A Study of Ductile Damage and Failure of Pure Copper Part II: Analysis of the Deep Drawing Process of a Cylindrical Shell

Transcription

1 TECHNISCHE MECHANIK, 3, 6, (1), sumitte: Ferury, 11 A Stuy of Ductile Dmge n Filure of Pure Copper Prt II: Anlysis of the Deep Drwing Process of Cylinricl Shell M. Zpr, N. Tutyshkin, W.H. Müller, R. Wille The nlysis of the stress-strin stte n strin inuce mge of cylinricl shell me of copper uring the process of eep rwing is presente. The stresses on the contct surfce of operting tools (punch n ie) re ssigne implicitly, which les to the mixe ounry-vlue prolem. The results re otine on the sis of the solution of the constitutive ifferentil utions presenting plne plstic flow in curviliner chrcteristic coorintes. The mteril functions ruire for the nlysis of eep rwing were otine y experimentl stuies of uctile mge n filure of pure copper (cf., Zpr et l., 11). The process of eformtion with iscontinuities of the tngentil velocities t the plstic zone ounries is iscusse. An estimte of the locl strins n mge in the mteril is given oth for the plstic zone n for its ounries. The istriutions of strins n of the mge within the wll of finishe prt re etermine. These istriutions strongly ffect the strength properties of shell. The moeling of uctile mge in mteril uring eep rwing is se on experimentl results n consierly extens them for wier rnge of stress trixilities. It is shown tht the use of rwing ie with cone ngle of o o les to noticele shift of the stress trixilities into rnge of negtive vlues s compre to eep rwing with ie of lrger ngle ( ). The moeling revels smoother increse n ecrese in mge of the finishe prt in cse of the smller cone ngle s well s the sence of voi colescence. This fct is very importnt when mnufcturing such proucts t high operting spees. The otine results in comintion with similr ones from the literture cn e pplie to the nlysis of metl forming processes with ominnt tensile eformtion (e.g., rwing, eep rwing, striprwing, ironing). 1 Introuction Drwing opertions re wiely use in Metl Forming (MF). Applie plsticity theory is use for the evelopment of MF processes n llows us to etermine the stress-strin fiels n relte process-epenent prmeters s well s to preict structurl n mechnicl properties of finishe prts. The relile preiction of strin-inuce mge incluing the limit stte of workpiece is very importnt prolem in metl forming. Dmge in the zones with intensive eformtion epens essentilly on the strin history, the stress stte, n the effect of locl het relese relte to the energy of plstic issiption. In this pper we stuy eep rwing of copper cylinricl shell (Fig. 1). The process force, P, is trnsmitte from the punch to the ottom prt of shell n gives rise to the tensile stresses in its wlls. Thus tensile plstic strins ominte in the mteril uring eep rwing. This fct llows us to preict strin inuce mge in eep-rwn prts y using the experimentl results otine from unixil tension tests. Bonor et l. (5) show possiility in orer to ssess uctile mge n filure in metls in processes with non-proportionl loing n chnging stress trixilities y using mteril prmeters etermine experimentlly from simple tests. Numericl methos re successfully use to solve MF prolems in mny cses. These methos offer mny constntly incresing possiilities in the moeling of plstic eformtion. One of the first numericl nlyses of sheet MF processes ws one y Woo (1965), who consiere the mteril s isotropic n use finite ifference metho to solve the cse of xisymmetric punch stretching n rwing. An elstic-plstic finite element (FE) formultion of stretch forming for punch n ie of ritrry shpe ws introuce y Wng n Buinsky (1978). Oh et l. (1979) pplie the moifie Cockcroft n Lthm tensile plstic work criterion n moifie McClintock s voi growth criterion in orer to preict the occurrence of centrl ursting in xisymmetric rwing n extrusion y using the rigi-plstic FE formultion. The criticl vlues of mteril prmeters ppering in the frcture criteri were otine from tension/compression experiments. Mny reserchers (cf., e.g., Gelin, et l., 1985; Arvs, 1986; Bennni n Ouin, 1995) use Gurson s plstic potentil (cf., Gurson, 1977) for escription of uctile mge in vrious metl forming processes. All of them me use of criticl voi volume frction s frcture initition criterion. Mthur n Dwson (1987) use constitutive ution with n internl stte vrile for moeling the plstic flow n mge evelopment in metls uring o o 631

