Thin-Walled Tube Extension by Rigid Curved Punch

Transcription

1 Engineeing,, 3, doi:.436/eng..355 Publihed Online May ( Thin-Walled Tube Extenion by Rigid Cuved Punch Abtact Rotilav I. Nepehin Platic Defomation Sytem Depatment, Mocow State Univeity of Technology STANKIN, Mocow, Ruia Received Febuay, ; evied Mach 6, ; accepted Mach 3, Compute model i developed fo non-teady and teady-tate poce of thin-walled tube extenion by the igid punch with cuved pofile. Rigid-platic membane hell theoy with quadatic yield citeion i ued. Tube mateial nomal aniotopy, wok hadening, wall thickne vaiation and fiction effect ae conideed. FORTRAN pogam of the model pedict ditibution of the thickne, meidian te, yield te and peue along cuved geneato of defomed tube and the tube extenion foce veu punch diplacement elation. Model pediction ae coelated with expeimental data. Keywod: Tube Extenion, Rigid-Platic Membane Shell, Cuved Rigid Punch, Nomal Aniotopy, Wok Hadening, Wall Thickne Vaiation. Intoduction Thin-walled tube extenion by igid punch with cuved geneato can be ued fo foming of pecified pofile at the tube end defined by it function in machine deign. Appoximate analyi of the thin-walled tube extenion by igid punch i contained, mainly, by conical fom with appoximate yield citeion of iotopic mateial, appoximate wall thickne vaiation and wok hadening effect [-4]. Pactical technology of the tube extenion by cone punch eveal cuved geneato foming at tanition egion fom cone to cylinde pat of the tube [-4], which ae difficult to contol. But cuved geneato can be pecified by hydo- o aeodynamic paamete of the tube pofile. Thin-walled metal tube can eveal nomal aniotopy induced by cold olling technology with the eult of te-tain and wall thickne effect duing tube extenion. So, development of compute model of the tube extenion poce by igid punch with cuved geneato and mateial nomal aniotopy conideation, deem, i impotant engineeing poblem. Simulation of non-teady tube platic foming by the finite element method (FEM) i limited by difficult poblem of lage matix Equation accuate olution fo nonlinea platic mateial model with wok hadening effect, te-tain elation, vaiable hell thickne and cuved tool bounday. Nonlinea platic poblem in commecial FEM code i teated a non-linea elaticity o vicou olid without finite yield te, with the eult of non-accuate te tate calculation. Moe accuate imulation of thin-walled tube platic foming, deem, hould be made by coect numeical olution of odinay diffeential Equation deived fom exact equilibium Equation of membane igid-platic thin-walled hell model with Mie yield citeion, including wok hadening and nomal aniotopy effect. Thi appoach wa ued uccefully fo non-teady model of platic hell dawing, which good coelated with expeiment [5-8], and fo thin-walled tube eduction by matix with cuved pofile [9]. Peented model of the thin-walled tube extenion by igid punch with cuved geneato i baed on membane theoy of the igid-platic hell of evolution, with mateial nomal aniotopy, wok hadening, wall thickne vaiation and fiction effect included. FORTRAN pogam ae witten fo numeical olution of the poblem diffeential Equation. Numeical eult of compute imulation ae given fo the S- mode coine, double cicula and cone-cicula punch pofile. Peented model fo coine punch pofile and elated model [9] fo the tube eduction by cuved matix ae eaonable coelated with expeimental data.. Poblem Fomulation Scheme of the thin-walled tube extenion by the punch with cuved geneato i hown in Figue. Cylindical co-odinate, z, θ ae elated with fixed punch, while Copyight SciRe.

