he Manufacuring Engineering Division of ASME for publicaion in he JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscrip received June 25, 2008; final manuscrip received December 31, 2008; published online March 18, Review conduced by Jian Cao. is measured via a load cell conneced o he upper half of he die. Applying Coulomb s law, he inerface fricion coefficien can be deermined by Eq. 1. Fricion force, F f, and normal load, F c, are boh measured by load cells conneced o he sysem. To faciliae measuremen of F c, spli die sysem is used. = F f 1 F C This es requires careful design of he ooling o ensure ha energy flow in he sysem componens does no resul in significan errors in he measuremen of he normal load. Varian II was originally developed a he Engineering Research Cener for Ne Shape Manufacuring a he Ohio Sae Universiy 13,14. In his es, he verical shaf ha holds he ube is conneced o a load cell ha measures he fricional force, F f. Unlike varian I, he normal load in his es is deermined indirecly using he inernal pressure of he ube and he properies of he ube maerial. The fricion coefficien is deermined by cf f = 2 P i DiL where F f is he fricion force, P i is he inernal pressure in he ube, Di is he inernal diameef he ube, L is he effecive lengh of he ube, and c is a consan. Accurae deerminaion of c is criical. For higher pressure and ubular specimens wih higher diameer-o-wall hickness raios, he value of c will approach 1. Varian III, developed a he Universiy of Paderborn 15,16, differs from varian II in how he fricion force is deermined. In his es, wo punches are conneced o he ube ends. These punches are conneced o separae load cells o measure he fricion force. This es allows emulaion of he compression of a ube being pushed hrough a die. The fricion force is deermined by aking he difference in he fricion loads measured by he wo load cells. The fricion coefficien can be deermined by Eq. 3. The es assumes ha he inernal pressure is equal o he pressure Journal of Manufacuring Science and Engineering APRIL 2009, Vol. 131 / Copyrigh 2009 by ASME

2 Fig. 1 a Fricion zone in THF; b ypical THF ooling a he ool-workpiece inerface. In realiy, he inerface pressure may vary drasically depending on he fricion level and geomeric variables. = F 1 F 2 3 P i DiL While all of he es varians discussed above can be used o rank lubricans effecively, hey canno accuraely deermine inerface pressure disribuion, which is a vial parameer for accurae deerminaion of fricion coefficien as well as for sudying oher ribological aspecs a he inerface, such as he wear rae of he ools. 2 Objecives and Approach The objecives of his paper are o a esablish a closed-form soluion ha describes he principal sress sae and srain sae of a ube ha is pushed hrough a die under inernal pressure loading, and b formulae a closed-form soluion ha will faciliae deerminaion of conac pressure a he ool-ube inerface and in urn make possible o accuraely deermine inerface fricion based on Coulomb s law. The derived equaions should also be valuable in designing guiding zones where variaions in he inerface pressure loading are a funcion of inerface fricion and he raio beween ube lengh and ube diameer. The analyical model will also bring abou beer undersanding of he severiy of deformaion a he ool-workpiece inerface as he sae of sress/srain varies along he ube lengh. The derivaion commences by esablishing expressions for he principal sresses and srains acing on a plasically deformed ubular maerial ha is being pushed along a die under consan inernal pressure P i. These are radial sress r, longiudinal sress z, hoop sress h, longiudinal srain z, and radial srain r. Hoop srain,, is assumed o be zero because he ube is consrained by he die. Effecive sress and effecive srain are esablished based on von Mises yield crieria, and he deforming ubular maerial is assumed o follow he power law, =K n, where is he flow sress, K is he srengh coefficien, is he effecive srain, and n is he srain hardening exponen. Afer he sae of sress is esablished, he expression for inerface fricion assuming Coulomb s law is derived. The analyical model is verified by a comparison wih finie elemen resuls. Finally, he poenial applicaions of he developed analyical model are discussed. 