2 rwing opertion. They use voi growth mge moel propose y Cocks n Ashy (198) to preict the evolution of internl porosity. Dmge ccumultion ue to the cretion of new vois is neglecte. Koyshi et l. (1989) hs worke on severl sheet forming prolems incluing rwing, ulging, stretch forming n ening using n incrementl strin theory; in their work the mteril ws ssume s rigi-plstic n workhrening chrcteristics were inclue. Alerti et l. (1993) crrie out n nlysis of rwing process y employing the plstic potentils propose y Gurson (1977), Neelemn n Tvergr (1984), n Oyne et l. (198) for the preiction of the onset of the centrl urst. The criticl vlue of voi volume frction t frcture initition ws otine from tension test. Their numericl preictions re in goo greement with the experimentl results. Alerti et l. (1994) use Oyne s criterion to preict centrl ursting in rwing with the help of rigi-plstic FE formultion. The mteril-epenent constnts in the criterion were etermine from compression tests. They lso performe experiments y using n ultrsonic control system for etecting the efects. Tng et l. (1994) use the FE-coe DEFORM to preict chevron efects in the rwing opertion long with Cockcroft n Lthm s criterion. Hsu n Chu (1995) employe totl Lgrngin formultion n virtul work theory for the nlysis of xisymmetric sheet metl forming processes. We shoul note tht only few importnt pulictions re mentione ove, otherwise the list coul e very long n coul e suject of the inepenent pper. Intereste reers cn e referre to Rey et l. () for etile survey of the pulictions on this prolem. We shoul note severl importnt ppers tht were pulishe within lst ten yers n el specificlly with the moeling of eep rwing. Rey et l. () evelop continuum mge mechnics moel for voi growth n micro-crck initition se on the microscopic phenomen of voi nucletion, growth, n colescence n use this moel to preict the frcture initition in eep rwing (i.e., centrl ursting). They lso otin generlize strin n mge istriution y using the criticl mge criteri. Hu et l. () incorporte voi nucletion n growth moels into the FE-coe ABAQUS together with n nisotropic fourth-orer strin rte potentil so tht the mge evolution uring eep rwing of texture luminum sheets cn e nlyze. They revel the influence of plstic nisotropy together with tht of voi nucletion n growth on mge evolution for col rolle s well s col rolle n nnele luminum sheets. A mge mechnics moel propose y Chow et l. (3) for the preiction of loclize necking n filure of sheet metls hs een incorporte into the FE-coe LS-DYNA. The moifie progrm hs een successfully pplie to simulte oth single-step n two-step forming processes, incluing eep rwing. The forming limit strins re preicte for oth proportionl n non-proportionl loing conitions, n they re foun to e in greement with the test results. Guo et l. (4) evelop the Pseuo-Inverse Approch (PIA) y tking into ccount the loing history: Some relistic intermeite configurtions re etermine without contct tretment in orer to consier the loing history. A more ccurte mge rte moel presente in their pper llows strong coupling etween plsticity n mge. The otine numericl results show the fesiility of mge consiertion in PIA n the influence of mge effects on the sheet formility uring eep rwing. Khelif n Oujene (8) present n efficient mge moel se on the strong coupling of oth nisotropic elsto-plsticity n isotropic uctile mge. The evelope moel is couple with FEM y using ABAQUS. A comprison etween numericl n experimentl results is presente in the context of the squre cup eep rwing enchmrk test of Numisheet The results emonstrte the cpility of the moel to preict where n when the mge zones will pper in the workpiece uring the forming opertion. Bouif et l. (9) introuce micromechnicl mge moel se on crystl plsticity theory n continuum mge mechnics. They pply this moel in ABAQUS to simulte the mge occurrence in eep rwing of FCC polycrystlline sheet n then present mcroscopic uivlent plstic strin n mge mps for ifferent vlues of the punch isplcement. However, other non-fe methos closely relte to physicl mechnism of plstic eformtion hve lwys een pplie to n in-epth stuy of forming processes. For exmple, the metho of slip lines is successfully use to nlyze the processes with plstic flow (Hill, 195). It ws experimentlly shown tht lrge efects (vois) generte microscopic slip ns (Yokoori, 1968). Therefore, etermintion of the slip line (or slip n) fiel is useful when preicting strin-inuce mge n the limit stte of mteril efore its mcroscopic filure. In this pper we use clcultion technique for the strins ccumulte uring isplcement of mteril prticles long trjectories in the plstic zone. This metho is se on mpping of flow lines onto plne of flow velocities. The technique llows us to etermine mge istriution in the wll of finishe prt. In orer to clculte the stresses (n stress trixilities) in the plstic zone noes mixe ounry-vlue prolem shoul e solve, i.e., the stresses on the contct surfce of operting tools (punch n ie) re ssigne implicitly s the reltion etween sher n norml stresses. The moeling results revel tht for eep-rwn o shells it is pproprite to use ies with cone ngles in orer to ecrese the mge n to increse the process strin. The nlysis of eep rwing of cylinricl shell me of copper inclues the following steps: (i) formultion of the constitutive utions n ounry conitions (Section ); (ii) etermintion of stresses o 63

3 n plstic flow velocities (Section 3); (iii) clcultion of the ccumulte strins, uctile mge n uctile filure (Sections 4 n 5); n (iv) Section 6 contins iscussion of the results. Figure 1. Deep rwing of cylinricl shell Constitutive Equtions n Bounry Conitions An xisymmetric stress-strin stte occurs in the generl cse uring eep rwing of cylinricl prts. In orer to represent the xisymmetric stress-strin stte completely it is sufficient to etermine the fiels of stresses n plstic flow velocities in the meriin cross-section of eforme soli. The strin stte is plne uring the eep rwing process of thin-wlle cylinricl shell with the rtio s. 5 etween its thickness (s) n imeter ( ) since the hoop strin is infinitely smll (cf., Hill, 195). This conition llows us to reuce n xisymmetric strin prolem to prticulr plne strin prolem n to escrie the stress-strin stte of shell in plne Crtesin coorintes x, y (Fig. ) y the ifferentil utions of uilirium s x t xy + x y the von Mises yiel surfce - s y ) + 4t rz t xy s y, +, (1) x y y ( s x 4t, () the conition of coxility of the evitoric strin rtes, e& ij, n the evitoric stresses, s ij u u x x x - u y + u y y y s x - s x t xy y, (3) n the incompressiility conition 633