2 R. I. NEPERSHIN 453 the tube i moved in poitive z diection. The punch cuved pofile i pecified by diffeentiable function = (z) on the axial length H with continue tangent angle, defined by deivative d dz = tg. The S- mode pofile i conideed with continue conjunction with cylinde uface of the tube with inne adiu at the point A, and the punch with maximal adiu R at the point C. The angle = at the point A and C, and = α at the maximum point d dz =. The pofile cuvatue available hould atify condition of continue punch-tube contact with poitive peue p, defined by olution of diffeential equilibium Equation with platic yield citeion. Platic foming of the tube i geneated by axial diplacement of the tube igid pat with initial wall thickne h. Diplacement define defomed tube egment AB. If = l then point B of the tube edge i coincided with the final punch pofile point C, whee the tube platic tain inceae i topped. Tube extenion poce i non-teady at the diplacement inteval l with the platic tain inceae and the wall thickne h deceae on the cuved tube egment AB. If l then teady-tate extenion poce begin, with accuacy of fiction effect defined by lip of cylindical tube egment with adiu R on the punch uface. Tube extenion atio R i contained by the limit platic tain e p * of tenion in cicula diection θ of the tube font edge which lead to local inceae of the platic tain followed by factue of the tube edge. The e p * value i defined by mateial wok hadening behavio []. Limit atio R, defined by the platic tain e p *, i.-.3 fo high platic teel tube extenion by cone punch [,4]. Inceae of the metal platicity by heat of the defomed tube lead to eential inceae of extenion atio [4]. In the cae of tube extenion at elevated tempeatue the ideal platic mateial model can be ued with the yield te etimation fo mean tain ate and tempeatue value. Second containt of the limit tube extenion atio i buckling of thin-walled initial tube induced by compeion meidional te σ A at the ection z =. Detailed expeimental invetigation of cylindical tube buckling ae given in Ref [,]. Appoximate etimation of the citical elation A S fo the tube extenion by cone punch ae given in Ref [4]. Citical buckling atio of the tube can be inceaed eentially by kinematical containt of the tube wall in tube extenion die deign [4]. 3. Ste Stain Relation In the cae of thin-walled tube extenion defomed mateial element of the tube middle uface i loaded by membane pincipal tee σ = σ θ > in cicula diection θ, σ = < in meidian diection tangent to the punch pofile, and σ 3 = in nomal diection to the punch pofile. Genealized Mie yield citeion fo the pincipal tee in the cae of nomal aniotopy in diection of the tube wall thickne can be witten a follow [7] a () a Coefficient of nomal aniotopy a i atio of the width to thickne platic tain defined by axial tenion of heet metal pecimen []. Mateial yield te σ i defined by accumulated platic tain e p uing wok hadening elation n CeP () Platic flow ule aociated with the yield citeion () define incement of the platic tain in and θ diection a a de cde, c (3) a a Accumulated effective platic tain incement de p i defined by the and de incement uing platic incompeibility condition a dep de de dede (4) aa Subtitution elation de θ = d and Equation (3) into Equation (4) define de p a the function of the te tate and cicula platic tain incement a d dep сс (5) aa Platic incompeibility condition and Equation (3) define diffeential elation fo the wall thickne veu cicula platic tain incement and te tate coefficient c dh d c (6) h At the tube edge B (Figue ) we have =, c = a ( a), and wall thickne i found by integation of Equation (6) h hb h exp ln (7) a B hb Extenion poce can be ued to fom of hot ing with initial dimenion, h, l to final dimenion R, h, l, with the wall thickne h defined by Equation (7) h h exp ln a R h h (8) Copyight SciRe.