3 Sress and Srain Analysis of he Guiding Zone Triboes 3.1 Assumpions. The loading condiions in he guiding zone where he maerial is fed oward he die caviy are dicaed by he inernal pressure and axial feeding. The axial feeding is generally done a a relaively lower fluid pressure as compared wih he calibraion pressure, which is acivaed a he end of he THF process. The loading process in he guiding zone can be considered o be approximaely proporional along he ube lengh. Thus, Hencky s deformaion heory can be applied in he analysis. I has been shown ha under proporional loading, he magniudes of resuling srains are pah independen and reduce he governing equaions o Hencky s oal deformaion heory 17. Researchers have also shown ha he deformaion heory of plasiciy may be used for a range of loading pahs oher han he proporion loading wihou violaing he general requiremens for physical soundness of plasiciy heory 18,19. Varma e al. 20 found ha by changing he loading condiions during hydroforming, one can subjec he ube o nonproporional or proporional srain pahs. In heir sudies, hey found ha non- Fig. 2 Varians of guiding zone riboess / Vol. 131, APRIL 2009 Transacions of he ASME

3 Fig. 3 Scheme of guiding zone riboes Fig. 5 Sress sae of ube covered by secions I and II proporional condiions occurs when fluid pressure along wih axial end feed is prescribed, whereas he proporional condiion is observed when he fluid volume flow rae is specified in conjuncion wih axial conracion. In his sudy, Hencky s deformaion heory is adoped. Various simplificaions and assumpions used hroughou he derivaions are as follows. The ube is considered as a hin wall srucure. Longiudinal sress hrough he hickness direcion is assumed o be uniform. Shear sress hrough he hickness direcion induced by fricion force is negleced. Tubular maerial is assumed o be isoropic. Throughou he process. he maerial is considered o be in plasic sae of deformaion. 3.2 Sress Analysis. The scheme of he es is shown in Fig. 3. Figure 4 shows he sress sae on he elemen cu from he ube wall. In he guiding zone riboes, he ube is firs pressurized. When he yielding pressure is aained, he ube expands unil i esablishes conac wih he die. The gap beween he die and he ube before pressurizaion sars is of he ordef 1% of he ube diameer. Since he ube is pushed agains a fricional surface, he inernal pressure P i has forced he ube maerial o conform o he die surface such ha srain in he hoop direcion =0, achieving plane srain condiion. Based on Hencky s oal deformaion heory 17 and he condiion =0, Eq. 4 holds. = d d Z + r /2 =0 = Z + r /2 Taking he balance of forces in he r- and r-z plane, where he elemen widh is 1 uni, Eqs. 5 and 6 can be obained. 4 Piri + rr0 2 sin 2 =0 lim sin = 0 2 = r r i / 5 2 Z + d Z 2 + d Z z2 + f 2 dz =0 6 In he guiding zone riboes, radial sress is compressive sress, i.e., r 0; hus he fricion law can be expressed as dz = r Z d dz dz d = d r = d Z f = r = r d Z dz = d z r + Z dz In he guiding zone riboes, here are wo secions along he longiudinal direcion of he ube Fig. 5. a a 7 Secion I: The secion near he ube end which is pushed a by punch and characerized by he condiions ha z r and z 0. Secion II: The secion near he free ube end characerized by he condiions ha r z and z 0. In oher words, secion I will exhibi compressive longiudinal sress while secion II will exhibi ensile longiudinal sress. I should be noed ha he lenghs l 1 and l 2 will vary based on he locaion along he ube where r = z Deails on he deerminaion of l 1 and l 2 are given in Sec Using Von Misses yield crieria, he effecive sress and srain under plane srain condiion, =0, can be expressed by Eqs. 8 and 9 for secions I and II, respec- Fig. 4 Sresses acing on he elemen: a r- plane and b r-z plane Journal of Manufacuring Science and Engineering APRIL 2009, Vol. 131 /

4 volume consan condiion. Subsiuing hickness expression ino Eq. 15 and inegraing i, Eq. 16 can be obained. d = dfkn+1 z z n = dz dz K n+1 n z n 1d z dz 14 Fig. 6 ively. Flow sress of mos meals used in meal forming closely obey he power =K n. Subsiuing power law ino Eqs. 8 and 9 leads o Eq. 10. = 1 r Z, = z, = 2 3 = 1 Z r, = z, = 2 3 r Z = K n+1 z n secion I Z r = K n+1 n z secion II 10 Equaions 4, 5, 7, and 10 give he iner-relaionship of sresses along he ube lengh. To have a complee descripion of he sae of sress along he ube, he longiudinal srain z needs o be deermined. Also, hoop sress and longiudinal sress need o be expressed in erms of inerface fricion and oher known quaniies. 