4 u x u y + x y, (4) where s x, s y, t xy re nonzero components of the stress tensor ij n u x, u y re the components of the vector of plstic flow velocity, u. s, t 3 is the yiel stress t sher y s y Figure. Plstic zone (ounry conitions, slip lines n movement trjectories of mteril prticles) for the o following initil t: s s 1. 6, j 18, m p. 1, m. 9 The prmetric representtion of stresses for perfectly rigi-plstic mteril s x s H - t y sin, s y s H + t y sin, t xy t y cos (5) les to the following form of Eqs. (1-4) n s x Ê ˆ t Á cos + sin Ë x y H - y s, H Ê ˆ - t y Á sin - cos, (6) y Ë x y u u Ê u x y u ˆ Á x y u tn, x u y +. (7) x y Ë y x x y These four Equtions (6) n (7) re of the hyperolic type n hve two pir-wise coincient sets of mutully orthogonl chrcteristic lines y y tn ( - chrcteristics), -cot, ( - chrcteristics), (8) x x where is the ngle etween the x-xis n the tngent to the chrcteristic line in ech mteril point of the plstic zone. The chrcteristic lines coincie with the slip lines n form gri of mutully orthogonl curves, n, in the plstic zone. Eqs. (6) n (7) cn e rewritten in the curviliner coorintes n s follows: s - t (long the lines ), s + t (long the lines ), (9) H y H y 634

5 u - u (long the lines ), u + u (long the lines ). (1) Eqs. (9) cn e rewritten in the integrl form H t y - x( ) (long the lines ), s t y + h( ) s H (long the lines ) (11) n re known s the Hencky plsticity integrls (cf., Hencky, 193) while Eqs. (1) re known s the Geiringer h in Eq. (11) cn e etermine from ounry correltions (cf., Geiringer, 193). The prmeters x ( ) n ( ) conitions. For exmple, if the hyrosttic stress s HC n the prmetric ngle C re known in the ounry point C on the contct with punch (Fig. ) then for slip lines (lines AC ) n (lines CDB ) which pss through the point C we my write: x( ) s H t y - C, h ( ) s t y + C C H. The verge sher yiel stress t y cn e erive from the conition of constnt plstic strin energy ensity for hrene mteril n from the ssume moel of perfectly rigi-plstic mteril with the yiel stress s y m (Fig. 3). The re s e e s efines n increment of plstic strin energy element w p y ( ) uner the stress-strin curve y ( e ) ensity for hrene mteril e 1 Ú s y ( e ) e s y m e, s y ( ) y s e e m Ú e C s y, t m y. (1) e 3 Uner comine (isproportionl) loing the uivlent ccumulte strin, foun y integrtion w.r.t. the strin pth ( cf., e.g., Hill, 195) e, (or Oqvist prmeter) cn e e Ú e& t Úe. (13) s OM s OM Figure 3. True stress-strin curve for pure copper In orer to clculte stresses n strins it is necessry to plot the fiel of slip lines (chrcteristics) in the plstic zone y using Eqs. (8), to fin the stress prmeters s t n in noe points y Eqs. (11), n then to etermine flow velocities u n u y Eqs. (1). The solution of the utions (8) n (11) shoul stisfy the ounry conitions n les, thus, to the solution of ounry-vlue prolems. The ounry conitions t the contct etween the processe mteril n the working tools re cuse y friction n cn e set y Coulom s lw H t c ms n, (14) where t c n s n enotes the sher stress n the norml pressure on the contct, respectively, the inex c stns for contct, m is the friction coefficient. y 635

