Inverse Problem of Failure Mechanics for a Drawing Die Strengthened with a Holder

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1 ca Pychca Huaca V. N. Ivs Pbm f Fau chacs f a Daw D Sh wh a H Vaf. samv baja chca Uvsy Bau baja E-ma: m-vaf@ma.u Fa E. Vyv Isu f ahmacs a chacs f NS f baja Bau baja E-ma: s@ma.u bsac: pa pbm f fau mchacs s cs f cccay a cys. I s assum ha h aw (a cy) s av awac sh by mas f a cy (h) a a h sufac f h aw h a N abay ca ca cacs f h (=...N). hca aayss f h av awac pv mma f fau paams (sss sy facs) f aw was ca u mma c. smpf mh f mma h fau paams f a ha ay aw was spaay cs. Kyws: ha-ay aw ; fc cy; av awac; cacs; sss sy facs; mma f aw fau paams Iuc Epc shws [] h a aby a uaby f mucmp csucs cmpa hmus s. ps sawch csucs a wy us usy a. Wh s hh pssu appaaus a ccu f av awac cc mucmp u a pssu s f us. sma ccu s mpm aw-ma wh aw h ws a s f aua css sc. h aw s a pcss wh a w a a pp s v a af huh h h f a spca sum (aw ) ha has sm ss sc ha h a w pc.

2 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H h aw s a maufacu fm ha ays usa ams ( ma h s) s ( aw s a a sc pps). h ha ays a am a mb s ha cu fy a aw h a u fm h pps s. h s cauh by a acv mchasm [] f a awbch ha vs h a af huh a aw bc a subjcs fma.. uc a aw. h pc f h aw usy shws ha [] h fau f ha ay aw s wh fc s (h) ccus bcaus f cac ppaa as h buay f h w a caba s f h aw. I hs cc a h sa s f w csucs f aw s s cssay pfm m aayss f h aw m ha h wu-b a cacs aa ufavaby w w sasus ss a caus fau h cus f a f. h s f h a mma cac shu b cs as a s chaacsc f h maa. h cu sa f vpm f h pma s f h mach pas pv cas h svcaby s f a mpac. hf h pma s f cmps (mucmp) csucs cass mpac. cas h aw s svcaby may b subsaay c by us s-chca mhs pacua by my av awac f h cc f a aw a a h. h su f a pbm f mchacs h ma f such av awac f a aw a fc u whch h sss f ca by hs s cu sw w h cac ppaa h aw whch s f pacua s. Fmua f h Pbm L us cs a sss-sa sa a ha ay aw fc wh a h u h ac f as ma a aa h cu. I s accp ha h cu f h aw fc s cs aua. s s w h a sufac f h s v absuy smh a aways has mc macscpc uas f a chca chaac. I sp f cpay sma ss f h uvss ha a uhss has a ssa ffc vaus paa pps f s [-6]. I s assum ha a ha ay aw s av awac fc wh h hp f a aua (h) ma f ma cab s. h awac fuc s w bfha a shu b f. L a av awac fc asc aw wh a u cy () hav N ca cacs f h (=...N). h c f h cacs ca h f ca ca sysms O y whs as ccs wh h s f cacs

3 ca Pychca Huaca V. N. a mas h a α wh h as (F. ). I s assum ha h cacs ps a f fm a as. f h w-cmp h pa ca θ sysm f hav chs h a h c f ccc ccs L L L wh a (F. ) spcvy. y L L L O y O α L E μ E μ Fu Cacua schm f vs pbm f fau mchacs f a aw wh fc cy Cs sm aa f h uh sufac f h aw. W w assum ha h pa sss sa c s fuf. I h aa ccup by h w-cmp (aw a h) h sss s cmps σ σ θ σ θ shu sasfy h ffa quas f pa hy f ascy [7]. D by E μ a E μ h Yu muus a Pss a f h aw a fc spcvy. h buay f h cu L s ps h fm: ( ) ( ) ()