3.3 Derivaion of Srain. The longiudinal srain z in secion I will be derived from Eqs. 4, 5, 7, and 10. The ube lengh of secion I is se o l 1 and he locaion of z equal o r is se as he original poin of he z-axis, which lead o he boundary condiions z =0 a z=0. The radial sress and longiudinal sress given in Eqs. 11 and 12 are obained by combining Eqs. 4, 5, and 10. Subsiuing he radius o hickness raio,, and f as a funcion of ino Eq. 12, Eq. 13 can be obained. z = Relaionship beween f and K n+1 z n 2r i r = K n+1 z n r i z = fk n+1 z n 13 1 where r i /= /, = /, and f = 2 / 1. As he ube is pushed hrough he die, he variable hickness of ube varies along he z-axis during deformaion. Figure 6 shows he change in parameer f wih respec o. I can be observed ha when =5, and f = As increases beyond 5, f decreases asympoically o 1. Thus for values of greaer han 5, he variaion in parameer f can be ignored. Thus, f can be aken as a consan in he longiudinal direcion. Equaion 14 is obained by aking a derivaive of z peraining o Eq. 13 wih respec o dz. Subsiuing Eqs. 7, 11, and 12 ino Eq. 14, Eq. 15 can be obained. Wih axial load, he ube hickness will increase and can be compued as = 0 e z from 0 z 1 2K n+1 n z n 1d z dz = 2 Kn+1 z n r i 1 2K n+1 z n r i d z dz 15 0 e 1 z 2K n+1 n z n 1 + z n r i 0 e z d z = z 2 0 e Kn+1 z z n r i 0 e z 2 16 Equaion 16 can be solved numerically o obain longiudinal srain, z, which can be subsiued ino Eqs. 11 and 12 o compue longiudinal sress, z, and radial sress, r, respecively. When longiudinal srain z is small, i is reasonable o assume ha he deformed ube hickness approximae he original hickness 0. Thus, Eq. 7 can be reduced o d Z /dz=/ r. Thus he second erm on he lef hand side of Eq. 15 can be ignored. By subsiuing / o = and r i / o = 1 ino Eq. 15, Eq. 17 can be obained. Inegraing Eq. 17, and considering he boundary condiions ha z =0 a z=0, an explici expression for z can be obained as given in Eq. 18. Deails on he inegraion of Eq. 17 are given in Appendix C. z 1 2K 0 n+1 n z n 1 d z = z 17 K n+1 z n z = K n+1 p z iexp 11/n 18 By subsiuing Eq. 18 ino Eqs. 11 and 12, he radial and longiudinal sresses can be obained as funcions of inerface fricion, inernal pressure, wall hickness, and raio of ube ouer radius o ube wall hickness. Equaion 19 gives a complee descripion of he sresses and srains of secion I. z r = p i exp z z = 2 1exp 1p i + p i 2 1 z = exp 1p i 2 + 2exp 1 z +1p i / Vol. 131, APRIL 2009 Transacions of he ASME

5 L 0 =0 L e rdz, l 2 = L, l 1 =0 23 Fig z = K n+1 p z iexp 11/n, r = z, =0 19 By he same derivaion process, Eq. 20 can be obained o describe he sresses and srains of secion II. The deailed derivaion of Eq. 20 is given in Appendix A. z r = p i exp z z = p i p i 2 11 exp = 1 2 p z i1 + exp exp z p i z = 2 1 K n+1 Sress sae of ube fully covered by secion II p z i1 exp, r = z, =0 1/n Deerminaion of Deformed Lengh L, and Lenghs l 1 and l 2 for Secions I and II Deerminaion of Deformed Lengh L When he Tube is Fully Covered by Secion II. In orde deermine wheher he ube is fully covered by secion II, he boundary condiions are firs examined. The boundary condiion of his riboes is such ha one end of he ube is free, for which z is zero. This mees he condiions of secion II ha z r. Subsiuing z =0 ino Eq. 20 l 2 can be deermined. l 2 p i p i 2 11 exp =0 l 2 = 2 1 ln From Eq. 21, i can be seen ha when is very small, l 2 can 1 heoreically be greaer han L. Therefore, when ln2 1/2 2L, he ube is fully covered by secion II, as shown in Fig. 7. Since his condiion occurs under lower fricion, he change in ube lengh due o deformaion is insignifican. Thus he final ube lengh, L, can be approximaed o he original ube lengh L 0. Alernaively, we can deermine, L, by volume consancy condiion, as given in Eq. 22. where V o is he original volume of he ube and V is he volume of he ube afer es. Because he ube is fully covered by secion II, only Eq. 20 is used o calculae he sress and srain. L can be calculaed from Eq. 23, where r is aken from Eq. 20. Thus, when he ube is fully covered by secion II, he deformed lengh L, and lenghs l 1 and l 2 for secions I and II, respecively, can be deermined by Eq Deerminaion of l 1 and l 2 When he Tube is Covered by Secions I and II. The sae of sress when he ube is covered by secions I and II is shown in Fig. 5. The lengh, L, is calculaed by volume consancy using Eq. 22, where he volume of he ube is calculaed by summing he respecive volumes for secions I and II, respecively, as shown in Eq. 24. l 1 and l 2 are finally