6 The ounry conition (14) les to complicte ounry-vlue prolem since the stresses on the contct ounries re specifie implicitly. It is impossile to plot the slip line fiel irectly y using Eqs. (8) since the stresses n the ngle etween the slip lines n the contct ounries re unknown. A principlly possile solution of similr ounry-vlue prolems for plne plstic flow y using mppings of the ounry conitions (of the type shown in Eq. (14)) onto the stress plne s n, t ws initilly shown y Prger n Hoge (1951). In our pper the ounry-vlue prolem of tht type is solve y mpping of the ounry conitions (1) onto the plne s H, t (or onto the reuce coorintes s H t y, t t y ). The use of the mpping plne s H, t (or s H t y, t t y ) inste of the plne s n, t, llows us to fin stress trixilities (ST) in the mppe slip line noes: ST s H s s H t y 3, where s H enotes the hyrosttic stress. Moreover the filure strin e f tht ppers in the constitutive utions for uctile mge ( cf., Zpr et l., 11) cn e etermine y the filure locus ( ( ST) e ). f In our prolem the ounry conitions in terms of the stresses (14) re specifie on the contct surfce of the punch, t c mps n, n on the ie, t c ms n. By consiering Eqs. (5),3 we my otin t c t xy t y cos n s s s t sin on the contct surfce of the punch. In wht follows the reltion ( ) n y H + contct with the punch res y ( - s ) - H s t for the mp È t c t 1 + m p 1 H. (15) 1 + m ÍÎ p For the contct surfce of the ie (incline t the ngle j to the x -xis) we my write, respectively t m c t È 1 + m 1 + m ÍÎ Í1 + m ÍÎ c s H È m ( 1 - s H ) - s H cosj È Í 1 + m ( 1 - s H ) - s H sin j In view of Eq. (5) 3 the Eqs. (11) llow mpping the slip lines onto the plne t t Ès Í ÍÎ ty H H cos -x( ) (the slip lines ), - + h( ) y t t Î s H, t s follows. (16) È s cos Í (the slip lines ). (17) ÍÎ ty y 3 Determintion of Stresses n Plstic Flow Velocities All prmeters of eep rwing re clculte y using the following input t ( cf., Fig. 1): The mteril of the illet is pure sheet copper (99.97 %); the outer imeter n the wll thickness of the initil illet n the finishe prt re.8 mm, s 1. mm n mm ( ), s - p ( ).6 mm, respectively; the working imeters of the punch n the ie re p 18.7 mm n mm, respectively; the ie cone ngle is o j 36 ; the friction coefficients on the contct surfces of the punch n the ie re m p. 1 n m. 9, respectively. The illet is sujecte to so-clle preforming. This opertion forms conicl shpe of the initil curvture etween wll n ottom (cf., Fig. 1). Preforming llows eliminting the pek force of eformtion which might pper in the curvture uring susuent eep rwing opertion, n, therefore, mkes the seprte nlysis of eformtion in the ner-ottom region irrelevnt. Fig. 4 shows mpping of the plstic flow fiel (the ounry conitions (15) n (16)) n the slip lines (17) onto the stress plne s H t y, t t y for the given input t. Tle 1 presents the mounts of the hyrosttic stress ivie y the yiel stress t sher ( s H t y ), Stress Trixility (ST) n the prmetric ngle ( ) in the noes locte on the plstic zone ounry. Fig. shows the corresponing slip lines incluing the plstic zone ounries. Accoring to Eqs. (11) the hyrosttic stress is constnt long rectiliner segments of the slip lines. Thus, s HC s H A n s H D s H B (cf., Tle 1). 636

7 Figure 4. Mpping of the ounry conitions (15) n (16) n the slip lines (17) onto the stress plne s H t y, t t y Tle 1. Prmeters of stresses n plstic flow velocities in ounry noes Noes s H t y ST u x u u y u u u u u A B C D In orer to etermine the yiel stress t sher n pproximte y n exponentil function s ( e ) Be y t y y Eqs. (1) the true stress-strin igrm (Fig. 3) cn e. For the input t given we hve fin fin 1 t y Ú Be e e n e n fin B e n MP, where the true finl strin t eep rwing is ( 3) ln( F F ) ( 3) ln( ). 58 e. fin The etermine fiel of the slip lines n llows us to efine the ounries AC n BDC of the plstic zone in the meriin cross-section of the prt. In orer to clculte the hyrosttic stress y Eqs. (11) it is first of ll necessry to fin its vlue in one noe point of the plstic zone. For this purpose, we use conition of uilirium of the mteril t the ounry CDB (tht coincies with the slip line ) ( sin + t cos ) Ú s H mx s, (18) CDB where t mx t y enotes the mximum sher stress (cting long the slip lines); s is the ifferentil of n rc of the line. The solution of Eq. (18) llows fining the hyrosttic stress in the ounry point C (Fig. ) : s H C. MP ( s H C t y. 1 ). The etermine fiel of slip lines (i.e., the mximum sher stress trjectories t ) llows us to clculte the fiel of plstic flow velocities (cf., Fig. 5) y using Eqs. (1). The prmetric ngle ppering in Eqs. (1) is,. The fiel of plstic flow velocities represents the noes of gri known function of coorintes: ( ) of the mteril lines which coincie with the slip lines n in the coorinte system u r, uz. Such representtion llows us to fin the velocity vector u r n its components u x, u y (or u, u ) for ech noe of the plstic zone. For exmple the rius vector of the represente point m is the velocity vector u r M of the mteril point tht coincies with the point M of the slip line fiel, n its components re the velocity 637

8 components u, u. The velocity fiel meets oth the ounry conitions on the contct with the ie: M M x, u -u sinj u -u cosj u u cos D cos j - D enoting sclr of the mteril velocity vector on the contct with the ie) n the mteril continuity conition for the ccepte scheme of plne eformtion: u ( s s) up y (with ( ). Figure 5. The fiel of plstic flow velocities The ounry lines BDC n AC re lines of velocity iscontinuities. When the mteril prticles cross the - ounry lines their velocities chnge ruptly n the tngentil velocity jumps: + + Δ ut ut -ut, where u t n - u t re the vlues of the tngentil velocity on oth sies of the ounry. Physiclly speking, the velocity iscontinuity lines represent thin uctile lyer where the tngentil velocity u t vries strongly through the lyer thickness n the norml velocity u n is constnt (since its chnge woul le to the crck initition). The tngentil velocity iscontinuity t the ounry lines BDC n AC cn e efine y the length of the segments (or c ) n c (Fig. 5) sinj Δ u u, cos - ( j ) BDC D Δ sinj. (19) u AC u cos D - ( j - ) tg( ) The etermine fiel of flow velocities llows us to plot the movement trjectories of mteril prticles ccoring to the ifferentil utions written in terms of, - coorintes: s u (,, t) t, s u (,, t), where s, s re projections of the vector s r tr which efines irecte increment of the trjectory s r tr over time perio t. Since the eep rwing process is sttionry the velocity fiel in the plstic zone oes not chnge uring the eformtion n the velocity components tht pper in the ifferentil utions for trjectories epen only on coorintes: u u (, ), u u (, ). This mens tht the movement trjectories of mteril prticles coincie with the corresponing flow lines s fl n cn e plotte y using the following ution: s s u u. An ritrry flow line s fl (which crosses the point M on the ounry BDC ) cn e mppe onto the plne of velocities y mens of its hoogrph mc. The point efines the velocity u of prticle efore it enters the plstic zone, the point m efines its velocity fter crossing the ounry BDC, the segment mc correspons to continuous chnge in the velocity uring prticle isplcement within the centere fn ABC, n the point efines the velocity u fter crossing the ounry AC. The hoogrph of flow lines llows us p to clculte the strin ccumulte y mteril prticles uring their movement in the plstic zone. C 638