3 454 R. I. NEPERSHIN and length l defined by contant volume condition l l h h h R h Ste equilibium Equation conideed below will be olved uing the yield citeion () to wite poitive cicula te σ θ a the function of meidian te and yield te σ a a a) () a 4. Tube Ste State Ste tate of defomed tube i calculated uing membane theoy of the igid-platic hell with yield citeion (), wok hadening (), wall thickne vaiation (6) and Coulomb fiction coefficient f at the punch contact bounday pecified by it geneato. The hell element equilibium equation in nomal diection to the hell middle uface define elation of the nomal peue p veu tee σ φ, σ θ and the hell cuvatue p h R R (9) () If punch geneato i pecified by the function = (z) then cuvatue adii ae defined by the Equation R 3/ d tg z h d, tg () d dz R h (3) co Fom Equation () it i follow, that contact peue p i poitive if R R (4) If inequality (4) i not atified, then the tube i depated fom the punch pofile, and cuved fee bounday of the tube i geneated. Ste tate of extended tube atifie inequalitie > and <. Hence, the inequality (4) i defined by the tube te tate, value and ign of the punch pofile adii R and R. Tube te tate hould atify equilibium Equation in meidian diection to the middle uface, which fo the cae of vaiable wall thickne and Coulomb contact fiction i witten a follow: d dh pf (5) d h d hin Subtitution Equation () and () into Equation (5) give diffeential Equation fo the meidian te, with pecified ditibution of the h and σ on the tube middle uface. If the punch pofile i pecified by the Equation = (z) with continue cuvatue adii defined by Equation () and (3), then integation of Equation (5) can be pefomed uing z vaiable. In thi cae Equation (5) can be witten in the fom: d tg f dh f tg (6) dz R co h dz whee σ θ and R ae defined by Equation () and (). If concave and convex punch pofile egment ae pecified by cicle adii and, then vaiable i eaonable fo integation of Equation (5). Uing elation d = in d, d = in d and Equation (), diffeential Equation (5) can be witten in the fom a Equation (7) on the concave pofile egment, and Equation (8) on the convex pofile egment. If the punch pofile i cone, conjugated with cicle adii and, then Equation (7) and (8) ae ued on the cuved pofile, while the length l of cone geneato, inclined at the angle α to the z axi, i ued fo integation of Equation (5), which take the fom: d in dh in f co dl h dl (9) Simulation of non-teady tube extenion by igid punch i pefomed by numeical integation of Equation (6)-(9) by econd ode Runge method with pecified punch pofile, and the tube font edge B moving fom the point A to the point C (Figue ). Integation ae pefomed along the pofile fom the point B, whee the bounday condition ae pecified e e h h, p, p ln B B, () d dh h in f in f co, d h d (7) d dh h in f in f co, d h d (8) Copyight SciRe.

4 R. I. NEPERSHIN 455 and thickne h B i defined by Equation (7), to the point A, uing ditibution of the h, e p and σ known fo peviou poition of the point B, which i coincided with the point A at the initial poce tage. Effective platic tain incement de p ae calculated fom Equation (3) and (5) with mateial point diplacement to the neighbouing node of the tube egment AB. Summeing of the effective platic tain of mateial point and calculation of the σ and h fom Equation (), (6), define ditibution of the σ and h fo the next poce tage. Calculation of non-teady tage ae teminated when the point B i coincided with the point C, and teady-tate tube extenion begin. Diplacement of initial tube i elated with the edge point B poition and wall thickne ditibution by integal incompeibility condition B h hdz h h () Aco Extenion foce P veu i defined by maximum compeion te σ A () = () at the point A, which i found by integation of Equation (6)-(9) when point B i moving fom the point A to the point C. P h h () A FORTRAN pogam ae witten fo compute imulation of the tube extenion fom initial adiu to the final adiu R with thee punch pofile pecified on the length H of the z axi (Figue ). Figue. Tube extenion by the igid cuved punch. 5. Punch Pofile Coine pofile i pecified by the function R z coπ, z H (3) H Fit and econd deivative of the function (3), which define the tangent angle and cuvatue adiu R by Equation (), ae a follow tg π R z in π (4) H H R π z co π z H H d d (5) Equation (3)-(5) ae ued fo numeical integation of Equation (6) with contant tep dz = H N, whee N i numbe of node on the punch pofile. Double cicula pofile i pecified by cicle adii on the concave and on the convex egment with tangent angle α at the bend point. The pofile paamete, and α atify the following elation R (6) ( co ) in (7) H The angle α i found fom Equation (6) and (7) in the fom d R in, d (8) d H If adiu i pecified and atify the inequality H, (9) in then adiu can be found fom Equation (7), and vice vea. So, double cicula punch pofile i defined by the paamete H, R, and o. The pofile co-odinate ae pecified in paametic fom veu tangent angle co, z in, (3) On the concave egment, and R co, z H in, (3) On the convex egment. Equation (3) and (3) ae ued fo numeical integation of Equation (7) and (8) with contant tep d = N, whee N i node numbe on the each cicula pofile. Cone pofile with cicula conjunction i pecified by adii and on the cuved concave and convex egment, length L and angle α of the cone egment. The value,, L, α ae elated with H, R, by the Equa- Copyight SciRe.