4 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H L h u cu L b f fm as. h buay cs f h cs pbm a f h fm: p fp f ( ) () f () v v v v ( ) f (4) h cac s facs H v v θ a aa a aa csus f h vcs f h spacm ps f h cu L; (θ) s h s awac fuc; f s h fc fac f h aw -w pa; ; p s h pssu h sufac f h aw. h mpau f h sufac ays f h aw cass u aw u h ac f cac fc. By aw h sufac f h aw h aa f cac fc wh w (w ) h acs a sufac ha suc ha caus by h u fc. aa fcs τ=fp pm h as f ha h cac aa f h a h w h aw pcss. h a amu f ha a m u s ppa h pw f h fc fcs a h amu f h ha as a h p h cac wh ca θ w b qua Q Vfp wh V s h ma spacm vcy f h w wh spc h aw (aw vcy). h a amu f ha Q(θ) w b csum as fws: ha fw h aw Q (θ) a sma ha fw Q (θ) f cas h w ha. I h cas f say ha cha h f f h mpau f h aw a aua h may b uc h su f buay vau pbm f ha-cucvy h aw (5) h fc f : (6) f ( ) : Q ( ) h cac aa f : ( ) 4

5 ca Pychca Huaca V. N. H s h mpau a asc spc aw ; s h mpau h fc cy; λ λ a h hma cucvy cffcs f h aw a h spcvy; s Lapac s pa; s h mpau f vm h a sufac f h h; α s h ha cha fm h u cyca sufac f h h wh a mum; Q fpv s h sy f h sufac ha suc f a aw ; α s a cffc f ha fw spaa f a aw. F f h awac fuc h sam f h pbm shu b cmpm by a c (c) ha aws us m h s av awac. cc h Iv-Ova hy [8] f quasb facu h sss sy fac s a paam chaac h sss sa h vcy f h cac. Csquy h mama vau quay f h sss sy cffc a h cac p s spsb f h fau f h aw s maa. Ivsa h basc fau paams a h fuc f awac f h aw s juc a fc h maa pps a h facs hm w ca subsaay c h fau by schca mhs pacua by vay h av awac (h fuc (θ)). Fuh w accp mma f quay f mama sss sy facs h vcy f h cac ps h aw. h mma f h mama vau f h sss sy cffc w pm a cas h svcaby f h aw f h aw. hus s qu m h juc av awac (θ) such ha h sss f ca by h a pcss pvs h cac fm ppaa. N s h ay f h sa pbm s accp ha h s awac fuc (θ) may b ps as a Fu ss. Csquy h cffcs h pas f h s awac fuc shu b maa s ha h mma f h mama sss sy facs a pv. hs aa c aws m h s fuc (θ). h Cas f a S Cac I sv h sa vs pbm s cssay sv a pbm f fau mchacs f h aw a fc h pa. ps h buay f h a cu L f h aw h fm: H 5

6 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H H a cs b s wh / s a sma paam s h as hh f h bu f ma ma uay f h sufac fc a H(θ) s a fuc p f a w paam. Us a pfm h masums hav b ma f a a sufac f h aw a h appma vaus Fu cffcs f h fuc H(θ) scb ach pf f h a aw sufac hav b cacua f h fuc H(θ). W f mpaus h sss s cmps a h spacms h aw a h h fm f pass sma paam ε: (7) () () () () () () v v v... v v v... ( ) ( ) ( )... wh h ms wh ε f hh a c f smpfca. H a appma mpaus; a fs appma mpaus spcvy; a appma ssss; a a fs appma ssss; v v a aa a aa spacms a a appma; a v v a fs appma spacms. Each f h abv appmas sasfs h sysm f ffa quas f h pa hy f ascy [7]. Epa ss h psss f mpau ssss a spacms h vcy = w ba h vaus f csus f mpau sss ss a spacm cmps f =ρ(θ). Us h pubas mh wh a wha has b sa w av a h squc f buay cs f h buay vau pbms f facu mchacs f a aw a fc cy a a appma f f f Q ; (8) f p; fp 6