9 4 Accumulte Strins n Ductile Dmge Mteril prticles move long their trjectories, n the strins re ccumulte uring this process. The hoogrph of flow lines (Fig. 5) llows us to clculte the ccumulte strin y using Eq. (13) s follows. The nonzero evitoric strin increments cn e written in terms of plne curviliner coorintes, e Ê u Á Ë s -u ˆ t, s e Ê u Á Ë s + u ˆ Ê ˆ t, Á u u g t s -u + + u () s s s s Ë or, ccoring to rule of ifferentition of the vector components in curviliner coorintes, we my hve u u u u e t, e t, s u s u Ê u u ˆ u u g Á + t + Ë s s u u. (1) As it follows from the Geiringer reltions (1), the liner strin rte in the irection of the slip lines is e & e& n the uivlent strin rte is e & L& 3, where L & g& enotes the uivlent sher strin rte (cf., e.g. Kchnov, 1986). Eq. (13) tkes on the following form Ê ˆ e L 3, Á u u L Ú +, () Ë u u sfl where L is the uivlent sher strin n s fl is the flow line. Following Kchnov (4) the uivlent sher strin correspons to sher mechnism of the plstic eformtion n is pplicle when stuying the processes with plstic flow of metls y using the slip lines (Hill, 195) n the evolution of mge long the slip ns (Yokoori, 1968). The uivlent strins e n L re correlte y simple ution ( e L 3 ) n re uivlent. The mteril ccumultes the uctile strin uner eep rwing s provie y the hoogrph of flow lines. Initilly, when the mteril crosses the ounry line BDC of the velocity iscontinuity, the strin increses ruptly ΔL BDC Δu BDC sinj, (3) u cos BDC ( j - ) cos D where, in ccornce with the hoogrph, for ny prticle crossing the ounry BDC in the point M we my write Δ u u( sinj cos( j - )), u u u cos BDC D BDC M. Then isplcement of the prticle within the plstic zone ABC is plotte y segment mc of the hoogrph of flow lines (Fig. 5). It llows us to etermine n increse of the ccumulte strin in the plstic zone ABC (up to intersection with the ounry AC ) where C u ABC sinj ΔLABC Ú u Ú, (4) sinj - cos - sin u u sfl ABC M ( j ) ABC Δu u ( sinj cos( j - D )) n BDC Δ u + u sin (- ) u (( ( ) ) ( )) ABC BDC j - cos j - D sin cos j - D D sin. Finlly, when the mteril prticles cross the ounry line AC of the velocity iscontinuity n leve the plstic zone, the strin increses ruptly gin ΔL AC AC ctg( - C ) ( j - D ) sinc Δu AC sinj, (5) u sinj - cos 639

10 where Δu u ( sinj cos( j - ) (- )), u u (( j - cos( j - ) sin ) ( j - )) AC D tg C The overll uivlent strin ccumulte y the mteril prticles is given y ( ΔL + ΔL ΔL ) 3 AC D C cos sin. e L 3 +. (6) BDC ABC AC An increse of the strins long 3 movement trjectories s fl 1, sfl, sfl3 (cf., Fig. ) of the prticles with 3 ifferent originl coorintes y s is shown in Fig. 6. As the origin of these trjectories ( s fl s ) we ssume their point on the ounry line BDC (cf., Fig. ). There is lwys jump of the strin t the plstic zone ounries (i.e., t the velocity iscontinuity lines). D Figure 6. An increse of the strins uring the mteril isplcement within the plstic zone (for prticles with the originl coorinte: 1 - y s. ; - y s. 3 ; 3 - y s. 67 ) Strin-inuce mge of the finishe prt shoul e preicte y mens of numericl integrtion of the constitutive utions for the normlize mesures of mge ( cf., Zpr et l., 11 for etils) where w [ e ( e )] kk Ú e e ( e ) e 1 f f, w e [ e ( e )] e e ( e ) 3, (7) Ú f f e kk n e enote plstic ilttion n the uivlent evitoric strin t meso-level (for meso-shells with vois); e ( e ) + ( 9 )( e ) ) 1 kk is the mgnitue of the filure vector O z f (see Fig. 3 in Zpr et f f f l., 11); e kk f n ( 3 ) e re the coorintes of the vector O z f f ; e f is the uivlent filure strin t mcro-level; the sh refers to ifferentition w.r.t. the prmeter tht governs the process of eformtion (in this cse w.r.t. e ). The etermintion of the mteril functions for s-elivere pure copper ws the min experimentl prolem in the previous pper of the uthors (Zpr et l., 11). The limit uivlent strin e epens on Stress Trixility (ST) n cn e etermine y using the experimentl igrm of plsticity ( ST) f e (or, frcture locus) plotte for n s-elivere mteril stuie uner specifie temperture-rte conitions n fter recrystlliztion nneling (Fig. 7). The frcture locus for pure copper is se on the empiricl t otine for unixil tension uner superimpose hyrosttic pressure y Brigmn (1964) n Pugh (197). The ST e B exp - c ST, empiricl curves e f ( ) cn e stisfctorily escrie y n exponentil function, [ ( )] f where B n c re prmeters efine y chrcteristic points on the curve e ( ST). f f 64