5 456 R. I. NEPERSHIN tion in co Lin R (3) Lco H (33) If the angle α atify inequality in co R H, (34) then L and adii um ae found fom Equation (3) and (33) in L R H co H Lco (35) (36) in So, cone punch pofile with cicula conjunction i defined by H, R,, α and adiu o. Co-odinate of the pofile ae pecified in paametic fom veu by Equation (3) and (3) on the cuved egment, and veu l on the cone egment co lin, z in lco, l L (37) Equation (3), (3) ae ued fo numeical integation of Equation (7) and (8) with tep d and node numbe N on the each cuved egment. Equation (37) ae ued fo numeical integation of Equation (9) on the cone egment with tep dl = LN, whee N i node numbe on the cone pofile egment. (a) coine, (b) double cicula and (c) cone with cicula conjunction punch pofile. Cloe P max value ae explained by equal length H, extenion atio R and cloe cuvatue adii of the cuved punch pofile egment. Ditibution of accumulated effective platic tain e p, meidian te, contact peue, wall thickne hh and yield te R along defomed tube geneato at the final extenion tage ae hown in Figue 3 fo (a) coine, (b) double cicula and (c) cone with cicula conjunction punch pofile. Value e p, h/h, and p at the font tube edge z = H, whee =, ae defined by the final adiu R fo all punch pofile, with e p =.8, hh =.868, =.543 and p = 3.3. Ditibution of e p, hh, ae contained by pecified initial and final value, and ae cloe fo thee punch pofile conideed. Contact peue p ditibution ae eentially 6. Numeical Reult Numeical eult ae peented fo the tube extenion imulation fom initial adiu = 3 mm with wall thickne h = mm to final adiu R = 4 mm by thee cuved punch pofile on the length H = 5 mm. Tube mateial i low cabon teel with initial yield te σ = 3 N/mm and wok hadening paamete C =.7, n =.67. Coine punch pofile, defined by Equation (3), ha continue cuvatue vaiation with adii = =.67 mm at the point z = and z = H and tangent angle α =.56 at the middle bend point. Double cicula pofile i pecified by adii = 5. mm, =.5 mm and α =.76 at the bend point. Cone pofile with cicula conjugation i pecified by =. mm, = 4. mm, α =.5 and L = 4.6 mm of cone egment. Foce P veu diplacement elation ae hown in Figue fo thee punch pofile with a = and f =., up to the teady tate tube extenion onet. Relation P() ae S-mode cuve with P max value 8.6, 7.9, 7.83 kn at the final value l = 8.75, 3.33, 9.94 mm fo Figue. Extenion foce P veu diplacement fo (a) coine; (b) double cicula and (c) cone with cicula conjunction punch pofile. Copyight SciRe.