deermined by Eq. 25. L 0 =0L ln2 1/2 2 exp Z dz ln2 1/2 2 exp Z dz 2 l 2 = ln l 1 = L l , 4 Derivaion of Expression for Fricion Coefficien As discussed in Sec. 1, mos of he exising riboes varians for he guiding zone assume ha he inerface pressure beween he ube and he die is approximaely he same as he inernal ube fluid pressure 16,21. The radial sress derived in Sec. 3 Eqs. 19 and 20 shows clearly ha he inerface pressure may vary drasically depending on he coefficien of fricion and oher geomeric variables. In his secion, we will derive he expression for inerface fricion. This derivaion will include parameers ha can be obained in he guiding zone riboes experimens, such as fricion force F f measured by he load cell refe Fig. 2 and fluid pressure Pisuppliedinsideheube,andhemeasuredoriginalubelenghL o and deformed ube lengh L. In mos of he guiding zone riboes, he longiudinal srain is small and he radial pressures can be deermined by Eqs. 19 and 20, which give he conac pressures a he ool-ube inerface. The wo equaions represen wo conac scenarios ha may occur during he es. One of he conac condiions is ha he ube is full covered by secion II, and he oher is ha he ube is covered by secions I and II. When he ube is covered by secions I and II, Eqs. 19 and 20 are boh used o calculae he conac pressure. The sae of sress under his condiion is shown in Fig. 5. Fricion force F f is now expressed by Eq. 26. Equaion 27 is obained by subsiuing Eqs. 19 and 20 ino Eq. 26. Subsiuing Eq. 25 ino Eq. 28, Eq. 29 is obained. From Eq. 29, he coefficien of fricion can be expressed as shown in Eq. 30. F f =0 F f =0 l 2 2 r dz 2 +0 l 1 2 r dz 1 l 2 z 2 p i exp 2 2 dz 26 L V 0 = V 2 L 0 0 =2r 0 e rdz o l 1 2 p i z 1 1 dz 27 Journal of Manufacuring Science and Engineering APRIL 2009, Vol. 131 /

6 provides punch load a he commencemen of sliding, i.e., he iniial sage is no accouned in his model. I should also be noed ha he load buil-up duraion before he ube sa slide depends on he prescribed inerface fricion, as can be observed in Fig. 9f. Fig. 8 Schemaic of FEA model for he guiding zone es l 1 l 2 F f =2 p iexp exp 28 L F f =4 1 0 p iexp 1 29 = F ln f L 4 p i I should be noed ha he derivaion ha lead o Eq. 30 focused on expressing Coulomb fricion as a funcion of inerface pressure and fricion sress. During he experimen he fricional force, F f, can be measured. Since he normal load canno easily be measured, he analyical model provides he normal sress, r. The influence of ube surface roughness and speed of he punch is no considered. 5 Resuls and Discussion 5.1 Comparison Beween Analyical Model and Finie Elemen Simulaion Resuls. Finie elemen simulaions for he guiding zone were carried ou in orde validae he closed-form soluions developed for conac pressure, longiudinal sress, effecive sress and srain, longiudinal srain, given in Eqs. 11, 12, 16, and 20, and he soluions for fricion force given in Eq. 26. The finie elemen simulaions were carried ou by commercial rigid plasic implici finie elemen analysis FEA code, DEFORM 2D. Various case sudies were simulaed, wo of which are presened in his paper. Figure 8 shows he FEA model used. The die and punch were reaed as rigid bodies and he ube was discreized by 2000 quadrilaeral elemens wih five elemens across he wall hickness. The power law flow sress equaion was used wih srengh coefficien K=500 MPa and srain hardening exponen n=0.3. The forming duraion was se a 10 s. The die was fixed while he punch was assigned a longiudinal velociy of 15 mm/s. A pressure of 30 MPa was applied on he inner surface of he ube as inernal fluid pressure. The inerface fricion a he die-ube inerface was prescribed by assuming Coulombs fricion law. Figure 9 presens sae variables wih wo fricion condiion: one for fricion coefficien of =0.05 and anoher for =0.2. Figure 9 shows he comparison beween he analyical model and he FEA model for conac pressure, effecive sress, and effecive srain disribuion. The analyical model agrees well wih he FEA resuls for boh fricion condiions. Figure 9f show he punch load predicion comparison for =0.05 and =0.2. The punch load for =0.05 from analyical model is 50 kn. When fricion coefficien of =0.2 was used, he load increased o 240 kn. The corresponding FEA resuls for he punch load are 44 kn and 210 kn for =0.05 and =0.2, respecively. These values are close o ha obained using he analyical model. The punch load hisory from he finie elemen simulaion