11 Figure 7. The frcture locus for pure copper s-elivere n fter recrystlliztion nneling As stte ove, the mteril ccumultes uctile mge t eep rwing s follows: initilly, when it crosses the velocity iscontinuity line BDC n the uivlent strin increses ruptly ( e ΔL 3 ); then, within the plstic zone ABC ( Δe leves the plstic zone ( Δe AC ABC Δ BDC ) n, finlly, when it crosses the velocity iscontinuity line AC n ). Thus, the ccumulte uivlent mge cn e presente s elow ( w + w ) 1 ( ( w + Δw + Δw + w ) + ( w + w + w w )) 1 w BDC 1ABC Δ 1AC BDC where w 1,w re the initil vlues of mge mesures for the s-elivere mteril. ABC BDC AC, (8) Figures 8 n 9 show how the mge epens on the uivlent strin for ifferent movement trjectories of mteril prticles with the following originl coorintes: y s. ;.3;.67. Pure sheet copper revels lrger uctility fter recrystlliztion nneling s compre to its s-elivere stte. Consuently, the finishe shell me of nnele copper hs the smller uivlent mge ( w ) thn the shell me of selivere copper ( w ). It is importnt to notice tht the mge ccumultes irregulrly n with vrious intensity long ifferent trjectories of mteril prticles. This fct les to nonuniform istriution of the mge through the wll thickness of finishe prts. Figure 8. The mge mesures w1 n w vs. the uivlent strin e (for prticles with the originl coorinte: 1 - y s. ; - y s. 3 ; 3 - y s. 67 ) 641

12 Figure 9. The uivlent mge w vs. the uivlent strin e (for prticles with the originl coorinte: 1 - y s. ; - y s. 3 ; 3 - y s. 67 ) At this point we shoul iscuss the possiility of n onset of voi colescence within the plstic zone. Fig. 1 shows how the uivlent mge t the onset of voi colescence ( w c ) epens on stress trixility for the cse of tension of sheet pure copper with vrious initil voi volume frction ( f ). The curves w c ( ST ) were otine from the experimentl t on the uctile mge of copper specimens ( cf., Zpr et l. (11) n Proen et l., 1998). The notche specimens re use in orer to ttin high positive stress trixilities ST Π[ 1 3; 3] (cf., Proen et l., 1998), while negtive stress trixilities ST [-1 3; ] y tension with superimpose hyrosttic pressure, s (cf., e.g., Brigmn, 1964). H Πcn e reche Figure 1. The uivlent mge t the onset of voi colescence ( w c ) vs. stress trixility (ST) in cse of tension of sheet pure copper with vrious initil voi volume frctions ( f ): 1, specimens without rtificil efects with f. n.3, respectively; n etween the curves 3 n 4 - specimens with rtificil efects (pre-rille holes) n with f From the limite experimentl evience ville for pure copper it my e speculte tht colescence evolves from microscopic plstic sher locliztion to n intervoi necking moe followe y plstic tensile locliztion. The colescence moe y microscopic sher ning cn e oserve y in situ SEM tensile testing (Proen et l., 1998). This fct proves the known nlyticl criterion for the onset of voi colescence se on n explicit micro-mechnicl point of view n evelope y Brown n Emury (1974). The epenencies w c ( ST) cn e quite stisfctorily pproximte y the following function 64