7 ca Pychca Huaca V. N. f (9) v v v v f ; h cac facs y ; y f y h fs appma f H f ; () () () f f N; f () v v v v f ; h cac facs y ; y f y H vm f c ; a cssv mpaus; c s mpau f N H c H () H H. ach appma h su f h buay vau pbm f ha cucvy hy s suh by h mh f spaa f vaabs. W f mpaus f a aw a f a fc cy h fm C C / () C C4 / C C C C cs s 7

8 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H C C4 C C4 cs h csas 4 s. C C C C4 C C C C4 C C C C4 4 a m fm h buay cs f h hma cucvy hy pbm (8) (). Bcaus f h h h csp fmua a ps h. sv h hmascy pbm w w us h hmasc spacm pa [9]. I h cs pbm h hmasc spacm pa f a aw F a fc cy F s m a ach appma by h su f h fw ffa quas j j j j F F (j=) (4) H α α a h cffcs f a mpau pas f a aw a fc h spcvy a μ μ a h Pss a f a aw a fc cy maa. W w s a su f quas (4) h fm: F f cs f s F f cs f s (5) ach appma f h fucs f f f f w ba ay ffa quas whs sus a fu by h mh f vaa f h csas. f m h hmasc spacm pas f a aw a fc cy us w w fmua j j j j j [9] w cacua h ssss a spacms v v f a j j j j j aw a as a v v f a fc cy ha csp h hmasc spacm pas a ach appma. h fu ssss a spacms f a aw a fc cy w sasfy buay cs (9) () spcvy. ach appma s cssay f h sc sss sa sa: j j j j j v v f a aw a v v f a fc cy s ha h buay cs (9) () b fuf. j j j j j Csquy f m h sc sss-sa sa a a appma f a aw a fc cy w hav h fw buay cs: f p fp (6) f f 8

9 ca Pychca Huaca V. N. 9 f v v v v f (7) f y y y y y wh f v v v v f. W ca w h buay cs f pbm (6)-(7) by mas f h Ksv ushshv fmua [7] h fm f a buay vau pbm f f w pas f cmp pas: f h aw f h fc cy. W w s h cmp pas h fm [7 ] (8) c (9) () b a () H ; a h s fucs chaac h spacm scuy acss h cac v v u u G () 4 h cs cas.

10 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H Us (8)-() f f cmp pas a w ps h buay cs h fm: p f () (4) f f f4 ( ) ( ) G G f f (5) f 4 f 5 f6 æ G ; p p. f 5 f 6 W h f-ha s f h buay c (4) by h fuc h w hav f f f (6) 4 W assum ha h fuc whch s a sf-baac sysm f fcs ac h fc cy as vw fm h aw ca b pa h ccua cu L (τ=p(θ)) a cmp Fu ss (7) F m h cmp pas (6) h cu L a h c a w hav c (8) h cu p( ) L.

11 ca Pychca Huaca V. N. h fucs a a aayca h f h asvs css sc f h fc cy a may b ps [7] by h ss (). W us h pw ss mh [7] f h cffcs a f h pas a. b F m h s uw quas w cs h su f h pbm f a aw. f sm asfmas f h cmp pas a pm ps h buay cs f h fucs a h fm () a (9) () G a G G G G f f f a 5 b 6 a a b. F h fucs ' f ( f f ) ( f f w w assum ) ha hy ca b pa Fu ss ' f f 6 D f f4 B f ' ' 5. H h cffcs D a B p h s fuc m by su hy. a a h buay cs () (9) a us m h cffcs c a h buay c () s us m h quas D. s a su w f: ' ' ' p f c ()

12 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H ' 4 4 (=±± ) f p c ' c ' '' D G D G '' D G ''. h h-ha ss f h fmuas f m h cffcs c b a a ca h cffcs f pass f h awac fuc a as h as f h s fuc. Sasfy by h fucs (8) h buay c (7) h cac facs w ba a cmp sua a qua wh spc h uw fuc f S () ' f y y. H h vaabs a msss quas f S a m [] by h fmua S