shown in Fig. 9f shows wo disinc sages. The punch load increases gradually wih an increase in he forming ime up o a maximum load; hereafer a consan load is mainained for he whole forming duraion. The iniial sage shows load buildup before he ube sars o slide agains he die. The analyical model 5.2 Poenial Areas of Applicaion for he Esablished Closed-Form Soluions. The esablished equaions can provide local disribuion of field variables along he ube lengh in erms of principal sresses and srains as a funcion of fricion coefficien, fluid pressure, and geomeric variables. These equaions can faciliae beer undersanding of ribological aspecs in he guiding zone and can be helpful in he design of ooling for THF. Tribological aspecs in THF. Mos fricional daa obained from exising riboess for he guiding zone have been deermined by assuming ha he inernal fluid pressure is he same as he inerface pressure. Figures 9a show ha he inerface pressure can increase dramaically beyond he inernal fluid pressure P i. Figure 9a shows ha a a fricion coefficien of =0.2, he inerface pressure varied from 30 MPa o 62 MPa when he inernal fluid pressure was 30 MPa. While coefficien of fricion may be reasonably approximaed by assuming ha inernal fluid pressure equals inerface pressure, using he inernal pressure for sudying oher ribological aspecs such as die wear may lead o significan error. Figure 9c shows he seep gradiens for he longiudinal sress of he ordef MPa. High compressive sresses dicaed by z can provide useful informaion on how cerain lubricans may perform, as well as giving insigh on possible changes in he ube surface morphology. Furhermore, knowing he conac sress a he ool-ube inerface ogeher wih longiudinal sress disribuions should provide informaion o he ribologis/ lubrican formulaon wheher microplasohydrodynamic or microplasohydrosaic lubricaion mechanisms are likely o occur. For highly srain hardening maerials, he local srain disribuion may provide insigh as o wha ypes of lubrican chemisries may be applicable on surfaces ha exhibi significan hardening during deformaion. As discussed in Sec. 1, mos of he guiding zone riboess developed o dae assume ha he inerface pressure o be equal o he inernal fluid pressure. From he derived equaions for conac pressure, see Fig. 9a, he approximaion of inernal pressure will be reasonable only when he fricion level is small and when he ube sample is shorer. As seen in Fig. 9a, up o a ube lengh of 75 mm, he conac pressure is equal o he inernal fluid pressure when =0.05. The experimenal resuls for he guiding zone from Hwang e al. 21 were compared wih he resuls from he derived analyical model. Hwang e al. 21 carried ou he experimens using 70 mm long ube wih an ouer diameef 72 mm and a wall hickness of 3 mm. The inernal pressure of 20 MPa was assumed o be equal o he conac pressure. Using lubrican R68, hey obained a fricion load of 14.5 kn, which yielded a fricion coefficien = From he expression given in Eq. 30, where = / o =12, he inerface fricion can be deermined as follows: ,500 = 3 ln +1 = The fricion value obained from he analyical expression is almos idenical o wha was obained by Hwang e al. 21. As discussed above his should be he case wih low fricion levels. Tube hydroforming ooling design. The guiding zone is an inegral paf THF ooling. The purpose of he guiding zone is o hold and align he ube. I is where he ube is compleely sealed so ha high forming pressure can be aained. The guiding zone also serves as a gaeway for he maerial o be fed o he die caviy. Since he highes sliding velociy is encounered in his zone, he die is suscepible o failure due o wear. Figure 1b shows a ypical THF ooling wih feeding die insers a he guiding zone. Due o high sliding velociy and fricion sress, he guid / Vol. 131, APRIL 2009 Transacions of he ASME