13 1 w c rccot( K( ST) - C), (9) π where prmeters K n C re the functions of voi volume frction ( f ) for s-elivere mteril ( m + m1 f ) K Bf, ( n + n1 f ) C Af (3) with B , m , m , A , n , n The sctter of the vlues of B, A, m, m1, n, n1 les to scttern of the vlues of K n C. Figure 11. Prmeters K n C vs. voi volume frction ( f ) for sheet pure copper s-elivere The function (9) hs properties tht re sustntil in terms of etection of the onset of colescence: It hs two symptotes w c 1, w c n the inflexion point ST C K with the extremum for the erivtive w c ( ST ). Asymptotic ehvior of the function (9) within the rnge of negtive stress trixilities correspons, from physicl point of view, to heling of microscopic efects uner hyrosttic pressure ( s H < ). The process of voi heling prevents their colescence. Another limit of the function (9) oserve within the rnge of positive stress trixilities correspons to the counter physicl process relte to n intensive growth of vois uner tensile stresses. At high vlues of the initil voi volume frction this process cn le to voi colescence t the eginning of plstic eformtion. Fining the loction of the inflexion point for the function (9) ruires w ST. further experiments ime to extremum seeking for the erivtive ( ) c In the cse for eep rwing of sheet pure copper with the initil voi volume frction f. n with stress trixilities ST the onset of colescence occurs when the uivlent mge ecomes w c Preicte vlues of the uivlent mge in the finishe prt re s follows: for pure copper in s-elivere conition n for pure nnele copper (cf., Fig. 1). One cn see tht they re less thn w c Therefore, voi colescence oes not occur in the stuie process n, consuently, the finishe shell will hve microscopic structure of high qulity without lrge cvernous efects tht coul rise from voi colescence. Next, we consier possiility of reucing the strin-inuce mge of the finishe shell y using rwing o o ie with the smller cone ngle ( j ). In this cse the plstic zone hs consierly lrger extent in the re of eformtion. It les to shift of the stress stte into rnge of negtive stress trixilities n, ccoringly, to the lrger limit strin ( e ) s compre to eep rwing through ie with the lrger cone f ngle. The fiel of plstic flow velocities is more uniform tht enles more proportionl incresing of the ccumulte strins. Fig. 1 shows how the uivlent mge epens on the uivlent strin. The use of rwing ie with the smller cone ngle reuces the uivlent mge of the finishe shell me of s-elivere copper from 643

14 w own to w This les in the cse of nnele copper to miniml vlues of the uivlent mge: w It is ovious tht voi colescence oes not occur here s well s in previous cse of the lrger cone ngle n the finishe shell will lso hve the microscopic structure of rther high qulity. Figure 1. The uivlent mge w vs. the uivlent strin e (for prticles with the originl coorinte: 1 - y s. ; - y s. 3 ; 3 - y s. 67 ) 5 Preiction of Ductile Filure Preiction of uctile filure in metl forming processes is ruire for the forming limit nlysis. The following exmple shows the possiility to rw shell with thinner wll ( s.41 mm inste of s.6 mm). Due to the lrger finl eformtion cone ngle of the ie is increse up to j 4. The stress-strin stte incluing the slip line fiel is etermine within the plstic zone (Fig. 13) using the methos presente in Sections, 3, n 4. The corresponing rise of the uivlent strin long the movement trjectories s, s s (cf., Fig. 13) of the mteril prticles with the initil coorintes y s.;.3;. 67 is shown fl 1 fl, fl3 o in Fig. 14 (left). The uivlent mge epenent on the uivlent strin is presente in Fig. 14 (right). For these trjectories mge rises up to the limit vlue ( w 1) when the uivlent strins re ul to e.68;.75;.79, which re the filure strins e f for these trjectories. The points e s in Fig. 14 (left). They re locte on stright segments of the plots which correspon to curves ( ) s fl e f re shown in the jump of the strin increments. The mteril prticles re sujecte to this jump ( D e AF ) when crossing the velocity iscontinuity line AF coinciing with the slip line 3. Thus, the mteril reches the mximum mge t the line AF which is expecte to e the mcro-filure trjectory outgoing from the vertex ( A) of the ie ngle. A nturl sctter of the process prmeters les to the fct tht the preicte zone of filure is represente y the mteril lyer in neighorhoo of the line AF. For exmple, the stochstic nture of contct friction results in sctter of the prmetric ngle F t the point F on the contct with punch n, ccoring to the prmetric representtion (3) of the von Mises yiel conition (), in scttern of the velocity iscontinuity line AF (Fig. 13, she re). The coincience of the preicte zone of filure with the slip ns (i.e., Lüers ns) correspons to the morphology of uctile filure which is minly governe y voi nucletion, growth, n susuent colescence in strongly eforme region ( cf., e.g., Yokoori, 1968). 644

15 Figure 13. Plstic zone (ounry conitions, slip lines n movement trjectories of mteril prticles) for the o following initil t: s s. 4, j, m p. 1, m. 9 Figure 14. An increse of the uivlent strins (left) n the uivlent mge (right) uring the mteril isplcement within the plstic zone (for prticles with the originl coorinte: 1 - y s. ; - y s. 3 ; 3 - y s. 67 ) 6 Discussion The min prolem of this stuy is to preict the strin-inuce mge in the finishe prt fter eep rwing. The constitutive utions of tensoril theory for uctile mge re use for the moeling. Stresses n flow velocities which pper in these utions re etermine from the solution of mixe ounry prolem ( i.e., when the sher stress is implicitly specifie on the contct ounries with n operting tool). The mteril functions were experimentlly etermine n presente y Zpr et l. (11). The moeling results. The igrms for the mge mesures epenent on the uivlent strin ccumulte uring isplcement of mteril prticles long the trjectories within the plstic zone re plotte. An 645