6 R. I. NEPERSHIN 457 diffeent fo conideed punch pofile. In the cae of coine pofile (Figue 3(a)) adiu R on meidian plane and peue p ae continue. Minimal peue value.68 i at the pofile bend point with inceae of the peue to maximal value 6.8 at the point z = on concave pofile egment. In the cae of double cicula pofile (Figue 3(b)) thee i peue dicontinue change fom.3 to 4.9 at the pofile bend point, a the eult the value and ign dicontinuity of adiu R in Equation (). In the cae of cone with cicula conjunction pofile (Figue 3(c)) adiu R i dicontinue at the point of cone conjunction with con- 5 p cave and convex pofile egment, whee minimal peue ae.46 and.45. Peue value ae deceaed on convex pofile egment with mall compeion te and deceae of cicula te σ θ, a can ee in Equation (). But module of negative adiu R ae lage on convex pofile egment, with the eult of poitive peue p along all pofile without deviation of defomed tube fom the punch contact boundaie. Ditibution of compeive meidian te along pofile ae continue inceaed cuve with maximal value.488,.485 and.484 at the point z = fo pofile (a), (b) and (c) accodingly. Effect of diffeent peue ditibution on the and othe vaiable i negligible, becaue peue and fiction coefficient in Equation (5)-(9) ae mall. Inceae of aniotopy coefficient a fom to lead to inceae of minimal wall thickne of the tube font edge, defined by Equation (8), at 4.%; with deceae of maximal compeion te max and tube extenion foce P max at.7% fo conideed punch pofile. Fiction effect on maximal compeion meidian te and extenion foce P value i given in Table fo iotopic tube extenion to the final adiu R and thee punch pofile (a), (b) and (c) conideed above. Inceae of the fiction coefficient f fom to.5 lead to inceae of maximal and P value at 4% fo all punch pofile, with a mall diffeence of the pofile length. 5 p 7. Expeiment Expeimental veification of the thin-walled tube extenion theoy ha been pefomed uing device fo equential thin-walled ing extenion by the punch with S-mode cuved pofile hown in Figue 4 [3]. Fit ing 4 with initial dimenion L, h, d i fixed in uppot 3 goove (left ide in Figue 4). Punch puhed by the od, and fit ing i extended up to the middle of the punch Table. Fiction and punch pofile effect on maximal value of the meidian te and extenion foce. 5 p Figue 3. Ditibution of platic tain e p, meidian te, wall thickne hh, yield te and contact peue p along the tube geneato fo (a) coine, (b) double cicula and (c) cone with cicula conjunction punch pofile coine pofile P, kn double cicula pofile P, kn cone with cicula conjunction pofile P, kn f Copyight SciRe.

7 458 R. I. NEPERSHIN pofile (ight ide in Figue 4). Second ing 5 i fixed in uppot 3. Duing econd toke of the punch ing 5 i extended to the middle of the punch pofile, and ing 4 i extended to the uppe end of the punch. Then ing 6 i fixed in uppot 3 followed by the punch thid toke, and fit ing 4 i puhed fom the punch with final dimenion L, h, d. Dimenion L, h, d of pecimen fo extenion by the punch wee obtained by the ing eduction though the matix with S-mode pofile hown in Figue 5 [4] and ued fo veification of the thin-walled tube eduction model [9]. Fit ing 4 with initial dimenion L, h, d i fixed in cylindical pat of the matix with diamete d and puhed to the middle of the matix pofile by the punch. Second ing 5 i fixed in the matix and puhed to the middle of the matix by econd toke of the punch, while fit ing i puhed to the end of the matix pofile. Finally, ing 6 i fixed in the matix followed by it puhing to the middle of the matix, ing 5 i puhed to the end of the matix duing thid punch toke, and fit ing 4 i puhed out of the matix