7 Fig. 9 Influence of inerface fricion on conac pressure, effecive sress, and effecive srain disribuion ing zone may be subjeced o severe wear. Using he esablished equaions, ool designers can quickly plo a fricion hill envelope o aid in deermining suiable geomeric parameers, load requiremens for press and axial cylinder acuaors, ec. From Eq. 30, we saw ha he conac pressure is formulaed based on he coordinae sysem CS1 depiced again in Fig. 10 where he origin a he cenerline poin mees he condiion z = r. If a new coordinae sysem CS2 Fig. 10 is esablished a he free end of he ube, hen poin P1 in secion I has wo ses of coordinae z 1,r and z,r for CS1 and CS2, respecively, while poin P2 in he secion II has wo ses of coordinae z 2,r and z,r for CS1 and CS2, respecively. By he geomerical relaion, Eq. 31 can be obained. Combining Eqs. 19 and 20 lead o Eq. 32. Z = Z 1 + L 2 L 2 = Z 2 + Z Z 1 = Z L 2 Z 2 = L 2 Z 31 z 1 r = p i exp, 0 z 1 l 1 r = p i exp z 2, 0 z 2 l 2 32 Subsiuing Eq. 31 ino Eq. 32, Eq. 33 is obained, which can be simplified o Eq. 34. By subsiuing Eq. 25 ino Eq. 34, he conac pressure in he form given in Eq. 35 is obained. Fig. 10 Coordinae sysems of ube r = p i exp Z L 2, L 2 z L Journal of Manufacuring Science and Engineering APRIL 2009, Vol. 131 /

8 Fig. 11 Fricion hill envelope for THF guiding zone Fig. 12 Variaion of maximum R p wih L o /D o raio for various fricion condiions r = p i exp r = p i exp Z L 2, 0 z L 2 33 r = p i exp Z L Z 2 2 z = i exp 2 1p 2 1 ln Equaion 35 can be used o plo a fricion hill envelope for he guiding zone. Figure 11 shows an example of a fricion hill envelope for a guiding zone wih 200 mm ube lengh, 50 mm ube diameer, and 2 mm wall hickness a an inernal pressure P i.as seen in Fig. 11, when =0.2, he maximum pressure acing on he ube-die inerface is over wo imes higher han he fluid pressure. A more generic form, however, is a dimensionless scheme ha shows he variaion in maximum die conac pressure o fluid pressure raio r / P i wih ube lengh o diameer raio L o /D o for various fricion condiions. The variable R p is defined as raio of conac pressure, r, o inernal pressure, P i. By considering he ube as a hin wall srucure where 10, he raio R p can be expressed as a funcion of inerface fricion and geomeric variables. From Eq. 36, i can be seen ha he maximum R p occurs a he loaded end of he ube where Z=L o, as expressed in Eq. 37. R p = r 2 2 z p i = exp 2 1exp z 36 where 2 2 1, 2 1 D o =2 1 0 D o MaxR p exp L o 37 D o Equaion 37 shows ha he maximum conac pressure along he ube lengh depends on he fricion coefficien and he raio of ube lengh L o o he ube ouer diameer D o. Figure 12 shows he variaion in R p wih he L o /D o raio. I can be seen ha maximum conac pressure increases rapidly wih he increase in L o /D o raio. Thus, he raio, L o /D o, is a criical parameer in he design of he guiding zone for ube hydroforming sysems. The fricion hill envelope given in Fig. 11 and variaion in Max R p versus L o /D o given in Fig. 12 have been esablished based on Eq. 35, which is valid for low srain range. Figures 11 and 12 can herefore be applied when he ube lengh o diameer raio L o /D o does no exceed 4 and a maximum coefficien of fricion of =0.2. Appendix B gives deails on derivaion of expressions for deermining sae variables a higher srain level z = An example of ube-die conac pressure disribuion a higher srain level is given in Fig. 13. These curves were generaed by he derived expression given in Appendix B. Figure 13 shows he maximum pressure ha can be generaed when an inernal pressure of 60 MPa is applied on a ubular maerial wih a srain hardening exponen, n=0.3, and a srengh coefficien, K=500 MPa. As can be seen from Fig. 13, an original ube lengh of 500 mm L o /D o =10 resul in a conac pressure of up o 450 MPa when a coefficien of fricion of =0.4 is exhibied a he inerface. This pressure is over seven imes higher han he forming pressure. The figure also shows ha he increase in inerface pressure is highly influenced by he increase in L o /D o raio. Figure 13 also shows he comparison beween he analyical model and FEA for ube lenghs of 200 mm and 400 mm. Wih 200 mm ubing, FEA shows conac pressure a he ube-die inerface of 86 MPa, 113 MPa, and 138 MPa for =0.1, =0.2, and =0.3, respecively, whereas he analyical model exhibied conac pressures of 87 MPa, 109 MPa, and 133 MPa for =0.1, =0.2, and =0.3 respecively. As he ube is pushed in he guiding zone, ube hickening will occur and he ube lengh will be shorened. For an iniial ube lengh of 200 mm, FEA shows ha final deformed ube lenghs were 181 mm, 147 mm, and 127 mm for =0.1, =0.2, and =0.3, respecively. The analyical model resuled in final ube lenghs of 181 mm, 150 mm, and 130 mm for =0.1, =0.2, and =0.3, respecively. Good agreemen can be observed beween analyical model and FEA resuls. Fig. 13 Evoluion of fricion hill envelop and maximum conac pressure for a ubular maerial wih K=500 MPa, n=0.3, and a forming pressure of 60 MPa / Vol. 131, APRIL 2009 Transacions of he ASME