13 ca Pychca Huaca V. N. ; ; F h cac h sua a qua w shu a aa quay pss h spacm uquss c ac h cac cu () U h aa c () h sua a qua () by mas f abaa pcu (s h pp []) s uc h sysm f abac quas f m uws m m... : m m m S m f m m m m m cs m... cs (4) If (4) w pass h cmp cjua vaus w w aa abac quas. h ba sysms f quas wh spc a b c m m... pm f a v awac (θ) f h sss-sa sa f a aw a fc cy h psc f a cac h aw a a appma. I h sa pma s... a b m. pbm h cffcs Csquy h ba u abac sysm s s cs. F h sss sy facs a h cac ps a a appma w hav: K K m c a h h v (5) m 4 m I KII m m a a h f v m 4 m I KII m F csuc h mss quas w qu h mma f mama vau f h sss sy fac pma K m (6)

14 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H wh a scs cc wh cay capacy ha saby f a aw uavaaby f pasc fmas a as h fac ha pma K K h wh K s h chaacsc f h hsh vau f h aw maa h facu uhss whch s m pmay. a appma h pma pbm s uc h f f h cffcs (h c paams) f h pas f h awac fuc h Fu ss. h quas m ay p h cffcs f h Fu ss f h awac fuc. Csquy h quay f h sss sy cffc (5) (h bjcv fuc) as ay ps h c paams (c vaabs). hus us h mma c a a appma h cs pbm s uc a a pamm pbm. Numca cacuas a pfm by h smp ahm. I pas f h awac fuc w w cf sv ms. h cacuas w cuc cfmy h fm f h aw N []: =65mm; =95 mm; =57 mm. h mchaca chaacscs f h aw maa (ha ay BK6;) a hs (ma cab s) w accp b qua E=68 5 Pa; ν= a E =6 5 Pa; ν =8. h a pssu p cha wh 9-96 Pa. f f s quas f h appma w ca csuc h su f h pbm a a fs appma. h fucs N a a m h bas f h ba su f =. h buay cs () may b w h fm f a buay vau pbm f f cmp pas a whch a suh h fm sma (8)-() wh bvus chas. h fuh cus f h su s h sam as a a appma. h ba sua a qua wh spc u aa c f yp () by mas f h abaa mh s uc h sysm f abac quas f m uws m m.... h s cffcs b c a a ca h h-ha s f hs sysm. I h sa pma s pbm h cffcs (=±± ) shu b f. Csquy h ba u abac sysm s s cs. F sss sy cffcs h vcy f h cac ps a a fs appma w hav: K m c a h h v (7) m 4 m I KII m 4

15 ca Pychca Huaca V. N. K m a a h f v m 4 m I KII m F csuc h mss quas w us h mma c K pma m aw f h abv-m scs. h quas ss f h awac fuc m ay p h cffcs. f h Fu Csquy h quay f sss sy fac (7) (h bjcv fuc) as ay ps c paams (c vaabs). hus h pma pbm a a fs appma may as b uc a a pamm pbm. Numca cacua was ca u by a symp ahm. h cacua sus f h awac fuc (h cffcs a v mm) a v / ab f h cas ; / ; ; p= Pa. ab h vaus f Fu cffcs f pma awac α α α α α 4 α 5 α 6 α β β β β 4 β 5 β 6 β If f h cac s h a sufac f h aw h a ach appma h quay () s pac by a aa c ha psss fss f ssss a h cac f =. 4 h Cas f a bay Numb f Cacs Nw w assum ha h asc aw a h fc sufac a N ca cacs f h (= N) (F. ). W cs h pma s pbm m acy a pbm f h f f such a awac fuc f h juc f h aw a h fc cy ha mma f mama vau f h sss sy facs a h cac ps. I hs cas h pbm s sv by aay wh h pbm f a s cac. j j j j h cmp pas a j a a h 5