with final dimenion L, h, d. Extenion of educed ing by the cuved punch wa ued to inceae platic tain and popetie of the ing. Sample fo eduction and extenion expeiment with dimenion d = 39.9 mm, h =.98 mm, L = 7.3 mm L Figue 4. Device fo equential ing extenion by cuved punch. -puh od, -punch, 3-uppot, 4-6-extended ing. L Figue 5. Device fo equential ing eduction by cuved matix. -punch, -matix, 3-uppot, 4-6-educed ing. (Figue 5) wee tuned fom hot olled tube of cabon teel St 3 (Ruian metallugy tandad). Wok hadening cuve σ (e p ) wa found by compeion tet of hot ing pecimen, tuned fom the tube, by lubicated mooth flat die. Mateial paamete of the appoximation () ae σ = 3 Hmm, C =.8 and n =.4. Io- topic mateial i aumed, with a = in Equation (3)-(). Coine pofile (3) wee ued fo the matix and punch manufactued on machine tool with digital contol pogam, followed by the heat teatment fo pecified hadne. Pofile (3) paamete ae R = 8 mm, = 3.74 mm, H = 3mm fo the punch (Figue 4), and R = mm, = 6 mm, H = 3 mm fo the matix (Figue 5). Reduction and extenion expeiment wee pefomed on tandad hydaulic teting machine with bough the foce and diplacement contol. Ring pecimen, punch and matix pofile wee lubicated by oil-gaphite upenion. Coulomb fiction coefficient f fo the tool oughne with lubication wa aumed in the ange Compaion of pedicted elation P() fo the ing eduction by cuved coine matix [9] with expeimental data i hown in Figue 6. Non-teady and teady tate expeimental data ae eaonable coelated with pedicted elation in the ange of poible Coulomb fiction coefficient value. Pedicted final ing dimenion afte eduction [9,4] h =.6 mm, L = 9.5 mm ae good coelated with meaued dimenion h =.3 mm, L = 9.3 mm of the educed ing. Device fo the ing extenion by the cuved coine Copyight SciRe.

8 R. I. NEPERSHIN 459 punch (Figue 4) ha been manufactued eveal month late afte ing eduction device with the eult of educed wok hadening effect elaxation. Compaion of pedicted elation P() fo the educed ing extenion by the cuved coine punch of the peent model with expeimental data i hown in Figue 7. Non-teady expeimental data ae good coelated with the model in the ange.5-.8 of the fiction coefficient f. Fit ing extenion to the middle of the punch pofile i cloe to pedicted cuve with f =.8, while futhe two ing extenion to the end of the punch pofile i cloe to pedicted cuve with f =.5. Pedicted final ing dimenion afte extenion h = mm, L = 7.4 mm ae good coelated with meaued dimenion h =.95 mm, L = 7.3 mm of the final ing. 8. Concluion Model of thin-walled tube extenion by cuved igid punch baed on membane igid-platic theoy ae developed by numeical olution of diffeential Equation along pecified cuved punch geneato with conideation of mateial nomal aniotopy, wok hadening, contact fiction and wall thickne vaiation, defined by genealized Mie flow ule. FORTRAN pogam of the model pedict extenion foce veu diplacement and ditibution of effective platic tain, yield te, meidian te and contact peue along the tube geneato at pecified punch o tube diplacement up to teady tate poce beginning. Poitive contact peue, defined by pecified S-mode punch pofile, i eential condition fo the tube foming without deviation fom contact bounday with the punch. Numeical example of mild teel tube extenion by S-mode coine, double cicula and cone with cicula conjunction punch pofile fo extenion atio R = Figue 6. Pedicted (olid line) and expeimental ( non-teady, teady tate) elation P() fo thin-walled ing eduction by cuved coine matix. Figue 7. Pedicted (olid line) and expeimental ( non-teady, teady tate ) elation P() fo thin-walled ing extenion by cuved coine punch..3 with nomal aniotopy vaiation fom to how inceae of minimal thickne of the tube font edge at 4.