9 6 Conclusions Closed-form soluions ha characerize he guiding zone riboes for ube hydroforming were esablished based on a mechanisic approach. From he derived analyical model field, variables can be compued along he ube. These variables include a conac pressure disribuion, b effecive sress and srain disribuion, c longiudinal sress and srain disribuion, and d hoop sress disribuion. Based on he derived equaion for he conac pressure a he ool-ube inerface, an expression for deermining he coefficien of fricion for he guiding zone riboes was esablished. This expression is a funcion of ube geomeric variables, inernal pressure, and fricion force obained from riboes experimen. Through his sudy, fricion hill envelopes for he guiding zone were esablished. These fricion hill envelopes show he variaion in conac pressure along he ube lengh as a funcion of fricion coefficien. The sudy has also shown ha he maximum conac pressure a he ool-ube inerface increase rapidly wih he increase in L o /D o raio. Thus, he raio L o /D o is a criical parameer in he design of he guiding zone for ube hydroforming sysems. Acknowledgmen The auhors would like o acknowledge he Naional Science Foundaion, hrough which his work was funded under Projec No. DMI Nomenclaure fricion coefficien F f fricion force F c normal load F,F 1,F 2 ube end load P i inernal pressure Di inner diameer D o ouer diameer r i inner radius ouer radius L deformed ube lengh L o iniial ube lengh l 1 lengh of secion I l 2 lengh of secion II deformed ube hickness 0 original ube hickness V volume of deformed ube V 0 volume of iniial ube raio of ubeuer radius o iniial ube wall hickness D i inernal diameef he ube raio of ubeuer radius o insananeous ube wall hickness R p raio of conac pressure o inernal pressure effecive sress effecive srain K srengh coefficien n srain hardening exponen z longiudinal sress r radial sress h hoop sress z longiudinal srain r radial srain hoop srain Z,Z 1,Z 2 longiudinal coordinae of he ube Appendix A: Derivaion of Srain ε z in Secion II of he Tube in he Guiding Zone The longiudinal srain z of secion II can be derived from Eqs. 4, 5, 7, and 10. We se he ube lengh l 2 and he original poin on he z-axis o where z = r refer Fig. 5. Because secion II is near he free ube end, he longiudinal srain z is small. Thus he erm d in Eq. 6 can be negleced and Eq. 6 can be reduced o Eq. A1. The radial sress and longiudinal sress given in Eqs. A2 and A3 are obained by combining Eqs. 4, 5, and 10. From Eq. A3, Eq. A4 can be derived. Subsiuing Eqs. A1 and A2 ino Eq. A4 yields Eq. A5. d Z dz = r A1 z = r = d z dz = P i 2r i K n+1 z n 2 2 P i r i 1 2K n+1 z n K n+1 n z n 1 d z dz A2 A3 A4 1 2K n+1 n 1 n d z z dz = r i P i 2 Kn+1 n z A5 Secion II is near he free ube end and he axial load from he punch drops significanly in his secion, which resuls in small axial srain z. Thus, i is reasonable o assume ha he deformed ube hickness approximae original lengh 0. Le / o =, r i / o = 1, and subsiue ino Eq. A5 o obain Eq. A6. Inegraing Eq. A6, and invoking he boundary condiions ha z =0 a z=0, an explici expression for z can be obained as given in Eq. A7. By subsiuing Eq. A7 ino Eqs. A2 and A3, he radial and longiudinal sress can be obained as funcions of inerface fricion, inernal pressure, wall hickness, and he raio of ube ouer radius o ube wall hickness. 1 2K n+1 n 1 n d z z dz = P i 1 2 Kn+1 n z 2 1 z = K n+1 p i1 exp z 2 11/n A6 A7 Appendix B: Explici formula for Evaluaion of Sae Variables a High Srain Range Figure 14 shows a ypical characerisic curve Z versus z obained from Eq. 16. Figure 14 shows ha longiudinal srain z varies approximaely linearly wih longiudinal ube lengh z a high srain range of z = 0.1 z = 2.0. I is herefore possible o ge an explici formula for evaluaing he sae variables such as conac pressure a high srain level. From he linear relaionship, Eq. 16 can be simplified o Eq. B1. z = z Slp z B1 where Slp is he slope of he curve and Z 0.1 is he longiudinal lengh where he longiudinal srain is z = 0.1. The slope of he curve can be compued approximaely by subsiuing z = 1.0 ino he inegrand a he lef side of Eq. 16 which leads o Eq. B2. Z 0.1 can be obained by subsiuing z = 1.0 ino Eq. 18, as shown in Eq. B3. Noe ha he slope, Slp, is a funcion of K, n,, o, Pi, and. Journal of Manufacuring Science and Engineering APRIL 2009, Vol. 131 /