16 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H cas f abay may cacs. By sasfy h buay cs h cac s a ach appma w ba h sysm f N sua a j quas f h fucs (= N). ach appma h sysm f sua a quas f h cacs w shu a h aa cs j (j=) (8) ach appma u h abv m cs by mas f h abaa pcss h sysm f sua a quas s uc h j sysm f N abac quas f m N uws (j=; = N; m= ) m N m j. m j j j m m m S m f m (9) Csuc f h mss quas s a a ach appma sma h cas f cac. F h sss sy facs h vcy f h cac a ach appma w K K m c a h h v (4) m 4 m I KII m m a a h f v m 4 j j m j I KII m (j=; = N; m= ) Us h mma c h cas f a abay umb f cacs h cs pbm s uc a a pamm pbm aw f h m scs. h umca cacua was ca u wh a symp ahm f h sam fm f h aw. I s assum ha h aw s pv wh h cacs: 45 / 5 /8 5 ; / 5 ; / / / h sus f cacuas f h awac fuc (h cffcs a v mm) a c ab. 6

17 ca Pychca Huaca V. N. ab h vaus f Fu cffcs f pma fc f h cas f h cacs α α α α α 4 α 5 α 6 α β β β β 4 β 5 β 6 β h pma su.. h fu cffcs α a β pm a cas h cay capacy f a aw (aw ). 5 Smpf h f Sv h Ivs Pbm I h cas f sva cacs h amu f cacuas cass. W cs a smpf mh f sv h pbm f h f f h pma av awac f h juc f h aw a h fc cy. Epa h s awac fuc h Fu ss wh ha amu f ms qua h umb f cacs ps. I h cas f N a cacs h aw w w b sc by N cffcs f h awac fuc pas h Fu ss. W qu ha a ach appma h sss sy facs h vcy f cacs s b qua. N cmp a abac quas h ma sv quas (hy w scuss abv) w a cs abac sysm f f a h uws j j cu h cffcs (j=) f h pas f h awac fuc h Fu ss. W assum ha N cacs hav f a pa h a sufac f h aw. h h umb f h cacs vcs w qua (N-N ). I hs cas wh a pa f h cacs s a sufac cac h pas f h s awac fuc h Fu ss w w us (N-N ) cffcs. qu ha h sss sy facs a vcs b qua. (N-N ) a abac quas h ma sv quas w hs cas a cs abac sysm f quas f m a h uws. I s apppa appy h smpf mh f sv a pbm h mma f facu paams f a aw vv a a amu f cacs wh h us f h symp mh causs a a vum f cacuas. F h umca su f h ba sysm f quas w us h Gauss mh wh a chc f h pcpa m. hus h sus mhs f h mma f fau paams cmp ach h. 7

18 V.. samv a. Ivs Pbm f Fau chacs f a Daw D Sh wh a H Ccuss h ma quas ba h pap aw f h v av awac by umca cacuas wh h hp f h f f h sss sy facs pc h wh f cacs s h aw a sabsh h amssb v f fccy a h mama vaus f pa as pv a suffc aby ma. h s sa h su f h pma s pbm f h f f h av awac h juc f h aw a h fc cy pms f h pma mc paams f h aw a h fc cy whch sus a cas h cay capacy. I shu b ha h ba sus a appcab h cas f a b facu. fcs [] shv D. N.: Sa a cy f ach Pas Dvpm Vs ashsyya N. pp. -5 [] P I. L. Ema. Z.: Daw hy auya scw 97 [] Zha E. samv V..: Nuca f a Cac u I Cmpss f Cyca Bs ca Pychca Huaca V. 9 N. pp [4] hmas..: uh Sufacs Lma L 98 [5] yu Ş.: Sufac uhss Pc ach Casam aa Us NN ca Pychca Huaca V. 8 N. pp. - [6] us. us K.: Pascy a Cp f as Sp Va B Hb [7] ushshv N. I.: Sm Basc Pbms f ahmaca hy Eascy msam Kuw 977 [8] Chpav G. P.: chacs f B Facu c Gaw H Nw Y 979 [9] Paus H.: Isaa Wams-Pau Sp W 959 [] Paasyu V. V. Savu. P. Dasyshy. P.: Ga h f Su f w-dmsa Pbms h hy f Cacs E. Fac. ch. N. pp [] samv V..: N-O-Dmsa Easpasc Pbms Naua scw 987 [] Samv V. S. Echmas E. F. Favsy V.. a hs.: a- W Ha y fc b ashsyya scw 958 8