% with deceae of maximal extenion foce at.7%. Inceae of the fiction coefficient fom to.5 lead to datic gowth of extenion foce and maximal compeion meidian te, which can be contained by the tube buckling. Model of thin-walled tube extenion by cuved punch and thin-walled tube eduction by cuved matix [9] ae ued in patent [3,4] fo thin-walled ing platic foming. Pedicted foce-diplacement elation and dimenion of thin-walled cabon teel ing foming by lubicated matix and punch ae eaonable coelated with expeimental data. 9. Refeence [] V. P. Romanovky, Cold Stamping Handbook, Mahinotoenie, Leningad, 979. [] E. A. Popov, Bai of Sheet Stamping Theoy, Mahinotoenie, Mocow, 977. [3] E. A Popov, V. G. Kovalev and I. N. Shubin, Technology and Automation of Sheet Stamping, Baumann Univeity Pe, Mocow, 3. [4] Ju. A. Avekiev and A. Ju. Avekiev, Cold Stamping Technology, Mahinotoenie, Mocow, 989. [5] R. I. Nepehin, Simulation of Thin-Walled Axiymmetical Shell Dawing with Flat Flange, Kuznechno- Shtampovochnoe Poizvodtvo: Obabotka Mateialov Davleniem, No. 6, 8, pp [6] R. I. Nepehin, Simulation of thin-walled Axiymmetical Shell Dawing with Flat Flange (Continuation), Kuznechno-Shtampovochnoe Poizvodtvo: Obabotka Mateialov Davleniem, No.7, 8, pp [7] R. I. Nepehin, Simulation of Thin-Walled Axiymmetical Shell Dawing by Complex Fom Punch with Nomal Aniotopy and wok hadening of Wokpiece Mate- Copyight SciRe.

9 46 R. I. NEPERSHIN ial Conideation, Kuznechno-Shtampovochnoe Poizvodtvo: Obabotka Mateialov Davleniem, No. 3, 9, pp [8] R. I. Nepehin, Thin-Walled Conical Shell Dawing fom a Plane Blank, Mechanic of Solid, Vol. 45. No.,, pp. -. [9] R. I. Nepehin, Peing of Thin-Walled Tube by a Cuvilinea Matix, Jounal of Machiney Manufactue and Reliability, Vol. 38, No. 3,, pp [] A. D. Tomlenov, Theoy of Metal Platic Defomation, Metallugy, Mocow, 97. [] K. R. F. Andew, G. L. England and E. Ghani, Claification of the Axial Collape of Cylindical Tube unde Quai-Static Loading, Intenational Jounal of Mechanical Science, Vol. 5, No. 9-, 983, pp doi:.6/-743(83)976- [] A. G. Mamali and W. Johnon, The Quai-Static Cumpling of Thin-Walled Cicula Cylinde and Futa unde Axial Compeion, Intenational Jounal of Mechanical Science, Vol. 5, No. 9-, 983, pp doi:.6/-743(83)978-4 [3] R. I. Nepehin, Device fo Extenion of Thin-Walled Cylindical Ring, Ruian Fedeation Patent, No , Regiteed in Augut 9. [4] R. I. Nepehin, Device fo eduction of Thin-Walled Cylindical Ring, Ruian Fedeation Patent No. 958, Regiteed in 7 June.. Notation o R o h o minimal punch adiu and initial tube inne adiu (mm) maximal punch adiu and final tube inne adiu (mm) initial thickne of the tube wall (mm) h vaiable thickne of the tube wall (mm) tube diplacement elative the fixed punch (mm) l o final tube diplacement at the end of non-teady poce (mm) f Coulomb fiction coefficient, z, θ cylindical co-odinate H o length of cuved punch pofile along the z axi (mm) P tube extenion foce (kn) R cuvatue adiu of the tube middle uface on meidian plane (mm) R cuvatue adiu of the tube middle uface on nomal plane (mm) tangent angle of the punch pofile and tube middle uface with the z axi σ initial yield te of the tube mateial ( Hmm ) e p accumulated effective platic tain σ yield te of the tube mateial defined by e p ( Hmm ) a nomal aniotopy paamete elation of the width to thickne platic tain meaued duing tenion tet of the heet pecimen σ, σ membane pincipal tee meidian te of the tube middle uface element σ θ cicula te of the tube middle uface element p nomal peue on the punch pofile τ fiction te on the punch pofile Copyight SciRe.