10 subsiuing Eq. B4 ino Eqs. 11 and 12, he conac pressure and longiudinal sress can be expressed explicily. The deformed ube lengh L can be evaluaed by he volume consan condiion as given in Eq. B5. Subsiuing Eq. B4 ino Eq. 11, he explici form for conac pressure evaluaion is obained as Eq. B6. The expression for maximum pressure can be obained by subsiuing Z=L o ino Eq. B6. z = bz c Fig. 14 Characerisic curve of Eq. 16 ek n+1 +2P i e b = 2 en +1K n+1 0 e +2P i 0 e e 0 e 1 2K n+1 0 e n +1 0 e Slp = r i 2 0 e Kn+1 0 e 2 = 1 2 en +1K n+1 0 e +2P i 0 e e ek n P i e 2 1 = K n+1 p z ie /n z 0.1 = B2 ln Kn n 2 1p i +1 B3 Subsiuing Eqs. B2 and B3 ino Eq. B1, an explici expression for he longiudinal srain can be obained, Eq. B4. By c = ln Kn n 2 1p i +1 B4 L V 0 = V 2r 0 0 L 0 2r 0 0 exp0.1 + bz cdz =0 L = lnbl 0 + exp0.1 bc 0.1 b + c B5 r = Kn bz c n 0 e 0.1+bz c 2 0 e 0.1+bz c 2 0 e 0.1+bz c B6 Appendix C: Inegraion of Equaion (17) dz = 1 2K n+1 n z n 1 d z K n+1 z n Le a = 1 2K n+1, b = 1 2 Kn+1, c = an dz = z n 1 d z b z n = a + c b db z n + c b z n + c P i C1 C2 C3 a b lnb z n + c + c 0 = z C4 The boundary condiions are such ha when z=0, z = r.by subsiuing z =0 a z=0 ino Eq. C4, he consan c 0 can be deermined. References c 0 = a b lnc, ln z = a b z n + c b c z = bexp c bz 11/n a 2 1 = K n+1 p z 11/n iexp 2 1 C5 1 Koc, M., and Alan, T., 2001, An Overall Review of he Tube Hydroforming THF Technology, J. Maer. Process. Technol., 108, pp Sieger, K., Häussermann, M., Lösch, B., and Rieger, R., 2000, Recen Developmens in Hydroforming Technology, J. Maer. Process. Technol., 982, pp Dohmann, F., and Harl, Ch., 1996, Hydroforming A Mehod o Manufacure Lighweigh Pars, J. Maer. Process. Technol., 60, pp Nguyen, B. N., Johson, K., and Khaleel, M. A., 2003, Analysis of Tube Hydroforming by Means of an Inverse Approach, ASME J. Manuf. Sci. Eng., 125, pp Asnafi, N., and Skogsgårdh, A., 2000, Theoreical and Experimenal Analysis of Sroke-Conrolled Tube Hydroforming, Maer. Sci. Eng., A, , pp Asnafi, N., 1999, Analyical Modelling of Tube Hydroforming, Thin-Walled Sruc., 344, pp / Vol. 131, APRIL 2009 Transacions of he ASME

11 7 Ngaile, G., Jaeger, S., and Alan, T., 2004, Lubricaion in Tube Hydroforming THF Par I: Lubricaion Mechanisms and Developmen of Model Tess o Evaluae Lubricans and Die Coaings in he Transiion and Expansion Zones, J. Maer. Process. Technol., 146, pp Prier, M., and Schmoeckel, D., 1999, Tribology of Inernal High Pressure Forming, MAT-INFO Werkssoff-Informaionosgesellschaf mbh, Humburger Allee 26, D Frankfur, pp Ngaile, G., Jaeger, S., and Alan, T., 2004, Lubricaion in Tube Hydroforming THF Par II: Performance Evaluaion of Lubricans using LDH Tes and Pear Shaped Tube Expansion Tes, J. Maer. Process. Technol., 146, pp Dalon, G., 1999, The Role of Lubricans in Hydroforming, Proceedings of he Auomoive Tube Conference, Dearborn, MI, Apr Koc, M., 2003, Tribological Issues in he Tube Hydroforming Process Selecion of a Lubrican for Robus Process Condiions for an Auomoive Srucural Frame, ASME J. Manuf. Sci. Eng., 125, pp Gariey, M., 2003, Enhancemen of Tribological Condiions in Tube Hydroforming and he Viabiliy of Twis Compression Tes for Screening Tube Hydroforming Lubricans, MS hesis, The Ohio Sae Universiy, Columbus. 13 Ngaile, G., Federico, V., Tibari, K., and Alan, T., 2001, Lubricaion in Tube Hydroforming THF, Trans. NAMRI/SME, XXIX, pp Ngaile, G., and Alan, T, 2001, Pracical Mehods for Evaluaion Lubricans for Tube Hydroforming, Hydroforming Journal, pp Plancak, M., Vollersen, F., and Woischig, J., 2005, Analysis, Finie Elemen Simulaion and Experimenal Invesigaion of Fricion in Tube Hydroforming, J. Maer. Process. Technol., 170, pp Vollersen, F., and Plancak, M., 2002, On Possibiliies for he Deerminaion of he Coefficien of Fricion in Hydroforming of Tubes, J. Maer. Process. Technol., , pp Imaninejad, M., and Subhash, G., 2005, Proporional Loading of Thick- Walled Cylinders, In. J. Pressure Vessels Piping, 82, pp Jahed, H., Lamber, S. B., and Dubey, R. N., 1998, Toal Deformaion Theory for Non-Proporional Loading, In. J. Pressure Vessels Piping, 75, pp Budiansky, B., 1959, A reassessmen of Deformaion Theories of Plasiciy, ASME J. Appl. Mech., 26, pp Varma, N. S. P., and Narasimhan, R., 2008, A Numerical Sudy of he Effec of Loading Condiions on Tubular Hydroforming, J. Maer. Process. Technol., 196, pp Hwang, Y. M. and Huang, L. S., 2005, Fricion Tess in Tube Hydroforming, Proc. Ins. Mech. Eng., Par B, 219, pp Journal of Manufacuring Science and Engineering APRIL 2009, Vol. 131 /