16 pprecile effect of the ie cone ngle on the mount of mge in finishe cylinricl shell is scertine. o o Reuction of this ngle from j own to o o j les to shift of stress trixilities into rnge of negtive vlues n, ccoring to the frcture locus, to lrger limit strin n lower mge. However, o o strong reuction of the ie cone ngle ( j ) my le to higher pressure from uctile mteril on the contct surfce of punch n ie n, ccoring to the friction lw (13), to higher tngentil friction forces. The use of nnele pure copper (inste of s-elivere) noticely reuces mge of the finishe shell s well. We shoul note tht the Eqs. (7) llow us to preict mge in view of chnging stress trixilities for n ritrrily complex strin pth (Fig. 6). In this cse the prmeters of stresses n strins which pper in Eqs. (7) shoul e etermine with high ccurcy for stuie MF process. The nlysis revele tht voi colescence oes not occur in the stuie process of eep rwing. Voi colescence cn le to lrge cvernous efects n clusters n, consuently, to rop in operting performnce of metllic components sujecte to high tempertures, pressures, n strin rtes. Such proucts n components re wiely use in erospce, utomotive n energy engineering. The evelopment of such metl forming proceures tht cn provie the sence of voi colescence (i.e., with w < wc ) is very importnt n, in some cses, necessry for metlwre inustry. The prolem of relile preiction of uctile filure rises when stuying the forming limit of the processe e ST mterils. Eqs. (7) n (8) llow us to etermine for the known stress-strin stte n filure locus ( ) set of points in the plstic region (filure zone) where the uivlent mge reches its pek vlue ( w 1 ). In cse of eep rwing of the shell with very thin wll ( s.41 mm ) the mteril reches the limit mge t the velocity iscontinuity line which strts from the vertex of the ie ngle n which is expecte to e trjectory of mcroscopic filure. The sustntil fctor is nturl sctter of the process prmeters (the stochstic nture of contct friction, sctter of physicl n mechnicl properties of the mteril, etc.). This les to trnsformtion of the preicte filure trjectory into the zone of filure ( i.e., set of possile positions of this trjectory). Drwing ies with concve-convex genertrix of their operting contours cn e use in orer to reuce the extent of mge in finishe prts pprecily. Devenpeck n Richmon (1965) performe numer of comprtive tests for specimens which were cut from the rs rwn through ifferent ies with conicl, convex, concve, n sigmoil profiles. Richmon n Devenpeck term such ies with concve-convex genertrix s sigmoil ies since plne curve with one inflexion point n two prllel symptotes is known s sigmoi. The specimens rwn through sigmoil ie revele the longest ftigue life. The nlysis of mge ccumulte uring the process of eep rwing through sigmoil ie is one of the vnce prolems in this reserch re. Acknowlegement The present work ws supporte y Deutsche Forschungsgemeinschft (DFG) through the reserch project MU 175/5-1. f References Alerti, N., Brcellon, A., Cnnizzro, L., Micri, F.: Introuction of uctile frcture in metl forming processes: An pproch se on the mge mechnics. Annls of the CIRP, 43, (1994), 7-1 Alerti, N., Brcellon, A., Msnt, A., Micri, F.: Centrl ursting efects in rwing n extrusion: numericl n ultrsonic evlution. Annls of the CIRP, 4, (1993) Arvs, N.: The nlysis of voi growth tht les to centrl ursts uring extrusion, J. Mech. Phys. Solis, 34, (1986), Bennni, B., Ouin, J.: Bckwr cn extrusion of steels: effects of punch esign on flow moe n voi volume frction. Int. J. Mch. Tools n Mnuf., 33, (1995),

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18 Oyne, M., Sto, T., Okimoto, K., Shim, S.: Criteri for uctile frcture n their pplictions, J. Mech. Work. Tech., 4, (198), Proen, T., Doghri, I., Delnny, F.: Experimentl n numericl comprison of voi growth moels n voi colescence criteri for the preiction of uctile frcture in copper rs. Act Mter., 46, (1998), Prger, W., Hoge, P. G.: Theory of Perfectly Plstic Solis. New York, Wiley (1951) Pugh, H.L.D.: Mechnicl Behviour of Mterils uner Pressure. Elsevier, Lonon (197) Rey, N.V., Dixit, P.M., Ll, G.K.: Ductile frcture criteri n its preiction in xisymmetric rwing. Int. J. Mch. Tools & Mnuf., 4, (), Tng, J., Wu, W.T., Wlters, J.: Recent evelopments n pplictions of finite element metho in metl forming, J. Mter. Proc. Tech., 46, (1994), Wng, N.M., Buinsky, B.: Anlysis of sheet metl stmping y finite element metho. ASME J. Appl. Mech., 1, (1978), 73-8 Woo, D.M.: The stretching forming test. The Engineer,, (1965), Yokoori, T.: An Interisciplinry Approch to Frcture n Strength of Solis. Goron & Brech, Pu. New York (1968) Zpr, M.A., Tutyshkin, N.D., Müller, W.H., Wille, R.: A Stuy of Ductile Dmge n Filure of Pure Copper Prt I: Constitutive Equtions n Experiments. Tech. Mech., 31 (), (11), Aresses: Dr. M. Zpr, Prof. Dr. W. H. Müller, Dr. R. Wille, Lehrstuhl für Kontinuumsmechnik un Mteriltheorie, Technische Universität Berlin, Sekretrit MS, Einsteinufer 5, D-1587 Berlin, Germny. E-mil: Mksim.Zpr@tu-erlin.e, Wolfgng.H.Mueller@tu-erlin.e, Rlf.Wille@tu-erlin.e, Prof. Dr. N. Tutyshkin, Lehrstuhl für Buwesen, Bustoffe un Bukonstruktionen, Universität Tul, 36 Tul, Russln. E-mil: tutyshkin@mil.ru 648